[{"data":1,"prerenderedAt":21442},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-08-04-the-representations-in-llm":3},{"post":4,"nextPost":5002,"prevPost":5171},{"id":5,"title":6,"body":7,"description":4987,"draft":4988,"enableComment":4989,"extension":4990,"image":4976,"important":4988,"location":4991,"meta":4992,"navigation":4989,"ogImage":4991,"onday":4993,"path":4994,"seo":4995,"stem":4996,"summary":4997,"tags":4998,"__hash__":5001},"blog\u002Fblog\u002F2026\u002F2026-08-04-the-representations-in-llm.md","Representation Manifolds of LLM",{"type":8,"value":9,"toc":4975},"minimark",[10,25,33,44,49,56,62,69,410,760,866,870,876,1024,1329,1335,1338,1952,2593,2600,2604,2607,2633,2636,2642,2646,2649,3249,3256,3337,3341,3344,3351,3791,3996,4206,4468,4545,4548,4551,4555,4558,4604,4675,4764,4771,4775,4782,4793,4803,4806,4830,4848,4858,4862,4865,4872,4875,4882,4955,4962,4969],[11,12,13,14,24],"p",{},"If you have followed the progress in mechanistic interpretability over the past two years, you have likely encountered a certain kind of figure: when the activation vectors of a particular layer in a large language model are projected down to two or three dimensions and plotted, the representations of concepts such as \"months,\" \"days of the week,\" \"years,\" and \"colors\" do not scatter chaotically through space, but instead arrange themselves along an elegant curve—sometimes even a closed circle or a torus. Researchers such as Chris Olah, Josh Batson at Anthropic, and Engels et al. have all demonstrated this phenomenon in works including ",[15,16,17],"em",{},[18,19,23],"a",{"href":20,"rel":21},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2405.14860",[22],"nofollow","Not All Language Model Features Are One-Dimensionally Linear",".",[11,26,27,28,32],{},"These discoveries are captivating, but they leave behind an awkward lacuna: ",[29,30,31],"strong",{},"we see the manifold, yet we cannot articulate precisely what relationship holds between the manifold and the \"concept\" it represents."," Why would \"year\"—something that ought to be a straight line—be twisted inside the model into a curved，meandering curve through high-dimensional space? Is the circular arrangement of \"colors\" a coincidence, or a necessity? Can the cosine similarity between vectors actually tell us how semantically close two concepts are?",[11,34,35,36,43],{},"Three mathematicians—Alexander Modell, Patrick Rubin-Delanchy, and Nick Whiteley, from Imperial College London, the University of Edinburgh, and the University of Bristol, respectively—attempt, in their paper ",[15,37,38],{},[18,39,42],{"href":40,"rel":41},"https:\u002F\u002Farxiv.org\u002Fpdf\u002F2505.18235",[22],"The Origins of Representation Manifolds in Large Language Models",", to provide, for the first time, a \"minimally viable\" mathematical theory to answer these questions. This post traces the arc of their argument and offers my own reflections.",[45,46,48],"h2",{"id":47},"the-linear-representation-hypothesis","The Linear Representation Hypothesis",[11,50,51,52,55],{},"The field of mechanistic interpretability has long been anchored by a central conviction known as the ",[29,53,54],{},"Linear Representation Hypothesis (LRH)",": a model encodes human-interpretable \"features\"—such as \"possesses fluffy ears,\" \"mentions the Eiffel Tower,\" or \"is in Arabic\"—as a set of nearly orthogonal direction vectors in representation space. The representation of a given input is then a sparse linear combination of these direction vectors, weighted by whether and to what degree each feature is present. The Sparse Autoencoder (SAE), a widely adopted interpretability tool today, is built directly on this hypothesis: one trains an autoencoder with a sparsity penalty to approximate these \"dictionary vectors.\"",[11,57,58,59],{},"Yet mounting evidence suggests that this \"black-or-white, one-feature-one-direction\" model cannot account for certain phenomena. The authors cite an extensive body of literature: digits in modular addition tasks are encoded as circles; ring-like structures appear in multilingual models; \"dates\" and \"days of the week\" exhibit distorted toroidal geometries; fractal geometries even emerge in simulated hidden Markov models. These examples share a common thread: ",[29,60,61],{},"a feature is no longer a single line, but an entire, continuous, potentially nonlinearly curved manifold.",[11,63,64,65,68],{},"The field has therefore proposed a generalized version of LRH—the ",[29,66,67],{},"Multi-Dimensional Linear Representation Hypothesis",":",[70,71,75],"span",{"className":72,"translate":74},[73],"katex-display","no",[70,76,79,168],{"className":77,"translate":74},[78],"katex",[70,80,83],{"className":81},[82],"katex-mathml",[84,85,88],"math",{"xmlns":86,"display":87},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[89,90,91,163],"semantics",{},[92,93,94,99,104,107,110,113,136,144,146,148,150,157,159,161],"mrow",{},[95,96,98],"mi",{"mathvariant":97},"normal","Ψ",[100,101,103],"mo",{"stretchy":102},"false","(",[95,105,106],{},"x",[100,108,109],{"stretchy":102},")",[100,111,112],{},"=",[114,115,116,119],"munder",{},[100,117,118],{},"∑",[92,120,121,124,127,130,132,134],{},[95,122,123],{},"f",[100,125,126],{},"∈",[95,128,129],{},"F",[100,131,103],{"stretchy":102},[95,133,106],{},[100,135,109],{"stretchy":102},[137,138,139,142],"msub",{},[95,140,141],{},"ρ",[95,143,123],{},[100,145,103],{"stretchy":102},[95,147,106],{},[100,149,109],{"stretchy":102},[137,151,152,155],{},[95,153,154],{},"v",[95,156,123],{},[100,158,103],{"stretchy":102},[95,160,106],{},[100,162,109],{"stretchy":102},[164,165,167],"annotation",{"encoding":166},"application\u002Fx-tex","\\Psi(x) = \\sum_{f \\in F(x)} \\rho_f(x) v_f(x)",[70,169,173,210],{"className":170,"ariaHidden":172},[171],"katex-html","true",[70,174,177,182,186,190,194,198,203,207],{"className":175},[176],"base",[70,178],{"className":179,"style":181},[180],"strut","height:1em;vertical-align:-0.25em;",[70,183,98],{"className":184},[185],"mord",[70,187,103],{"className":188},[189],"mopen",[70,191,106],{"className":192},[185,193],"mathnormal",[70,195,109],{"className":196},[197],"mclose",[70,199],{"className":200,"style":202},[201],"mspace","margin-right:0.2778em;",[70,204,112],{"className":205},[206],"mrel",[70,208],{"className":209,"style":202},[201],[70,211,213,217,301,305,350,353,356,359,401,404,407],{"className":212},[176],[70,214],{"className":215,"style":216},[180],"height:2.566em;vertical-align:-1.516em;",[70,218,222],{"className":219},[220,221],"mop","op-limits",[70,223,227,292],{"className":224},[225,226],"vlist-t","vlist-t2",[70,228,231,287],{"className":229},[230],"vlist-r",[70,232,236,274],{"className":233,"style":235},[234],"vlist","height:1.05em;",[70,237,239,244],{"style":238},"top:-1.809em;margin-left:0em;",[70,240],{"className":241,"style":243},[242],"pstrut","height:3.05em;",[70,245,251],{"className":246},[247,248,249,250],"sizing","reset-size6","size3","mtight",[70,252,254,258,261,265,268,271],{"className":253},[185,250],[70,255,123],{"className":256,"style":257},[185,193,250],"margin-right:0.1076em;",[70,259,126],{"className":260},[206,250],[70,262,129],{"className":263,"style":264},[185,193,250],"margin-right:0.1389em;",[70,266,103],{"className":267},[189,250],[70,269,106],{"className":270},[185,193,250],[70,272,109],{"className":273},[197,250],[70,275,277,280],{"style":276},"top:-3.05em;",[70,278],{"className":279,"style":243},[242],[70,281,282],{},[70,283,118],{"className":284},[220,285,286],"op-symbol","large-op",[70,288,291],{"className":289},[290],"vlist-s","​",[70,293,295],{"className":294},[230],[70,296,299],{"className":297,"style":298},[234],"height:1.516em;",[70,300],{},[70,302],{"className":303,"style":304},[201],"margin-right:0.1667em;",[70,306,308,311],{"className":307},[185],[70,309,141],{"className":310},[185,193],[70,312,315],{"className":313},[314],"msupsub",[70,316,318,341],{"className":317},[225,226],[70,319,321,338],{"className":320},[230],[70,322,325],{"className":323,"style":324},[234],"height:0.3361em;",[70,326,328,332],{"style":327},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[70,329],{"className":330,"style":331},[242],"height:2.7em;",[70,333,335],{"className":334},[247,248,249,250],[70,336,123],{"className":337,"style":257},[185,193,250],[70,339,291],{"className":340},[290],[70,342,344],{"className":343},[230],[70,345,348],{"className":346,"style":347},[234],"height:0.2861em;",[70,349],{},[70,351,103],{"className":352},[189],[70,354,106],{"className":355},[185,193],[70,357,109],{"className":358},[197],[70,360,362,366],{"className":361},[185],[70,363,154],{"className":364,"style":365},[185,193],"margin-right:0.0359em;",[70,367,369],{"className":368},[314],[70,370,372,393],{"className":371},[225,226],[70,373,375,390],{"className":374},[230],[70,376,378],{"className":377,"style":324},[234],[70,379,381,384],{"style":380},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[70,382],{"className":383,"style":331},[242],[70,385,387],{"className":386},[247,248,249,250],[70,388,123],{"className":389,"style":257},[185,193,250],[70,391,291],{"className":392},[290],[70,394,396],{"className":395},[230],[70,397,399],{"className":398,"style":347},[234],[70,400],{},[70,402,103],{"className":403},[189],[70,405,106],{"className":406},[185,193],[70,408,109],{"className":409},[197],[11,411,412,413,499,500,529,530,604,605,689,690,759],{},"Here ",[70,414,416,440],{"className":415,"translate":74},[78],[70,417,419],{"className":418},[82],[84,420,421],{"xmlns":86},[89,422,423,437],{},[92,424,425,431,433,435],{},[137,426,427,429],{},[95,428,154],{},[95,430,123],{},[100,432,103],{"stretchy":102},[95,434,106],{},[100,436,109],{"stretchy":102},[164,438,439],{"encoding":166},"v_f(x)",[70,441,443],{"className":442,"ariaHidden":172},[171],[70,444,446,450,490,493,496],{"className":445},[176],[70,447],{"className":448,"style":449},[180],"height:1.0361em;vertical-align:-0.2861em;",[70,451,453,456],{"className":452},[185],[70,454,154],{"className":455,"style":365},[185,193],[70,457,459],{"className":458},[314],[70,460,462,482],{"className":461},[225,226],[70,463,465,479],{"className":464},[230],[70,466,468],{"className":467,"style":324},[234],[70,469,470,473],{"style":380},[70,471],{"className":472,"style":331},[242],[70,474,476],{"className":475},[247,248,249,250],[70,477,123],{"className":478,"style":257},[185,193,250],[70,480,291],{"className":481},[290],[70,483,485],{"className":484},[230],[70,486,488],{"className":487,"style":347},[234],[70,489],{},[70,491,103],{"className":492},[189],[70,494,106],{"className":495},[185,193],[70,497,109],{"className":498},[197]," is no longer a fixed direction but can vary continuously with the input ",[70,501,503,516],{"className":502,"translate":74},[78],[70,504,506],{"className":505},[82],[84,507,508],{"xmlns":86},[89,509,510,514],{},[92,511,512],{},[95,513,106],{},[164,515,106],{"encoding":166},[70,517,519],{"className":518,"ariaHidden":172},[171],[70,520,522,526],{"className":521},[176],[70,523],{"className":524,"style":525},[180],"height:0.4306em;",[70,527,106],{"className":528},[185,193]," within some subspace ",[70,531,533,552],{"className":532,"translate":74},[78],[70,534,536],{"className":535},[82],[84,537,538],{"xmlns":86},[89,539,540,549],{},[92,541,542],{},[137,543,544,547],{},[95,545,546],{},"V",[95,548,123],{},[164,550,551],{"encoding":166},"V_f",[70,553,555],{"className":554,"ariaHidden":172},[171],[70,556,558,562],{"className":557},[176],[70,559],{"className":560,"style":561},[180],"height:0.9694em;vertical-align:-0.2861em;",[70,563,565,569],{"className":564},[185],[70,566,546],{"className":567,"style":568},[185,193],"margin-right:0.2222em;",[70,570,572],{"className":571},[314],[70,573,575,596],{"className":574},[225,226],[70,576,578,593],{"className":577},[230],[70,579,581],{"className":580,"style":324},[234],[70,582,584,587],{"style":583},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[70,585],{"className":586,"style":331},[242],[70,588,590],{"className":589},[247,248,249,250],[70,591,123],{"className":592,"style":257},[185,193,250],[70,594,291],{"className":595},[290],[70,597,599],{"className":598},[230],[70,600,602],{"className":601,"style":347},[234],[70,603],{},". The standard LRH is merely the special case where ",[70,606,608,631],{"className":607,"translate":74},[78],[70,609,611],{"className":610},[82],[84,612,613],{"xmlns":86},[89,614,615,629],{},[92,616,617,623,625,627],{},[137,618,619,621],{},[95,620,154],{},[95,622,123],{},[100,624,103],{"stretchy":102},[95,626,106],{},[100,628,109],{"stretchy":102},[164,630,439],{"encoding":166},[70,632,634],{"className":633,"ariaHidden":172},[171],[70,635,637,640,680,683,686],{"className":636},[176],[70,638],{"className":639,"style":449},[180],[70,641,643,646],{"className":642},[185],[70,644,154],{"className":645,"style":365},[185,193],[70,647,649],{"className":648},[314],[70,650,652,672],{"className":651},[225,226],[70,653,655,669],{"className":654},[230],[70,656,658],{"className":657,"style":324},[234],[70,659,660,663],{"style":380},[70,661],{"className":662,"style":331},[242],[70,664,666],{"className":665},[247,248,249,250],[70,667,123],{"className":668,"style":257},[185,193,250],[70,670,291],{"className":671},[290],[70,673,675],{"className":674},[230],[70,676,678],{"className":677,"style":347},[234],[70,679],{},[70,681,103],{"className":682},[189],[70,684,106],{"className":685},[185,193],[70,687,109],{"className":688},[197]," is constant and ",[70,691,693,710],{"className":692,"translate":74},[78],[70,694,696],{"className":695},[82],[84,697,698],{"xmlns":86},[89,699,700,708],{},[92,701,702],{},[137,703,704,706],{},[95,705,546],{},[95,707,123],{},[164,709,551],{"encoding":166},[70,711,713],{"className":712,"ariaHidden":172},[171],[70,714,716,719],{"className":715},[176],[70,717],{"className":718,"style":561},[180],[70,720,722,725],{"className":721},[185],[70,723,546],{"className":724,"style":568},[185,193],[70,726,728],{"className":727},[314],[70,729,731,751],{"className":730},[225,226],[70,732,734,748],{"className":733},[230],[70,735,737],{"className":736,"style":324},[234],[70,738,739,742],{"style":583},[70,740],{"className":741,"style":331},[242],[70,743,745],{"className":744},[247,248,249,250],[70,746,123],{"className":747,"style":257},[185,193,250],[70,749,291],{"className":750},[290],[70,752,754],{"className":753},[230],[70,755,757],{"className":756,"style":347},[234],[70,758],{}," is one-dimensional.",[11,761,762,763],{},"What this paper sets out to address is precisely the most central and yet most ambiguous part of this generalized hypothesis: ",[29,764,765,766,795,796,865],{},"in what manifold form does feature ",[70,767,769,782],{"className":768,"translate":74},[78],[70,770,772],{"className":771},[82],[84,773,774],{"xmlns":86},[89,775,776,780],{},[92,777,778],{},[95,779,123],{},[164,781,123],{"encoding":166},[70,783,785],{"className":784,"ariaHidden":172},[171],[70,786,788,792],{"className":787},[176],[70,789],{"className":790,"style":791},[180],"height:0.8889em;vertical-align:-0.1944em;",[70,793,123],{"className":794,"style":257},[185,193]," manifest within subspace ",[70,797,799,816],{"className":798,"translate":74},[78],[70,800,802],{"className":801},[82],[84,803,804],{"xmlns":86},[89,805,806,814],{},[92,807,808],{},[137,809,810,812],{},[95,811,546],{},[95,813,123],{},[164,815,551],{"encoding":166},[70,817,819],{"className":818,"ariaHidden":172},[171],[70,820,822,825],{"className":821},[176],[70,823],{"className":824,"style":561},[180],[70,826,828,831],{"className":827},[185],[70,829,546],{"className":830,"style":568},[185,193],[70,832,834],{"className":833},[314],[70,835,837,857],{"className":836},[225,226],[70,838,840,854],{"className":839},[230],[70,841,843],{"className":842,"style":324},[234],[70,844,845,848],{"style":583},[70,846],{"className":847,"style":331},[242],[70,849,851],{"className":850},[247,248,249,250],[70,852,123],{"className":853,"style":257},[185,193,250],[70,855,291],{"className":856},[290],[70,858,860],{"className":859},[230],[70,861,863],{"className":862,"style":347},[234],[70,864],{},", and what is the relationship between this manifold and the \"feature\" itself?",[45,867,869],{"id":868},"defining-features-as-metric-spaces","Defining \"Features\" as Metric Spaces",[11,871,872,873],{},"The most elegant move in the paper is to first resolve a question that sounds somewhat philosophical but is in fact critically important: ",[29,874,875],{},"what exactly is \"a feature\"?",[11,877,878,879,882,883,1023],{},"The answer the authors provide is unexpectedly succinct: ",[29,880,881],{},"a feature is a metric space"," ",[70,884,886,919],{"className":885,"translate":74},[78],[70,887,889],{"className":888},[82],[84,890,891],{"xmlns":86},[89,892,893,916],{},[92,894,895,897,904,907,914],{},[100,896,103],{"stretchy":102},[137,898,899,902],{},[95,900,901],{},"Z",[95,903,123],{},[100,905,906],{"separator":172},",",[137,908,909,912],{},[95,910,911],{},"d",[95,913,123],{},[100,915,109],{"stretchy":102},[164,917,918],{"encoding":166},"(Z_f, d_f)",[70,920,922],{"className":921,"ariaHidden":172},[171],[70,923,925,928,931,973,977,980,1020],{"className":924},[176],[70,926],{"className":927,"style":449},[180],[70,929,103],{"className":930},[189],[70,932,934,938],{"className":933},[185],[70,935,901],{"className":936,"style":937},[185,193],"margin-right:0.0715em;",[70,939,941],{"className":940},[314],[70,942,944,965],{"className":943},[225,226],[70,945,947,962],{"className":946},[230],[70,948,950],{"className":949,"style":324},[234],[70,951,953,956],{"style":952},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[70,954],{"className":955,"style":331},[242],[70,957,959],{"className":958},[247,248,249,250],[70,960,123],{"className":961,"style":257},[185,193,250],[70,963,291],{"className":964},[290],[70,966,968],{"className":967},[230],[70,969,971],{"className":970,"style":347},[234],[70,972],{},[70,974,906],{"className":975},[976],"mpunct",[70,978],{"className":979,"style":304},[201],[70,981,983,986],{"className":982},[185],[70,984,911],{"className":985},[185,193],[70,987,989],{"className":988},[314],[70,990,992,1012],{"className":991},[225,226],[70,993,995,1009],{"className":994},[230],[70,996,998],{"className":997,"style":324},[234],[70,999,1000,1003],{"style":327},[70,1001],{"className":1002,"style":331},[242],[70,1004,1006],{"className":1005},[247,248,249,250],[70,1007,123],{"className":1008,"style":257},[185,193,250],[70,1010,291],{"className":1011},[290],[70,1013,1015],{"className":1014},[230],[70,1016,1018],{"className":1017,"style":347},[234],[70,1019],{},[70,1021,109],{"className":1022},[197],"—a set together with a notion of \"distance\" defined on that set. This definition, though seemingly plain, turns out to be remarkably expressive:",[1025,1026,1027,1105,1253],"ul",{},[1028,1029,1030,1033,1034,1104],"li",{},[29,1031,1032],{},"Atomic features"," (presence or absence of a cat): ",[70,1035,1037,1055],{"className":1036,"translate":74},[78],[70,1038,1040],{"className":1039},[82],[84,1041,1042],{"xmlns":86},[89,1043,1044,1052],{},[92,1045,1046],{},[137,1047,1048,1050],{},[95,1049,901],{},[95,1051,123],{},[164,1053,1054],{"encoding":166},"Z_f",[70,1056,1058],{"className":1057,"ariaHidden":172},[171],[70,1059,1061,1064],{"className":1060},[176],[70,1062],{"className":1063,"style":561},[180],[70,1065,1067,1070],{"className":1066},[185],[70,1068,901],{"className":1069,"style":937},[185,193],[70,1071,1073],{"className":1072},[314],[70,1074,1076,1096],{"className":1075},[225,226],[70,1077,1079,1093],{"className":1078},[230],[70,1080,1082],{"className":1081,"style":324},[234],[70,1083,1084,1087],{"style":952},[70,1085],{"className":1086,"style":331},[242],[70,1088,1090],{"className":1089},[247,248,249,250],[70,1091,123],{"className":1092,"style":257},[185,193,250],[70,1094,291],{"className":1095},[290],[70,1097,1099],{"className":1098},[230],[70,1100,1102],{"className":1101,"style":347},[234],[70,1103],{}," is a singleton set;",[1028,1106,1107,1110,1111,1180,1181,1252],{},[29,1108,1109],{},"Hierarchical features"," (a taxonomic tree of animals): ",[70,1112,1114,1131],{"className":1113,"translate":74},[78],[70,1115,1117],{"className":1116},[82],[84,1118,1119],{"xmlns":86},[89,1120,1121,1129],{},[92,1122,1123],{},[137,1124,1125,1127],{},[95,1126,901],{},[95,1128,123],{},[164,1130,1054],{"encoding":166},[70,1132,1134],{"className":1133,"ariaHidden":172},[171],[70,1135,1137,1140],{"className":1136},[176],[70,1138],{"className":1139,"style":561},[180],[70,1141,1143,1146],{"className":1142},[185],[70,1144,901],{"className":1145,"style":937},[185,193],[70,1147,1149],{"className":1148},[314],[70,1150,1152,1172],{"className":1151},[225,226],[70,1153,1155,1169],{"className":1154},[230],[70,1156,1158],{"className":1157,"style":324},[234],[70,1159,1160,1163],{"style":952},[70,1161],{"className":1162,"style":331},[242],[70,1164,1166],{"className":1165},[247,248,249,250],[70,1167,123],{"className":1168,"style":257},[185,193,250],[70,1170,291],{"className":1171},[290],[70,1173,1175],{"className":1174},[230],[70,1176,1178],{"className":1177,"style":347},[234],[70,1179],{}," is a discrete set, and ",[70,1182,1184,1202],{"className":1183,"translate":74},[78],[70,1185,1187],{"className":1186},[82],[84,1188,1189],{"xmlns":86},[89,1190,1191,1199],{},[92,1192,1193],{},[137,1194,1195,1197],{},[95,1196,911],{},[95,1198,123],{},[164,1200,1201],{"encoding":166},"d_f",[70,1203,1205],{"className":1204,"ariaHidden":172},[171],[70,1206,1208,1212],{"className":1207},[176],[70,1209],{"className":1210,"style":1211},[180],"height:0.9805em;vertical-align:-0.2861em;",[70,1213,1215,1218],{"className":1214},[185],[70,1216,911],{"className":1217},[185,193],[70,1219,1221],{"className":1220},[314],[70,1222,1224,1244],{"className":1223},[225,226],[70,1225,1227,1241],{"className":1226},[230],[70,1228,1230],{"className":1229,"style":324},[234],[70,1231,1232,1235],{"style":327},[70,1233],{"className":1234,"style":331},[242],[70,1236,1238],{"className":1237},[247,248,249,250],[70,1239,123],{"className":1240,"style":257},[185,193,250],[70,1242,291],{"className":1243},[290],[70,1245,1247],{"className":1246},[230],[70,1248,1250],{"className":1249,"style":347},[234],[70,1251],{}," is the tree-distance on it;",[1028,1254,1255,1258,1259,1328],{},[29,1256,1257],{},"Continuous features"," (color hue, day of the year, calendar year): ",[70,1260,1262,1279],{"className":1261,"translate":74},[78],[70,1263,1265],{"className":1264},[82],[84,1266,1267],{"xmlns":86},[89,1268,1269,1277],{},[92,1270,1271],{},[137,1272,1273,1275],{},[95,1274,901],{},[95,1276,123],{},[164,1278,1054],{"encoding":166},[70,1280,1282],{"className":1281,"ariaHidden":172},[171],[70,1283,1285,1288],{"className":1284},[176],[70,1286],{"className":1287,"style":561},[180],[70,1289,1291,1294],{"className":1290},[185],[70,1292,901],{"className":1293,"style":937},[185,193],[70,1295,1297],{"className":1296},[314],[70,1298,1300,1320],{"className":1299},[225,226],[70,1301,1303,1317],{"className":1302},[230],[70,1304,1306],{"className":1305,"style":324},[234],[70,1307,1308,1311],{"style":952},[70,1309],{"className":1310,"style":331},[242],[70,1312,1314],{"className":1313},[247,248,249,250],[70,1315,123],{"className":1316,"style":257},[185,193,250],[70,1318,291],{"className":1319},[290],[70,1321,1323],{"className":1322},[230],[70,1324,1326],{"className":1325,"style":347},[234],[70,1327],{}," can be an interval, a circle, or a higher-dimensional Euclidean space.",[11,1330,1331,1332],{},"This framework is more flexible than the \"Euclidean space\" or \"hypersphere\" assumptions common in learning theory, yet substantially simpler—and more tractable—than the Riemannian manifolds with group structure found in the disentanglement literature. It is a characteristically mathematician's choice: ",[29,1333,1334],{},"strike the right balance between expressive power and tractability.",[11,1336,1337],{},"With the definition of \"feature as metric space\" in hand, the authors state the first core hypothesis of the paper:",[1339,1340,1341],"blockquote",{},[11,1342,1343,1346,1347,1435,1436,1520,1521,1734,1735,24],{},[29,1344,1345],{},"Hypothesis 1 (Continuous Correspondence Hypothesis)."," There exists a continuous, invertible, one-to-one correspondence between feature values ",[70,1348,1350,1375],{"className":1349,"translate":74},[78],[70,1351,1353],{"className":1352},[82],[84,1354,1355],{"xmlns":86},[89,1356,1357,1372],{},[92,1358,1359,1366,1368,1370],{},[137,1360,1361,1364],{},[95,1362,1363],{},"z",[95,1365,123],{},[100,1367,103],{"stretchy":102},[95,1369,106],{},[100,1371,109],{"stretchy":102},[164,1373,1374],{"encoding":166},"z_f(x)",[70,1376,1378],{"className":1377,"ariaHidden":172},[171],[70,1379,1381,1384,1426,1429,1432],{"className":1380},[176],[70,1382],{"className":1383,"style":449},[180],[70,1385,1387,1391],{"className":1386},[185],[70,1388,1363],{"className":1389,"style":1390},[185,193],"margin-right:0.044em;",[70,1392,1394],{"className":1393},[314],[70,1395,1397,1418],{"className":1396},[225,226],[70,1398,1400,1415],{"className":1399},[230],[70,1401,1403],{"className":1402,"style":324},[234],[70,1404,1406,1409],{"style":1405},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[70,1407],{"className":1408,"style":331},[242],[70,1410,1412],{"className":1411},[247,248,249,250],[70,1413,123],{"className":1414,"style":257},[185,193,250],[70,1416,291],{"className":1417},[290],[70,1419,1421],{"className":1420},[230],[70,1422,1424],{"className":1423,"style":347},[234],[70,1425],{},[70,1427,103],{"className":1428},[189],[70,1430,106],{"className":1431},[185,193],[70,1433,109],{"className":1434},[197]," and representation directions ",[70,1437,1439,1462],{"className":1438,"translate":74},[78],[70,1440,1442],{"className":1441},[82],[84,1443,1444],{"xmlns":86},[89,1445,1446,1460],{},[92,1447,1448,1454,1456,1458],{},[137,1449,1450,1452],{},[95,1451,154],{},[95,1453,123],{},[100,1455,103],{"stretchy":102},[95,1457,106],{},[100,1459,109],{"stretchy":102},[164,1461,439],{"encoding":166},[70,1463,1465],{"className":1464,"ariaHidden":172},[171],[70,1466,1468,1471,1511,1514,1517],{"className":1467},[176],[70,1469],{"className":1470,"style":449},[180],[70,1472,1474,1477],{"className":1473},[185],[70,1475,154],{"className":1476,"style":365},[185,193],[70,1478,1480],{"className":1479},[314],[70,1481,1483,1503],{"className":1482},[225,226],[70,1484,1486,1500],{"className":1485},[230],[70,1487,1489],{"className":1488,"style":324},[234],[70,1490,1491,1494],{"style":380},[70,1492],{"className":1493,"style":331},[242],[70,1495,1497],{"className":1496},[247,248,249,250],[70,1498,123],{"className":1499,"style":257},[185,193,250],[70,1501,291],{"className":1502},[290],[70,1504,1506],{"className":1505},[230],[70,1507,1509],{"className":1508,"style":347},[234],[70,1510],{},[70,1512,103],{"className":1513},[189],[70,1515,106],{"className":1516},[185,193],[70,1518,109],{"className":1519},[197],"; that is, there exists a continuous map ",[70,1522,1524,1572],{"className":1523,"translate":74},[78],[70,1525,1527],{"className":1526},[82],[84,1528,1529],{"xmlns":86},[89,1530,1531,1569],{},[92,1532,1533,1540,1542,1548,1551],{},[137,1534,1535,1538],{},[95,1536,1537],{},"ϕ",[95,1539,123],{},[100,1541,68],{},[137,1543,1544,1546],{},[95,1545,901],{},[95,1547,123],{},[100,1549,1550],{},"→",[1552,1553,1554,1557],"msup",{},[95,1555,1556],{},"S",[92,1558,1559,1562,1565],{},[95,1560,1561],{},"D",[100,1563,1564],{},"−",[1566,1567,1568],"mn",{},"1",[164,1570,1571],{"encoding":166},"\\phi_f: Z_f \\to S^{D-1}",[70,1573,1575,1630,1685],{"className":1574,"ariaHidden":172},[171],[70,1576,1578,1581,1621,1624,1627],{"className":1577},[176],[70,1579],{"className":1580,"style":1211},[180],[70,1582,1584,1587],{"className":1583},[185],[70,1585,1537],{"className":1586},[185,193],[70,1588,1590],{"className":1589},[314],[70,1591,1593,1613],{"className":1592},[225,226],[70,1594,1596,1610],{"className":1595},[230],[70,1597,1599],{"className":1598,"style":324},[234],[70,1600,1601,1604],{"style":327},[70,1602],{"className":1603,"style":331},[242],[70,1605,1607],{"className":1606},[247,248,249,250],[70,1608,123],{"className":1609,"style":257},[185,193,250],[70,1611,291],{"className":1612},[290],[70,1614,1616],{"className":1615},[230],[70,1617,1619],{"className":1618,"style":347},[234],[70,1620],{},[70,1622],{"className":1623,"style":202},[201],[70,1625,68],{"className":1626},[206],[70,1628],{"className":1629,"style":202},[201],[70,1631,1633,1636,1676,1679,1682],{"className":1632},[176],[70,1634],{"className":1635,"style":561},[180],[70,1637,1639,1642],{"className":1638},[185],[70,1640,901],{"className":1641,"style":937},[185,193],[70,1643,1645],{"className":1644},[314],[70,1646,1648,1668],{"className":1647},[225,226],[70,1649,1651,1665],{"className":1650},[230],[70,1652,1654],{"className":1653,"style":324},[234],[70,1655,1656,1659],{"style":952},[70,1657],{"className":1658,"style":331},[242],[70,1660,1662],{"className":1661},[247,248,249,250],[70,1663,123],{"className":1664,"style":257},[185,193,250],[70,1666,291],{"className":1667},[290],[70,1669,1671],{"className":1670},[230],[70,1672,1674],{"className":1673,"style":347},[234],[70,1675],{},[70,1677],{"className":1678,"style":202},[201],[70,1680,1550],{"className":1681},[206],[70,1683],{"className":1684,"style":202},[201],[70,1686,1688,1692],{"className":1687},[176],[70,1689],{"className":1690,"style":1691},[180],"height:0.8413em;",[70,1693,1695,1699],{"className":1694},[185],[70,1696,1556],{"className":1697,"style":1698},[185,193],"margin-right:0.0576em;",[70,1700,1702],{"className":1701},[314],[70,1703,1705],{"className":1704},[225],[70,1706,1708],{"className":1707},[230],[70,1709,1711],{"className":1710,"style":1691},[234],[70,1712,1714,1717],{"style":1713},"top:-3.063em;margin-right:0.05em;",[70,1715],{"className":1716,"style":331},[242],[70,1718,1720],{"className":1719},[247,248,249,250],[70,1721,1723,1727,1731],{"className":1722},[185,250],[70,1724,1561],{"className":1725,"style":1726},[185,193,250],"margin-right:0.0278em;",[70,1728,1564],{"className":1729},[1730,250],"mbin",[70,1732,1568],{"className":1733},[185,250]," (into the unit hypersphere) such that ",[70,1736,1738,1786],{"className":1737,"translate":74},[78],[70,1739,1741],{"className":1740},[82],[84,1742,1743],{"xmlns":86},[89,1744,1745,1783],{},[92,1746,1747,1753,1755,1757,1759,1761,1767,1769,1775,1777,1779,1781],{},[137,1748,1749,1751],{},[95,1750,154],{},[95,1752,123],{},[100,1754,103],{"stretchy":102},[95,1756,106],{},[100,1758,109],{"stretchy":102},[100,1760,112],{},[137,1762,1763,1765],{},[95,1764,1537],{},[95,1766,123],{},[100,1768,103],{"stretchy":102},[137,1770,1771,1773],{},[95,1772,1363],{},[95,1774,123],{},[100,1776,103],{"stretchy":102},[95,1778,106],{},[100,1780,109],{"stretchy":102},[100,1782,109],{"stretchy":102},[164,1784,1785],{"encoding":166},"v_f(x) = \\phi_f(z_f(x))",[70,1787,1789,1853],{"className":1788,"ariaHidden":172},[171],[70,1790,1792,1795,1835,1838,1841,1844,1847,1850],{"className":1791},[176],[70,1793],{"className":1794,"style":449},[180],[70,1796,1798,1801],{"className":1797},[185],[70,1799,154],{"className":1800,"style":365},[185,193],[70,1802,1804],{"className":1803},[314],[70,1805,1807,1827],{"className":1806},[225,226],[70,1808,1810,1824],{"className":1809},[230],[70,1811,1813],{"className":1812,"style":324},[234],[70,1814,1815,1818],{"style":380},[70,1816],{"className":1817,"style":331},[242],[70,1819,1821],{"className":1820},[247,248,249,250],[70,1822,123],{"className":1823,"style":257},[185,193,250],[70,1825,291],{"className":1826},[290],[70,1828,1830],{"className":1829},[230],[70,1831,1833],{"className":1832,"style":347},[234],[70,1834],{},[70,1836,103],{"className":1837},[189],[70,1839,106],{"className":1840},[185,193],[70,1842,109],{"className":1843},[197],[70,1845],{"className":1846,"style":202},[201],[70,1848,112],{"className":1849},[206],[70,1851],{"className":1852,"style":202},[201],[70,1854,1856,1859,1899,1902,1942,1945,1948],{"className":1855},[176],[70,1857],{"className":1858,"style":449},[180],[70,1860,1862,1865],{"className":1861},[185],[70,1863,1537],{"className":1864},[185,193],[70,1866,1868],{"className":1867},[314],[70,1869,1871,1891],{"className":1870},[225,226],[70,1872,1874,1888],{"className":1873},[230],[70,1875,1877],{"className":1876,"style":324},[234],[70,1878,1879,1882],{"style":327},[70,1880],{"className":1881,"style":331},[242],[70,1883,1885],{"className":1884},[247,248,249,250],[70,1886,123],{"className":1887,"style":257},[185,193,250],[70,1889,291],{"className":1890},[290],[70,1892,1894],{"className":1893},[230],[70,1895,1897],{"className":1896,"style":347},[234],[70,1898],{},[70,1900,103],{"className":1901},[189],[70,1903,1905,1908],{"className":1904},[185],[70,1906,1363],{"className":1907,"style":1390},[185,193],[70,1909,1911],{"className":1910},[314],[70,1912,1914,1934],{"className":1913},[225,226],[70,1915,1917,1931],{"className":1916},[230],[70,1918,1920],{"className":1919,"style":324},[234],[70,1921,1922,1925],{"style":1405},[70,1923],{"className":1924,"style":331},[242],[70,1926,1928],{"className":1927},[247,248,249,250],[70,1929,123],{"className":1930,"style":257},[185,193,250],[70,1932,291],{"className":1933},[290],[70,1935,1937],{"className":1936},[230],[70,1938,1940],{"className":1939,"style":347},[234],[70,1941],{},[70,1943,103],{"className":1944},[189],[70,1946,106],{"className":1947},[185,193],[70,1949,1951],{"className":1950},[197],"))",[11,1953,1954,1955,2024,2025,2098,2099,2172,2173,2242,2243,2312,2313,2382,2383,2452,2453,2522,2523,2592],{},"Coupled with the technical premise that ",[70,1956,1958,1975],{"className":1957,"translate":74},[78],[70,1959,1961],{"className":1960},[82],[84,1962,1963],{"xmlns":86},[89,1964,1965,1973],{},[92,1966,1967],{},[137,1968,1969,1971],{},[95,1970,901],{},[95,1972,123],{},[164,1974,1054],{"encoding":166},[70,1976,1978],{"className":1977,"ariaHidden":172},[171],[70,1979,1981,1984],{"className":1980},[176],[70,1982],{"className":1983,"style":561},[180],[70,1985,1987,1990],{"className":1986},[185],[70,1988,901],{"className":1989,"style":937},[185,193],[70,1991,1993],{"className":1992},[314],[70,1994,1996,2016],{"className":1995},[225,226],[70,1997,1999,2013],{"className":1998},[230],[70,2000,2002],{"className":2001,"style":324},[234],[70,2003,2004,2007],{"style":952},[70,2005],{"className":2006,"style":331},[242],[70,2008,2010],{"className":2009},[247,248,249,250],[70,2011,123],{"className":2012,"style":257},[185,193,250],[70,2014,291],{"className":2015},[290],[70,2017,2019],{"className":2018},[230],[70,2020,2022],{"className":2021,"style":347},[234],[70,2023],{}," is compact, this hypothesis immediately yields a clean corollary (Proposition 1): ",[29,2026,2027,2097],{},[70,2028,2030,2048],{"className":2029,"translate":74},[78],[70,2031,2033],{"className":2032},[82],[84,2034,2035],{"xmlns":86},[89,2036,2037,2045],{},[92,2038,2039],{},[137,2040,2041,2043],{},[95,2042,1537],{},[95,2044,123],{},[164,2046,2047],{"encoding":166},"\\phi_f",[70,2049,2051],{"className":2050,"ariaHidden":172},[171],[70,2052,2054,2057],{"className":2053},[176],[70,2055],{"className":2056,"style":1211},[180],[70,2058,2060,2063],{"className":2059},[185],[70,2061,1537],{"className":2062},[185,193],[70,2064,2066],{"className":2065},[314],[70,2067,2069,2089],{"className":2068},[225,226],[70,2070,2072,2086],{"className":2071},[230],[70,2073,2075],{"className":2074,"style":324},[234],[70,2076,2077,2080],{"style":327},[70,2078],{"className":2079,"style":331},[242],[70,2081,2083],{"className":2082},[247,248,249,250],[70,2084,123],{"className":2085,"style":257},[185,193,250],[70,2087,291],{"className":2088},[290],[70,2090,2092],{"className":2091},[230],[70,2093,2095],{"className":2094,"style":347},[234],[70,2096],{}," is a homeomorphism."," In other words, the representation manifold ",[70,2100,2102,2121],{"className":2101,"translate":74},[78],[70,2103,2105],{"className":2104},[82],[84,2106,2107],{"xmlns":86},[89,2108,2109,2118],{},[92,2110,2111],{},[137,2112,2113,2116],{},[95,2114,2115],{},"M",[95,2117,123],{},[164,2119,2120],{"encoding":166},"M_f",[70,2122,2124],{"className":2123,"ariaHidden":172},[171],[70,2125,2127,2130],{"className":2126},[176],[70,2128],{"className":2129,"style":561},[180],[70,2131,2133,2137],{"className":2132},[185],[70,2134,2115],{"className":2135,"style":2136},[185,193],"margin-right:0.109em;",[70,2138,2140],{"className":2139},[314],[70,2141,2143,2164],{"className":2142},[225,226],[70,2144,2146,2161],{"className":2145},[230],[70,2147,2149],{"className":2148,"style":324},[234],[70,2150,2152,2155],{"style":2151},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[70,2153],{"className":2154,"style":331},[242],[70,2156,2158],{"className":2157},[247,248,249,250],[70,2159,123],{"className":2160,"style":257},[185,193,250],[70,2162,291],{"className":2163},[290],[70,2165,2167],{"className":2166},[230],[70,2168,2170],{"className":2169,"style":347},[234],[70,2171],{}," (the image of ",[70,2174,2176,2193],{"className":2175,"translate":74},[78],[70,2177,2179],{"className":2178},[82],[84,2180,2181],{"xmlns":86},[89,2182,2183,2191],{},[92,2184,2185],{},[137,2186,2187,2189],{},[95,2188,1537],{},[95,2190,123],{},[164,2192,2047],{"encoding":166},[70,2194,2196],{"className":2195,"ariaHidden":172},[171],[70,2197,2199,2202],{"className":2198},[176],[70,2200],{"className":2201,"style":1211},[180],[70,2203,2205,2208],{"className":2204},[185],[70,2206,1537],{"className":2207},[185,193],[70,2209,2211],{"className":2210},[314],[70,2212,2214,2234],{"className":2213},[225,226],[70,2215,2217,2231],{"className":2216},[230],[70,2218,2220],{"className":2219,"style":324},[234],[70,2221,2222,2225],{"style":327},[70,2223],{"className":2224,"style":331},[242],[70,2226,2228],{"className":2227},[247,248,249,250],[70,2229,123],{"className":2230,"style":257},[185,193,250],[70,2232,291],{"className":2233},[290],[70,2235,2237],{"className":2236},[230],[70,2238,2240],{"className":2239,"style":347},[234],[70,2241],{},") is topologically \"the same shape\" as the original feature space ",[70,2244,2246,2263],{"className":2245,"translate":74},[78],[70,2247,2249],{"className":2248},[82],[84,2250,2251],{"xmlns":86},[89,2252,2253,2261],{},[92,2254,2255],{},[137,2256,2257,2259],{},[95,2258,901],{},[95,2260,123],{},[164,2262,1054],{"encoding":166},[70,2264,2266],{"className":2265,"ariaHidden":172},[171],[70,2267,2269,2272],{"className":2268},[176],[70,2270],{"className":2271,"style":561},[180],[70,2273,2275,2278],{"className":2274},[185],[70,2276,901],{"className":2277,"style":937},[185,193],[70,2279,2281],{"className":2280},[314],[70,2282,2284,2304],{"className":2283},[225,226],[70,2285,2287,2301],{"className":2286},[230],[70,2288,2290],{"className":2289,"style":324},[234],[70,2291,2292,2295],{"style":952},[70,2293],{"className":2294,"style":331},[242],[70,2296,2298],{"className":2297},[247,248,249,250],[70,2299,123],{"className":2300,"style":257},[185,193,250],[70,2302,291],{"className":2303},[290],[70,2305,2307],{"className":2306},[230],[70,2308,2310],{"className":2309,"style":347},[234],[70,2311],{},": if ",[70,2314,2316,2333],{"className":2315,"translate":74},[78],[70,2317,2319],{"className":2318},[82],[84,2320,2321],{"xmlns":86},[89,2322,2323,2331],{},[92,2324,2325],{},[137,2326,2327,2329],{},[95,2328,901],{},[95,2330,123],{},[164,2332,1054],{"encoding":166},[70,2334,2336],{"className":2335,"ariaHidden":172},[171],[70,2337,2339,2342],{"className":2338},[176],[70,2340],{"className":2341,"style":561},[180],[70,2343,2345,2348],{"className":2344},[185],[70,2346,901],{"className":2347,"style":937},[185,193],[70,2349,2351],{"className":2350},[314],[70,2352,2354,2374],{"className":2353},[225,226],[70,2355,2357,2371],{"className":2356},[230],[70,2358,2360],{"className":2359,"style":324},[234],[70,2361,2362,2365],{"style":952},[70,2363],{"className":2364,"style":331},[242],[70,2366,2368],{"className":2367},[247,248,249,250],[70,2369,123],{"className":2370,"style":257},[185,193,250],[70,2372,291],{"className":2373},[290],[70,2375,2377],{"className":2376},[230],[70,2378,2380],{"className":2379,"style":347},[234],[70,2381],{}," is an interval, ",[70,2384,2386,2403],{"className":2385,"translate":74},[78],[70,2387,2389],{"className":2388},[82],[84,2390,2391],{"xmlns":86},[89,2392,2393,2401],{},[92,2394,2395],{},[137,2396,2397,2399],{},[95,2398,2115],{},[95,2400,123],{},[164,2402,2120],{"encoding":166},[70,2404,2406],{"className":2405,"ariaHidden":172},[171],[70,2407,2409,2412],{"className":2408},[176],[70,2410],{"className":2411,"style":561},[180],[70,2413,2415,2418],{"className":2414},[185],[70,2416,2115],{"className":2417,"style":2136},[185,193],[70,2419,2421],{"className":2420},[314],[70,2422,2424,2444],{"className":2423},[225,226],[70,2425,2427,2441],{"className":2426},[230],[70,2428,2430],{"className":2429,"style":324},[234],[70,2431,2432,2435],{"style":2151},[70,2433],{"className":2434,"style":331},[242],[70,2436,2438],{"className":2437},[247,248,249,250],[70,2439,123],{"className":2440,"style":257},[185,193,250],[70,2442,291],{"className":2443},[290],[70,2445,2447],{"className":2446},[230],[70,2448,2450],{"className":2449,"style":347},[234],[70,2451],{}," is a curve; if ",[70,2454,2456,2473],{"className":2455,"translate":74},[78],[70,2457,2459],{"className":2458},[82],[84,2460,2461],{"xmlns":86},[89,2462,2463,2471],{},[92,2464,2465],{},[137,2466,2467,2469],{},[95,2468,901],{},[95,2470,123],{},[164,2472,1054],{"encoding":166},[70,2474,2476],{"className":2475,"ariaHidden":172},[171],[70,2477,2479,2482],{"className":2478},[176],[70,2480],{"className":2481,"style":561},[180],[70,2483,2485,2488],{"className":2484},[185],[70,2486,901],{"className":2487,"style":937},[185,193],[70,2489,2491],{"className":2490},[314],[70,2492,2494,2514],{"className":2493},[225,226],[70,2495,2497,2511],{"className":2496},[230],[70,2498,2500],{"className":2499,"style":324},[234],[70,2501,2502,2505],{"style":952},[70,2503],{"className":2504,"style":331},[242],[70,2506,2508],{"className":2507},[247,248,249,250],[70,2509,123],{"className":2510,"style":257},[185,193,250],[70,2512,291],{"className":2513},[290],[70,2515,2517],{"className":2516},[230],[70,2518,2520],{"className":2519,"style":347},[234],[70,2521],{}," is a circle, ",[70,2524,2526,2543],{"className":2525,"translate":74},[78],[70,2527,2529],{"className":2528},[82],[84,2530,2531],{"xmlns":86},[89,2532,2533,2541],{},[92,2534,2535],{},[137,2536,2537,2539],{},[95,2538,2115],{},[95,2540,123],{},[164,2542,2120],{"encoding":166},[70,2544,2546],{"className":2545,"ariaHidden":172},[171],[70,2547,2549,2552],{"className":2548},[176],[70,2550],{"className":2551,"style":561},[180],[70,2553,2555,2558],{"className":2554},[185],[70,2556,2115],{"className":2557,"style":2136},[185,193],[70,2559,2561],{"className":2560},[314],[70,2562,2564,2584],{"className":2563},[225,226],[70,2565,2567,2581],{"className":2566},[230],[70,2568,2570],{"className":2569,"style":324},[234],[70,2571,2572,2575],{"style":2151},[70,2573],{"className":2574,"style":331},[242],[70,2576,2578],{"className":2577},[247,248,249,250],[70,2579,123],{"className":2580,"style":257},[185,193,250],[70,2582,291],{"className":2583},[290],[70,2585,2587],{"className":2586},[230],[70,2588,2590],{"className":2589,"style":347},[234],[70,2591],{}," is a loop; connected components, holes, and branch points—all these topological properties are faithfully preserved.",[11,2594,2595,2596,2599],{},"The significance of this step is that ",[29,2597,2598],{},"for the first time, it ties \"what the manifold looks like\" to \"the structure of the concept itself\" in rigorous mathematical language",", rather than stopping at the intuitive description of \"it looks like a circle.\"",[45,2601,2603],{"id":2602},"empirical-validation-with-real-data","Empirical Validation with Real Data",[11,2605,2606],{},"A theory, however elegant, must answer to data. The authors selected three representative case studies:",[2608,2609,2610,2621,2627],"ol",{},[1028,2611,2612,2615,2616,2620],{},[29,2613,2614],{},"Colors",": 3072-dimensional embeddings were generated from English color names using OpenAI's ",[2617,2618,2619],"code",{},"text-embedding-large-3",", then reduced to three dimensions via PCA for visualization;",[1028,2622,2623,2626],{},[29,2624,2625],{},"Years",": the \"20th-century year\" feature was extracted from layer 7 of GPT-2-small using SAEs, following Engels et al. (2025);",[1028,2628,2629,2632],{},[29,2630,2631],{},"Dates",": embeddings were generated from prompts such as \"January 1st\" through \"December 31st,\" again using OpenAI's embedding model.",[11,2634,2635],{},"The results are striking: the color embeddings arrange themselves along a ring, with the hue ordering (red → purple → blue → green → yellow → orange → red) matching the standard color wheel exactly. The token activations for years trace out a curve that winds through three-dimensional space, faintly evocative of the human intuition of a \"timeline.\" The authors further estimated the ordinal structure along the manifold using a k-nearest-neighbor graph and computed rank correlations with the true years, obtaining a Kendall correlation coefficient of 0.97 and a Spearman correlation exceeding 0.99—an almost perfectly monotonic correspondence, providing strong support for the homeomorphism prediction.",[11,2637,2638,2639],{},"The paper also contains a particularly revealing \"gotcha\" detail: in the original Engels et al. work, the representations of \"days of the week\" and \"months,\" when projected onto the first two principal components, appeared as tidy circles. The authors point out, however, that once the third principal component is examined, one finds that this \"circle\" is in fact continuously twisting and weaving in the third dimension; the clean circle compressed into a two-dimensional plane is a visual artifact (see their Figure 2). This observation serves as a cautionary note: ",[29,2640,2641],{},"low-dimensional projections, while indispensable as visualization tools in interpretability research, can also mislead.",[45,2643,2645],{"id":2644},"an-explanation-in-terms-of-computational-expressivity","An Explanation in Terms of Computational Expressivity",[11,2647,2648],{},"At this point a natural question arises: since \"year\" is fundamentally a one-dimensional quantity, why doesn't the model simply encode it as a straight line segment in representation space, rather than twisting it into a curve through high dimensions?",[11,2650,2651,2652,2655,2656,2684,2685,2761,2762,2890,2891,2919,2920,2981,2982,3141,3142,3186,3187,3248],{},"The authors offer an answer that is both intuitive and characteristically mathematical: ",[29,2653,2654],{},"expressivity",". If the goal were merely to \"read out\" ",[70,2657,2659,2672],{"className":2658,"translate":74},[78],[70,2660,2662],{"className":2661},[82],[84,2663,2664],{"xmlns":86},[89,2665,2666,2670],{},[92,2667,2668],{},[95,2669,1363],{},[164,2671,1363],{"encoding":166},[70,2673,2675],{"className":2674,"ariaHidden":172},[171],[70,2676,2678,2681],{"className":2677},[176],[70,2679],{"className":2680,"style":525},[180],[70,2682,1363],{"className":2683,"style":1390},[185,193]," itself through a linear projection (i.e., to make the identity function ",[70,2686,2688,2717],{"className":2687,"translate":74},[78],[70,2689,2691],{"className":2690},[82],[84,2692,2693],{"xmlns":86},[89,2694,2695,2714],{},[92,2696,2697,2704,2706,2708,2710,2712],{},[92,2698,2699,2702],{},[95,2700,2701],{"mathvariant":97},"i",[95,2703,911],{"mathvariant":97},[100,2705,103],{"stretchy":102},[95,2707,1363],{},[100,2709,109],{"stretchy":102},[100,2711,112],{},[95,2713,1363],{},[164,2715,2716],{"encoding":166},"\\mathrm{id}(z) = z",[70,2718,2720,2752],{"className":2719,"ariaHidden":172},[171],[70,2721,2723,2726,2734,2737,2740,2743,2746,2749],{"className":2722},[176],[70,2724],{"className":2725,"style":181},[180],[70,2727,2729],{"className":2728},[185],[70,2730,2733],{"className":2731},[185,2732],"mathrm","id",[70,2735,103],{"className":2736},[189],[70,2738,1363],{"className":2739,"style":1390},[185,193],[70,2741,109],{"className":2742},[197],[70,2744],{"className":2745,"style":202},[201],[70,2747,112],{"className":2748},[206],[70,2750],{"className":2751,"style":202},[201],[70,2753,2755,2758],{"className":2754},[176],[70,2756],{"className":2757,"style":525},[180],[70,2759,1363],{"className":2760,"style":1390},[185,193]," computable via a single linear operation), two orthogonal directions ",[70,2763,2765,2792],{"className":2764,"translate":74},[78],[70,2766,2768],{"className":2767},[82],[84,2769,2770],{"xmlns":86},[89,2771,2772,2789],{},[92,2773,2774,2781,2783],{},[137,2775,2776,2778],{},[95,2777,154],{},[1566,2779,2780],{},"0",[100,2782,906],{"separator":172},[137,2784,2785,2787],{},[95,2786,154],{},[1566,2788,1568],{},[164,2790,2791],{"encoding":166},"v_0, v_1",[70,2793,2795],{"className":2794,"ariaHidden":172},[171],[70,2796,2798,2802,2844,2847,2850],{"className":2797},[176],[70,2799],{"className":2800,"style":2801},[180],"height:0.625em;vertical-align:-0.1944em;",[70,2803,2805,2808],{"className":2804},[185],[70,2806,154],{"className":2807,"style":365},[185,193],[70,2809,2811],{"className":2810},[314],[70,2812,2814,2835],{"className":2813},[225,226],[70,2815,2817,2832],{"className":2816},[230],[70,2818,2821],{"className":2819,"style":2820},[234],"height:0.3011em;",[70,2822,2823,2826],{"style":380},[70,2824],{"className":2825,"style":331},[242],[70,2827,2829],{"className":2828},[247,248,249,250],[70,2830,2780],{"className":2831},[185,250],[70,2833,291],{"className":2834},[290],[70,2836,2838],{"className":2837},[230],[70,2839,2842],{"className":2840,"style":2841},[234],"height:0.15em;",[70,2843],{},[70,2845,906],{"className":2846},[976],[70,2848],{"className":2849,"style":304},[201],[70,2851,2853,2856],{"className":2852},[185],[70,2854,154],{"className":2855,"style":365},[185,193],[70,2857,2859],{"className":2858},[314],[70,2860,2862,2882],{"className":2861},[225,226],[70,2863,2865,2879],{"className":2864},[230],[70,2866,2868],{"className":2867,"style":2820},[234],[70,2869,2870,2873],{"style":380},[70,2871],{"className":2872,"style":331},[242],[70,2874,2876],{"className":2875},[247,248,249,250],[70,2877,1568],{"className":2878},[185,250],[70,2880,291],{"className":2881},[290],[70,2883,2885],{"className":2884},[230],[70,2886,2888],{"className":2887,"style":2841},[234],[70,2889],{}," would suffice. But if one also wishes to make higher-order polynomials of ",[70,2892,2894,2907],{"className":2893,"translate":74},[78],[70,2895,2897],{"className":2896},[82],[84,2898,2899],{"xmlns":86},[89,2900,2901,2905],{},[92,2902,2903],{},[95,2904,1363],{},[164,2906,1363],{"encoding":166},[70,2908,2910],{"className":2909,"ariaHidden":172},[171],[70,2911,2913,2916],{"className":2912},[176],[70,2914],{"className":2915,"style":525},[180],[70,2917,1363],{"className":2918,"style":1390},[185,193]," (e.g., ",[70,2921,2923,2942],{"className":2922,"translate":74},[78],[70,2924,2926],{"className":2925},[82],[84,2927,2928],{"xmlns":86},[89,2929,2930,2939],{},[92,2931,2932],{},[1552,2933,2934,2936],{},[95,2935,1363],{},[1566,2937,2938],{},"2",[164,2940,2941],{"encoding":166},"z^2",[70,2943,2945],{"className":2944,"ariaHidden":172},[171],[70,2946,2948,2952],{"className":2947},[176],[70,2949],{"className":2950,"style":2951},[180],"height:0.8141em;",[70,2953,2955,2958],{"className":2954},[185],[70,2956,1363],{"className":2957,"style":1390},[185,193],[70,2959,2961],{"className":2960},[314],[70,2962,2964],{"className":2963},[225],[70,2965,2967],{"className":2966},[230],[70,2968,2970],{"className":2969,"style":2951},[234],[70,2971,2972,2975],{"style":1713},[70,2973],{"className":2974,"style":331},[242],[70,2976,2978],{"className":2977},[247,248,249,250],[70,2979,2938],{"className":2980},[185,250],") linearly readable, then more orthogonal directions ",[70,2983,2985,3023],{"className":2984,"translate":74},[78],[70,2986,2988],{"className":2987},[82],[84,2989,2990],{"xmlns":86},[89,2991,2992,3020],{},[92,2993,2994,3000,3002,3005,3007],{},[137,2995,2996,2998],{},[95,2997,154],{},[1566,2999,2780],{},[100,3001,906],{"separator":172},[100,3003,3004],{},"…",[100,3006,906],{"separator":172},[137,3008,3009,3011],{},[95,3010,154],{},[92,3012,3013,3015,3018],{},[95,3014,11],{},[100,3016,3017],{},"+",[1566,3019,1568],{},[164,3021,3022],{"encoding":166},"v_0, \\dots, v_{p+1}",[70,3024,3026],{"className":3025,"ariaHidden":172},[171],[70,3027,3029,3033,3073,3076,3079,3083,3086,3089,3092],{"className":3028},[176],[70,3030],{"className":3031,"style":3032},[180],"height:0.7167em;vertical-align:-0.2861em;",[70,3034,3036,3039],{"className":3035},[185],[70,3037,154],{"className":3038,"style":365},[185,193],[70,3040,3042],{"className":3041},[314],[70,3043,3045,3065],{"className":3044},[225,226],[70,3046,3048,3062],{"className":3047},[230],[70,3049,3051],{"className":3050,"style":2820},[234],[70,3052,3053,3056],{"style":380},[70,3054],{"className":3055,"style":331},[242],[70,3057,3059],{"className":3058},[247,248,249,250],[70,3060,2780],{"className":3061},[185,250],[70,3063,291],{"className":3064},[290],[70,3066,3068],{"className":3067},[230],[70,3069,3071],{"className":3070,"style":2841},[234],[70,3072],{},[70,3074,906],{"className":3075},[976],[70,3077],{"className":3078,"style":304},[201],[70,3080,3004],{"className":3081},[3082],"minner",[70,3084],{"className":3085,"style":304},[201],[70,3087,906],{"className":3088},[976],[70,3090],{"className":3091,"style":304},[201],[70,3093,3095,3098],{"className":3094},[185],[70,3096,154],{"className":3097,"style":365},[185,193],[70,3099,3101],{"className":3100},[314],[70,3102,3104,3133],{"className":3103},[225,226],[70,3105,3107,3130],{"className":3106},[230],[70,3108,3110],{"className":3109,"style":2820},[234],[70,3111,3112,3115],{"style":380},[70,3113],{"className":3114,"style":331},[242],[70,3116,3118],{"className":3117},[247,248,249,250],[70,3119,3121,3124,3127],{"className":3120},[185,250],[70,3122,11],{"className":3123},[185,193,250],[70,3125,3017],{"className":3126},[1730,250],[70,3128,1568],{"className":3129},[185,250],[70,3131,291],{"className":3132},[290],[70,3134,3136],{"className":3135},[230],[70,3137,3139],{"className":3138,"style":347},[234],[70,3140],{}," are required, and the path traced by ",[70,3143,3145,3165],{"className":3144,"translate":74},[78],[70,3146,3148],{"className":3147},[82],[84,3149,3150],{"xmlns":86},[89,3151,3152,3162],{},[92,3153,3154,3156,3158,3160],{},[95,3155,1537],{},[100,3157,103],{"stretchy":102},[95,3159,1363],{},[100,3161,109],{"stretchy":102},[164,3163,3164],{"encoding":166},"\\phi(z)",[70,3166,3168],{"className":3167,"ariaHidden":172},[171],[70,3169,3171,3174,3177,3180,3183],{"className":3170},[176],[70,3172],{"className":3173,"style":181},[180],[70,3175,1537],{"className":3176},[185,193],[70,3178,103],{"className":3179},[189],[70,3181,1363],{"className":3182,"style":1390},[185,193],[70,3184,109],{"className":3185},[197]," must accordingly bend through a ",[70,3188,3190,3212],{"className":3189,"translate":74},[78],[70,3191,3193],{"className":3192},[82],[84,3194,3195],{"xmlns":86},[89,3196,3197,3209],{},[92,3198,3199,3201,3203,3205,3207],{},[100,3200,103],{"stretchy":102},[95,3202,11],{},[100,3204,3017],{},[1566,3206,2938],{},[100,3208,109],{"stretchy":102},[164,3210,3211],{"encoding":166},"(p+2)",[70,3213,3215,3236],{"className":3214,"ariaHidden":172},[171],[70,3216,3218,3221,3224,3227,3230,3233],{"className":3217},[176],[70,3219],{"className":3220,"style":181},[180],[70,3222,103],{"className":3223},[189],[70,3225,11],{"className":3226},[185,193],[70,3228],{"className":3229,"style":568},[201],[70,3231,3017],{"className":3232},[1730],[70,3234],{"className":3235,"style":568},[201],[70,3237,3239,3242,3245],{"className":3238},[176],[70,3240],{"className":3241,"style":181},[180],[70,3243,2938],{"className":3244},[185],[70,3246,109],{"className":3247},[197],"-dimensional subspace.",[11,3250,3251,3252,3255],{},"In other words: ",[29,3253,3254],{},"the degree to which a manifold is \"twisted\" through high-dimensional space encodes, in some sense, how many distinct (nonlinear) functions over this feature can be read out directly by subsequent network layers through a simple linear projection."," This provides a functionalist explanation for why the model encodes a simple one-dimensional concept as a complex high-dimensional manifold—not because the model \"can't help itself,\" but because doing so facilitates downstream computation.",[11,3257,3258,3259,3262,3263,3332,3333,3336],{},"The authors then connect this perspective to the ",[29,3260,3261],{},"superposition hypothesis",": since the representation dimension is far smaller than the number of latent features, the model has little choice but to have most features share the same representation space sparsely and near-orthogonally. Under this constraint, the fact that ",[70,3264,3266,3283],{"className":3265,"translate":74},[78],[70,3267,3269],{"className":3268},[82],[84,3270,3271],{"xmlns":86},[89,3272,3273,3281],{},[92,3274,3275],{},[137,3276,3277,3279],{},[95,3278,1537],{},[95,3280,123],{},[164,3282,2047],{"encoding":166},[70,3284,3286],{"className":3285,"ariaHidden":172},[171],[70,3287,3289,3292],{"className":3288},[176],[70,3290],{"className":3291,"style":1211},[180],[70,3293,3295,3298],{"className":3294},[185],[70,3296,1537],{"className":3297},[185,193],[70,3299,3301],{"className":3300},[314],[70,3302,3304,3324],{"className":3303},[225,226],[70,3305,3307,3321],{"className":3306},[230],[70,3308,3310],{"className":3309,"style":324},[234],[70,3311,3312,3315],{"style":327},[70,3313],{"className":3314,"style":331},[242],[70,3316,3318],{"className":3317},[247,248,249,250],[70,3319,123],{"className":3320,"style":257},[185,193,250],[70,3322,291],{"className":3323},[290],[70,3325,3327],{"className":3326},[230],[70,3328,3330],{"className":3329,"style":347},[234],[70,3331],{},"—the direction encoding for a given feature—does indeed contain the \"linearly readable\" component of the identity function also goes some way toward explaining why simple ",[29,3334,3335],{},"linear probes"," are often surprisingly effective at \"fishing out\" a specific feature from superposed representations in practice.",[45,3338,3340],{"id":3339},"cosine-similarity","Cosine Similarity",[11,3342,3343],{},"If the preceding sections constitute the scaffolding, then this section is the paper's true \"hardcore\" contribution and, in my view, the part most worth remembering.",[11,3345,3346,3347,3350],{},"The authors advance ",[29,3348,3349],{},"Hypothesis 2 (Cosine Similarity Reflects Distance)",": locally, the cosine similarity between representations is some decreasing function of the squared distance between feature values:",[70,3352,3354],{"className":3353,"translate":74},[73],[70,3355,3357,3460],{"className":3356,"translate":74},[78],[70,3358,3360],{"className":3359},[82],[84,3361,3362],{"xmlns":86,"display":87},[89,3363,3364,3457],{},[92,3365,3366,3384,3386,3392,3394,3396,3398,3400,3406,3408,3416,3418,3420,3422,3429,3431,3437,3439,3441,3443,3449,3455],{},[92,3367,3368,3371,3374,3377,3379,3381],{},[95,3369,3370],{"mathvariant":97},"C",[95,3372,3373],{"mathvariant":97},"o",[95,3375,3376],{"mathvariant":97},"s",[95,3378,1556],{"mathvariant":97},[95,3380,2701],{"mathvariant":97},[95,3382,3383],{"mathvariant":97},"m",[100,3385,103],{"stretchy":102},[137,3387,3388,3390],{},[95,3389,1537],{},[95,3391,123],{},[100,3393,103],{"stretchy":102},[95,3395,1363],{},[100,3397,109],{"stretchy":102},[100,3399,906],{"separator":172},[137,3401,3402,3404],{},[95,3403,1537],{},[95,3405,123],{},[100,3407,103],{"stretchy":102},[1552,3409,3410,3412],{},[95,3411,1363],{},[100,3413,3415],{"mathvariant":97,"lspace":3414,"rspace":3414},"0em","′",[100,3417,109],{"stretchy":102},[100,3419,109],{"stretchy":102},[100,3421,112],{},[137,3423,3424,3427],{},[95,3425,3426],{},"g",[95,3428,123],{},[100,3430,103],{"stretchy":102},[137,3432,3433,3435],{},[95,3434,911],{},[95,3436,123],{},[100,3438,103],{"stretchy":102},[95,3440,1363],{},[100,3442,906],{"separator":172},[1552,3444,3445,3447],{},[95,3446,1363],{},[100,3448,3415],{"mathvariant":97,"lspace":3414,"rspace":3414},[1552,3450,3451,3453],{},[100,3452,109],{"stretchy":102},[1566,3454,2938],{},[100,3456,109],{"stretchy":102},[164,3458,3459],{"encoding":166},"\\mathrm{CosSim}(\\phi_f(z), \\phi_f(z')) = g_f(d_f(z, z')^2)",[70,3461,3463,3624],{"className":3462,"ariaHidden":172},[171],[70,3464,3466,3470,3477,3480,3520,3523,3526,3529,3532,3535,3575,3578,3612,3615,3618,3621],{"className":3465},[176],[70,3467],{"className":3468,"style":3469},[180],"height:1.088em;vertical-align:-0.2861em;",[70,3471,3473],{"className":3472},[185],[70,3474,3476],{"className":3475},[185,2732],"CosSim",[70,3478,103],{"className":3479},[189],[70,3481,3483,3486],{"className":3482},[185],[70,3484,1537],{"className":3485},[185,193],[70,3487,3489],{"className":3488},[314],[70,3490,3492,3512],{"className":3491},[225,226],[70,3493,3495,3509],{"className":3494},[230],[70,3496,3498],{"className":3497,"style":324},[234],[70,3499,3500,3503],{"style":327},[70,3501],{"className":3502,"style":331},[242],[70,3504,3506],{"className":3505},[247,248,249,250],[70,3507,123],{"className":3508,"style":257},[185,193,250],[70,3510,291],{"className":3511},[290],[70,3513,3515],{"className":3514},[230],[70,3516,3518],{"className":3517,"style":347},[234],[70,3519],{},[70,3521,103],{"className":3522},[189],[70,3524,1363],{"className":3525,"style":1390},[185,193],[70,3527,109],{"className":3528},[197],[70,3530,906],{"className":3531},[976],[70,3533],{"className":3534,"style":304},[201],[70,3536,3538,3541],{"className":3537},[185],[70,3539,1537],{"className":3540},[185,193],[70,3542,3544],{"className":3543},[314],[70,3545,3547,3567],{"className":3546},[225,226],[70,3548,3550,3564],{"className":3549},[230],[70,3551,3553],{"className":3552,"style":324},[234],[70,3554,3555,3558],{"style":327},[70,3556],{"className":3557,"style":331},[242],[70,3559,3561],{"className":3560},[247,248,249,250],[70,3562,123],{"className":3563,"style":257},[185,193,250],[70,3565,291],{"className":3566},[290],[70,3568,3570],{"className":3569},[230],[70,3571,3573],{"className":3572,"style":347},[234],[70,3574],{},[70,3576,103],{"className":3577},[189],[70,3579,3581,3584],{"className":3580},[185],[70,3582,1363],{"className":3583,"style":1390},[185,193],[70,3585,3587],{"className":3586},[314],[70,3588,3590],{"className":3589},[225],[70,3591,3593],{"className":3592},[230],[70,3594,3597],{"className":3595,"style":3596},[234],"height:0.8019em;",[70,3598,3600,3603],{"style":3599},"top:-3.113em;margin-right:0.05em;",[70,3601],{"className":3602,"style":331},[242],[70,3604,3606],{"className":3605},[247,248,249,250],[70,3607,3609],{"className":3608},[185,250],[70,3610,3415],{"className":3611},[185,250],[70,3613,1951],{"className":3614},[197],[70,3616],{"className":3617,"style":202},[201],[70,3619,112],{"className":3620},[206],[70,3622],{"className":3623,"style":202},[201],[70,3625,3627,3631,3671,3674,3714,3717,3720,3723,3726,3758,3788],{"className":3626},[176],[70,3628],{"className":3629,"style":3630},[180],"height:1.1502em;vertical-align:-0.2861em;",[70,3632,3634,3637],{"className":3633},[185],[70,3635,3426],{"className":3636,"style":365},[185,193],[70,3638,3640],{"className":3639},[314],[70,3641,3643,3663],{"className":3642},[225,226],[70,3644,3646,3660],{"className":3645},[230],[70,3647,3649],{"className":3648,"style":324},[234],[70,3650,3651,3654],{"style":380},[70,3652],{"className":3653,"style":331},[242],[70,3655,3657],{"className":3656},[247,248,249,250],[70,3658,123],{"className":3659,"style":257},[185,193,250],[70,3661,291],{"className":3662},[290],[70,3664,3666],{"className":3665},[230],[70,3667,3669],{"className":3668,"style":347},[234],[70,3670],{},[70,3672,103],{"className":3673},[189],[70,3675,3677,3680],{"className":3676},[185],[70,3678,911],{"className":3679},[185,193],[70,3681,3683],{"className":3682},[314],[70,3684,3686,3706],{"className":3685},[225,226],[70,3687,3689,3703],{"className":3688},[230],[70,3690,3692],{"className":3691,"style":324},[234],[70,3693,3694,3697],{"style":327},[70,3695],{"className":3696,"style":331},[242],[70,3698,3700],{"className":3699},[247,248,249,250],[70,3701,123],{"className":3702,"style":257},[185,193,250],[70,3704,291],{"className":3705},[290],[70,3707,3709],{"className":3708},[230],[70,3710,3712],{"className":3711,"style":347},[234],[70,3713],{},[70,3715,103],{"className":3716},[189],[70,3718,1363],{"className":3719,"style":1390},[185,193],[70,3721,906],{"className":3722},[976],[70,3724],{"className":3725,"style":304},[201],[70,3727,3729,3732],{"className":3728},[185],[70,3730,1363],{"className":3731,"style":1390},[185,193],[70,3733,3735],{"className":3734},[314],[70,3736,3738],{"className":3737},[225],[70,3739,3741],{"className":3740},[230],[70,3742,3744],{"className":3743,"style":3596},[234],[70,3745,3746,3749],{"style":3599},[70,3747],{"className":3748,"style":331},[242],[70,3750,3752],{"className":3751},[247,248,249,250],[70,3753,3755],{"className":3754},[185,250],[70,3756,3415],{"className":3757},[185,250],[70,3759,3761,3764],{"className":3760},[197],[70,3762,109],{"className":3763},[197],[70,3765,3767],{"className":3766},[314],[70,3768,3770],{"className":3769},[225],[70,3771,3773],{"className":3772},[230],[70,3774,3777],{"className":3775,"style":3776},[234],"height:0.8641em;",[70,3778,3779,3782],{"style":3599},[70,3780],{"className":3781,"style":331},[242],[70,3783,3785],{"className":3784},[247,248,249,250],[70,3786,2938],{"className":3787},[185,250],[70,3789,109],{"className":3790},[197],[11,3792,3793,3794,3864,3865,3995],{},"The only requirements are that ",[70,3795,3797,3815],{"className":3796,"translate":74},[78],[70,3798,3800],{"className":3799},[82],[84,3801,3802],{"xmlns":86},[89,3803,3804,3812],{},[92,3805,3806],{},[137,3807,3808,3810],{},[95,3809,3426],{},[95,3811,123],{},[164,3813,3814],{"encoding":166},"g_f",[70,3816,3818],{"className":3817,"ariaHidden":172},[171],[70,3819,3821,3824],{"className":3820},[176],[70,3822],{"className":3823,"style":3032},[180],[70,3825,3827,3830],{"className":3826},[185],[70,3828,3426],{"className":3829,"style":365},[185,193],[70,3831,3833],{"className":3832},[314],[70,3834,3836,3856],{"className":3835},[225,226],[70,3837,3839,3853],{"className":3838},[230],[70,3840,3842],{"className":3841,"style":324},[234],[70,3843,3844,3847],{"style":380},[70,3845],{"className":3846,"style":331},[242],[70,3848,3850],{"className":3849},[247,248,249,250],[70,3851,123],{"className":3852,"style":257},[185,193,250],[70,3854,291],{"className":3855},[290],[70,3857,3859],{"className":3858},[230],[70,3860,3862],{"className":3861,"style":347},[234],[70,3863],{}," be twice differentiable near 0 and satisfy ",[70,3866,3868,3900],{"className":3867,"translate":74},[78],[70,3869,3871],{"className":3870},[82],[84,3872,3873],{"xmlns":86},[89,3874,3875,3897],{},[92,3876,3877,3886,3888,3890,3892,3895],{},[3878,3879,3880,3882,3884],"msubsup",{},[95,3881,3426],{},[95,3883,123],{},[100,3885,3415],{"mathvariant":97,"lspace":3414,"rspace":3414},[100,3887,103],{"stretchy":102},[1566,3889,2780],{},[100,3891,109],{"stretchy":102},[100,3893,3894],{},"\u003C",[1566,3896,2780],{},[164,3898,3899],{"encoding":166},"g_f'(0) \u003C 0",[70,3901,3903,3985],{"className":3902,"ariaHidden":172},[171],[70,3904,3906,3910,3967,3970,3973,3976,3979,3982],{"className":3905},[176],[70,3907],{"className":3908,"style":3909},[180],"height:1.1711em;vertical-align:-0.4192em;",[70,3911,3913,3916],{"className":3912},[185],[70,3914,3426],{"className":3915,"style":365},[185,193],[70,3917,3919],{"className":3918},[314],[70,3920,3922,3958],{"className":3921},[225,226],[70,3923,3925,3955],{"className":3924},[230],[70,3926,3929,3941],{"className":3927,"style":3928},[234],"height:0.7519em;",[70,3930,3932,3935],{"style":3931},"top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;",[70,3933],{"className":3934,"style":331},[242],[70,3936,3938],{"className":3937},[247,248,249,250],[70,3939,123],{"className":3940,"style":257},[185,193,250],[70,3942,3943,3946],{"style":1713},[70,3944],{"className":3945,"style":331},[242],[70,3947,3949],{"className":3948},[247,248,249,250],[70,3950,3952],{"className":3951},[185,250],[70,3953,3415],{"className":3954},[185,250],[70,3956,291],{"className":3957},[290],[70,3959,3961],{"className":3960},[230],[70,3962,3965],{"className":3963,"style":3964},[234],"height:0.4192em;",[70,3966],{},[70,3968,103],{"className":3969},[189],[70,3971,2780],{"className":3972},[185],[70,3974,109],{"className":3975},[197],[70,3977],{"className":3978,"style":202},[201],[70,3980,3894],{"className":3981},[206],[70,3983],{"className":3984,"style":202},[201],[70,3986,3988,3992],{"className":3987},[176],[70,3989],{"className":3990,"style":3991},[180],"height:0.6444em;",[70,3993,2780],{"className":3994},[185],"; beyond these conditions, no further restrictions are imposed.",[11,3997,3998,3999,4002,4003,4033,4034,4103,4104,4135,4136,4205],{},"Under the joint validity of Hypotheses 1 and 2, the authors prove the paper's central result, ",[29,4000,4001],{},"Theorem 1",": Let ",[70,4004,4006,4021],{"className":4005,"translate":74},[78],[70,4007,4009],{"className":4008},[82],[84,4010,4011],{"xmlns":86},[89,4012,4013,4018],{},[92,4014,4015],{},[95,4016,4017],{},"η",[164,4019,4020],{"encoding":166},"\\eta",[70,4022,4024],{"className":4023,"ariaHidden":172},[171],[70,4025,4027,4030],{"className":4026},[176],[70,4028],{"className":4029,"style":2801},[180],[70,4031,4017],{"className":4032,"style":365},[185,193]," be a finite-length path in the feature space ",[70,4035,4037,4054],{"className":4036,"translate":74},[78],[70,4038,4040],{"className":4039},[82],[84,4041,4042],{"xmlns":86},[89,4043,4044,4052],{},[92,4045,4046],{},[137,4047,4048,4050],{},[95,4049,901],{},[95,4051,123],{},[164,4053,1054],{"encoding":166},[70,4055,4057],{"className":4056,"ariaHidden":172},[171],[70,4058,4060,4063],{"className":4059},[176],[70,4061],{"className":4062,"style":561},[180],[70,4064,4066,4069],{"className":4065},[185],[70,4067,901],{"className":4068,"style":937},[185,193],[70,4070,4072],{"className":4071},[314],[70,4073,4075,4095],{"className":4074},[225,226],[70,4076,4078,4092],{"className":4077},[230],[70,4079,4081],{"className":4080,"style":324},[234],[70,4082,4083,4086],{"style":952},[70,4084],{"className":4085,"style":331},[242],[70,4087,4089],{"className":4088},[247,248,249,250],[70,4090,123],{"className":4091,"style":257},[185,193,250],[70,4093,291],{"className":4094},[290],[70,4096,4098],{"className":4097},[230],[70,4099,4101],{"className":4100,"style":347},[234],[70,4102],{},", and let ",[70,4105,4107,4122],{"className":4106,"translate":74},[78],[70,4108,4110],{"className":4109},[82],[84,4111,4112],{"xmlns":86},[89,4113,4114,4119],{},[92,4115,4116],{},[95,4117,4118],{},"γ",[164,4120,4121],{"encoding":166},"\\gamma",[70,4123,4125],{"className":4124,"ariaHidden":172},[171],[70,4126,4128,4131],{"className":4127},[176],[70,4129],{"className":4130,"style":2801},[180],[70,4132,4118],{"className":4133,"style":4134},[185,193],"margin-right:0.0556em;"," be its corresponding path on the manifold ",[70,4137,4139,4156],{"className":4138,"translate":74},[78],[70,4140,4142],{"className":4141},[82],[84,4143,4144],{"xmlns":86},[89,4145,4146,4154],{},[92,4147,4148],{},[137,4149,4150,4152],{},[95,4151,2115],{},[95,4153,123],{},[164,4155,2120],{"encoding":166},[70,4157,4159],{"className":4158,"ariaHidden":172},[171],[70,4160,4162,4165],{"className":4161},[176],[70,4163],{"className":4164,"style":561},[180],[70,4166,4168,4171],{"className":4167},[185],[70,4169,2115],{"className":4170,"style":2136},[185,193],[70,4172,4174],{"className":4173},[314],[70,4175,4177,4197],{"className":4176},[225,226],[70,4178,4180,4194],{"className":4179},[230],[70,4181,4183],{"className":4182,"style":324},[234],[70,4184,4185,4188],{"style":2151},[70,4186],{"className":4187,"style":331},[242],[70,4189,4191],{"className":4190},[247,248,249,250],[70,4192,123],{"className":4193,"style":257},[185,193,250],[70,4195,291],{"className":4196},[290],[70,4198,4200],{"className":4199},[230],[70,4201,4203],{"className":4202,"style":347},[234],[70,4204],{},". Then",[70,4207,4209],{"className":4208,"translate":74},[73],[70,4210,4212,4269],{"className":4211,"translate":74},[78],[70,4213,4215],{"className":4214},[82],[84,4216,4217],{"xmlns":86,"display":87},[89,4218,4219,4266],{},[92,4220,4221,4224,4226,4228,4230,4232,4255,4258,4260,4262,4264],{},[95,4222,4223],{},"L",[100,4225,103],{"stretchy":102},[95,4227,4118],{},[100,4229,109],{"stretchy":102},[100,4231,112],{},[4233,4234,4235],"msqrt",{},[92,4236,4237,4239,4241,4249,4251,4253],{},[100,4238,1564],{},[1566,4240,2938],{},[3878,4242,4243,4245,4247],{},[95,4244,3426],{},[95,4246,123],{},[100,4248,3415],{"mathvariant":97,"lspace":3414,"rspace":3414},[100,4250,103],{"stretchy":102},[1566,4252,2780],{},[100,4254,109],{"stretchy":102},[100,4256,4257],{},"⋅",[95,4259,4223],{},[100,4261,103],{"stretchy":102},[95,4263,4017],{},[100,4265,109],{"stretchy":102},[164,4267,4268],{"encoding":166},"L(\\gamma) = \\sqrt{-2g_f'(0)} \\cdot L(\\eta)",[70,4270,4272,4299,4450],{"className":4271,"ariaHidden":172},[171],[70,4273,4275,4278,4281,4284,4287,4290,4293,4296],{"className":4274},[176],[70,4276],{"className":4277,"style":181},[180],[70,4279,4223],{"className":4280},[185,193],[70,4282,103],{"className":4283},[189],[70,4285,4118],{"className":4286,"style":4134},[185,193],[70,4288,109],{"className":4289},[197],[70,4291],{"className":4292,"style":202},[201],[70,4294,112],{"className":4295},[206],[70,4297],{"className":4298,"style":202},[201],[70,4300,4302,4306,4441,4444,4447],{"className":4301},[176],[70,4303],{"className":4304,"style":4305},[180],"height:1.84em;vertical-align:-0.6498em;",[70,4307,4310],{"className":4308},[185,4309],"sqrt",[70,4311,4313,4432],{"className":4312},[225,226],[70,4314,4316,4429],{"className":4315},[230],[70,4317,4320,4406],{"className":4318,"style":4319},[234],"height:1.1902em;",[70,4321,4325,4329],{"className":4322,"style":4324},[4323],"svg-align","top:-3.8em;",[70,4326],{"className":4327,"style":4328},[242],"height:3.8em;",[70,4330,4333,4336,4339,4397,4400,4403],{"className":4331,"style":4332},[185],"padding-left:1em;",[70,4334,1564],{"className":4335},[185],[70,4337,2938],{"className":4338},[185],[70,4340,4342,4345],{"className":4341},[185],[70,4343,3426],{"className":4344,"style":365},[185,193],[70,4346,4348],{"className":4347},[314],[70,4349,4351,4388],{"className":4350},[225,226],[70,4352,4354,4385],{"className":4353},[230],[70,4355,4358,4370],{"className":4356,"style":4357},[234],"height:0.7337em;",[70,4359,4361,4364],{"style":4360},"top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;",[70,4362],{"className":4363,"style":331},[242],[70,4365,4367],{"className":4366},[247,248,249,250],[70,4368,123],{"className":4369,"style":257},[185,193,250],[70,4371,4373,4376],{"style":4372},"top:-3.0448em;margin-right:0.05em;",[70,4374],{"className":4375,"style":331},[242],[70,4377,4379],{"className":4378},[247,248,249,250],[70,4380,4382],{"className":4381},[185,250],[70,4383,3415],{"className":4384},[185,250],[70,4386,291],{"className":4387},[290],[70,4389,4391],{"className":4390},[230],[70,4392,4395],{"className":4393,"style":4394},[234],"height:0.4374em;",[70,4396],{},[70,4398,103],{"className":4399},[189],[70,4401,2780],{"className":4402},[185],[70,4404,109],{"className":4405},[197],[70,4407,4409,4412],{"style":4408},"top:-3.1502em;",[70,4410],{"className":4411,"style":4328},[242],[70,4413,4417],{"className":4414,"style":4416},[4415],"hide-tail","min-width:1.02em;height:1.88em;",[4418,4419,4425],"svg",{"xmlns":4420,"width":4421,"height":4422,"viewBox":4423,"preserveAspectRatio":4424},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.88em","0 0 400000 1944","xMinYMin slice",[4426,4427],"path",{"d":4428},"M983 90\nl0 -0\nc4,-6.7,10,-10,18,-10 H400000v40\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM1001 80h400000v40h-400000z",[70,4430,291],{"className":4431},[290],[70,4433,4435],{"className":4434},[230],[70,4436,4439],{"className":4437,"style":4438},[234],"height:0.6498em;",[70,4440],{},[70,4442],{"className":4443,"style":568},[201],[70,4445,4257],{"className":4446},[1730],[70,4448],{"className":4449,"style":568},[201],[70,4451,4453,4456,4459,4462,4465],{"className":4452},[176],[70,4454],{"className":4455,"style":181},[180],[70,4457,4223],{"className":4458},[185,193],[70,4460,103],{"className":4461},[189],[70,4463,4017],{"className":4464,"style":365},[185,193],[70,4466,109],{"className":4467},[197],[11,4469,4470,4471,4474,4475,4544],{},"In plain language: ",[29,4472,4473],{},"the length of the shortest path (geodesic) along the manifold is strictly proportional to the length of the corresponding shortest path in the feature space."," That is, even without knowing the exact form of ",[70,4476,4478,4495],{"className":4477,"translate":74},[78],[70,4479,4481],{"className":4480},[82],[84,4482,4483],{"xmlns":86},[89,4484,4485,4493],{},[92,4486,4487],{},[137,4488,4489,4491],{},[95,4490,3426],{},[95,4492,123],{},[164,4494,3814],{"encoding":166},[70,4496,4498],{"className":4497,"ariaHidden":172},[171],[70,4499,4501,4504],{"className":4500},[176],[70,4502],{"className":4503,"style":3032},[180],[70,4505,4507,4510],{"className":4506},[185],[70,4508,3426],{"className":4509,"style":365},[185,193],[70,4511,4513],{"className":4512},[314],[70,4514,4516,4536],{"className":4515},[225,226],[70,4517,4519,4533],{"className":4518},[230],[70,4520,4522],{"className":4521,"style":324},[234],[70,4523,4524,4527],{"style":380},[70,4525],{"className":4526,"style":331},[242],[70,4528,4530],{"className":4529},[247,248,249,250],[70,4531,123],{"className":4532,"style":257},[185,193,250],[70,4534,291],{"className":4535},[290],[70,4537,4539],{"className":4538},[230],[70,4540,4542],{"className":4541,"style":347},[234],[70,4543],{},", so long as the local cosine similarity is some smooth, decreasing function of the squared distance, the \"geodesic distance\" along the representation manifold—traveling along the manifold itself—recovers the true distance in concept space, exactly, up to a single uniform scaling constant.",[11,4546,4547],{},"This result addresses a hanging conjecture posed by Olah and Batson in 2024: they had written that \"the idea that feature manifolds are embedded in more complex ways than their topology alone would require, possibly in order to realize a specific distance metric, may be quite deep and important.\" This paper, in effect, uses the rigorous language of metric geometry to crystallize that intuitive conjecture into a theorem that can be both proved and falsified.",[11,4549,4550],{},"The proof itself draws on the classical definition of \"path length\" in metric geometry (the supremum of polygonal-approximation sums), with the central technique being a Taylor expansion of cosine similarity near zero, followed by a series of careful triangle-inequality arguments to bound the error terms—a solid piece of analytic reasoning (see the appendix for details).",[45,4552,4554],{"id":4553},"homeomorphism-is-easy-isometry-is-hard","Homeomorphism Is Easy; Isometry Is Hard",[11,4556,4557],{},"Theory established, the authors return to the three empirical case studies—colors, years, and dates—to test Hypothesis 2 and Theorem 1. They design two diagnostic approaches:",[2608,4559,4560,4598],{},[1028,4561,4562,4565,4566,4597],{},[29,4563,4564],{},"Direct method",": plot \"cosine similarity vs. squared distance\" as a scatter plot to see whether a locally decreasing trend emerges, using Chatterjee's correlation coefficient ",[70,4567,4569,4584],{"className":4568,"translate":74},[78],[70,4570,4572],{"className":4571},[82],[84,4573,4574],{"xmlns":86},[89,4575,4576,4581],{},[92,4577,4578],{},[95,4579,4580],{},"ξ",[164,4582,4583],{"encoding":166},"\\xi",[70,4585,4587],{"className":4586,"ariaHidden":172},[171],[70,4588,4590,4593],{"className":4589},[176],[70,4591],{"className":4592,"style":791},[180],[70,4594,4580],{"className":4595,"style":4596},[185,193],"margin-right:0.046em;"," to quantify the overall degree of functional dependence;",[1028,4599,4600,4603],{},[29,4601,4602],{},"Indirect method",": test Theorem 1 itself—estimate geodesic distances along the manifold using a k-nearest-neighbor graph, check whether they are proportional to geodesic distances in the feature space, and use the Pearson correlation coefficient to assess linearity.",[11,4605,4606,4607,4610,4611,4674],{},"The results are intriguing. For \"colors\" and \"dates\"—both of which carry a natural periodic structure—a simple circular metric (hue angle; day of the year) yields reasonably strong support for isometry (dates achieve a Pearson correlation of 0.97). ",[29,4608,4609],{},"\"Years,\" however, stumbles:"," if one assumes that the distance between years is simply ",[70,4612,4614,4638],{"className":4613,"translate":74},[78],[70,4615,4617],{"className":4616},[82],[84,4618,4619],{"xmlns":86},[89,4620,4621,4635],{},[92,4622,4623,4626,4628,4630,4633],{},[95,4624,4625],{"mathvariant":97},"∣",[95,4627,106],{},[100,4629,1564],{},[95,4631,4632],{},"y",[95,4634,4625],{"mathvariant":97},[164,4636,4637],{"encoding":166},"|x-y|",[70,4639,4641,4662],{"className":4640,"ariaHidden":172},[171],[70,4642,4644,4647,4650,4653,4656,4659],{"className":4643},[176],[70,4645],{"className":4646,"style":181},[180],[70,4648,4625],{"className":4649},[185],[70,4651,106],{"className":4652},[185,193],[70,4654],{"className":4655,"style":568},[201],[70,4657,1564],{"className":4658},[1730],[70,4660],{"className":4661,"style":568},[201],[70,4663,4665,4668,4671],{"className":4664},[176],[70,4666],{"className":4667,"style":181},[180],[70,4669,4632],{"className":4670,"style":365},[185,193],[70,4672,4625],{"className":4673},[185]," (e.g., 1990 and 2000 are 10 units apart), the empirical data do not support this hypothesis at all. Figure 4 reveals that years closer to \"the present\" (the paper takes GPT-2's release year of 2019 as the reference point) are stretched further apart on the manifold—their distances are \"magnified.\"",[11,4676,4677,4678,4759,4760,4763],{},"The authors therefore make a very elegant correction: they replace the metric space of years with Euclidean distance on ",[70,4679,4681,4712],{"className":4680,"translate":74},[78],[70,4682,4684],{"className":4683},[82],[84,4685,4686],{"xmlns":86},[89,4687,4688,4709],{},[92,4689,4690,4693,4696,4698,4701,4703,4707],{},[95,4691,4692],{},"log",[100,4694,4695],{},"⁡",[100,4697,103],{"stretchy":102},[1566,4699,4700],{},"2019",[100,4702,1564],{},[4704,4705,4706],"mtext",{},"year",[100,4708,109],{"stretchy":102},[164,4710,4711],{"encoding":166},"\\log(2019 - \\text{year})",[70,4713,4715,4743],{"className":4714,"ariaHidden":172},[171],[70,4716,4718,4721,4728,4731,4734,4737,4740],{"className":4717},[176],[70,4719],{"className":4720,"style":181},[180],[70,4722,4724,4725],{"className":4723},[220],"lo",[70,4726,3426],{"style":4727},"margin-right:0.0139em;",[70,4729,103],{"className":4730},[189],[70,4732,4700],{"className":4733},[185],[70,4735],{"className":4736,"style":568},[201],[70,4738,1564],{"className":4739},[1730],[70,4741],{"className":4742,"style":568},[201],[70,4744,4746,4749,4756],{"className":4745},[176],[70,4747],{"className":4748,"style":181},[180],[70,4750,4753],{"className":4751},[185,4752],"text",[70,4754,4706],{"className":4755},[185],[70,4757,109],{"className":4758},[197],"—that is, what the model may be encoding is not \"calendar year\" per se, but ",[29,4761,4762],{},"\"how long ago\" on a logarithmic scale",". This new metric space is topologically equivalent to the original year interval (homeomorphism is preserved, with rank correlations still near 1), but when isometry is tested under this new metric, the results flip from \"unsupported\" to \"strongly supported\" (Chatterjee coefficient 0.84, Pearson correlation 0.99).",[11,4765,4766,4767,4770],{},"This section is, I think, the most insightful empirical part of the entire paper: ",[29,4768,4769],{},"it demonstrates with clarity that \"homeomorphism\" and \"isometry\" are two entirely different levels of geometric fidelity."," The former asks only \"is the shape right?\"; the latter asks \"are the numerical distance relations right?\" A manifold can be topologically perfectly consistent with the concept space you envision, yet geometrically (in the specific functional form of the distance) be entirely different. And the process of deciphering \"how exactly is distance encoded\" is itself an act of reconnaissance into how the model \"understands\" the concept of time. That GPT-2 encodes \"more recent years\" as further apart from one another hints, in a certain sense, that years closer to the model's training cutoff may carry denser semantic texture—news and events are more densely packed—and are therefore allotted a larger \"representational space budget.\" This is a rather enchanting conjecture, though the paper itself does not delve deeply into a causal explanation for this pattern.",[45,4772,4774],{"id":4773},"the-platonic-representation-hypothesis","The Platonic Representation Hypothesis",[11,4776,4777,4778,4781],{},"While reading this paper, I found myself unable to resist drawing comparisons to another position paper, also from 2024, that provoked substantial discussion in the machine learning community—the ",[29,4779,4780],{},"Platonic Representation Hypothesis (PRH)"," proposed by Huh, Cheung, Wang, and Isola (ICML 2024). Although the two papers differ in their specific objects of study and methodology almost entirely, reading them side by side reveals that they operate at two different scales on the same overarching question, and that they are in fact complementary—arguably even two successive links in a causal chain.",[11,4783,4784,4785,4788,4789,4792],{},"The core claim of PRH is this: deep networks with different architectures, different modalities (vision, language), and different training objectives tend, as they scale up in size and task diversity, to ",[29,4786,4787],{},"converge"," their internal representations toward a shared geometric structure. The authors liken this hypothesized representation space—progressively approximated by more and more models—to the Platonic \"ideal reality\" (the realm of Forms) that exists independently of and transcends any particular concrete entity: the representation learned by each specific model is merely a noisy,biased projection of this \"Platonic representation.\" Their empirical evidence includes the finding that, when processing paired image-text data, the way image models and language models measure the distances between data points becomes increasingly similar as the models grow larger; and that across an expanding array of vision models with different architectures and training regimes, the pairwise representational similarity systematically rises. They further advance an explanatory conjecture: ",[29,4790,4791],{},"the driving force behind this convergence is that models, in the course of learning, are compelled to approximate the common, truth statistical structure of the data-generating process (\"reality\" itself)","—and the more diverse the tasks, the stronger this convergence pressure becomes.",[11,4794,4795,4796,4799,4800],{},"If PRH is concerned with \"",[29,4797,4798],{},"across different models",", whether the representation space as a whole is converging toward a single geometry,\" then the paper by Modell, Rubin-Delanchy, and Whiteley is concerned with a smaller and more specific scale: ",[29,4801,4802],{},"within a single model, why does the submanifold corresponding to a single feature (color, year, date) exhibit a particular geometric shape, and what is the precise mathematical relationship between this shape and the \"concept\" it corresponds to?",[11,4804,4805],{},"Taken together, the two papers can be read as tracing a rather coherent logical chain:",[2608,4807,4808,4814,4824],{},[1028,4809,4810,4811],{},"PRH advances a macroscopic conjecture—that the representation spaces of different models are converging toward a common geometry that reflects the \"true statistical structure of the world,\" and that the degree of this convergence can be quantified by the \"consistency in how models measure distances between data points\" (the PRH paper uses precisely tools such as kernel alignment and representational similarity, which assess the consistency of pairwise-distance patterns). But PRH itself remains at the level of \"existence\" and \"convergence trends\"; it does not delve deeply into a more foundational question: ",[29,4812,4813],{},"how, within the representation space of a single model, does distance itself correspond to the \"true\" distance in concept space?",[1028,4815,4816,4817,4820,4821],{},"The present paper, ",[15,4818,4819],{},"The Origins of Representation Manifolds",", answers precisely this more foundational, more microscopic question. Theorem 1 tells us: so long as there exists a smooth functional relationship between local cosine similarity and feature distance (Hypothesis 2), the geodesic distance along the representation manifold will recover the \"true\" distance in concept space, exactly, up to a single proportionality constant. This supplies, for PRH's claim that \"models learn to measure distances between data points in some consistent way,\" an explanation at the level of geometric mechanism: ",[29,4822,4823],{},"the model does not simply \"memorize\" distances; rather, by mapping features onto a manifold of a particular shape and curvature, it implicitly encodes the distance structure through the geodesic paths on that manifold.",[1028,4825,4826,4827],{},"Conversely, the circular structures observed for colors and dates in this paper—and the specific finding that \"colors are arranged in the hue order of the standard color wheel\"—can be seen as a microscopic corroboration of PRH: hue itself is a periodic structure that exists objectively in the physical world (the continuous spectrum of visible-light wavelengths, together with the response curves of the three types of cone cells in the human retina, jointly determining the color space). If different image models and different language models, independently and without coordination, all learn that \"hue is a circle,\" this is precisely an instance—concrete, isolatable, and testable—of the PRH phenomenon that different models converge to a shared representation reflecting the true statistical structure of the world. In other words, ",[29,4828,4829],{},"PRH provides the macroscopic narrative of \"why convergence happens\"; this paper provides the microscopic characterization of \"what the geometry looks like after convergence, and how this geometry precisely encodes semantic distance.\"",[11,4831,4832,4833,4836,4837,4839,4840,4843,4844,4847],{},"Of course, there exists an evident tension between the two works, and it is worth acknowledging honestly. The convergence claim of PRH rests, to a large extent, on ",[29,4834,4835],{},"cross-model comparisons","—using tools such as canonical correlation analysis and kernel alignment to compare how different models measure the pairwise-distance patterns of the same set of data points, which belongs to the statistics of \"relations between representations.\" ",[15,4838,4819],{},", by contrast, discusses from start to finish the geometric structure ",[29,4841,4842],{},"within a single model",", within the subspace of a single feature, and barely touches on the question of whether representations from different models are comparable to one another. That is to say, even if the theorem relating cosine similarity to geodesic distance holds for a particular model (e.g., GPT-2), it does not directly imply that another model with a completely different architecture would encode the feature \"year\" in the same way—with the same metric space, the same logarithmic scale. Indeed, the specific finding that \"years are encoded on a logarithmic scale, with distances increasingly stretched for years closer to the reference year\" carries, in itself, a considerable degree of ",[29,4845,4846],{},"model-specificity",". That the reference point for GPT-2 is naturally set as its own release year (2019) suggests that this encoding scheme is likely strongly tied to the temporal distribution of this particular model's training corpus, rather than being a universal, \"Platonic\" distance encoding to which all language models converge. If one were to repeat the same experiment on a model with a different training cutoff and a different corpus distribution, would the logarithmic scale and the reference point shift accordingly? This is, in fact, a question eminently worthy of direct investigation in follow-up work—and in a certain sense, it constitutes a 天然 experimental design for subjecting PRH's convergence claim to an empirical test at the finer granularity of \"the specific geometric parameters of a single feature.\"",[11,4849,4850,4851,4854,4855,4857],{},"In my assessment, reading these two papers together is more illuminating than reading either alone: ",[29,4852,4853],{},"PRH tells us that \"everyone is converging toward the same direction,\" while this paper supplies the mathematical language for characterizing what exactly that convergence converges to, and how such convergence can be rigorously measured and falsified."," If, in the future, someone wishes to genuinely test whether PRH holds at the level of specific features—for instance, whether the circular structure of hue or the logarithmic encoding of time reappear consistently across models with different architectures and training data—then the homeomorphism tests, the isometry tests (Chatterjee coefficient, Pearson correlation of KNN-based geodesic distances) proposed in ",[15,4856,4819],{}," constitute an almost ready-made, standardized toolbox. This may well be this paper's most broadly reusable \"by-product,\" beyond its own mathematical results.",[45,4859,4861],{"id":4860},"concluding-remarks","Concluding Remarks",[11,4863,4864],{},"Having read the paper, I would like to offer my assessment from several angles.",[11,4866,4867,4868,4871],{},"First, this is a rare effort to ",[29,4869,4870],{},"rigorize a vague intuition",", and its primary value lies in providing language and tools rather than a definitive answer. The field of mechanistic interpretability presently abounds in discoveries that amount to \"looking at pictures and describing them\"—\"oh, this feature looks like a circle,\" \"this one looks like a tree structure\"—but lacks a unified mathematical language with which to organize these observations. Defining features as metric spaces, and cleanly separating the notions of \"homeomorphism\" and \"isometry,\" is itself an important conceptual tool: it forces researchers to translate the vague intuition of \"I think this manifold corresponds to that concept\" into concrete, falsifiable hypotheses (\"is this the right metric space?\"), and provides specific statistical testing procedures (Chatterjee coefficient, geodesic distances via k-nearest-neighbor graphs). This kind of rigor is currently in relatively short supply in the field.",[11,4873,4874],{},"Second, Theorem 1 is elegant mathematics, but its \"probative force\" depends, to some degree, on a rather strong premise—that a smooth functional relationship exists between cosine similarity and distance. This hypothesis itself is not derived from model training dynamics or architectural design considerations, but is instead placed on the table as a \"reasonable guess\" and then verified empirically. In other words, the paper reads more like \"if this hypothesis holds, what beautiful corollaries follow?\" than \"why the representation space necessarily exhibits this structure\" (this is the sense in which the word \"Origins\" in the title is perhaps a touch wishful—the paper does not actually derive, from optimization objectives or the dynamics of gradient descent, why manifolds \"emerge\"; it largely operates under the premise that they already exist and proceeds to characterize what they should look like). This is not entirely a criticism, however: mechanistic interpretability as a whole is still at the stage of \"observing phenomena and constructing descriptive theories,\" and very few works genuinely derive representation geometry from first principles. This paper at least executes the \"descriptive theory\" step with greater rigor than most comparable efforts.",[11,4876,4877,4878,4881],{},"Third, the logarithmic-scale discovery for years is the single most impressive—and most thought-provoking—empirical result in the entire paper. ",[29,4879,4880],{},"It hints at a methodological trap worth guarding against: the homeomorphism test (is the topology right?) has a very low bar; it is easy to get something that \"looks right,\" but this is far from sufficient to demonstrate that we genuinely understand the model's encoding scheme."," Had the authors stopped at \"years are arranged along a curve whose order matches the true chronological order, with a rank correlation of 0.97, indicating that the model has learned the ordering of years,\" this conclusion would be rather a hole—virtually any monotonic mapping could achieve this. What carries genuine information is the non-uniform stretching of distances revealed by the isometry test, and the detailed hypothesis behind it: encoding \"time since the reference point\" on a logarithmic scale. This reminds us that future interpretability research of a similar kind should not stop at \"does the shape look similar?\"; rather, it should, as this paper does, press further to ask \"is the distance metric correct?\"—for only then can we excavate the implicit assumptions that the model has truly encoded.",[11,4883,4884,4885,4954],{},"Fourth, from an applied perspective, this paper carries direct methodological implications for steering research, but remains some distance from a truly operational tool. The paper concludes by noting that, if one could learn ",[70,4886,4888,4905],{"className":4887,"translate":74},[78],[70,4889,4891],{"className":4890},[82],[84,4892,4893],{"xmlns":86},[89,4894,4895,4903],{},[92,4896,4897],{},[137,4898,4899,4901],{},[95,4900,1537],{},[95,4902,123],{},[164,4904,2047],{"encoding":166},[70,4906,4908],{"className":4907,"ariaHidden":172},[171],[70,4909,4911,4914],{"className":4910},[176],[70,4912],{"className":4913,"style":1211},[180],[70,4915,4917,4920],{"className":4916},[185],[70,4918,1537],{"className":4919},[185,193],[70,4921,4923],{"className":4922},[314],[70,4924,4926,4946],{"className":4925},[225,226],[70,4927,4929,4943],{"className":4928},[230],[70,4930,4932],{"className":4931,"style":324},[234],[70,4933,4934,4937],{"style":327},[70,4935],{"className":4936,"style":331},[242],[70,4938,4940],{"className":4939},[247,248,249,250],[70,4941,123],{"className":4942,"style":257},[185,193,250],[70,4944,291],{"className":4945},[290],[70,4947,4949],{"className":4948},[230],[70,4950,4952],{"className":4951,"style":347},[234],[70,4953],{}," (the map from features to the manifold), one could in principle perform more refined representation editing that respects the intrinsic geometric structure of the concept—for example, to \"shift\" a date feature forward by half a year, one should travel along the manifold's geodesic rather than simply adding a vector in Euclidean space. This is a promising direction, and it echoes the authors' call for \"manifold-aware SAEs.\" At present, however, this remains at the conceptual level: how to robustly estimate a high-dimensional, noisy manifold is still an unsolved statistical problem, as the authors themselves frankly acknowledge in the limitations section.",[11,4956,4957,4958,4961],{},"Fifth, a contribution that may be underappreciated is the paper's implications for text embedding services. Many practitioners working on RAG, semantic search, and recommendation systems unthinkingly use cosine similarity as a measure of \"semantic proximity,\" rarely pausing to ask what geometric structure actually lies beneath that numerical similarity score. This paper serves as a reminder: ",[29,4959,4960],{},"cosine similarity can indeed faithfully reflect distances \"along the intrinsic geometric paths of concept space\"—but this is a local property, contingent on Hypothesis 2 holding, and the geometric shape differs entirely from one feature to another (circle, line segment, logarithmic line segment)."," Treating it as a universal, cross-feature \"semantic distance\" metric may obscure a great deal of important nonlinear structure (as the color example illustrates, where \"cosine similarity at large distances manifestly deviates from an isometric relationship\"). This is a caution worth lodging in the minds of engineers tuning embedding-based retrieval systems and anomaly detection pipelines.",[11,4963,4964,4965,4968],{},"This paper does not attempt to explain all representational-geometric phenomena, nor does it deliver a tool ready for production deployment. It simply does \"a mathematician's job\": ",[29,4966,4967],{},"for a phenomenon that has been widely observed yet lacks precise definition, erect a minimal but rigorous theoretical framework, and honestly take it to the data to test where its boundaries lie."," In a field that increasingly relies on \"alchemical,\" empirically driven observation, this kind of work—returning to first principles, willing to state its assumptions clearly, write out its proofs completely, and honestly its limitations openly—is intrinsically worth being seen by more people.",[11,4970,4971,4972],{},"If I had to name the single deepest impression this paper left on me, it would probably be the specific finding that \"years are encoded as logarithmic time-distance.\" Through one minimal example, it shows that behind the seemingly plain geometry of vector spaces, there may truly reside something akin to a human cognitive intuition: recent events are sharper, more finely discriminated; distant events are compressed in memory. This is perhaps the most captivating thing about mechanistic interpretability research: ",[29,4973,4974],{},"not proving that the model is a black-box piece of magic, but, little by little, translating what it is truly \"thinking\" on the inside into a language we can read.",{"title":4976,"searchDepth":4977,"depth":4977,"links":4978},"",2,[4979,4980,4981,4982,4983,4984,4985,4986],{"id":47,"depth":4977,"text":48},{"id":868,"depth":4977,"text":869},{"id":2602,"depth":4977,"text":2603},{"id":2644,"depth":4977,"text":2645},{"id":3339,"depth":4977,"text":3340},{"id":4553,"depth":4977,"text":4554},{"id":4773,"depth":4977,"text":4774},{"id":4860,"depth":4977,"text":4861},"If you have followed the progress in mechanistic interpretability over the past two years, you have likely encountered a certain kind of figure: when the activation vectors of a particular layer in a large language model are projected down to two or three dimensions and plotted, the representations of concepts such as \"months,\" \"days of the week,\" \"years,\" and \"colors\" do not scatter chaotically through space, but instead arrange themselves along an elegant curve—sometimes even a closed circle or a torus. Researchers such as Chris Olah, Josh Batson at Anthropic, and Engels et al. have all demonstrated this phenomenon in works including Not All Language Model Features Are One-Dimensionally Linear.",false,true,"md",null,{},"2026-08-04","\u002Fblog\u002F2026\u002F2026-08-04-the-representations-in-llm",{"title":6,"description":4987},"blog\u002F2026\u002F2026-08-04-the-representations-in-llm","Formalizes LLM representation manifolds by defining features as metric spaces, proving cosine similarity encodes geodesic feature distance, then validating homeomorphism and isometry empirically on colors, dates, and years.",[4999,5000],"machine-learning","ai","XOOXSccYekvL-iDyMBWQV5yyBGYpJ28WpGhxDSsSo00",{"id":5003,"title":5004,"body":5005,"description":5009,"draft":4988,"enableComment":4989,"extension":4990,"image":4976,"important":4988,"location":5161,"meta":5162,"navigation":4989,"ogImage":4991,"onday":5163,"path":5164,"seo":5165,"stem":5166,"summary":5167,"tags":5168,"__hash__":5170},"blog\u002Fblog\u002F2026\u002F2026-08-13-xie-zuo-yu-yan-de-zhuan-bian.md","写作语言的转变",{"type":8,"value":5006,"toc":5159},[5007,5010,5013,5016,5141,5144,5147,5150,5153,5156],[11,5008,5009],{},"长期以来，我对自己的博客定位从来不是分享，而更多是给自己看的「笔记」。很多情况是：记录一个刚刚搞懂的东西 → 写下来 → 以后自己查 → 写完就结束。我的博客基本都很短，基本三分钟以内就能读完。大约在 2023 年开始，同时为了学习英语，锻炼技术写作能力，我在疯狂背单词的同时，也可以保持用英语进行技术写作。现在坚持了快两年。所以长期以来，利用英语写作，对我来说压力并不大。写作时不需要追求完整论证，只需要把自己的思路编码下来。哪怕句子稍微生硬一点，只要自己以后能看懂，就已经达成目的。",[11,5011,5012],{},"过去两年的英语写作之所以轻松，是因为它服务于「笔记」功能——短、线性、结论明确。这种写作不需要复杂的逻辑嵌套，不需要反复回看和修改结构，英语的线性特征反而成了一种约束，帮你把想法压缩成清晰的短句。",[11,5014,5015],{},"但是后来，情况发生了变化，我的博客从「给自己看的笔记」的定位，开始转移到了「表达与推演」。当写作从记录变成表达与推演，语言就从工具变成了负担。写作负荷不是恒定的，它会随着结构复杂度和语言熟练度的乘积而非线性增长。",[11,5017,5018,5019,5140],{},"文章越来越长，关于技术写作的部分，有的文章我需要插入大量的 ",[70,5020,5022,5037],{"className":5021,"translate":74},[78],[70,5023,5025],{"className":5024},[82],[84,5026,5027],{"xmlns":86},[89,5028,5029,5034],{},[92,5030,5031],{},[4704,5032,5033],{},"LaTeX",[164,5035,5036],{"encoding":166},"\\LaTeX",[70,5038,5040],{"className":5039,"ariaHidden":172},[171],[70,5041,5043,5047],{"className":5042},[176],[70,5044],{"className":5045,"style":5046},[180],"height:0.8988em;vertical-align:-0.2155em;",[70,5048,5050,5054,5058,5081,5085],{"className":5049},[185,4752],[70,5051,4223],{"className":5052},[185,5053],"textrm",[70,5055],{"className":5056,"style":5057},[201],"margin-right:-0.36em;",[70,5059,5061],{"className":5060},[225],[70,5062,5064],{"className":5063},[230],[70,5065,5068],{"className":5066,"style":5067},[234],"height:0.6833em;",[70,5069,5071,5074],{"style":5070},"top:-2.905em;",[70,5072],{"className":5073,"style":331},[242],[70,5075,5077],{"className":5076},[185],[70,5078,5080],{"className":5079},[185,5053,250,247,248,249],"A",[70,5082],{"className":5083,"style":5084},[201],"margin-right:-0.15em;",[70,5086,5088,5092,5096,5132,5136],{"className":5087},[185,4752],[70,5089,5091],{"className":5090},[185,5053],"T",[70,5093],{"className":5094,"style":5095},[201],"margin-right:-0.1667em;",[70,5097,5099,5123],{"className":5098},[225,226],[70,5100,5102,5120],{"className":5101},[230],[70,5103,5106],{"className":5104,"style":5105},[234],"height:0.4678em;",[70,5107,5109,5113],{"style":5108},"top:-2.7845em;",[70,5110],{"className":5111,"style":5112},[242],"height:3em;",[70,5114,5116],{"className":5115},[185],[70,5117,5119],{"className":5118},[185,5053],"E",[70,5121,291],{"className":5122},[290],[70,5124,5126],{"className":5125},[230],[70,5127,5130],{"className":5128,"style":5129},[234],"height:0.2155em;",[70,5131],{},[70,5133],{"className":5134,"style":5135},[201],"margin-right:-0.125em;",[70,5137,5139],{"className":5138},[185,5053],"X"," 公式、推导和演绎过程。此时再使用非母语进行技术写作时心智负担显著增大。短文里，一句话如果不知道怎么写，一般情况可以用简单句子绕一下，问题不大。但是到了长篇技术文章，可能连续几页都在描述一个复杂的思想。这时候需要同时控制数学符号、技术概念、论证结构、前后术语一致性、句法，还有段落之间的衔接。于是工作记忆里面同时跑着很多东西。",[11,5142,5143],{},"而且，长篇博客并不是一次写完的，很可能需要花上几个星期慢慢打磨，或者经常回顾我自己的博客的思想。写作的时候，我大多数时间都在思考：「这个理论到底应该怎么解释？」。但是，结果脑子里却不断出现：「这里应该用 which 还是 that？」、「这个东西应该叫 representation 还是 formulation？」、「这个词性是否合适，有没有对应的名词形式？」之类细枝末节的语法问题。久而久之，写作起来会非常累。在短文本写作里，这种差异几乎感觉不到。但当文本从几百字增长到几千、几万字以后，语言的视觉结构、信息密度、词法形态、定位效率和工作记忆负担都会开始成为写作系统的一部分。",[11,5145,5146],{},"另外一种原因，英文是线形文字，阅读效率天然就很低：想在长文中定位到某一块位置，必须要从每一段从头开始逐行扫过，无法像汉字那样逐块扫描。很多时候我在长文写作中，需要不断地往回头看。而英语的语法线性强，从句嵌套多了以后，读者和作者都容易迷失。因为汉字同时包含视觉图像和声音两种信息，它信息密度和视觉辨识特征，使中文文本非常适合视觉扫描。而英文单词之间存在大量空格，真正有语义重量的东西被拆成了很多视觉单元。所以当文章达到几千甚至上万字以后，回来看自己几周前写的东西，会出现一种很奇怪的体验：中文是在「看结构」，英文更容易变成「读句子」。所以，就导致语义单元和视觉单元不对齐——一个概念可能要三四个单词才能表达，视觉上要扫过更长的距离才能抓住一个完整意群。",[11,5148,5149],{},"其实不止是我，很多技术写作者都会遇到一个问题：语言不是中性的容器，它会反过来塑造你思考的形状。",[11,5151,5152],{},"笔记型写作的特点，思维已经完成。只是在编码一个已经清晰的结论，供未来的自己检索。表达与推演型写作则不同，写作本身就是思维过程。你在写的过程中推演、发现、修正。文字不是思维的镜像，而是思维的工具。当写作成为思考工具时，语言就不再只是输出端的问题，而是输入端的问题。非常需要语言来帮助你组织尚未成形的想法，来试探逻辑的边界，来连接不同的概念。",[11,5154,5155],{},"从此以后，我的博客功能，从以前的博客 externalized memory（外部记忆），到现在逐渐变成了 externalized thinking（外部化思考）。",[11,5157,5158],{},"也许从英语写作到中文写作，是在为思维本身让路。让认知资源从语言操作中解放出来，还给真正的思考。",{"title":4976,"searchDepth":4977,"depth":4977,"links":5160},[],"河南郑州",{},"2026-08-13","\u002Fblog\u002F2026\u002F2026-08-13-xie-zuo-yu-yan-de-zhuan-bian",{"title":5004,"description":5009},"blog\u002F2026\u002F2026-08-13-xie-zuo-yu-yan-de-zhuan-bian","长期以来，为了学习英语并增强熟练度，在技术写作时，我都会刻意使用英语来写作。但是后来我发现使用英语的阅读和思考心智负担非常大。从笔记到表达与推演，写作目标发生变化后，认知资源也需要实现对应的重新分配。",[5169],"thoughts","GgbL6w_I6tbQihpjDly03_3pFg9J3kM5x2weDCk5vHw",{"id":5172,"title":5173,"body":5174,"description":21430,"draft":4988,"enableComment":4989,"extension":4990,"image":4976,"important":4988,"location":4991,"meta":21431,"navigation":4989,"ogImage":4991,"onday":21432,"path":21433,"seo":21434,"stem":21435,"summary":21436,"tags":21437,"__hash__":21441},"blog\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en.md","The Diagonal Argument: A Proof Technique That Transformed Mathematics and Computer Science",{"type":8,"value":5175,"toc":21419},[5176,5183,5186,5190,5193,5323,5463,6018,6183,6277,6280,7200,7203,7207,7268,7323,7329,7484,8922,9126,9378,9798,10034,10069,10253,10634,10943,11038,11193,11277,11462,11741,12877,12880,12884,12887,13195,13200,13207,13370,13374,13377,13607,13694,13699,13879,14006,14196,14199,14450,14505,15380,15556,15563,16410,16734,16737,17049,17053,17059,17155,17159,17166,17288,17295,17300,17600,17603,17637,17971,18097,18147,18564,18603,18800,19642,19823,19826,20064,20068,20074,20088,20092,20095,20103,20106,20694,20697,20938,20941,20945,20948,21322,21325,21412,21415],[11,5177,5178,5179,5182],{},"In 1891, the German mathematician Georg Cantor published a paper of merely a few pages, proving a conclusion that appeared counterintuitive at the time: ",[29,5180,5181],{},"there exist infinities of different \"sizes.\""," The proof technique he employed — the diagonal argument — not only resolved the problem at hand, but went on to become one of the most formidable weapons in the arsenal of mathematical logic, computability theory, and even philosophy over the ensuing century and more.",[11,5184,5185],{},"From \"there are more real numbers than natural numbers,\" to Turing's proof that \"the halting problem is undecidable,\" to Gödel's demonstration that \"any sufficiently strong formal system contains undecidable propositions\" — these results, seemingly disparate in subject matter, are at their core variants of the same idea. This article will guide the reader from the most elementary foundations of set theory, deriving the complete process of the diagonal argument step by step, elucidating why it possesses such power, and rendering the abstract proof into a concrete process that can be executed and observed through code.",[45,5187,5189],{"id":5188},"equal-in-size","Equal in Size",[11,5191,5192],{},"Before discussing infinite sets, we must first clarify a preliminary question: how does one compare the sizes of two sets?",[11,5194,5195,5196,5322],{},"For finite sets, the answer is straightforward: simply count the number of elements. But the set of natural numbers ",[70,5197,5199,5245],{"className":5198,"translate":74},[78],[70,5200,5202],{"className":5201},[82],[84,5203,5204],{"xmlns":86},[89,5205,5206,5242],{},[92,5207,5208,5212,5214,5217,5219,5221,5223,5225,5227,5229,5232,5234,5236,5239],{},[95,5209,5211],{"mathvariant":5210},"double-struck","N",[100,5213,112],{},[100,5215,5216],{"stretchy":102},"{",[1566,5218,2780],{},[100,5220,906],{"separator":172},[1566,5222,1568],{},[100,5224,906],{"separator":172},[1566,5226,2938],{},[100,5228,906],{"separator":172},[1566,5230,5231],{},"3",[100,5233,906],{"separator":172},[100,5235,3004],{},[4704,5237,5238],{}," ",[100,5240,5241],{"stretchy":102},"}",[164,5243,5244],{"encoding":166},"\\mathbb{N} = \\{0, 1, 2, 3, \\dots\\}",[70,5246,5248,5268],{"className":5247,"ariaHidden":172},[171],[70,5249,5251,5255,5259,5262,5265],{"className":5250},[176],[70,5252],{"className":5253,"style":5254},[180],"height:0.6889em;",[70,5256,5211],{"className":5257},[185,5258],"mathbb",[70,5260],{"className":5261,"style":202},[201],[70,5263,112],{"className":5264},[206],[70,5266],{"className":5267,"style":202},[201],[70,5269,5271,5274,5277,5280,5283,5286,5289,5292,5295,5298,5301,5304,5307,5310,5313,5316,5319],{"className":5270},[176],[70,5272],{"className":5273,"style":181},[180],[70,5275,5216],{"className":5276},[189],[70,5278,2780],{"className":5279},[185],[70,5281,906],{"className":5282},[976],[70,5284],{"className":5285,"style":304},[201],[70,5287,1568],{"className":5288},[185],[70,5290,906],{"className":5291},[976],[70,5293],{"className":5294,"style":304},[201],[70,5296,2938],{"className":5297},[185],[70,5299,906],{"className":5300},[976],[70,5302],{"className":5303,"style":304},[201],[70,5305,5231],{"className":5306},[185],[70,5308,906],{"className":5309},[976],[70,5311],{"className":5312,"style":304},[201],[70,5314,3004],{"className":5315},[3082],[70,5317],{"className":5318,"style":304},[201],[70,5320,5241],{"className":5321},[197]," is infinite and cannot be \"counted to completion\"; we therefore require a new instrument.",[11,5324,5325,5328,5329,5357,5358,5388,5389,5462],{},[29,5326,5327],{},"Definition (bijection)."," Given two sets ",[70,5330,5332,5345],{"className":5331,"translate":74},[78],[70,5333,5335],{"className":5334},[82],[84,5336,5337],{"xmlns":86},[89,5338,5339,5343],{},[92,5340,5341],{},[95,5342,5080],{},[164,5344,5080],{"encoding":166},[70,5346,5348],{"className":5347,"ariaHidden":172},[171],[70,5349,5351,5354],{"className":5350},[176],[70,5352],{"className":5353,"style":5067},[180],[70,5355,5080],{"className":5356},[185,193]," and ",[70,5359,5361,5375],{"className":5360,"translate":74},[78],[70,5362,5364],{"className":5363},[82],[84,5365,5366],{"xmlns":86},[89,5367,5368,5373],{},[92,5369,5370],{},[95,5371,5372],{},"B",[164,5374,5372],{"encoding":166},[70,5376,5378],{"className":5377,"ariaHidden":172},[171],[70,5379,5381,5384],{"className":5380},[176],[70,5382],{"className":5383,"style":5067},[180],[70,5385,5372],{"className":5386,"style":5387},[185,193],"margin-right:0.0502em;",", if there exists a function ",[70,5390,5392,5414],{"className":5391,"translate":74},[78],[70,5393,5395],{"className":5394},[82],[84,5396,5397],{"xmlns":86},[89,5398,5399,5411],{},[92,5400,5401,5403,5405,5407,5409],{},[95,5402,123],{},[100,5404,68],{},[95,5406,5080],{},[100,5408,1550],{},[95,5410,5372],{},[164,5412,5413],{"encoding":166},"f: A \\to B",[70,5415,5417,5435,5453],{"className":5416,"ariaHidden":172},[171],[70,5418,5420,5423,5426,5429,5432],{"className":5419},[176],[70,5421],{"className":5422,"style":791},[180],[70,5424,123],{"className":5425,"style":257},[185,193],[70,5427],{"className":5428,"style":202},[201],[70,5430,68],{"className":5431},[206],[70,5433],{"className":5434,"style":202},[201],[70,5436,5438,5441,5444,5447,5450],{"className":5437},[176],[70,5439],{"className":5440,"style":5067},[180],[70,5442,5080],{"className":5443},[185,193],[70,5445],{"className":5446,"style":202},[201],[70,5448,1550],{"className":5449},[206],[70,5451],{"className":5452,"style":202},[201],[70,5454,5456,5459],{"className":5455},[176],[70,5457],{"className":5458,"style":5067},[180],[70,5460,5372],{"className":5461,"style":5387},[185,193]," satisfying:",[1025,5464,5465,5837],{},[1028,5466,5467,5470,5471,5836],{},[29,5468,5469],{},"Injective",": ",[70,5472,5474,5530],{"className":5473,"translate":74},[78],[70,5475,5477],{"className":5476},[82],[84,5478,5479],{"xmlns":86},[89,5480,5481,5527],{},[92,5482,5483,5489,5492,5498,5501,5503,5505,5511,5513,5515,5517,5519,5525],{},[137,5484,5485,5487],{},[95,5486,18],{},[1566,5488,1568],{},[100,5490,5491],{"mathvariant":97},"≠",[137,5493,5494,5496],{},[95,5495,18],{},[1566,5497,2938],{},[100,5499,5500],{},"⇒",[95,5502,123],{},[100,5504,103],{"stretchy":102},[137,5506,5507,5509],{},[95,5508,18],{},[1566,5510,1568],{},[100,5512,109],{"stretchy":102},[100,5514,5491],{"mathvariant":97},[95,5516,123],{},[100,5518,103],{"stretchy":102},[137,5520,5521,5523],{},[95,5522,18],{},[1566,5524,2938],{},[100,5526,109],{"stretchy":102},[164,5528,5529],{"encoding":166},"a_1 \\neq a_2 \\Rightarrow f(a_1) \\neq f(a_2)",[70,5531,5533,5628,5684,5781],{"className":5532,"ariaHidden":172},[171],[70,5534,5536,5539,5579,5582,5625],{"className":5535},[176],[70,5537],{"className":5538,"style":791},[180],[70,5540,5542,5545],{"className":5541},[185],[70,5543,18],{"className":5544},[185,193],[70,5546,5548],{"className":5547},[314],[70,5549,5551,5571],{"className":5550},[225,226],[70,5552,5554,5568],{"className":5553},[230],[70,5555,5557],{"className":5556,"style":2820},[234],[70,5558,5559,5562],{"style":327},[70,5560],{"className":5561,"style":331},[242],[70,5563,5565],{"className":5564},[247,248,249,250],[70,5566,1568],{"className":5567},[185,250],[70,5569,291],{"className":5570},[290],[70,5572,5574],{"className":5573},[230],[70,5575,5577],{"className":5576,"style":2841},[234],[70,5578],{},[70,5580],{"className":5581,"style":202},[201],[70,5583,5585,5618,5622],{"className":5584},[206],[70,5586,5588],{"className":5587},[206],[70,5589,5592],{"className":5590},[185,5591],"vbox",[70,5593,5596],{"className":5594},[5595],"thinbox",[70,5597,5600,5603,5614],{"className":5598},[5599],"rlap",[70,5601],{"className":5602,"style":791},[180],[70,5604,5607],{"className":5605},[5606],"inner",[70,5608,5610],{"className":5609},[185],[70,5611,5613],{"className":5612},[206],"",[70,5615],{"className":5616},[5617],"fix",[70,5619],{"className":5620},[201,5621],"nobreak",[70,5623,112],{"className":5624},[206],[70,5626],{"className":5627,"style":202},[201],[70,5629,5631,5635,5675,5678,5681],{"className":5630},[176],[70,5632],{"className":5633,"style":5634},[180],"height:0.5806em;vertical-align:-0.15em;",[70,5636,5638,5641],{"className":5637},[185],[70,5639,18],{"className":5640},[185,193],[70,5642,5644],{"className":5643},[314],[70,5645,5647,5667],{"className":5646},[225,226],[70,5648,5650,5664],{"className":5649},[230],[70,5651,5653],{"className":5652,"style":2820},[234],[70,5654,5655,5658],{"style":327},[70,5656],{"className":5657,"style":331},[242],[70,5659,5661],{"className":5660},[247,248,249,250],[70,5662,2938],{"className":5663},[185,250],[70,5665,291],{"className":5666},[290],[70,5668,5670],{"className":5669},[230],[70,5671,5673],{"className":5672,"style":2841},[234],[70,5674],{},[70,5676],{"className":5677,"style":202},[201],[70,5679,5500],{"className":5680},[206],[70,5682],{"className":5683,"style":202},[201],[70,5685,5687,5690,5693,5696,5736,5739,5742,5778],{"className":5686},[176],[70,5688],{"className":5689,"style":181},[180],[70,5691,123],{"className":5692,"style":257},[185,193],[70,5694,103],{"className":5695},[189],[70,5697,5699,5702],{"className":5698},[185],[70,5700,18],{"className":5701},[185,193],[70,5703,5705],{"className":5704},[314],[70,5706,5708,5728],{"className":5707},[225,226],[70,5709,5711,5725],{"className":5710},[230],[70,5712,5714],{"className":5713,"style":2820},[234],[70,5715,5716,5719],{"style":327},[70,5717],{"className":5718,"style":331},[242],[70,5720,5722],{"className":5721},[247,248,249,250],[70,5723,1568],{"className":5724},[185,250],[70,5726,291],{"className":5727},[290],[70,5729,5731],{"className":5730},[230],[70,5732,5734],{"className":5733,"style":2841},[234],[70,5735],{},[70,5737,109],{"className":5738},[197],[70,5740],{"className":5741,"style":202},[201],[70,5743,5745,5772,5775],{"className":5744},[206],[70,5746,5748],{"className":5747},[206],[70,5749,5751],{"className":5750},[185,5591],[70,5752,5754],{"className":5753},[5595],[70,5755,5757,5760,5769],{"className":5756},[5599],[70,5758],{"className":5759,"style":791},[180],[70,5761,5763],{"className":5762},[5606],[70,5764,5766],{"className":5765},[185],[70,5767,5613],{"className":5768},[206],[70,5770],{"className":5771},[5617],[70,5773],{"className":5774},[201,5621],[70,5776,112],{"className":5777},[206],[70,5779],{"className":5780,"style":202},[201],[70,5782,5784,5787,5790,5793,5833],{"className":5783},[176],[70,5785],{"className":5786,"style":181},[180],[70,5788,123],{"className":5789,"style":257},[185,193],[70,5791,103],{"className":5792},[189],[70,5794,5796,5799],{"className":5795},[185],[70,5797,18],{"className":5798},[185,193],[70,5800,5802],{"className":5801},[314],[70,5803,5805,5825],{"className":5804},[225,226],[70,5806,5808,5822],{"className":5807},[230],[70,5809,5811],{"className":5810,"style":2820},[234],[70,5812,5813,5816],{"style":327},[70,5814],{"className":5815,"style":331},[242],[70,5817,5819],{"className":5818},[247,248,249,250],[70,5820,2938],{"className":5821},[185,250],[70,5823,291],{"className":5824},[290],[70,5826,5828],{"className":5827},[230],[70,5829,5831],{"className":5830,"style":2841},[234],[70,5832],{},[70,5834,109],{"className":5835},[197]," (distinct inputs yield distinct outputs)",[1028,5838,5839,5842,5843,5896,5897,5949,5950,6017],{},[29,5840,5841],{},"Surjective",": for every ",[70,5844,5846,5865],{"className":5845,"translate":74},[78],[70,5847,5849],{"className":5848},[82],[84,5850,5851],{"xmlns":86},[89,5852,5853,5862],{},[92,5854,5855,5858,5860],{},[95,5856,5857],{},"b",[100,5859,126],{},[95,5861,5372],{},[164,5863,5864],{"encoding":166},"b \\in B",[70,5866,5868,5887],{"className":5867,"ariaHidden":172},[171],[70,5869,5871,5875,5878,5881,5884],{"className":5870},[176],[70,5872],{"className":5873,"style":5874},[180],"height:0.7335em;vertical-align:-0.0391em;",[70,5876,5857],{"className":5877},[185,193],[70,5879],{"className":5880,"style":202},[201],[70,5882,126],{"className":5883},[206],[70,5885],{"className":5886,"style":202},[201],[70,5888,5890,5893],{"className":5889},[176],[70,5891],{"className":5892,"style":5067},[180],[70,5894,5372],{"className":5895,"style":5387},[185,193],", there exists ",[70,5898,5900,5918],{"className":5899,"translate":74},[78],[70,5901,5903],{"className":5902},[82],[84,5904,5905],{"xmlns":86},[89,5906,5907,5915],{},[92,5908,5909,5911,5913],{},[95,5910,18],{},[100,5912,126],{},[95,5914,5080],{},[164,5916,5917],{"encoding":166},"a \\in A",[70,5919,5921,5940],{"className":5920,"ariaHidden":172},[171],[70,5922,5924,5928,5931,5934,5937],{"className":5923},[176],[70,5925],{"className":5926,"style":5927},[180],"height:0.5782em;vertical-align:-0.0391em;",[70,5929,18],{"className":5930},[185,193],[70,5932],{"className":5933,"style":202},[201],[70,5935,126],{"className":5936},[206],[70,5938],{"className":5939,"style":202},[201],[70,5941,5943,5946],{"className":5942},[176],[70,5944],{"className":5945,"style":5067},[180],[70,5947,5080],{"className":5948},[185,193]," such that ",[70,5951,5953,5977],{"className":5952,"translate":74},[78],[70,5954,5956],{"className":5955},[82],[84,5957,5958],{"xmlns":86},[89,5959,5960,5974],{},[92,5961,5962,5964,5966,5968,5970,5972],{},[95,5963,123],{},[100,5965,103],{"stretchy":102},[95,5967,18],{},[100,5969,109],{"stretchy":102},[100,5971,112],{},[95,5973,5857],{},[164,5975,5976],{"encoding":166},"f(a) = b",[70,5978,5980,6007],{"className":5979,"ariaHidden":172},[171],[70,5981,5983,5986,5989,5992,5995,5998,6001,6004],{"className":5982},[176],[70,5984],{"className":5985,"style":181},[180],[70,5987,123],{"className":5988,"style":257},[185,193],[70,5990,103],{"className":5991},[189],[70,5993,18],{"className":5994},[185,193],[70,5996,109],{"className":5997},[197],[70,5999],{"className":6000,"style":202},[201],[70,6002,112],{"className":6003},[206],[70,6005],{"className":6006,"style":202},[201],[70,6008,6010,6014],{"className":6009},[176],[70,6011],{"className":6012,"style":6013},[180],"height:0.6944em;",[70,6015,5857],{"className":6016},[185,193]," (every output is covered)",[11,6019,6020,6021,6049,6050,5357,6078,6106,6107,6110,6111,6182],{},"then ",[70,6022,6024,6037],{"className":6023,"translate":74},[78],[70,6025,6027],{"className":6026},[82],[84,6028,6029],{"xmlns":86},[89,6030,6031,6035],{},[92,6032,6033],{},[95,6034,123],{},[164,6036,123],{"encoding":166},[70,6038,6040],{"className":6039,"ariaHidden":172},[171],[70,6041,6043,6046],{"className":6042},[176],[70,6044],{"className":6045,"style":791},[180],[70,6047,123],{"className":6048,"style":257},[185,193]," is a bijection, and we say that ",[70,6051,6053,6066],{"className":6052,"translate":74},[78],[70,6054,6056],{"className":6055},[82],[84,6057,6058],{"xmlns":86},[89,6059,6060,6064],{},[92,6061,6062],{},[95,6063,5080],{},[164,6065,5080],{"encoding":166},[70,6067,6069],{"className":6068,"ariaHidden":172},[171],[70,6070,6072,6075],{"className":6071},[176],[70,6073],{"className":6074,"style":5067},[180],[70,6076,5080],{"className":6077},[185,193],[70,6079,6081,6094],{"className":6080,"translate":74},[78],[70,6082,6084],{"className":6083},[82],[84,6085,6086],{"xmlns":86},[89,6087,6088,6092],{},[92,6089,6090],{},[95,6091,5372],{},[164,6093,5372],{"encoding":166},[70,6095,6097],{"className":6096,"ariaHidden":172},[171],[70,6098,6100,6103],{"className":6099},[176],[70,6101],{"className":6102,"style":5067},[180],[70,6104,5372],{"className":6105,"style":5387},[185,193]," are ",[29,6108,6109],{},"equinumerous"," (denoted ",[70,6112,6114,6140],{"className":6113,"translate":74},[78],[70,6115,6117],{"className":6116},[82],[84,6118,6119],{"xmlns":86},[89,6120,6121,6137],{},[92,6122,6123,6125,6127,6129,6131,6133,6135],{},[95,6124,4625],{"mathvariant":97},[95,6126,5080],{},[95,6128,4625],{"mathvariant":97},[100,6130,112],{},[95,6132,4625],{"mathvariant":97},[95,6134,5372],{},[95,6136,4625],{"mathvariant":97},[164,6138,6139],{"encoding":166},"|A| = |B|",[70,6141,6143,6167],{"className":6142,"ariaHidden":172},[171],[70,6144,6146,6149,6152,6155,6158,6161,6164],{"className":6145},[176],[70,6147],{"className":6148,"style":181},[180],[70,6150,4625],{"className":6151},[185],[70,6153,5080],{"className":6154},[185,193],[70,6156,4625],{"className":6157},[185],[70,6159],{"className":6160,"style":202},[201],[70,6162,112],{"className":6163},[206],[70,6165],{"className":6166,"style":202},[201],[70,6168,6170,6173,6176,6179],{"className":6169},[176],[70,6171],{"className":6172,"style":181},[180],[70,6174,4625],{"className":6175},[185],[70,6177,5372],{"className":6178,"style":5387},[185,193],[70,6180,4625],{"className":6181},[185],") — that is, they are \"equal in size.\"",[11,6184,6185,6188,6189,6217,6218,6247,6248,6276],{},[29,6186,6187],{},"Definition (countable set)."," A set ",[70,6190,6192,6205],{"className":6191,"translate":74},[78],[70,6193,6195],{"className":6194},[82],[84,6196,6197],{"xmlns":86},[89,6198,6199,6203],{},[92,6200,6201],{},[95,6202,5080],{},[164,6204,5080],{"encoding":166},[70,6206,6208],{"className":6207,"ariaHidden":172},[171],[70,6209,6211,6214],{"className":6210},[176],[70,6212],{"className":6213,"style":5067},[180],[70,6215,5080],{"className":6216},[185,193]," is said to be countable if it can be placed in bijection with the natural numbers ",[70,6219,6221,6235],{"className":6220,"translate":74},[78],[70,6222,6224],{"className":6223},[82],[84,6225,6226],{"xmlns":86},[89,6227,6228,6232],{},[92,6229,6230],{},[95,6231,5211],{"mathvariant":5210},[164,6233,6234],{"encoding":166},"\\mathbb{N}",[70,6236,6238],{"className":6237,"ariaHidden":172},[171],[70,6239,6241,6244],{"className":6240},[176],[70,6242],{"className":6243,"style":5254},[180],[70,6245,5211],{"className":6246},[185,5258]," (or if ",[70,6249,6251,6264],{"className":6250,"translate":74},[78],[70,6252,6254],{"className":6253},[82],[84,6255,6256],{"xmlns":86},[89,6257,6258,6262],{},[92,6259,6260],{},[95,6261,5080],{},[164,6263,5080],{"encoding":166},[70,6265,6267],{"className":6266,"ariaHidden":172},[171],[70,6268,6270,6273],{"className":6269},[176],[70,6271],{"className":6272,"style":5067},[180],[70,6274,5080],{"className":6275},[185,193]," is finite).",[11,6278,6279],{},"A few examples will reveal that the concept of \"countable\" is considerably more permissive than intuition would suggest:",[1025,6281,6282,6458,7122],{},[1028,6283,6284,882,6287,6385,6386,24],{},[29,6285,6286],{},"The even numbers",[70,6288,6290,6328],{"className":6289,"translate":74},[78],[70,6291,6293],{"className":6292},[82],[84,6294,6295],{"xmlns":86},[89,6296,6297,6325],{},[92,6298,6299,6301,6303,6305,6307,6309,6312,6314,6317,6319,6321,6323],{},[100,6300,5216],{"stretchy":102},[1566,6302,2780],{},[100,6304,906],{"separator":172},[1566,6306,2938],{},[100,6308,906],{"separator":172},[1566,6310,6311],{},"4",[100,6313,906],{"separator":172},[1566,6315,6316],{},"6",[100,6318,906],{"separator":172},[100,6320,3004],{},[4704,6322,5238],{},[100,6324,5241],{"stretchy":102},[164,6326,6327],{"encoding":166},"\\{0, 2, 4, 6, \\dots\\}",[70,6329,6331],{"className":6330,"ariaHidden":172},[171],[70,6332,6334,6337,6340,6343,6346,6349,6352,6355,6358,6361,6364,6367,6370,6373,6376,6379,6382],{"className":6333},[176],[70,6335],{"className":6336,"style":181},[180],[70,6338,5216],{"className":6339},[189],[70,6341,2780],{"className":6342},[185],[70,6344,906],{"className":6345},[976],[70,6347],{"className":6348,"style":304},[201],[70,6350,2938],{"className":6351},[185],[70,6353,906],{"className":6354},[976],[70,6356],{"className":6357,"style":304},[201],[70,6359,6311],{"className":6360},[185],[70,6362,906],{"className":6363},[976],[70,6365],{"className":6366,"style":304},[201],[70,6368,6316],{"className":6369},[185],[70,6371,906],{"className":6372},[976],[70,6374],{"className":6375,"style":304},[201],[70,6377,3004],{"className":6378},[3082],[70,6380],{"className":6381,"style":304},[201],[70,6383,5241],{"className":6384},[197]," are countable: the bijection is ",[70,6387,6389,6416],{"className":6388,"translate":74},[78],[70,6390,6392],{"className":6391},[82],[84,6393,6394],{"xmlns":86},[89,6395,6396,6413],{},[92,6397,6398,6400,6402,6405,6407,6409,6411],{},[95,6399,123],{},[100,6401,103],{"stretchy":102},[95,6403,6404],{},"n",[100,6406,109],{"stretchy":102},[100,6408,112],{},[1566,6410,2938],{},[95,6412,6404],{},[164,6414,6415],{"encoding":166},"f(n) = 2n",[70,6417,6419,6446],{"className":6418,"ariaHidden":172},[171],[70,6420,6422,6425,6428,6431,6434,6437,6440,6443],{"className":6421},[176],[70,6423],{"className":6424,"style":181},[180],[70,6426,123],{"className":6427,"style":257},[185,193],[70,6429,103],{"className":6430},[189],[70,6432,6404],{"className":6433},[185,193],[70,6435,109],{"className":6436},[197],[70,6438],{"className":6439,"style":202},[201],[70,6441,112],{"className":6442},[206],[70,6444],{"className":6445,"style":202},[201],[70,6447,6449,6452,6455],{"className":6448},[176],[70,6450],{"className":6451,"style":3991},[180],[70,6453,2938],{"className":6454},[185],[70,6456,6404],{"className":6457},[185,193],[1028,6459,6460,882,6463,6620,6621,6921,6922,7017,7018,24],{},[29,6461,6462],{},"The integers",[70,6464,6466,6518],{"className":6465,"translate":74},[78],[70,6467,6469],{"className":6468},[82],[84,6470,6471],{"xmlns":86},[89,6472,6473,6515],{},[92,6474,6475,6477,6479,6481,6483,6485,6487,6489,6491,6493,6495,6497,6499,6501,6503,6505,6507,6509,6511,6513],{},[95,6476,901],{"mathvariant":5210},[100,6478,112],{},[100,6480,5216],{"stretchy":102},[100,6482,3004],{},[100,6484,906],{"separator":172},[100,6486,1564],{},[1566,6488,2938],{},[100,6490,906],{"separator":172},[100,6492,1564],{},[1566,6494,1568],{},[100,6496,906],{"separator":172},[1566,6498,2780],{},[100,6500,906],{"separator":172},[1566,6502,1568],{},[100,6504,906],{"separator":172},[1566,6506,2938],{},[100,6508,906],{"separator":172},[100,6510,3004],{},[4704,6512,5238],{},[100,6514,5241],{"stretchy":102},[164,6516,6517],{"encoding":166},"\\mathbb{Z} = \\{\\dots, -2, -1, 0, 1, 2, \\dots\\}",[70,6519,6521,6539],{"className":6520,"ariaHidden":172},[171],[70,6522,6524,6527,6530,6533,6536],{"className":6523},[176],[70,6525],{"className":6526,"style":5254},[180],[70,6528,901],{"className":6529},[185,5258],[70,6531],{"className":6532,"style":202},[201],[70,6534,112],{"className":6535},[206],[70,6537],{"className":6538,"style":202},[201],[70,6540,6542,6545,6548,6551,6554,6557,6560,6563,6566,6569,6572,6575,6578,6581,6584,6587,6590,6593,6596,6599,6602,6605,6608,6611,6614,6617],{"className":6541},[176],[70,6543],{"className":6544,"style":181},[180],[70,6546,5216],{"className":6547},[189],[70,6549,3004],{"className":6550},[3082],[70,6552],{"className":6553,"style":304},[201],[70,6555,906],{"className":6556},[976],[70,6558],{"className":6559,"style":304},[201],[70,6561,1564],{"className":6562},[185],[70,6564,2938],{"className":6565},[185],[70,6567,906],{"className":6568},[976],[70,6570],{"className":6571,"style":304},[201],[70,6573,1564],{"className":6574},[185],[70,6576,1568],{"className":6577},[185],[70,6579,906],{"className":6580},[976],[70,6582],{"className":6583,"style":304},[201],[70,6585,2780],{"className":6586},[185],[70,6588,906],{"className":6589},[976],[70,6591],{"className":6592,"style":304},[201],[70,6594,1568],{"className":6595},[185],[70,6597,906],{"className":6598},[976],[70,6600],{"className":6601,"style":304},[201],[70,6603,2938],{"className":6604},[185],[70,6606,906],{"className":6607},[976],[70,6609],{"className":6610,"style":304},[201],[70,6612,3004],{"className":6613},[3082],[70,6615],{"className":6616,"style":304},[201],[70,6618,5241],{"className":6619},[197]," are countable: one may use ",[70,6622,6624,6720],{"className":6623,"translate":74},[78],[70,6625,6627],{"className":6626},[82],[84,6628,6629],{"xmlns":86},[89,6630,6631,6717],{},[92,6632,6633,6635,6637,6639,6641,6643],{},[95,6634,123],{},[100,6636,103],{"stretchy":102},[95,6638,6404],{},[100,6640,109],{"stretchy":102},[100,6642,112],{},[92,6644,6645,6647],{},[100,6646,5216],{"fence":172},[6648,6649,6653,6682],"mtable",{"rowspacing":6650,"columnalign":6651,"columnspacing":6652},"0.36em","left left","1em",[6654,6655,6656,6671],"mtr",{},[6657,6658,6659],"mtd",{},[6660,6661,6662],"mstyle",{"scriptlevel":2780,"displaystyle":102},[92,6663,6664,6666,6669],{},[95,6665,6404],{},[95,6667,6668],{"mathvariant":97},"\u002F",[1566,6670,2938],{},[6657,6672,6673],{},[6660,6674,6675],{"scriptlevel":2780,"displaystyle":102},[92,6676,6677,6679],{},[95,6678,6404],{},[4704,6680,6681],{}," even",[6654,6683,6684,6706],{},[6657,6685,6686],{},[6660,6687,6688],{"scriptlevel":2780,"displaystyle":102},[92,6689,6690,6692,6694,6696,6698,6700,6702,6704],{},[100,6691,1564],{},[100,6693,103],{"stretchy":102},[95,6695,6404],{},[100,6697,3017],{},[1566,6699,1568],{},[100,6701,109],{"stretchy":102},[95,6703,6668],{"mathvariant":97},[1566,6705,2938],{},[6657,6707,6708],{},[6660,6709,6710],{"scriptlevel":2780,"displaystyle":102},[92,6711,6712,6714],{},[95,6713,6404],{},[4704,6715,6716],{}," odd",[164,6718,6719],{"encoding":166},"f(n) = \\begin{cases} n\u002F2 & n \\text{ even} \\\\ -(n+1)\u002F2 & n \\text{ odd} \\end{cases}",[70,6721,6723,6750],{"className":6722,"ariaHidden":172},[171],[70,6724,6726,6729,6732,6735,6738,6741,6744,6747],{"className":6725},[176],[70,6727],{"className":6728,"style":181},[180],[70,6730,123],{"className":6731,"style":257},[185,193],[70,6733,103],{"className":6734},[189],[70,6736,6404],{"className":6737},[185,193],[70,6739,109],{"className":6740},[197],[70,6742],{"className":6743,"style":202},[201],[70,6745,112],{"className":6746},[206],[70,6748],{"className":6749,"style":202},[201],[70,6751,6753,6757],{"className":6752},[176],[70,6754],{"className":6755,"style":6756},[180],"height:3em;vertical-align:-1.25em;",[70,6758,6760,6770,6917],{"className":6759},[3082],[70,6761,6765],{"className":6762,"style":6764},[189,6763],"delimcenter","top:0em;",[70,6766,5216],{"className":6767},[6768,6769],"delimsizing","size4",[70,6771,6773],{"className":6772},[185],[70,6774,6776,6855,6860],{"className":6775},[6648],[70,6777,6780],{"className":6778},[6779],"col-align-l",[70,6781,6783,6846],{"className":6782},[225,226],[70,6784,6786,6843],{"className":6785},[230],[70,6787,6790,6807],{"className":6788,"style":6789},[234],"height:1.69em;",[70,6791,6793,6797],{"style":6792},"top:-3.69em;",[70,6794],{"className":6795,"style":6796},[242],"height:3.008em;",[70,6798,6800,6803],{"className":6799},[185],[70,6801,6404],{"className":6802},[185,193],[70,6804,6806],{"className":6805},[185],"\u002F2",[70,6808,6810,6813],{"style":6809},"top:-2.25em;",[70,6811],{"className":6812,"style":6796},[242],[70,6814,6816,6819,6822,6825,6828,6831,6834,6837,6840],{"className":6815},[185],[70,6817,1564],{"className":6818},[185],[70,6820,103],{"className":6821},[189],[70,6823,6404],{"className":6824},[185,193],[70,6826],{"className":6827,"style":568},[201],[70,6829,3017],{"className":6830},[1730],[70,6832],{"className":6833,"style":568},[201],[70,6835,1568],{"className":6836},[185],[70,6838,109],{"className":6839},[197],[70,6841,6806],{"className":6842},[185],[70,6844,291],{"className":6845},[290],[70,6847,6849],{"className":6848},[230],[70,6850,6853],{"className":6851,"style":6852},[234],"height:1.19em;",[70,6854],{},[70,6856],{"className":6857,"style":6859},[6858],"arraycolsep","width:1em;",[70,6861,6863],{"className":6862},[6779],[70,6864,6866,6909],{"className":6865},[225,226],[70,6867,6869,6906],{"className":6868},[230],[70,6870,6872,6889],{"className":6871,"style":6789},[234],[70,6873,6874,6877],{"style":6792},[70,6875],{"className":6876,"style":6796},[242],[70,6878,6880,6883],{"className":6879},[185],[70,6881,6404],{"className":6882},[185,193],[70,6884,6886],{"className":6885},[185,4752],[70,6887,6681],{"className":6888},[185],[70,6890,6891,6894],{"style":6809},[70,6892],{"className":6893,"style":6796},[242],[70,6895,6897,6900],{"className":6896},[185],[70,6898,6404],{"className":6899},[185,193],[70,6901,6903],{"className":6902},[185,4752],[70,6904,6716],{"className":6905},[185],[70,6907,291],{"className":6908},[290],[70,6910,6912],{"className":6911},[230],[70,6913,6915],{"className":6914,"style":6852},[234],[70,6916],{},[70,6918],{"className":6919},[197,6920],"nulldelimiter"," to map ",[70,6923,6925,6959],{"className":6924,"translate":74},[78],[70,6926,6928],{"className":6927},[82],[84,6929,6930],{"xmlns":86},[89,6931,6932,6956],{},[92,6933,6934,6936,6938,6940,6942,6944,6946,6948,6950,6952,6954],{},[1566,6935,2780],{},[100,6937,906],{"separator":172},[1566,6939,1568],{},[100,6941,906],{"separator":172},[1566,6943,2938],{},[100,6945,906],{"separator":172},[1566,6947,5231],{},[100,6949,906],{"separator":172},[1566,6951,6311],{},[100,6953,906],{"separator":172},[100,6955,3004],{},[164,6957,6958],{"encoding":166},"0,1,2,3,4,\\dots",[70,6960,6962],{"className":6961,"ariaHidden":172},[171],[70,6963,6965,6969,6972,6975,6978,6981,6984,6987,6990,6993,6996,6999,7002,7005,7008,7011,7014],{"className":6964},[176],[70,6966],{"className":6967,"style":6968},[180],"height:0.8389em;vertical-align:-0.1944em;",[70,6970,2780],{"className":6971},[185],[70,6973,906],{"className":6974},[976],[70,6976],{"className":6977,"style":304},[201],[70,6979,1568],{"className":6980},[185],[70,6982,906],{"className":6983},[976],[70,6985],{"className":6986,"style":304},[201],[70,6988,2938],{"className":6989},[185],[70,6991,906],{"className":6992},[976],[70,6994],{"className":6995,"style":304},[201],[70,6997,5231],{"className":6998},[185],[70,7000,906],{"className":7001},[976],[70,7003],{"className":7004,"style":304},[201],[70,7006,6311],{"className":7007},[185],[70,7009,906],{"className":7010},[976],[70,7012],{"className":7013,"style":304},[201],[70,7015,3004],{"className":7016},[3082]," onto ",[70,7019,7021,7059],{"className":7020,"translate":74},[78],[70,7022,7024],{"className":7023},[82],[84,7025,7026],{"xmlns":86},[89,7027,7028,7056],{},[92,7029,7030,7032,7034,7036,7038,7040,7042,7044,7046,7048,7050,7052,7054],{},[1566,7031,2780],{},[100,7033,906],{"separator":172},[100,7035,1564],{},[1566,7037,1568],{},[100,7039,906],{"separator":172},[1566,7041,1568],{},[100,7043,906],{"separator":172},[100,7045,1564],{},[1566,7047,2938],{},[100,7049,906],{"separator":172},[1566,7051,2938],{},[100,7053,906],{"separator":172},[100,7055,3004],{},[164,7057,7058],{"encoding":166},"0,-1,1,-2,2,\\dots",[70,7060,7062],{"className":7061,"ariaHidden":172},[171],[70,7063,7065,7068,7071,7074,7077,7080,7083,7086,7089,7092,7095,7098,7101,7104,7107,7110,7113,7116,7119],{"className":7064},[176],[70,7066],{"className":7067,"style":6968},[180],[70,7069,2780],{"className":7070},[185],[70,7072,906],{"className":7073},[976],[70,7075],{"className":7076,"style":304},[201],[70,7078,1564],{"className":7079},[185],[70,7081,1568],{"className":7082},[185],[70,7084,906],{"className":7085},[976],[70,7087],{"className":7088,"style":304},[201],[70,7090,1568],{"className":7091},[185],[70,7093,906],{"className":7094},[976],[70,7096],{"className":7097,"style":304},[201],[70,7099,1564],{"className":7100},[185],[70,7102,2938],{"className":7103},[185],[70,7105,906],{"className":7106},[976],[70,7108],{"className":7109,"style":304},[201],[70,7111,2938],{"className":7112},[185],[70,7114,906],{"className":7115},[976],[70,7117],{"className":7118,"style":304},[201],[70,7120,3004],{"className":7121},[3082],[1028,7123,7124,882,7127,7158,7159,7199],{},[29,7125,7126],{},"The rational numbers",[70,7128,7130,7145],{"className":7129,"translate":74},[78],[70,7131,7133],{"className":7132},[82],[84,7134,7135],{"xmlns":86},[89,7136,7137,7142],{},[92,7138,7139],{},[95,7140,7141],{"mathvariant":5210},"Q",[164,7143,7144],{"encoding":166},"\\mathbb{Q}",[70,7146,7148],{"className":7147,"ariaHidden":172},[171],[70,7149,7151,7155],{"className":7150},[176],[70,7152],{"className":7153,"style":7154},[180],"height:0.8556em;vertical-align:-0.1667em;",[70,7156,7141],{"className":7157},[185,5258]," are likewise countable (one may employ the \"diagonal enumeration method\" to arrange all fractions ",[70,7160,7162,7181],{"className":7161,"translate":74},[78],[70,7163,7165],{"className":7164},[82],[84,7166,7167],{"xmlns":86},[89,7168,7169,7178],{},[92,7170,7171,7173,7175],{},[95,7172,11],{},[95,7174,6668],{"mathvariant":97},[95,7176,7177],{},"q",[164,7179,7180],{"encoding":166},"p\u002Fq",[70,7182,7184],{"className":7183,"ariaHidden":172},[171],[70,7185,7187,7190,7193,7196],{"className":7186},[176],[70,7188],{"className":7189,"style":181},[180],[70,7191,11],{"className":7192},[185,193],[70,7194,6668],{"className":7195},[185],[70,7197,7177],{"className":7198,"style":365},[185,193]," into a sequence — this is another classical result of Cantor's, sharing the name \"diagonal\" with the argument treated in this article, though the usage differs; the reader is cautioned against conflating the two).",[11,7201,7202],{},"It would appear that all infinite sets can be accommodated within the framework of the natural numbers — until we encounter the real numbers.",[45,7204,7206],{"id":7205},"the-uncountability-of-the-real-numbers","The Uncountability of the Real Numbers",[1339,7208,7209],{},[11,7210,7211,7214,7215,7267],{},[29,7212,7213],{},"Theorem (Cantor, 1891)."," The set of real numbers in the interval ",[70,7216,7218,7240],{"className":7217,"translate":74},[78],[70,7219,7221],{"className":7220},[82],[84,7222,7223],{"xmlns":86},[89,7224,7225,7237],{},[92,7226,7227,7229,7231,7233,7235],{},[100,7228,103],{"stretchy":102},[1566,7230,2780],{},[100,7232,906],{"separator":172},[1566,7234,1568],{},[100,7236,109],{"stretchy":102},[164,7238,7239],{"encoding":166},"(0,1)",[70,7241,7243],{"className":7242,"ariaHidden":172},[171],[70,7244,7246,7249,7252,7255,7258,7261,7264],{"className":7245},[176],[70,7247],{"className":7248,"style":181},[180],[70,7250,103],{"className":7251},[189],[70,7253,2780],{"className":7254},[185],[70,7256,906],{"className":7257},[976],[70,7259],{"className":7260,"style":304},[201],[70,7262,1568],{"className":7263},[185],[70,7265,109],{"className":7266},[197]," is uncountable.",[11,7269,7270,7271,7322],{},"This means: there exists no method whatsoever by which the real numbers in ",[70,7272,7274,7295],{"className":7273,"translate":74},[78],[70,7275,7277],{"className":7276},[82],[84,7278,7279],{"xmlns":86},[89,7280,7281,7293],{},[92,7282,7283,7285,7287,7289,7291],{},[100,7284,103],{"stretchy":102},[1566,7286,2780],{},[100,7288,906],{"separator":172},[1566,7290,1568],{},[100,7292,109],{"stretchy":102},[164,7294,7239],{"encoding":166},[70,7296,7298],{"className":7297,"ariaHidden":172},[171],[70,7299,7301,7304,7307,7310,7313,7316,7319],{"className":7300},[176],[70,7302],{"className":7303,"style":181},[180],[70,7305,103],{"className":7306},[189],[70,7308,2780],{"className":7309},[185],[70,7311,906],{"className":7312},[976],[70,7314],{"className":7315,"style":304},[201],[70,7317,1568],{"className":7318},[185],[70,7320,109],{"className":7321},[197]," can be placed in one-to-one correspondence with the natural numbers. Put differently, the infinity of the real numbers is \"larger\" than the infinity of the natural numbers.",[11,7324,7325,7326,24],{},"We proceed by ",[15,7327,7328],{},"reductio ad absurdum",[11,7330,7331,7334,7335,7386,7387,7483],{},[29,7332,7333],{},"Assumption",": the real numbers in ",[70,7336,7338,7359],{"className":7337,"translate":74},[78],[70,7339,7341],{"className":7340},[82],[84,7342,7343],{"xmlns":86},[89,7344,7345,7357],{},[92,7346,7347,7349,7351,7353,7355],{},[100,7348,103],{"stretchy":102},[1566,7350,2780],{},[100,7352,906],{"separator":172},[1566,7354,1568],{},[100,7356,109],{"stretchy":102},[164,7358,7239],{"encoding":166},[70,7360,7362],{"className":7361,"ariaHidden":172},[171],[70,7363,7365,7368,7371,7374,7377,7380,7383],{"className":7364},[176],[70,7366],{"className":7367,"style":181},[180],[70,7369,103],{"className":7370},[189],[70,7372,2780],{"className":7373},[185],[70,7375,906],{"className":7376},[976],[70,7378],{"className":7379,"style":304},[201],[70,7381,1568],{"className":7382},[185],[70,7384,109],{"className":7385},[197]," are countable. That is, there exists a bijection ",[70,7388,7390,7420],{"className":7389,"translate":74},[78],[70,7391,7393],{"className":7392},[82],[84,7394,7395],{"xmlns":86},[89,7396,7397,7417],{},[92,7398,7399,7401,7403,7405,7407,7409,7411,7413,7415],{},[95,7400,123],{},[100,7402,68],{},[95,7404,5211],{"mathvariant":5210},[100,7406,1550],{},[100,7408,103],{"stretchy":102},[1566,7410,2780],{},[100,7412,906],{"separator":172},[1566,7414,1568],{},[100,7416,109],{"stretchy":102},[164,7418,7419],{"encoding":166},"f: \\mathbb{N} \\to (0,1)",[70,7421,7423,7441,7459],{"className":7422,"ariaHidden":172},[171],[70,7424,7426,7429,7432,7435,7438],{"className":7425},[176],[70,7427],{"className":7428,"style":791},[180],[70,7430,123],{"className":7431,"style":257},[185,193],[70,7433],{"className":7434,"style":202},[201],[70,7436,68],{"className":7437},[206],[70,7439],{"className":7440,"style":202},[201],[70,7442,7444,7447,7450,7453,7456],{"className":7443},[176],[70,7445],{"className":7446,"style":5254},[180],[70,7448,5211],{"className":7449},[185,5258],[70,7451],{"className":7452,"style":202},[201],[70,7454,1550],{"className":7455},[206],[70,7457],{"className":7458,"style":202},[201],[70,7460,7462,7465,7468,7471,7474,7477,7480],{"className":7461},[176],[70,7463],{"className":7464,"style":181},[180],[70,7466,103],{"className":7467},[189],[70,7469,2780],{"className":7470},[185],[70,7472,906],{"className":7473},[976],[70,7475],{"className":7476,"style":304},[201],[70,7478,1568],{"className":7479},[185],[70,7481,109],{"className":7482},[197],", permitting us to arrange all such real numbers into an infinite list:",[70,7485,7487],{"className":7486,"translate":74},[73],[70,7488,7490,7776],{"className":7489,"translate":74},[78],[70,7491,7493],{"className":7492},[82],[84,7494,7495],{"xmlns":86,"display":87},[89,7496,7497,7773],{},[6648,7498,7501,7562,7622,7682,7742],{"rowspacing":7499,"columnalign":7500,"columnspacing":3414},"0.25em","right left",[6654,7502,7503,7513],{},[6657,7504,7505],{},[6660,7506,7507],{"scriptlevel":2780,"displaystyle":172},[137,7508,7509,7511],{},[95,7510,106],{},[1566,7512,1568],{},[6657,7514,7515],{},[6660,7516,7517],{"scriptlevel":2780,"displaystyle":172},[92,7518,7519,7521,7523,7526,7533,7535,7542,7544,7551,7553,7560],{},[92,7520],{},[100,7522,112],{},[1566,7524,7525],{},"0.",[137,7527,7528,7530],{},[95,7529,911],{},[1566,7531,7532],{},"11",[4704,7534,5238],{},[137,7536,7537,7539],{},[95,7538,911],{},[1566,7540,7541],{},"12",[4704,7543,5238],{},[137,7545,7546,7548],{},[95,7547,911],{},[1566,7549,7550],{},"13",[4704,7552,5238],{},[137,7554,7555,7557],{},[95,7556,911],{},[1566,7558,7559],{},"14",[100,7561,3004],{},[6654,7563,7564,7574],{},[6657,7565,7566],{},[6660,7567,7568],{"scriptlevel":2780,"displaystyle":172},[137,7569,7570,7572],{},[95,7571,106],{},[1566,7573,2938],{},[6657,7575,7576],{},[6660,7577,7578],{"scriptlevel":2780,"displaystyle":172},[92,7579,7580,7582,7584,7586,7593,7595,7602,7604,7611,7613,7620],{},[92,7581],{},[100,7583,112],{},[1566,7585,7525],{},[137,7587,7588,7590],{},[95,7589,911],{},[1566,7591,7592],{},"21",[4704,7594,5238],{},[137,7596,7597,7599],{},[95,7598,911],{},[1566,7600,7601],{},"22",[4704,7603,5238],{},[137,7605,7606,7608],{},[95,7607,911],{},[1566,7609,7610],{},"23",[4704,7612,5238],{},[137,7614,7615,7617],{},[95,7616,911],{},[1566,7618,7619],{},"24",[100,7621,3004],{},[6654,7623,7624,7634],{},[6657,7625,7626],{},[6660,7627,7628],{"scriptlevel":2780,"displaystyle":172},[137,7629,7630,7632],{},[95,7631,106],{},[1566,7633,5231],{},[6657,7635,7636],{},[6660,7637,7638],{"scriptlevel":2780,"displaystyle":172},[92,7639,7640,7642,7644,7646,7653,7655,7662,7664,7671,7673,7680],{},[92,7641],{},[100,7643,112],{},[1566,7645,7525],{},[137,7647,7648,7650],{},[95,7649,911],{},[1566,7651,7652],{},"31",[4704,7654,5238],{},[137,7656,7657,7659],{},[95,7658,911],{},[1566,7660,7661],{},"32",[4704,7663,5238],{},[137,7665,7666,7668],{},[95,7667,911],{},[1566,7669,7670],{},"33",[4704,7672,5238],{},[137,7674,7675,7677],{},[95,7676,911],{},[1566,7678,7679],{},"34",[100,7681,3004],{},[6654,7683,7684,7694],{},[6657,7685,7686],{},[6660,7687,7688],{"scriptlevel":2780,"displaystyle":172},[137,7689,7690,7692],{},[95,7691,106],{},[1566,7693,6311],{},[6657,7695,7696],{},[6660,7697,7698],{"scriptlevel":2780,"displaystyle":172},[92,7699,7700,7702,7704,7706,7713,7715,7722,7724,7731,7733,7740],{},[92,7701],{},[100,7703,112],{},[1566,7705,7525],{},[137,7707,7708,7710],{},[95,7709,911],{},[1566,7711,7712],{},"41",[4704,7714,5238],{},[137,7716,7717,7719],{},[95,7718,911],{},[1566,7720,7721],{},"42",[4704,7723,5238],{},[137,7725,7726,7728],{},[95,7727,911],{},[1566,7729,7730],{},"43",[4704,7732,5238],{},[137,7734,7735,7737],{},[95,7736,911],{},[1566,7738,7739],{},"44",[100,7741,3004],{},[6654,7743,7744,7750],{},[6657,7745,7746],{},[6660,7747,7748],{"scriptlevel":2780,"displaystyle":172},[92,7749],{},[6657,7751,7752],{},[6660,7753,7754],{"scriptlevel":2780,"displaystyle":172},[92,7755,7756,7758,7761],{},[92,7757],{},[4704,7759,7760],{},"  ",[92,7762,7763,7766],{},[95,7764,7765],{"mathvariant":97},"⋮",[7767,7768,7769],"mpadded",{"height":3414,"voffset":3414},[201,7770],{"mathbackground":7771,"width":3414,"height":7772},"black","1.5em",[164,7774,7775],{"encoding":166},"\\begin{aligned}\nx_1 &= 0.d_{11}\\, d_{12}\\, d_{13}\\, d_{14} \\dots \\\\\nx_2 &= 0.d_{21}\\, d_{22}\\, d_{23}\\, d_{24} \\dots \\\\\nx_3 &= 0.d_{31}\\, d_{32}\\, d_{33}\\, d_{34} \\dots \\\\\nx_4 &= 0.d_{41}\\, d_{42}\\, d_{43}\\, d_{44} \\dots \\\\\n&\\ \\ 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denotes the ",[70,9008,9010,9023],{"className":9009,"translate":74},[78],[70,9011,9013],{"className":9012},[82],[84,9014,9015],{"xmlns":86},[89,9016,9017,9021],{},[92,9018,9019],{},[95,9020,8947],{},[164,9022,8947],{"encoding":166},[70,9024,9026],{"className":9025,"ariaHidden":172},[171],[70,9027,9029,9033],{"className":9028},[176],[70,9030],{"className":9031,"style":9032},[180],"height:0.854em;vertical-align:-0.1944em;",[70,9034,8947],{"className":9035,"style":8993},[185,193],"-th decimal digit of the ",[70,9038,9040,9053],{"className":9039,"translate":74},[78],[70,9041,9043],{"className":9042},[82],[84,9044,9045],{"xmlns":86},[89,9046,9047,9051],{},[92,9048,9049],{},[95,9050,2701],{},[164,9052,2701],{"encoding":166},[70,9054,9056],{"className":9055,"ariaHidden":172},[171],[70,9057,9059,9063],{"className":9058},[176],[70,9060],{"className":9061,"style":9062},[180],"height:0.6595em;",[70,9064,2701],{"className":9065},[185,193],"-th number (a digit from ",[70,9068,9070,9083],{"className":9069,"translate":74},[78],[70,9071,9073],{"className":9072},[82],[84,9074,9075],{"xmlns":86},[89,9076,9077,9081],{},[92,9078,9079],{},[1566,9080,2780],{},[164,9082,2780],{"encoding":166},[70,9084,9086],{"className":9085,"ariaHidden":172},[171],[70,9087,9089,9092],{"className":9088},[176],[70,9090],{"className":9091,"style":3991},[180],[70,9093,2780],{"className":9094},[185]," to ",[70,9097,9099,9113],{"className":9098,"translate":74},[78],[70,9100,9102],{"className":9101},[82],[84,9103,9104],{"xmlns":86},[89,9105,9106,9111],{},[92,9107,9108],{},[1566,9109,9110],{},"9",[164,9112,9110],{"encoding":166},[70,9114,9116],{"className":9115,"ariaHidden":172},[171],[70,9117,9119,9122],{"className":9118},[176],[70,9120],{"className":9121,"style":3991},[180],[70,9123,9110],{"className":9124},[185],").",[11,9127,9128,9131,9132,9377],{},[29,9129,9130],{},"Constructing the diagonal number."," We now proceed along the \"diagonal\" — that is, we extract the 1st digit of the 1st number, the 2nd digit of the 2nd number, the 3rd digit of the 3rd number, and so forth — and from these digits construct a new number ",[70,9133,9135,9180],{"className":9134,"translate":74},[78],[70,9136,9138],{"className":9137},[82],[84,9139,9140],{"xmlns":86},[89,9141,9142,9177],{},[92,9143,9144,9146,9148,9150,9157,9163,9169,9175],{},[95,9145,4632],{},[100,9147,112],{},[1566,9149,7525],{},[137,9151,9152,9155],{},[95,9153,9154],{},"e",[1566,9156,1568],{},[137,9158,9159,9161],{},[95,9160,9154],{},[1566,9162,2938],{},[137,9164,9165,9167],{},[95,9166,9154],{},[1566,9168,5231],{},[137,9170,9171,9173],{},[95,9172,9154],{},[1566,9174,6311],{},[100,9176,3004],{},[164,9178,9179],{"encoding":166},"y = 0.e_1 e_2 e_3 e_4 \\dots",[70,9181,9183,9201],{"className":9182,"ariaHidden":172},[171],[70,9184,9186,9189,9192,9195,9198],{"className":9185},[176],[70,9187],{"className":9188,"style":2801},[180],[70,9190,4632],{"className":9191,"style":365},[185,193],[70,9193],{"className":9194,"style":202},[201],[70,9196,112],{"className":9197},[206],[70,9199],{"className":9200,"style":202},[201],[70,9202,9204,9208,9211,9251,9291,9331,9371,9374],{"className":9203},[176],[70,9205],{"className":9206,"style":9207},[180],"height:0.7944em;vertical-align:-0.15em;",[70,9209,7525],{"className":9210},[185],[70,9212,9214,9217],{"className":9213},[185],[70,9215,9154],{"className":9216},[185,193],[70,9218,9220],{"className":9219},[314],[70,9221,9223,9243],{"className":9222},[225,226],[70,9224,9226,9240],{"className":9225},[230],[70,9227,9229],{"className":9228,"style":2820},[234],[70,9230,9231,9234],{"style":327},[70,9232],{"className":9233,"style":331},[242],[70,9235,9237],{"className":9236},[247,248,249,250],[70,9238,1568],{"className":9239},[185,250],[70,9241,291],{"className":9242},[290],[70,9244,9246],{"className":9245},[230],[70,9247,9249],{"className":9248,"style":2841},[234],[70,9250],{},[70,9252,9254,9257],{"className":9253},[185],[70,9255,9154],{"className":9256},[185,193],[70,9258,9260],{"className":9259},[314],[70,9261,9263,9283],{"className":9262},[225,226],[70,9264,9266,9280],{"className":9265},[230],[70,9267,9269],{"className":9268,"style":2820},[234],[70,9270,9271,9274],{"style":327},[70,9272],{"className":9273,"style":331},[242],[70,9275,9277],{"className":9276},[247,248,249,250],[70,9278,2938],{"className":9279},[185,250],[70,9281,291],{"className":9282},[290],[70,9284,9286],{"className":9285},[230],[70,9287,9289],{"className":9288,"style":2841},[234],[70,9290],{},[70,9292,9294,9297],{"className":9293},[185],[70,9295,9154],{"className":9296},[185,193],[70,9298,9300],{"className":9299},[314],[70,9301,9303,9323],{"className":9302},[225,226],[70,9304,9306,9320],{"className":9305},[230],[70,9307,9309],{"className":9308,"style":2820},[234],[70,9310,9311,9314],{"style":327},[70,9312],{"className":9313,"style":331},[242],[70,9315,9317],{"className":9316},[247,248,249,250],[70,9318,5231],{"className":9319},[185,250],[70,9321,291],{"className":9322},[290],[70,9324,9326],{"className":9325},[230],[70,9327,9329],{"className":9328,"style":2841},[234],[70,9330],{},[70,9332,9334,9337],{"className":9333},[185],[70,9335,9154],{"className":9336},[185,193],[70,9338,9340],{"className":9339},[314],[70,9341,9343,9363],{"className":9342},[225,226],[70,9344,9346,9360],{"className":9345},[230],[70,9347,9349],{"className":9348,"style":2820},[234],[70,9350,9351,9354],{"style":327},[70,9352],{"className":9353,"style":331},[242],[70,9355,9357],{"className":9356},[247,248,249,250],[70,9358,6311],{"className":9359},[185,250],[70,9361,291],{"className":9362},[290],[70,9364,9366],{"className":9365},[230],[70,9367,9369],{"className":9368,"style":2841},[234],[70,9370],{},[70,9372],{"className":9373,"style":304},[201],[70,9375,3004],{"className":9376},[3082]," according to the following rule:",[70,9379,9381],{"className":9380,"translate":74},[73],[70,9382,9384,9472],{"className":9383,"translate":74},[78],[70,9385,9387],{"className":9386},[82],[84,9388,9389],{"xmlns":86,"display":87},[89,9390,9391,9469],{},[92,9392,9393,9399,9401],{},[137,9394,9395,9397],{},[95,9396,9154],{},[95,9398,6404],{},[100,9400,112],{},[92,9402,9403,9405],{},[100,9404,5216],{"fence":172},[6648,9406,9407,9439],{"rowspacing":6650,"columnalign":6651,"columnspacing":6652},[6654,9408,9409,9416],{},[6657,9410,9411],{},[6660,9412,9413],{"scriptlevel":2780,"displaystyle":102},[1566,9414,9415],{},"5",[6657,9417,9418],{},[6660,9419,9420],{"scriptlevel":2780,"displaystyle":102},[92,9421,9422,9425,9435,9437],{},[4704,9423,9424],{},"if ",[137,9426,9427,9429],{},[95,9428,911],{},[92,9430,9431,9433],{},[95,9432,6404],{},[95,9434,6404],{},[100,9436,5491],{"mathvariant":97},[1566,9438,9415],{},[6654,9440,9441,9447],{},[6657,9442,9443],{},[6660,9444,9445],{"scriptlevel":2780,"displaystyle":102},[1566,9446,6316],{},[6657,9448,9449],{},[6660,9450,9451],{"scriptlevel":2780,"displaystyle":102},[92,9452,9453,9455,9465,9467],{},[4704,9454,9424],{},[137,9456,9457,9459],{},[95,9458,911],{},[92,9460,9461,9463],{},[95,9462,6404],{},[95,9464,6404],{},[100,9466,112],{},[1566,9468,9415],{},[164,9470,9471],{"encoding":166},"e_n = \\begin{cases} 5 & \\text{if } d_{nn} \\neq 5 \\\\ 6 & \\text{if } d_{nn} = 5 \\end{cases}",[70,9473,9475,9531],{"className":9474,"ariaHidden":172},[171],[70,9476,9478,9481,9522,9525,9528],{"className":9477},[176],[70,9479],{"className":9480,"style":5634},[180],[70,9482,9484,9487],{"className":9483},[185],[70,9485,9154],{"className":9486},[185,193],[70,9488,9490],{"className":9489},[314],[70,9491,9493,9514],{"className":9492},[225,226],[70,9494,9496,9511],{"className":9495},[230],[70,9497,9500],{"className":9498,"style":9499},[234],"height:0.1514em;",[70,9501,9502,9505],{"style":327},[70,9503],{"className":9504,"style":331},[242],[70,9506,9508],{"className":9507},[247,248,249,250],[70,9509,6404],{"className":9510},[185,193,250],[70,9512,291],{"className":9513},[290],[70,9515,9517],{"className":9516},[230],[70,9518,9520],{"className":9519,"style":2841},[234],[70,9521],{},[70,9523],{"className":9524,"style":202},[201],[70,9526,112],{"className":9527},[206],[70,9529],{"className":9530,"style":202},[201],[70,9532,9534,9537],{"className":9533},[176],[70,9535],{"className":9536,"style":6756},[180],[70,9538,9540,9546,9795],{"className":9539},[3082],[70,9541,9543],{"className":9542,"style":6764},[189,6763],[70,9544,5216],{"className":9545},[6768,6769],[70,9547,9549],{"className":9548},[185],[70,9550,9552,9597,9600],{"className":9551},[6648],[70,9553,9555],{"className":9554},[6779],[70,9556,9558,9589],{"className":9557},[225,226],[70,9559,9561,9586],{"className":9560},[230],[70,9562,9564,9575],{"className":9563,"style":6789},[234],[70,9565,9566,9569],{"style":6792},[70,9567],{"className":9568,"style":6796},[242],[70,9570,9572],{"className":9571},[185],[70,9573,9415],{"className":9574},[185],[70,9576,9577,9580],{"style":6809},[70,9578],{"className":9579,"style":6796},[242],[70,9581,9583],{"className":9582},[185],[70,9584,6316],{"className":9585},[185],[70,9587,291],{"className":9588},[290],[70,9590,9592],{"className":9591},[230],[70,9593,9595],{"className":9594,"style":6852},[234],[70,9596],{},[70,9598],{"className":9599,"style":6859},[6858],[70,9601,9603],{"className":9602},[6779],[70,9604,9606,9787],{"className":9605},[225,226],[70,9607,9609,9784],{"className":9608},[230],[70,9610,9612,9715],{"className":9611,"style":6789},[234],[70,9613,9614,9617],{"style":6792},[70,9615],{"className":9616,"style":6796},[242],[70,9618,9620,9626,9670,9673,9709,9712],{"className":9619},[185],[70,9621,9623],{"className":9622},[185,4752],[70,9624,9424],{"className":9625},[185],[70,9627,9629,9632],{"className":9628},[185],[70,9630,911],{"className":9631},[185,193],[70,9633,9635],{"className":9634},[314],[70,9636,9638,9662],{"className":9637},[225,226],[70,9639,9641,9659],{"className":9640},[230],[70,9642,9644],{"className":9643,"style":9499},[234],[70,9645,9646,9649],{"style":327},[70,9647],{"className":9648,"style":331},[242],[70,9650,9652],{"className":9651},[247,248,249,250],[70,9653,9655],{"className":9654},[185,250],[70,9656,9658],{"className":9657},[185,193,250],"nn",[70,9660,291],{"className":9661},[290],[70,9663,9665],{"className":9664},[230],[70,9666,9668],{"className":9667,"style":2841},[234],[70,9669],{},[70,9671],{"className":9672,"style":202},[201],[70,9674,9676,9703,9706],{"className":9675},[206],[70,9677,9679],{"className":9678},[206],[70,9680,9682],{"className":9681},[185,5591],[70,9683,9685],{"className":9684},[5595],[70,9686,9688,9691,9700],{"className":9687},[5599],[70,9689],{"className":9690,"style":791},[180],[70,9692,9694],{"className":9693},[5606],[70,9695,9697],{"className":9696},[185],[70,9698,5613],{"className":9699},[206],[70,9701],{"className":9702},[5617],[70,9704],{"className":9705},[201,5621],[70,9707,112],{"className":9708},[206],[70,9710],{"className":9711,"style":202},[201],[70,9713,9415],{"className":9714},[185],[70,9716,9717,9720],{"style":6809},[70,9718],{"className":9719,"style":6796},[242],[70,9721,9723,9729,9772,9775,9778,9781],{"className":9722},[185],[70,9724,9726],{"className":9725},[185,4752],[70,9727,9424],{"className":9728},[185],[70,9730,9732,9735],{"className":9731},[185],[70,9733,911],{"className":9734},[185,193],[70,9736,9738],{"className":9737},[314],[70,9739,9741,9764],{"className":9740},[225,226],[70,9742,9744,9761],{"className":9743},[230],[70,9745,9747],{"className":9746,"style":9499},[234],[70,9748,9749,9752],{"style":327},[70,9750],{"className":9751,"style":331},[242],[70,9753,9755],{"className":9754},[247,248,249,250],[70,9756,9758],{"className":9757},[185,250],[70,9759,9658],{"className":9760},[185,193,250],[70,9762,291],{"className":9763},[290],[70,9765,9767],{"className":9766},[230],[70,9768,9770],{"className":9769,"style":2841},[234],[70,9771],{},[70,9773],{"className":9774,"style":202},[201],[70,9776,112],{"className":9777},[206],[70,9779],{"className":9780,"style":202},[201],[70,9782,9415],{"className":9783},[185],[70,9785,291],{"className":9786},[290],[70,9788,9790],{"className":9789},[230],[70,9791,9793],{"className":9792,"style":6852},[234],[70,9794],{},[70,9796],{"className":9797},[197,6920],[11,9799,9800,9801,9975,9976,10004,10005,10033],{},"(The sole purpose of this rule is to ensure ",[70,9802,9804,9834],{"className":9803,"translate":74},[78],[70,9805,9807],{"className":9806},[82],[84,9808,9809],{"xmlns":86},[89,9810,9811,9831],{},[92,9812,9813,9819,9821],{},[137,9814,9815,9817],{},[95,9816,9154],{},[95,9818,6404],{},[100,9820,5491],{"mathvariant":97},[137,9822,9823,9825],{},[95,9824,911],{},[92,9826,9827,9829],{},[95,9828,6404],{},[95,9830,6404],{},[164,9832,9833],{"encoding":166},"e_n \\neq d_{nn}",[70,9835,9837,9925],{"className":9836,"ariaHidden":172},[171],[70,9838,9840,9843,9883,9886,9922],{"className":9839},[176],[70,9841],{"className":9842,"style":791},[180],[70,9844,9846,9849],{"className":9845},[185],[70,9847,9154],{"className":9848},[185,193],[70,9850,9852],{"className":9851},[314],[70,9853,9855,9875],{"className":9854},[225,226],[70,9856,9858,9872],{"className":9857},[230],[70,9859,9861],{"className":9860,"style":9499},[234],[70,9862,9863,9866],{"style":327},[70,9864],{"className":9865,"style":331},[242],[70,9867,9869],{"className":9868},[247,248,249,250],[70,9870,6404],{"className":9871},[185,193,250],[70,9873,291],{"className":9874},[290],[70,9876,9878],{"className":9877},[230],[70,9879,9881],{"className":9880,"style":2841},[234],[70,9882],{},[70,9884],{"className":9885,"style":202},[201],[70,9887,9889,9916,9919],{"className":9888},[206],[70,9890,9892],{"className":9891},[206],[70,9893,9895],{"className":9894},[185,5591],[70,9896,9898],{"className":9897},[5595],[70,9899,9901,9904,9913],{"className":9900},[5599],[70,9902],{"className":9903,"style":791},[180],[70,9905,9907],{"className":9906},[5606],[70,9908,9910],{"className":9909},[185],[70,9911,5613],{"className":9912},[206],[70,9914],{"className":9915},[5617],[70,9917],{"className":9918},[201,5621],[70,9920,112],{"className":9921},[206],[70,9923],{"className":9924,"style":202},[201],[70,9926,9928,9932],{"className":9927},[176],[70,9929],{"className":9930,"style":9931},[180],"height:0.8444em;vertical-align:-0.15em;",[70,9933,9935,9938],{"className":9934},[185],[70,9936,911],{"className":9937},[185,193],[70,9939,9941],{"className":9940},[314],[70,9942,9944,9967],{"className":9943},[225,226],[70,9945,9947,9964],{"className":9946},[230],[70,9948,9950],{"className":9949,"style":9499},[234],[70,9951,9952,9955],{"style":327},[70,9953],{"className":9954,"style":331},[242],[70,9956,9958],{"className":9957},[247,248,249,250],[70,9959,9961],{"className":9960},[185,250],[70,9962,9658],{"className":9963},[185,193,250],[70,9965,291],{"className":9966},[290],[70,9968,9970],{"className":9969},[230],[70,9971,9973],{"className":9972,"style":2841},[234],[70,9974],{},"; the specific digits chosen are immaterial, so long as each digit deliberately \"avoids\" the corresponding diagonal digit, while also steering clear of all-",[70,9977,9979,9992],{"className":9978,"translate":74},[78],[70,9980,9982],{"className":9981},[82],[84,9983,9984],{"xmlns":86},[89,9985,9986,9990],{},[92,9987,9988],{},[1566,9989,2780],{},[164,9991,2780],{"encoding":166},[70,9993,9995],{"className":9994,"ariaHidden":172},[171],[70,9996,9998,10001],{"className":9997},[176],[70,9999],{"className":10000,"style":3991},[180],[70,10002,2780],{"className":10003},[185]," or all-",[70,10006,10008,10021],{"className":10007,"translate":74},[78],[70,10009,10011],{"className":10010},[82],[84,10012,10013],{"xmlns":86},[89,10014,10015,10019],{},[92,10016,10017],{},[1566,10018,9110],{},[164,10020,9110],{"encoding":166},[70,10022,10024],{"className":10023,"ariaHidden":172},[171],[70,10025,10027,10030],{"className":10026},[176],[70,10028],{"className":10029,"style":3991},[180],[70,10031,9110],{"className":10032},[185]," representations, which would introduce ambiguity in decimal notation.)",[11,10035,10036,10037],{},"The critical question now presents itself: ",[29,10038,10039,10040,10068],{},"does ",[70,10041,10043,10056],{"className":10042,"translate":74},[78],[70,10044,10046],{"className":10045},[82],[84,10047,10048],{"xmlns":86},[89,10049,10050,10054],{},[92,10051,10052],{},[95,10053,4632],{},[164,10055,4632],{"encoding":166},[70,10057,10059],{"className":10058,"ariaHidden":172},[171],[70,10060,10062,10065],{"className":10061},[176],[70,10063],{"className":10064,"style":2801},[180],[70,10066,4632],{"className":10067,"style":365},[185,193]," appear in the original list?",[11,10070,10071,10072,10100,10101,10129,10130,10224,10225,9125],{},"Suppose ",[70,10073,10075,10088],{"className":10074,"translate":74},[78],[70,10076,10078],{"className":10077},[82],[84,10079,10080],{"xmlns":86},[89,10081,10082,10086],{},[92,10083,10084],{},[95,10085,4632],{},[164,10087,4632],{"encoding":166},[70,10089,10091],{"className":10090,"ariaHidden":172},[171],[70,10092,10094,10097],{"className":10093},[176],[70,10095],{"className":10096,"style":2801},[180],[70,10098,4632],{"className":10099,"style":365},[185,193]," does appear in the list; then ",[70,10102,10104,10117],{"className":10103,"translate":74},[78],[70,10105,10107],{"className":10106},[82],[84,10108,10109],{"xmlns":86},[89,10110,10111,10115],{},[92,10112,10113],{},[95,10114,4632],{},[164,10116,4632],{"encoding":166},[70,10118,10120],{"className":10119,"ariaHidden":172},[171],[70,10121,10123,10126],{"className":10122},[176],[70,10124],{"className":10125,"style":2801},[180],[70,10127,4632],{"className":10128,"style":365},[185,193]," must equal one of the enumerated numbers, say ",[70,10131,10133,10156],{"className":10132,"translate":74},[78],[70,10134,10136],{"className":10135},[82],[84,10137,10138],{"xmlns":86},[89,10139,10140,10153],{},[92,10141,10142,10144,10146],{},[95,10143,4632],{},[100,10145,112],{},[137,10147,10148,10150],{},[95,10149,106],{},[95,10151,10152],{},"k",[164,10154,10155],{"encoding":166},"y = x_k",[70,10157,10159,10177],{"className":10158,"ariaHidden":172},[171],[70,10160,10162,10165,10168,10171,10174],{"className":10161},[176],[70,10163],{"className":10164,"style":2801},[180],[70,10166,4632],{"className":10167,"style":365},[185,193],[70,10169],{"className":10170,"style":202},[201],[70,10172,112],{"className":10173},[206],[70,10175],{"className":10176,"style":202},[201],[70,10178,10180,10183],{"className":10179},[176],[70,10181],{"className":10182,"style":5634},[180],[70,10184,10186,10189],{"className":10185},[185],[70,10187,106],{"className":10188},[185,193],[70,10190,10192],{"className":10191},[314],[70,10193,10195,10216],{"className":10194},[225,226],[70,10196,10198,10213],{"className":10197},[230],[70,10199,10201],{"className":10200,"style":324},[234],[70,10202,10203,10206],{"style":327},[70,10204],{"className":10205,"style":331},[242],[70,10207,10209],{"className":10208},[247,248,249,250],[70,10210,10152],{"className":10211,"style":10212},[185,193,250],"margin-right:0.0315em;",[70,10214,291],{"className":10215},[290],[70,10217,10219],{"className":10218},[230],[70,10220,10222],{"className":10221,"style":2841},[234],[70,10223],{}," (for some specific index ",[70,10226,10228,10241],{"className":10227,"translate":74},[78],[70,10229,10231],{"className":10230},[82],[84,10232,10233],{"xmlns":86},[89,10234,10235,10239],{},[92,10236,10237],{},[95,10238,10152],{},[164,10240,10152],{"encoding":166},[70,10242,10244],{"className":10243,"ariaHidden":172},[171],[70,10245,10247,10250],{"className":10246},[176],[70,10248],{"className":10249,"style":6013},[180],[70,10251,10152],{"className":10252,"style":10212},[185,193],[11,10254,10255,10256,10284,10285,10313,10314,10384,10385,10454,10455,10535,10536,10284,10564,9125],{},"But, by our construction rule, the ",[70,10257,10259,10272],{"className":10258,"translate":74},[78],[70,10260,10262],{"className":10261},[82],[84,10263,10264],{"xmlns":86},[89,10265,10266,10270],{},[92,10267,10268],{},[95,10269,10152],{},[164,10271,10152],{"encoding":166},[70,10273,10275],{"className":10274,"ariaHidden":172},[171],[70,10276,10278,10281],{"className":10277},[176],[70,10279],{"className":10280,"style":6013},[180],[70,10282,10152],{"className":10283,"style":10212},[185,193],"-th decimal digit of ",[70,10286,10288,10301],{"className":10287,"translate":74},[78],[70,10289,10291],{"className":10290},[82],[84,10292,10293],{"xmlns":86},[89,10294,10295,10299],{},[92,10296,10297],{},[95,10298,4632],{},[164,10300,4632],{"encoding":166},[70,10302,10304],{"className":10303,"ariaHidden":172},[171],[70,10305,10307,10310],{"className":10306},[176],[70,10308],{"className":10309,"style":2801},[180],[70,10311,4632],{"className":10312,"style":365},[185,193]," is ",[70,10315,10317,10335],{"className":10316,"translate":74},[78],[70,10318,10320],{"className":10319},[82],[84,10321,10322],{"xmlns":86},[89,10323,10324,10332],{},[92,10325,10326],{},[137,10327,10328,10330],{},[95,10329,9154],{},[95,10331,10152],{},[164,10333,10334],{"encoding":166},"e_k",[70,10336,10338],{"className":10337,"ariaHidden":172},[171],[70,10339,10341,10344],{"className":10340},[176],[70,10342],{"className":10343,"style":5634},[180],[70,10345,10347,10350],{"className":10346},[185],[70,10348,9154],{"className":10349},[185,193],[70,10351,10353],{"className":10352},[314],[70,10354,10356,10376],{"className":10355},[225,226],[70,10357,10359,10373],{"className":10358},[230],[70,10360,10362],{"className":10361,"style":324},[234],[70,10363,10364,10367],{"style":327},[70,10365],{"className":10366,"style":331},[242],[70,10368,10370],{"className":10369},[247,248,249,250],[70,10371,10152],{"className":10372,"style":10212},[185,193,250],[70,10374,291],{"className":10375},[290],[70,10377,10379],{"className":10378},[230],[70,10380,10382],{"className":10381,"style":2841},[234],[70,10383],{},", and ",[70,10386,10388,10405],{"className":10387,"translate":74},[78],[70,10389,10391],{"className":10390},[82],[84,10392,10393],{"xmlns":86},[89,10394,10395,10403],{},[92,10396,10397],{},[137,10398,10399,10401],{},[95,10400,9154],{},[95,10402,10152],{},[164,10404,10334],{"encoding":166},[70,10406,10408],{"className":10407,"ariaHidden":172},[171],[70,10409,10411,10414],{"className":10410},[176],[70,10412],{"className":10413,"style":5634},[180],[70,10415,10417,10420],{"className":10416},[185],[70,10418,9154],{"className":10419},[185,193],[70,10421,10423],{"className":10422},[314],[70,10424,10426,10446],{"className":10425},[225,226],[70,10427,10429,10443],{"className":10428},[230],[70,10430,10432],{"className":10431,"style":324},[234],[70,10433,10434,10437],{"style":327},[70,10435],{"className":10436,"style":331},[242],[70,10438,10440],{"className":10439},[247,248,249,250],[70,10441,10152],{"className":10442,"style":10212},[185,193,250],[70,10444,291],{"className":10445},[290],[70,10447,10449],{"className":10448},[230],[70,10450,10452],{"className":10451,"style":2841},[234],[70,10453],{}," was deliberately constructed to differ from ",[70,10456,10458,10480],{"className":10457,"translate":74},[78],[70,10459,10461],{"className":10460},[82],[84,10462,10463],{"xmlns":86},[89,10464,10465,10477],{},[92,10466,10467],{},[137,10468,10469,10471],{},[95,10470,911],{},[92,10472,10473,10475],{},[95,10474,10152],{},[95,10476,10152],{},[164,10478,10479],{"encoding":166},"d_{kk}",[70,10481,10483],{"className":10482,"ariaHidden":172},[171],[70,10484,10486,10489],{"className":10485},[176],[70,10487],{"className":10488,"style":9931},[180],[70,10490,10492,10495],{"className":10491},[185],[70,10493,911],{"className":10494},[185,193],[70,10496,10498],{"className":10497},[314],[70,10499,10501,10527],{"className":10500},[225,226],[70,10502,10504,10524],{"className":10503},[230],[70,10505,10507],{"className":10506,"style":324},[234],[70,10508,10509,10512],{"style":327},[70,10510],{"className":10511,"style":331},[242],[70,10513,10515],{"className":10514},[247,248,249,250],[70,10516,10518,10521],{"className":10517},[185,250],[70,10519,10152],{"className":10520,"style":10212},[185,193,250],[70,10522,10152],{"className":10523,"style":10212},[185,193,250],[70,10525,291],{"className":10526},[290],[70,10528,10530],{"className":10529},[230],[70,10531,10533],{"className":10532,"style":2841},[234],[70,10534],{}," (i.e., the ",[70,10537,10539,10552],{"className":10538,"translate":74},[78],[70,10540,10542],{"className":10541},[82],[84,10543,10544],{"xmlns":86},[89,10545,10546,10550],{},[92,10547,10548],{},[95,10549,10152],{},[164,10551,10152],{"encoding":166},[70,10553,10555],{"className":10554,"ariaHidden":172},[171],[70,10556,10558,10561],{"className":10557},[176],[70,10559],{"className":10560,"style":6013},[180],[70,10562,10152],{"className":10563,"style":10212},[185,193],[70,10565,10567,10585],{"className":10566,"translate":74},[78],[70,10568,10570],{"className":10569},[82],[84,10571,10572],{"xmlns":86},[89,10573,10574,10582],{},[92,10575,10576],{},[137,10577,10578,10580],{},[95,10579,106],{},[95,10581,10152],{},[164,10583,10584],{"encoding":166},"x_k",[70,10586,10588],{"className":10587,"ariaHidden":172},[171],[70,10589,10591,10594],{"className":10590},[176],[70,10592],{"className":10593,"style":5634},[180],[70,10595,10597,10600],{"className":10596},[185],[70,10598,106],{"className":10599},[185,193],[70,10601,10603],{"className":10602},[314],[70,10604,10606,10626],{"className":10605},[225,226],[70,10607,10609,10623],{"className":10608},[230],[70,10610,10612],{"className":10611,"style":324},[234],[70,10613,10614,10617],{"style":327},[70,10615],{"className":10616,"style":331},[242],[70,10618,10620],{"className":10619},[247,248,249,250],[70,10621,10152],{"className":10622,"style":10212},[185,193,250],[70,10624,291],{"className":10625},[290],[70,10627,10629],{"className":10628},[230],[70,10630,10632],{"className":10631,"style":2841},[234],[70,10633],{},[70,10635,10637],{"className":10636,"translate":74},[73],[70,10638,10640,10688],{"className":10639,"translate":74},[78],[70,10641,10643],{"className":10642},[82],[84,10644,10645],{"xmlns":86,"display":87},[89,10646,10647,10685],{},[92,10648,10649,10655,10657,10667,10670,10673,10675,10677,10679],{},[137,10650,10651,10653],{},[95,10652,9154],{},[95,10654,10152],{},[100,10656,5491],{"mathvariant":97},[137,10658,10659,10661],{},[95,10660,911],{},[92,10662,10663,10665],{},[95,10664,10152],{},[95,10666,10152],{},[4704,10668,10669],{},"  ",[100,10671,10672],{},"⟹",[4704,10674,10669],{},[95,10676,4632],{},[100,10678,5491],{"mathvariant":97},[137,10680,10681,10683],{},[95,10682,106],{},[95,10684,10152],{},[164,10686,10687],{"encoding":166},"e_k \\neq d_{kk} \\implies y \\neq x_k",[70,10689,10691,10779,10846,10897],{"className":10690,"ariaHidden":172},[171],[70,10692,10694,10697,10737,10740,10776],{"className":10693},[176],[70,10695],{"className":10696,"style":791},[180],[70,10698,10700,10703],{"className":10699},[185],[70,10701,9154],{"className":10702},[185,193],[70,10704,10706],{"className":10705},[314],[70,10707,10709,10729],{"className":10708},[225,226],[70,10710,10712,10726],{"className":10711},[230],[70,10713,10715],{"className":10714,"style":324},[234],[70,10716,10717,10720],{"style":327},[70,10718],{"className":10719,"style":331},[242],[70,10721,10723],{"className":10722},[247,248,249,250],[70,10724,10152],{"className":10725,"style":10212},[185,193,250],[70,10727,291],{"className":10728},[290],[70,10730,10732],{"className":10731},[230],[70,10733,10735],{"className":10734,"style":2841},[234],[70,10736],{},[70,10738],{"className":10739,"style":202},[201],[70,10741,10743,10770,10773],{"className":10742},[206],[70,10744,10746],{"className":10745},[206],[70,10747,10749],{"className":10748},[185,5591],[70,10750,10752],{"className":10751},[5595],[70,10753,10755,10758,10767],{"className":10754},[5599],[70,10756],{"className":10757,"style":791},[180],[70,10759,10761],{"className":10760},[5606],[70,10762,10764],{"className":10763},[185],[70,10765,5613],{"className":10766},[206],[70,10768],{"className":10769},[5617],[70,10771],{"className":10772},[201,5621],[70,10774,112],{"className":10775},[206],[70,10777],{"className":10778,"style":202},[201],[70,10780,10782,10785,10831,10834,10837,10840,10843],{"className":10781},[176],[70,10783],{"className":10784,"style":9931},[180],[70,10786,10788,10791],{"className":10787},[185],[70,10789,911],{"className":10790},[185,193],[70,10792,10794],{"className":10793},[314],[70,10795,10797,10823],{"className":10796},[225,226],[70,10798,10800,10820],{"className":10799},[230],[70,10801,10803],{"className":10802,"style":324},[234],[70,10804,10805,10808],{"style":327},[70,10806],{"className":10807,"style":331},[242],[70,10809,10811],{"className":10810},[247,248,249,250],[70,10812,10814,10817],{"className":10813},[185,250],[70,10815,10152],{"className":10816,"style":10212},[185,193,250],[70,10818,10152],{"className":10819,"style":10212},[185,193,250],[70,10821,291],{"className":10822},[290],[70,10824,10826],{"className":10825},[230],[70,10827,10829],{"className":10828,"style":2841},[234],[70,10830],{},[70,10832],{"className":10833,"style":202},[201],[70,10835],{"className":10836,"style":202},[201],[70,10838,10672],{"className":10839},[206],[70,10841],{"className":10842,"style":202},[201],[70,10844],{"className":10845,"style":202},[201],[70,10847,10849,10852,10855,10858,10894],{"className":10848},[176],[70,10850],{"className":10851,"style":791},[180],[70,10853,4632],{"className":10854,"style":365},[185,193],[70,10856],{"className":10857,"style":202},[201],[70,10859,10861,10888,10891],{"className":10860},[206],[70,10862,10864],{"className":10863},[206],[70,10865,10867],{"className":10866},[185,5591],[70,10868,10870],{"className":10869},[5595],[70,10871,10873,10876,10885],{"className":10872},[5599],[70,10874],{"className":10875,"style":791},[180],[70,10877,10879],{"className":10878},[5606],[70,10880,10882],{"className":10881},[185],[70,10883,5613],{"className":10884},[206],[70,10886],{"className":10887},[5617],[70,10889],{"className":10890},[201,5621],[70,10892,112],{"className":10893},[206],[70,10895],{"className":10896,"style":202},[201],[70,10898,10900,10903],{"className":10899},[176],[70,10901],{"className":10902,"style":5634},[180],[70,10904,10906,10909],{"className":10905},[185],[70,10907,106],{"className":10908},[185,193],[70,10910,10912],{"className":10911},[314],[70,10913,10915,10935],{"className":10914},[225,226],[70,10916,10918,10932],{"className":10917},[230],[70,10919,10921],{"className":10920,"style":324},[234],[70,10922,10923,10926],{"style":327},[70,10924],{"className":10925,"style":331},[242],[70,10927,10929],{"className":10928},[247,248,249,250],[70,10930,10152],{"className":10931,"style":10212},[185,193,250],[70,10933,291],{"className":10934},[290],[70,10936,10938],{"className":10937},[230],[70,10939,10941],{"className":10940,"style":2841},[234],[70,10942],{},[11,10944,10945,10946,11037],{},"This contradicts the supposition that ",[70,10947,10949,10970],{"className":10948,"translate":74},[78],[70,10950,10952],{"className":10951},[82],[84,10953,10954],{"xmlns":86},[89,10955,10956,10968],{},[92,10957,10958,10960,10962],{},[95,10959,4632],{},[100,10961,112],{},[137,10963,10964,10966],{},[95,10965,106],{},[95,10967,10152],{},[164,10969,10155],{"encoding":166},[70,10971,10973,10991],{"className":10972,"ariaHidden":172},[171],[70,10974,10976,10979,10982,10985,10988],{"className":10975},[176],[70,10977],{"className":10978,"style":2801},[180],[70,10980,4632],{"className":10981,"style":365},[185,193],[70,10983],{"className":10984,"style":202},[201],[70,10986,112],{"className":10987},[206],[70,10989],{"className":10990,"style":202},[201],[70,10992,10994,10997],{"className":10993},[176],[70,10995],{"className":10996,"style":5634},[180],[70,10998,11000,11003],{"className":10999},[185],[70,11001,106],{"className":11002},[185,193],[70,11004,11006],{"className":11005},[314],[70,11007,11009,11029],{"className":11008},[225,226],[70,11010,11012,11026],{"className":11011},[230],[70,11013,11015],{"className":11014,"style":324},[234],[70,11016,11017,11020],{"style":327},[70,11018],{"className":11019,"style":331},[242],[70,11021,11023],{"className":11022},[247,248,249,250],[70,11024,10152],{"className":11025,"style":10212},[185,193,250],[70,11027,291],{"className":11028},[290],[70,11030,11032],{"className":11031},[230],[70,11033,11035],{"className":11034,"style":2841},[234],[70,11036],{},"!",[11,11039,11040,11041,882,11044,11072,11073,11101,11102,11130,11131,11159,11160,11188,11189,11192],{},"Moreover, this contradiction holds for ",[29,11042,11043],{},"every",[70,11045,11047,11060],{"className":11046,"translate":74},[78],[70,11048,11050],{"className":11049},[82],[84,11051,11052],{"xmlns":86},[89,11053,11054,11058],{},[92,11055,11056],{},[95,11057,10152],{},[164,11059,10152],{"encoding":166},[70,11061,11063],{"className":11062,"ariaHidden":172},[171],[70,11064,11066,11069],{"className":11065},[176],[70,11067],{"className":11068,"style":6013},[180],[70,11070,10152],{"className":11071,"style":10212},[185,193]," — ",[70,11074,11076,11089],{"className":11075,"translate":74},[78],[70,11077,11079],{"className":11078},[82],[84,11080,11081],{"xmlns":86},[89,11082,11083,11087],{},[92,11084,11085],{},[95,11086,4632],{},[164,11088,4632],{"encoding":166},[70,11090,11092],{"className":11091,"ariaHidden":172},[171],[70,11093,11095,11098],{"className":11094},[176],[70,11096],{"className":11097,"style":2801},[180],[70,11099,4632],{"className":11100,"style":365},[185,193]," differs from the 1st number at the 1st digit, from the 2nd number at the 2nd digit, from the ",[70,11103,11105,11118],{"className":11104,"translate":74},[78],[70,11106,11108],{"className":11107},[82],[84,11109,11110],{"xmlns":86},[89,11111,11112,11116],{},[92,11113,11114],{},[95,11115,6404],{},[164,11117,6404],{"encoding":166},[70,11119,11121],{"className":11120,"ariaHidden":172},[171],[70,11122,11124,11127],{"className":11123},[176],[70,11125],{"className":11126,"style":525},[180],[70,11128,6404],{"className":11129},[185,193],"-th number at the ",[70,11132,11134,11147],{"className":11133,"translate":74},[78],[70,11135,11137],{"className":11136},[82],[84,11138,11139],{"xmlns":86},[89,11140,11141,11145],{},[92,11142,11143],{},[95,11144,6404],{},[164,11146,6404],{"encoding":166},[70,11148,11150],{"className":11149,"ariaHidden":172},[171],[70,11151,11153,11156],{"className":11152},[176],[70,11154],{"className":11155,"style":525},[180],[70,11157,6404],{"className":11158},[185,193],"-th digit... Hence ",[70,11161,11163,11176],{"className":11162,"translate":74},[78],[70,11164,11166],{"className":11165},[82],[84,11167,11168],{"xmlns":86},[89,11169,11170,11174],{},[92,11171,11172],{},[95,11173,4632],{},[164,11175,4632],{"encoding":166},[70,11177,11179],{"className":11178,"ariaHidden":172},[171],[70,11180,11182,11185],{"className":11181},[176],[70,11183],{"className":11184,"style":2801},[180],[70,11186,4632],{"className":11187,"style":365},[185,193]," differs from ",[29,11190,11191],{},"every single"," number in the list.",[11,11194,11195,11196,11224,11225,11276],{},"Yet ",[70,11197,11199,11212],{"className":11198,"translate":74},[78],[70,11200,11202],{"className":11201},[82],[84,11203,11204],{"xmlns":86},[89,11205,11206,11210],{},[92,11207,11208],{},[95,11209,4632],{},[164,11211,4632],{"encoding":166},[70,11213,11215],{"className":11214,"ariaHidden":172},[171],[70,11216,11218,11221],{"className":11217},[176],[70,11219],{"className":11220,"style":2801},[180],[70,11222,4632],{"className":11223,"style":365},[185,193]," is indisputably a real number in ",[70,11226,11228,11249],{"className":11227,"translate":74},[78],[70,11229,11231],{"className":11230},[82],[84,11232,11233],{"xmlns":86},[89,11234,11235,11247],{},[92,11236,11237,11239,11241,11243,11245],{},[100,11238,103],{"stretchy":102},[1566,11240,2780],{},[100,11242,906],{"separator":172},[1566,11244,1568],{},[100,11246,109],{"stretchy":102},[164,11248,7239],{"encoding":166},[70,11250,11252],{"className":11251,"ariaHidden":172},[171],[70,11253,11255,11258,11261,11264,11267,11270,11273],{"className":11254},[176],[70,11256],{"className":11257,"style":181},[180],[70,11259,103],{"className":11260},[189],[70,11262,2780],{"className":11263},[185],[70,11265,906],{"className":11266},[976],[70,11268],{"className":11269,"style":304},[201],[70,11271,1568],{"className":11272},[185],[70,11274,109],{"className":11275},[197]," (every one of its digits is a legitimate decimal digit).",[11,11278,11279,11282,11283,11311,11312,11407,11408,6106,11459,24],{},[29,11280,11281],{},"Conclusion",": regardless of how one constructs this \"list of all real numbers,\" one can always construct a real number ",[70,11284,11286,11299],{"className":11285,"translate":74},[78],[70,11287,11289],{"className":11288},[82],[84,11290,11291],{"xmlns":86},[89,11292,11293,11297],{},[92,11294,11295],{},[95,11296,4632],{},[164,11298,4632],{"encoding":166},[70,11300,11302],{"className":11301,"ariaHidden":172},[171],[70,11303,11305,11308],{"className":11304},[176],[70,11306],{"className":11307,"style":2801},[180],[70,11309,4632],{"className":11310,"style":365},[185,193]," that escapes the list. This demonstrates that the initial assumption — the existence of a bijection ",[70,11313,11315,11344],{"className":11314,"translate":74},[78],[70,11316,11318],{"className":11317},[82],[84,11319,11320],{"xmlns":86},[89,11321,11322,11342],{},[92,11323,11324,11326,11328,11330,11332,11334,11336,11338,11340],{},[95,11325,123],{},[100,11327,68],{},[95,11329,5211],{"mathvariant":5210},[100,11331,1550],{},[100,11333,103],{"stretchy":102},[1566,11335,2780],{},[100,11337,906],{"separator":172},[1566,11339,1568],{},[100,11341,109],{"stretchy":102},[164,11343,7419],{"encoding":166},[70,11345,11347,11365,11383],{"className":11346,"ariaHidden":172},[171],[70,11348,11350,11353,11356,11359,11362],{"className":11349},[176],[70,11351],{"className":11352,"style":791},[180],[70,11354,123],{"className":11355,"style":257},[185,193],[70,11357],{"className":11358,"style":202},[201],[70,11360,68],{"className":11361},[206],[70,11363],{"className":11364,"style":202},[201],[70,11366,11368,11371,11374,11377,11380],{"className":11367},[176],[70,11369],{"className":11370,"style":5254},[180],[70,11372,5211],{"className":11373},[185,5258],[70,11375],{"className":11376,"style":202},[201],[70,11378,1550],{"className":11379},[206],[70,11381],{"className":11382,"style":202},[201],[70,11384,11386,11389,11392,11395,11398,11401,11404],{"className":11385},[176],[70,11387],{"className":11388,"style":181},[180],[70,11390,103],{"className":11391},[189],[70,11393,2780],{"className":11394},[185],[70,11396,906],{"className":11397},[976],[70,11399],{"className":11400,"style":304},[201],[70,11402,1568],{"className":11403},[185],[70,11405,109],{"className":11406},[197]," — is false. The real numbers in ",[70,11409,11411,11432],{"className":11410,"translate":74},[78],[70,11412,11414],{"className":11413},[82],[84,11415,11416],{"xmlns":86},[89,11417,11418,11430],{},[92,11419,11420,11422,11424,11426,11428],{},[100,11421,103],{"stretchy":102},[1566,11423,2780],{},[100,11425,906],{"separator":172},[1566,11427,1568],{},[100,11429,109],{"stretchy":102},[164,11431,7239],{"encoding":166},[70,11433,11435],{"className":11434,"ariaHidden":172},[171],[70,11436,11438,11441,11444,11447,11450,11453,11456],{"className":11437},[176],[70,11439],{"className":11440,"style":181},[180],[70,11442,103],{"className":11443},[189],[70,11445,2780],{"className":11446},[185],[70,11448,906],{"className":11449},[976],[70,11451],{"className":11452,"style":304},[201],[70,11454,1568],{"className":11455},[185],[70,11457,109],{"className":11458},[197],[29,11460,11461],{},"uncountable",[11,11463,11464,11465,11665,11666,11736,11737,11740],{},"Why the name \"diagonal\"? The term derives from the manner in which the digits ",[70,11466,11468,11506],{"className":11467,"translate":74},[78],[70,11469,11471],{"className":11470},[82],[84,11472,11473],{"xmlns":86},[89,11474,11475,11503],{},[92,11476,11477,11483,11485,11491,11493,11499,11501],{},[137,11478,11479,11481],{},[95,11480,911],{},[1566,11482,7532],{},[100,11484,906],{"separator":172},[137,11486,11487,11489],{},[95,11488,911],{},[1566,11490,7601],{},[100,11492,906],{"separator":172},[137,11494,11495,11497],{},[95,11496,911],{},[1566,11498,7670],{},[100,11500,906],{"separator":172},[100,11502,3004],{},[164,11504,11505],{"encoding":166},"d_{11}, d_{22}, d_{33}, \\dots",[70,11507,11509],{"className":11508,"ariaHidden":172},[171],[70,11510,11512,11515,11558,11561,11564,11607,11610,11613,11656,11659,11662],{"className":11511},[176],[70,11513],{"className":11514,"style":791},[180],[70,11516,11518,11521],{"className":11517},[185],[70,11519,911],{"className":11520},[185,193],[70,11522,11524],{"className":11523},[314],[70,11525,11527,11550],{"className":11526},[225,226],[70,11528,11530,11547],{"className":11529},[230],[70,11531,11533],{"className":11532,"style":2820},[234],[70,11534,11535,11538],{"style":327},[70,11536],{"className":11537,"style":331},[242],[70,11539,11541],{"className":11540},[247,248,249,250],[70,11542,11544],{"className":11543},[185,250],[70,11545,7532],{"className":11546},[185,250],[70,11548,291],{"className":11549},[290],[70,11551,11553],{"className":11552},[230],[70,11554,11556],{"className":11555,"style":2841},[234],[70,11557],{},[70,11559,906],{"className":11560},[976],[70,11562],{"className":11563,"style":304},[201],[70,11565,11567,11570],{"className":11566},[185],[70,11568,911],{"className":11569},[185,193],[70,11571,11573],{"className":11572},[314],[70,11574,11576,11599],{"className":11575},[225,226],[70,11577,11579,11596],{"className":11578},[230],[70,11580,11582],{"className":11581,"style":2820},[234],[70,11583,11584,11587],{"style":327},[70,11585],{"className":11586,"style":331},[242],[70,11588,11590],{"className":11589},[247,248,249,250],[70,11591,11593],{"className":11592},[185,250],[70,11594,7601],{"className":11595},[185,250],[70,11597,291],{"className":11598},[290],[70,11600,11602],{"className":11601},[230],[70,11603,11605],{"className":11604,"style":2841},[234],[70,11606],{},[70,11608,906],{"className":11609},[976],[70,11611],{"className":11612,"style":304},[201],[70,11614,11616,11619],{"className":11615},[185],[70,11617,911],{"className":11618},[185,193],[70,11620,11622],{"className":11621},[314],[70,11623,11625,11648],{"className":11624},[225,226],[70,11626,11628,11645],{"className":11627},[230],[70,11629,11631],{"className":11630,"style":2820},[234],[70,11632,11633,11636],{"style":327},[70,11634],{"className":11635,"style":331},[242],[70,11637,11639],{"className":11638},[247,248,249,250],[70,11640,11642],{"className":11641},[185,250],[70,11643,7670],{"className":11644},[185,250],[70,11646,291],{"className":11647},[290],[70,11649,11651],{"className":11650},[230],[70,11652,11654],{"className":11653,"style":2841},[234],[70,11655],{},[70,11657,906],{"className":11658},[976],[70,11660],{"className":11661,"style":304},[201],[70,11663,3004],{"className":11664},[3082]," are selected in the proof — if one arranges all ",[70,11667,11669,11687],{"className":11668,"translate":74},[78],[70,11670,11672],{"className":11671},[82],[84,11673,11674],{"xmlns":86},[89,11675,11676,11684],{},[92,11677,11678],{},[137,11679,11680,11682],{},[95,11681,106],{},[95,11683,2701],{},[164,11685,11686],{"encoding":166},"x_i",[70,11688,11690],{"className":11689,"ariaHidden":172},[171],[70,11691,11693,11696],{"className":11692},[176],[70,11694],{"className":11695,"style":5634},[180],[70,11697,11699,11702],{"className":11698},[185],[70,11700,106],{"className":11701},[185,193],[70,11703,11705],{"className":11704},[314],[70,11706,11708,11728],{"className":11707},[225,226],[70,11709,11711,11725],{"className":11710},[230],[70,11712,11714],{"className":11713,"style":8978},[234],[70,11715,11716,11719],{"style":327},[70,11717],{"className":11718,"style":331},[242],[70,11720,11722],{"className":11721},[247,248,249,250],[70,11723,2701],{"className":11724},[185,193,250],[70,11726,291],{"className":11727},[290],[70,11729,11731],{"className":11730},[230],[70,11732,11734],{"className":11733,"style":2841},[234],[70,11735],{}," vertically into an infinite matrix, these positions constitute precisely the ",[29,11738,11739],{},"main diagonal"," of the matrix:",[70,11742,11744],{"className":11743,"translate":74},[73],[70,11745,11747,11953],{"className":11746,"translate":74},[78],[70,11748,11750],{"className":11749},[82],[84,11751,11752],{"xmlns":86,"display":87},[89,11753,11754,11950],{},[92,11755,11756,11758,11948],{},[100,11757,103],{"fence":172},[6648,11759,11762,11811,11857,11903],{"rowspacing":11760,"columnalign":11761,"columnspacing":6652},"0.16em","center center center 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& d_{12} & d_{13} & \\cdots \\\\\nd_{21} & \\boxed{d_{22}} & d_{23} & \\cdots \\\\\nd_{31} & d_{32} & \\boxed{d_{33}} & \\cdots \\\\\n\\vdots & \\vdots & \\vdots & 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10.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z",[70,12867,291],{"className":12868},[290],[70,12870,12872],{"className":12871},[230],[70,12873,12875],{"className":12874,"style":12010},[234],[70,12876],{},[11,12878,12879],{},"We \"flip\" each digit along the diagonal, constructing a new element that necessarily evades the entire list.",[45,12881,12883],{"id":12882},"constructing-an-escapee","Constructing an Escapee",[11,12885,12886],{},"The formulae alone may remain somewhat abstract; let us render the process concrete through code. The program below takes a sample list of ten \"real numbers\" and actually executes the diagonal construction, allowing the reader to see clearly how the new number \"avoids\" each of the original numbers.",[12888,12889,12893],"pre",{"className":12890,"code":12891,"language":12892,"meta":4976,"style":4976},"language-python shiki shiki-themes github-dark","def diagonal_construct(number_list, digits=10):\n    \"\"\"\n    number_list: list of strings, each of the form \"0.d1d2d3...\"\n    returns: a new number string constructed via the diagonal method\n    \"\"\"\n    new_digits = []\n    for i, num_str in enumerate(number_list):\n        decimal_part = num_str.split('.')[1].ljust(digits, '0')\n        d = int(decimal_part[i])       # i-th digit of the i-th number (the diagonal digit)\n        new_d = (d + 5) % 10           # rule: make the new digit differ from the original\n        new_digits.append(str(new_d))\n    return \"0.\" + \"\".join(new_digits)\n\n\nsample_list = [\n    \"0.1234567890\", \"0.9876543210\", \"0.5555555555\", \"0.1111111111\",\n    \"0.3141592653\", \"0.2718281828\", \"0.0000000001\", \"0.9999999998\",\n    \"0.4242424242\", \"0.6180339887\",\n]\n\nnew_number = diagonal_construct(sample_list)\nprint(\"Diagonal-constructed new number:\", new_number)\n","python",[2617,12894,12895,12920,12926,12932,12938,12943,12954,12972,13000,13018,13046,13058,13076,13082,13087,13098,13123,13146,13159,13165,13170,13181],{"__ignoreMap":4976},[70,12896,12899,12903,12907,12911,12913,12917],{"class":12897,"line":12898},"line",1,[70,12900,12902],{"class":12901},"snl16","def",[70,12904,12906],{"class":12905},"svObZ"," diagonal_construct",[70,12908,12910],{"class":12909},"s95oV","(number_list, digits",[70,12912,112],{"class":12901},[70,12914,12916],{"class":12915},"sDLfK","10",[70,12918,12919],{"class":12909},"):\n",[70,12921,12922],{"class":12897,"line":4977},[70,12923,12925],{"class":12924},"sU2Wk","    \"\"\"\n",[70,12927,12929],{"class":12897,"line":12928},3,[70,12930,12931],{"class":12924},"    number_list: list of strings, each of the form \"0.d1d2d3...\"\n",[70,12933,12935],{"class":12897,"line":12934},4,[70,12936,12937],{"class":12924},"    returns: a new number string constructed via the diagonal method\n",[70,12939,12941],{"class":12897,"line":12940},5,[70,12942,12925],{"class":12924},[70,12944,12946,12949,12951],{"class":12897,"line":12945},6,[70,12947,12948],{"class":12909},"    new_digits ",[70,12950,112],{"class":12901},[70,12952,12953],{"class":12909}," []\n",[70,12955,12957,12960,12963,12966,12969],{"class":12897,"line":12956},7,[70,12958,12959],{"class":12901},"    for",[70,12961,12962],{"class":12909}," i, num_str ",[70,12964,12965],{"class":12901},"in",[70,12967,12968],{"class":12915}," enumerate",[70,12970,12971],{"class":12909},"(number_list):\n",[70,12973,12975,12978,12980,12983,12986,12989,12991,12994,12997],{"class":12897,"line":12974},8,[70,12976,12977],{"class":12909},"        decimal_part ",[70,12979,112],{"class":12901},[70,12981,12982],{"class":12909}," num_str.split(",[70,12984,12985],{"class":12924},"'.'",[70,12987,12988],{"class":12909},")[",[70,12990,1568],{"class":12915},[70,12992,12993],{"class":12909},"].ljust(digits, ",[70,12995,12996],{"class":12924},"'0'",[70,12998,12999],{"class":12909},")\n",[70,13001,13003,13006,13008,13011,13014],{"class":12897,"line":13002},9,[70,13004,13005],{"class":12909},"        d ",[70,13007,112],{"class":12901},[70,13009,13010],{"class":12915}," int",[70,13012,13013],{"class":12909},"(decimal_part[i])       ",[70,13015,13017],{"class":13016},"sAwPA","# i-th digit of the i-th number (the diagonal digit)\n",[70,13019,13021,13024,13026,13029,13031,13034,13037,13040,13043],{"class":12897,"line":13020},10,[70,13022,13023],{"class":12909},"        new_d ",[70,13025,112],{"class":12901},[70,13027,13028],{"class":12909}," (d ",[70,13030,3017],{"class":12901},[70,13032,13033],{"class":12915}," 5",[70,13035,13036],{"class":12909},") ",[70,13038,13039],{"class":12901},"%",[70,13041,13042],{"class":12915}," 10",[70,13044,13045],{"class":13016},"           # rule: make the new digit differ from the original\n",[70,13047,13049,13052,13055],{"class":12897,"line":13048},11,[70,13050,13051],{"class":12909},"        new_digits.append(",[70,13053,13054],{"class":12915},"str",[70,13056,13057],{"class":12909},"(new_d))\n",[70,13059,13061,13064,13067,13070,13073],{"class":12897,"line":13060},12,[70,13062,13063],{"class":12901},"    return",[70,13065,13066],{"class":12924}," \"0.\"",[70,13068,13069],{"class":12901}," +",[70,13071,13072],{"class":12924}," \"\"",[70,13074,13075],{"class":12909},".join(new_digits)\n",[70,13077,13079],{"class":12897,"line":13078},13,[70,13080,13081],{"emptyLinePlaceholder":4989},"\n",[70,13083,13085],{"class":12897,"line":13084},14,[70,13086,13081],{"emptyLinePlaceholder":4989},[70,13088,13090,13093,13095],{"class":12897,"line":13089},15,[70,13091,13092],{"class":12909},"sample_list ",[70,13094,112],{"class":12901},[70,13096,13097],{"class":12909}," [\n",[70,13099,13101,13104,13107,13110,13112,13115,13117,13120],{"class":12897,"line":13100},16,[70,13102,13103],{"class":12924},"    \"0.1234567890\"",[70,13105,13106],{"class":12909},", ",[70,13108,13109],{"class":12924},"\"0.9876543210\"",[70,13111,13106],{"class":12909},[70,13113,13114],{"class":12924},"\"0.5555555555\"",[70,13116,13106],{"class":12909},[70,13118,13119],{"class":12924},"\"0.1111111111\"",[70,13121,13122],{"class":12909},",\n",[70,13124,13126,13129,13131,13134,13136,13139,13141,13144],{"class":12897,"line":13125},17,[70,13127,13128],{"class":12924},"    \"0.3141592653\"",[70,13130,13106],{"class":12909},[70,13132,13133],{"class":12924},"\"0.2718281828\"",[70,13135,13106],{"class":12909},[70,13137,13138],{"class":12924},"\"0.0000000001\"",[70,13140,13106],{"class":12909},[70,13142,13143],{"class":12924},"\"0.9999999998\"",[70,13145,13122],{"class":12909},[70,13147,13149,13152,13154,13157],{"class":12897,"line":13148},18,[70,13150,13151],{"class":12924},"    \"0.4242424242\"",[70,13153,13106],{"class":12909},[70,13155,13156],{"class":12924},"\"0.6180339887\"",[70,13158,13122],{"class":12909},[70,13160,13162],{"class":12897,"line":13161},19,[70,13163,13164],{"class":12909},"]\n",[70,13166,13168],{"class":12897,"line":13167},20,[70,13169,13081],{"emptyLinePlaceholder":4989},[70,13171,13173,13176,13178],{"class":12897,"line":13172},21,[70,13174,13175],{"class":12909},"new_number ",[70,13177,112],{"class":12901},[70,13179,13180],{"class":12909}," diagonal_construct(sample_list)\n",[70,13182,13184,13187,13189,13192],{"class":12897,"line":13183},22,[70,13185,13186],{"class":12915},"print",[70,13188,103],{"class":12909},[70,13190,13191],{"class":12924},"\"Diagonal-constructed new number:\"",[70,13193,13194],{"class":12909},", new_number)\n",[11,13196,13197],{},[29,13198,13199],{},"Output:",[12888,13201,13205],{"className":13202,"code":13204,"language":4752},[13203],"language-text","Original list:\n  No. 1: 0.1234567890   (1st diagonal digit: 1)\n  No. 2: 0.9876543210   (2nd diagonal digit: 8)\n  No. 3: 0.5555555555   (3rd diagonal digit: 5)\n  No. 4: 0.1111111111   (4th diagonal digit: 1)\n  No. 5: 0.3141592653   (5th diagonal digit: 5)\n  No. 6: 0.2718281828   (6th diagonal digit: 8)\n  No. 7: 0.0000000001   (7th diagonal digit: 0)\n  No. 8: 0.9999999998   (8th diagonal digit: 9)\n  No. 9: 0.4242424242   (9th diagonal digit: 4)\n  No. 10: 0.6180339887  (10th diagonal digit: 7)\n\nDiagonal-constructed new number: 0.6306035492\n\nVerifying that the new number differs from each number in the list:\n  Differs from No. 1 at digit 1: original=1, new=6, differs=True\n  Differs from No. 2 at digit 2: original=8, new=3, differs=True\n  Differs from No. 3 at digit 3: original=5, new=0, differs=True\n  Differs from No. 4 at digit 4: original=1, new=6, differs=True\n  Differs from No. 5 at digit 5: original=5, new=0, differs=True\n  Differs from No. 6 at digit 6: original=8, new=3, differs=True\n  Differs from No. 7 at digit 7: original=0, new=5, differs=True\n  Differs from No. 8 at digit 8: original=9, new=4, differs=True\n  Differs from No. 9 at digit 9: original=4, new=9, differs=True\n  Differs from No. 10 at digit 10: original=7, new=2, differs=True\n",[2617,13206,13204],{"__ignoreMap":4976},[11,13208,13209,13210,11188,13213,13215,13216,13106,13244,13369],{},"One may observe with clarity that the newly constructed number ",[2617,13211,13212],{},"0.6306035492",[29,13214,11043],{}," number in the list precisely at the corresponding digit position. Naturally, only the finite case of ten numbers is demonstrated here (a genuinely infinite list cannot, after all, be stored in a computer), but this is precisely the intuitive instantiation of the proof's core logical step — that for arbitrary ",[70,13217,13219,13232],{"className":13218,"translate":74},[78],[70,13220,13222],{"className":13221},[82],[84,13223,13224],{"xmlns":86},[89,13225,13226,13230],{},[92,13227,13228],{},[95,13229,10152],{},[164,13231,10152],{"encoding":166},[70,13233,13235],{"className":13234,"ariaHidden":172},[171],[70,13236,13238,13241],{"className":13237},[176],[70,13239],{"className":13240,"style":6013},[180],[70,13242,10152],{"className":13243,"style":10212},[185,193],[70,13245,13247,13269],{"className":13246,"translate":74},[78],[70,13248,13250],{"className":13249},[82],[84,13251,13252],{"xmlns":86},[89,13253,13254,13266],{},[92,13255,13256,13258,13260],{},[95,13257,4632],{},[100,13259,5491],{"mathvariant":97},[137,13261,13262,13264],{},[95,13263,106],{},[95,13265,10152],{},[164,13267,13268],{"encoding":166},"y \\neq x_k",[70,13270,13272,13323],{"className":13271,"ariaHidden":172},[171],[70,13273,13275,13278,13281,13284,13320],{"className":13274},[176],[70,13276],{"className":13277,"style":791},[180],[70,13279,4632],{"className":13280,"style":365},[185,193],[70,13282],{"className":13283,"style":202},[201],[70,13285,13287,13314,13317],{"className":13286},[206],[70,13288,13290],{"className":13289},[206],[70,13291,13293],{"className":13292},[185,5591],[70,13294,13296],{"className":13295},[5595],[70,13297,13299,13302,13311],{"className":13298},[5599],[70,13300],{"className":13301,"style":791},[180],[70,13303,13305],{"className":13304},[5606],[70,13306,13308],{"className":13307},[185],[70,13309,5613],{"className":13310},[206],[70,13312],{"className":13313},[5617],[70,13315],{"className":13316},[201,5621],[70,13318,112],{"className":13319},[206],[70,13321],{"className":13322,"style":202},[201],[70,13324,13326,13329],{"className":13325},[176],[70,13327],{"className":13328,"style":5634},[180],[70,13330,13332,13335],{"className":13331},[185],[70,13333,106],{"className":13334},[185,193],[70,13336,13338],{"className":13337},[314],[70,13339,13341,13361],{"className":13340},[225,226],[70,13342,13344,13358],{"className":13343},[230],[70,13345,13347],{"className":13346,"style":324},[234],[70,13348,13349,13352],{"style":327},[70,13350],{"className":13351,"style":331},[242],[70,13353,13355],{"className":13354},[247,248,249,250],[70,13356,10152],{"className":13357,"style":10212},[185,193,250],[70,13359,291],{"className":13360},[290],[70,13362,13364],{"className":13363},[230],[70,13365,13367],{"className":13366,"style":2841},[234],[70,13368],{},". Regardless of the length of the list, this construction rule guarantees that the new number evades every single entry.",[45,13371,13373],{"id":13372},"cantors-theorem","Cantor's Theorem",[11,13375,13376],{},"The diagonal argument serves not merely to prove the uncountability of the reals; it possesses a more general and more powerful formulation: Cantor's Theorem.",[1339,13378,13379],{},[11,13380,13381,13384,13385,13413,13414,13462,13463,13491,13492,13520,13521,24],{},[29,13382,13383],{},"Cantor's Theorem."," For any set ",[70,13386,13388,13401],{"className":13387,"translate":74},[78],[70,13389,13391],{"className":13390},[82],[84,13392,13393],{"xmlns":86},[89,13394,13395,13399],{},[92,13396,13397],{},[95,13398,5080],{},[164,13400,5080],{"encoding":166},[70,13402,13404],{"className":13403,"ariaHidden":172},[171],[70,13405,13407,13410],{"className":13406},[176],[70,13408],{"className":13409,"style":5067},[180],[70,13411,5080],{"className":13412},[185,193],", the power set ",[70,13415,13417,13439],{"className":13416,"translate":74},[78],[70,13418,13420],{"className":13419},[82],[84,13421,13422],{"xmlns":86},[89,13423,13424,13436],{},[92,13425,13426,13430,13432,13434],{},[95,13427,13429],{"mathvariant":13428},"script","P",[100,13431,103],{"stretchy":102},[95,13433,5080],{},[100,13435,109],{"stretchy":102},[164,13437,13438],{"encoding":166},"\\mathcal{P}(A)",[70,13440,13442],{"className":13441,"ariaHidden":172},[171],[70,13443,13445,13448,13453,13456,13459],{"className":13444},[176],[70,13446],{"className":13447,"style":181},[180],[70,13449,13429],{"className":13450,"style":13452},[185,13451],"mathcal","margin-right:0.0822em;",[70,13454,103],{"className":13455},[189],[70,13457,5080],{"className":13458},[185,193],[70,13460,109],{"className":13461},[197]," (i.e., the set of all subsets of ",[70,13464,13466,13479],{"className":13465,"translate":74},[78],[70,13467,13469],{"className":13468},[82],[84,13470,13471],{"xmlns":86},[89,13472,13473,13477],{},[92,13474,13475],{},[95,13476,5080],{},[164,13478,5080],{"encoding":166},[70,13480,13482],{"className":13481,"ariaHidden":172},[171],[70,13483,13485,13488],{"className":13484},[176],[70,13486],{"className":13487,"style":5067},[180],[70,13489,5080],{"className":13490},[185,193],") has strictly greater cardinality than ",[70,13493,13495,13508],{"className":13494,"translate":74},[78],[70,13496,13498],{"className":13497},[82],[84,13499,13500],{"xmlns":86},[89,13501,13502,13506],{},[92,13503,13504],{},[95,13505,5080],{},[164,13507,5080],{"encoding":166},[70,13509,13511],{"className":13510,"ariaHidden":172},[171],[70,13512,13514,13517],{"className":13513},[176],[70,13515],{"className":13516,"style":5067},[180],[70,13518,5080],{"className":13519},[185,193]," itself; that is, ",[70,13522,13524,13556],{"className":13523,"translate":74},[78],[70,13525,13527],{"className":13526},[82],[84,13528,13529],{"xmlns":86},[89,13530,13531,13553],{},[92,13532,13533,13535,13537,13539,13541,13543,13545,13547,13549,13551],{},[95,13534,4625],{"mathvariant":97},[95,13536,5080],{},[95,13538,4625],{"mathvariant":97},[100,13540,3894],{},[95,13542,4625],{"mathvariant":97},[95,13544,13429],{"mathvariant":13428},[100,13546,103],{"stretchy":102},[95,13548,5080],{},[100,13550,109],{"stretchy":102},[95,13552,4625],{"mathvariant":97},[164,13554,13555],{"encoding":166},"|A| \u003C |\\mathcal{P}(A)|",[70,13557,13559,13583],{"className":13558,"ariaHidden":172},[171],[70,13560,13562,13565,13568,13571,13574,13577,13580],{"className":13561},[176],[70,13563],{"className":13564,"style":181},[180],[70,13566,4625],{"className":13567},[185],[70,13569,5080],{"className":13570},[185,193],[70,13572,4625],{"className":13573},[185],[70,13575],{"className":13576,"style":202},[201],[70,13578,3894],{"className":13579},[206],[70,13581],{"className":13582,"style":202},[201],[70,13584,13586,13589,13592,13595,13598,13601,13604],{"className":13585},[176],[70,13587],{"className":13588,"style":181},[180],[70,13590,4625],{"className":13591},[185],[70,13593,13429],{"className":13594,"style":13452},[185,13451],[70,13596,103],{"className":13597},[189],[70,13599,5080],{"className":13600},[185,193],[70,13602,109],{"className":13603},[197],[70,13605,4625],{"className":13606},[185],[11,13608,13609,13610,5357,13613,13616,13617,13620,13621,13649,13650,13693],{},"It should be noted that this theorem holds for both ",[29,13611,13612],{},"finite sets",[29,13614,13615],{},"infinite sets",", and for infinite sets in particular, it reveals a startling fact: ",[29,13618,13619],{},"there is no \"largest infinity.\""," For however large ",[70,13622,13624,13637],{"className":13623,"translate":74},[78],[70,13625,13627],{"className":13626},[82],[84,13628,13629],{"xmlns":86},[89,13630,13631,13635],{},[92,13632,13633],{},[95,13634,5080],{},[164,13636,5080],{"encoding":166},[70,13638,13640],{"className":13639,"ariaHidden":172},[171],[70,13641,13643,13646],{"className":13642},[176],[70,13644],{"className":13645,"style":5067},[180],[70,13647,5080],{"className":13648},[185,193]," may be, ",[70,13651,13653,13672],{"className":13652,"translate":74},[78],[70,13654,13656],{"className":13655},[82],[84,13657,13658],{"xmlns":86},[89,13659,13660,13670],{},[92,13661,13662,13664,13666,13668],{},[95,13663,13429],{"mathvariant":13428},[100,13665,103],{"stretchy":102},[95,13667,5080],{},[100,13669,109],{"stretchy":102},[164,13671,13438],{"encoding":166},[70,13673,13675],{"className":13674,"ariaHidden":172},[171],[70,13676,13678,13681,13684,13687,13690],{"className":13677},[176],[70,13679],{"className":13680,"style":181},[180],[70,13682,13429],{"className":13683,"style":13452},[185,13451],[70,13685,103],{"className":13686},[189],[70,13688,5080],{"className":13689},[185,193],[70,13691,109],{"className":13692},[197]," is always larger; one may iterate the power set operation indefinitely, obtaining an ever-ascending hierarchy of infinities.",[13695,13696,13698],"h3",{"id":13697},"proof","Proof",[11,13700,13701,13704,13705,13793,13794,9125],{},[29,13702,13703],{},"Assume",", for the sake of contradiction, that there exists a surjection ",[70,13706,13708,13736],{"className":13707,"translate":74},[78],[70,13709,13711],{"className":13710},[82],[84,13712,13713],{"xmlns":86},[89,13714,13715,13733],{},[92,13716,13717,13719,13721,13723,13725,13727,13729,13731],{},[95,13718,123],{},[100,13720,68],{},[95,13722,5080],{},[100,13724,1550],{},[95,13726,13429],{"mathvariant":13428},[100,13728,103],{"stretchy":102},[95,13730,5080],{},[100,13732,109],{"stretchy":102},[164,13734,13735],{"encoding":166},"f: A \\to \\mathcal{P}(A)",[70,13737,13739,13757,13775],{"className":13738,"ariaHidden":172},[171],[70,13740,13742,13745,13748,13751,13754],{"className":13741},[176],[70,13743],{"className":13744,"style":791},[180],[70,13746,123],{"className":13747,"style":257},[185,193],[70,13749],{"className":13750,"style":202},[201],[70,13752,68],{"className":13753},[206],[70,13755],{"className":13756,"style":202},[201],[70,13758,13760,13763,13766,13769,13772],{"className":13759},[176],[70,13761],{"className":13762,"style":5067},[180],[70,13764,5080],{"className":13765},[185,193],[70,13767],{"className":13768,"style":202},[201],[70,13770,1550],{"className":13771},[206],[70,13773],{"className":13774,"style":202},[201],[70,13776,13778,13781,13784,13787,13790],{"className":13777},[176],[70,13779],{"className":13780,"style":181},[180],[70,13782,13429],{"className":13783,"style":13452},[185,13451],[70,13785,103],{"className":13786},[189],[70,13788,5080],{"className":13789},[185,193],[70,13791,109],{"className":13792},[197]," (we shall demonstrate that no such surjection can exist, whence ",[70,13795,13797,13828],{"className":13796,"translate":74},[78],[70,13798,13800],{"className":13799},[82],[84,13801,13802],{"xmlns":86},[89,13803,13804,13826],{},[92,13805,13806,13808,13810,13812,13814,13816,13818,13820,13822,13824],{},[95,13807,4625],{"mathvariant":97},[95,13809,5080],{},[95,13811,4625],{"mathvariant":97},[100,13813,3894],{},[95,13815,4625],{"mathvariant":97},[95,13817,13429],{"mathvariant":13428},[100,13819,103],{"stretchy":102},[95,13821,5080],{},[100,13823,109],{"stretchy":102},[95,13825,4625],{"mathvariant":97},[164,13827,13555],{"encoding":166},[70,13829,13831,13855],{"className":13830,"ariaHidden":172},[171],[70,13832,13834,13837,13840,13843,13846,13849,13852],{"className":13833},[176],[70,13835],{"className":13836,"style":181},[180],[70,13838,4625],{"className":13839},[185],[70,13841,5080],{"className":13842},[185,193],[70,13844,4625],{"className":13845},[185],[70,13847],{"className":13848,"style":202},[201],[70,13850,3894],{"className":13851},[206],[70,13853],{"className":13854,"style":202},[201],[70,13856,13858,13861,13864,13867,13870,13873,13876],{"className":13857},[176],[70,13859],{"className":13860,"style":181},[180],[70,13862,4625],{"className":13863},[185],[70,13865,13429],{"className":13866,"style":13452},[185,13451],[70,13868,103],{"className":13869},[189],[70,13871,5080],{"className":13872},[185,193],[70,13874,109],{"className":13875},[197],[70,13877,4625],{"className":13878},[185],[11,13880,13881,13882,13106,13932,13976,13977,14005],{},"For each element ",[70,13883,13885,13902],{"className":13884,"translate":74},[78],[70,13886,13888],{"className":13887},[82],[84,13889,13890],{"xmlns":86},[89,13891,13892,13900],{},[92,13893,13894,13896,13898],{},[95,13895,18],{},[100,13897,126],{},[95,13899,5080],{},[164,13901,5917],{"encoding":166},[70,13903,13905,13923],{"className":13904,"ariaHidden":172},[171],[70,13906,13908,13911,13914,13917,13920],{"className":13907},[176],[70,13909],{"className":13910,"style":5927},[180],[70,13912,18],{"className":13913},[185,193],[70,13915],{"className":13916,"style":202},[201],[70,13918,126],{"className":13919},[206],[70,13921],{"className":13922,"style":202},[201],[70,13924,13926,13929],{"className":13925},[176],[70,13927],{"className":13928,"style":5067},[180],[70,13930,5080],{"className":13931},[185,193],[70,13933,13935,13955],{"className":13934,"translate":74},[78],[70,13936,13938],{"className":13937},[82],[84,13939,13940],{"xmlns":86},[89,13941,13942,13952],{},[92,13943,13944,13946,13948,13950],{},[95,13945,123],{},[100,13947,103],{"stretchy":102},[95,13949,18],{},[100,13951,109],{"stretchy":102},[164,13953,13954],{"encoding":166},"f(a)",[70,13956,13958],{"className":13957,"ariaHidden":172},[171],[70,13959,13961,13964,13967,13970,13973],{"className":13960},[176],[70,13962],{"className":13963,"style":181},[180],[70,13965,123],{"className":13966,"style":257},[185,193],[70,13968,103],{"className":13969},[189],[70,13971,18],{"className":13972},[185,193],[70,13974,109],{"className":13975},[197]," is a subset of ",[70,13978,13980,13993],{"className":13979,"translate":74},[78],[70,13981,13983],{"className":13982},[82],[84,13984,13985],{"xmlns":86},[89,13986,13987,13991],{},[92,13988,13989],{},[95,13990,5080],{},[164,13992,5080],{"encoding":166},[70,13994,13996],{"className":13995,"ariaHidden":172},[171],[70,13997,13999,14002],{"className":13998},[176],[70,14000],{"className":14001,"style":5067},[180],[70,14003,5080],{"className":14004},[185,193],". We now construct a special subset:",[70,14007,14009],{"className":14008,"translate":74},[73],[70,14010,14012,14057],{"className":14011,"translate":74},[78],[70,14013,14015],{"className":14014},[82],[84,14016,14017],{"xmlns":86,"display":87},[89,14018,14019,14054],{},[92,14020,14021,14023,14025,14027,14029,14031,14033,14035,14037,14039,14042,14044,14046,14048,14050,14052],{},[95,14022,1561],{},[100,14024,112],{},[100,14026,5216],{"stretchy":102},[4704,14028,5238],{},[95,14030,18],{},[100,14032,126],{},[95,14034,5080],{},[100,14036,68],{},[95,14038,18],{},[100,14040,14041],{"mathvariant":97},"∉",[95,14043,123],{},[100,14045,103],{"stretchy":102},[95,14047,18],{},[100,14049,109],{"stretchy":102},[4704,14051,5238],{},[100,14053,5241],{"stretchy":102},[164,14055,14056],{"encoding":166},"D = \\{\\, a \\in A : a \\notin f(a) \\,\\}",[70,14058,14060,14078,14102,14120,14172],{"className":14059,"ariaHidden":172},[171],[70,14061,14063,14066,14069,14072,14075],{"className":14062},[176],[70,14064],{"className":14065,"style":5067},[180],[70,14067,1561],{"className":14068,"style":1726},[185,193],[70,14070],{"className":14071,"style":202},[201],[70,14073,112],{"className":14074},[206],[70,14076],{"className":14077,"style":202},[201],[70,14079,14081,14084,14087,14090,14093,14096,14099],{"className":14080},[176],[70,14082],{"className":14083,"style":181},[180],[70,14085,5216],{"className":14086},[189],[70,14088],{"className":14089,"style":304},[201],[70,14091,18],{"className":14092},[185,193],[70,14094],{"className":14095,"style":202},[201],[70,14097,126],{"className":14098},[206],[70,14100],{"className":14101,"style":202},[201],[70,14103,14105,14108,14111,14114,14117],{"className":14104},[176],[70,14106],{"className":14107,"style":5067},[180],[70,14109,5080],{"className":14110},[185,193],[70,14112],{"className":14113,"style":202},[201],[70,14115,68],{"className":14116},[206],[70,14118],{"className":14119,"style":202},[201],[70,14121,14123,14126,14129,14132,14169],{"className":14122},[176],[70,14124],{"className":14125,"style":181},[180],[70,14127,18],{"className":14128},[185,193],[70,14130],{"className":14131,"style":202},[201],[70,14133,14135,14141],{"className":14134},[206],[70,14136,14138],{"className":14137},[185],[70,14139,126],{"className":14140},[206],[70,14142,14144],{"className":14143},[185,5591],[70,14145,14147],{"className":14146},[5595],[70,14148,14151,14154,14166],{"className":14149},[14150],"llap",[70,14152],{"className":14153,"style":181},[180],[70,14155,14157],{"className":14156},[5606],[70,14158,14160,14163],{"className":14159},[185],[70,14161,6668],{"className":14162},[185],[70,14164],{"className":14165,"style":4134},[201],[70,14167],{"className":14168},[5617],[70,14170],{"className":14171,"style":202},[201],[70,14173,14175,14178,14181,14184,14187,14190,14193],{"className":14174},[176],[70,14176],{"className":14177,"style":181},[180],[70,14179,123],{"className":14180,"style":257},[185,193],[70,14182,103],{"className":14183},[189],[70,14185,18],{"className":14186},[185,193],[70,14188,109],{"className":14189},[197],[70,14191],{"className":14192,"style":304},[201],[70,14194,5241],{"className":14195},[197],[11,14197,14198],{},"This is the \"diagonal set\" — it collects all elements that \"do not belong to the subset they are mapped to.\"",[11,14200,14201,14202,14230,14231,14259,14260,14288,14289,14332,14333,5949,14384,24],{},"Since ",[70,14203,14205,14218],{"className":14204,"translate":74},[78],[70,14206,14208],{"className":14207},[82],[84,14209,14210],{"xmlns":86},[89,14211,14212,14216],{},[92,14213,14214],{},[95,14215,123],{},[164,14217,123],{"encoding":166},[70,14219,14221],{"className":14220,"ariaHidden":172},[171],[70,14222,14224,14227],{"className":14223},[176],[70,14225],{"className":14226,"style":791},[180],[70,14228,123],{"className":14229,"style":257},[185,193]," is surjective, ",[70,14232,14234,14247],{"className":14233,"translate":74},[78],[70,14235,14237],{"className":14236},[82],[84,14238,14239],{"xmlns":86},[89,14240,14241,14245],{},[92,14242,14243],{},[95,14244,1561],{},[164,14246,1561],{"encoding":166},[70,14248,14250],{"className":14249,"ariaHidden":172},[171],[70,14251,14253,14256],{"className":14252},[176],[70,14254],{"className":14255,"style":5067},[180],[70,14257,1561],{"className":14258,"style":1726},[185,193]," (being a subset of ",[70,14261,14263,14276],{"className":14262,"translate":74},[78],[70,14264,14266],{"className":14265},[82],[84,14267,14268],{"xmlns":86},[89,14269,14270,14274],{},[92,14271,14272],{},[95,14273,5080],{},[164,14275,5080],{"encoding":166},[70,14277,14279],{"className":14278,"ariaHidden":172},[171],[70,14280,14282,14285],{"className":14281},[176],[70,14283],{"className":14284,"style":5067},[180],[70,14286,5080],{"className":14287},[185,193],", hence an element of ",[70,14290,14292,14311],{"className":14291,"translate":74},[78],[70,14293,14295],{"className":14294},[82],[84,14296,14297],{"xmlns":86},[89,14298,14299,14309],{},[92,14300,14301,14303,14305,14307],{},[95,14302,13429],{"mathvariant":13428},[100,14304,103],{"stretchy":102},[95,14306,5080],{},[100,14308,109],{"stretchy":102},[164,14310,13438],{"encoding":166},[70,14312,14314],{"className":14313,"ariaHidden":172},[171],[70,14315,14317,14320,14323,14326,14329],{"className":14316},[176],[70,14318],{"className":14319,"style":181},[180],[70,14321,13429],{"className":14322,"style":13452},[185,13451],[70,14324,103],{"className":14325},[189],[70,14327,5080],{"className":14328},[185,193],[70,14330,109],{"className":14331},[197],") must possess a preimage; that is, there exists some ",[70,14334,14336,14354],{"className":14335,"translate":74},[78],[70,14337,14339],{"className":14338},[82],[84,14340,14341],{"xmlns":86},[89,14342,14343,14351],{},[92,14344,14345,14347,14349],{},[95,14346,911],{},[100,14348,126],{},[95,14350,5080],{},[164,14352,14353],{"encoding":166},"d \\in A",[70,14355,14357,14375],{"className":14356,"ariaHidden":172},[171],[70,14358,14360,14363,14366,14369,14372],{"className":14359},[176],[70,14361],{"className":14362,"style":5874},[180],[70,14364,911],{"className":14365},[185,193],[70,14367],{"className":14368,"style":202},[201],[70,14370,126],{"className":14371},[206],[70,14373],{"className":14374,"style":202},[201],[70,14376,14378,14381],{"className":14377},[176],[70,14379],{"className":14380,"style":5067},[180],[70,14382,5080],{"className":14383},[185,193],[70,14385,14387,14411],{"className":14386,"translate":74},[78],[70,14388,14390],{"className":14389},[82],[84,14391,14392],{"xmlns":86},[89,14393,14394,14408],{},[92,14395,14396,14398,14400,14402,14404,14406],{},[95,14397,123],{},[100,14399,103],{"stretchy":102},[95,14401,911],{},[100,14403,109],{"stretchy":102},[100,14405,112],{},[95,14407,1561],{},[164,14409,14410],{"encoding":166},"f(d) = D",[70,14412,14414,14441],{"className":14413,"ariaHidden":172},[171],[70,14415,14417,14420,14423,14426,14429,14432,14435,14438],{"className":14416},[176],[70,14418],{"className":14419,"style":181},[180],[70,14421,123],{"className":14422,"style":257},[185,193],[70,14424,103],{"className":14425},[189],[70,14427,911],{"className":14428},[185,193],[70,14430,109],{"className":14431},[197],[70,14433],{"className":14434,"style":202},[201],[70,14436,112],{"className":14437},[206],[70,14439],{"className":14440,"style":202},[201],[70,14442,14444,14447],{"className":14443},[176],[70,14445],{"className":14446,"style":5067},[180],[70,14448,1561],{"className":14449,"style":1726},[185,193],[11,14451,14452,14453,14504],{},"Now we ask: is ",[70,14454,14456,14474],{"className":14455,"translate":74},[78],[70,14457,14459],{"className":14458},[82],[84,14460,14461],{"xmlns":86},[89,14462,14463,14471],{},[92,14464,14465,14467,14469],{},[95,14466,911],{},[100,14468,126],{},[95,14470,1561],{},[164,14472,14473],{"encoding":166},"d \\in D",[70,14475,14477,14495],{"className":14476,"ariaHidden":172},[171],[70,14478,14480,14483,14486,14489,14492],{"className":14479},[176],[70,14481],{"className":14482,"style":5874},[180],[70,14484,911],{"className":14485},[185,193],[70,14487],{"className":14488,"style":202},[201],[70,14490,126],{"className":14491},[206],[70,14493],{"className":14494,"style":202},[201],[70,14496,14498,14501],{"className":14497},[176],[70,14499],{"className":14500,"style":5067},[180],[70,14502,1561],{"className":14503,"style":1726},[185,193],"?",[1025,14506,14507,14952],{},[1028,14508,14509,14562,14563,13106,14591,14746,14747,14812,14813,14951],{},[29,14510,14511,14512],{},"If ",[70,14513,14515,14532],{"className":14514,"translate":74},[78],[70,14516,14518],{"className":14517},[82],[84,14519,14520],{"xmlns":86},[89,14521,14522,14530],{},[92,14523,14524,14526,14528],{},[95,14525,911],{},[100,14527,126],{},[95,14529,1561],{},[164,14531,14473],{"encoding":166},[70,14533,14535,14553],{"className":14534,"ariaHidden":172},[171],[70,14536,14538,14541,14544,14547,14550],{"className":14537},[176],[70,14539],{"className":14540,"style":5874},[180],[70,14542,911],{"className":14543},[185,193],[70,14545],{"className":14546,"style":202},[201],[70,14548,126],{"className":14549},[206],[70,14551],{"className":14552,"style":202},[201],[70,14554,14556,14559],{"className":14555},[176],[70,14557],{"className":14558,"style":5067},[180],[70,14560,1561],{"className":14561,"style":1726},[185,193],": by the definition of ",[70,14564,14566,14579],{"className":14565,"translate":74},[78],[70,14567,14569],{"className":14568},[82],[84,14570,14571],{"xmlns":86},[89,14572,14573,14577],{},[92,14574,14575],{},[95,14576,1561],{},[164,14578,1561],{"encoding":166},[70,14580,14582],{"className":14581,"ariaHidden":172},[171],[70,14583,14585,14588],{"className":14584},[176],[70,14586],{"className":14587,"style":5067},[180],[70,14589,1561],{"className":14590,"style":1726},[185,193],[70,14592,14594,14631],{"className":14593,"translate":74},[78],[70,14595,14597],{"className":14596},[82],[84,14598,14599],{"xmlns":86},[89,14600,14601,14628],{},[92,14602,14603,14605,14607,14609,14611,14614,14616,14618,14620,14622,14624,14626],{},[95,14604,911],{},[100,14606,126],{},[95,14608,1561],{},[4704,14610,10669],{},[100,14612,14613],{},"⟺",[4704,14615,10669],{},[95,14617,911],{},[100,14619,14041],{"mathvariant":97},[95,14621,123],{},[100,14623,103],{"stretchy":102},[95,14625,911],{},[100,14627,109],{"stretchy":102},[164,14629,14630],{"encoding":166},"d \\in D \\iff d \\notin f(d)",[70,14632,14634,14652,14677,14728],{"className":14633,"ariaHidden":172},[171],[70,14635,14637,14640,14643,14646,14649],{"className":14636},[176],[70,14638],{"className":14639,"style":5874},[180],[70,14641,911],{"className":14642},[185,193],[70,14644],{"className":14645,"style":202},[201],[70,14647,126],{"className":14648},[206],[70,14650],{"className":14651,"style":202},[201],[70,14653,14655,14659,14662,14665,14668,14671,14674],{"className":14654},[176],[70,14656],{"className":14657,"style":14658},[180],"height:0.7073em;vertical-align:-0.024em;",[70,14660,1561],{"className":14661,"style":1726},[185,193],[70,14663],{"className":14664,"style":202},[201],[70,14666],{"className":14667,"style":202},[201],[70,14669,14613],{"className":14670},[206],[70,14672],{"className":14673,"style":202},[201],[70,14675],{"className":14676,"style":202},[201],[70,14678,14680,14683,14686,14689,14725],{"className":14679},[176],[70,14681],{"className":14682,"style":181},[180],[70,14684,911],{"className":14685},[185,193],[70,14687],{"className":14688,"style":202},[201],[70,14690,14692,14698],{"className":14691},[206],[70,14693,14695],{"className":14694},[185],[70,14696,126],{"className":14697},[206],[70,14699,14701],{"className":14700},[185,5591],[70,14702,14704],{"className":14703},[5595],[70,14705,14707,14710,14722],{"className":14706},[14150],[70,14708],{"className":14709,"style":181},[180],[70,14711,14713],{"className":14712},[5606],[70,14714,14716,14719],{"className":14715},[185],[70,14717,6668],{"className":14718},[185],[70,14720],{"className":14721,"style":4134},[201],[70,14723],{"className":14724},[5617],[70,14726],{"className":14727,"style":202},[201],[70,14729,14731,14734,14737,14740,14743],{"className":14730},[176],[70,14732],{"className":14733,"style":181},[180],[70,14735,123],{"className":14736,"style":257},[185,193],[70,14738,103],{"className":14739},[189],[70,14741,911],{"className":14742},[185,193],[70,14744,109],{"className":14745},[197],". Since ",[70,14748,14750,14773],{"className":14749,"translate":74},[78],[70,14751,14753],{"className":14752},[82],[84,14754,14755],{"xmlns":86},[89,14756,14757,14771],{},[92,14758,14759,14761,14763,14765,14767,14769],{},[95,14760,123],{},[100,14762,103],{"stretchy":102},[95,14764,911],{},[100,14766,109],{"stretchy":102},[100,14768,112],{},[95,14770,1561],{},[164,14772,14410],{"encoding":166},[70,14774,14776,14803],{"className":14775,"ariaHidden":172},[171],[70,14777,14779,14782,14785,14788,14791,14794,14797,14800],{"className":14778},[176],[70,14780],{"className":14781,"style":181},[180],[70,14783,123],{"className":14784,"style":257},[185,193],[70,14786,103],{"className":14787},[189],[70,14789,911],{"className":14790},[185,193],[70,14792,109],{"className":14793},[197],[70,14795],{"className":14796,"style":202},[201],[70,14798,112],{"className":14799},[206],[70,14801],{"className":14802,"style":202},[201],[70,14804,14806,14809],{"className":14805},[176],[70,14807],{"className":14808,"style":5067},[180],[70,14810,1561],{"className":14811,"style":1726},[185,193],", we have ",[70,14814,14816,14846],{"className":14815,"translate":74},[78],[70,14817,14819],{"className":14818},[82],[84,14820,14821],{"xmlns":86},[89,14822,14823,14843],{},[92,14824,14825,14827,14829,14831,14833,14835,14837,14839,14841],{},[95,14826,911],{},[100,14828,126],{},[95,14830,1561],{},[4704,14832,10669],{},[100,14834,14613],{},[4704,14836,10669],{},[95,14838,911],{},[100,14840,14041],{"mathvariant":97},[95,14842,1561],{},[164,14844,14845],{"encoding":166},"d \\in D \\iff d \\notin D",[70,14847,14849,14867,14891,14942],{"className":14848,"ariaHidden":172},[171],[70,14850,14852,14855,14858,14861,14864],{"className":14851},[176],[70,14853],{"className":14854,"style":5874},[180],[70,14856,911],{"className":14857},[185,193],[70,14859],{"className":14860,"style":202},[201],[70,14862,126],{"className":14863},[206],[70,14865],{"className":14866,"style":202},[201],[70,14868,14870,14873,14876,14879,14882,14885,14888],{"className":14869},[176],[70,14871],{"className":14872,"style":14658},[180],[70,14874,1561],{"className":14875,"style":1726},[185,193],[70,14877],{"className":14878,"style":202},[201],[70,14880],{"className":14881,"style":202},[201],[70,14883,14613],{"className":14884},[206],[70,14886],{"className":14887,"style":202},[201],[70,14889],{"className":14890,"style":202},[201],[70,14892,14894,14897,14900,14903,14939],{"className":14893},[176],[70,14895],{"className":14896,"style":181},[180],[70,14898,911],{"className":14899},[185,193],[70,14901],{"className":14902,"style":202},[201],[70,14904,14906,14912],{"className":14905},[206],[70,14907,14909],{"className":14908},[185],[70,14910,126],{"className":14911},[206],[70,14913,14915],{"className":14914},[185,5591],[70,14916,14918],{"className":14917},[5595],[70,14919,14921,14924,14936],{"className":14920},[14150],[70,14922],{"className":14923,"style":181},[180],[70,14925,14927],{"className":14926},[5606],[70,14928,14930,14933],{"className":14929},[185],[70,14931,6668],{"className":14932},[185],[70,14934],{"className":14935,"style":4134},[201],[70,14937],{"className":14938},[5617],[70,14940],{"className":14941,"style":202},[201],[70,14943,14945,14948],{"className":14944},[176],[70,14946],{"className":14947,"style":5067},[180],[70,14949,1561],{"className":14950,"style":1726},[185,193],". Contradiction!",[1028,14953,14954,15040,15041,15240,15241,15329,15330,14951],{},[29,14955,14511,14956],{},[70,14957,14959,14977],{"className":14958,"translate":74},[78],[70,14960,14962],{"className":14961},[82],[84,14963,14964],{"xmlns":86},[89,14965,14966,14974],{},[92,14967,14968,14970,14972],{},[95,14969,911],{},[100,14971,14041],{"mathvariant":97},[95,14973,1561],{},[164,14975,14976],{"encoding":166},"d \\notin D",[70,14978,14980,15031],{"className":14979,"ariaHidden":172},[171],[70,14981,14983,14986,14989,14992,15028],{"className":14982},[176],[70,14984],{"className":14985,"style":181},[180],[70,14987,911],{"className":14988},[185,193],[70,14990],{"className":14991,"style":202},[201],[70,14993,14995,15001],{"className":14994},[206],[70,14996,14998],{"className":14997},[185],[70,14999,126],{"className":15000},[206],[70,15002,15004],{"className":15003},[185,5591],[70,15005,15007],{"className":15006},[5595],[70,15008,15010,15013,15025],{"className":15009},[14150],[70,15011],{"className":15012,"style":181},[180],[70,15014,15016],{"className":15015},[5606],[70,15017,15019,15022],{"className":15018},[185],[70,15020,6668],{"className":15021},[185],[70,15023],{"className":15024,"style":4134},[201],[70,15026],{"className":15027},[5617],[70,15029],{"className":15030,"style":202},[201],[70,15032,15034,15037],{"className":15033},[176],[70,15035],{"className":15036,"style":5067},[180],[70,15038,1561],{"className":15039,"style":1726},[185,193],": likewise, ",[70,15042,15044,15087],{"className":15043,"translate":74},[78],[70,15045,15047],{"className":15046},[82],[84,15048,15049],{"xmlns":86},[89,15050,15051,15084],{},[92,15052,15053,15055,15057,15059,15061,15063,15065,15068,15070,15072,15074,15076,15078,15080,15082],{},[95,15054,911],{},[100,15056,14041],{"mathvariant":97},[95,15058,1561],{},[4704,15060,10669],{},[100,15062,14613],{},[4704,15064,10669],{},[95,15066,15067],{"mathvariant":97},"¬",[100,15069,103],{"stretchy":102},[95,15071,911],{},[100,15073,14041],{"mathvariant":97},[95,15075,123],{},[100,15077,103],{"stretchy":102},[95,15079,911],{},[100,15081,109],{"stretchy":102},[100,15083,109],{"stretchy":102},[164,15085,15086],{"encoding":166},"d \\notin D \\iff \\neg(d \\notin f(d))",[70,15088,15090,15141,15165,15222],{"className":15089,"ariaHidden":172},[171],[70,15091,15093,15096,15099,15102,15138],{"className":15092},[176],[70,15094],{"className":15095,"style":181},[180],[70,15097,911],{"className":15098},[185,193],[70,15100],{"className":15101,"style":202},[201],[70,15103,15105,15111],{"className":15104},[206],[70,15106,15108],{"className":15107},[185],[70,15109,126],{"className":15110},[206],[70,15112,15114],{"className":15113},[185,5591],[70,15115,15117],{"className":15116},[5595],[70,15118,15120,15123,15135],{"className":15119},[14150],[70,15121],{"className":15122,"style":181},[180],[70,15124,15126],{"className":15125},[5606],[70,15127,15129,15132],{"className":15128},[185],[70,15130,6668],{"className":15131},[185],[70,15133],{"className":15134,"style":4134},[201],[70,15136],{"className":15137},[5617],[70,15139],{"className":15140,"style":202},[201],[70,15142,15144,15147,15150,15153,15156,15159,15162],{"className":15143},[176],[70,15145],{"className":15146,"style":14658},[180],[70,15148,1561],{"className":15149,"style":1726},[185,193],[70,15151],{"className":15152,"style":202},[201],[70,15154],{"className":15155,"style":202},[201],[70,15157,14613],{"className":15158},[206],[70,15160],{"className":15161,"style":202},[201],[70,15163],{"className":15164,"style":202},[201],[70,15166,15168,15171,15174,15177,15180,15183,15219],{"className":15167},[176],[70,15169],{"className":15170,"style":181},[180],[70,15172,15067],{"className":15173},[185],[70,15175,103],{"className":15176},[189],[70,15178,911],{"className":15179},[185,193],[70,15181],{"className":15182,"style":202},[201],[70,15184,15186,15192],{"className":15185},[206],[70,15187,15189],{"className":15188},[185],[70,15190,126],{"className":15191},[206],[70,15193,15195],{"className":15194},[185,5591],[70,15196,15198],{"className":15197},[5595],[70,15199,15201,15204,15216],{"className":15200},[14150],[70,15202],{"className":15203,"style":181},[180],[70,15205,15207],{"className":15206},[5606],[70,15208,15210,15213],{"className":15209},[185],[70,15211,6668],{"className":15212},[185],[70,15214],{"className":15215,"style":4134},[201],[70,15217],{"className":15218},[5617],[70,15220],{"className":15221,"style":202},[201],[70,15223,15225,15228,15231,15234,15237],{"className":15224},[176],[70,15226],{"className":15227,"style":181},[180],[70,15229,123],{"className":15230,"style":257},[185,193],[70,15232,103],{"className":15233},[189],[70,15235,911],{"className":15236},[185,193],[70,15238,1951],{"className":15239},[197],", i.e., ",[70,15242,15244,15272],{"className":15243,"translate":74},[78],[70,15245,15247],{"className":15246},[82],[84,15248,15249],{"xmlns":86},[89,15250,15251,15269],{},[92,15252,15253,15255,15257,15259,15261,15263,15265,15267],{},[95,15254,911],{},[100,15256,126],{},[95,15258,123],{},[100,15260,103],{"stretchy":102},[95,15262,911],{},[100,15264,109],{"stretchy":102},[100,15266,112],{},[95,15268,1561],{},[164,15270,15271],{"encoding":166},"d \\in f(d) = D",[70,15273,15275,15293,15320],{"className":15274,"ariaHidden":172},[171],[70,15276,15278,15281,15284,15287,15290],{"className":15277},[176],[70,15279],{"className":15280,"style":5874},[180],[70,15282,911],{"className":15283},[185,193],[70,15285],{"className":15286,"style":202},[201],[70,15288,126],{"className":15289},[206],[70,15291],{"className":15292,"style":202},[201],[70,15294,15296,15299,15302,15305,15308,15311,15314,15317],{"className":15295},[176],[70,15297],{"className":15298,"style":181},[180],[70,15300,123],{"className":15301,"style":257},[185,193],[70,15303,103],{"className":15304},[189],[70,15306,911],{"className":15307},[185,193],[70,15309,109],{"className":15310},[197],[70,15312],{"className":15313,"style":202},[201],[70,15315,112],{"className":15316},[206],[70,15318],{"className":15319,"style":202},[201],[70,15321,15323,15326],{"className":15322},[176],[70,15324],{"className":15325,"style":5067},[180],[70,15327,1561],{"className":15328,"style":1726},[185,193],", which yields ",[70,15331,15333,15350],{"className":15332,"translate":74},[78],[70,15334,15336],{"className":15335},[82],[84,15337,15338],{"xmlns":86},[89,15339,15340,15348],{},[92,15341,15342,15344,15346],{},[95,15343,911],{},[100,15345,126],{},[95,15347,1561],{},[164,15349,14473],{"encoding":166},[70,15351,15353,15371],{"className":15352,"ariaHidden":172},[171],[70,15354,15356,15359,15362,15365,15368],{"className":15355},[176],[70,15357],{"className":15358,"style":5874},[180],[70,15360,911],{"className":15361},[185,193],[70,15363],{"className":15364,"style":202},[201],[70,15366,126],{"className":15367},[206],[70,15369],{"className":15370,"style":202},[201],[70,15372,15374,15377],{"className":15373},[176],[70,15375],{"className":15376,"style":5067},[180],[70,15378,1561],{"className":15379,"style":1726},[185,193],[11,15381,15382,15383,15470,15471,24],{},"Both cases lead to self-contradiction, demonstrating that the assumption \"there exists a surjection ",[70,15384,15386,15413],{"className":15385,"translate":74},[78],[70,15387,15389],{"className":15388},[82],[84,15390,15391],{"xmlns":86},[89,15392,15393,15411],{},[92,15394,15395,15397,15399,15401,15403,15405,15407,15409],{},[95,15396,123],{},[100,15398,68],{},[95,15400,5080],{},[100,15402,1550],{},[95,15404,13429],{"mathvariant":13428},[100,15406,103],{"stretchy":102},[95,15408,5080],{},[100,15410,109],{"stretchy":102},[164,15412,13735],{"encoding":166},[70,15414,15416,15434,15452],{"className":15415,"ariaHidden":172},[171],[70,15417,15419,15422,15425,15428,15431],{"className":15418},[176],[70,15420],{"className":15421,"style":791},[180],[70,15423,123],{"className":15424,"style":257},[185,193],[70,15426],{"className":15427,"style":202},[201],[70,15429,68],{"className":15430},[206],[70,15432],{"className":15433,"style":202},[201],[70,15435,15437,15440,15443,15446,15449],{"className":15436},[176],[70,15438],{"className":15439,"style":5067},[180],[70,15441,5080],{"className":15442},[185,193],[70,15444],{"className":15445,"style":202},[201],[70,15447,1550],{"className":15448},[206],[70,15450],{"className":15451,"style":202},[201],[70,15453,15455,15458,15461,15464,15467],{"className":15454},[176],[70,15456],{"className":15457,"style":181},[180],[70,15459,13429],{"className":15460,"style":13452},[185,13451],[70,15462,103],{"className":15463},[189],[70,15465,5080],{"className":15466},[185,193],[70,15468,109],{"className":15469},[197],"\" is untenable. Hence ",[70,15472,15474,15505],{"className":15473,"translate":74},[78],[70,15475,15477],{"className":15476},[82],[84,15478,15479],{"xmlns":86},[89,15480,15481,15503],{},[92,15482,15483,15485,15487,15489,15491,15493,15495,15497,15499,15501],{},[95,15484,4625],{"mathvariant":97},[95,15486,5080],{},[95,15488,4625],{"mathvariant":97},[100,15490,3894],{},[95,15492,4625],{"mathvariant":97},[95,15494,13429],{"mathvariant":13428},[100,15496,103],{"stretchy":102},[95,15498,5080],{},[100,15500,109],{"stretchy":102},[95,15502,4625],{"mathvariant":97},[164,15504,13555],{"encoding":166},[70,15506,15508,15532],{"className":15507,"ariaHidden":172},[171],[70,15509,15511,15514,15517,15520,15523,15526,15529],{"className":15510},[176],[70,15512],{"className":15513,"style":181},[180],[70,15515,4625],{"className":15516},[185],[70,15518,5080],{"className":15519},[185,193],[70,15521,4625],{"className":15522},[185],[70,15524],{"className":15525,"style":202},[201],[70,15527,3894],{"className":15528},[206],[70,15530],{"className":15531,"style":202},[201],[70,15533,15535,15538,15541,15544,15547,15550,15553],{"className":15534},[176],[70,15536],{"className":15537,"style":181},[180],[70,15539,4625],{"className":15540},[185],[70,15542,13429],{"className":15543,"style":13452},[185,13451],[70,15545,103],{"className":15546},[189],[70,15548,5080],{"className":15549},[185,193],[70,15551,109],{"className":15552},[197],[70,15554,4625],{"className":15555},[185],[11,15557,15558,15559,15562],{},"The attentive reader may already have discerned that the structure of this proof is ",[29,15560,15561],{},"identical"," to that of the uncountability proof presented above:",[15564,15565,15566,15578],"table",{},[15567,15568,15569],"thead",{},[15570,15571,15572,15576],"tr",{},[15573,15574,15575],"th",{},"Uncountability of the Reals",[15573,15577,13373],{},[15579,15580,15581,15772,16059,16346],"tbody",{},[15570,15582,15583,15682],{},[15584,15585,15586,15587],"td",{},"Assume a bijection ",[70,15588,15590,15619],{"className":15589,"translate":74},[78],[70,15591,15593],{"className":15592},[82],[84,15594,15595],{"xmlns":86},[89,15596,15597,15617],{},[92,15598,15599,15601,15603,15605,15607,15609,15611,15613,15615],{},[95,15600,123],{},[100,15602,68],{},[95,15604,5211],{"mathvariant":5210},[100,15606,1550],{},[100,15608,103],{"stretchy":102},[1566,15610,2780],{},[100,15612,906],{"separator":172},[1566,15614,1568],{},[100,15616,109],{"stretchy":102},[164,15618,7419],{"encoding":166},[70,15620,15622,15640,15658],{"className":15621,"ariaHidden":172},[171],[70,15623,15625,15628,15631,15634,15637],{"className":15624},[176],[70,15626],{"className":15627,"style":791},[180],[70,15629,123],{"className":15630,"style":257},[185,193],[70,15632],{"className":15633,"style":202},[201],[70,15635,68],{"className":15636},[206],[70,15638],{"className":15639,"style":202},[201],[70,15641,15643,15646,15649,15652,15655],{"className":15642},[176],[70,15644],{"className":15645,"style":5254},[180],[70,15647,5211],{"className":15648},[185,5258],[70,15650],{"className":15651,"style":202},[201],[70,15653,1550],{"className":15654},[206],[70,15656],{"className":15657,"style":202},[201],[70,15659,15661,15664,15667,15670,15673,15676,15679],{"className":15660},[176],[70,15662],{"className":15663,"style":181},[180],[70,15665,103],{"className":15666},[189],[70,15668,2780],{"className":15669},[185],[70,15671,906],{"className":15672},[976],[70,15674],{"className":15675,"style":304},[201],[70,15677,1568],{"className":15678},[185],[70,15680,109],{"className":15681},[197],[15584,15683,15684,15685],{},"Assume a surjection ",[70,15686,15688,15715],{"className":15687,"translate":74},[78],[70,15689,15691],{"className":15690},[82],[84,15692,15693],{"xmlns":86},[89,15694,15695,15713],{},[92,15696,15697,15699,15701,15703,15705,15707,15709,15711],{},[95,15698,123],{},[100,15700,68],{},[95,15702,5080],{},[100,15704,1550],{},[95,15706,13429],{"mathvariant":13428},[100,15708,103],{"stretchy":102},[95,15710,5080],{},[100,15712,109],{"stretchy":102},[164,15714,13735],{"encoding":166},[70,15716,15718,15736,15754],{"className":15717,"ariaHidden":172},[171],[70,15719,15721,15724,15727,15730,15733],{"className":15720},[176],[70,15722],{"className":15723,"style":791},[180],[70,15725,123],{"className":15726,"style":257},[185,193],[70,15728],{"className":15729,"style":202},[201],[70,15731,68],{"className":15732},[206],[70,15734],{"className":15735,"style":202},[201],[70,15737,15739,15742,15745,15748,15751],{"className":15738},[176],[70,15740],{"className":15741,"style":5067},[180],[70,15743,5080],{"className":15744},[185,193],[70,15746],{"className":15747,"style":202},[201],[70,15749,1550],{"className":15750},[206],[70,15752],{"className":15753,"style":202},[201],[70,15755,15757,15760,15763,15766,15769],{"className":15756},[176],[70,15758],{"className":15759,"style":181},[180],[70,15761,13429],{"className":15762,"style":13452},[185,13451],[70,15764,103],{"className":15765},[189],[70,15767,5080],{"className":15768},[185,193],[70,15770,109],{"className":15771},[197],[15570,15773,15774,15905],{},[15584,15775,15776,15777,15805,15806,15834,15835],{},"Construct the diagonal number ",[70,15778,15780,15793],{"className":15779,"translate":74},[78],[70,15781,15783],{"className":15782},[82],[84,15784,15785],{"xmlns":86},[89,15786,15787,15791],{},[92,15788,15789],{},[95,15790,4632],{},[164,15792,4632],{"encoding":166},[70,15794,15796],{"className":15795,"ariaHidden":172},[171],[70,15797,15799,15802],{"className":15798},[176],[70,15800],{"className":15801,"style":2801},[180],[70,15803,4632],{"className":15804,"style":365},[185,193],", each digit differing from the ",[70,15807,15809,15822],{"className":15808,"translate":74},[78],[70,15810,15812],{"className":15811},[82],[84,15813,15814],{"xmlns":86},[89,15815,15816,15820],{},[92,15817,15818],{},[95,15819,6404],{},[164,15821,6404],{"encoding":166},[70,15823,15825],{"className":15824,"ariaHidden":172},[171],[70,15826,15828,15831],{"className":15827},[176],[70,15829],{"className":15830,"style":525},[180],[70,15832,6404],{"className":15833},[185,193],"-th digit of ",[70,15836,15838,15856],{"className":15837,"translate":74},[78],[70,15839,15841],{"className":15840},[82],[84,15842,15843],{"xmlns":86},[89,15844,15845,15853],{},[92,15846,15847],{},[137,15848,15849,15851],{},[95,15850,106],{},[95,15852,6404],{},[164,15854,15855],{"encoding":166},"x_n",[70,15857,15859],{"className":15858,"ariaHidden":172},[171],[70,15860,15862,15865],{"className":15861},[176],[70,15863],{"className":15864,"style":5634},[180],[70,15866,15868,15871],{"className":15867},[185],[70,15869,106],{"className":15870},[185,193],[70,15872,15874],{"className":15873},[314],[70,15875,15877,15897],{"className":15876},[225,226],[70,15878,15880,15894],{"className":15879},[230],[70,15881,15883],{"className":15882,"style":9499},[234],[70,15884,15885,15888],{"style":327},[70,15886],{"className":15887,"style":331},[242],[70,15889,15891],{"className":15890},[247,248,249,250],[70,15892,6404],{"className":15893},[185,193,250],[70,15895,291],{"className":15896},[290],[70,15898,15900],{"className":15899},[230],[70,15901,15903],{"className":15902,"style":2841},[234],[70,15904],{},[15584,15906,15907,15908],{},"Construct the diagonal set ",[70,15909,15911,15947],{"className":15910,"translate":74},[78],[70,15912,15914],{"className":15913},[82],[84,15915,15916],{"xmlns":86},[89,15917,15918,15944],{},[92,15919,15920,15922,15924,15926,15928,15930,15932,15934,15936,15938,15940,15942],{},[95,15921,1561],{},[100,15923,112],{},[100,15925,5216],{"stretchy":102},[95,15927,18],{},[100,15929,68],{},[95,15931,18],{},[100,15933,14041],{"mathvariant":97},[95,15935,123],{},[100,15937,103],{"stretchy":102},[95,15939,18],{},[100,15941,109],{"stretchy":102},[100,15943,5241],{"stretchy":102},[164,15945,15946],{"encoding":166},"D = \\{a : a \\notin f(a)\\}",[70,15948,15950,15968,15989,16040],{"className":15949,"ariaHidden":172},[171],[70,15951,15953,15956,15959,15962,15965],{"className":15952},[176],[70,15954],{"className":15955,"style":5067},[180],[70,15957,1561],{"className":15958,"style":1726},[185,193],[70,15960],{"className":15961,"style":202},[201],[70,15963,112],{"className":15964},[206],[70,15966],{"className":15967,"style":202},[201],[70,15969,15971,15974,15977,15980,15983,15986],{"className":15970},[176],[70,15972],{"className":15973,"style":181},[180],[70,15975,5216],{"className":15976},[189],[70,15978,18],{"className":15979},[185,193],[70,15981],{"className":15982,"style":202},[201],[70,15984,68],{"className":15985},[206],[70,15987],{"className":15988,"style":202},[201],[70,15990,15992,15995,15998,16001,16037],{"className":15991},[176],[70,15993],{"className":15994,"style":181},[180],[70,15996,18],{"className":15997},[185,193],[70,15999],{"className":16000,"style":202},[201],[70,16002,16004,16010],{"className":16003},[206],[70,16005,16007],{"className":16006},[185],[70,16008,126],{"className":16009},[206],[70,16011,16013],{"className":16012},[185,5591],[70,16014,16016],{"className":16015},[5595],[70,16017,16019,16022,16034],{"className":16018},[14150],[70,16020],{"className":16021,"style":181},[180],[70,16023,16025],{"className":16024},[5606],[70,16026,16028,16031],{"className":16027},[185],[70,16029,6668],{"className":16030},[185],[70,16032],{"className":16033,"style":4134},[201],[70,16035],{"className":16036},[5617],[70,16038],{"className":16039,"style":202},[201],[70,16041,16043,16046,16049,16052,16055],{"className":16042},[176],[70,16044],{"className":16045,"style":181},[180],[70,16047,123],{"className":16048,"style":257},[185,193],[70,16050,103],{"className":16051},[189],[70,16053,18],{"className":16054},[185,193],[70,16056,16058],{"className":16057},[197],")}",[15570,16060,16061,16217],{},[15584,16062,16063,16188,16189],{},[70,16064,16066,16088],{"className":16065,"translate":74},[78],[70,16067,16069],{"className":16068},[82],[84,16070,16071],{"xmlns":86},[89,16072,16073,16085],{},[92,16074,16075,16077,16079],{},[95,16076,4632],{},[100,16078,5491],{"mathvariant":97},[137,16080,16081,16083],{},[95,16082,106],{},[95,16084,6404],{},[164,16086,16087],{"encoding":166},"y \\neq x_n",[70,16089,16091,16142],{"className":16090,"ariaHidden":172},[171],[70,16092,16094,16097,16100,16103,16139],{"className":16093},[176],[70,16095],{"className":16096,"style":791},[180],[70,16098,4632],{"className":16099,"style":365},[185,193],[70,16101],{"className":16102,"style":202},[201],[70,16104,16106,16133,16136],{"className":16105},[206],[70,16107,16109],{"className":16108},[206],[70,16110,16112],{"className":16111},[185,5591],[70,16113,16115],{"className":16114},[5595],[70,16116,16118,16121,16130],{"className":16117},[5599],[70,16119],{"className":16120,"style":791},[180],[70,16122,16124],{"className":16123},[5606],[70,16125,16127],{"className":16126},[185],[70,16128,5613],{"className":16129},[206],[70,16131],{"className":16132},[5617],[70,16134],{"className":16135},[201,5621],[70,16137,112],{"className":16138},[206],[70,16140],{"className":16141,"style":202},[201],[70,16143,16145,16148],{"className":16144},[176],[70,16146],{"className":16147,"style":5634},[180],[70,16149,16151,16154],{"className":16150},[185],[70,16152,106],{"className":16153},[185,193],[70,16155,16157],{"className":16156},[314],[70,16158,16160,16180],{"className":16159},[225,226],[70,16161,16163,16177],{"className":16162},[230],[70,16164,16166],{"className":16165,"style":9499},[234],[70,16167,16168,16171],{"style":327},[70,16169],{"className":16170,"style":331},[242],[70,16172,16174],{"className":16173},[247,248,249,250],[70,16175,6404],{"className":16176},[185,193,250],[70,16178,291],{"className":16179},[290],[70,16181,16183],{"className":16182},[230],[70,16184,16186],{"className":16185,"style":2841},[234],[70,16187],{}," for all ",[70,16190,16192,16205],{"className":16191,"translate":74},[78],[70,16193,16195],{"className":16194},[82],[84,16196,16197],{"xmlns":86},[89,16198,16199,16203],{},[92,16200,16201],{},[95,16202,6404],{},[164,16204,6404],{"encoding":166},[70,16206,16208],{"className":16207,"ariaHidden":172},[171],[70,16209,16211,16214],{"className":16210},[176],[70,16212],{"className":16213,"style":525},[180],[70,16215,6404],{"className":16216},[185,193],[15584,16218,16219,16188,16318],{},[70,16220,16222,16246],{"className":16221,"translate":74},[78],[70,16223,16225],{"className":16224},[82],[84,16226,16227],{"xmlns":86},[89,16228,16229,16243],{},[92,16230,16231,16233,16235,16237,16239,16241],{},[95,16232,1561],{},[100,16234,5491],{"mathvariant":97},[95,16236,123],{},[100,16238,103],{"stretchy":102},[95,16240,18],{},[100,16242,109],{"stretchy":102},[164,16244,16245],{"encoding":166},"D \\neq f(a)",[70,16247,16249,16300],{"className":16248,"ariaHidden":172},[171],[70,16250,16252,16255,16258,16261,16297],{"className":16251},[176],[70,16253],{"className":16254,"style":791},[180],[70,16256,1561],{"className":16257,"style":1726},[185,193],[70,16259],{"className":16260,"style":202},[201],[70,16262,16264,16291,16294],{"className":16263},[206],[70,16265,16267],{"className":16266},[206],[70,16268,16270],{"className":16269},[185,5591],[70,16271,16273],{"className":16272},[5595],[70,16274,16276,16279,16288],{"className":16275},[5599],[70,16277],{"className":16278,"style":791},[180],[70,16280,16282],{"className":16281},[5606],[70,16283,16285],{"className":16284},[185],[70,16286,5613],{"className":16287},[206],[70,16289],{"className":16290},[5617],[70,16292],{"className":16293},[201,5621],[70,16295,112],{"className":16296},[206],[70,16298],{"className":16299,"style":202},[201],[70,16301,16303,16306,16309,16312,16315],{"className":16302},[176],[70,16304],{"className":16305,"style":181},[180],[70,16307,123],{"className":16308,"style":257},[185,193],[70,16310,103],{"className":16311},[189],[70,16313,18],{"className":16314},[185,193],[70,16316,109],{"className":16317},[197],[70,16319,16321,16334],{"className":16320,"translate":74},[78],[70,16322,16324],{"className":16323},[82],[84,16325,16326],{"xmlns":86},[89,16327,16328,16332],{},[92,16329,16330],{},[95,16331,18],{},[164,16333,18],{"encoding":166},[70,16335,16337],{"className":16336,"ariaHidden":172},[171],[70,16338,16340,16343],{"className":16339},[176],[70,16341],{"className":16342,"style":525},[180],[70,16344,18],{"className":16345},[185,193],[15570,16347,16348,16379],{},[15584,16349,16350,16378],{},[70,16351,16353,16366],{"className":16352,"translate":74},[78],[70,16354,16356],{"className":16355},[82],[84,16357,16358],{"xmlns":86},[89,16359,16360,16364],{},[92,16361,16362],{},[95,16363,4632],{},[164,16365,4632],{"encoding":166},[70,16367,16369],{"className":16368,"ariaHidden":172},[171],[70,16370,16372,16375],{"className":16371},[176],[70,16373],{"className":16374,"style":2801},[180],[70,16376,4632],{"className":16377,"style":365},[185,193]," cannot be covered by the list; contradiction",[15584,16380,16381,16409],{},[70,16382,16384,16397],{"className":16383,"translate":74},[78],[70,16385,16387],{"className":16386},[82],[84,16388,16389],{"xmlns":86},[89,16390,16391,16395],{},[92,16392,16393],{},[95,16394,1561],{},[164,16396,1561],{"encoding":166},[70,16398,16400],{"className":16399,"ariaHidden":172},[171],[70,16401,16403,16406],{"className":16402},[176],[70,16404],{"className":16405,"style":5067},[180],[70,16407,1561],{"className":16408,"style":1726},[185,193]," cannot be covered by the mapping; contradiction",[11,16411,16412,16413,16464,16465,16552,16553,16637,16638,16682,16683,24],{},"In fact, the real numbers in ",[70,16414,16416,16437],{"className":16415,"translate":74},[78],[70,16417,16419],{"className":16418},[82],[84,16420,16421],{"xmlns":86},[89,16422,16423,16435],{},[92,16424,16425,16427,16429,16431,16433],{},[100,16426,103],{"stretchy":102},[1566,16428,2780],{},[100,16430,906],{"separator":172},[1566,16432,1568],{},[100,16434,109],{"stretchy":102},[164,16436,7239],{"encoding":166},[70,16438,16440],{"className":16439,"ariaHidden":172},[171],[70,16441,16443,16446,16449,16452,16455,16458,16461],{"className":16442},[176],[70,16444],{"className":16445,"style":181},[180],[70,16447,103],{"className":16448},[189],[70,16450,2780],{"className":16451},[185],[70,16453,906],{"className":16454},[976],[70,16456],{"className":16457,"style":304},[201],[70,16459,1568],{"className":16460},[185],[70,16462,109],{"className":16463},[197]," can be placed in approximate correspondence with ",[70,16466,16468,16494],{"className":16467,"translate":74},[78],[70,16469,16471],{"className":16470},[82],[84,16472,16473],{"xmlns":86},[89,16474,16475,16491],{},[92,16476,16477,16479,16481,16483,16485],{},[100,16478,5216],{"stretchy":102},[1566,16480,2780],{},[100,16482,906],{"separator":172},[1566,16484,1568],{},[1552,16486,16487,16489],{},[100,16488,5241],{"stretchy":102},[95,16490,5211],{"mathvariant":5210},[164,16492,16493],{"encoding":166},"\\{0,1\\}^{\\mathbb{N}}",[70,16495,16497],{"className":16496,"ariaHidden":172},[171],[70,16498,16500,16504,16507,16510,16513,16516,16519],{"className":16499},[176],[70,16501],{"className":16502,"style":16503},[180],"height:1.0952em;vertical-align:-0.25em;",[70,16505,5216],{"className":16506},[189],[70,16508,2780],{"className":16509},[185],[70,16511,906],{"className":16512},[976],[70,16514],{"className":16515,"style":304},[201],[70,16517,1568],{"className":16518},[185],[70,16520,16522,16525],{"className":16521},[197],[70,16523,5241],{"className":16524},[197],[70,16526,16528],{"className":16527},[314],[70,16529,16531],{"className":16530},[225],[70,16532,16534],{"className":16533},[230],[70,16535,16538],{"className":16536,"style":16537},[234],"height:0.8452em;",[70,16539,16540,16543],{"style":1713},[70,16541],{"className":16542,"style":331},[242],[70,16544,16546],{"className":16545},[247,248,249,250],[70,16547,16549],{"className":16548},[185,250],[70,16550,5211],{"className":16551},[185,5258,250]," (the set of infinite binary sequences of 0s and 1s), and ",[70,16554,16556,16581],{"className":16555,"translate":74},[78],[70,16557,16559],{"className":16558},[82],[84,16560,16561],{"xmlns":86},[89,16562,16563,16579],{},[92,16564,16565,16567,16569,16571,16573],{},[100,16566,5216],{"stretchy":102},[1566,16568,2780],{},[100,16570,906],{"separator":172},[1566,16572,1568],{},[1552,16574,16575,16577],{},[100,16576,5241],{"stretchy":102},[95,16578,5211],{"mathvariant":5210},[164,16580,16493],{"encoding":166},[70,16582,16584],{"className":16583,"ariaHidden":172},[171],[70,16585,16587,16590,16593,16596,16599,16602,16605],{"className":16586},[176],[70,16588],{"className":16589,"style":16503},[180],[70,16591,5216],{"className":16592},[189],[70,16594,2780],{"className":16595},[185],[70,16597,906],{"className":16598},[976],[70,16600],{"className":16601,"style":304},[201],[70,16603,1568],{"className":16604},[185],[70,16606,16608,16611],{"className":16607},[197],[70,16609,5241],{"className":16610},[197],[70,16612,16614],{"className":16613},[314],[70,16615,16617],{"className":16616},[225],[70,16618,16620],{"className":16619},[230],[70,16621,16623],{"className":16622,"style":16537},[234],[70,16624,16625,16628],{"style":1713},[70,16626],{"className":16627,"style":331},[242],[70,16629,16631],{"className":16630},[247,248,249,250],[70,16632,16634],{"className":16633},[185,250],[70,16635,5211],{"className":16636},[185,5258,250]," is, in essence, ",[70,16639,16641,16661],{"className":16640,"translate":74},[78],[70,16642,16644],{"className":16643},[82],[84,16645,16646],{"xmlns":86},[89,16647,16648,16658],{},[92,16649,16650,16652,16654,16656],{},[95,16651,13429],{"mathvariant":13428},[100,16653,103],{"stretchy":102},[95,16655,5211],{"mathvariant":5210},[100,16657,109],{"stretchy":102},[164,16659,16660],{"encoding":166},"\\mathcal{P}(\\mathbb{N})",[70,16662,16664],{"className":16663,"ariaHidden":172},[171],[70,16665,16667,16670,16673,16676,16679],{"className":16666},[176],[70,16668],{"className":16669,"style":181},[180],[70,16671,13429],{"className":16672,"style":13452},[185,13451],[70,16674,103],{"className":16675},[189],[70,16677,5211],{"className":16678},[185,5258],[70,16680,109],{"className":16681},[197]," (each subset corresponds to a 0\u002F1 sequence indicating whether a given position belongs to the subset). Thus, \"the reals are uncountable\" is, in truth, a concrete instantiation of Cantor's Theorem for the special case ",[70,16684,16686,16704],{"className":16685,"translate":74},[78],[70,16687,16689],{"className":16688},[82],[84,16690,16691],{"xmlns":86},[89,16692,16693,16701],{},[92,16694,16695,16697,16699],{},[95,16696,5080],{},[100,16698,112],{},[95,16700,5211],{"mathvariant":5210},[164,16702,16703],{"encoding":166},"A = \\mathbb{N}",[70,16705,16707,16725],{"className":16706,"ariaHidden":172},[171],[70,16708,16710,16713,16716,16719,16722],{"className":16709},[176],[70,16711],{"className":16712,"style":5067},[180],[70,16714,5080],{"className":16715},[185,193],[70,16717],{"className":16718,"style":202},[201],[70,16720,112],{"className":16721},[206],[70,16723],{"className":16724,"style":202},[201],[70,16726,16728,16731],{"className":16727},[176],[70,16729],{"className":16730,"style":5254},[180],[70,16732,5211],{"className":16733},[185,5258],[11,16735,16736],{},"While we cannot represent infinite sets inside a computer, we can verify the logical skeleton of the proof using a finite set:",[12888,16738,16740],{"className":12890,"code":16739,"language":12892,"meta":4976,"style":4976},"def cantor_diagonal_demo(A):\n    # Construct an \"arbitrary\" mapping f: A -> P(A) for demonstration\n    f = {}\n    for i in A:\n        f[i] = frozenset(j for j in A if (i + j) % 2 == 0)\n\n    print(\"Set A =\", set(A))\n    for i in A:\n        print(f\"  f({i}) = {set(f[i])}\")\n\n    # Diagonal construction: D = { i in A : i not in f(i) }\n    D = frozenset(i for i in A if i not in f[i])\n    print(f\"Diagonal-constructed set D = {set(D)}\")\n\n    matches = [i for i in A if f[i] == D]\n    print(f\"Does there exist i such that f(i) = D? -> {matches if matches else 'Does not exist (contradiction established)'}\")\n\ncantor_diagonal_demo([0, 1, 2, 3])\n",[2617,16741,16742,16752,16757,16767,16779,16827,16831,16849,16859,16893,16897,16902,16935,16957,16961,16990,17023,17027],{"__ignoreMap":4976},[70,16743,16744,16746,16749],{"class":12897,"line":12898},[70,16745,12902],{"class":12901},[70,16747,16748],{"class":12905}," cantor_diagonal_demo",[70,16750,16751],{"class":12909},"(A):\n",[70,16753,16754],{"class":12897,"line":4977},[70,16755,16756],{"class":13016},"    # Construct an \"arbitrary\" mapping f: A -> P(A) for demonstration\n",[70,16758,16759,16762,16764],{"class":12897,"line":12928},[70,16760,16761],{"class":12909},"    f ",[70,16763,112],{"class":12901},[70,16765,16766],{"class":12909}," {}\n",[70,16768,16769,16771,16774,16776],{"class":12897,"line":12934},[70,16770,12959],{"class":12901},[70,16772,16773],{"class":12909}," i ",[70,16775,12965],{"class":12901},[70,16777,16778],{"class":12909}," A:\n",[70,16780,16781,16784,16786,16789,16792,16795,16798,16800,16803,16806,16809,16811,16814,16816,16819,16822,16825],{"class":12897,"line":12940},[70,16782,16783],{"class":12909},"        f[i] ",[70,16785,112],{"class":12901},[70,16787,16788],{"class":12915}," frozenset",[70,16790,16791],{"class":12909},"(j ",[70,16793,16794],{"class":12901},"for",[70,16796,16797],{"class":12909}," j ",[70,16799,12965],{"class":12901},[70,16801,16802],{"class":12909}," A ",[70,16804,16805],{"class":12901},"if",[70,16807,16808],{"class":12909}," (i ",[70,16810,3017],{"class":12901},[70,16812,16813],{"class":12909}," j) ",[70,16815,13039],{"class":12901},[70,16817,16818],{"class":12915}," 2",[70,16820,16821],{"class":12901}," ==",[70,16823,16824],{"class":12915}," 0",[70,16826,12999],{"class":12909},[70,16828,16829],{"class":12897,"line":12945},[70,16830,13081],{"emptyLinePlaceholder":4989},[70,16832,16833,16836,16838,16841,16843,16846],{"class":12897,"line":12956},[70,16834,16835],{"class":12915},"    print",[70,16837,103],{"class":12909},[70,16839,16840],{"class":12924},"\"Set A =\"",[70,16842,13106],{"class":12909},[70,16844,16845],{"class":12915},"set",[70,16847,16848],{"class":12909},"(A))\n",[70,16850,16851,16853,16855,16857],{"class":12897,"line":12974},[70,16852,12959],{"class":12901},[70,16854,16773],{"class":12909},[70,16856,12965],{"class":12901},[70,16858,16778],{"class":12909},[70,16860,16861,16864,16866,16868,16871,16873,16875,16877,16880,16883,16886,16888,16891],{"class":12897,"line":13002},[70,16862,16863],{"class":12915},"        print",[70,16865,103],{"class":12909},[70,16867,123],{"class":12901},[70,16869,16870],{"class":12924},"\"  f(",[70,16872,5216],{"class":12915},[70,16874,2701],{"class":12909},[70,16876,5241],{"class":12915},[70,16878,16879],{"class":12924},") = ",[70,16881,16882],{"class":12915},"{set",[70,16884,16885],{"class":12909},"(f[i])",[70,16887,5241],{"class":12915},[70,16889,16890],{"class":12924},"\"",[70,16892,12999],{"class":12909},[70,16894,16895],{"class":12897,"line":13020},[70,16896,13081],{"emptyLinePlaceholder":4989},[70,16898,16899],{"class":12897,"line":13048},[70,16900,16901],{"class":13016},"    # Diagonal construction: D = { i in A : i not in f(i) }\n",[70,16903,16904,16907,16909,16911,16914,16916,16918,16920,16922,16924,16926,16929,16932],{"class":12897,"line":13060},[70,16905,16906],{"class":12909},"    D ",[70,16908,112],{"class":12901},[70,16910,16788],{"class":12915},[70,16912,16913],{"class":12909},"(i ",[70,16915,16794],{"class":12901},[70,16917,16773],{"class":12909},[70,16919,12965],{"class":12901},[70,16921,16802],{"class":12909},[70,16923,16805],{"class":12901},[70,16925,16773],{"class":12909},[70,16927,16928],{"class":12901},"not",[70,16930,16931],{"class":12901}," in",[70,16933,16934],{"class":12909}," f[i])\n",[70,16936,16937,16939,16941,16943,16946,16948,16951,16953,16955],{"class":12897,"line":13078},[70,16938,16835],{"class":12915},[70,16940,103],{"class":12909},[70,16942,123],{"class":12901},[70,16944,16945],{"class":12924},"\"Diagonal-constructed set D = ",[70,16947,16882],{"class":12915},[70,16949,16950],{"class":12909},"(D)",[70,16952,5241],{"class":12915},[70,16954,16890],{"class":12924},[70,16956,12999],{"class":12909},[70,16958,16959],{"class":12897,"line":13084},[70,16960,13081],{"emptyLinePlaceholder":4989},[70,16962,16963,16966,16968,16971,16973,16975,16977,16979,16981,16984,16987],{"class":12897,"line":13089},[70,16964,16965],{"class":12909},"    matches ",[70,16967,112],{"class":12901},[70,16969,16970],{"class":12909}," [i ",[70,16972,16794],{"class":12901},[70,16974,16773],{"class":12909},[70,16976,12965],{"class":12901},[70,16978,16802],{"class":12909},[70,16980,16805],{"class":12901},[70,16982,16983],{"class":12909}," f[i] ",[70,16985,16986],{"class":12901},"==",[70,16988,16989],{"class":12909}," D]\n",[70,16991,16992,16994,16996,16998,17001,17003,17006,17008,17011,17014,17017,17019,17021],{"class":12897,"line":13100},[70,16993,16835],{"class":12915},[70,16995,103],{"class":12909},[70,16997,123],{"class":12901},[70,16999,17000],{"class":12924},"\"Does there exist i such that f(i) = D? -> ",[70,17002,5216],{"class":12915},[70,17004,17005],{"class":12909},"matches ",[70,17007,16805],{"class":12901},[70,17009,17010],{"class":12909}," matches ",[70,17012,17013],{"class":12901},"else",[70,17015,17016],{"class":12924}," 'Does not exist (contradiction established)'",[70,17018,5241],{"class":12915},[70,17020,16890],{"class":12924},[70,17022,12999],{"class":12909},[70,17024,17025],{"class":12897,"line":13125},[70,17026,13081],{"emptyLinePlaceholder":4989},[70,17028,17029,17032,17034,17036,17038,17040,17042,17044,17046],{"class":12897,"line":13148},[70,17030,17031],{"class":12909},"cantor_diagonal_demo([",[70,17033,2780],{"class":12915},[70,17035,13106],{"class":12909},[70,17037,1568],{"class":12915},[70,17039,13106],{"class":12909},[70,17041,2938],{"class":12915},[70,17043,13106],{"class":12909},[70,17045,5231],{"class":12915},[70,17047,17048],{"class":12909},"])\n",[11,17050,17051],{},[29,17052,13199],{},[12888,17054,17057],{"className":17055,"code":17056,"language":4752},[13203],"Set A = {0, 1, 2, 3}\nAssumed mapping f: A -> P(A):\n  f(0) = {0, 2}\n  f(1) = {1, 3}\n  f(2) = {0, 2}\n  f(3) = {1, 3}\n\nDiagonal-constructed set D = { i : i ∉ f(i) } = set()\nDoes there exist i such that f(i) = D? -> Does not exist (contradiction established)\n\nVerifying f(i) ≠ D for each i:\n  i=0: i∉D but i∈f(0), hence D ≠ f(0)\n  i=1: i∉D but i∈f(1), hence D ≠ f(1)\n  i=2: i∉D but i∈f(2), hence D ≠ f(2)\n  i=3: i∉D but i∈f(3), hence D ≠ f(3)\n",[2617,17058,17056],{"__ignoreMap":4976},[11,17060,17061,17062,17090,17091,17094,17095,17123,17124,882,17127,24],{},"Here, the ",[70,17063,17065,17078],{"className":17064,"translate":74},[78],[70,17066,17068],{"className":17067},[82],[84,17069,17070],{"xmlns":86},[89,17071,17072,17076],{},[92,17073,17074],{},[95,17075,1561],{},[164,17077,1561],{"encoding":166},[70,17079,17081],{"className":17080,"ariaHidden":172},[171],[70,17082,17084,17087],{"className":17083},[176],[70,17085],{"className":17086,"style":5067},[180],[70,17088,1561],{"className":17089,"style":1726},[185,193]," constructed happens to be the ",[29,17092,17093],{},"empty set",", and this particular mapping ",[70,17096,17098,17111],{"className":17097,"translate":74},[78],[70,17099,17101],{"className":17100},[82],[84,17102,17103],{"xmlns":86},[89,17104,17105,17109],{},[92,17106,17107],{},[95,17108,123],{},[164,17110,123],{"encoding":166},[70,17112,17114],{"className":17113,"ariaHidden":172},[171],[70,17115,17117,17120],{"className":17116},[176],[70,17118],{"className":17119,"style":791},[180],[70,17121,123],{"className":17122,"style":257},[185,193]," happens not to map any element to the empty set — a direct manifestation of the diagonal argument's central idea of \"deliberately constructing an escapee.\" Naturally, this is merely one concrete example of a mapping; the genuine proof is a general argument that holds for ",[29,17125,17126],{},"every possible mapping",[70,17128,17130,17143],{"className":17129,"translate":74},[78],[70,17131,17133],{"className":17132},[82],[84,17134,17135],{"xmlns":86},[89,17136,17137,17141],{},[92,17138,17139],{},[95,17140,123],{},[164,17142,123],{"encoding":166},[70,17144,17146],{"className":17145,"ariaHidden":172},[171],[70,17147,17149,17152],{"className":17148},[176],[70,17150],{"className":17151,"style":791},[180],[70,17153,123],{"className":17154,"style":257},[185,193],[45,17156,17158],{"id":17157},"the-undecidability-of-the-halting-problem","The Undecidability of the Halting Problem",[11,17160,17161,17162,17165],{},"One of the most striking applications of the diagonal argument appears in Alan Turing's 1936 proof of the ",[29,17163,17164],{},"undecidability of the halting problem",". This result directly laid the foundation for the entire field of computability theory.",[11,17167,17168,17171,17172,17200,17201,17229,17230,17258,17259,17287],{},[29,17169,17170],{},"The Halting Problem."," Given a program ",[70,17173,17175,17188],{"className":17174,"translate":74},[78],[70,17176,17178],{"className":17177},[82],[84,17179,17180],{"xmlns":86},[89,17181,17182,17186],{},[92,17183,17184],{},[95,17185,13429],{},[164,17187,13429],{"encoding":166},[70,17189,17191],{"className":17190,"ariaHidden":172},[171],[70,17192,17194,17197],{"className":17193},[176],[70,17195],{"className":17196,"style":5067},[180],[70,17198,13429],{"className":17199,"style":264},[185,193]," and an input ",[70,17202,17204,17217],{"className":17203,"translate":74},[78],[70,17205,17207],{"className":17206},[82],[84,17208,17209],{"xmlns":86},[89,17210,17211,17215],{},[92,17212,17213],{},[95,17214,106],{},[164,17216,106],{"encoding":166},[70,17218,17220],{"className":17219,"ariaHidden":172},[171],[70,17221,17223,17226],{"className":17222},[176],[70,17224],{"className":17225,"style":525},[180],[70,17227,106],{"className":17228},[185,193],", does there exist a universal algorithm capable of determining whether ",[70,17231,17233,17246],{"className":17232,"translate":74},[78],[70,17234,17236],{"className":17235},[82],[84,17237,17238],{"xmlns":86},[89,17239,17240,17244],{},[92,17241,17242],{},[95,17243,13429],{},[164,17245,13429],{"encoding":166},[70,17247,17249],{"className":17248,"ariaHidden":172},[171],[70,17250,17252,17255],{"className":17251},[176],[70,17253],{"className":17254,"style":5067},[180],[70,17256,13429],{"className":17257,"style":264},[185,193],", when run on input ",[70,17260,17262,17275],{"className":17261,"translate":74},[78],[70,17263,17265],{"className":17264},[82],[84,17266,17267],{"xmlns":86},[89,17268,17269,17273],{},[92,17270,17271],{},[95,17272,106],{},[164,17274,106],{"encoding":166},[70,17276,17278],{"className":17277,"ariaHidden":172},[171],[70,17279,17281,17284],{"className":17280},[176],[70,17282],{"className":17283,"style":525},[180],[70,17285,106],{"className":17286},[185,193],", will eventually halt (and return a result) or will run forever (enter an infinite loop)?",[11,17289,17290,17291,17294],{},"Turing proved: ",[29,17292,17293],{},"no such universal algorithm exists"," (a variant of the diagonal argument).",[11,17296,17297,17299],{},[29,17298,13703],{}," the existence of such a \"universal halting decider,\" which we formalize as a function:",[70,17301,17303],{"className":17302,"translate":74},[73],[70,17304,17306,17393],{"className":17305,"translate":74},[78],[70,17307,17309],{"className":17308},[82],[84,17310,17311],{"xmlns":86,"display":87},[89,17312,17313,17390],{},[92,17314,17315,17318,17320,17322,17324,17326,17328,17330],{},[4704,17316,17317],{},"halts",[100,17319,103],{"stretchy":102},[95,17321,13429],{},[100,17323,906],{"separator":172},[95,17325,106],{},[100,17327,109],{"stretchy":102},[100,17329,112],{},[92,17331,17332,17334],{},[100,17333,5216],{"fence":172},[6648,17335,17336,17363],{"rowspacing":6650,"columnalign":6651,"columnspacing":6652},[6654,17337,17338,17344],{},[6657,17339,17340],{},[6660,17341,17342],{"scriptlevel":2780,"displaystyle":102},[4704,17343,172],{},[6657,17345,17346],{},[6660,17347,17348],{"scriptlevel":2780,"displaystyle":102},[92,17349,17350,17352,17354,17356,17358,17360],{},[4704,17351,9424],{},[95,17353,13429],{},[100,17355,103],{"stretchy":102},[95,17357,106],{},[100,17359,109],{"stretchy":102},[4704,17361,17362],{}," halts",[6654,17364,17365,17371],{},[6657,17366,17367],{},[6660,17368,17369],{"scriptlevel":2780,"displaystyle":102},[4704,17370,102],{},[6657,17372,17373],{},[6660,17374,17375],{"scriptlevel":2780,"displaystyle":102},[92,17376,17377,17379,17381,17383,17385,17387],{},[4704,17378,9424],{},[95,17380,13429],{},[100,17382,103],{"stretchy":102},[95,17384,106],{},[100,17386,109],{"stretchy":102},[4704,17388,17389],{}," does not halt",[164,17391,17392],{"encoding":166},"\\text{halts}(P, x) = \\begin{cases} \\text{true} & \\text{if } P(x) \\text{ halts} \\\\ \\text{false} & \\text{if } P(x) \\text{ does not halt} \\end{cases}",[70,17394,17396,17435],{"className":17395,"ariaHidden":172},[171],[70,17397,17399,17402,17408,17411,17414,17417,17420,17423,17426,17429,17432],{"className":17398},[176],[70,17400],{"className":17401,"style":181},[180],[70,17403,17405],{"className":17404},[185,4752],[70,17406,17317],{"className":17407},[185],[70,17409,103],{"className":17410},[189],[70,17412,13429],{"className":17413,"style":264},[185,193],[70,17415,906],{"className":17416},[976],[70,17418],{"className":17419,"style":304},[201],[70,17421,106],{"className":17422},[185,193],[70,17424,109],{"className":17425},[197],[70,17427],{"className":17428,"style":202},[201],[70,17430,112],{"className":17431},[206],[70,17433],{"className":17434,"style":202},[201],[70,17436,17438,17441],{"className":17437},[176],[70,17439],{"className":17440,"style":6756},[180],[70,17442,17444,17450,17597],{"className":17443},[3082],[70,17445,17447],{"className":17446,"style":6764},[189,6763],[70,17448,5216],{"className":17449},[6768,6769],[70,17451,17453],{"className":17452},[185],[70,17454,17456,17507,17510],{"className":17455},[6648],[70,17457,17459],{"className":17458},[6779],[70,17460,17462,17499],{"className":17461},[225,226],[70,17463,17465,17496],{"className":17464},[230],[70,17466,17468,17482],{"className":17467,"style":6789},[234],[70,17469,17470,17473],{"style":6792},[70,17471],{"className":17472,"style":6796},[242],[70,17474,17476],{"className":17475},[185],[70,17477,17479],{"className":17478},[185,4752],[70,17480,172],{"className":17481},[185],[70,17483,17484,17487],{"style":6809},[70,17485],{"className":17486,"style":6796},[242],[70,17488,17490],{"className":17489},[185],[70,17491,17493],{"className":17492},[185,4752],[70,17494,102],{"className":17495},[185],[70,17497,291],{"className":17498},[290],[70,17500,17502],{"className":17501},[230],[70,17503,17505],{"className":17504,"style":6852},[234],[70,17506],{},[70,17508],{"className":17509,"style":6859},[6858],[70,17511,17513],{"className":17512},[6779],[70,17514,17516,17589],{"className":17515},[225,226],[70,17517,17519,17586],{"className":17518},[230],[70,17520,17522,17554],{"className":17521,"style":6789},[234],[70,17523,17524,17527],{"style":6792},[70,17525],{"className":17526,"style":6796},[242],[70,17528,17530,17536,17539,17542,17545,17548],{"className":17529},[185],[70,17531,17533],{"className":17532},[185,4752],[70,17534,9424],{"className":17535},[185],[70,17537,13429],{"className":17538,"style":264},[185,193],[70,17540,103],{"className":17541},[189],[70,17543,106],{"className":17544},[185,193],[70,17546,109],{"className":17547},[197],[70,17549,17551],{"className":17550},[185,4752],[70,17552,17362],{"className":17553},[185],[70,17555,17556,17559],{"style":6809},[70,17557],{"className":17558,"style":6796},[242],[70,17560,17562,17568,17571,17574,17577,17580],{"className":17561},[185],[70,17563,17565],{"className":17564},[185,4752],[70,17566,9424],{"className":17567},[185],[70,17569,13429],{"className":17570,"style":264},[185,193],[70,17572,103],{"className":17573},[189],[70,17575,106],{"className":17576},[185,193],[70,17578,109],{"className":17579},[197],[70,17581,17583],{"className":17582},[185,4752],[70,17584,17389],{"className":17585},[185],[70,17587,291],{"className":17588},[290],[70,17590,17592],{"className":17591},[230],[70,17593,17595],{"className":17594,"style":6852},[234],[70,17596],{},[70,17598],{"className":17599},[197,6920],[11,17601,17602],{},"This function itself must always return a correct answer within finite time (this is the meaning of \"universal\").",[11,17604,17605,882,17608,17636],{},[29,17606,17607],{},"Construct a diagonal program",[70,17609,17611,17624],{"className":17610,"translate":74},[78],[70,17612,17614],{"className":17613},[82],[84,17615,17616],{"xmlns":86},[89,17617,17618,17622],{},[92,17619,17620],{},[95,17621,1561],{},[164,17623,1561],{"encoding":166},[70,17625,17627],{"className":17626,"ariaHidden":172},[171],[70,17628,17630,17633],{"className":17629},[176],[70,17631],{"className":17632,"style":5067},[180],[70,17634,1561],{"className":17635,"style":1726},[185,193],", whose behavior is defined as follows:",[70,17638,17640],{"className":17639,"translate":74},[73],[70,17641,17643,17734],{"className":17642,"translate":74},[78],[70,17644,17646],{"className":17645},[82],[84,17647,17648],{"xmlns":86,"display":87},[89,17649,17650,17731],{},[92,17651,17652,17654,17656,17658,17660,17662],{},[95,17653,1561],{},[100,17655,103],{"stretchy":102},[95,17657,13429],{},[100,17659,109],{"stretchy":102},[100,17661,112],{},[92,17663,17664,17666],{},[100,17665,5216],{"fence":172},[6648,17667,17668,17700],{"rowspacing":6650,"columnalign":6651,"columnspacing":6652},[6654,17669,17670,17677],{},[6657,17671,17672],{},[6660,17673,17674],{"scriptlevel":2780,"displaystyle":102},[4704,17675,17676],{},"infinite loop",[6657,17678,17679],{},[6660,17680,17681],{"scriptlevel":2780,"displaystyle":102},[92,17682,17683,17686,17688,17690,17692,17694,17696,17698],{},[4704,17684,17685],{},"if halts",[100,17687,103],{"stretchy":102},[95,17689,13429],{},[100,17691,906],{"separator":172},[95,17693,13429],{},[100,17695,109],{"stretchy":102},[100,17697,112],{},[4704,17699,172],{},[6654,17701,17702,17709],{},[6657,17703,17704],{},[6660,17705,17706],{"scriptlevel":2780,"displaystyle":102},[4704,17707,17708],{},"halt immediately",[6657,17710,17711],{},[6660,17712,17713],{"scriptlevel":2780,"displaystyle":102},[92,17714,17715,17717,17719,17721,17723,17725,17727,17729],{},[4704,17716,17685],{},[100,17718,103],{"stretchy":102},[95,17720,13429],{},[100,17722,906],{"separator":172},[95,17724,13429],{},[100,17726,109],{"stretchy":102},[100,17728,112],{},[4704,17730,102],{},[164,17732,17733],{"encoding":166},"D(P) = \\begin{cases} \\text{infinite loop} & \\text{if } \\text{halts}(P, P) = \\text{true} \\\\ \\text{halt immediately} & \\text{if } \\text{halts}(P, P) = \\text{false} \\end{cases}",[70,17735,17737,17764],{"className":17736,"ariaHidden":172},[171],[70,17738,17740,17743,17746,17749,17752,17755,17758,17761],{"className":17739},[176],[70,17741],{"className":17742,"style":181},[180],[70,17744,1561],{"className":17745,"style":1726},[185,193],[70,17747,103],{"className":17748},[189],[70,17750,13429],{"className":17751,"style":264},[185,193],[70,17753,109],{"className":17754},[197],[70,17756],{"className":17757,"style":202},[201],[70,17759,112],{"className":17760},[206],[70,17762],{"className":17763,"style":202},[201],[70,17765,17767,17770],{"className":17766},[176],[70,17768],{"className":17769,"style":6756},[180],[70,17771,17773,17779,17968],{"className":17772},[3082],[70,17774,17776],{"className":17775,"style":6764},[189,6763],[70,17777,5216],{"className":17778},[6768,6769],[70,17780,17782],{"className":17781},[185],[70,17783,17785,17836,17839],{"className":17784},[6648],[70,17786,17788],{"className":17787},[6779],[70,17789,17791,17828],{"className":17790},[225,226],[70,17792,17794,17825],{"className":17793},[230],[70,17795,17797,17811],{"className":17796,"style":6789},[234],[70,17798,17799,17802],{"style":6792},[70,17800],{"className":17801,"style":6796},[242],[70,17803,17805],{"className":17804},[185],[70,17806,17808],{"className":17807},[185,4752],[70,17809,17676],{"className":17810},[185],[70,17812,17813,17816],{"style":6809},[70,17814],{"className":17815,"style":6796},[242],[70,17817,17819],{"className":17818},[185],[70,17820,17822],{"className":17821},[185,4752],[70,17823,17708],{"className":17824},[185],[70,17826,291],{"className":17827},[290],[70,17829,17831],{"className":17830},[230],[70,17832,17834],{"className":17833,"style":6852},[234],[70,17835],{},[70,17837],{"className":17838,"style":6859},[6858],[70,17840,17842],{"className":17841},[6779],[70,17843,17845,17960],{"className":17844},[225,226],[70,17846,17848,17957],{"className":17847},[230],[70,17849,17851,17904],{"className":17850,"style":6789},[234],[70,17852,17853,17856],{"style":6792},[70,17854],{"className":17855,"style":6796},[242],[70,17857,17859,17865,17871,17874,17877,17880,17883,17886,17889,17892,17895,17898],{"className":17858},[185],[70,17860,17862],{"className":17861},[185,4752],[70,17863,9424],{"className":17864},[185],[70,17866,17868],{"className":17867},[185,4752],[70,17869,17317],{"className":17870},[185],[70,17872,103],{"className":17873},[189],[70,17875,13429],{"className":17876,"style":264},[185,193],[70,17878,906],{"className":17879},[976],[70,17881],{"className":17882,"style":304},[201],[70,17884,13429],{"className":17885,"style":264},[185,193],[70,17887,109],{"className":17888},[197],[70,17890],{"className":17891,"style":202},[201],[70,17893,112],{"className":17894},[206],[70,17896],{"className":17897,"style":202},[201],[70,17899,17901],{"className":17900},[185,4752],[70,17902,172],{"className":17903},[185],[70,17905,17906,17909],{"style":6809},[70,17907],{"className":17908,"style":6796},[242],[70,17910,17912,17918,17924,17927,17930,17933,17936,17939,17942,17945,17948,17951],{"className":17911},[185],[70,17913,17915],{"className":17914},[185,4752],[70,17916,9424],{"className":17917},[185],[70,17919,17921],{"className":17920},[185,4752],[70,17922,17317],{"className":17923},[185],[70,17925,103],{"className":17926},[189],[70,17928,13429],{"className":17929,"style":264},[185,193],[70,17931,906],{"className":17932},[976],[70,17934],{"className":17935,"style":304},[201],[70,17937,13429],{"className":17938,"style":264},[185,193],[70,17940,109],{"className":17941},[197],[70,17943],{"className":17944,"style":202},[201],[70,17946,112],{"className":17947},[206],[70,17949],{"className":17950,"style":202},[201],[70,17952,17954],{"className":17953},[185,4752],[70,17955,102],{"className":17956},[185],[70,17958,291],{"className":17959},[290],[70,17961,17963],{"className":17962},[230],[70,17964,17966],{"className":17965,"style":6852},[234],[70,17967],{},[70,17969],{"className":17970},[197,6920],[11,17972,17973,17974,17977,17978,18038,18039,18067,18068,18096],{},"Note the critical operation here: ",[29,17975,17976],{},"feeding a program to itself as its own input"," (",[70,17979,17981,18005],{"className":17980,"translate":74},[78],[70,17982,17984],{"className":17983},[82],[84,17985,17986],{"xmlns":86},[89,17987,17988,18002],{},[92,17989,17990,17992,17994,17996,17998,18000],{},[4704,17991,17317],{},[100,17993,103],{"stretchy":102},[95,17995,13429],{},[100,17997,906],{"separator":172},[95,17999,13429],{},[100,18001,109],{"stretchy":102},[164,18003,18004],{"encoding":166},"\\text{halts}(P, P)",[70,18006,18008],{"className":18007,"ariaHidden":172},[171],[70,18009,18011,18014,18020,18023,18026,18029,18032,18035],{"className":18010},[176],[70,18012],{"className":18013,"style":181},[180],[70,18015,18017],{"className":18016},[185,4752],[70,18018,17317],{"className":18019},[185],[70,18021,103],{"className":18022},[189],[70,18024,13429],{"className":18025,"style":264},[185,193],[70,18027,906],{"className":18028},[976],[70,18030],{"className":18031,"style":304},[201],[70,18033,13429],{"className":18034,"style":264},[185,193],[70,18036,109],{"className":18037},[197],"). This is the direct analogue of the diagonal argument's operation of \"taking the ",[70,18040,18042,18055],{"className":18041,"translate":74},[78],[70,18043,18045],{"className":18044},[82],[84,18046,18047],{"xmlns":86},[89,18048,18049,18053],{},[92,18050,18051],{},[95,18052,6404],{},[164,18054,6404],{"encoding":166},[70,18056,18058],{"className":18057,"ariaHidden":172},[171],[70,18059,18061,18064],{"className":18060},[176],[70,18062],{"className":18063,"style":525},[180],[70,18065,6404],{"className":18066},[185,193],"-th digit of the ",[70,18069,18071,18084],{"className":18070,"translate":74},[78],[70,18072,18074],{"className":18073},[82],[84,18075,18076],{"xmlns":86},[89,18077,18078,18082],{},[92,18079,18080],{},[95,18081,6404],{},[164,18083,6404],{"encoding":166},[70,18085,18087],{"className":18086,"ariaHidden":172},[171],[70,18088,18090,18093],{"className":18089},[176],[70,18091],{"className":18092,"style":525},[180],[70,18094,6404],{"className":18095},[185,193],"-th number\" — the self-referential move of making a program \"examine\" itself.",[11,18098,18099,18102,18103,24],{},[29,18100,18101],{},"Deriving the contradiction."," We now examine the concrete execution ",[70,18104,18106,18126],{"className":18105,"translate":74},[78],[70,18107,18109],{"className":18108},[82],[84,18110,18111],{"xmlns":86},[89,18112,18113,18123],{},[92,18114,18115,18117,18119,18121],{},[95,18116,1561],{},[100,18118,103],{"stretchy":102},[95,18120,1561],{},[100,18122,109],{"stretchy":102},[164,18124,18125],{"encoding":166},"D(D)",[70,18127,18129],{"className":18128,"ariaHidden":172},[171],[70,18130,18132,18135,18138,18141,18144],{"className":18131},[176],[70,18133],{"className":18134,"style":181},[180],[70,18136,1561],{"className":18137,"style":1726},[185,193],[70,18139,103],{"className":18140},[189],[70,18142,1561],{"className":18143,"style":1726},[185,193],[70,18145,109],{"className":18146},[197],[1025,18148,18149,18358],{},[1028,18150,14511,18151,18237,18238,18281,18282,13106,18310,18353,18354,18357],{},[70,18152,18154,18182],{"className":18153,"translate":74},[78],[70,18155,18157],{"className":18156},[82],[84,18158,18159],{"xmlns":86},[89,18160,18161,18179],{},[92,18162,18163,18165,18167,18169,18171,18173,18175,18177],{},[4704,18164,17317],{},[100,18166,103],{"stretchy":102},[95,18168,1561],{},[100,18170,906],{"separator":172},[95,18172,1561],{},[100,18174,109],{"stretchy":102},[100,18176,112],{},[4704,18178,172],{},[164,18180,18181],{"encoding":166},"\\text{halts}(D, D) = \\text{true}",[70,18183,18185,18224],{"className":18184,"ariaHidden":172},[171],[70,18186,18188,18191,18197,18200,18203,18206,18209,18212,18215,18218,18221],{"className":18187},[176],[70,18189],{"className":18190,"style":181},[180],[70,18192,18194],{"className":18193},[185,4752],[70,18195,17317],{"className":18196},[185],[70,18198,103],{"className":18199},[189],[70,18201,1561],{"className":18202,"style":1726},[185,193],[70,18204,906],{"className":18205},[976],[70,18207],{"className":18208,"style":304},[201],[70,18210,1561],{"className":18211,"style":1726},[185,193],[70,18213,109],{"className":18214},[197],[70,18216],{"className":18217,"style":202},[201],[70,18219,112],{"className":18220},[206],[70,18222],{"className":18223,"style":202},[201],[70,18225,18227,18231],{"className":18226},[176],[70,18228],{"className":18229,"style":18230},[180],"height:0.6151em;",[70,18232,18234],{"className":18233},[185,4752],[70,18235,172],{"className":18236},[185]," (i.e., it is judged that \"",[70,18239,18241,18260],{"className":18240,"translate":74},[78],[70,18242,18244],{"className":18243},[82],[84,18245,18246],{"xmlns":86},[89,18247,18248,18258],{},[92,18249,18250,18252,18254,18256],{},[95,18251,1561],{},[100,18253,103],{"stretchy":102},[95,18255,1561],{},[100,18257,109],{"stretchy":102},[164,18259,18125],{"encoding":166},[70,18261,18263],{"className":18262,"ariaHidden":172},[171],[70,18264,18266,18269,18272,18275,18278],{"className":18265},[176],[70,18267],{"className":18268,"style":181},[180],[70,18270,1561],{"className":18271,"style":1726},[185,193],[70,18273,103],{"className":18274},[189],[70,18276,1561],{"className":18277,"style":1726},[185,193],[70,18279,109],{"className":18280},[197]," halts\"), then by the definition of ",[70,18283,18285,18298],{"className":18284,"translate":74},[78],[70,18286,18288],{"className":18287},[82],[84,18289,18290],{"xmlns":86},[89,18291,18292,18296],{},[92,18293,18294],{},[95,18295,1561],{},[164,18297,1561],{"encoding":166},[70,18299,18301],{"className":18300,"ariaHidden":172},[171],[70,18302,18304,18307],{"className":18303},[176],[70,18305],{"className":18306,"style":5067},[180],[70,18308,1561],{"className":18309,"style":1726},[185,193],[70,18311,18313,18332],{"className":18312,"translate":74},[78],[70,18314,18316],{"className":18315},[82],[84,18317,18318],{"xmlns":86},[89,18319,18320,18330],{},[92,18321,18322,18324,18326,18328],{},[95,18323,1561],{},[100,18325,103],{"stretchy":102},[95,18327,1561],{},[100,18329,109],{"stretchy":102},[164,18331,18125],{"encoding":166},[70,18333,18335],{"className":18334,"ariaHidden":172},[171],[70,18336,18338,18341,18344,18347,18350],{"className":18337},[176],[70,18339],{"className":18340,"style":181},[180],[70,18342,1561],{"className":18343,"style":1726},[185,193],[70,18345,103],{"className":18346},[189],[70,18348,1561],{"className":18349,"style":1726},[185,193],[70,18351,109],{"className":18352},[197]," should ",[29,18355,18356],{},"loop infinitely",". This contradicts \"it halts.\"",[1028,18359,14511,18360,18237,18445,18488,18489,13106,18517,18353,18560,18563],{},[70,18361,18363,18391],{"className":18362,"translate":74},[78],[70,18364,18366],{"className":18365},[82],[84,18367,18368],{"xmlns":86},[89,18369,18370,18388],{},[92,18371,18372,18374,18376,18378,18380,18382,18384,18386],{},[4704,18373,17317],{},[100,18375,103],{"stretchy":102},[95,18377,1561],{},[100,18379,906],{"separator":172},[95,18381,1561],{},[100,18383,109],{"stretchy":102},[100,18385,112],{},[4704,18387,102],{},[164,18389,18390],{"encoding":166},"\\text{halts}(D, D) = \\text{false}",[70,18392,18394,18433],{"className":18393,"ariaHidden":172},[171],[70,18395,18397,18400,18406,18409,18412,18415,18418,18421,18424,18427,18430],{"className":18396},[176],[70,18398],{"className":18399,"style":181},[180],[70,18401,18403],{"className":18402},[185,4752],[70,18404,17317],{"className":18405},[185],[70,18407,103],{"className":18408},[189],[70,18410,1561],{"className":18411,"style":1726},[185,193],[70,18413,906],{"className":18414},[976],[70,18416],{"className":18417,"style":304},[201],[70,18419,1561],{"className":18420,"style":1726},[185,193],[70,18422,109],{"className":18423},[197],[70,18425],{"className":18426,"style":202},[201],[70,18428,112],{"className":18429},[206],[70,18431],{"className":18432,"style":202},[201],[70,18434,18436,18439],{"className":18435},[176],[70,18437],{"className":18438,"style":6013},[180],[70,18440,18442],{"className":18441},[185,4752],[70,18443,102],{"className":18444},[185],[70,18446,18448,18467],{"className":18447,"translate":74},[78],[70,18449,18451],{"className":18450},[82],[84,18452,18453],{"xmlns":86},[89,18454,18455,18465],{},[92,18456,18457,18459,18461,18463],{},[95,18458,1561],{},[100,18460,103],{"stretchy":102},[95,18462,1561],{},[100,18464,109],{"stretchy":102},[164,18466,18125],{"encoding":166},[70,18468,18470],{"className":18469,"ariaHidden":172},[171],[70,18471,18473,18476,18479,18482,18485],{"className":18472},[176],[70,18474],{"className":18475,"style":181},[180],[70,18477,1561],{"className":18478,"style":1726},[185,193],[70,18480,103],{"className":18481},[189],[70,18483,1561],{"className":18484,"style":1726},[185,193],[70,18486,109],{"className":18487},[197]," does not halt\"), then by the definition of ",[70,18490,18492,18505],{"className":18491,"translate":74},[78],[70,18493,18495],{"className":18494},[82],[84,18496,18497],{"xmlns":86},[89,18498,18499,18503],{},[92,18500,18501],{},[95,18502,1561],{},[164,18504,1561],{"encoding":166},[70,18506,18508],{"className":18507,"ariaHidden":172},[171],[70,18509,18511,18514],{"className":18510},[176],[70,18512],{"className":18513,"style":5067},[180],[70,18515,1561],{"className":18516,"style":1726},[185,193],[70,18518,18520,18539],{"className":18519,"translate":74},[78],[70,18521,18523],{"className":18522},[82],[84,18524,18525],{"xmlns":86},[89,18526,18527,18537],{},[92,18528,18529,18531,18533,18535],{},[95,18530,1561],{},[100,18532,103],{"stretchy":102},[95,18534,1561],{},[100,18536,109],{"stretchy":102},[164,18538,18125],{"encoding":166},[70,18540,18542],{"className":18541,"ariaHidden":172},[171],[70,18543,18545,18548,18551,18554,18557],{"className":18544},[176],[70,18546],{"className":18547,"style":181},[180],[70,18549,1561],{"className":18550,"style":1726},[185,193],[70,18552,103],{"className":18553},[189],[70,18555,1561],{"className":18556,"style":1726},[185,193],[70,18558,109],{"className":18559},[197],[29,18561,18562],{},"halt immediately",". This contradicts \"it does not halt.\"",[11,18565,18566,18567,18599,18600],{},"Both cases lead to self-contradiction. Hence the assumption that \"a universal halting decider ",[70,18568,18570,18584],{"className":18569,"translate":74},[78],[70,18571,18573],{"className":18572},[82],[84,18574,18575],{"xmlns":86},[89,18576,18577,18581],{},[92,18578,18579],{},[4704,18580,17317],{},[164,18582,18583],{"encoding":166},"\\text{halts}",[70,18585,18587],{"className":18586,"ariaHidden":172},[171],[70,18588,18590,18593],{"className":18589},[176],[70,18591],{"className":18592,"style":6013},[180],[70,18594,18596],{"className":18595},[185,4752],[70,18597,17317],{"className":18598},[185]," exists\" is false. ",[29,18601,18602],{},"The halting problem is undecidable.",[11,18604,18605,18606,18799],{},"If one enumerates all possible programs according to some rule (",[70,18607,18609,18647],{"className":18608,"translate":74},[78],[70,18610,18612],{"className":18611},[82],[84,18613,18614],{"xmlns":86},[89,18615,18616,18644],{},[92,18617,18618,18624,18626,18632,18634,18640,18642],{},[137,18619,18620,18622],{},[95,18621,13429],{},[1566,18623,1568],{},[100,18625,906],{"separator":172},[137,18627,18628,18630],{},[95,18629,13429],{},[1566,18631,2938],{},[100,18633,906],{"separator":172},[137,18635,18636,18638],{},[95,18637,13429],{},[1566,18639,5231],{},[100,18641,906],{"separator":172},[100,18643,3004],{},[164,18645,18646],{"encoding":166},"P_1, P_2, P_3, \\dots",[70,18648,18650],{"className":18649,"ariaHidden":172},[171],[70,18651,18653,18657,18698,18701,18704,18744,18747,18750,18790,18793,18796],{"className":18652},[176],[70,18654],{"className":18655,"style":18656},[180],"height:0.8778em;vertical-align:-0.1944em;",[70,18658,18660,18663],{"className":18659},[185],[70,18661,13429],{"className":18662,"style":264},[185,193],[70,18664,18666],{"className":18665},[314],[70,18667,18669,18690],{"className":18668},[225,226],[70,18670,18672,18687],{"className":18671},[230],[70,18673,18675],{"className":18674,"style":2820},[234],[70,18676,18678,18681],{"style":18677},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[70,18679],{"className":18680,"style":331},[242],[70,18682,18684],{"className":18683},[247,248,249,250],[70,18685,1568],{"className":18686},[185,250],[70,18688,291],{"className":18689},[290],[70,18691,18693],{"className":18692},[230],[70,18694,18696],{"className":18695,"style":2841},[234],[70,18697],{},[70,18699,906],{"className":18700},[976],[70,18702],{"className":18703,"style":304},[201],[70,18705,18707,18710],{"className":18706},[185],[70,18708,13429],{"className":18709,"style":264},[185,193],[70,18711,18713],{"className":18712},[314],[70,18714,18716,18736],{"className":18715},[225,226],[70,18717,18719,18733],{"className":18718},[230],[70,18720,18722],{"className":18721,"style":2820},[234],[70,18723,18724,18727],{"style":18677},[70,18725],{"className":18726,"style":331},[242],[70,18728,18730],{"className":18729},[247,248,249,250],[70,18731,2938],{"className":18732},[185,250],[70,18734,291],{"className":18735},[290],[70,18737,18739],{"className":18738},[230],[70,18740,18742],{"className":18741,"style":2841},[234],[70,18743],{},[70,18745,906],{"className":18746},[976],[70,18748],{"className":18749,"style":304},[201],[70,18751,18753,18756],{"className":18752},[185],[70,18754,13429],{"className":18755,"style":264},[185,193],[70,18757,18759],{"className":18758},[314],[70,18760,18762,18782],{"className":18761},[225,226],[70,18763,18765,18779],{"className":18764},[230],[70,18766,18768],{"className":18767,"style":2820},[234],[70,18769,18770,18773],{"style":18677},[70,18771],{"className":18772,"style":331},[242],[70,18774,18776],{"className":18775},[247,248,249,250],[70,18777,5231],{"className":18778},[185,250],[70,18780,291],{"className":18781},[290],[70,18783,18785],{"className":18784},[230],[70,18786,18788],{"className":18787,"style":2841},[234],[70,18789],{},[70,18791,906],{"className":18792},[976],[70,18794],{"className":18795,"style":304},[201],[70,18797,3004],{"className":18798},[3082]," — this is feasible, since programs are essentially finite-length strings) and imagines an infinite table:",[70,18801,18803],{"className":18802,"translate":74},[73],[70,18804,18806,18971],{"className":18805,"translate":74},[78],[70,18807,18809],{"className":18808},[82],[84,18810,18811],{"xmlns":86,"display":87},[89,18812,18813,18968],{},[92,18814,18815,18817,18966],{},[100,18816,103],{"fence":172},[6648,18818,18820,18880,18940],{"rowspacing":11760,"columnalign":18819,"columnspacing":6652},"center center center",[6654,18821,18822,18848,18874],{},[6657,18823,18824],{},[6660,18825,18826],{"scriptlevel":2780,"displaystyle":102},[92,18827,18828,18830,18832,18838,18840,18846],{},[4704,18829,17317],{},[100,18831,103],{"stretchy":102},[137,18833,18834,18836],{},[95,18835,13429],{},[1566,18837,1568],{},[100,18839,906],{"separator":172},[137,18841,18842,18844],{},[95,18843,13429],{},[1566,18845,1568],{},[100,18847,109],{"stretchy":102},[6657,18849,18850],{},[6660,18851,18852],{"scriptlevel":2780,"displaystyle":102},[92,18853,18854,18856,18858,18864,18866,18872],{},[4704,18855,17317],{},[100,18857,103],{"stretchy":102},[137,18859,18860,18862],{},[95,18861,13429],{},[1566,18863,1568],{},[100,18865,906],{"separator":172},[137,18867,18868,18870],{},[95,18869,13429],{},[1566,18871,2938],{},[100,18873,109],{"stretchy":102},[6657,18875,18876],{},[6660,18877,18878],{"scriptlevel":2780,"displaystyle":102},[100,18879,11810],{"lspace":3414,"rspace":3414},[6654,18881,18882,18908,18934],{},[6657,18883,18884],{},[6660,18885,18886],{"scriptlevel":2780,"displaystyle":102},[92,18887,18888,18890,18892,18898,18900,18906],{},[4704,18889,17317],{},[100,18891,103],{"stretchy":102},[137,18893,18894,18896],{},[95,18895,13429],{},[1566,18897,2938],{},[100,18899,906],{"separator":172},[137,18901,18902,18904],{},[95,18903,13429],{},[1566,18905,1568],{},[100,18907,109],{"stretchy":102},[6657,18909,18910],{},[6660,18911,18912],{"scriptlevel":2780,"displaystyle":102},[92,18913,18914,18916,18918,18924,18926,18932],{},[4704,18915,17317],{},[100,18917,103],{"stretchy":102},[137,18919,18920,18922],{},[95,18921,13429],{},[1566,18923,2938],{},[100,18925,906],{"separator":172},[137,18927,18928,18930],{},[95,18929,13429],{},[1566,18931,2938],{},[100,18933,109],{"stretchy":102},[6657,18935,18936],{},[6660,18937,18938],{"scriptlevel":2780,"displaystyle":102},[100,18939,11810],{"lspace":3414,"rspace":3414},[6654,18941,18942,18954,18960],{},[6657,18943,18944],{},[6660,18945,18946],{"scriptlevel":2780,"displaystyle":102},[92,18947,18948,18950],{},[95,18949,7765],{"mathvariant":97},[7767,18951,18952],{"height":3414,"voffset":3414},[201,18953],{"mathbackground":7771,"width":3414,"height":7772},[6657,18955,18956],{},[6660,18957,18958],{"scriptlevel":2780,"displaystyle":102},[92,18959],{},[6657,18961,18962],{},[6660,18963,18964],{"scriptlevel":2780,"displaystyle":102},[100,18965,11947],{"lspace":3414,"rspace":3414},[100,18967,109],{"fence":172},[164,18969,18970],{"encoding":166},"\\begin{pmatrix}\n\\text{halts}(P_1, P_1) & \\text{halts}(P_1, P_2) & \\cdots \\\\\n\\text{halts}(P_2, P_1) & \\text{halts}(P_2, P_2) & \\cdots \\\\\n\\vdots & & \\ddots\n\\end{pmatrix}",[70,18972,18974],{"className":18973,"ariaHidden":172},[171],[70,18975,18977,18981],{"className":18976},[176],[70,18978],{"className":18979,"style":18980},[180],"height:4.26em;vertical-align:-1.88em;",[70,18982,18984,19029,19604],{"className":18983},[3082],[70,18985,18987],{"className":18986},[189],[70,18988,18990],{"className":18989},[6768,11972],[70,18991,18993,19020],{"className":18992},[225,226],[70,18994,18996,19017],{"className":18995},[230],[70,18997,19000],{"className":18998,"style":18999},[234],"height:2.35em;",[70,19001,19003,19007],{"style":19002},"top:-4.35em;",[70,19004],{"className":19005,"style":19006},[242],"height:6.2em;",[70,19008,19010],{"style":19009},"width:0.875em;height:4.2em;",[4418,19011,19014],{"xmlns":4420,"width":11995,"height":19012,"viewBox":19013},"4.2em","0 0 875 4200",[4426,19015],{"d":19016},"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,684c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-692c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z",[70,19018,291],{"className":19019},[290],[70,19021,19023],{"className":19022},[230],[70,19024,19027],{"className":19025,"style":19026},[234],"height:1.85em;",[70,19028],{},[70,19030,19032],{"className":19031},[185],[70,19033,19035,19291,19294,19297,19542,19545,19548],{"className":19034},[6648],[70,19036,19038],{"className":19037},[12022],[70,19039,19041,19282],{"className":19040},[225,226],[70,19042,19044,19279],{"className":19043},[230],[70,19045,19048,19155,19261],{"className":19046,"style":19047},[234],"height:2.38em;",[70,19049,19051,19054],{"style":19050},"top:-5.2275em;",[70,19052],{"className":19053,"style":8042},[242],[70,19055,19057,19063,19066,19106,19109,19112,19152],{"className":19056},[185],[70,19058,19060],{"className":19059},[185,4752],[70,19061,17317],{"className":19062},[185],[70,19064,103],{"className":19065},[189],[70,19067,19069,19072],{"className":19068},[185],[70,19070,13429],{"className":19071,"style":264},[185,193],[70,19073,19075],{"className":19074},[314],[70,19076,19078,19098],{"className":19077},[225,226],[70,19079,19081,19095],{"className":19080},[230],[70,19082,19084],{"className":19083,"style":2820},[234],[70,19085,19086,19089],{"style":18677},[70,19087],{"className":19088,"style":331},[242],[70,19090,19092],{"className":19091},[247,248,249,250],[70,19093,1568],{"className":19094},[185,250],[70,19096,291],{"className":19097},[290],[70,19099,19101],{"className":19100},[230],[70,19102,19104],{"className":19103,"style":2841},[234],[70,19105],{},[70,19107,906],{"className":19108},[976],[70,19110],{"className":19111,"style":304},[201],[70,19113,19115,19118],{"className":19114},[185],[70,19116,13429],{"className":19117,"style":264},[185,193],[70,19119,19121],{"className":19120},[314],[70,19122,19124,19144],{"className":19123},[225,226],[70,19125,19127,19141],{"className":19126},[230],[70,19128,19130],{"className":19129,"style":2820},[234],[70,19131,19132,19135],{"style":18677},[70,19133],{"className":19134,"style":331},[242],[70,19136,19138],{"className":19137},[247,248,249,250],[70,19139,1568],{"className":19140},[185,250],[70,19142,291],{"className":19143},[290],[70,19145,19147],{"className":19146},[230],[70,19148,19150],{"className":19149,"style":2841},[234],[70,19151],{},[70,19153,109],{"className":19154},[197],[70,19156,19157,19160],{"style":8461},[70,19158],{"className":19159,"style":8042},[242],[70,19161,19163,19169,19172,19212,19215,19218,19258],{"className":19162},[185],[70,19164,19166],{"className":19165},[185,4752],[70,19167,17317],{"className":19168},[185],[70,19170,103],{"className":19171},[189],[70,19173,19175,19178],{"className":19174},[185],[70,19176,13429],{"className":19177,"style":264},[185,193],[70,19179,19181],{"className":19180},[314],[70,19182,19184,19204],{"className":19183},[225,226],[70,19185,19187,19201],{"className":19186},[230],[70,19188,19190],{"className":19189,"style":2820},[234],[70,19191,19192,19195],{"style":18677},[70,19193],{"className":19194,"style":331},[242],[70,19196,19198],{"className":19197},[247,248,249,250],[70,19199,2938],{"className":19200},[185,250],[70,19202,291],{"className":19203},[290],[70,19205,19207],{"className":19206},[230],[70,19208,19210],{"className":19209,"style":2841},[234],[70,19211],{},[70,19213,906],{"className":19214},[976],[70,19216],{"className":19217,"style":304},[201],[70,19219,19221,19224],{"className":19220},[185],[70,19222,13429],{"className":19223,"style":264},[185,193],[70,19225,19227],{"className":19226},[314],[70,19228,19230,19250],{"className":19229},[225,226],[70,19231,19233,19247],{"className":19232},[230],[70,19234,19236],{"className":19235,"style":2820},[234],[70,19237,19238,19241],{"style":18677},[70,19239],{"className":19240,"style":331},[242],[70,19242,19244],{"className":19243},[247,248,249,250],[70,19245,1568],{"className":19246},[185,250],[70,19248,291],{"className":19249},[290],[70,19251,19253],{"className":19252},[230],[70,19254,19256],{"className":19255,"style":2841},[234],[70,19257],{},[70,19259,109],{"className":19260},[197],[70,19262,19264,19267],{"style":19263},"top:-2.1675em;",[70,19265],{"className":19266,"style":8042},[242],[70,19268,19270],{"className":19269},[185],[70,19271,19273,19276],{"className":19272},[185],[70,19274,7765],{"className":19275},[185],[70,19277],{"className":19278,"style":8910},[185,8909],[70,19280,291],{"className":19281},[290],[70,19283,19285],{"className":19284},[230],[70,19286,19289],{"className":19287,"style":19288},[234],"height:1.88em;",[70,19290],{},[70,19292],{"className":19293,"style":12276},[6858],[70,19295],{"className":19296,"style":12276},[6858],[70,19298,19300],{"className":19299},[12022],[70,19301,19303,19534],{"className":19302},[225,226],[70,19304,19306,19531],{"className":19305},[230],[70,19307,19309,19416,19522],{"className":19308,"style":19047},[234],[70,19310,19312,19315],{"style":19311},"top:-5.04em;",[70,19313],{"className":19314,"style":7812},[242],[70,19316,19318,19324,19327,19367,19370,19373,19413],{"className":19317},[185],[70,19319,19321],{"className":19320},[185,4752],[70,19322,17317],{"className":19323},[185],[70,19325,103],{"className":19326},[189],[70,19328,19330,19333],{"className":19329},[185],[70,19331,13429],{"className":19332,"style":264},[185,193],[70,19334,19336],{"className":19335},[314],[70,19337,19339,19359],{"className":19338},[225,226],[70,19340,19342,19356],{"className":19341},[230],[70,19343,19345],{"className":19344,"style":2820},[234],[70,19346,19347,19350],{"style":18677},[70,19348],{"className":19349,"style":331},[242],[70,19351,19353],{"className":19352},[247,248,249,250],[70,19354,1568],{"className":19355},[185,250],[70,19357,291],{"className":19358},[290],[70,19360,19362],{"className":19361},[230],[70,19363,19365],{"className":19364,"style":2841},[234],[70,19366],{},[70,19368,906],{"className":19369},[976],[70,19371],{"className":19372,"style":304},[201],[70,19374,19376,19379],{"className":19375},[185],[70,19377,13429],{"className":19378,"style":264},[185,193],[70,19380,19382],{"className":19381},[314],[70,19383,19385,19405],{"className":19384},[225,226],[70,19386,19388,19402],{"className":19387},[230],[70,19389,19391],{"className":19390,"style":2820},[234],[70,19392,19393,19396],{"style":18677},[70,19394],{"className":19395,"style":331},[242],[70,19397,19399],{"className":19398},[247,248,249,250],[70,19400,2938],{"className":19401},[185,250],[70,19403,291],{"className":19404},[290],[70,19406,19408],{"className":19407},[230],[70,19409,19411],{"className":19410,"style":2841},[234],[70,19412],{},[70,19414,109],{"className":19415},[197],[70,19417,19418,19421],{"style":7907},[70,19419],{"className":19420,"style":7812},[242],[70,19422,19424,19430,19433,19473,19476,19479,19519],{"className":19423},[185],[70,19425,19427],{"className":19426},[185,4752],[70,19428,17317],{"className":19429},[185],[70,19431,103],{"className":19432},[189],[70,19434,19436,19439],{"className":19435},[185],[70,19437,13429],{"className":19438,"style":264},[185,193],[70,19440,19442],{"className":19441},[314],[70,19443,19445,19465],{"className":19444},[225,226],[70,19446,19448,19462],{"className":19447},[230],[70,19449,19451],{"className":19450,"style":2820},[234],[70,19452,19453,19456],{"style":18677},[70,19454],{"className":19455,"style":331},[242],[70,19457,19459],{"className":19458},[247,248,249,250],[70,19460,2938],{"className":19461},[185,250],[70,19463,291],{"className":19464},[290],[70,19466,19468],{"className":19467},[230],[70,19469,19471],{"className":19470,"style":2841},[234],[70,19472],{},[70,19474,906],{"className":19475},[976],[70,19477],{"className":19478,"style":304},[201],[70,19480,19482,19485],{"className":19481},[185],[70,19483,13429],{"className":19484,"style":264},[185,193],[70,19486,19488],{"className":19487},[314],[70,19489,19491,19511],{"className":19490},[225,226],[70,19492,19494,19508],{"className":19493},[230],[70,19495,19497],{"className":19496,"style":2820},[234],[70,19498,19499,19502],{"style":18677},[70,19500],{"className":19501,"style":331},[242],[70,19503,19505],{"className":19504},[247,248,249,250],[70,19506,2938],{"className":19507},[185,250],[70,19509,291],{"className":19510},[290],[70,19512,19514],{"className":19513},[230],[70,19515,19517],{"className":19516,"style":2841},[234],[70,19518],{},[70,19520,109],{"className":19521},[197],[70,19523,19525,19528],{"style":19524},"top:-1.98em;",[70,19526],{"className":19527,"style":7812},[242],[70,19529],{"className":19530},[185],[70,19532,291],{"className":19533},[290],[70,19535,19537],{"className":19536},[230],[70,19538,19540],{"className":19539,"style":19288},[234],[70,19541],{},[70,19543],{"className":19544,"style":12276},[6858],[70,19546],{"className":19547,"style":12276},[6858],[70,19549,19551],{"className":19550},[12022],[70,19552,19554,19596],{"className":19553},[225,226],[70,19555,19557,19593],{"className":19556},[230],[70,19558,19560,19571,19582],{"className":19559,"style":19047},[234],[70,19561,19562,19565],{"style":19311},[70,19563],{"className":19564,"style":7812},[242],[70,19566,19568],{"className":19567},[185],[70,19569,11810],{"className":19570},[3082],[70,19572,19573,19576],{"style":7907},[70,19574],{"className":19575,"style":7812},[242],[70,19577,19579],{"className":19578},[185],[70,19580,11810],{"className":19581},[3082],[70,19583,19584,19587],{"style":19524},[70,19585],{"className":19586,"style":7812},[242],[70,19588,19590],{"className":19589},[185],[70,19591,11947],{"className":19592},[3082],[70,19594,291],{"className":19595},[290],[70,19597,19599],{"className":19598},[230],[70,19600,19602],{"className":19601,"style":19288},[234],[70,19603],{},[70,19605,19607],{"className":19606},[197],[70,19608,19610],{"className":19609},[6768,11972],[70,19611,19613,19634],{"className":19612},[225,226],[70,19614,19616,19631],{"className":19615},[230],[70,19617,19619],{"className":19618,"style":18999},[234],[70,19620,19621,19624],{"style":19002},[70,19622],{"className":19623,"style":19006},[242],[70,19625,19626],{"style":19009},[4418,19627,19628],{"xmlns":4420,"width":11995,"height":19012,"viewBox":19013},[4426,19629],{"d":19630},"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,609\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-744c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z",[70,19632,291],{"className":19633},[290],[70,19635,19637],{"className":19636},[230],[70,19638,19640],{"className":19639,"style":19026},[234],[70,19641],{},[11,19643,19644,19645,19673,19674,882,19676,19818,19819,19822],{},"The behavior of the diagonal program ",[70,19646,19648,19661],{"className":19647,"translate":74},[78],[70,19649,19651],{"className":19650},[82],[84,19652,19653],{"xmlns":86},[89,19654,19655,19659],{},[92,19656,19657],{},[95,19658,1561],{},[164,19660,1561],{"encoding":166},[70,19662,19664],{"className":19663,"ariaHidden":172},[171],[70,19665,19667,19670],{"className":19666},[176],[70,19668],{"className":19669,"style":5067},[180],[70,19671,1561],{"className":19672,"style":1726},[185,193]," depends precisely on the ",[29,19675,11739],{},[70,19677,19679,19711],{"className":19678,"translate":74},[78],[70,19680,19682],{"className":19681},[82],[84,19683,19684],{"xmlns":86},[89,19685,19686,19708],{},[92,19687,19688,19690,19692,19698,19700,19706],{},[4704,19689,17317],{},[100,19691,103],{"stretchy":102},[137,19693,19694,19696],{},[95,19695,13429],{},[95,19697,2701],{},[100,19699,906],{"separator":172},[137,19701,19702,19704],{},[95,19703,13429],{},[95,19705,2701],{},[100,19707,109],{"stretchy":102},[164,19709,19710],{"encoding":166},"\\text{halts}(P_i, P_i)",[70,19712,19714],{"className":19713,"ariaHidden":172},[171],[70,19715,19717,19720,19726,19729,19769,19772,19775,19815],{"className":19716},[176],[70,19718],{"className":19719,"style":181},[180],[70,19721,19723],{"className":19722},[185,4752],[70,19724,17317],{"className":19725},[185],[70,19727,103],{"className":19728},[189],[70,19730,19732,19735],{"className":19731},[185],[70,19733,13429],{"className":19734,"style":264},[185,193],[70,19736,19738],{"className":19737},[314],[70,19739,19741,19761],{"className":19740},[225,226],[70,19742,19744,19758],{"className":19743},[230],[70,19745,19747],{"className":19746,"style":8978},[234],[70,19748,19749,19752],{"style":18677},[70,19750],{"className":19751,"style":331},[242],[70,19753,19755],{"className":19754},[247,248,249,250],[70,19756,2701],{"className":19757},[185,193,250],[70,19759,291],{"className":19760},[290],[70,19762,19764],{"className":19763},[230],[70,19765,19767],{"className":19766,"style":2841},[234],[70,19768],{},[70,19770,906],{"className":19771},[976],[70,19773],{"className":19774,"style":304},[201],[70,19776,19778,19781],{"className":19777},[185],[70,19779,13429],{"className":19780,"style":264},[185,193],[70,19782,19784],{"className":19783},[314],[70,19785,19787,19807],{"className":19786},[225,226],[70,19788,19790,19804],{"className":19789},[230],[70,19791,19793],{"className":19792,"style":8978},[234],[70,19794,19795,19798],{"style":18677},[70,19796],{"className":19797,"style":331},[242],[70,19799,19801],{"className":19800},[247,248,249,250],[70,19802,2701],{"className":19803},[185,193,250],[70,19805,291],{"className":19806},[290],[70,19808,19810],{"className":19809},[230],[70,19811,19813],{"className":19812,"style":2841},[234],[70,19814],{},[70,19816,109],{"className":19817},[197]," of this table, and it deliberately behaves in the opposite manner. This is ",[29,19820,19821],{},"structurally identical"," to Cantor's construction of a real number \"each of whose digits differs from the corresponding diagonal digit.\"",[11,19824,19825],{},"We cannot actually implement a universal halting decider (since none exists!), but we can demonstrate through code how this logical contradiction concretely arises:",[12888,19827,19829],{"className":12890,"code":19828,"language":12892,"meta":4976,"style":4976},"import sys\nsys.setrecursionlimit(50)\n\ndef hypothetical_halts(f, x, fuel=10):\n    \"\"\"Hypothetical halting decider (for paradox demonstration only; not a genuinely feasible implementation)\"\"\"\n    try:\n        result = f(x, fuel)\n        return result is not None\n    except RecursionError:\n        return False\n\ndef D(x, fuel=10):\n    if fuel \u003C= 0:\n        raise RecursionError(\"fuel exhausted, simulating infinite loop\")\n    will_halt = hypothetical_halts(x, x, fuel - 1)\n    if will_halt:\n        raise RecursionError(\"simulating infinite loop: halts says it halts, so D deliberately does not halt\")\n    else:\n        return \"D halted\"\n\noutcome = D(D, fuel=8)\nprint(\"Execution result of D(D):\", outcome)\n",[2617,19830,19831,19839,19849,19853,19869,19874,19882,19892,19909,19919,19926,19930,19946,19961,19975,19993,20000,20013,20020,20027,20031,20052],{"__ignoreMap":4976},[70,19832,19833,19836],{"class":12897,"line":12898},[70,19834,19835],{"class":12901},"import",[70,19837,19838],{"class":12909}," sys\n",[70,19840,19841,19844,19847],{"class":12897,"line":4977},[70,19842,19843],{"class":12909},"sys.setrecursionlimit(",[70,19845,19846],{"class":12915},"50",[70,19848,12999],{"class":12909},[70,19850,19851],{"class":12897,"line":12928},[70,19852,13081],{"emptyLinePlaceholder":4989},[70,19854,19855,19857,19860,19863,19865,19867],{"class":12897,"line":12934},[70,19856,12902],{"class":12901},[70,19858,19859],{"class":12905}," hypothetical_halts",[70,19861,19862],{"class":12909},"(f, x, fuel",[70,19864,112],{"class":12901},[70,19866,12916],{"class":12915},[70,19868,12919],{"class":12909},[70,19870,19871],{"class":12897,"line":12940},[70,19872,19873],{"class":12924},"    \"\"\"Hypothetical halting decider (for paradox demonstration only; not a genuinely feasible implementation)\"\"\"\n",[70,19875,19876,19879],{"class":12897,"line":12945},[70,19877,19878],{"class":12901},"    try",[70,19880,19881],{"class":12909},":\n",[70,19883,19884,19887,19889],{"class":12897,"line":12956},[70,19885,19886],{"class":12909},"        result ",[70,19888,112],{"class":12901},[70,19890,19891],{"class":12909}," f(x, fuel)\n",[70,19893,19894,19897,19900,19903,19906],{"class":12897,"line":12974},[70,19895,19896],{"class":12901},"        return",[70,19898,19899],{"class":12909}," result ",[70,19901,19902],{"class":12901},"is",[70,19904,19905],{"class":12901}," not",[70,19907,19908],{"class":12915}," None\n",[70,19910,19911,19914,19917],{"class":12897,"line":13002},[70,19912,19913],{"class":12901},"    except",[70,19915,19916],{"class":12915}," RecursionError",[70,19918,19881],{"class":12909},[70,19920,19921,19923],{"class":12897,"line":13020},[70,19922,19896],{"class":12901},[70,19924,19925],{"class":12915}," False\n",[70,19927,19928],{"class":12897,"line":13048},[70,19929,13081],{"emptyLinePlaceholder":4989},[70,19931,19932,19934,19937,19940,19942,19944],{"class":12897,"line":13060},[70,19933,12902],{"class":12901},[70,19935,19936],{"class":12905}," D",[70,19938,19939],{"class":12909},"(x, fuel",[70,19941,112],{"class":12901},[70,19943,12916],{"class":12915},[70,19945,12919],{"class":12909},[70,19947,19948,19951,19954,19957,19959],{"class":12897,"line":13078},[70,19949,19950],{"class":12901},"    if",[70,19952,19953],{"class":12909}," fuel ",[70,19955,19956],{"class":12901},"\u003C=",[70,19958,16824],{"class":12915},[70,19960,19881],{"class":12909},[70,19962,19963,19966,19968,19970,19973],{"class":12897,"line":13084},[70,19964,19965],{"class":12901},"        raise",[70,19967,19916],{"class":12915},[70,19969,103],{"class":12909},[70,19971,19972],{"class":12924},"\"fuel exhausted, simulating infinite loop\"",[70,19974,12999],{"class":12909},[70,19976,19977,19980,19982,19985,19988,19991],{"class":12897,"line":13089},[70,19978,19979],{"class":12909},"    will_halt ",[70,19981,112],{"class":12901},[70,19983,19984],{"class":12909}," hypothetical_halts(x, x, fuel ",[70,19986,19987],{"class":12901},"-",[70,19989,19990],{"class":12915}," 1",[70,19992,12999],{"class":12909},[70,19994,19995,19997],{"class":12897,"line":13100},[70,19996,19950],{"class":12901},[70,19998,19999],{"class":12909}," will_halt:\n",[70,20001,20002,20004,20006,20008,20011],{"class":12897,"line":13125},[70,20003,19965],{"class":12901},[70,20005,19916],{"class":12915},[70,20007,103],{"class":12909},[70,20009,20010],{"class":12924},"\"simulating infinite loop: halts says it halts, so D deliberately does not halt\"",[70,20012,12999],{"class":12909},[70,20014,20015,20018],{"class":12897,"line":13148},[70,20016,20017],{"class":12901},"    else",[70,20019,19881],{"class":12909},[70,20021,20022,20024],{"class":12897,"line":13161},[70,20023,19896],{"class":12901},[70,20025,20026],{"class":12924}," \"D halted\"\n",[70,20028,20029],{"class":12897,"line":13167},[70,20030,13081],{"emptyLinePlaceholder":4989},[70,20032,20033,20036,20038,20041,20045,20047,20050],{"class":12897,"line":13172},[70,20034,20035],{"class":12909},"outcome ",[70,20037,112],{"class":12901},[70,20039,20040],{"class":12909}," D(D, ",[70,20042,20044],{"class":20043},"s9osk","fuel",[70,20046,112],{"class":12901},[70,20048,20049],{"class":12915},"8",[70,20051,12999],{"class":12909},[70,20053,20054,20056,20058,20061],{"class":12897,"line":13183},[70,20055,13186],{"class":12915},[70,20057,103],{"class":12909},[70,20059,20060],{"class":12924},"\"Execution result of D(D):\"",[70,20062,20063],{"class":12909},", outcome)\n",[11,20065,20066],{},[29,20067,13199],{},[12888,20069,20072],{"className":20070,"code":20071,"language":4752},[13203],"=== Diagonal Paradox Derivation ===\n\nAssume halts(f, x) is a perfect halting decider (correctly judges any program)\nConstruct D(x): if halts(x,x)==halts then D loops infinitely; if halts(x,x)==does-not-halt then D halts immediately\n\nCase A: if halts(D,D) judges 'halts' -> D's definition makes D loop infinitely -> contradiction\nCase B: if halts(D,D) judges 'does not halt' -> D's definition makes D halt immediately -> contradiction\n\n=> Either judgment leads to self-contradiction, demonstrating that a universal halts function cannot exist.\n\n=== Code Instantiation (low fuel for rapid demonstration) ===\nRecursion exception triggered, simulating the self-contradictory execution state: simulating infinite loop: halts says it halts, so D deliberately does not halt\n",[2617,20073,20071],{"__ignoreMap":4976},[11,20075,20076,20077,20080,20081,20083,20084,20087],{},"The ",[2617,20078,20079],{},"hypothetical_halts"," in this code is plainly not a genuinely \"universal\" decider — it operates only within a finite ",[2617,20082,20044],{}," (step budget), which is the sole reason we are able to execute it on a real computer. This, in fact, corroborates Turing's conclusion: ",[29,20085,20086],{},"a truly universal halting decider, one that always returns the correct answer, cannot possibly exist",". Any \"simulated version\" we can write necessarily suffers from limitations of one kind or another (such as the step ceiling employed here).",[45,20089,20091],{"id":20090},"gödels-incompleteness-theorems","Gödel's Incompleteness Theorems",[11,20093,20094],{},"In 1931, Kurt Gödel deployed an even more ingenious variant of the diagonal argument to prove one of the most profound results in the history of mathematical logic:",[1339,20096,20097],{},[11,20098,20099,20102],{},[29,20100,20101],{},"Gödel's First Incompleteness Theorem."," Any consistent (non-self-contradictory) formal system that contains elementary arithmetic possesses a proposition that can be neither proved nor disproved within the system.",[11,20104,20105],{},"Gödel's proof proceeds in three broad steps:",[2608,20107,20108,20114,20469],{},[1028,20109,20110,20113],{},[29,20111,20112],{},"Gödel numbering."," Encode every formula and every proof within the formal system as a natural number (this encoding technique is itself a stroke of genius, now known as \"Gödel numbers\"). In this way, \"whether a given proof proves a given formula\" becomes an arithmetical relation over natural numbers, expressible within the system itself.",[1028,20115,20116,20119,20120,20149,20150,20225,20228,20229,20232,20233,20277,20278,20308,20309,20395,20396,20439,20440,20468],{},[29,20117,20118],{},"Constructing a self-referential proposition."," Exploiting the encoding technique, one constructs a proposition ",[70,20121,20123,20137],{"className":20122,"translate":74},[78],[70,20124,20126],{"className":20125},[82],[84,20127,20128],{"xmlns":86},[89,20129,20130,20135],{},[92,20131,20132],{},[95,20133,20134],{},"G",[164,20136,20134],{"encoding":166},[70,20138,20140],{"className":20139,"ariaHidden":172},[171],[70,20141,20143,20146],{"className":20142},[176],[70,20144],{"className":20145,"style":5067},[180],[70,20147,20134],{"className":20148},[185,193]," whose essential content is:",[70,20151,20153],{"className":20152,"translate":74},[73],[70,20154,20156,20182],{"className":20155,"translate":74},[78],[70,20157,20159],{"className":20158},[82],[84,20160,20161],{"xmlns":86,"display":87},[89,20162,20163,20179],{},[92,20164,20165,20167,20169,20171,20174,20176],{},[95,20166,20134],{},[100,20168,68],{},[100,20170,112],{},[4704,20172,20173],{},"\"Proposition ",[95,20175,20134],{},[4704,20177,20178],{}," itself is unprovable within this system\"",[164,20180,20181],{"encoding":166},"G := \\text{\"Proposition } G \\text{ itself is unprovable within this system\"}",[70,20183,20185,20204],{"className":20184,"ariaHidden":172},[171],[70,20186,20188,20191,20194,20197,20201],{"className":20187},[176],[70,20189],{"className":20190,"style":5067},[180],[70,20192,20134],{"className":20193},[185,193],[70,20195],{"className":20196,"style":202},[201],[70,20198,20200],{"className":20199},[206],":=",[70,20202],{"className":20203,"style":202},[201],[70,20205,20207,20210,20216,20219],{"className":20206},[176],[70,20208],{"className":20209,"style":791},[180],[70,20211,20213],{"className":20212},[185,4752],[70,20214,20173],{"className":20215},[185],[70,20217,20134],{"className":20218},[185,193],[70,20220,20222],{"className":20221},[185,4752],[70,20223,20178],{"className":20224},[185],[20226,20227],"br",{},"This is a highly self-referential construction, whose mathematical foundation is known as the ",[29,20230,20231],{},"Diagonal Lemma"," (or fixed-point lemma): for any formula ",[70,20234,20236,20256],{"className":20235,"translate":74},[78],[70,20237,20239],{"className":20238},[82],[84,20240,20241],{"xmlns":86},[89,20242,20243,20253],{},[92,20244,20245,20247,20249,20251],{},[95,20246,1537],{},[100,20248,103],{"stretchy":102},[95,20250,106],{},[100,20252,109],{"stretchy":102},[164,20254,20255],{"encoding":166},"\\phi(x)",[70,20257,20259],{"className":20258,"ariaHidden":172},[171],[70,20260,20262,20265,20268,20271,20274],{"className":20261},[176],[70,20263],{"className":20264,"style":181},[180],[70,20266,1537],{"className":20267},[185,193],[70,20269,103],{"className":20270},[189],[70,20272,106],{"className":20273},[185,193],[70,20275,109],{"className":20276},[197]," with a single free variable, one can construct a sentence ",[70,20279,20281,20296],{"className":20280,"translate":74},[78],[70,20282,20284],{"className":20283},[82],[84,20285,20286],{"xmlns":86},[89,20287,20288,20293],{},[92,20289,20290],{},[95,20291,20292],{},"ψ",[164,20294,20295],{"encoding":166},"\\psi",[70,20297,20299],{"className":20298,"ariaHidden":172},[171],[70,20300,20302,20305],{"className":20301},[176],[70,20303],{"className":20304,"style":791},[180],[70,20306,20292],{"className":20307,"style":365},[185,193]," such that the system proves ",[70,20310,20312,20347],{"className":20311,"translate":74},[78],[70,20313,20315],{"className":20314},[82],[84,20316,20317],{"xmlns":86},[89,20318,20319,20344],{},[92,20320,20321,20323,20326,20328,20330,20335,20337,20342],{},[95,20322,20292],{},[100,20324,20325],{},"↔",[95,20327,1537],{},[100,20329,103],{"stretchy":102},[100,20331,20332],{},[95,20333,20334],{"mathvariant":97},"⌜",[95,20336,20292],{},[100,20338,20339],{},[95,20340,20341],{"mathvariant":97},"⌝",[100,20343,109],{"stretchy":102},[164,20345,20346],{"encoding":166},"\\psi \\leftrightarrow \\phi(\\ulcorner \\psi \\urcorner)",[70,20348,20350,20368],{"className":20349,"ariaHidden":172},[171],[70,20351,20353,20356,20359,20362,20365],{"className":20352},[176],[70,20354],{"className":20355,"style":791},[180],[70,20357,20292],{"className":20358,"style":365},[185,193],[70,20360],{"className":20361,"style":202},[201],[70,20363,20325],{"className":20364},[206],[70,20366],{"className":20367,"style":202},[201],[70,20369,20371,20374,20377,20380,20385,20388,20392],{"className":20370},[176],[70,20372],{"className":20373,"style":181},[180],[70,20375,1537],{"className":20376},[185,193],[70,20378,103],{"className":20379},[189],[70,20381,20384],{"className":20382},[189,20383],"amsrm","┌",[70,20386,20292],{"className":20387,"style":365},[185,193],[70,20389,20391],{"className":20390},[197,20383],"┐",[70,20393,109],{"className":20394},[197]," (where ",[70,20397,20399,20421],{"className":20398,"translate":74},[78],[70,20400,20402],{"className":20401},[82],[84,20403,20404],{"xmlns":86},[89,20405,20406,20418],{},[92,20407,20408,20412,20414],{},[100,20409,20410],{},[95,20411,20334],{"mathvariant":97},[95,20413,20292],{},[100,20415,20416],{},[95,20417,20341],{"mathvariant":97},[164,20419,20420],{"encoding":166},"\\ulcorner \\psi \\urcorner",[70,20422,20424],{"className":20423,"ariaHidden":172},[171],[70,20425,20427,20430,20433,20436],{"className":20426},[176],[70,20428],{"className":20429,"style":791},[180],[70,20431,20384],{"className":20432},[189,20383],[70,20434,20292],{"className":20435,"style":365},[185,193],[70,20437,20391],{"className":20438},[197,20383]," denotes the Gödel number of ",[70,20441,20443,20456],{"className":20442,"translate":74},[78],[70,20444,20446],{"className":20445},[82],[84,20447,20448],{"xmlns":86},[89,20449,20450,20454],{},[92,20451,20452],{},[95,20453,20292],{},[164,20455,20295],{"encoding":166},[70,20457,20459],{"className":20458,"ariaHidden":172},[171],[70,20460,20462,20465],{"className":20461},[176],[70,20463],{"className":20464,"style":791},[180],[70,20466,20292],{"className":20467,"style":365},[185,193],"). This lemma is precisely the abstraction of the \"diagonal\" idea within logic: it enables a proposition to \"speak about\" a property of its own code.",[1028,20470,20471,20474],{},[29,20472,20473],{},"Deriving a contradictory dilemma.",[1025,20475,20476,20566,20662],{},[1028,20477,20478,20479,20507,20508,20536,20537,20565],{},"If the system can prove ",[70,20480,20482,20495],{"className":20481,"translate":74},[78],[70,20483,20485],{"className":20484},[82],[84,20486,20487],{"xmlns":86},[89,20488,20489,20493],{},[92,20490,20491],{},[95,20492,20134],{},[164,20494,20134],{"encoding":166},[70,20496,20498],{"className":20497,"ariaHidden":172},[171],[70,20499,20501,20504],{"className":20500},[176],[70,20502],{"className":20503,"style":5067},[180],[70,20505,20134],{"className":20506},[185,193],", then, by the meaning of ",[70,20509,20511,20524],{"className":20510,"translate":74},[78],[70,20512,20514],{"className":20513},[82],[84,20515,20516],{"xmlns":86},[89,20517,20518,20522],{},[92,20519,20520],{},[95,20521,20134],{},[164,20523,20134],{"encoding":166},[70,20525,20527],{"className":20526,"ariaHidden":172},[171],[70,20528,20530,20533],{"className":20529},[176],[70,20531],{"className":20532,"style":5067},[180],[70,20534,20134],{"className":20535},[185,193]," (\"",[70,20538,20540,20553],{"className":20539,"translate":74},[78],[70,20541,20543],{"className":20542},[82],[84,20544,20545],{"xmlns":86},[89,20546,20547,20551],{},[92,20548,20549],{},[95,20550,20134],{},[164,20552,20134],{"encoding":166},[70,20554,20556],{"className":20555,"ariaHidden":172},[171],[70,20557,20559,20562],{"className":20558},[176],[70,20560],{"className":20561,"style":5067},[180],[70,20563,20134],{"className":20564},[185,193]," is unprovable\"), the system would simultaneously prove a false proposition — contradicting the consistency of the system.",[1028,20567,20568,20569,20597,20598,20632,20633,20661],{},"If the system can disprove ",[70,20570,20572,20585],{"className":20571,"translate":74},[78],[70,20573,20575],{"className":20574},[82],[84,20576,20577],{"xmlns":86},[89,20578,20579,20583],{},[92,20580,20581],{},[95,20582,20134],{},[164,20584,20134],{"encoding":166},[70,20586,20588],{"className":20587,"ariaHidden":172},[171],[70,20589,20591,20594],{"className":20590},[176],[70,20592],{"className":20593,"style":5067},[180],[70,20595,20134],{"className":20596},[185,193]," (i.e., prove ",[70,20599,20601,20617],{"className":20600,"translate":74},[78],[70,20602,20604],{"className":20603},[82],[84,20605,20606],{"xmlns":86},[89,20607,20608,20614],{},[92,20609,20610,20612],{},[95,20611,15067],{"mathvariant":97},[95,20613,20134],{},[164,20615,20616],{"encoding":166},"\\neg G",[70,20618,20620],{"className":20619,"ariaHidden":172},[171],[70,20621,20623,20626,20629],{"className":20622},[176],[70,20624],{"className":20625,"style":5067},[180],[70,20627,15067],{"className":20628},[185],[70,20630,20134],{"className":20631},[185,193],", which amounts to proving \"",[70,20634,20636,20649],{"className":20635,"translate":74},[78],[70,20637,20639],{"className":20638},[82],[84,20640,20641],{"xmlns":86},[89,20642,20643,20647],{},[92,20644,20645],{},[95,20646,20134],{},[164,20648,20134],{"encoding":166},[70,20650,20652],{"className":20651,"ariaHidden":172},[171],[70,20653,20655,20658],{"className":20654},[176],[70,20656],{"className":20657,"style":5067},[180],[70,20659,20134],{"className":20660},[185,193]," is provable\"), then, under reasonable supplementary conditions, this likewise leads to contradiction.",[1028,20663,20664,20665,20693],{},"Hence ",[70,20666,20668,20681],{"className":20667,"translate":74},[78],[70,20669,20671],{"className":20670},[82],[84,20672,20673],{"xmlns":86},[89,20674,20675,20679],{},[92,20676,20677],{},[95,20678,20134],{},[164,20680,20134],{"encoding":166},[70,20682,20684],{"className":20683,"ariaHidden":172},[171],[70,20685,20687,20690],{"className":20686},[176],[70,20688],{"className":20689,"style":5067},[180],[70,20691,20134],{"className":20692},[185,193]," can be neither proved nor disproved — it is an \"undecidable proposition\" within the system.",[11,20695,20696],{},"Commonality with the preceding two proofs:",[15564,20698,20699,20829],{},[15567,20700,20701],{},[15570,20702,20703,20705,20736,20767,20798],{},[15573,20704],{},[15573,20706,20707,20708],{},"Diagonal Number ",[70,20709,20711,20724],{"className":20710,"translate":74},[78],[70,20712,20714],{"className":20713},[82],[84,20715,20716],{"xmlns":86},[89,20717,20718,20722],{},[92,20719,20720],{},[95,20721,4632],{},[164,20723,4632],{"encoding":166},[70,20725,20727],{"className":20726,"ariaHidden":172},[171],[70,20728,20730,20733],{"className":20729},[176],[70,20731],{"className":20732,"style":2801},[180],[70,20734,4632],{"className":20735,"style":365},[185,193],[15573,20737,20738,20739],{},"Diagonal Set ",[70,20740,20742,20755],{"className":20741,"translate":74},[78],[70,20743,20745],{"className":20744},[82],[84,20746,20747],{"xmlns":86},[89,20748,20749,20753],{},[92,20750,20751],{},[95,20752,1561],{},[164,20754,1561],{"encoding":166},[70,20756,20758],{"className":20757,"ariaHidden":172},[171],[70,20759,20761,20764],{"className":20760},[176],[70,20762],{"className":20763,"style":5067},[180],[70,20765,1561],{"className":20766,"style":1726},[185,193],[15573,20768,20769,20770],{},"Diagonal Program ",[70,20771,20773,20786],{"className":20772,"translate":74},[78],[70,20774,20776],{"className":20775},[82],[84,20777,20778],{"xmlns":86},[89,20779,20780,20784],{},[92,20781,20782],{},[95,20783,1561],{},[164,20785,1561],{"encoding":166},[70,20787,20789],{"className":20788,"ariaHidden":172},[171],[70,20790,20792,20795],{"className":20791},[176],[70,20793],{"className":20794,"style":5067},[180],[70,20796,1561],{"className":20797,"style":1726},[185,193],[15573,20799,20800,20801],{},"Gödel Sentence ",[70,20802,20804,20817],{"className":20803,"translate":74},[78],[70,20805,20807],{"className":20806},[82],[84,20808,20809],{"xmlns":86},[89,20810,20811,20815],{},[92,20812,20813],{},[95,20814,20134],{},[164,20816,20134],{"encoding":166},[70,20818,20820],{"className":20819,"ariaHidden":172},[171],[70,20821,20823,20826],{"className":20822},[176],[70,20824],{"className":20825,"style":5067},[180],[70,20827,20134],{"className":20828},[185,193],[15579,20830,20831,20848,20865],{},[15570,20832,20833,20836,20839,20842,20845],{},[15584,20834,20835],{},"Self-referential object",[15584,20837,20838],{},"Each digit \"avoids\" itself",[15584,20840,20841],{},"Collects elements \"not belonging to themselves\"",[15584,20843,20844],{},"Uses its own behavior to negate the judgment about itself",[15584,20846,20847],{},"Asserts \"I am unprovable\"",[15570,20849,20850,20853,20856,20859,20862],{},[15584,20851,20852],{},"Mechanism relied upon",[15584,20854,20855],{},"Diagonal of the decimal expansion",[15584,20857,20858],{},"Self-reference in set membership",[15584,20860,20861],{},"A program fed to itself as input",[15584,20863,20864],{},"Self-reference realized via Gödel numbering",[15570,20866,20867,20870,20901,20932,20935],{},[15584,20868,20869],{},"Source of contradiction",[15584,20871,20872,20873],{},"The assumed bijection cannot cover ",[70,20874,20876,20889],{"className":20875,"translate":74},[78],[70,20877,20879],{"className":20878},[82],[84,20880,20881],{"xmlns":86},[89,20882,20883,20887],{},[92,20884,20885],{},[95,20886,4632],{},[164,20888,4632],{"encoding":166},[70,20890,20892],{"className":20891,"ariaHidden":172},[171],[70,20893,20895,20898],{"className":20894},[176],[70,20896],{"className":20897,"style":2801},[180],[70,20899,4632],{"className":20900,"style":365},[185,193],[15584,20902,20903,20904],{},"The assumed surjection cannot cover ",[70,20905,20907,20920],{"className":20906,"translate":74},[78],[70,20908,20910],{"className":20909},[82],[84,20911,20912],{"xmlns":86},[89,20913,20914,20918],{},[92,20915,20916],{},[95,20917,1561],{},[164,20919,1561],{"encoding":166},[70,20921,20923],{"className":20922,"ariaHidden":172},[171],[70,20924,20926,20929],{"className":20925},[176],[70,20927],{"className":20928,"style":5067},[180],[70,20930,1561],{"className":20931,"style":1726},[185,193],[15584,20933,20934],{},"The assumed decider yields paradox",[15584,20936,20937],{},"The assumed provability yields paradox",[11,20939,20940],{},"These four proofs, in their essence, perform the same operation: exploiting a \"self-referential\" construction to fabricate an object that necessarily contradicts the assumed \"complete coverage.\" It is precisely this that makes the diagonal argument — spanning the three great domains of set theory, computability theory, and mathematical logic — one of the most penetrating proof ideas of the twentieth century.",[45,20942,20944],{"id":20943},"further-extensions","Further Extensions",[11,20946,20947],{},"Beyond the three classical applications discussed above, the diagonal argument and its underlying idea permeate numerous corners of computer science and mathematics:",[1025,20949,20950,20956,21019,21025],{},[1028,20951,20952,20955],{},[29,20953,20954],{},"Kolmogorov complexity."," The diagonal argument can be used to prove the existence of \"incompressible\" strings — i.e., strings that cannot be generated by any program shorter than the string itself — and, moreover, that such strings constitute the overwhelming majority.",[1028,20957,20958,20961,20962,9125],{},[29,20959,20960],{},"The time and space hierarchy theorems in complexity theory."," The diagonal argument is employed to prove that endowing a Turing machine with additional time or space budget genuinely enables it to solve strictly more problems (this underlies hierarchy relations such as ",[70,20963,20965,20985],{"className":20964,"translate":74},[78],[70,20966,20968],{"className":20967},[82],[84,20969,20970],{"xmlns":86},[89,20971,20972,20982],{},[92,20973,20974,20976,20979],{},[95,20975,13429],{},[100,20977,20978],{},"⊊",[4704,20980,20981],{},"EXP",[164,20983,20984],{"encoding":166},"P \\subsetneq \\text{EXP}",[70,20986,20988,21007],{"className":20987,"ariaHidden":172},[171],[70,20989,20991,20995,20998,21001,21004],{"className":20990},[176],[70,20992],{"className":20993,"style":20994},[180],"height:0.8193em;vertical-align:-0.136em;",[70,20996,13429],{"className":20997,"style":264},[185,193],[70,20999],{"className":21000,"style":202},[201],[70,21002,20978],{"className":21003},[206,20383],[70,21005],{"className":21006,"style":202},[201],[70,21008,21010,21013],{"className":21009},[176],[70,21011],{"className":21012,"style":5067},[180],[70,21014,21016],{"className":21015},[185,4752],[70,21017,20981],{"className":21018},[185],[1028,21020,21021,21024],{},[29,21022,21023],{},"Tarski's undefinability theorem."," This result demonstrates that the concept of \"truth\" cannot be fully defined within a sufficiently strong formal system; the proof structure bears a close resemblance to Gödel's theorem.",[1028,21026,21027,21030,21031,21171,21172,21321],{},[29,21028,21029],{},"Russell's paradox."," Although historically predating the widespread recognition of Cantor's diagonal argument in its modern formulation, the paradox of \"the set of all sets that do not contain themselves,\" ",[70,21032,21034,21065],{"className":21033,"translate":74},[78],[70,21035,21037],{"className":21036},[82],[84,21038,21039],{"xmlns":86},[89,21040,21041,21062],{},[92,21042,21043,21046,21048,21050,21052,21054,21056,21058,21060],{},[95,21044,21045],{},"R",[100,21047,112],{},[100,21049,5216],{"stretchy":102},[95,21051,106],{},[100,21053,68],{},[95,21055,106],{},[100,21057,14041],{"mathvariant":97},[95,21059,106],{},[100,21061,5241],{"stretchy":102},[164,21063,21064],{"encoding":166},"R = \\{x : x \\notin x\\}",[70,21066,21068,21087,21108,21159],{"className":21067,"ariaHidden":172},[171],[70,21069,21071,21074,21078,21081,21084],{"className":21070},[176],[70,21072],{"className":21073,"style":5067},[180],[70,21075,21045],{"className":21076,"style":21077},[185,193],"margin-right:0.0077em;",[70,21079],{"className":21080,"style":202},[201],[70,21082,112],{"className":21083},[206],[70,21085],{"className":21086,"style":202},[201],[70,21088,21090,21093,21096,21099,21102,21105],{"className":21089},[176],[70,21091],{"className":21092,"style":181},[180],[70,21094,5216],{"className":21095},[189],[70,21097,106],{"className":21098},[185,193],[70,21100],{"className":21101,"style":202},[201],[70,21103,68],{"className":21104},[206],[70,21106],{"className":21107,"style":202},[201],[70,21109,21111,21114,21117,21120,21156],{"className":21110},[176],[70,21112],{"className":21113,"style":181},[180],[70,21115,106],{"className":21116},[185,193],[70,21118],{"className":21119,"style":202},[201],[70,21121,21123,21129],{"className":21122},[206],[70,21124,21126],{"className":21125},[185],[70,21127,126],{"className":21128},[206],[70,21130,21132],{"className":21131},[185,5591],[70,21133,21135],{"className":21134},[5595],[70,21136,21138,21141,21153],{"className":21137},[14150],[70,21139],{"className":21140,"style":181},[180],[70,21142,21144],{"className":21143},[5606],[70,21145,21147,21150],{"className":21146},[185],[70,21148,6668],{"className":21149},[185],[70,21151],{"className":21152,"style":4134},[201],[70,21154],{"className":21155},[5617],[70,21157],{"className":21158,"style":202},[201],[70,21160,21162,21165,21168],{"className":21161},[176],[70,21163],{"className":21164,"style":181},[180],[70,21166,106],{"className":21167},[185,193],[70,21169,5241],{"className":21170},[197],", shares an identical structure with the set ",[70,21173,21175,21210],{"className":21174,"translate":74},[78],[70,21176,21178],{"className":21177},[82],[84,21179,21180],{"xmlns":86},[89,21181,21182,21208],{},[92,21183,21184,21186,21188,21190,21192,21194,21196,21198,21200,21202,21204,21206],{},[95,21185,1561],{},[100,21187,112],{},[100,21189,5216],{"stretchy":102},[95,21191,18],{},[100,21193,68],{},[95,21195,18],{},[100,21197,14041],{"mathvariant":97},[95,21199,123],{},[100,21201,103],{"stretchy":102},[95,21203,18],{},[100,21205,109],{"stretchy":102},[100,21207,5241],{"stretchy":102},[164,21209,15946],{"encoding":166},[70,21211,21213,21231,21252,21303],{"className":21212,"ariaHidden":172},[171],[70,21214,21216,21219,21222,21225,21228],{"className":21215},[176],[70,21217],{"className":21218,"style":5067},[180],[70,21220,1561],{"className":21221,"style":1726},[185,193],[70,21223],{"className":21224,"style":202},[201],[70,21226,112],{"className":21227},[206],[70,21229],{"className":21230,"style":202},[201],[70,21232,21234,21237,21240,21243,21246,21249],{"className":21233},[176],[70,21235],{"className":21236,"style":181},[180],[70,21238,5216],{"className":21239},[189],[70,21241,18],{"className":21242},[185,193],[70,21244],{"className":21245,"style":202},[201],[70,21247,68],{"className":21248},[206],[70,21250],{"className":21251,"style":202},[201],[70,21253,21255,21258,21261,21264,21300],{"className":21254},[176],[70,21256],{"className":21257,"style":181},[180],[70,21259,18],{"className":21260},[185,193],[70,21262],{"className":21263,"style":202},[201],[70,21265,21267,21273],{"className":21266},[206],[70,21268,21270],{"className":21269},[185],[70,21271,126],{"className":21272},[206],[70,21274,21276],{"className":21275},[185,5591],[70,21277,21279],{"className":21278},[5595],[70,21280,21282,21285,21297],{"className":21281},[14150],[70,21283],{"className":21284,"style":181},[180],[70,21286,21288],{"className":21287},[5606],[70,21289,21291,21294],{"className":21290},[185],[70,21292,6668],{"className":21293},[185],[70,21295],{"className":21296,"style":4134},[201],[70,21298],{"className":21299},[5617],[70,21301],{"className":21302,"style":202},[201],[70,21304,21306,21309,21312,21315,21318],{"className":21305},[176],[70,21307],{"className":21308,"style":181},[180],[70,21310,123],{"className":21311,"style":257},[185,193],[70,21313,103],{"className":21314},[189],[70,21316,18],{"className":21317},[185,193],[70,21319,16058],{"className":21320},[197]," in the proof of Cantor's Theorem — both are products of \"self-reference + self-exclusion.\"",[11,21323,21324],{},"If one were to distill the diagonal argument into a reusable \"template of thought,\" the outline would run approximately as follows:",[2608,21326,21327,21333,21339,21406],{},[1028,21328,21329,21332],{},[29,21330,21331],{},"Point of departure (reductio)."," Assume the existence of a \"perfect coverage\" or \"universal judgment\" — a bijection, a surjection, a decider, a proof system.",[1028,21334,21335,21338],{},[29,21336,21337],{},"Construct a self-referential object."," Exploit the \"numbering\" or \"mapping\" capacity provided by the assumption itself to construct a new object deliberately \"at odds\" with every item in the assumed coverage (a diagonal number, a diagonal set, a diagonal program, a self-referential proposition).",[1028,21340,21341,21344,21345,21347,21348,21376,21377,21405],{},[29,21342,21343],{},"Verify its evasive character."," Prove that this new object differs systematically from ",[29,21346,11043],{}," item within the scope of the assumed coverage (typically through the pattern of \"the ",[70,21349,21351,21364],{"className":21350,"translate":74},[78],[70,21352,21354],{"className":21353},[82],[84,21355,21356],{"xmlns":86},[89,21357,21358,21362],{},[92,21359,21360],{},[95,21361,6404],{},[164,21363,6404],{"encoding":166},[70,21365,21367],{"className":21366,"ariaHidden":172},[171],[70,21368,21370,21373],{"className":21369},[176],[70,21371],{"className":21372,"style":525},[180],[70,21374,6404],{"className":21375},[185,193],"-th item differs at the ",[70,21378,21380,21393],{"className":21379,"translate":74},[78],[70,21381,21383],{"className":21382},[82],[84,21384,21385],{"xmlns":86},[89,21386,21387,21391],{},[92,21388,21389],{},[95,21390,6404],{},[164,21392,6404],{"encoding":166},[70,21394,21396],{"className":21395,"ariaHidden":172},[171],[70,21397,21399,21402],{"className":21398},[176],[70,21400],{"className":21401,"style":525},[180],[70,21403,6404],{"className":21404},[185,193],"-th position\").",[1028,21407,21408,21411],{},[29,21409,21410],{},"Derive a contradiction."," The new object ought, by the assumption, to be covered — yet the construction procedure guarantees that it necessarily escapes coverage. Contradiction ensues; the original assumption is refuted.",[11,21413,21414],{},"Once this template is grasped, it becomes apparent that it functions much like a master key, capable of unlocking numerous impossibility results in mathematics and computer science that may initially appear forbiddingly abstruse. It is for this reason that the diagonal argument is frequently described as \"one of the most elegant proof techniques of the twentieth century\" — employing a logic of the utmost simplicity, it reveals the boundaries of such profound concepts as infinity, computability, and formal systems.",[21416,21417,21418],"style",{},"html pre.shiki code .snl16, html code.shiki .snl16{--shiki-default:#F97583}html pre.shiki code .svObZ, html code.shiki .svObZ{--shiki-default:#B392F0}html pre.shiki code .s95oV, html code.shiki .s95oV{--shiki-default:#E1E4E8}html pre.shiki code .sDLfK, html code.shiki .sDLfK{--shiki-default:#79B8FF}html pre.shiki code .sU2Wk, html code.shiki .sU2Wk{--shiki-default:#9ECBFF}html pre.shiki code .sAwPA, html code.shiki .sAwPA{--shiki-default:#6A737D}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .s9osk, html code.shiki .s9osk{--shiki-default:#FFAB70}",{"title":4976,"searchDepth":4977,"depth":4977,"links":21420},[21421,21422,21423,21424,21427,21428,21429],{"id":5188,"depth":4977,"text":5189},{"id":7205,"depth":4977,"text":7206},{"id":12882,"depth":4977,"text":12883},{"id":13372,"depth":4977,"text":13373,"children":21425},[21426],{"id":13697,"depth":12928,"text":13698},{"id":17157,"depth":4977,"text":17158},{"id":20090,"depth":4977,"text":20091},{"id":20943,"depth":4977,"text":20944},"In 1891, the German mathematician Georg Cantor published a paper of merely a few pages, proving a conclusion that appeared counterintuitive at the time: there exist infinities of different \"sizes.\" The proof technique he employed — the diagonal argument — not only resolved the problem at hand, but went on to become one of the most formidable weapons in the arsenal of mathematical logic, computability theory, and even philosophy over the ensuing century and more.",{},"2026-07-31","\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en",{"title":5173,"description":21430},"blog\u002F2026\u002F2026-07-31-the-diagonal-argument-en","The diagonal argument, introduced by Cantor in 1891, proves some infinities are strictly larger than others by constructing an object that evades any proposed complete list or mapping. This same self-referential trick underlies Cantor's theorem, Turing's halting problem, and Gödel's incompleteness theorem—one idea unifying set theory, computability, and logic.",[21438,21439,21440],"mathematics","cs","computation","z4HhqEEMHNtXvMVNiRvPZlDoaIKEYvh2uuh-HONpMJA",1786880520438]