[{"data":1,"prerenderedAt":16390},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en":3},{"id":4,"title":5,"body":6,"cardimage":16375,"description":16376,"draft":16377,"enableComment":8015,"extension":16378,"image":7824,"meta":16379,"navigation":8015,"onday":16380,"path":16381,"seo":16382,"stem":16383,"summary":16384,"tags":16385,"__hash__":16389},"blog\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en.md","The Diagonal Argument: A Proof Technique That Transformed Mathematics and Computer Science",{"type":7,"value":8,"toc":16364},"minimark",[9,18,21,26,29,194,342,921,1087,1181,1184,2114,2117,2121,2183,2238,2245,2400,3843,4050,4304,4724,4960,4995,5180,5561,5870,5965,6121,6205,6390,6669,7808,7811,7815,7818,8130,8135,8142,8305,8309,8312,8543,8630,8635,8815,8942,9135,9138,9389,9444,10320,10496,10503,11350,11676,11679,11991,11995,12001,12097,12101,12108,12231,12238,12243,12543,12546,12580,12914,13040,13090,13507,13546,13743,14585,14766,14769,15007,15011,15017,15031,15035,15038,15046,15049,15639,15642,15883,15886,15890,15893,16267,16270,16357,16360],[10,11,12,13,17],"p",{},"In 1891, the German mathematician Georg Cantor published a paper of merely a few pages, proving a conclusion that appeared counterintuitive at the time: ",[14,15,16],"strong",{},"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.",[10,19,20],{},"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.",[22,23,25],"h2",{"id":24},"equal-in-size","Equal in Size",[10,27,28],{},"Before discussing infinite sets, we must first clarify a preliminary question: how does one compare the sizes of two sets?",[10,30,31,32,193],{},"For finite sets, the answer is straightforward: simply count the number of elements. But the set of natural numbers ",[33,34,38,103],"span",{"className":35,"translate":37},[36],"katex","no",[33,39,42],{"className":40},[41],"katex-mathml",[43,44,46],"math",{"xmlns":45},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[47,48,49,98],"semantics",{},[50,51,52,57,61,65,69,73,76,78,81,83,86,88,91,95],"mrow",{},[53,54,56],"mi",{"mathvariant":55},"double-struck","N",[58,59,60],"mo",{},"=",[58,62,64],{"stretchy":63},"false","{",[66,67,68],"mn",{},"0",[58,70,72],{"separator":71},"true",",",[66,74,75],{},"1",[58,77,72],{"separator":71},[66,79,80],{},"2",[58,82,72],{"separator":71},[66,84,85],{},"3",[58,87,72],{"separator":71},[58,89,90],{},"…",[92,93,94],"mtext",{}," ",[58,96,97],{"stretchy":63},"}",[99,100,102],"annotation",{"encoding":101},"application\u002Fx-tex","\\mathbb{N} = \\{0, 1, 2, 3, \\dots\\}",[33,104,107,133],{"className":105,"ariaHidden":71},[106],"katex-html",[33,108,111,116,121,126,130],{"className":109},[110],"base",[33,112],{"className":113,"style":115},[114],"strut","height:0.6889em;",[33,117,56],{"className":118},[119,120],"mord","mathbb",[33,122],{"className":123,"style":125},[124],"mspace","margin-right:0.2778em;",[33,127,60],{"className":128},[129],"mrel",[33,131],{"className":132,"style":125},[124],[33,134,136,140,144,147,151,155,158,161,164,167,170,173,176,179,182,186,189],{"className":135},[110],[33,137],{"className":138,"style":139},[114],"height:1em;vertical-align:-0.25em;",[33,141,64],{"className":142},[143],"mopen",[33,145,68],{"className":146},[119],[33,148,72],{"className":149},[150],"mpunct",[33,152],{"className":153,"style":154},[124],"margin-right:0.1667em;",[33,156,75],{"className":157},[119],[33,159,72],{"className":160},[150],[33,162],{"className":163,"style":154},[124],[33,165,80],{"className":166},[119],[33,168,72],{"className":169},[150],[33,171],{"className":172,"style":154},[124],[33,174,85],{"className":175},[119],[33,177,72],{"className":178},[150],[33,180],{"className":181,"style":154},[124],[33,183,90],{"className":184},[185],"minner",[33,187],{"className":188,"style":154},[124],[33,190,97],{"className":191},[192],"mclose"," is infinite and cannot be \"counted to completion\"; we therefore require a new instrument.",[10,195,196,199,200,231,232,262,263,341],{},[14,197,198],{},"Definition (bijection)."," Given two sets ",[33,201,203,217],{"className":202,"translate":37},[36],[33,204,206],{"className":205},[41],[43,207,208],{"xmlns":45},[47,209,210,215],{},[50,211,212],{},[53,213,214],{},"A",[99,216,214],{"encoding":101},[33,218,220],{"className":219,"ariaHidden":71},[106],[33,221,223,227],{"className":222},[110],[33,224],{"className":225,"style":226},[114],"height:0.6833em;",[33,228,214],{"className":229},[119,230],"mathnormal"," and ",[33,233,235,249],{"className":234,"translate":37},[36],[33,236,238],{"className":237},[41],[43,239,240],{"xmlns":45},[47,241,242,247],{},[50,243,244],{},[53,245,246],{},"B",[99,248,246],{"encoding":101},[33,250,252],{"className":251,"ariaHidden":71},[106],[33,253,255,258],{"className":254},[110],[33,256],{"className":257,"style":226},[114],[33,259,246],{"className":260,"style":261},[119,230],"margin-right:0.0502em;",", if there exists a function ",[33,264,266,291],{"className":265,"translate":37},[36],[33,267,269],{"className":268},[41],[43,270,271],{"xmlns":45},[47,272,273,288],{},[50,274,275,278,281,283,286],{},[53,276,277],{},"f",[58,279,280],{},":",[53,282,214],{},[58,284,285],{},"→",[53,287,246],{},[99,289,290],{"encoding":101},"f: A \\to B",[33,292,294,314,332],{"className":293,"ariaHidden":71},[106],[33,295,297,301,305,308,311],{"className":296},[110],[33,298],{"className":299,"style":300},[114],"height:0.8889em;vertical-align:-0.1944em;",[33,302,277],{"className":303,"style":304},[119,230],"margin-right:0.1076em;",[33,306],{"className":307,"style":125},[124],[33,309,280],{"className":310},[129],[33,312],{"className":313,"style":125},[124],[33,315,317,320,323,326,329],{"className":316},[110],[33,318],{"className":319,"style":226},[114],[33,321,214],{"className":322},[119,230],[33,324],{"className":325,"style":125},[124],[33,327,285],{"className":328},[129],[33,330],{"className":331,"style":125},[124],[33,333,335,338],{"className":334},[110],[33,336],{"className":337,"style":226},[114],[33,339,246],{"className":340,"style":261},[119,230]," satisfying:",[343,344,345,739],"ul",{},[346,347,348,351,352,738],"li",{},[14,349,350],{},"Injective",": ",[33,353,355,416],{"className":354,"translate":37},[36],[33,356,358],{"className":357},[41],[43,359,360],{"xmlns":45},[47,361,362,413],{},[50,363,364,372,376,382,385,387,390,396,399,401,403,405,411],{},[365,366,367,370],"msub",{},[53,368,369],{},"a",[66,371,75],{},[58,373,375],{"mathvariant":374},"normal","≠",[365,377,378,380],{},[53,379,369],{},[66,381,80],{},[58,383,384],{},"⇒",[53,386,277],{},[58,388,389],{"stretchy":63},"(",[365,391,392,394],{},[53,393,369],{},[66,395,75],{},[58,397,398],{"stretchy":63},")",[58,400,375],{"mathvariant":374},[53,402,277],{},[58,404,389],{"stretchy":63},[365,406,407,409],{},[53,408,369],{},[66,410,80],{},[58,412,398],{"stretchy":63},[99,414,415],{"encoding":101},"a_1 \\neq a_2 \\Rightarrow f(a_1) \\neq f(a_2)",[33,417,419,530,586,683],{"className":418,"ariaHidden":71},[106],[33,420,422,425,481,484,527],{"className":421},[110],[33,423],{"className":424,"style":300},[114],[33,426,428,431],{"className":427},[119],[33,429,369],{"className":430},[119,230],[33,432,435],{"className":433},[434],"msupsub",[33,436,440,472],{"className":437},[438,439],"vlist-t","vlist-t2",[33,441,444,467],{"className":442},[443],"vlist-r",[33,445,449],{"className":446,"style":448},[447],"vlist","height:0.3011em;",[33,450,452,457],{"style":451},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[33,453],{"className":454,"style":456},[455],"pstrut","height:2.7em;",[33,458,464],{"className":459},[460,461,462,463],"sizing","reset-size6","size3","mtight",[33,465,75],{"className":466},[119,463],[33,468,471],{"className":469},[470],"vlist-s","​",[33,473,475],{"className":474},[443],[33,476,479],{"className":477,"style":478},[447],"height:0.15em;",[33,480],{},[33,482],{"className":483,"style":125},[124],[33,485,487,520,524],{"className":486},[129],[33,488,490],{"className":489},[129],[33,491,494],{"className":492},[119,493],"vbox",[33,495,498],{"className":496},[497],"thinbox",[33,499,502,505,516],{"className":500},[501],"rlap",[33,503],{"className":504,"style":300},[114],[33,506,509],{"className":507},[508],"inner",[33,510,512],{"className":511},[119],[33,513,515],{"className":514},[129],"",[33,517],{"className":518},[519],"fix",[33,521],{"className":522},[124,523],"nobreak",[33,525,60],{"className":526},[129],[33,528],{"className":529,"style":125},[124],[33,531,533,537,577,580,583],{"className":532},[110],[33,534],{"className":535,"style":536},[114],"height:0.5806em;vertical-align:-0.15em;",[33,538,540,543],{"className":539},[119],[33,541,369],{"className":542},[119,230],[33,544,546],{"className":545},[434],[33,547,549,569],{"className":548},[438,439],[33,550,552,566],{"className":551},[443],[33,553,555],{"className":554,"style":448},[447],[33,556,557,560],{"style":451},[33,558],{"className":559,"style":456},[455],[33,561,563],{"className":562},[460,461,462,463],[33,564,80],{"className":565},[119,463],[33,567,471],{"className":568},[470],[33,570,572],{"className":571},[443],[33,573,575],{"className":574,"style":478},[447],[33,576],{},[33,578],{"className":579,"style":125},[124],[33,581,384],{"className":582},[129],[33,584],{"className":585,"style":125},[124],[33,587,589,592,595,598,638,641,644,680],{"className":588},[110],[33,590],{"className":591,"style":139},[114],[33,593,277],{"className":594,"style":304},[119,230],[33,596,389],{"className":597},[143],[33,599,601,604],{"className":600},[119],[33,602,369],{"className":603},[119,230],[33,605,607],{"className":606},[434],[33,608,610,630],{"className":609},[438,439],[33,611,613,627],{"className":612},[443],[33,614,616],{"className":615,"style":448},[447],[33,617,618,621],{"style":451},[33,619],{"className":620,"style":456},[455],[33,622,624],{"className":623},[460,461,462,463],[33,625,75],{"className":626},[119,463],[33,628,471],{"className":629},[470],[33,631,633],{"className":632},[443],[33,634,636],{"className":635,"style":478},[447],[33,637],{},[33,639,398],{"className":640},[192],[33,642],{"className":643,"style":125},[124],[33,645,647,674,677],{"className":646},[129],[33,648,650],{"className":649},[129],[33,651,653],{"className":652},[119,493],[33,654,656],{"className":655},[497],[33,657,659,662,671],{"className":658},[501],[33,660],{"className":661,"style":300},[114],[33,663,665],{"className":664},[508],[33,666,668],{"className":667},[119],[33,669,515],{"className":670},[129],[33,672],{"className":673},[519],[33,675],{"className":676},[124,523],[33,678,60],{"className":679},[129],[33,681],{"className":682,"style":125},[124],[33,684,686,689,692,695,735],{"className":685},[110],[33,687],{"className":688,"style":139},[114],[33,690,277],{"className":691,"style":304},[119,230],[33,693,389],{"className":694},[143],[33,696,698,701],{"className":697},[119],[33,699,369],{"className":700},[119,230],[33,702,704],{"className":703},[434],[33,705,707,727],{"className":706},[438,439],[33,708,710,724],{"className":709},[443],[33,711,713],{"className":712,"style":448},[447],[33,714,715,718],{"style":451},[33,716],{"className":717,"style":456},[455],[33,719,721],{"className":720},[460,461,462,463],[33,722,80],{"className":723},[119,463],[33,725,471],{"className":726},[470],[33,728,730],{"className":729},[443],[33,731,733],{"className":732,"style":478},[447],[33,734],{},[33,736,398],{"className":737},[192]," (distinct inputs yield distinct outputs)",[346,740,741,744,745,799,800,852,853,920],{},[14,742,743],{},"Surjective",": for every ",[33,746,748,768],{"className":747,"translate":37},[36],[33,749,751],{"className":750},[41],[43,752,753],{"xmlns":45},[47,754,755,765],{},[50,756,757,760,763],{},[53,758,759],{},"b",[58,761,762],{},"∈",[53,764,246],{},[99,766,767],{"encoding":101},"b \\in B",[33,769,771,790],{"className":770,"ariaHidden":71},[106],[33,772,774,778,781,784,787],{"className":773},[110],[33,775],{"className":776,"style":777},[114],"height:0.7335em;vertical-align:-0.0391em;",[33,779,759],{"className":780},[119,230],[33,782],{"className":783,"style":125},[124],[33,785,762],{"className":786},[129],[33,788],{"className":789,"style":125},[124],[33,791,793,796],{"className":792},[110],[33,794],{"className":795,"style":226},[114],[33,797,246],{"className":798,"style":261},[119,230],", there exists ",[33,801,803,821],{"className":802,"translate":37},[36],[33,804,806],{"className":805},[41],[43,807,808],{"xmlns":45},[47,809,810,818],{},[50,811,812,814,816],{},[53,813,369],{},[58,815,762],{},[53,817,214],{},[99,819,820],{"encoding":101},"a \\in A",[33,822,824,843],{"className":823,"ariaHidden":71},[106],[33,825,827,831,834,837,840],{"className":826},[110],[33,828],{"className":829,"style":830},[114],"height:0.5782em;vertical-align:-0.0391em;",[33,832,369],{"className":833},[119,230],[33,835],{"className":836,"style":125},[124],[33,838,762],{"className":839},[129],[33,841],{"className":842,"style":125},[124],[33,844,846,849],{"className":845},[110],[33,847],{"className":848,"style":226},[114],[33,850,214],{"className":851},[119,230]," such that ",[33,854,856,880],{"className":855,"translate":37},[36],[33,857,859],{"className":858},[41],[43,860,861],{"xmlns":45},[47,862,863,877],{},[50,864,865,867,869,871,873,875],{},[53,866,277],{},[58,868,389],{"stretchy":63},[53,870,369],{},[58,872,398],{"stretchy":63},[58,874,60],{},[53,876,759],{},[99,878,879],{"encoding":101},"f(a) = b",[33,881,883,910],{"className":882,"ariaHidden":71},[106],[33,884,886,889,892,895,898,901,904,907],{"className":885},[110],[33,887],{"className":888,"style":139},[114],[33,890,277],{"className":891,"style":304},[119,230],[33,893,389],{"className":894},[143],[33,896,369],{"className":897},[119,230],[33,899,398],{"className":900},[192],[33,902],{"className":903,"style":125},[124],[33,905,60],{"className":906},[129],[33,908],{"className":909,"style":125},[124],[33,911,913,917],{"className":912},[110],[33,914],{"className":915,"style":916},[114],"height:0.6944em;",[33,918,759],{"className":919},[119,230]," (every output is covered)",[10,922,923,924,952,953,231,981,1009,1010,1013,1014,1086],{},"then ",[33,925,927,940],{"className":926,"translate":37},[36],[33,928,930],{"className":929},[41],[43,931,932],{"xmlns":45},[47,933,934,938],{},[50,935,936],{},[53,937,277],{},[99,939,277],{"encoding":101},[33,941,943],{"className":942,"ariaHidden":71},[106],[33,944,946,949],{"className":945},[110],[33,947],{"className":948,"style":300},[114],[33,950,277],{"className":951,"style":304},[119,230]," is a bijection, and we say that ",[33,954,956,969],{"className":955,"translate":37},[36],[33,957,959],{"className":958},[41],[43,960,961],{"xmlns":45},[47,962,963,967],{},[50,964,965],{},[53,966,214],{},[99,968,214],{"encoding":101},[33,970,972],{"className":971,"ariaHidden":71},[106],[33,973,975,978],{"className":974},[110],[33,976],{"className":977,"style":226},[114],[33,979,214],{"className":980},[119,230],[33,982,984,997],{"className":983,"translate":37},[36],[33,985,987],{"className":986},[41],[43,988,989],{"xmlns":45},[47,990,991,995],{},[50,992,993],{},[53,994,246],{},[99,996,246],{"encoding":101},[33,998,1000],{"className":999,"ariaHidden":71},[106],[33,1001,1003,1006],{"className":1002},[110],[33,1004],{"className":1005,"style":226},[114],[33,1007,246],{"className":1008,"style":261},[119,230]," are ",[14,1011,1012],{},"equinumerous"," (denoted ",[33,1015,1017,1044],{"className":1016,"translate":37},[36],[33,1018,1020],{"className":1019},[41],[43,1021,1022],{"xmlns":45},[47,1023,1024,1041],{},[50,1025,1026,1029,1031,1033,1035,1037,1039],{},[53,1027,1028],{"mathvariant":374},"∣",[53,1030,214],{},[53,1032,1028],{"mathvariant":374},[58,1034,60],{},[53,1036,1028],{"mathvariant":374},[53,1038,246],{},[53,1040,1028],{"mathvariant":374},[99,1042,1043],{"encoding":101},"|A| = |B|",[33,1045,1047,1071],{"className":1046,"ariaHidden":71},[106],[33,1048,1050,1053,1056,1059,1062,1065,1068],{"className":1049},[110],[33,1051],{"className":1052,"style":139},[114],[33,1054,1028],{"className":1055},[119],[33,1057,214],{"className":1058},[119,230],[33,1060,1028],{"className":1061},[119],[33,1063],{"className":1064,"style":125},[124],[33,1066,60],{"className":1067},[129],[33,1069],{"className":1070,"style":125},[124],[33,1072,1074,1077,1080,1083],{"className":1073},[110],[33,1075],{"className":1076,"style":139},[114],[33,1078,1028],{"className":1079},[119],[33,1081,246],{"className":1082,"style":261},[119,230],[33,1084,1028],{"className":1085},[119],") — that is, they are \"equal in size.\"",[10,1088,1089,1092,1093,1121,1122,1151,1152,1180],{},[14,1090,1091],{},"Definition (countable set)."," A set ",[33,1094,1096,1109],{"className":1095,"translate":37},[36],[33,1097,1099],{"className":1098},[41],[43,1100,1101],{"xmlns":45},[47,1102,1103,1107],{},[50,1104,1105],{},[53,1106,214],{},[99,1108,214],{"encoding":101},[33,1110,1112],{"className":1111,"ariaHidden":71},[106],[33,1113,1115,1118],{"className":1114},[110],[33,1116],{"className":1117,"style":226},[114],[33,1119,214],{"className":1120},[119,230]," is said to be countable if it can be placed in bijection with the natural numbers ",[33,1123,1125,1139],{"className":1124,"translate":37},[36],[33,1126,1128],{"className":1127},[41],[43,1129,1130],{"xmlns":45},[47,1131,1132,1136],{},[50,1133,1134],{},[53,1135,56],{"mathvariant":55},[99,1137,1138],{"encoding":101},"\\mathbb{N}",[33,1140,1142],{"className":1141,"ariaHidden":71},[106],[33,1143,1145,1148],{"className":1144},[110],[33,1146],{"className":1147,"style":115},[114],[33,1149,56],{"className":1150},[119,120]," (or if ",[33,1153,1155,1168],{"className":1154,"translate":37},[36],[33,1156,1158],{"className":1157},[41],[43,1159,1160],{"xmlns":45},[47,1161,1162,1166],{},[50,1163,1164],{},[53,1165,214],{},[99,1167,214],{"encoding":101},[33,1169,1171],{"className":1170,"ariaHidden":71},[106],[33,1172,1174,1177],{"className":1173},[110],[33,1175],{"className":1176,"style":226},[114],[33,1178,214],{"className":1179},[119,230]," is finite).",[10,1182,1183],{},"A few examples will reveal that the concept of \"countable\" is considerably more permissive than intuition would suggest:",[343,1185,1186,1365,2035],{},[346,1187,1188,1191,1192,1290,1291,1364],{},[14,1189,1190],{},"The even numbers"," ",[33,1193,1195,1233],{"className":1194,"translate":37},[36],[33,1196,1198],{"className":1197},[41],[43,1199,1200],{"xmlns":45},[47,1201,1202,1230],{},[50,1203,1204,1206,1208,1210,1212,1214,1217,1219,1222,1224,1226,1228],{},[58,1205,64],{"stretchy":63},[66,1207,68],{},[58,1209,72],{"separator":71},[66,1211,80],{},[58,1213,72],{"separator":71},[66,1215,1216],{},"4",[58,1218,72],{"separator":71},[66,1220,1221],{},"6",[58,1223,72],{"separator":71},[58,1225,90],{},[92,1227,94],{},[58,1229,97],{"stretchy":63},[99,1231,1232],{"encoding":101},"\\{0, 2, 4, 6, \\dots\\}",[33,1234,1236],{"className":1235,"ariaHidden":71},[106],[33,1237,1239,1242,1245,1248,1251,1254,1257,1260,1263,1266,1269,1272,1275,1278,1281,1284,1287],{"className":1238},[110],[33,1240],{"className":1241,"style":139},[114],[33,1243,64],{"className":1244},[143],[33,1246,68],{"className":1247},[119],[33,1249,72],{"className":1250},[150],[33,1252],{"className":1253,"style":154},[124],[33,1255,80],{"className":1256},[119],[33,1258,72],{"className":1259},[150],[33,1261],{"className":1262,"style":154},[124],[33,1264,1216],{"className":1265},[119],[33,1267,72],{"className":1268},[150],[33,1270],{"className":1271,"style":154},[124],[33,1273,1221],{"className":1274},[119],[33,1276,72],{"className":1277},[150],[33,1279],{"className":1280,"style":154},[124],[33,1282,90],{"className":1283},[185],[33,1285],{"className":1286,"style":154},[124],[33,1288,97],{"className":1289},[192]," are countable: the bijection is ",[33,1292,1294,1321],{"className":1293,"translate":37},[36],[33,1295,1297],{"className":1296},[41],[43,1298,1299],{"xmlns":45},[47,1300,1301,1318],{},[50,1302,1303,1305,1307,1310,1312,1314,1316],{},[53,1304,277],{},[58,1306,389],{"stretchy":63},[53,1308,1309],{},"n",[58,1311,398],{"stretchy":63},[58,1313,60],{},[66,1315,80],{},[53,1317,1309],{},[99,1319,1320],{"encoding":101},"f(n) = 2n",[33,1322,1324,1351],{"className":1323,"ariaHidden":71},[106],[33,1325,1327,1330,1333,1336,1339,1342,1345,1348],{"className":1326},[110],[33,1328],{"className":1329,"style":139},[114],[33,1331,277],{"className":1332,"style":304},[119,230],[33,1334,389],{"className":1335},[143],[33,1337,1309],{"className":1338},[119,230],[33,1340,398],{"className":1341},[192],[33,1343],{"className":1344,"style":125},[124],[33,1346,60],{"className":1347},[129],[33,1349],{"className":1350,"style":125},[124],[33,1352,1354,1358,1361],{"className":1353},[110],[33,1355],{"className":1356,"style":1357},[114],"height:0.6444em;",[33,1359,80],{"className":1360},[119],[33,1362,1309],{"className":1363},[119,230],".",[346,1366,1367,1191,1370,1529,1530,1834,1835,1930,1931,1364],{},[14,1368,1369],{},"The integers",[33,1371,1373,1427],{"className":1372,"translate":37},[36],[33,1374,1376],{"className":1375},[41],[43,1377,1378],{"xmlns":45},[47,1379,1380,1424],{},[50,1381,1382,1385,1387,1389,1391,1393,1396,1398,1400,1402,1404,1406,1408,1410,1412,1414,1416,1418,1420,1422],{},[53,1383,1384],{"mathvariant":55},"Z",[58,1386,60],{},[58,1388,64],{"stretchy":63},[58,1390,90],{},[58,1392,72],{"separator":71},[58,1394,1395],{},"−",[66,1397,80],{},[58,1399,72],{"separator":71},[58,1401,1395],{},[66,1403,75],{},[58,1405,72],{"separator":71},[66,1407,68],{},[58,1409,72],{"separator":71},[66,1411,75],{},[58,1413,72],{"separator":71},[66,1415,80],{},[58,1417,72],{"separator":71},[58,1419,90],{},[92,1421,94],{},[58,1423,97],{"stretchy":63},[99,1425,1426],{"encoding":101},"\\mathbb{Z} = \\{\\dots, -2, -1, 0, 1, 2, \\dots\\}",[33,1428,1430,1448],{"className":1429,"ariaHidden":71},[106],[33,1431,1433,1436,1439,1442,1445],{"className":1432},[110],[33,1434],{"className":1435,"style":115},[114],[33,1437,1384],{"className":1438},[119,120],[33,1440],{"className":1441,"style":125},[124],[33,1443,60],{"className":1444},[129],[33,1446],{"className":1447,"style":125},[124],[33,1449,1451,1454,1457,1460,1463,1466,1469,1472,1475,1478,1481,1484,1487,1490,1493,1496,1499,1502,1505,1508,1511,1514,1517,1520,1523,1526],{"className":1450},[110],[33,1452],{"className":1453,"style":139},[114],[33,1455,64],{"className":1456},[143],[33,1458,90],{"className":1459},[185],[33,1461],{"className":1462,"style":154},[124],[33,1464,72],{"className":1465},[150],[33,1467],{"className":1468,"style":154},[124],[33,1470,1395],{"className":1471},[119],[33,1473,80],{"className":1474},[119],[33,1476,72],{"className":1477},[150],[33,1479],{"className":1480,"style":154},[124],[33,1482,1395],{"className":1483},[119],[33,1485,75],{"className":1486},[119],[33,1488,72],{"className":1489},[150],[33,1491],{"className":1492,"style":154},[124],[33,1494,68],{"className":1495},[119],[33,1497,72],{"className":1498},[150],[33,1500],{"className":1501,"style":154},[124],[33,1503,75],{"className":1504},[119],[33,1506,72],{"className":1507},[150],[33,1509],{"className":1510,"style":154},[124],[33,1512,80],{"className":1513},[119],[33,1515,72],{"className":1516},[150],[33,1518],{"className":1519,"style":154},[124],[33,1521,90],{"className":1522},[185],[33,1524],{"className":1525,"style":154},[124],[33,1527,97],{"className":1528},[192]," are countable: one may use ",[33,1531,1533,1630],{"className":1532,"translate":37},[36],[33,1534,1536],{"className":1535},[41],[43,1537,1538],{"xmlns":45},[47,1539,1540,1627],{},[50,1541,1542,1544,1546,1548,1550,1552],{},[53,1543,277],{},[58,1545,389],{"stretchy":63},[53,1547,1309],{},[58,1549,398],{"stretchy":63},[58,1551,60],{},[50,1553,1554,1556],{},[58,1555,64],{"fence":71},[1557,1558,1562,1591],"mtable",{"rowspacing":1559,"columnalign":1560,"columnspacing":1561},"0.36em","left left","1em",[1563,1564,1565,1580],"mtr",{},[1566,1567,1568],"mtd",{},[1569,1570,1571],"mstyle",{"scriptlevel":68,"displaystyle":63},[50,1572,1573,1575,1578],{},[53,1574,1309],{},[53,1576,1577],{"mathvariant":374},"\u002F",[66,1579,80],{},[1566,1581,1582],{},[1569,1583,1584],{"scriptlevel":68,"displaystyle":63},[50,1585,1586,1588],{},[53,1587,1309],{},[92,1589,1590],{}," even",[1563,1592,1593,1616],{},[1566,1594,1595],{},[1569,1596,1597],{"scriptlevel":68,"displaystyle":63},[50,1598,1599,1601,1603,1605,1608,1610,1612,1614],{},[58,1600,1395],{},[58,1602,389],{"stretchy":63},[53,1604,1309],{},[58,1606,1607],{},"+",[66,1609,75],{},[58,1611,398],{"stretchy":63},[53,1613,1577],{"mathvariant":374},[66,1615,80],{},[1566,1617,1618],{},[1569,1619,1620],{"scriptlevel":68,"displaystyle":63},[50,1621,1622,1624],{},[53,1623,1309],{},[92,1625,1626],{}," odd",[99,1628,1629],{"encoding":101},"f(n) = \\begin{cases} n\u002F2 & n \\text{ even} \\\\ -(n+1)\u002F2 & n \\text{ odd} \\end{cases}",[33,1631,1633,1660],{"className":1632,"ariaHidden":71},[106],[33,1634,1636,1639,1642,1645,1648,1651,1654,1657],{"className":1635},[110],[33,1637],{"className":1638,"style":139},[114],[33,1640,277],{"className":1641,"style":304},[119,230],[33,1643,389],{"className":1644},[143],[33,1646,1309],{"className":1647},[119,230],[33,1649,398],{"className":1650},[192],[33,1652],{"className":1653,"style":125},[124],[33,1655,60],{"className":1656},[129],[33,1658],{"className":1659,"style":125},[124],[33,1661,1663,1667],{"className":1662},[110],[33,1664],{"className":1665,"style":1666},[114],"height:3em;vertical-align:-1.25em;",[33,1668,1670,1680,1830],{"className":1669},[185],[33,1671,1675],{"className":1672,"style":1674},[143,1673],"delimcenter","top:0em;",[33,1676,64],{"className":1677},[1678,1679],"delimsizing","size4",[33,1681,1683],{"className":1682},[119],[33,1684,1686,1767,1772],{"className":1685},[1557],[33,1687,1690],{"className":1688},[1689],"col-align-l",[33,1691,1693,1758],{"className":1692},[438,439],[33,1694,1696,1755],{"className":1695},[443],[33,1697,1700,1717],{"className":1698,"style":1699},[447],"height:1.69em;",[33,1701,1703,1707],{"style":1702},"top:-3.69em;",[33,1704],{"className":1705,"style":1706},[455],"height:3.008em;",[33,1708,1710,1713],{"className":1709},[119],[33,1711,1309],{"className":1712},[119,230],[33,1714,1716],{"className":1715},[119],"\u002F2",[33,1718,1720,1723],{"style":1719},"top:-2.25em;",[33,1721],{"className":1722,"style":1706},[455],[33,1724,1726,1729,1732,1735,1739,1743,1746,1749,1752],{"className":1725},[119],[33,1727,1395],{"className":1728},[119],[33,1730,389],{"className":1731},[143],[33,1733,1309],{"className":1734},[119,230],[33,1736],{"className":1737,"style":1738},[124],"margin-right:0.2222em;",[33,1740,1607],{"className":1741},[1742],"mbin",[33,1744],{"className":1745,"style":1738},[124],[33,1747,75],{"className":1748},[119],[33,1750,398],{"className":1751},[192],[33,1753,1716],{"className":1754},[119],[33,1756,471],{"className":1757},[470],[33,1759,1761],{"className":1760},[443],[33,1762,1765],{"className":1763,"style":1764},[447],"height:1.19em;",[33,1766],{},[33,1768],{"className":1769,"style":1771},[1770],"arraycolsep","width:1em;",[33,1773,1775],{"className":1774},[1689],[33,1776,1778,1822],{"className":1777},[438,439],[33,1779,1781,1819],{"className":1780},[443],[33,1782,1784,1802],{"className":1783,"style":1699},[447],[33,1785,1786,1789],{"style":1702},[33,1787],{"className":1788,"style":1706},[455],[33,1790,1792,1795],{"className":1791},[119],[33,1793,1309],{"className":1794},[119,230],[33,1796,1799],{"className":1797},[119,1798],"text",[33,1800,1590],{"className":1801},[119],[33,1803,1804,1807],{"style":1719},[33,1805],{"className":1806,"style":1706},[455],[33,1808,1810,1813],{"className":1809},[119],[33,1811,1309],{"className":1812},[119,230],[33,1814,1816],{"className":1815},[119,1798],[33,1817,1626],{"className":1818},[119],[33,1820,471],{"className":1821},[470],[33,1823,1825],{"className":1824},[443],[33,1826,1828],{"className":1827,"style":1764},[447],[33,1829],{},[33,1831],{"className":1832},[192,1833],"nulldelimiter"," to map ",[33,1836,1838,1872],{"className":1837,"translate":37},[36],[33,1839,1841],{"className":1840},[41],[43,1842,1843],{"xmlns":45},[47,1844,1845,1869],{},[50,1846,1847,1849,1851,1853,1855,1857,1859,1861,1863,1865,1867],{},[66,1848,68],{},[58,1850,72],{"separator":71},[66,1852,75],{},[58,1854,72],{"separator":71},[66,1856,80],{},[58,1858,72],{"separator":71},[66,1860,85],{},[58,1862,72],{"separator":71},[66,1864,1216],{},[58,1866,72],{"separator":71},[58,1868,90],{},[99,1870,1871],{"encoding":101},"0,1,2,3,4,\\dots",[33,1873,1875],{"className":1874,"ariaHidden":71},[106],[33,1876,1878,1882,1885,1888,1891,1894,1897,1900,1903,1906,1909,1912,1915,1918,1921,1924,1927],{"className":1877},[110],[33,1879],{"className":1880,"style":1881},[114],"height:0.8389em;vertical-align:-0.1944em;",[33,1883,68],{"className":1884},[119],[33,1886,72],{"className":1887},[150],[33,1889],{"className":1890,"style":154},[124],[33,1892,75],{"className":1893},[119],[33,1895,72],{"className":1896},[150],[33,1898],{"className":1899,"style":154},[124],[33,1901,80],{"className":1902},[119],[33,1904,72],{"className":1905},[150],[33,1907],{"className":1908,"style":154},[124],[33,1910,85],{"className":1911},[119],[33,1913,72],{"className":1914},[150],[33,1916],{"className":1917,"style":154},[124],[33,1919,1216],{"className":1920},[119],[33,1922,72],{"className":1923},[150],[33,1925],{"className":1926,"style":154},[124],[33,1928,90],{"className":1929},[185]," onto ",[33,1932,1934,1972],{"className":1933,"translate":37},[36],[33,1935,1937],{"className":1936},[41],[43,1938,1939],{"xmlns":45},[47,1940,1941,1969],{},[50,1942,1943,1945,1947,1949,1951,1953,1955,1957,1959,1961,1963,1965,1967],{},[66,1944,68],{},[58,1946,72],{"separator":71},[58,1948,1395],{},[66,1950,75],{},[58,1952,72],{"separator":71},[66,1954,75],{},[58,1956,72],{"separator":71},[58,1958,1395],{},[66,1960,80],{},[58,1962,72],{"separator":71},[66,1964,80],{},[58,1966,72],{"separator":71},[58,1968,90],{},[99,1970,1971],{"encoding":101},"0,-1,1,-2,2,\\dots",[33,1973,1975],{"className":1974,"ariaHidden":71},[106],[33,1976,1978,1981,1984,1987,1990,1993,1996,1999,2002,2005,2008,2011,2014,2017,2020,2023,2026,2029,2032],{"className":1977},[110],[33,1979],{"className":1980,"style":1881},[114],[33,1982,68],{"className":1983},[119],[33,1985,72],{"className":1986},[150],[33,1988],{"className":1989,"style":154},[124],[33,1991,1395],{"className":1992},[119],[33,1994,75],{"className":1995},[119],[33,1997,72],{"className":1998},[150],[33,2000],{"className":2001,"style":154},[124],[33,2003,75],{"className":2004},[119],[33,2006,72],{"className":2007},[150],[33,2009],{"className":2010,"style":154},[124],[33,2012,1395],{"className":2013},[119],[33,2015,80],{"className":2016},[119],[33,2018,72],{"className":2019},[150],[33,2021],{"className":2022,"style":154},[124],[33,2024,80],{"className":2025},[119],[33,2027,72],{"className":2028},[150],[33,2030],{"className":2031,"style":154},[124],[33,2033,90],{"className":2034},[185],[346,2036,2037,1191,2040,2071,2072,2113],{},[14,2038,2039],{},"The rational numbers",[33,2041,2043,2058],{"className":2042,"translate":37},[36],[33,2044,2046],{"className":2045},[41],[43,2047,2048],{"xmlns":45},[47,2049,2050,2055],{},[50,2051,2052],{},[53,2053,2054],{"mathvariant":55},"Q",[99,2056,2057],{"encoding":101},"\\mathbb{Q}",[33,2059,2061],{"className":2060,"ariaHidden":71},[106],[33,2062,2064,2068],{"className":2063},[110],[33,2065],{"className":2066,"style":2067},[114],"height:0.8556em;vertical-align:-0.1667em;",[33,2069,2054],{"className":2070},[119,120]," are likewise countable (one may employ the \"diagonal enumeration method\" to arrange all fractions ",[33,2073,2075,2094],{"className":2074,"translate":37},[36],[33,2076,2078],{"className":2077},[41],[43,2079,2080],{"xmlns":45},[47,2081,2082,2091],{},[50,2083,2084,2086,2088],{},[53,2085,10],{},[53,2087,1577],{"mathvariant":374},[53,2089,2090],{},"q",[99,2092,2093],{"encoding":101},"p\u002Fq",[33,2095,2097],{"className":2096,"ariaHidden":71},[106],[33,2098,2100,2103,2106,2109],{"className":2099},[110],[33,2101],{"className":2102,"style":139},[114],[33,2104,10],{"className":2105},[119,230],[33,2107,1577],{"className":2108},[119],[33,2110,2090],{"className":2111,"style":2112},[119,230],"margin-right:0.0359em;"," 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).",[10,2115,2116],{},"It would appear that all infinite sets can be accommodated within the framework of the natural numbers — until we encounter the real numbers.",[22,2118,2120],{"id":2119},"the-uncountability-of-the-real-numbers","The Uncountability of the Real Numbers",[2122,2123,2124],"blockquote",{},[10,2125,2126,2129,2130,2182],{},[14,2127,2128],{},"Theorem (Cantor, 1891)."," The set of real numbers in the interval ",[33,2131,2133,2155],{"className":2132,"translate":37},[36],[33,2134,2136],{"className":2135},[41],[43,2137,2138],{"xmlns":45},[47,2139,2140,2152],{},[50,2141,2142,2144,2146,2148,2150],{},[58,2143,389],{"stretchy":63},[66,2145,68],{},[58,2147,72],{"separator":71},[66,2149,75],{},[58,2151,398],{"stretchy":63},[99,2153,2154],{"encoding":101},"(0,1)",[33,2156,2158],{"className":2157,"ariaHidden":71},[106],[33,2159,2161,2164,2167,2170,2173,2176,2179],{"className":2160},[110],[33,2162],{"className":2163,"style":139},[114],[33,2165,389],{"className":2166},[143],[33,2168,68],{"className":2169},[119],[33,2171,72],{"className":2172},[150],[33,2174],{"className":2175,"style":154},[124],[33,2177,75],{"className":2178},[119],[33,2180,398],{"className":2181},[192]," is uncountable.",[10,2184,2185,2186,2237],{},"This means: there exists no method whatsoever by which the real numbers in ",[33,2187,2189,2210],{"className":2188,"translate":37},[36],[33,2190,2192],{"className":2191},[41],[43,2193,2194],{"xmlns":45},[47,2195,2196,2208],{},[50,2197,2198,2200,2202,2204,2206],{},[58,2199,389],{"stretchy":63},[66,2201,68],{},[58,2203,72],{"separator":71},[66,2205,75],{},[58,2207,398],{"stretchy":63},[99,2209,2154],{"encoding":101},[33,2211,2213],{"className":2212,"ariaHidden":71},[106],[33,2214,2216,2219,2222,2225,2228,2231,2234],{"className":2215},[110],[33,2217],{"className":2218,"style":139},[114],[33,2220,389],{"className":2221},[143],[33,2223,68],{"className":2224},[119],[33,2226,72],{"className":2227},[150],[33,2229],{"className":2230,"style":154},[124],[33,2232,75],{"className":2233},[119],[33,2235,398],{"className":2236},[192]," 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.",[10,2239,2240,2241,1364],{},"We proceed by ",[2242,2243,2244],"em",{},"reductio ad absurdum",[10,2246,2247,2250,2251,2302,2303,2399],{},[14,2248,2249],{},"Assumption",": the real numbers in ",[33,2252,2254,2275],{"className":2253,"translate":37},[36],[33,2255,2257],{"className":2256},[41],[43,2258,2259],{"xmlns":45},[47,2260,2261,2273],{},[50,2262,2263,2265,2267,2269,2271],{},[58,2264,389],{"stretchy":63},[66,2266,68],{},[58,2268,72],{"separator":71},[66,2270,75],{},[58,2272,398],{"stretchy":63},[99,2274,2154],{"encoding":101},[33,2276,2278],{"className":2277,"ariaHidden":71},[106],[33,2279,2281,2284,2287,2290,2293,2296,2299],{"className":2280},[110],[33,2282],{"className":2283,"style":139},[114],[33,2285,389],{"className":2286},[143],[33,2288,68],{"className":2289},[119],[33,2291,72],{"className":2292},[150],[33,2294],{"className":2295,"style":154},[124],[33,2297,75],{"className":2298},[119],[33,2300,398],{"className":2301},[192]," are countable. That is, there exists a bijection ",[33,2304,2306,2336],{"className":2305,"translate":37},[36],[33,2307,2309],{"className":2308},[41],[43,2310,2311],{"xmlns":45},[47,2312,2313,2333],{},[50,2314,2315,2317,2319,2321,2323,2325,2327,2329,2331],{},[53,2316,277],{},[58,2318,280],{},[53,2320,56],{"mathvariant":55},[58,2322,285],{},[58,2324,389],{"stretchy":63},[66,2326,68],{},[58,2328,72],{"separator":71},[66,2330,75],{},[58,2332,398],{"stretchy":63},[99,2334,2335],{"encoding":101},"f: \\mathbb{N} \\to (0,1)",[33,2337,2339,2357,2375],{"className":2338,"ariaHidden":71},[106],[33,2340,2342,2345,2348,2351,2354],{"className":2341},[110],[33,2343],{"className":2344,"style":300},[114],[33,2346,277],{"className":2347,"style":304},[119,230],[33,2349],{"className":2350,"style":125},[124],[33,2352,280],{"className":2353},[129],[33,2355],{"className":2356,"style":125},[124],[33,2358,2360,2363,2366,2369,2372],{"className":2359},[110],[33,2361],{"className":2362,"style":115},[114],[33,2364,56],{"className":2365},[119,120],[33,2367],{"className":2368,"style":125},[124],[33,2370,285],{"className":2371},[129],[33,2373],{"className":2374,"style":125},[124],[33,2376,2378,2381,2384,2387,2390,2393,2396],{"className":2377},[110],[33,2379],{"className":2380,"style":139},[114],[33,2382,389],{"className":2383},[143],[33,2385,68],{"className":2386},[119],[33,2388,72],{"className":2389},[150],[33,2391],{"className":2392,"style":154},[124],[33,2394,75],{"className":2395},[119],[33,2397,398],{"className":2398},[192],", permitting us to arrange all such real numbers into an infinite list:",[33,2401,2404],{"className":2402,"translate":37},[2403],"katex-display",[33,2405,2407,2697],{"className":2406,"translate":37},[36],[33,2408,2410],{"className":2409},[41],[43,2411,2413],{"xmlns":45,"display":2412},"block",[47,2414,2415,2694],{},[1557,2416,2420,2483,2543,2603,2663],{"rowspacing":2417,"columnalign":2418,"columnspacing":2419},"0.25em","right left","0em",[1563,2421,2422,2433],{},[1566,2423,2424],{},[1569,2425,2426],{"scriptlevel":68,"displaystyle":71},[365,2427,2428,2431],{},[53,2429,2430],{},"x",[66,2432,75],{},[1566,2434,2435],{},[1569,2436,2437],{"scriptlevel":68,"displaystyle":71},[50,2438,2439,2441,2443,2446,2454,2456,2463,2465,2472,2474,2481],{},[50,2440],{},[58,2442,60],{},[66,2444,2445],{},"0.",[365,2447,2448,2451],{},[53,2449,2450],{},"d",[66,2452,2453],{},"11",[92,2455,94],{},[365,2457,2458,2460],{},[53,2459,2450],{},[66,2461,2462],{},"12",[92,2464,94],{},[365,2466,2467,2469],{},[53,2468,2450],{},[66,2470,2471],{},"13",[92,2473,94],{},[365,2475,2476,2478],{},[53,2477,2450],{},[66,2479,2480],{},"14",[58,2482,90],{},[1563,2484,2485,2495],{},[1566,2486,2487],{},[1569,2488,2489],{"scriptlevel":68,"displaystyle":71},[365,2490,2491,2493],{},[53,2492,2430],{},[66,2494,80],{},[1566,2496,2497],{},[1569,2498,2499],{"scriptlevel":68,"displaystyle":71},[50,2500,2501,2503,2505,2507,2514,2516,2523,2525,2532,2534,2541],{},[50,2502],{},[58,2504,60],{},[66,2506,2445],{},[365,2508,2509,2511],{},[53,2510,2450],{},[66,2512,2513],{},"21",[92,2515,94],{},[365,2517,2518,2520],{},[53,2519,2450],{},[66,2521,2522],{},"22",[92,2524,94],{},[365,2526,2527,2529],{},[53,2528,2450],{},[66,2530,2531],{},"23",[92,2533,94],{},[365,2535,2536,2538],{},[53,2537,2450],{},[66,2539,2540],{},"24",[58,2542,90],{},[1563,2544,2545,2555],{},[1566,2546,2547],{},[1569,2548,2549],{"scriptlevel":68,"displaystyle":71},[365,2550,2551,2553],{},[53,2552,2430],{},[66,2554,85],{},[1566,2556,2557],{},[1569,2558,2559],{"scriptlevel":68,"displaystyle":71},[50,2560,2561,2563,2565,2567,2574,2576,2583,2585,2592,2594,2601],{},[50,2562],{},[58,2564,60],{},[66,2566,2445],{},[365,2568,2569,2571],{},[53,2570,2450],{},[66,2572,2573],{},"31",[92,2575,94],{},[365,2577,2578,2580],{},[53,2579,2450],{},[66,2581,2582],{},"32",[92,2584,94],{},[365,2586,2587,2589],{},[53,2588,2450],{},[66,2590,2591],{},"33",[92,2593,94],{},[365,2595,2596,2598],{},[53,2597,2450],{},[66,2599,2600],{},"34",[58,2602,90],{},[1563,2604,2605,2615],{},[1566,2606,2607],{},[1569,2608,2609],{"scriptlevel":68,"displaystyle":71},[365,2610,2611,2613],{},[53,2612,2430],{},[66,2614,1216],{},[1566,2616,2617],{},[1569,2618,2619],{"scriptlevel":68,"displaystyle":71},[50,2620,2621,2623,2625,2627,2634,2636,2643,2645,2652,2654,2661],{},[50,2622],{},[58,2624,60],{},[66,2626,2445],{},[365,2628,2629,2631],{},[53,2630,2450],{},[66,2632,2633],{},"41",[92,2635,94],{},[365,2637,2638,2640],{},[53,2639,2450],{},[66,2641,2642],{},"42",[92,2644,94],{},[365,2646,2647,2649],{},[53,2648,2450],{},[66,2650,2651],{},"43",[92,2653,94],{},[365,2655,2656,2658],{},[53,2657,2450],{},[66,2659,2660],{},"44",[58,2662,90],{},[1563,2664,2665,2671],{},[1566,2666,2667],{},[1569,2668,2669],{"scriptlevel":68,"displaystyle":71},[50,2670],{},[1566,2672,2673],{},[1569,2674,2675],{"scriptlevel":68,"displaystyle":71},[50,2676,2677,2679,2682],{},[50,2678],{},[92,2680,2681],{},"  ",[50,2683,2684,2687],{},[53,2685,2686],{"mathvariant":374},"⋮",[2688,2689,2690],"mpadded",{"height":2419,"voffset":2419},[124,2691],{"mathbackground":2692,"width":2419,"height":2693},"black","1.5em",[99,2695,2696],{"encoding":101},"\\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&\\ \\ \\vdots\n\\end{aligned}",[33,2698,2700],{"className":2699,"ariaHidden":71},[106],[33,2701,2703,2707],{"className":2702},[110],[33,2704],{"className":2705,"style":2706},[114],"height:7.86em;vertical-align:-3.68em;",[33,2708,2710],{"className":2709},[119],[33,2711,2713,2945],{"className":2712},[1557],[33,2714,2717],{"className":2715},[2716],"col-align-r",[33,2718,2720,2936],{"className":2719},[438,439],[33,2721,2723,2933],{"className":2722},[443],[33,2724,2727,2777,2826,2875,2924],{"className":2725,"style":2726},[447],"height:4.18em;",[33,2728,2730,2734],{"style":2729},"top:-6.84em;",[33,2731],{"className":2732,"style":2733},[455],"height:3.5em;",[33,2735,2737],{"className":2736},[119],[33,2738,2740,2743],{"className":2739},[119],[33,2741,2430],{"className":2742},[119,230],[33,2744,2746],{"className":2745},[434],[33,2747,2749,2769],{"className":2748},[438,439],[33,2750,2752,2766],{"className":2751},[443],[33,2753,2755],{"className":2754,"style":448},[447],[33,2756,2757,2760],{"style":451},[33,2758],{"className":2759,"style":456},[455],[33,2761,2763],{"className":2762},[460,461,462,463],[33,2764,75],{"className":2765},[119,463],[33,2767,471],{"className":2768},[470],[33,2770,2772],{"className":2771},[443],[33,2773,2775],{"className":2774,"style":478},[447],[33,2776],{},[33,2778,2780,2783],{"style":2779},"top:-5.34em;",[33,2781],{"className":2782,"style":2733},[455],[33,2784,2786],{"className":2785},[119],[33,2787,2789,2792],{"className":2788},[119],[33,2790,2430],{"className":2791},[119,230],[33,2793,2795],{"className":2794},[434],[33,2796,2798,2818],{"className":2797},[438,439],[33,2799,2801,2815],{"className":2800},[443],[33,2802,2804],{"className":2803,"style":448},[447],[33,2805,2806,2809],{"style":451},[33,2807],{"className":2808,"style":456},[455],[33,2810,2812],{"className":2811},[460,461,462,463],[33,2813,80],{"className":2814},[119,463],[33,2816,471],{"className":2817},[470],[33,2819,2821],{"className":2820},[443],[33,2822,2824],{"className":2823,"style":478},[447],[33,2825],{},[33,2827,2829,2832],{"style":2828},"top:-3.84em;",[33,2830],{"className":2831,"style":2733},[455],[33,2833,2835],{"className":2834},[119],[33,2836,2838,2841],{"className":2837},[119],[33,2839,2430],{"className":2840},[119,230],[33,2842,2844],{"className":2843},[434],[33,2845,2847,2867],{"className":2846},[438,439],[33,2848,2850,2864],{"className":2849},[443],[33,2851,2853],{"className":2852,"style":448},[447],[33,2854,2855,2858],{"style":451},[33,2856],{"className":2857,"style":456},[455],[33,2859,2861],{"className":2860},[460,461,462,463],[33,2862,85],{"className":2863},[119,463],[33,2865,471],{"className":2866},[470],[33,2868,2870],{"className":2869},[443],[33,2871,2873],{"className":2872,"style":478},[447],[33,2874],{},[33,2876,2878,2881],{"style":2877},"top:-2.34em;",[33,2879],{"className":2880,"style":2733},[455],[33,2882,2884],{"className":2883},[119],[33,2885,2887,2890],{"className":2886},[119],[33,2888,2430],{"className":2889},[119,230],[33,2891,2893],{"className":2892},[434],[33,2894,2896,2916],{"className":2895},[438,439],[33,2897,2899,2913],{"className":2898},[443],[33,2900,2902],{"className":2901,"style":448},[447],[33,2903,2904,2907],{"style":451},[33,2905],{"className":2906,"style":456},[455],[33,2908,2910],{"className":2909},[460,461,462,463],[33,2911,1216],{"className":2912},[119,463],[33,2914,471],{"className":2915},[470],[33,2917,2919],{"className":2918},[443],[33,2920,2922],{"className":2921,"style":478},[447],[33,2923],{},[33,2925,2927,2930],{"style":2926},"top:-0.18em;",[33,2928],{"className":2929,"style":2733},[455],[33,2931],{"className":2932},[119],[33,2934,471],{"className":2935},[470],[33,2937,2939],{"className":2938},[443],[33,2940,2943],{"className":2941,"style":2942},[447],"height:3.68em;",[33,2944],{},[33,2946,2948],{"className":2947},[1689],[33,2949,2951,3835],{"className":2950},[438,439],[33,2952,2954,3832],{"className":2953},[443],[33,2955,2957,3169,3380,3591,3802],{"className":2956,"style":2726},[447],[33,2958,2960,2964],{"style":2959},"top:-7.0275em;",[33,2961],{"className":2962,"style":2963},[455],"height:3.6875em;",[33,2965,2967,2970,2973,2976,2979,2982,3025,3028,3071,3074,3117,3120,3163,3166],{"className":2966},[119],[33,2968],{"className":2969},[119],[33,2971],{"className":2972,"style":125},[124],[33,2974,60],{"className":2975},[129],[33,2977],{"className":2978,"style":125},[124],[33,2980,2445],{"className":2981},[119],[33,2983,2985,2988],{"className":2984},[119],[33,2986,2450],{"className":2987},[119,230],[33,2989,2991],{"className":2990},[434],[33,2992,2994,3017],{"className":2993},[438,439],[33,2995,2997,3014],{"className":2996},[443],[33,2998,3000],{"className":2999,"style":448},[447],[33,3001,3002,3005],{"style":451},[33,3003],{"className":3004,"style":456},[455],[33,3006,3008],{"className":3007},[460,461,462,463],[33,3009,3011],{"className":3010},[119,463],[33,3012,2453],{"className":3013},[119,463],[33,3015,471],{"className":3016},[470],[33,3018,3020],{"className":3019},[443],[33,30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denotes the ",[33,3932,3934,3947],{"className":3933,"translate":37},[36],[33,3935,3937],{"className":3936},[41],[43,3938,3939],{"xmlns":45},[47,3940,3941,3945],{},[50,3942,3943],{},[53,3944,3869],{},[99,3946,3869],{"encoding":101},[33,3948,3950],{"className":3949,"ariaHidden":71},[106],[33,3951,3953,3957],{"className":3952},[110],[33,3954],{"className":3955,"style":3956},[114],"height:0.854em;vertical-align:-0.1944em;",[33,3958,3869],{"className":3959,"style":3916},[119,230],"-th decimal digit of the ",[33,3962,3964,3977],{"className":3963,"translate":37},[36],[33,3965,3967],{"className":3966},[41],[43,3968,3969],{"xmlns":45},[47,3970,3971,3975],{},[50,3972,3973],{},[53,3974,3866],{},[99,3976,3866],{"encoding":101},[33,3978,3980],{"className":3979,"ariaHidden":71},[106],[33,3981,3983,3987],{"className":3982},[110],[33,3984],{"className":3985,"style":3986},[114],"height:0.6595em;",[33,3988,3866],{"className":3989},[119,230],"-th number (a digit from ",[33,3992,3994,4007],{"className":3993,"translate":37},[36],[33,3995,3997],{"className":3996},[41],[43,3998,3999],{"xmlns":45},[47,4000,4001,4005],{},[50,4002,4003],{},[66,4004,68],{},[99,4006,68],{"encoding":101},[33,4008,4010],{"className":4009,"ariaHidden":71},[106],[33,4011,4013,4016],{"className":4012},[110],[33,4014],{"className":4015,"style":1357},[114],[33,4017,68],{"className":4018},[119]," to ",[33,4021,4023,4037],{"className":4022,"translate":37},[36],[33,4024,4026],{"className":4025},[41],[43,4027,4028],{"xmlns":45},[47,4029,4030,4035],{},[50,4031,4032],{},[66,4033,4034],{},"9",[99,4036,4034],{"encoding":101},[33,4038,4040],{"className":4039,"ariaHidden":71},[106],[33,4041,4043,4046],{"className":4042},[110],[33,4044],{"className":4045,"style":1357},[114],[33,4047,4034],{"className":4048},[119],").",[10,4051,4052,4055,4056,4303],{},[14,4053,4054],{},"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 ",[33,4057,4059,4105],{"className":4058,"translate":37},[36],[33,4060,4062],{"className":4061},[41],[43,4063,4064],{"xmlns":45},[47,4065,4066,4102],{},[50,4067,4068,4071,4073,4075,4082,4088,4094,4100],{},[53,4069,4070],{},"y",[58,4072,60],{},[66,4074,2445],{},[365,4076,4077,4080],{},[53,4078,4079],{},"e",[66,4081,75],{},[365,4083,4084,4086],{},[53,4085,4079],{},[66,4087,80],{},[365,4089,4090,4092],{},[53,4091,4079],{},[66,4093,85],{},[365,4095,4096,4098],{},[53,4097,4079],{},[66,4099,1216],{},[58,4101,90],{},[99,4103,4104],{"encoding":101},"y = 0.e_1 e_2 e_3 e_4 \\dots",[33,4106,4108,4127],{"className":4107,"ariaHidden":71},[106],[33,4109,4111,4115,4118,4121,4124],{"className":4110},[110],[33,4112],{"className":4113,"style":4114},[114],"height:0.625em;vertical-align:-0.1944em;",[33,4116,4070],{"className":4117,"style":2112},[119,230],[33,4119],{"className":4120,"style":125},[124],[33,4122,60],{"className":4123},[129],[33,4125],{"className":4126,"style":125},[124],[33,4128,4130,4134,4137,4177,4217,4257,4297,4300],{"className":4129},[110],[33,4131],{"className":4132,"style":4133},[114],"height:0.7944em;vertical-align:-0.15em;",[33,4135,2445],{"className":4136},[119],[33,4138,4140,4143],{"className":4139},[119],[33,4141,4079],{"className":4142},[119,230],[33,4144,4146],{"className":4145},[434],[33,4147,4149,4169],{"className":4148},[438,439],[33,4150,4152,4166],{"className":4151},[443],[33,4153,4155],{"className":4154,"style":448},[447],[33,4156,4157,4160],{"style":451},[33,4158],{"className":4159,"style":456},[455],[33,4161,4163],{"className":4162},[460,461,462,463],[33,4164,75],{"className":4165},[119,463],[33,4167,471],{"className":4168},[470],[33,4170,4172],{"className":4171},[443],[33,4173,4175],{"className":4174,"style":478},[447],[33,4176],{},[33,4178,4180,4183],{"className":4179},[119],[33,4181,4079],{"className":4182},[119,230],[33,4184,4186],{"className":4185},[434],[33,4187,4189,4209],{"className":4188},[438,439],[33,4190,4192,4206],{"className":4191},[443],[33,4193,4195],{"className":4194,"style":448},[447],[33,4196,4197,4200],{"style":451},[33,4198],{"className":4199,"style":456},[455],[33,4201,4203],{"className":4202},[460,461,462,463],[33,4204,80],{"className":4205},[119,463],[33,4207,471],{"className":4208},[470],[33,4210,4212],{"className":4211},[443],[33,4213,4215],{"className":4214,"style":478},[447],[33,4216],{},[33,4218,4220,4223],{"className":4219},[119],[33,4221,4079],{"className":4222},[119,230],[33,4224,4226],{"className":4225},[434],[33,4227,4229,4249],{"className":4228},[438,439],[33,4230,4232,4246],{"className":4231},[443],[33,4233,4235],{"className":4234,"style":448},[447],[33,4236,4237,4240],{"style":451},[33,4238],{"className":4239,"style":456},[455],[33,4241,4243],{"className":4242},[460,461,462,463],[33,4244,85],{"className":4245},[119,463],[33,4247,471],{"className":4248},[470],[33,4250,4252],{"className":4251},[443],[33,4253,4255],{"className":4254,"style":478},[447],[33,4256],{},[33,4258,4260,4263],{"className":4259},[119],[33,4261,4079],{"className":4262},[119,230],[33,4264,4266],{"className":4265},[434],[33,4267,4269,4289],{"className":4268},[438,439],[33,4270,4272,4286],{"className":4271},[443],[33,4273,4275],{"className":4274,"style":448},[447],[33,4276,4277,4280],{"style":451},[33,4278],{"className":4279,"style":456},[455],[33,4281,4283],{"className":4282},[460,461,462,463],[33,4284,1216],{"className":4285},[119,463],[33,4287,471],{"className":4288},[470],[33,4290,4292],{"className":4291},[443],[33,4293,4295],{"className":4294,"style":478},[447],[33,4296],{},[33,4298],{"className":4299,"style":154},[124],[33,4301,90],{"className":4302},[185]," according to the following rule:",[33,4305,4307],{"className":4306,"translate":37},[2403],[33,4308,4310,4398],{"className":4309,"translate":37},[36],[33,4311,4313],{"className":4312},[41],[43,4314,4315],{"xmlns":45,"display":2412},[47,4316,4317,4395],{},[50,4318,4319,4325,4327],{},[365,4320,4321,4323],{},[53,4322,4079],{},[53,4324,1309],{},[58,4326,60],{},[50,4328,4329,4331],{},[58,4330,64],{"fence":71},[1557,4332,4333,4365],{"rowspacing":1559,"columnalign":1560,"columnspacing":1561},[1563,4334,4335,4342],{},[1566,4336,4337],{},[1569,4338,4339],{"scriptlevel":68,"displaystyle":63},[66,4340,4341],{},"5",[1566,4343,4344],{},[1569,4345,4346],{"scriptlevel":68,"displaystyle":63},[50,4347,4348,4351,4361,4363],{},[92,4349,4350],{},"if ",[365,4352,4353,4355],{},[53,4354,2450],{},[50,4356,4357,4359],{},[53,4358,1309],{},[53,4360,1309],{},[58,4362,375],{"mathvariant":374},[66,4364,4341],{},[1563,4366,4367,4373],{},[1566,4368,4369],{},[1569,4370,4371],{"scriptlevel":68,"displaystyle":63},[66,4372,1221],{},[1566,4374,4375],{},[1569,4376,4377],{"scriptlevel":68,"displaystyle":63},[50,4378,4379,4381,4391,4393],{},[92,4380,4350],{},[365,4382,4383,4385],{},[53,4384,2450],{},[50,4386,4387,4389],{},[53,4388,1309],{},[53,4390,1309],{},[58,4392,60],{},[66,4394,4341],{},[99,4396,4397],{"encoding":101},"e_n = \\begin{cases} 5 & \\text{if } d_{nn} \\neq 5 \\\\ 6 & \\text{if } d_{nn} = 5 \\end{cases}",[33,4399,4401,4457],{"className":4400,"ariaHidden":71},[106],[33,4402,4404,4407,4448,4451,4454],{"className":4403},[110],[33,4405],{"className":4406,"style":536},[114],[33,4408,4410,4413],{"className":4409},[119],[33,4411,4079],{"className":4412},[119,230],[33,4414,4416],{"className":4415},[434],[33,4417,4419,4440],{"className":4418},[438,439],[33,4420,4422,4437],{"className":4421},[443],[33,4423,4426],{"className":4424,"style":4425},[447],"height:0.1514em;",[33,4427,4428,4431],{"style":451},[33,4429],{"className":4430,"style":456},[455],[33,4432,4434],{"className":4433},[460,461,462,463],[33,4435,1309],{"className":4436},[119,230,463],[33,4438,471],{"className":4439},[470],[33,4441,4443],{"className":4442},[443],[33,4444,4446],{"className":4445,"style":478},[447],[33,4447],{},[33,4449],{"className":4450,"style":125},[124],[33,4452,60],{"className":4453},[129],[33,4455],{"className":4456,"style":125},[124],[33,4458,4460,4463],{"className":4459},[110],[33,4461],{"className":4462,"style":1666},[114],[33,4464,4466,4472,4721],{"className":4465},[185],[33,4467,4469],{"className":4468,"style":1674},[143,1673],[33,4470,64],{"className":4471},[1678,1679],[33,4473,4475],{"className":4474},[119],[33,4476,4478,4523,4526],{"className":4477},[1557],[33,4479,4481],{"className":4480},[1689],[33,4482,4484,4515],{"className":4483},[438,439],[33,4485,4487,4512],{"className":4486},[443],[33,4488,4490,4501],{"className":4489,"style":1699},[447],[33,4491,4492,4495],{"style":1702},[33,4493],{"className":4494,"style":1706},[455],[33,4496,4498],{"className":4497},[119],[33,4499,4341],{"className":4500},[119],[33,4502,4503,4506],{"style":1719},[33,4504],{"className":4505,"style":1706},[455],[33,4507,4509],{"className":4508},[119],[33,4510,1221],{"className":4511},[119],[33,4513,471],{"className":4514},[470],[33,4516,4518],{"className":4517},[443],[33,4519,4521],{"className":4520,"style":1764},[447],[33,4522],{},[33,4524],{"className":4525,"style":1771},[1770],[33,4527,4529],{"className":4528},[1689],[33,4530,4532,4713],{"className":4531},[438,439],[33,4533,4535,4710],{"className":4534},[443],[33,4536,4538,4641],{"className":4537,"style":1699},[447],[33,4539,4540,4543],{"style":1702},[33,4541],{"className":4542,"style":1706},[455],[33,4544,4546,4552,4596,4599,4635,4638],{"className":4545},[119],[33,4547,4549],{"className":4548},[119,1798],[33,4550,4350],{"className":4551},[119],[33,4553,4555,4558],{"className":4554},[119],[33,4556,2450],{"className":4557},[119,230],[33,4559,4561],{"className":4560},[434],[33,4562,4564,4588],{"className":4563},[438,439],[33,4565,4567,4585],{"className":4566},[443],[33,4568,4570],{"className":4569,"style":4425},[447],[33,4571,4572,4575],{"style":451},[33,4573],{"className":4574,"style":456},[455],[33,4576,4578],{"className":4577},[460,461,462,463],[33,4579,4581],{"className":4580},[119,463],[33,4582,4584],{"className":4583},[119,230,463],"nn",[33,4586,471],{"className":4587},[470],[33,4589,4591],{"className":4590},[443],[33,4592,4594],{"className":4593,"style":478},[447],[33,4595],{},[33,4597],{"className":4598,"style":125},[124],[33,4600,4602,4629,4632],{"className":4601},[129],[33,4603,4605],{"className":4604},[129],[33,4606,4608],{"className":4607},[119,493],[33,4609,4611],{"className":4610},[497],[33,4612,4614,4617,4626],{"className":4613},[501],[33,4615],{"className":4616,"style":300},[114],[33,4618,4620],{"className":4619},[508],[33,4621,4623],{"className":4622},[119],[33,4624,515],{"className":4625},[129],[33,4627],{"className":4628},[519],[33,4630],{"className":4631},[124,523],[33,4633,60],{"className":4634},[129],[33,4636],{"className":4637,"style":125},[124],[33,4639,4341],{"className":4640},[119],[33,4642,4643,4646],{"style":1719},[33,4644],{"className":4645,"style":1706},[455],[33,4647,4649,4655,4698,4701,4704,4707],{"className":4648},[119],[33,4650,4652],{"className":4651},[119,1798],[33,4653,4350],{"className":4654},[119],[33,4656,4658,4661],{"className":4657},[119],[33,4659,2450],{"className":4660},[119,230],[33,4662,4664],{"className":4663},[434],[33,4665,4667,4690],{"className":4666},[438,439],[33,4668,4670,4687],{"className":4669},[443],[33,4671,4673],{"className":4672,"style":4425},[447],[33,4674,4675,4678],{"style":451},[33,4676],{"className":4677,"style":456},[455],[33,4679,4681],{"className":4680},[460,461,462,463],[33,4682,4684],{"className":4683},[119,463],[33,4685,4584],{"className":4686},[119,230,463],[33,4688,471],{"className":4689},[470],[33,4691,4693],{"className":4692},[443],[33,4694,4696],{"className":4695,"style":478},[447],[33,4697],{},[33,4699],{"className":4700,"style":125},[124],[33,4702,60],{"className":4703},[129],[33,4705],{"className":4706,"style":125},[124],[33,4708,4341],{"className":4709},[119],[33,4711,471],{"className":4712},[470],[33,4714,4716],{"className":4715},[443],[33,4717,4719],{"className":4718,"style":1764},[447],[33,4720],{},[33,4722],{"className":4723},[192,1833],[10,4725,4726,4727,4901,4902,4930,4931,4959],{},"(The sole purpose of this rule is to ensure ",[33,4728,4730,4760],{"className":4729,"translate":37},[36],[33,4731,4733],{"className":4732},[41],[43,4734,4735],{"xmlns":45},[47,4736,4737,4757],{},[50,4738,4739,4745,4747],{},[365,4740,4741,4743],{},[53,4742,4079],{},[53,4744,1309],{},[58,4746,375],{"mathvariant":374},[365,4748,4749,4751],{},[53,4750,2450],{},[50,4752,4753,4755],{},[53,4754,1309],{},[53,4756,1309],{},[99,4758,4759],{"encoding":101},"e_n \\neq d_{nn}",[33,4761,4763,4851],{"className":4762,"ariaHidden":71},[106],[33,4764,4766,4769,4809,4812,4848],{"className":4765},[110],[33,4767],{"className":4768,"style":300},[114],[33,4770,4772,4775],{"className":4771},[119],[33,4773,4079],{"className":4774},[119,230],[33,4776,4778],{"className":4777},[434],[33,4779,4781,4801],{"className":4780},[438,439],[33,4782,4784,4798],{"className":4783},[443],[33,4785,4787],{"className":4786,"style":4425},[447],[33,4788,4789,4792],{"style":451},[33,4790],{"className":4791,"style":456},[455],[33,4793,4795],{"className":4794},[460,461,462,463],[33,4796,1309],{"className":4797},[119,230,463],[33,4799,471],{"className":4800},[470],[33,4802,4804],{"className":4803},[443],[33,4805,4807],{"className":4806,"style":478},[447],[33,4808],{},[33,4810],{"className":4811,"style":125},[124],[33,4813,4815,4842,4845],{"className":4814},[129],[33,4816,4818],{"className":4817},[129],[33,4819,4821],{"className":4820},[119,493],[33,4822,4824],{"className":4823},[497],[33,4825,4827,4830,4839],{"className":4826},[501],[33,4828],{"className":4829,"style":300},[114],[33,4831,4833],{"className":4832},[508],[33,4834,4836],{"className":4835},[119],[33,4837,515],{"className":4838},[129],[33,4840],{"className":4841},[519],[33,4843],{"className":4844},[124,523],[33,4846,60],{"className":4847},[129],[33,4849],{"className":4850,"style":125},[124],[33,4852,4854,4858],{"className":4853},[110],[33,4855],{"className":4856,"style":4857},[114],"height:0.8444em;vertical-align:-0.15em;",[33,4859,4861,4864],{"className":4860},[119],[33,4862,2450],{"className":4863},[119,230],[33,4865,4867],{"className":4866},[434],[33,4868,4870,4893],{"className":4869},[438,439],[33,4871,4873,4890],{"className":4872},[443],[33,4874,4876],{"className":4875,"style":4425},[447],[33,4877,4878,4881],{"style":451},[33,4879],{"className":4880,"style":456},[455],[33,4882,4884],{"className":4883},[460,461,462,463],[33,4885,4887],{"className":4886},[119,463],[33,4888,4584],{"className":4889},[119,230,463],[33,4891,471],{"className":4892},[470],[33,4894,4896],{"className":4895},[443],[33,4897,4899],{"className":4898,"style":478},[447],[33,4900],{},"; the specific digits chosen are immaterial, so long as each digit deliberately \"avoids\" the corresponding diagonal digit, while also steering clear of all-",[33,4903,4905,4918],{"className":4904,"translate":37},[36],[33,4906,4908],{"className":4907},[41],[43,4909,4910],{"xmlns":45},[47,4911,4912,4916],{},[50,4913,4914],{},[66,4915,68],{},[99,4917,68],{"encoding":101},[33,4919,4921],{"className":4920,"ariaHidden":71},[106],[33,4922,4924,4927],{"className":4923},[110],[33,4925],{"className":4926,"style":1357},[114],[33,4928,68],{"className":4929},[119]," or all-",[33,4932,4934,4947],{"className":4933,"translate":37},[36],[33,4935,4937],{"className":4936},[41],[43,4938,4939],{"xmlns":45},[47,4940,4941,4945],{},[50,4942,4943],{},[66,4944,4034],{},[99,4946,4034],{"encoding":101},[33,4948,4950],{"className":4949,"ariaHidden":71},[106],[33,4951,4953,4956],{"className":4952},[110],[33,4954],{"className":4955,"style":1357},[114],[33,4957,4034],{"className":4958},[119]," representations, which would introduce ambiguity in decimal notation.)",[10,4961,4962,4963],{},"The critical question now presents itself: ",[14,4964,4965,4966,4994],{},"does ",[33,4967,4969,4982],{"className":4968,"translate":37},[36],[33,4970,4972],{"className":4971},[41],[43,4973,4974],{"xmlns":45},[47,4975,4976,4980],{},[50,4977,4978],{},[53,4979,4070],{},[99,4981,4070],{"encoding":101},[33,4983,4985],{"className":4984,"ariaHidden":71},[106],[33,4986,4988,4991],{"className":4987},[110],[33,4989],{"className":4990,"style":4114},[114],[33,4992,4070],{"className":4993,"style":2112},[119,230]," appear in the original list?",[10,4996,4997,4998,5026,5027,5055,5056,5151,5152,4049],{},"Suppose ",[33,4999,5001,5014],{"className":5000,"translate":37},[36],[33,5002,5004],{"className":5003},[41],[43,5005,5006],{"xmlns":45},[47,5007,5008,5012],{},[50,5009,5010],{},[53,5011,4070],{},[99,5013,4070],{"encoding":101},[33,5015,5017],{"className":5016,"ariaHidden":71},[106],[33,5018,5020,5023],{"className":5019},[110],[33,5021],{"className":5022,"style":4114},[114],[33,5024,4070],{"className":5025,"style":2112},[119,230]," does appear in the list; then ",[33,5028,5030,5043],{"className":5029,"translate":37},[36],[33,5031,5033],{"className":5032},[41],[43,5034,5035],{"xmlns":45},[47,5036,5037,5041],{},[50,5038,5039],{},[53,5040,4070],{},[99,5042,4070],{"encoding":101},[33,5044,5046],{"className":5045,"ariaHidden":71},[106],[33,5047,5049,5052],{"className":5048},[110],[33,5050],{"className":5051,"style":4114},[114],[33,5053,4070],{"className":5054,"style":2112},[119,230]," must equal one of the enumerated numbers, say ",[33,5057,5059,5082],{"className":5058,"translate":37},[36],[33,5060,5062],{"className":5061},[41],[43,5063,5064],{"xmlns":45},[47,5065,5066,5079],{},[50,5067,5068,5070,5072],{},[53,5069,4070],{},[58,5071,60],{},[365,5073,5074,5076],{},[53,5075,2430],{},[53,5077,5078],{},"k",[99,5080,5081],{"encoding":101},"y = x_k",[33,5083,5085,5103],{"className":5084,"ariaHidden":71},[106],[33,5086,5088,5091,5094,5097,5100],{"className":5087},[110],[33,5089],{"className":5090,"style":4114},[114],[33,5092,4070],{"className":5093,"style":2112},[119,230],[33,5095],{"className":5096,"style":125},[124],[33,5098,60],{"className":5099},[129],[33,5101],{"className":5102,"style":125},[124],[33,5104,5106,5109],{"className":5105},[110],[33,5107],{"className":5108,"style":536},[114],[33,5110,5112,5115],{"className":5111},[119],[33,5113,2430],{"className":5114},[119,230],[33,5116,5118],{"className":5117},[434],[33,5119,5121,5143],{"className":5120},[438,439],[33,5122,5124,5140],{"className":5123},[443],[33,5125,5128],{"className":5126,"style":5127},[447],"height:0.3361em;",[33,5129,5130,5133],{"style":451},[33,5131],{"className":5132,"style":456},[455],[33,5134,5136],{"className":5135},[460,461,462,463],[33,5137,5078],{"className":5138,"style":5139},[119,230,463],"margin-right:0.0315em;",[33,5141,471],{"className":5142},[470],[33,5144,5146],{"className":5145},[443],[33,5147,5149],{"className":5148,"style":478},[447],[33,5150],{}," (for some specific index ",[33,5153,5155,5168],{"className":5154,"translate":37},[36],[33,5156,5158],{"className":5157},[41],[43,5159,5160],{"xmlns":45},[47,5161,5162,5166],{},[50,5163,5164],{},[53,5165,5078],{},[99,5167,5078],{"encoding":101},[33,5169,5171],{"className":5170,"ariaHidden":71},[106],[33,5172,5174,5177],{"className":5173},[110],[33,5175],{"className":5176,"style":916},[114],[33,5178,5078],{"className":5179,"style":5139},[119,230],[10,5181,5182,5183,5211,5212,5240,5241,5311,5312,5381,5382,5462,5463,5211,5491,4049],{},"But, by our construction rule, the ",[33,5184,5186,5199],{"className":5185,"translate":37},[36],[33,5187,5189],{"className":5188},[41],[43,5190,5191],{"xmlns":45},[47,5192,5193,5197],{},[50,5194,5195],{},[53,5196,5078],{},[99,5198,5078],{"encoding":101},[33,5200,5202],{"className":5201,"ariaHidden":71},[106],[33,5203,5205,5208],{"className":5204},[110],[33,5206],{"className":5207,"style":916},[114],[33,5209,5078],{"className":5210,"style":5139},[119,230],"-th decimal digit of ",[33,5213,5215,5228],{"className":5214,"translate":37},[36],[33,5216,5218],{"className":5217},[41],[43,5219,5220],{"xmlns":45},[47,5221,5222,5226],{},[50,5223,5224],{},[53,5225,4070],{},[99,5227,4070],{"encoding":101},[33,5229,5231],{"className":5230,"ariaHidden":71},[106],[33,5232,5234,5237],{"className":5233},[110],[33,5235],{"className":5236,"style":4114},[114],[33,5238,4070],{"className":5239,"style":2112},[119,230]," is ",[33,5242,5244,5262],{"className":5243,"translate":37},[36],[33,5245,5247],{"className":5246},[41],[43,5248,5249],{"xmlns":45},[47,5250,5251,5259],{},[50,5252,5253],{},[365,5254,5255,5257],{},[53,5256,4079],{},[53,5258,5078],{},[99,5260,5261],{"encoding":101},"e_k",[33,5263,5265],{"className":5264,"ariaHidden":71},[106],[33,5266,5268,5271],{"className":5267},[110],[33,5269],{"className":5270,"style":536},[114],[33,5272,5274,5277],{"className":5273},[119],[33,5275,4079],{"className":5276},[119,230],[33,5278,5280],{"className":5279},[434],[33,5281,5283,5303],{"className":5282},[438,439],[33,5284,5286,5300],{"className":5285},[443],[33,5287,5289],{"className":5288,"style":5127},[447],[33,5290,5291,5294],{"style":451},[33,5292],{"className":5293,"style":456},[455],[33,5295,5297],{"className":5296},[460,461,462,463],[33,5298,5078],{"className":5299,"style":5139},[119,230,463],[33,5301,471],{"className":5302},[470],[33,5304,5306],{"className":5305},[443],[33,5307,5309],{"className":5308,"style":478},[447],[33,5310],{},", and ",[33,5313,5315,5332],{"className":5314,"translate":37},[36],[33,5316,5318],{"className":5317},[41],[43,5319,5320],{"xmlns":45},[47,5321,5322,5330],{},[50,5323,5324],{},[365,5325,5326,5328],{},[53,5327,4079],{},[53,5329,5078],{},[99,5331,5261],{"encoding":101},[33,5333,5335],{"className":5334,"ariaHidden":71},[106],[33,5336,5338,5341],{"className":5337},[110],[33,5339],{"className":5340,"style":536},[114],[33,5342,5344,5347],{"className":5343},[119],[33,5345,4079],{"className":5346},[119,230],[33,5348,5350],{"className":5349},[434],[33,5351,5353,5373],{"className":5352},[438,439],[33,5354,5356,5370],{"className":5355},[443],[33,5357,5359],{"className":5358,"style":5127},[447],[33,5360,5361,5364],{"style":451},[33,5362],{"className":5363,"style":456},[455],[33,5365,5367],{"className":5366},[460,461,462,463],[33,5368,5078],{"className":5369,"style":5139},[119,230,463],[33,5371,471],{"className":5372},[470],[33,5374,5376],{"className":5375},[443],[33,5377,5379],{"className":5378,"style":478},[447],[33,5380],{}," was deliberately constructed to differ from ",[33,5383,5385,5407],{"className":5384,"translate":37},[36],[33,5386,5388],{"className":5387},[41],[43,5389,5390],{"xmlns":45},[47,5391,5392,5404],{},[50,5393,5394],{},[365,5395,5396,5398],{},[53,5397,2450],{},[50,5399,5400,5402],{},[53,5401,5078],{},[53,5403,5078],{},[99,5405,5406],{"encoding":101},"d_{kk}",[33,5408,5410],{"className":5409,"ariaHidden":71},[106],[33,5411,5413,5416],{"className":5412},[110],[33,5414],{"className":5415,"style":4857},[114],[33,5417,5419,5422],{"className":5418},[119],[33,5420,2450],{"className":5421},[119,230],[33,5423,5425],{"className":5424},[434],[33,5426,5428,5454],{"className":5427},[438,439],[33,5429,5431,5451],{"className":5430},[443],[33,5432,5434],{"className":5433,"style":5127},[447],[33,5435,5436,5439],{"style":451},[33,5437],{"className":5438,"style":456},[455],[33,5440,5442],{"className":5441},[460,461,462,463],[33,5443,5445,5448],{"className":5444},[119,463],[33,5446,5078],{"className":5447,"style":5139},[119,230,463],[33,5449,5078],{"className":5450,"style":5139},[119,230,463],[33,5452,471],{"className":5453},[470],[33,5455,5457],{"className":5456},[443],[33,5458,5460],{"className":5459,"style":478},[447],[33,5461],{}," (i.e., the ",[33,5464,5466,5479],{"className":5465,"translate":37},[36],[33,5467,5469],{"className":5468},[41],[43,5470,5471],{"xmlns":45},[47,5472,5473,5477],{},[50,5474,5475],{},[53,5476,5078],{},[99,5478,5078],{"encoding":101},[33,5480,5482],{"className":5481,"ariaHidden":71},[106],[33,5483,5485,5488],{"className":5484},[110],[33,5486],{"className":5487,"style":916},[114],[33,5489,5078],{"className":5490,"style":5139},[119,230],[33,5492,5494,5512],{"className":5493,"translate":37},[36],[33,5495,5497],{"className":5496},[41],[43,5498,5499],{"xmlns":45},[47,5500,5501,5509],{},[50,5502,5503],{},[365,5504,5505,5507],{},[53,5506,2430],{},[53,5508,5078],{},[99,5510,5511],{"encoding":101},"x_k",[33,5513,5515],{"className":5514,"ariaHidden":71},[106],[33,5516,5518,5521],{"className":5517},[110],[33,5519],{"className":5520,"style":536},[114],[33,5522,5524,5527],{"className":5523},[119],[33,5525,2430],{"className":5526},[119,230],[33,5528,5530],{"className":5529},[434],[33,5531,5533,5553],{"className":5532},[438,439],[33,5534,5536,5550],{"className":5535},[443],[33,5537,5539],{"className":5538,"style":5127},[447],[33,5540,5541,5544],{"style":451},[33,5542],{"className":5543,"style":456},[455],[33,5545,5547],{"className":5546},[460,461,462,463],[33,5548,5078],{"className":5549,"style":5139},[119,230,463],[33,5551,471],{"className":5552},[470],[33,5554,5556],{"className":5555},[443],[33,5557,5559],{"className":5558,"style":478},[447],[33,5560],{},[33,5562,5564],{"className":5563,"translate":37},[2403],[33,5565,5567,5615],{"className":5566,"translate":37},[36],[33,5568,5570],{"className":5569},[41],[43,5571,5572],{"xmlns":45,"display":2412},[47,5573,5574,5612],{},[50,5575,5576,5582,5584,5594,5597,5600,5602,5604,5606],{},[365,5577,5578,5580],{},[53,5579,4079],{},[53,5581,5078],{},[58,5583,375],{"mathvariant":374},[365,5585,5586,5588],{},[53,5587,2450],{},[50,5589,5590,5592],{},[53,5591,5078],{},[53,5593,5078],{},[92,5595,5596],{},"  ",[58,5598,5599],{},"⟹",[92,5601,5596],{},[53,5603,4070],{},[58,5605,375],{"mathvariant":374},[365,5607,5608,5610],{},[53,5609,2430],{},[53,5611,5078],{},[99,5613,5614],{"encoding":101},"e_k \\neq d_{kk} \\implies y \\neq x_k",[33,5616,5618,5706,5773,5824],{"className":5617,"ariaHidden":71},[106],[33,5619,5621,5624,5664,5667,5703],{"className":5620},[110],[33,5622],{"className":5623,"style":300},[114],[33,5625,5627,5630],{"className":5626},[119],[33,5628,4079],{"className":5629},[119,230],[33,5631,5633],{"className":5632},[434],[33,5634,5636,5656],{"className":5635},[438,439],[33,5637,5639,5653],{"className":5638},[443],[33,5640,5642],{"className":5641,"style":5127},[447],[33,5643,5644,5647],{"style":451},[33,5645],{"className":5646,"style":456},[455],[33,5648,5650],{"className":5649},[460,461,462,463],[33,5651,5078],{"className":5652,"style":5139},[119,230,463],[33,5654,471],{"className":5655},[470],[33,5657,5659],{"className":5658},[443],[33,5660,5662],{"className":5661,"style":478},[447],[33,5663],{},[33,5665],{"className":5666,"style":125},[124],[33,5668,5670,5697,5700],{"className":5669},[129],[33,5671,5673],{"className":5672},[129],[33,5674,5676],{"className":5675},[119,493],[33,5677,5679],{"className":5678},[497],[33,5680,5682,5685,5694],{"className":5681},[501],[33,5683],{"className":5684,"style":300},[114],[33,5686,5688],{"className":5687},[508],[33,5689,5691],{"className":5690},[119],[33,5692,515],{"className":5693},[129],[33,5695],{"className":5696},[519],[33,5698],{"className":5699},[124,523],[33,5701,60],{"className":5702},[129],[33,5704],{"className":5705,"style":125},[124],[33,5707,5709,5712,5758,5761,5764,5767,5770],{"className":5708},[110],[33,5710],{"className":5711,"style":4857},[114],[33,5713,5715,5718],{"className":5714},[119],[33,5716,2450],{"className":5717},[119,230],[33,5719,5721],{"className":5720},[434],[33,5722,5724,5750],{"className":5723},[438,439],[33,5725,5727,5747],{"className":5726},[443],[33,5728,5730],{"className":5729,"style":5127},[447],[33,5731,5732,5735],{"style":451},[33,5733],{"className":5734,"style":456},[455],[33,5736,5738],{"className":5737},[460,461,462,463],[33,5739,5741,5744],{"className":5740},[119,463],[33,5742,5078],{"className":5743,"style":5139},[119,230,463],[33,5745,5078],{"className":5746,"style":5139},[119,230,463],[33,5748,471],{"className":5749},[470],[33,5751,5753],{"className":5752},[443],[33,5754,5756],{"className":5755,"style":478},[447],[33,5757],{},[33,5759],{"className":5760,"style":125},[124],[33,5762],{"className":5763,"style":125},[124],[33,5765,5599],{"className":5766},[129],[33,5768],{"className":5769,"style":125},[124],[33,5771],{"className":5772,"style":125},[124],[33,5774,5776,5779,5782,5785,5821],{"className":5775},[110],[33,5777],{"className":5778,"style":300},[114],[33,5780,4070],{"className":5781,"style":2112},[119,230],[33,5783],{"className":5784,"style":125},[124],[33,5786,5788,5815,5818],{"className":5787},[129],[33,5789,5791],{"className":5790},[129],[33,5792,5794],{"className":5793},[119,493],[33,5795,5797],{"className":5796},[497],[33,5798,5800,5803,5812],{"className":5799},[501],[33,5801],{"className":5802,"style":300},[114],[33,5804,5806],{"className":5805},[508],[33,5807,5809],{"className":5808},[119],[33,5810,515],{"className":5811},[129],[33,5813],{"className":5814},[519],[33,5816],{"className":5817},[124,523],[33,5819,60],{"className":5820},[129],[33,5822],{"className":5823,"style":125},[124],[33,5825,5827,5830],{"className":5826},[110],[33,5828],{"className":5829,"style":536},[114],[33,5831,5833,5836],{"className":5832},[119],[33,5834,2430],{"className":5835},[119,230],[33,5837,5839],{"className":5838},[434],[33,5840,5842,5862],{"className":5841},[438,439],[33,5843,5845,5859],{"className":5844},[443],[33,5846,5848],{"className":5847,"style":5127},[447],[33,5849,5850,5853],{"style":451},[33,5851],{"className":5852,"style":456},[455],[33,5854,5856],{"className":5855},[460,461,462,463],[33,5857,5078],{"className":5858,"style":5139},[119,230,463],[33,5860,471],{"className":5861},[470],[33,5863,5865],{"className":5864},[443],[33,5866,5868],{"className":5867,"style":478},[447],[33,5869],{},[10,5871,5872,5873,5964],{},"This contradicts the supposition that ",[33,5874,5876,5897],{"className":5875,"translate":37},[36],[33,5877,5879],{"className":5878},[41],[43,5880,5881],{"xmlns":45},[47,5882,5883,5895],{},[50,5884,5885,5887,5889],{},[53,5886,4070],{},[58,5888,60],{},[365,5890,5891,5893],{},[53,5892,2430],{},[53,5894,5078],{},[99,5896,5081],{"encoding":101},[33,5898,5900,5918],{"className":5899,"ariaHidden":71},[106],[33,5901,5903,5906,5909,5912,5915],{"className":5902},[110],[33,5904],{"className":5905,"style":4114},[114],[33,5907,4070],{"className":5908,"style":2112},[119,230],[33,5910],{"className":5911,"style":125},[124],[33,5913,60],{"className":5914},[129],[33,5916],{"className":5917,"style":125},[124],[33,5919,5921,5924],{"className":5920},[110],[33,5922],{"className":5923,"style":536},[114],[33,5925,5927,5930],{"className":5926},[119],[33,5928,2430],{"className":5929},[119,230],[33,5931,5933],{"className":5932},[434],[33,5934,5936,5956],{"className":5935},[438,439],[33,5937,5939,5953],{"className":5938},[443],[33,5940,5942],{"className":5941,"style":5127},[447],[33,5943,5944,5947],{"style":451},[33,5945],{"className":5946,"style":456},[455],[33,5948,5950],{"className":5949},[460,461,462,463],[33,5951,5078],{"className":5952,"style":5139},[119,230,463],[33,5954,471],{"className":5955},[470],[33,5957,5959],{"className":5958},[443],[33,5960,5962],{"className":5961,"style":478},[447],[33,5963],{},"!",[10,5966,5967,5968,1191,5971,5999,6000,6028,6029,6058,6059,6087,6088,6116,6117,6120],{},"Moreover, this contradiction holds for ",[14,5969,5970],{},"every",[33,5972,5974,5987],{"className":5973,"translate":37},[36],[33,5975,5977],{"className":5976},[41],[43,5978,5979],{"xmlns":45},[47,5980,5981,5985],{},[50,5982,5983],{},[53,5984,5078],{},[99,5986,5078],{"encoding":101},[33,5988,5990],{"className":5989,"ariaHidden":71},[106],[33,5991,5993,5996],{"className":5992},[110],[33,5994],{"className":5995,"style":916},[114],[33,5997,5078],{"className":5998,"style":5139},[119,230]," — ",[33,6001,6003,6016],{"className":6002,"translate":37},[36],[33,6004,6006],{"className":6005},[41],[43,6007,6008],{"xmlns":45},[47,6009,6010,6014],{},[50,6011,6012],{},[53,6013,4070],{},[99,6015,4070],{"encoding":101},[33,6017,6019],{"className":6018,"ariaHidden":71},[106],[33,6020,6022,6025],{"className":6021},[110],[33,6023],{"className":6024,"style":4114},[114],[33,6026,4070],{"className":6027,"style":2112},[119,230]," differs from the 1st number at the 1st digit, from the 2nd number at the 2nd digit, from the ",[33,6030,6032,6045],{"className":6031,"translate":37},[36],[33,6033,6035],{"className":6034},[41],[43,6036,6037],{"xmlns":45},[47,6038,6039,6043],{},[50,6040,6041],{},[53,6042,1309],{},[99,6044,1309],{"encoding":101},[33,6046,6048],{"className":6047,"ariaHidden":71},[106],[33,6049,6051,6055],{"className":6050},[110],[33,6052],{"className":6053,"style":6054},[114],"height:0.4306em;",[33,6056,1309],{"className":6057},[119,230],"-th number at the ",[33,6060,6062,6075],{"className":6061,"translate":37},[36],[33,6063,6065],{"className":6064},[41],[43,6066,6067],{"xmlns":45},[47,6068,6069,6073],{},[50,6070,6071],{},[53,6072,1309],{},[99,6074,1309],{"encoding":101},[33,6076,6078],{"className":6077,"ariaHidden":71},[106],[33,6079,6081,6084],{"className":6080},[110],[33,6082],{"className":6083,"style":6054},[114],[33,6085,1309],{"className":6086},[119,230],"-th digit... Hence ",[33,6089,6091,6104],{"className":6090,"translate":37},[36],[33,6092,6094],{"className":6093},[41],[43,6095,6096],{"xmlns":45},[47,6097,6098,6102],{},[50,6099,6100],{},[53,6101,4070],{},[99,6103,4070],{"encoding":101},[33,6105,6107],{"className":6106,"ariaHidden":71},[106],[33,6108,6110,6113],{"className":6109},[110],[33,6111],{"className":6112,"style":4114},[114],[33,6114,4070],{"className":6115,"style":2112},[119,230]," differs from ",[14,6118,6119],{},"every single"," number in the list.",[10,6122,6123,6124,6152,6153,6204],{},"Yet ",[33,6125,6127,6140],{"className":6126,"translate":37},[36],[33,6128,6130],{"className":6129},[41],[43,6131,6132],{"xmlns":45},[47,6133,6134,6138],{},[50,6135,6136],{},[53,6137,4070],{},[99,6139,4070],{"encoding":101},[33,6141,6143],{"className":6142,"ariaHidden":71},[106],[33,6144,6146,6149],{"className":6145},[110],[33,6147],{"className":6148,"style":4114},[114],[33,6150,4070],{"className":6151,"style":2112},[119,230]," is indisputably a real number in ",[33,6154,6156,6177],{"className":6155,"translate":37},[36],[33,6157,6159],{"className":6158},[41],[43,6160,6161],{"xmlns":45},[47,6162,6163,6175],{},[50,6164,6165,6167,6169,6171,6173],{},[58,6166,389],{"stretchy":63},[66,6168,68],{},[58,6170,72],{"separator":71},[66,6172,75],{},[58,6174,398],{"stretchy":63},[99,6176,2154],{"encoding":101},[33,6178,6180],{"className":6179,"ariaHidden":71},[106],[33,6181,6183,6186,6189,6192,6195,6198,6201],{"className":6182},[110],[33,6184],{"className":6185,"style":139},[114],[33,6187,389],{"className":6188},[143],[33,6190,68],{"className":6191},[119],[33,6193,72],{"className":6194},[150],[33,6196],{"className":6197,"style":154},[124],[33,6199,75],{"className":6200},[119],[33,6202,398],{"className":6203},[192]," (every one of its digits is a legitimate decimal digit).",[10,6206,6207,6210,6211,6239,6240,6335,6336,1009,6387,1364],{},[14,6208,6209],{},"Conclusion",": regardless of how one constructs this \"list of all real numbers,\" one can always construct a real number ",[33,6212,6214,6227],{"className":6213,"translate":37},[36],[33,6215,6217],{"className":6216},[41],[43,6218,6219],{"xmlns":45},[47,6220,6221,6225],{},[50,6222,6223],{},[53,6224,4070],{},[99,6226,4070],{"encoding":101},[33,6228,6230],{"className":6229,"ariaHidden":71},[106],[33,6231,6233,6236],{"className":6232},[110],[33,6234],{"className":6235,"style":4114},[114],[33,6237,4070],{"className":6238,"style":2112},[119,230]," that escapes the list. This demonstrates that the initial assumption — the existence of a bijection ",[33,6241,6243,6272],{"className":6242,"translate":37},[36],[33,6244,6246],{"className":6245},[41],[43,6247,6248],{"xmlns":45},[47,6249,6250,6270],{},[50,6251,6252,6254,6256,6258,6260,6262,6264,6266,6268],{},[53,6253,277],{},[58,6255,280],{},[53,6257,56],{"mathvariant":55},[58,6259,285],{},[58,6261,389],{"stretchy":63},[66,6263,68],{},[58,6265,72],{"separator":71},[66,6267,75],{},[58,6269,398],{"stretchy":63},[99,6271,2335],{"encoding":101},[33,6273,6275,6293,6311],{"className":6274,"ariaHidden":71},[106],[33,6276,6278,6281,6284,6287,6290],{"className":6277},[110],[33,6279],{"className":6280,"style":300},[114],[33,6282,277],{"className":6283,"style":304},[119,230],[33,6285],{"className":6286,"style":125},[124],[33,6288,280],{"className":6289},[129],[33,6291],{"className":6292,"style":125},[124],[33,6294,6296,6299,6302,6305,6308],{"className":6295},[110],[33,6297],{"className":6298,"style":115},[114],[33,6300,56],{"className":6301},[119,120],[33,6303],{"className":6304,"style":125},[124],[33,6306,285],{"className":6307},[129],[33,6309],{"className":6310,"style":125},[124],[33,6312,6314,6317,6320,6323,6326,6329,6332],{"className":6313},[110],[33,6315],{"className":6316,"style":139},[114],[33,6318,389],{"className":6319},[143],[33,6321,68],{"className":6322},[119],[33,6324,72],{"className":6325},[150],[33,6327],{"className":6328,"style":154},[124],[33,6330,75],{"className":6331},[119],[33,6333,398],{"className":6334},[192]," — is false. The real numbers in ",[33,6337,6339,6360],{"className":6338,"translate":37},[36],[33,6340,6342],{"className":6341},[41],[43,6343,6344],{"xmlns":45},[47,6345,6346,6358],{},[50,6347,6348,6350,6352,6354,6356],{},[58,6349,389],{"stretchy":63},[66,6351,68],{},[58,6353,72],{"separator":71},[66,6355,75],{},[58,6357,398],{"stretchy":63},[99,6359,2154],{"encoding":101},[33,6361,6363],{"className":6362,"ariaHidden":71},[106],[33,6364,6366,6369,6372,6375,6378,6381,6384],{"className":6365},[110],[33,6367],{"className":6368,"style":139},[114],[33,6370,389],{"className":6371},[143],[33,6373,68],{"className":6374},[119],[33,6376,72],{"className":6377},[150],[33,6379],{"className":6380,"style":154},[124],[33,6382,75],{"className":6383},[119],[33,6385,398],{"className":6386},[192],[14,6388,6389],{},"uncountable",[10,6391,6392,6393,6593,6594,6664,6665,6668],{},"Why the name \"diagonal\"? The term derives from the manner in which the digits ",[33,6394,6396,6434],{"className":6395,"translate":37},[36],[33,6397,6399],{"className":6398},[41],[43,6400,6401],{"xmlns":45},[47,6402,6403,6431],{},[50,6404,6405,6411,6413,6419,6421,6427,6429],{},[365,6406,6407,6409],{},[53,6408,2450],{},[66,6410,2453],{},[58,6412,72],{"separator":71},[365,6414,6415,6417],{},[53,6416,2450],{},[66,6418,2522],{},[58,6420,72],{"separator":71},[365,6422,6423,6425],{},[53,6424,2450],{},[66,6426,2591],{},[58,6428,72],{"separator":71},[58,6430,90],{},[99,6432,6433],{"encoding":101},"d_{11}, d_{22}, d_{33}, \\dots",[33,6435,6437],{"className":6436,"ariaHidden":71},[106],[33,6438,6440,6443,6486,6489,6492,6535,6538,6541,6584,6587,6590],{"className":6439},[110],[33,6441],{"className":6442,"style":300},[114],[33,6444,6446,6449],{"className":6445},[119],[33,6447,2450],{"className":6448},[119,230],[33,6450,6452],{"className":6451},[434],[33,6453,6455,6478],{"className":6454},[438,439],[33,6456,6458,6475],{"className":6457},[443],[33,6459,6461],{"className":6460,"style":448},[447],[33,6462,6463,6466],{"style":451},[33,6464],{"className":6465,"style":456},[455],[33,6467,6469],{"className":6468},[460,461,462,463],[33,6470,6472],{"className":6471},[119,463],[33,6473,2453],{"className":6474},[119,463],[33,6476,471],{"className":6477},[470],[33,6479,6481],{"className":6480},[443],[33,6482,6484],{"className":6483,"style":478},[447],[33,6485],{},[33,6487,72],{"className":6488},[150],[33,6490],{"className":6491,"style":154},[124],[33,6493,6495,6498],{"className":6494},[119],[33,6496,2450],{"className":6497},[119,230],[33,6499,6501],{"className":6500},[434],[33,6502,6504,6527],{"className":6503},[438,439],[33,6505,6507,6524],{"className":6506},[443],[33,6508,6510],{"className":6509,"style":448},[447],[33,6511,6512,6515],{"style":451},[33,6513],{"className":6514,"style":456},[455],[33,6516,6518],{"className":6517},[460,461,462,463],[33,6519,6521],{"className":6520},[119,463],[33,6522,2522],{"className":6523},[119,463],[33,6525,471],{"className":6526},[470],[33,6528,6530],{"className":6529},[443],[33,6531,6533],{"className":6532,"style":478},[447],[33,6534],{},[33,6536,72],{"className":6537},[150],[33,6539],{"className":6540,"style":154},[124],[33,6542,6544,6547],{"className":6543},[119],[33,6545,2450],{"className":6546},[119,230],[33,6548,6550],{"className":6549},[434],[33,6551,6553,6576],{"className":6552},[438,439],[33,6554,6556,6573],{"className":6555},[443],[33,6557,6559],{"className":6558,"style":448},[447],[33,6560,6561,6564],{"style":451},[33,6562],{"className":6563,"style":456},[455],[33,6565,6567],{"className":6566},[460,461,462,463],[33,6568,6570],{"className":6569},[119,463],[33,6571,2591],{"className":6572},[119,463],[33,6574,471],{"className":6575},[470],[33,6577,6579],{"className":6578},[443],[33,6580,6582],{"className":6581,"style":478},[447],[33,6583],{},[33,6585,72],{"className":6586},[150],[33,6588],{"className":6589,"style":154},[124],[33,6591,90],{"className":6592},[185]," are selected in the proof — if one arranges all 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vertically into an infinite matrix, these positions constitute precisely the ",[14,6666,6667],{},"main diagonal"," of the matrix:",[33,6670,6672],{"className":6671,"translate":37},[2403],[33,6673,6675,6881],{"className":6674,"translate":37},[36],[33,6676,6678],{"className":6677},[41],[43,6679,6680],{"xmlns":45,"display":2412},[47,6681,6682,6878],{},[50,6683,6684,6686,6876],{},[58,6685,389],{"fence":71},[1557,6687,6690,6739,6785,6831],{"rowspacing":6688,"columnalign":6689,"columnspacing":1561},"0.16em","center center center 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\"flip\" each digit along the diagonal, constructing a new element that necessarily evades the entire list.",[22,7812,7814],{"id":7813},"constructing-an-escapee","Constructing an Escapee",[10,7816,7817],{},"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.",[7819,7820,7825],"pre",{"className":7821,"code":7822,"language":7823,"meta":7824,"style":7824},"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","",[7826,7827,7828,7853,7860,7866,7872,7877,7888,7906,7934,7952,7980,7992,8010,8017,8022,8033,8058,8081,8094,8100,8105,8116],"code",{"__ignoreMap":7824},[33,7829,7832,7836,7840,7844,7846,7850],{"class":7830,"line":7831},"line",1,[33,7833,7835],{"class":7834},"snl16","def",[33,7837,7839],{"class":7838},"svObZ"," diagonal_construct",[33,7841,7843],{"class":7842},"s95oV","(number_list, digits",[33,7845,60],{"class":7834},[33,7847,7849],{"class":7848},"sDLfK","10",[33,7851,7852],{"class":7842},"):\n",[33,7854,7856],{"class":7830,"line":7855},2,[33,7857,7859],{"class":7858},"sU2Wk","    \"\"\"\n",[33,7861,7863],{"class":7830,"line":7862},3,[33,7864,7865],{"class":7858},"    number_list: list of strings, each of the form \"0.d1d2d3...\"\n",[33,7867,7869],{"class":7830,"line":7868},4,[33,7870,7871],{"class":7858},"    returns: a new number string constructed via the diagonal method\n",[33,7873,7875],{"class":7830,"line":7874},5,[33,7876,7859],{"class":7858},[33,7878,7880,7883,7885],{"class":7830,"line":7879},6,[33,7881,7882],{"class":7842},"    new_digits ",[33,7884,60],{"class":7834},[33,7886,7887],{"class":7842}," []\n",[33,7889,7891,7894,7897,7900,7903],{"class":7830,"line":7890},7,[33,7892,7893],{"class":7834},"    for",[33,7895,7896],{"class":7842}," i, num_str ",[33,7898,7899],{"class":7834},"in",[33,7901,7902],{"class":7848}," enumerate",[33,7904,7905],{"class":7842},"(number_list):\n",[33,7907,7909,7912,7914,7917,7920,7923,7925,7928,7931],{"class":7830,"line":7908},8,[33,7910,7911],{"class":7842},"        decimal_part ",[33,7913,60],{"class":7834},[33,7915,7916],{"class":7842}," num_str.split(",[33,7918,7919],{"class":7858},"'.'",[33,7921,7922],{"class":7842},")[",[33,7924,75],{"class":7848},[33,7926,7927],{"class":7842},"].ljust(digits, ",[33,7929,7930],{"class":7858},"'0'",[33,7932,7933],{"class":7842},")\n",[33,7935,7937,7940,7942,7945,7948],{"class":7830,"line":7936},9,[33,7938,7939],{"class":7842},"        d ",[33,7941,60],{"class":7834},[33,7943,7944],{"class":7848}," int",[33,7946,7947],{"class":7842},"(decimal_part[i])       ",[33,7949,7951],{"class":7950},"sAwPA","# i-th digit of the i-th number (the diagonal digit)\n",[33,7953,7955,7958,7960,7963,7965,7968,7971,7974,7977],{"class":7830,"line":7954},10,[33,7956,7957],{"class":7842},"        new_d ",[33,7959,60],{"class":7834},[33,7961,7962],{"class":7842}," (d ",[33,7964,1607],{"class":7834},[33,7966,7967],{"class":7848}," 5",[33,7969,7970],{"class":7842},") ",[33,7972,7973],{"class":7834},"%",[33,7975,7976],{"class":7848}," 10",[33,7978,7979],{"class":7950},"           # rule: make the new digit differ from the original\n",[33,7981,7983,7986,7989],{"class":7830,"line":7982},11,[33,7984,7985],{"class":7842},"        new_digits.append(",[33,7987,7988],{"class":7848},"str",[33,7990,7991],{"class":7842},"(new_d))\n",[33,7993,7995,7998,8001,8004,8007],{"class":7830,"line":7994},12,[33,7996,7997],{"class":7834},"    return",[33,7999,8000],{"class":7858}," \"0.\"",[33,8002,8003],{"class":7834}," +",[33,8005,8006],{"class":7858}," \"\"",[33,8008,8009],{"class":7842},".join(new_digits)\n",[33,8011,8013],{"class":7830,"line":8012},13,[33,8014,8016],{"emptyLinePlaceholder":8015},true,"\n",[33,8018,8020],{"class":7830,"line":8019},14,[33,8021,8016],{"emptyLinePlaceholder":8015},[33,8023,8025,8028,8030],{"class":7830,"line":8024},15,[33,8026,8027],{"class":7842},"sample_list ",[33,8029,60],{"class":7834},[33,8031,8032],{"class":7842}," [\n",[33,8034,8036,8039,8042,8045,8047,8050,8052,8055],{"class":7830,"line":8035},16,[33,8037,8038],{"class":7858},"    \"0.1234567890\"",[33,8040,8041],{"class":7842},", ",[33,8043,8044],{"class":7858},"\"0.9876543210\"",[33,8046,8041],{"class":7842},[33,8048,8049],{"class":7858},"\"0.5555555555\"",[33,8051,8041],{"class":7842},[33,8053,8054],{"class":7858},"\"0.1111111111\"",[33,8056,8057],{"class":7842},",\n",[33,8059,8061,8064,8066,8069,8071,8074,8076,8079],{"class":7830,"line":8060},17,[33,8062,8063],{"class":7858},"    \"0.3141592653\"",[33,8065,8041],{"class":7842},[33,8067,8068],{"class":7858},"\"0.2718281828\"",[33,8070,8041],{"class":7842},[33,8072,8073],{"class":7858},"\"0.0000000001\"",[33,8075,8041],{"class":7842},[33,8077,8078],{"class":7858},"\"0.9999999998\"",[33,8080,8057],{"class":7842},[33,8082,8084,8087,8089,8092],{"class":7830,"line":8083},18,[33,8085,8086],{"class":7858},"    \"0.4242424242\"",[33,8088,8041],{"class":7842},[33,8090,8091],{"class":7858},"\"0.6180339887\"",[33,8093,8057],{"class":7842},[33,8095,8097],{"class":7830,"line":8096},19,[33,8098,8099],{"class":7842},"]\n",[33,8101,8103],{"class":7830,"line":8102},20,[33,8104,8016],{"emptyLinePlaceholder":8015},[33,8106,8108,8111,8113],{"class":7830,"line":8107},21,[33,8109,8110],{"class":7842},"new_number ",[33,8112,60],{"class":7834},[33,8114,8115],{"class":7842}," diagonal_construct(sample_list)\n",[33,8117,8119,8122,8124,8127],{"class":7830,"line":8118},22,[33,8120,8121],{"class":7848},"print",[33,8123,389],{"class":7842},[33,8125,8126],{"class":7858},"\"Diagonal-constructed new number:\"",[33,8128,8129],{"class":7842},", new_number)\n",[10,8131,8132],{},[14,8133,8134],{},"Output:",[7819,8136,8140],{"className":8137,"code":8139,"language":1798},[8138],"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",[7826,8141,8139],{"__ignoreMap":7824},[10,8143,8144,8145,6116,8148,8150,8151,8041,8179,8304],{},"One may observe with clarity that the newly constructed number ",[7826,8146,8147],{},"0.6306035492",[14,8149,5970],{}," 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 ",[33,8152,8154,8167],{"className":8153,"translate":37},[36],[33,8155,8157],{"className":8156},[41],[43,8158,8159],{"xmlns":45},[47,8160,8161,8165],{},[50,8162,8163],{},[53,8164,5078],{},[99,8166,5078],{"encoding":101},[33,8168,8170],{"className":8169,"ariaHidden":71},[106],[33,8171,8173,8176],{"className":8172},[110],[33,8174],{"className":8175,"style":916},[114],[33,8177,5078],{"className":8178,"style":5139},[119,230],[33,8180,8182,8204],{"className":8181,"translate":37},[36],[33,8183,8185],{"className":8184},[41],[43,8186,8187],{"xmlns":45},[47,8188,8189,8201],{},[50,8190,8191,8193,8195],{},[53,8192,4070],{},[58,8194,375],{"mathvariant":374},[365,8196,8197,8199],{},[53,8198,2430],{},[53,8200,5078],{},[99,8202,8203],{"encoding":101},"y \\neq x_k",[33,8205,8207,8258],{"className":8206,"ariaHidden":71},[106],[33,8208,8210,8213,8216,8219,8255],{"className":8209},[110],[33,8211],{"className":8212,"style":300},[114],[33,8214,4070],{"className":8215,"style":2112},[119,230],[33,8217],{"className":8218,"style":125},[124],[33,8220,8222,8249,8252],{"className":8221},[129],[33,8223,8225],{"className":8224},[129],[33,8226,8228],{"className":8227},[119,493],[33,8229,8231],{"className":8230},[497],[33,8232,8234,8237,8246],{"className":8233},[501],[33,8235],{"className":8236,"style":300},[114],[33,8238,8240],{"className":8239},[508],[33,8241,8243],{"className":8242},[119],[33,8244,515],{"className":8245},[129],[33,8247],{"className":8248},[519],[33,8250],{"className":8251},[124,523],[33,8253,60],{"className":8254},[129],[33,8256],{"className":8257,"style":125},[124],[33,8259,8261,8264],{"className":8260},[110],[33,8262],{"className":8263,"style":536},[114],[33,8265,8267,8270],{"className":8266},[119],[33,8268,2430],{"className":8269},[119,230],[33,8271,8273],{"className":8272},[434],[33,8274,8276,8296],{"className":8275},[438,439],[33,8277,8279,8293],{"className":8278},[443],[33,8280,8282],{"className":8281,"style":5127},[447],[33,8283,8284,8287],{"style":451},[33,8285],{"className":8286,"style":456},[455],[33,8288,8290],{"className":8289},[460,461,462,463],[33,8291,5078],{"className":8292,"style":5139},[119,230,463],[33,8294,471],{"className":8295},[470],[33,8297,8299],{"className":8298},[443],[33,8300,8302],{"className":8301,"style":478},[447],[33,8303],{},". Regardless of the length of the list, this construction rule guarantees that the new number evades every single entry.",[22,8306,8308],{"id":8307},"cantors-theorem","Cantor's Theorem",[10,8310,8311],{},"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.",[2122,8313,8314],{},[10,8315,8316,8319,8320,8348,8349,8397,8398,8426,8427,8455,8456,1364],{},[14,8317,8318],{},"Cantor's Theorem."," For any set ",[33,8321,8323,8336],{"className":8322,"translate":37},[36],[33,8324,8326],{"className":8325},[41],[43,8327,8328],{"xmlns":45},[47,8329,8330,8334],{},[50,8331,8332],{},[53,8333,214],{},[99,8335,214],{"encoding":101},[33,8337,8339],{"className":8338,"ariaHidden":71},[106],[33,8340,8342,8345],{"className":8341},[110],[33,8343],{"className":8344,"style":226},[114],[33,8346,214],{"className":8347},[119,230],", the power set ",[33,8350,8352,8374],{"className":8351,"translate":37},[36],[33,8353,8355],{"className":8354},[41],[43,8356,8357],{"xmlns":45},[47,8358,8359,8371],{},[50,8360,8361,8365,8367,8369],{},[53,8362,8364],{"mathvariant":8363},"script","P",[58,8366,389],{"stretchy":63},[53,8368,214],{},[58,8370,398],{"stretchy":63},[99,8372,8373],{"encoding":101},"\\mathcal{P}(A)",[33,8375,8377],{"className":8376,"ariaHidden":71},[106],[33,8378,8380,8383,8388,8391,8394],{"className":8379},[110],[33,8381],{"className":8382,"style":139},[114],[33,8384,8364],{"className":8385,"style":8387},[119,8386],"mathcal","margin-right:0.0822em;",[33,8389,389],{"className":8390},[143],[33,8392,214],{"className":8393},[119,230],[33,8395,398],{"className":8396},[192]," (i.e., the set of all subsets of ",[33,8399,8401,8414],{"className":8400,"translate":37},[36],[33,8402,8404],{"className":8403},[41],[43,8405,8406],{"xmlns":45},[47,8407,8408,8412],{},[50,8409,8410],{},[53,8411,214],{},[99,8413,214],{"encoding":101},[33,8415,8417],{"className":8416,"ariaHidden":71},[106],[33,8418,8420,8423],{"className":8419},[110],[33,8421],{"className":8422,"style":226},[114],[33,8424,214],{"className":8425},[119,230],") has strictly greater cardinality than ",[33,8428,8430,8443],{"className":8429,"translate":37},[36],[33,8431,8433],{"className":8432},[41],[43,8434,8435],{"xmlns":45},[47,8436,8437,8441],{},[50,8438,8439],{},[53,8440,214],{},[99,8442,214],{"encoding":101},[33,8444,8446],{"className":8445,"ariaHidden":71},[106],[33,8447,8449,8452],{"className":8448},[110],[33,8450],{"className":8451,"style":226},[114],[33,8453,214],{"className":8454},[119,230]," itself; that is, ",[33,8457,8459,8492],{"className":8458,"translate":37},[36],[33,8460,8462],{"className":8461},[41],[43,8463,8464],{"xmlns":45},[47,8465,8466,8489],{},[50,8467,8468,8470,8472,8474,8477,8479,8481,8483,8485,8487],{},[53,8469,1028],{"mathvariant":374},[53,8471,214],{},[53,8473,1028],{"mathvariant":374},[58,8475,8476],{},"\u003C",[53,8478,1028],{"mathvariant":374},[53,8480,8364],{"mathvariant":8363},[58,8482,389],{"stretchy":63},[53,8484,214],{},[58,8486,398],{"stretchy":63},[53,8488,1028],{"mathvariant":374},[99,8490,8491],{"encoding":101},"|A| \u003C |\\mathcal{P}(A)|",[33,8493,8495,8519],{"className":8494,"ariaHidden":71},[106],[33,8496,8498,8501,8504,8507,8510,8513,8516],{"className":8497},[110],[33,8499],{"className":8500,"style":139},[114],[33,8502,1028],{"className":8503},[119],[33,8505,214],{"className":8506},[119,230],[33,8508,1028],{"className":8509},[119],[33,8511],{"className":8512,"style":125},[124],[33,8514,8476],{"className":8515},[129],[33,8517],{"className":8518,"style":125},[124],[33,8520,8522,8525,8528,8531,8534,8537,8540],{"className":8521},[110],[33,8523],{"className":8524,"style":139},[114],[33,8526,1028],{"className":8527},[119],[33,8529,8364],{"className":8530,"style":8387},[119,8386],[33,8532,389],{"className":8533},[143],[33,8535,214],{"className":8536},[119,230],[33,8538,398],{"className":8539},[192],[33,8541,1028],{"className":8542},[119],[10,8544,8545,8546,231,8549,8552,8553,8556,8557,8585,8586,8629],{},"It should be noted that this theorem holds for both ",[14,8547,8548],{},"finite sets",[14,8550,8551],{},"infinite sets",", and for infinite sets in particular, it reveals a startling fact: ",[14,8554,8555],{},"there is no \"largest infinity.\""," For however large ",[33,8558,8560,8573],{"className":8559,"translate":37},[36],[33,8561,8563],{"className":8562},[41],[43,8564,8565],{"xmlns":45},[47,8566,8567,8571],{},[50,8568,8569],{},[53,8570,214],{},[99,8572,214],{"encoding":101},[33,8574,8576],{"className":8575,"ariaHidden":71},[106],[33,8577,8579,8582],{"className":8578},[110],[33,8580],{"className":8581,"style":226},[114],[33,8583,214],{"className":8584},[119,230]," may be, ",[33,8587,8589,8608],{"className":8588,"translate":37},[36],[33,8590,8592],{"className":8591},[41],[43,8593,8594],{"xmlns":45},[47,8595,8596,8606],{},[50,8597,8598,8600,8602,8604],{},[53,8599,8364],{"mathvariant":8363},[58,8601,389],{"stretchy":63},[53,8603,214],{},[58,8605,398],{"stretchy":63},[99,8607,8373],{"encoding":101},[33,8609,8611],{"className":8610,"ariaHidden":71},[106],[33,8612,8614,8617,8620,8623,8626],{"className":8613},[110],[33,8615],{"className":8616,"style":139},[114],[33,8618,8364],{"className":8619,"style":8387},[119,8386],[33,8621,389],{"className":8622},[143],[33,8624,214],{"className":8625},[119,230],[33,8627,398],{"className":8628},[192]," is always larger; one may iterate the power set operation indefinitely, obtaining an ever-ascending hierarchy of infinities.",[8631,8632,8634],"h3",{"id":8633},"proof","Proof",[10,8636,8637,8640,8641,8729,8730,4049],{},[14,8638,8639],{},"Assume",", for the sake of contradiction, that there exists a surjection ",[33,8642,8644,8672],{"className":8643,"translate":37},[36],[33,8645,8647],{"className":8646},[41],[43,8648,8649],{"xmlns":45},[47,8650,8651,8669],{},[50,8652,8653,8655,8657,8659,8661,8663,8665,8667],{},[53,8654,277],{},[58,8656,280],{},[53,8658,214],{},[58,8660,285],{},[53,8662,8364],{"mathvariant":8363},[58,8664,389],{"stretchy":63},[53,8666,214],{},[58,8668,398],{"stretchy":63},[99,8670,8671],{"encoding":101},"f: A \\to \\mathcal{P}(A)",[33,8673,8675,8693,8711],{"className":8674,"ariaHidden":71},[106],[33,8676,8678,8681,8684,8687,8690],{"className":8677},[110],[33,8679],{"className":8680,"style":300},[114],[33,8682,277],{"className":8683,"style":304},[119,230],[33,8685],{"className":8686,"style":125},[124],[33,8688,280],{"className":8689},[129],[33,8691],{"className":8692,"style":125},[124],[33,8694,8696,8699,8702,8705,8708],{"className":8695},[110],[33,8697],{"className":8698,"style":226},[114],[33,8700,214],{"className":8701},[119,230],[33,8703],{"className":8704,"style":125},[124],[33,8706,285],{"className":8707},[129],[33,8709],{"className":8710,"style":125},[124],[33,8712,8714,8717,8720,8723,8726],{"className":8713},[110],[33,8715],{"className":8716,"style":139},[114],[33,8718,8364],{"className":8719,"style":8387},[119,8386],[33,8721,389],{"className":8722},[143],[33,8724,214],{"className":8725},[119,230],[33,8727,398],{"className":8728},[192]," (we shall demonstrate that no such surjection can exist, whence ",[33,8731,8733,8764],{"className":8732,"translate":37},[36],[33,8734,8736],{"className":8735},[41],[43,8737,8738],{"xmlns":45},[47,8739,8740,8762],{},[50,8741,8742,8744,8746,8748,8750,8752,8754,8756,8758,8760],{},[53,8743,1028],{"mathvariant":374},[53,8745,214],{},[53,8747,1028],{"mathvariant":374},[58,8749,8476],{},[53,8751,1028],{"mathvariant":374},[53,8753,8364],{"mathvariant":8363},[58,8755,389],{"stretchy":63},[53,8757,214],{},[58,8759,398],{"stretchy":63},[53,8761,1028],{"mathvariant":374},[99,8763,8491],{"encoding":101},[33,8765,8767,8791],{"className":8766,"ariaHidden":71},[106],[33,8768,8770,8773,8776,8779,8782,8785,8788],{"className":8769},[110],[33,8771],{"className":8772,"style":139},[114],[33,8774,1028],{"className":8775},[119],[33,8777,214],{"className":8778},[119,230],[33,8780,1028],{"className":8781},[119],[33,8783],{"className":8784,"style":125},[124],[33,8786,8476],{"className":8787},[129],[33,8789],{"className":8790,"style":125},[124],[33,8792,8794,8797,8800,8803,8806,8809,8812],{"className":8793},[110],[33,8795],{"className":8796,"style":139},[114],[33,8798,1028],{"className":8799},[119],[33,8801,8364],{"className":8802,"style":8387},[119,8386],[33,8804,389],{"className":8805},[143],[33,8807,214],{"className":8808},[119,230],[33,8810,398],{"className":8811},[192],[33,8813,1028],{"className":8814},[119],[10,8816,8817,8818,8041,8868,8912,8913,8941],{},"For each element ",[33,8819,8821,8838],{"className":8820,"translate":37},[36],[33,8822,8824],{"className":8823},[41],[43,8825,8826],{"xmlns":45},[47,8827,8828,8836],{},[50,8829,8830,8832,8834],{},[53,8831,369],{},[58,8833,762],{},[53,8835,214],{},[99,8837,820],{"encoding":101},[33,8839,8841,8859],{"className":8840,"ariaHidden":71},[106],[33,8842,8844,8847,8850,8853,8856],{"className":8843},[110],[33,8845],{"className":8846,"style":830},[114],[33,8848,369],{"className":8849},[119,230],[33,8851],{"className":8852,"style":125},[124],[33,8854,762],{"className":8855},[129],[33,8857],{"className":8858,"style":125},[124],[33,8860,8862,8865],{"className":8861},[110],[33,8863],{"className":8864,"style":226},[114],[33,8866,214],{"className":8867},[119,230],[33,8869,8871,8891],{"className":8870,"translate":37},[36],[33,8872,8874],{"className":8873},[41],[43,8875,8876],{"xmlns":45},[47,8877,8878,8888],{},[50,8879,8880,8882,8884,8886],{},[53,8881,277],{},[58,8883,389],{"stretchy":63},[53,8885,369],{},[58,8887,398],{"stretchy":63},[99,8889,8890],{"encoding":101},"f(a)",[33,8892,8894],{"className":8893,"ariaHidden":71},[106],[33,8895,8897,8900,8903,8906,8909],{"className":8896},[110],[33,8898],{"className":8899,"style":139},[114],[33,8901,277],{"className":8902,"style":304},[119,230],[33,8904,389],{"className":8905},[143],[33,8907,369],{"className":8908},[119,230],[33,8910,398],{"className":8911},[192]," is a subset of ",[33,8914,8916,8929],{"className":8915,"translate":37},[36],[33,8917,8919],{"className":8918},[41],[43,8920,8921],{"xmlns":45},[47,8922,8923,8927],{},[50,8924,8925],{},[53,8926,214],{},[99,8928,214],{"encoding":101},[33,8930,8932],{"className":8931,"ariaHidden":71},[106],[33,8933,8935,8938],{"className":8934},[110],[33,8936],{"className":8937,"style":226},[114],[33,8939,214],{"className":8940},[119,230],". We now construct a special subset:",[33,8943,8945],{"className":8944,"translate":37},[2403],[33,8946,8948,8994],{"className":8947,"translate":37},[36],[33,8949,8951],{"className":8950},[41],[43,8952,8953],{"xmlns":45,"display":2412},[47,8954,8955,8991],{},[50,8956,8957,8960,8962,8964,8966,8968,8970,8972,8974,8976,8979,8981,8983,8985,8987,8989],{},[53,8958,8959],{},"D",[58,8961,60],{},[58,8963,64],{"stretchy":63},[92,8965,94],{},[53,8967,369],{},[58,8969,762],{},[53,8971,214],{},[58,8973,280],{},[53,8975,369],{},[58,8977,8978],{"mathvariant":374},"∉",[53,8980,277],{},[58,8982,389],{"stretchy":63},[53,8984,369],{},[58,8986,398],{"stretchy":63},[92,8988,94],{},[58,8990,97],{"stretchy":63},[99,8992,8993],{"encoding":101},"D = \\{\\, a \\in A : a \\notin f(a) \\,\\}",[33,8995,8997,9016,9040,9058,9111],{"className":8996,"ariaHidden":71},[106],[33,8998,9000,9003,9007,9010,9013],{"className":8999},[110],[33,9001],{"className":9002,"style":226},[114],[33,9004,8959],{"className":9005,"style":9006},[119,230],"margin-right:0.0278em;",[33,9008],{"className":9009,"style":125},[124],[33,9011,60],{"className":9012},[129],[33,9014],{"className":9015,"style":125},[124],[33,9017,9019,9022,9025,9028,9031,9034,9037],{"className":9018},[110],[33,9020],{"className":9021,"style":139},[114],[33,9023,64],{"className":9024},[143],[33,9026],{"className":9027,"style":154},[124],[33,9029,369],{"className":9030},[119,230],[33,9032],{"className":9033,"style":125},[124],[33,9035,762],{"className":9036},[129],[33,9038],{"className":9039,"style":125},[124],[33,9041,9043,9046,9049,9052,9055],{"className":9042},[110],[33,9044],{"className":9045,"style":226},[114],[33,9047,214],{"className":9048},[119,230],[33,9050],{"className":9051,"style":125},[124],[33,9053,280],{"className":9054},[129],[33,9056],{"className":9057,"style":125},[124],[33,9059,9061,9064,9067,9070,9108],{"className":9060},[110],[33,9062],{"className":9063,"style":139},[114],[33,9065,369],{"className":9066},[119,230],[33,9068],{"className":9069,"style":125},[124],[33,9071,9073,9079],{"className":9072},[129],[33,9074,9076],{"className":9075},[119],[33,9077,762],{"className":9078},[129],[33,9080,9082],{"className":9081},[119,493],[33,9083,9085],{"className":9084},[497],[33,9086,9089,9092,9105],{"className":9087},[9088],"llap",[33,9090],{"className":9091,"style":139},[114],[33,9093,9095],{"className":9094},[508],[33,9096,9098,9101],{"className":9097},[119],[33,9099,1577],{"className":9100},[119],[33,9102],{"className":9103,"style":9104},[124],"margin-right:0.0556em;",[33,9106],{"className":9107},[519],[33,9109],{"className":9110,"style":125},[124],[33,9112,9114,9117,9120,9123,9126,9129,9132],{"className":9113},[110],[33,9115],{"className":9116,"style":139},[114],[33,9118,277],{"className":9119,"style":304},[119,230],[33,9121,389],{"className":9122},[143],[33,9124,369],{"className":9125},[119,230],[33,9127,398],{"className":9128},[192],[33,9130],{"className":9131,"style":154},[124],[33,9133,97],{"className":9134},[192],[10,9136,9137],{},"This is the \"diagonal set\" — it collects all elements that \"do not belong to the subset they are mapped to.\"",[10,9139,9140,9141,9169,9170,9198,9199,9227,9228,9271,9272,852,9323,1364],{},"Since ",[33,9142,9144,9157],{"className":9143,"translate":37},[36],[33,9145,9147],{"className":9146},[41],[43,9148,9149],{"xmlns":45},[47,9150,9151,9155],{},[50,9152,9153],{},[53,9154,277],{},[99,9156,277],{"encoding":101},[33,9158,9160],{"className":9159,"ariaHidden":71},[106],[33,9161,9163,9166],{"className":9162},[110],[33,9164],{"className":9165,"style":300},[114],[33,9167,277],{"className":9168,"style":304},[119,230]," is surjective, ",[33,9171,9173,9186],{"className":9172,"translate":37},[36],[33,9174,9176],{"className":9175},[41],[43,9177,9178],{"xmlns":45},[47,9179,9180,9184],{},[50,9181,9182],{},[53,9183,8959],{},[99,9185,8959],{"encoding":101},[33,9187,9189],{"className":9188,"ariaHidden":71},[106],[33,9190,9192,9195],{"className":9191},[110],[33,9193],{"className":9194,"style":226},[114],[33,9196,8959],{"className":9197,"style":9006},[119,230]," (being a subset of ",[33,9200,9202,9215],{"className":9201,"translate":37},[36],[33,9203,9205],{"className":9204},[41],[43,9206,9207],{"xmlns":45},[47,9208,9209,9213],{},[50,9210,9211],{},[53,9212,214],{},[99,9214,214],{"encoding":101},[33,9216,9218],{"className":9217,"ariaHidden":71},[106],[33,9219,9221,9224],{"className":9220},[110],[33,9222],{"className":9223,"style":226},[114],[33,9225,214],{"className":9226},[119,230],", hence an element of ",[33,9229,9231,9250],{"className":9230,"translate":37},[36],[33,9232,9234],{"className":9233},[41],[43,9235,9236],{"xmlns":45},[47,9237,9238,9248],{},[50,9239,9240,9242,9244,9246],{},[53,9241,8364],{"mathvariant":8363},[58,9243,389],{"stretchy":63},[53,9245,214],{},[58,9247,398],{"stretchy":63},[99,9249,8373],{"encoding":101},[33,9251,9253],{"className":9252,"ariaHidden":71},[106],[33,9254,9256,9259,9262,9265,9268],{"className":9255},[110],[33,9257],{"className":9258,"style":139},[114],[33,9260,8364],{"className":9261,"style":8387},[119,8386],[33,9263,389],{"className":9264},[143],[33,9266,214],{"className":9267},[119,230],[33,9269,398],{"className":9270},[192],") must possess a preimage; that is, there exists some ",[33,9273,9275,9293],{"className":9274,"translate":37},[36],[33,9276,9278],{"className":9277},[41],[43,9279,9280],{"xmlns":45},[47,9281,9282,9290],{},[50,9283,9284,9286,9288],{},[53,9285,2450],{},[58,9287,762],{},[53,9289,214],{},[99,9291,9292],{"encoding":101},"d \\in A",[33,9294,9296,9314],{"className":9295,"ariaHidden":71},[106],[33,9297,9299,9302,9305,9308,9311],{"className":9298},[110],[33,9300],{"className":9301,"style":777},[114],[33,9303,2450],{"className":9304},[119,230],[33,9306],{"className":9307,"style":125},[124],[33,9309,762],{"className":9310},[129],[33,9312],{"className":9313,"style":125},[124],[33,9315,9317,9320],{"className":9316},[110],[33,9318],{"className":9319,"style":226},[114],[33,9321,214],{"className":9322},[119,230],[33,9324,9326,9350],{"className":9325,"translate":37},[36],[33,9327,9329],{"className":9328},[41],[43,9330,9331],{"xmlns":45},[47,9332,9333,9347],{},[50,9334,9335,9337,9339,9341,9343,9345],{},[53,9336,277],{},[58,9338,389],{"stretchy":63},[53,9340,2450],{},[58,9342,398],{"stretchy":63},[58,9344,60],{},[53,9346,8959],{},[99,9348,9349],{"encoding":101},"f(d) = D",[33,9351,9353,9380],{"className":9352,"ariaHidden":71},[106],[33,9354,9356,9359,9362,9365,9368,9371,9374,9377],{"className":9355},[110],[33,9357],{"className":9358,"style":139},[114],[33,9360,277],{"className":9361,"style":304},[119,230],[33,9363,389],{"className":9364},[143],[33,9366,2450],{"className":9367},[119,230],[33,9369,398],{"className":9370},[192],[33,9372],{"className":9373,"style":125},[124],[33,9375,60],{"className":9376},[129],[33,9378],{"className":9379,"style":125},[124],[33,9381,9383,9386],{"className":9382},[110],[33,9384],{"className":9385,"style":226},[114],[33,9387,8959],{"className":9388,"style":9006},[119,230],[10,9390,9391,9392,9443],{},"Now we ask: is ",[33,9393,9395,9413],{"className":9394,"translate":37},[36],[33,9396,9398],{"className":9397},[41],[43,9399,9400],{"xmlns":45},[47,9401,9402,9410],{},[50,9403,9404,9406,9408],{},[53,9405,2450],{},[58,9407,762],{},[53,9409,8959],{},[99,9411,9412],{"encoding":101},"d \\in D",[33,9414,9416,9434],{"className":9415,"ariaHidden":71},[106],[33,9417,9419,9422,9425,9428,9431],{"className":9418},[110],[33,9420],{"className":9421,"style":777},[114],[33,9423,2450],{"className":9424},[119,230],[33,9426],{"className":9427,"style":125},[124],[33,9429,762],{"className":9430},[129],[33,9432],{"className":9433,"style":125},[124],[33,9435,9437,9440],{"className":9436},[110],[33,9438],{"className":9439,"style":226},[114],[33,9441,8959],{"className":9442,"style":9006},[119,230],"?",[343,9445,9446,9891],{},[346,9447,9448,9501,9502,8041,9530,9685,9686,9751,9752,9890],{},[14,9449,9450,9451],{},"If ",[33,9452,9454,9471],{"className":9453,"translate":37},[36],[33,9455,9457],{"className":9456},[41],[43,9458,9459],{"xmlns":45},[47,9460,9461,9469],{},[50,9462,9463,9465,9467],{},[53,9464,2450],{},[58,9466,762],{},[53,9468,8959],{},[99,9470,9412],{"encoding":101},[33,9472,9474,9492],{"className":9473,"ariaHidden":71},[106],[33,9475,9477,9480,9483,9486,9489],{"className":9476},[110],[33,9478],{"className":9479,"style":777},[114],[33,9481,2450],{"className":9482},[119,230],[33,9484],{"className":9485,"style":125},[124],[33,9487,762],{"className":9488},[129],[33,9490],{"className":9491,"style":125},[124],[33,9493,9495,9498],{"className":9494},[110],[33,9496],{"className":9497,"style":226},[114],[33,9499,8959],{"className":9500,"style":9006},[119,230],": by the definition of ",[33,9503,9505,9518],{"className":9504,"translate":37},[36],[33,9506,9508],{"className":9507},[41],[43,9509,9510],{"xmlns":45},[47,9511,9512,9516],{},[50,9513,9514],{},[53,9515,8959],{},[99,9517,8959],{"encoding":101},[33,9519,9521],{"className":9520,"ariaHidden":71},[106],[33,9522,9524,9527],{"className":9523},[110],[33,9525],{"className":9526,"style":226},[114],[33,9528,8959],{"className":9529,"style":9006},[119,230],[33,9531,9533,9570],{"className":9532,"translate":37},[36],[33,9534,9536],{"className":9535},[41],[43,9537,9538],{"xmlns":45},[47,9539,9540,9567],{},[50,9541,9542,9544,9546,9548,9550,9553,9555,9557,9559,9561,9563,9565],{},[53,9543,2450],{},[58,9545,762],{},[53,9547,8959],{},[92,9549,5596],{},[58,9551,9552],{},"⟺",[92,9554,5596],{},[53,9556,2450],{},[58,9558,8978],{"mathvariant":374},[53,9560,277],{},[58,9562,389],{"stretchy":63},[53,9564,2450],{},[58,9566,398],{"stretchy":63},[99,9568,9569],{"encoding":101},"d \\in D \\iff d \\notin f(d)",[33,9571,9573,9591,9616,9667],{"className":9572,"ariaHidden":71},[106],[33,9574,9576,9579,9582,9585,9588],{"className":9575},[110],[33,9577],{"className":9578,"style":777},[114],[33,9580,2450],{"className":9581},[119,230],[33,9583],{"className":9584,"style":125},[124],[33,9586,762],{"className":9587},[129],[33,9589],{"className":9590,"style":125},[124],[33,9592,9594,9598,9601,9604,9607,9610,9613],{"className":9593},[110],[33,9595],{"className":9596,"style":9597},[114],"height:0.7073em;vertical-align:-0.024em;",[33,9599,8959],{"className":9600,"style":9006},[119,230],[33,9602],{"className":9603,"style":125},[124],[33,9605],{"className":9606,"style":125},[124],[33,9608,9552],{"className":9609},[129],[33,9611],{"className":9612,"style":125},[124],[33,9614],{"className":9615,"style":125},[124],[33,9617,9619,9622,9625,9628,9664],{"className":9618},[110],[33,9620],{"className":9621,"style":139},[114],[33,9623,2450],{"className":9624},[119,230],[33,9626],{"className":9627,"style":125},[124],[33,9629,9631,9637],{"className":9630},[129],[33,9632,9634],{"className":9633},[119],[33,9635,762],{"className":9636},[129],[33,9638,9640],{"className":9639},[119,493],[33,9641,9643],{"className":9642},[497],[33,9644,9646,9649,9661],{"className":9645},[9088],[33,9647],{"className":9648,"style":139},[114],[33,9650,9652],{"className":9651},[508],[33,9653,9655,9658],{"className":9654},[119],[33,9656,1577],{"className":9657},[119],[33,9659],{"className":9660,"style":9104},[124],[33,9662],{"className":9663},[519],[33,9665],{"className":9666,"style":125},[124],[33,9668,9670,9673,9676,9679,9682],{"className":9669},[110],[33,9671],{"className":9672,"style":139},[114],[33,9674,277],{"className":9675,"style":304},[119,230],[33,9677,389],{"className":9678},[143],[33,9680,2450],{"className":9681},[119,230],[33,9683,398],{"className":9684},[192],". Since ",[33,9687,9689,9712],{"className":9688,"translate":37},[36],[33,9690,9692],{"className":9691},[41],[43,9693,9694],{"xmlns":45},[47,9695,9696,9710],{},[50,9697,9698,9700,9702,9704,9706,9708],{},[53,9699,277],{},[58,9701,389],{"stretchy":63},[53,9703,2450],{},[58,9705,398],{"stretchy":63},[58,9707,60],{},[53,9709,8959],{},[99,9711,9349],{"encoding":101},[33,9713,9715,9742],{"className":9714,"ariaHidden":71},[106],[33,9716,9718,9721,9724,9727,9730,9733,9736,9739],{"className":9717},[110],[33,9719],{"className":9720,"style":139},[114],[33,9722,277],{"className":9723,"style":304},[119,230],[33,9725,389],{"className":9726},[143],[33,9728,2450],{"className":9729},[119,230],[33,9731,398],{"className":9732},[192],[33,9734],{"className":9735,"style":125},[124],[33,9737,60],{"className":9738},[129],[33,9740],{"className":9741,"style":125},[124],[33,9743,9745,9748],{"className":9744},[110],[33,9746],{"className":9747,"style":226},[114],[33,9749,8959],{"className":9750,"style":9006},[119,230],", we have ",[33,9753,9755,9785],{"className":9754,"translate":37},[36],[33,9756,9758],{"className":9757},[41],[43,9759,9760],{"xmlns":45},[47,9761,9762,9782],{},[50,9763,9764,9766,9768,9770,9772,9774,9776,9778,9780],{},[53,9765,2450],{},[58,9767,762],{},[53,9769,8959],{},[92,9771,5596],{},[58,9773,9552],{},[92,9775,5596],{},[53,9777,2450],{},[58,9779,8978],{"mathvariant":374},[53,9781,8959],{},[99,9783,9784],{"encoding":101},"d \\in D \\iff d \\notin D",[33,9786,9788,9806,9830,9881],{"className":9787,"ariaHidden":71},[106],[33,9789,9791,9794,9797,9800,9803],{"className":9790},[110],[33,9792],{"className":9793,"style":777},[114],[33,9795,2450],{"className":9796},[119,230],[33,9798],{"className":9799,"style":125},[124],[33,9801,762],{"className":9802},[129],[33,9804],{"className":9805,"style":125},[124],[33,9807,9809,9812,9815,9818,9821,9824,9827],{"className":9808},[110],[33,9810],{"className":9811,"style":9597},[114],[33,9813,8959],{"className":9814,"style":9006},[119,230],[33,9816],{"className":9817,"style":125},[124],[33,9819],{"className":9820,"style":125},[124],[33,9822,9552],{"className":9823},[129],[33,9825],{"className":9826,"style":125},[124],[33,9828],{"className":9829,"style":125},[124],[33,9831,9833,9836,9839,9842,9878],{"className":9832},[110],[33,9834],{"className":9835,"style":139},[114],[33,9837,2450],{"className":9838},[119,230],[33,9840],{"className":9841,"style":125},[124],[33,9843,9845,9851],{"className":9844},[129],[33,9846,9848],{"className":9847},[119],[33,9849,762],{"className":9850},[129],[33,9852,9854],{"className":9853},[119,493],[33,9855,9857],{"className":9856},[497],[33,9858,9860,9863,9875],{"className":9859},[9088],[33,9861],{"className":9862,"style":139},[114],[33,9864,9866],{"className":9865},[508],[33,9867,9869,9872],{"className":9868},[119],[33,9870,1577],{"className":9871},[119],[33,9873],{"className":9874,"style":9104},[124],[33,9876],{"className":9877},[519],[33,9879],{"className":9880,"style":125},[124],[33,9882,9884,9887],{"className":9883},[110],[33,9885],{"className":9886,"style":226},[114],[33,9888,8959],{"className":9889,"style":9006},[119,230],". Contradiction!",[346,9892,9893,9979,9980,10180,10181,10269,10270,9890],{},[14,9894,9450,9895],{},[33,9896,9898,9916],{"className":9897,"translate":37},[36],[33,9899,9901],{"className":9900},[41],[43,9902,9903],{"xmlns":45},[47,9904,9905,9913],{},[50,9906,9907,9909,9911],{},[53,9908,2450],{},[58,9910,8978],{"mathvariant":374},[53,9912,8959],{},[99,9914,9915],{"encoding":101},"d \\notin D",[33,9917,9919,9970],{"className":9918,"ariaHidden":71},[106],[33,9920,9922,9925,9928,9931,9967],{"className":9921},[110],[33,9923],{"className":9924,"style":139},[114],[33,9926,2450],{"className":9927},[119,230],[33,9929],{"className":9930,"style":125},[124],[33,9932,9934,9940],{"className":9933},[129],[33,9935,9937],{"className":9936},[119],[33,9938,762],{"className":9939},[129],[33,9941,9943],{"className":9942},[119,493],[33,9944,9946],{"className":9945},[497],[33,9947,9949,9952,9964],{"className":9948},[9088],[33,9950],{"className":9951,"style":139},[114],[33,9953,9955],{"className":9954},[508],[33,9956,9958,9961],{"className":9957},[119],[33,9959,1577],{"className":9960},[119],[33,9962],{"className":9963,"style":9104},[124],[33,9965],{"className":9966},[519],[33,9968],{"className":9969,"style":125},[124],[33,9971,9973,9976],{"className":9972},[110],[33,9974],{"className":9975,"style":226},[114],[33,9977,8959],{"className":9978,"style":9006},[119,230],": likewise, ",[33,9981,9983,10026],{"className":9982,"translate":37},[36],[33,9984,9986],{"className":9985},[41],[43,9987,9988],{"xmlns":45},[47,9989,9990,10023],{},[50,9991,9992,9994,9996,9998,10000,10002,10004,10007,10009,10011,10013,10015,10017,10019,10021],{},[53,9993,2450],{},[58,9995,8978],{"mathvariant":374},[53,9997,8959],{},[92,9999,5596],{},[58,10001,9552],{},[92,10003,5596],{},[53,10005,10006],{"mathvariant":374},"¬",[58,10008,389],{"stretchy":63},[53,10010,2450],{},[58,10012,8978],{"mathvariant":374},[53,10014,277],{},[58,10016,389],{"stretchy":63},[53,10018,2450],{},[58,10020,398],{"stretchy":63},[58,10022,398],{"stretchy":63},[99,10024,10025],{"encoding":101},"d \\notin D \\iff \\neg(d \\notin f(d))",[33,10027,10029,10080,10104,10161],{"className":10028,"ariaHidden":71},[106],[33,10030,10032,10035,10038,10041,10077],{"className":10031},[110],[33,10033],{"className":10034,"style":139},[114],[33,10036,2450],{"className":10037},[119,230],[33,10039],{"className":10040,"style":125},[124],[33,10042,10044,10050],{"className":10043},[129],[33,10045,10047],{"className":10046},[119],[33,10048,762],{"className":10049},[129],[33,10051,10053],{"className":10052},[119,493],[33,10054,10056],{"className":10055},[497],[33,10057,10059,10062,10074],{"className":10058},[9088],[33,10060],{"className":10061,"style":139},[114],[33,10063,10065],{"className":10064},[508],[33,10066,10068,10071],{"className":10067},[119],[33,10069,1577],{"className":10070},[119],[33,10072],{"className":10073,"style":9104},[124],[33,10075],{"className":10076},[519],[33,10078],{"className":10079,"style":125},[124],[33,10081,10083,10086,10089,10092,10095,10098,10101],{"className":10082},[110],[33,10084],{"className":10085,"style":9597},[114],[33,10087,8959],{"className":10088,"style":9006},[119,230],[33,10090],{"className":10091,"style":125},[124],[33,10093],{"className":10094,"style":125},[124],[33,10096,9552],{"className":10097},[129],[33,10099],{"className":10100,"style":125},[124],[33,10102],{"className":10103,"style":125},[124],[33,10105,10107,10110,10113,10116,10119,10122,10158],{"className":10106},[110],[33,10108],{"className":10109,"style":139},[114],[33,10111,10006],{"className":10112},[119],[33,10114,389],{"className":10115},[143],[33,10117,2450],{"className":10118},[119,230],[33,10120],{"className":10121,"style":125},[124],[33,10123,10125,10131],{"className":10124},[129],[33,10126,10128],{"className":10127},[119],[33,10129,762],{"className":10130},[129],[33,10132,10134],{"className":10133},[119,493],[33,10135,10137],{"className":10136},[497],[33,10138,10140,10143,10155],{"className":10139},[9088],[33,10141],{"className":10142,"style":139},[114],[33,10144,10146],{"className":10145},[508],[33,10147,10149,10152],{"className":10148},[119],[33,10150,1577],{"className":10151},[119],[33,10153],{"className":10154,"style":9104},[124],[33,10156],{"className":10157},[519],[33,10159],{"className":10160,"style":125},[124],[33,10162,10164,10167,10170,10173,10176],{"className":10163},[110],[33,10165],{"className":10166,"style":139},[114],[33,10168,277],{"className":10169,"style":304},[119,230],[33,10171,389],{"className":10172},[143],[33,10174,2450],{"className":10175},[119,230],[33,10177,10179],{"className":10178},[192],"))",", i.e., ",[33,10182,10184,10212],{"className":10183,"translate":37},[36],[33,10185,10187],{"className":10186},[41],[43,10188,10189],{"xmlns":45},[47,10190,10191,10209],{},[50,10192,10193,10195,10197,10199,10201,10203,10205,10207],{},[53,10194,2450],{},[58,10196,762],{},[53,10198,277],{},[58,10200,389],{"stretchy":63},[53,10202,2450],{},[58,10204,398],{"stretchy":63},[58,10206,60],{},[53,10208,8959],{},[99,10210,10211],{"encoding":101},"d \\in f(d) = D",[33,10213,10215,10233,10260],{"className":10214,"ariaHidden":71},[106],[33,10216,10218,10221,10224,10227,10230],{"className":10217},[110],[33,10219],{"className":10220,"style":777},[114],[33,10222,2450],{"className":10223},[119,230],[33,10225],{"className":10226,"style":125},[124],[33,10228,762],{"className":10229},[129],[33,10231],{"className":10232,"style":125},[124],[33,10234,10236,10239,10242,10245,10248,10251,10254,10257],{"className":10235},[110],[33,10237],{"className":10238,"style":139},[114],[33,10240,277],{"className":10241,"style":304},[119,230],[33,10243,389],{"className":10244},[143],[33,10246,2450],{"className":10247},[119,230],[33,10249,398],{"className":10250},[192],[33,10252],{"className":10253,"style":125},[124],[33,10255,60],{"className":10256},[129],[33,10258],{"className":10259,"style":125},[124],[33,10261,10263,10266],{"className":10262},[110],[33,10264],{"className":10265,"style":226},[114],[33,10267,8959],{"className":10268,"style":9006},[119,230],", which yields ",[33,10271,10273,10290],{"className":10272,"translate":37},[36],[33,10274,10276],{"className":10275},[41],[43,10277,10278],{"xmlns":45},[47,10279,10280,10288],{},[50,10281,10282,10284,10286],{},[53,10283,2450],{},[58,10285,762],{},[53,10287,8959],{},[99,10289,9412],{"encoding":101},[33,10291,10293,10311],{"className":10292,"ariaHidden":71},[106],[33,10294,10296,10299,10302,10305,10308],{"className":10295},[110],[33,10297],{"className":10298,"style":777},[114],[33,10300,2450],{"className":10301},[119,230],[33,10303],{"className":10304,"style":125},[124],[33,10306,762],{"className":10307},[129],[33,10309],{"className":10310,"style":125},[124],[33,10312,10314,10317],{"className":10313},[110],[33,10315],{"className":10316,"style":226},[114],[33,10318,8959],{"className":10319,"style":9006},[119,230],[10,10321,10322,10323,10410,10411,1364],{},"Both cases lead to self-contradiction, demonstrating that the assumption \"there exists a surjection ",[33,10324,10326,10353],{"className":10325,"translate":37},[36],[33,10327,10329],{"className":10328},[41],[43,10330,10331],{"xmlns":45},[47,10332,10333,10351],{},[50,10334,10335,10337,10339,10341,10343,10345,10347,10349],{},[53,10336,277],{},[58,10338,280],{},[53,10340,214],{},[58,10342,285],{},[53,10344,8364],{"mathvariant":8363},[58,10346,389],{"stretchy":63},[53,10348,214],{},[58,10350,398],{"stretchy":63},[99,10352,8671],{"encoding":101},[33,10354,10356,10374,10392],{"className":10355,"ariaHidden":71},[106],[33,10357,10359,10362,10365,10368,10371],{"className":10358},[110],[33,10360],{"className":10361,"style":300},[114],[33,10363,277],{"className":10364,"style":304},[119,230],[33,10366],{"className":10367,"style":125},[124],[33,10369,280],{"className":10370},[129],[33,10372],{"className":10373,"style":125},[124],[33,10375,10377,10380,10383,10386,10389],{"className":10376},[110],[33,10378],{"className":10379,"style":226},[114],[33,10381,214],{"className":10382},[119,230],[33,10384],{"className":10385,"style":125},[124],[33,10387,285],{"className":10388},[129],[33,10390],{"className":10391,"style":125},[124],[33,10393,10395,10398,10401,10404,10407],{"className":10394},[110],[33,10396],{"className":10397,"style":139},[114],[33,10399,8364],{"className":10400,"style":8387},[119,8386],[33,10402,389],{"className":10403},[143],[33,10405,214],{"className":10406},[119,230],[33,10408,398],{"className":10409},[192],"\" is untenable. Hence ",[33,10412,10414,10445],{"className":10413,"translate":37},[36],[33,10415,10417],{"className":10416},[41],[43,10418,10419],{"xmlns":45},[47,10420,10421,10443],{},[50,10422,10423,10425,10427,10429,10431,10433,10435,10437,10439,10441],{},[53,10424,1028],{"mathvariant":374},[53,10426,214],{},[53,10428,1028],{"mathvariant":374},[58,10430,8476],{},[53,10432,1028],{"mathvariant":374},[53,10434,8364],{"mathvariant":8363},[58,10436,389],{"stretchy":63},[53,10438,214],{},[58,10440,398],{"stretchy":63},[53,10442,1028],{"mathvariant":374},[99,10444,8491],{"encoding":101},[33,10446,10448,10472],{"className":10447,"ariaHidden":71},[106],[33,10449,10451,10454,10457,10460,10463,10466,10469],{"className":10450},[110],[33,10452],{"className":10453,"style":139},[114],[33,10455,1028],{"className":10456},[119],[33,10458,214],{"className":10459},[119,230],[33,10461,1028],{"className":10462},[119],[33,10464],{"className":10465,"style":125},[124],[33,10467,8476],{"className":10468},[129],[33,10470],{"className":10471,"style":125},[124],[33,10473,10475,10478,10481,10484,10487,10490,10493],{"className":10474},[110],[33,10476],{"className":10477,"style":139},[114],[33,10479,1028],{"className":10480},[119],[33,10482,8364],{"className":10483,"style":8387},[119,8386],[33,10485,389],{"className":10486},[143],[33,10488,214],{"className":10489},[119,230],[33,10491,398],{"className":10492},[192],[33,10494,1028],{"className":10495},[119],[10,10497,10498,10499,10502],{},"The attentive reader may already have discerned that the structure of this proof is ",[14,10500,10501],{},"identical"," to that of the uncountability proof presented above:",[10504,10505,10506,10518],"table",{},[10507,10508,10509],"thead",{},[10510,10511,10512,10516],"tr",{},[10513,10514,10515],"th",{},"Uncountability of the Reals",[10513,10517,8308],{},[10519,10520,10521,10712,10999,11286],"tbody",{},[10510,10522,10523,10622],{},[10524,10525,10526,10527],"td",{},"Assume a bijection ",[33,10528,10530,10559],{"className":10529,"translate":37},[36],[33,10531,10533],{"className":10532},[41],[43,10534,10535],{"xmlns":45},[47,10536,10537,10557],{},[50,10538,10539,10541,10543,10545,10547,10549,10551,10553,10555],{},[53,10540,277],{},[58,10542,280],{},[53,10544,56],{"mathvariant":55},[58,10546,285],{},[58,10548,389],{"stretchy":63},[66,10550,68],{},[58,10552,72],{"separator":71},[66,10554,75],{},[58,10556,398],{"stretchy":63},[99,10558,2335],{"encoding":101},[33,10560,10562,10580,10598],{"className":10561,"ariaHidden":71},[106],[33,10563,10565,10568,10571,10574,10577],{"className":10564},[110],[33,10566],{"className":10567,"style":300},[114],[33,10569,277],{"className":10570,"style":304},[119,230],[33,10572],{"className":10573,"style":125},[124],[33,10575,280],{"className":10576},[129],[33,10578],{"className":10579,"style":125},[124],[33,10581,10583,10586,10589,10592,10595],{"className":10582},[110],[33,10584],{"className":10585,"style":115},[114],[33,10587,56],{"className":10588},[119,120],[33,10590],{"className":10591,"style":125},[124],[33,10593,285],{"className":10594},[129],[33,10596],{"className":10597,"style":125},[124],[33,10599,10601,10604,10607,10610,10613,10616,10619],{"className":10600},[110],[33,10602],{"className":10603,"style":139},[114],[33,10605,389],{"className":10606},[143],[33,10608,68],{"className":10609},[119],[33,10611,72],{"className":10612},[150],[33,10614],{"className":10615,"style":154},[124],[33,10617,75],{"className":10618},[119],[33,10620,398],{"className":10621},[192],[10524,10623,10624,10625],{},"Assume a surjection ",[33,10626,10628,10655],{"className":10627,"translate":37},[36],[33,10629,10631],{"className":10630},[41],[43,10632,10633],{"xmlns":45},[47,10634,10635,10653],{},[50,10636,10637,10639,10641,10643,10645,10647,10649,10651],{},[53,10638,277],{},[58,10640,280],{},[53,10642,214],{},[58,10644,285],{},[53,10646,8364],{"mathvariant":8363},[58,10648,389],{"stretchy":63},[53,10650,214],{},[58,10652,398],{"stretchy":63},[99,10654,8671],{"encoding":101},[33,10656,10658,10676,10694],{"className":10657,"ariaHidden":71},[106],[33,10659,10661,10664,10667,10670,10673],{"className":10660},[110],[33,10662],{"className":10663,"style":300},[114],[33,10665,277],{"className":10666,"style":304},[119,230],[33,10668],{"className":10669,"style":125},[124],[33,10671,280],{"className":10672},[129],[33,10674],{"className":10675,"style":125},[124],[33,10677,10679,10682,10685,10688,10691],{"className":10678},[110],[33,10680],{"className":10681,"style":226},[114],[33,10683,214],{"className":10684},[119,230],[33,10686],{"className":10687,"style":125},[124],[33,10689,285],{"className":10690},[129],[33,10692],{"className":10693,"style":125},[124],[33,10695,10697,10700,10703,10706,10709],{"className":10696},[110],[33,10698],{"className":10699,"style":139},[114],[33,10701,8364],{"className":10702,"style":8387},[119,8386],[33,10704,389],{"className":10705},[143],[33,10707,214],{"className":10708},[119,230],[33,10710,398],{"className":10711},[192],[10510,10713,10714,10845],{},[10524,10715,10716,10717,10745,10746,10774,10775],{},"Construct the diagonal number ",[33,10718,10720,10733],{"className":10719,"translate":37},[36],[33,10721,10723],{"className":10722},[41],[43,10724,10725],{"xmlns":45},[47,10726,10727,10731],{},[50,10728,10729],{},[53,10730,4070],{},[99,10732,4070],{"encoding":101},[33,10734,10736],{"className":10735,"ariaHidden":71},[106],[33,10737,10739,10742],{"className":10738},[110],[33,10740],{"className":10741,"style":4114},[114],[33,10743,4070],{"className":10744,"style":2112},[119,230],", each digit differing from the ",[33,10747,10749,10762],{"className":10748,"translate":37},[36],[33,10750,10752],{"className":10751},[41],[43,10753,10754],{"xmlns":45},[47,10755,10756,10760],{},[50,10757,10758],{},[53,10759,1309],{},[99,10761,1309],{"encoding":101},[33,10763,10765],{"className":10764,"ariaHidden":71},[106],[33,10766,10768,10771],{"className":10767},[110],[33,10769],{"className":10770,"style":6054},[114],[33,10772,1309],{"className":10773},[119,230],"-th digit of ",[33,10776,10778,10796],{"className":10777,"translate":37},[36],[33,10779,10781],{"className":10780},[41],[43,10782,10783],{"xmlns":45},[47,10784,10785,10793],{},[50,10786,10787],{},[365,10788,10789,10791],{},[53,10790,2430],{},[53,10792,1309],{},[99,10794,10795],{"encoding":101},"x_n",[33,10797,10799],{"className":10798,"ariaHidden":71},[106],[33,10800,10802,10805],{"className":10801},[110],[33,10803],{"className":10804,"style":536},[114],[33,10806,10808,10811],{"className":10807},[119],[33,10809,2430],{"className":10810},[119,230],[33,10812,10814],{"className":10813},[434],[33,10815,10817,10837],{"className":10816},[438,439],[33,10818,10820,10834],{"className":10819},[443],[33,10821,10823],{"className":10822,"style":4425},[447],[33,10824,10825,10828],{"style":451},[33,10826],{"className":10827,"style":456},[455],[33,10829,10831],{"className":10830},[460,461,462,463],[33,10832,1309],{"className":10833},[119,230,463],[33,10835,471],{"className":10836},[470],[33,10838,10840],{"className":10839},[443],[33,10841,10843],{"className":10842,"style":478},[447],[33,10844],{},[10524,10846,10847,10848],{},"Construct the diagonal set ",[33,10849,10851,10887],{"className":10850,"translate":37},[36],[33,10852,10854],{"className":10853},[41],[43,10855,10856],{"xmlns":45},[47,10857,10858,10884],{},[50,10859,10860,10862,10864,10866,10868,10870,10872,10874,10876,10878,10880,10882],{},[53,10861,8959],{},[58,10863,60],{},[58,10865,64],{"stretchy":63},[53,10867,369],{},[58,10869,280],{},[53,10871,369],{},[58,10873,8978],{"mathvariant":374},[53,10875,277],{},[58,10877,389],{"stretchy":63},[53,10879,369],{},[58,10881,398],{"stretchy":63},[58,10883,97],{"stretchy":63},[99,10885,10886],{"encoding":101},"D = \\{a : a \\notin f(a)\\}",[33,10888,10890,10908,10929,10980],{"className":10889,"ariaHidden":71},[106],[33,10891,10893,10896,10899,10902,10905],{"className":10892},[110],[33,10894],{"className":10895,"style":226},[114],[33,10897,8959],{"className":10898,"style":9006},[119,230],[33,10900],{"className":10901,"style":125},[124],[33,10903,60],{"className":10904},[129],[33,10906],{"className":10907,"style":125},[124],[33,10909,10911,10914,10917,10920,10923,10926],{"className":10910},[110],[33,10912],{"className":10913,"style":139},[114],[33,10915,64],{"className":10916},[143],[33,10918,369],{"className":10919},[119,230],[33,10921],{"className":10922,"style":125},[124],[33,10924,280],{"className":10925},[129],[33,10927],{"className":10928,"style":125},[124],[33,10930,10932,10935,10938,10941,10977],{"className":10931},[110],[33,10933],{"className":10934,"style":139},[114],[33,10936,369],{"className":10937},[119,230],[33,10939],{"className":10940,"style":125},[124],[33,10942,10944,10950],{"className":10943},[129],[33,10945,10947],{"className":10946},[119],[33,10948,762],{"className":10949},[129],[33,10951,10953],{"className":10952},[119,493],[33,10954,10956],{"className":10955},[497],[33,10957,10959,10962,10974],{"className":10958},[9088],[33,10960],{"className":10961,"style":139},[114],[33,10963,10965],{"className":10964},[508],[33,10966,10968,10971],{"className":10967},[119],[33,10969,1577],{"className":10970},[119],[33,10972],{"className":10973,"style":9104},[124],[33,10975],{"className":10976},[519],[33,10978],{"className":10979,"style":125},[124],[33,10981,10983,10986,10989,10992,10995],{"className":10982},[110],[33,10984],{"className":10985,"style":139},[114],[33,10987,277],{"className":10988,"style":304},[119,230],[33,10990,389],{"className":10991},[143],[33,10993,369],{"className":10994},[119,230],[33,10996,10998],{"className":10997},[192],")}",[10510,11000,11001,11157],{},[10524,11002,11003,11128,11129],{},[33,11004,11006,11028],{"className":11005,"translate":37},[36],[33,11007,11009],{"className":11008},[41],[43,11010,11011],{"xmlns":45},[47,11012,11013,11025],{},[50,11014,11015,11017,11019],{},[53,11016,4070],{},[58,11018,375],{"mathvariant":374},[365,11020,11021,11023],{},[53,11022,2430],{},[53,11024,1309],{},[99,11026,11027],{"encoding":101},"y \\neq x_n",[33,11029,11031,11082],{"className":11030,"ariaHidden":71},[106],[33,11032,11034,11037,11040,11043,11079],{"className":11033},[110],[33,11035],{"className":11036,"style":300},[114],[33,11038,4070],{"className":11039,"style":2112},[119,230],[33,11041],{"className":11042,"style":125},[124],[33,11044,11046,11073,11076],{"className":11045},[129],[33,11047,11049],{"className":11048},[129],[33,11050,11052],{"className":11051},[119,493],[33,11053,11055],{"className":11054},[497],[33,11056,11058,11061,11070],{"className":11057},[501],[33,11059],{"className":11060,"style":300},[114],[33,11062,11064],{"className":11063},[508],[33,11065,11067],{"className":11066},[119],[33,11068,515],{"className":11069},[129],[33,11071],{"className":11072},[519],[33,11074],{"className":11075},[124,523],[33,11077,60],{"className":11078},[129],[33,11080],{"className":11081,"style":125},[124],[33,11083,11085,11088],{"className":11084},[110],[33,11086],{"className":11087,"style":536},[114],[33,11089,11091,11094],{"className":11090},[119],[33,11092,2430],{"className":11093},[119,230],[33,11095,11097],{"className":11096},[434],[33,11098,11100,11120],{"className":11099},[438,439],[33,11101,11103,11117],{"className":11102},[443],[33,11104,11106],{"className":11105,"style":4425},[447],[33,11107,11108,11111],{"style":451},[33,11109],{"className":11110,"style":456},[455],[33,11112,11114],{"className":11113},[460,461,462,463],[33,11115,1309],{"className":11116},[119,230,463],[33,11118,471],{"className":11119},[470],[33,11121,11123],{"className":11122},[443],[33,11124,11126],{"className":11125,"style":478},[447],[33,11127],{}," for all ",[33,11130,11132,11145],{"className":11131,"translate":37},[36],[33,11133,11135],{"className":11134},[41],[43,11136,11137],{"xmlns":45},[47,11138,11139,11143],{},[50,11140,11141],{},[53,11142,1309],{},[99,11144,1309],{"encoding":101},[33,11146,11148],{"className":11147,"ariaHidden":71},[106],[33,11149,11151,11154],{"className":11150},[110],[33,11152],{"className":11153,"style":6054},[114],[33,11155,1309],{"className":11156},[119,230],[10524,11158,11159,11128,11258],{},[33,11160,11162,11186],{"className":11161,"translate":37},[36],[33,11163,11165],{"className":11164},[41],[43,11166,11167],{"xmlns":45},[47,11168,11169,11183],{},[50,11170,11171,11173,11175,11177,11179,11181],{},[53,11172,8959],{},[58,11174,375],{"mathvariant":374},[53,11176,277],{},[58,11178,389],{"stretchy":63},[53,11180,369],{},[58,11182,398],{"stretchy":63},[99,11184,11185],{"encoding":101},"D \\neq f(a)",[33,11187,11189,11240],{"className":11188,"ariaHidden":71},[106],[33,11190,11192,11195,11198,11201,11237],{"className":11191},[110],[33,11193],{"className":11194,"style":300},[114],[33,11196,8959],{"className":11197,"style":9006},[119,230],[33,11199],{"className":11200,"style":125},[124],[33,11202,11204,11231,11234],{"className":11203},[129],[33,11205,11207],{"className":11206},[129],[33,11208,11210],{"className":11209},[119,493],[33,11211,11213],{"className":11212},[497],[33,11214,11216,11219,11228],{"className":11215},[501],[33,11217],{"className":11218,"style":300},[114],[33,11220,11222],{"className":11221},[508],[33,11223,11225],{"className":11224},[119],[33,11226,515],{"className":11227},[129],[33,11229],{"className":11230},[519],[33,11232],{"className":11233},[124,523],[33,11235,60],{"className":11236},[129],[33,11238],{"className":11239,"style":125},[124],[33,11241,11243,11246,11249,11252,11255],{"className":11242},[110],[33,11244],{"className":11245,"style":139},[114],[33,11247,277],{"className":11248,"style":304},[119,230],[33,11250,389],{"className":11251},[143],[33,11253,369],{"className":11254},[119,230],[33,11256,398],{"className":11257},[192],[33,11259,11261,11274],{"className":11260,"translate":37},[36],[33,11262,11264],{"className":11263},[41],[43,11265,11266],{"xmlns":45},[47,11267,11268,11272],{},[50,11269,11270],{},[53,11271,369],{},[99,11273,369],{"encoding":101},[33,11275,11277],{"className":11276,"ariaHidden":71},[106],[33,11278,11280,11283],{"className":11279},[110],[33,11281],{"className":11282,"style":6054},[114],[33,11284,369],{"className":11285},[119,230],[10510,11287,11288,11319],{},[10524,11289,11290,11318],{},[33,11291,11293,11306],{"className":11292,"translate":37},[36],[33,11294,11296],{"className":11295},[41],[43,11297,11298],{"xmlns":45},[47,11299,11300,11304],{},[50,11301,11302],{},[53,11303,4070],{},[99,11305,4070],{"encoding":101},[33,11307,11309],{"className":11308,"ariaHidden":71},[106],[33,11310,11312,11315],{"className":11311},[110],[33,11313],{"className":11314,"style":4114},[114],[33,11316,4070],{"className":11317,"style":2112},[119,230]," cannot be covered by the list; contradiction",[10524,11320,11321,11349],{},[33,11322,11324,11337],{"className":11323,"translate":37},[36],[33,11325,11327],{"className":11326},[41],[43,11328,11329],{"xmlns":45},[47,11330,11331,11335],{},[50,11332,11333],{},[53,11334,8959],{},[99,11336,8959],{"encoding":101},[33,11338,11340],{"className":11339,"ariaHidden":71},[106],[33,11341,11343,11346],{"className":11342},[110],[33,11344],{"className":11345,"style":226},[114],[33,11347,8959],{"className":11348,"style":9006},[119,230]," cannot be covered by the mapping; contradiction",[10,11351,11352,11353,11404,11405,11494,11495,11579,11580,11624,11625,1364],{},"In fact, the real numbers in ",[33,11354,11356,11377],{"className":11355,"translate":37},[36],[33,11357,11359],{"className":11358},[41],[43,11360,11361],{"xmlns":45},[47,11362,11363,11375],{},[50,11364,11365,11367,11369,11371,11373],{},[58,11366,389],{"stretchy":63},[66,11368,68],{},[58,11370,72],{"separator":71},[66,11372,75],{},[58,11374,398],{"stretchy":63},[99,11376,2154],{"encoding":101},[33,11378,11380],{"className":11379,"ariaHidden":71},[106],[33,11381,11383,11386,11389,11392,11395,11398,11401],{"className":11382},[110],[33,11384],{"className":11385,"style":139},[114],[33,11387,389],{"className":11388},[143],[33,11390,68],{"className":11391},[119],[33,11393,72],{"className":11394},[150],[33,11396],{"className":11397,"style":154},[124],[33,11399,75],{"className":11400},[119],[33,11402,398],{"className":11403},[192]," can be placed in approximate correspondence with ",[33,11406,11408,11435],{"className":11407,"translate":37},[36],[33,11409,11411],{"className":11410},[41],[43,11412,11413],{"xmlns":45},[47,11414,11415,11432],{},[50,11416,11417,11419,11421,11423,11425],{},[58,11418,64],{"stretchy":63},[66,11420,68],{},[58,11422,72],{"separator":71},[66,11424,75],{},[11426,11427,11428,11430],"msup",{},[58,11429,97],{"stretchy":63},[53,11431,56],{"mathvariant":55},[99,11433,11434],{"encoding":101},"\\{0,1\\}^{\\mathbb{N}}",[33,11436,11438],{"className":11437,"ariaHidden":71},[106],[33,11439,11441,11445,11448,11451,11454,11457,11460],{"className":11440},[110],[33,11442],{"className":11443,"style":11444},[114],"height:1.0952em;vertical-align:-0.25em;",[33,11446,64],{"className":11447},[143],[33,11449,68],{"className":11450},[119],[33,11452,72],{"className":11453},[150],[33,11455],{"className":11456,"style":154},[124],[33,11458,75],{"className":11459},[119],[33,11461,11463,11466],{"className":11462},[192],[33,11464,97],{"className":11465},[192],[33,11467,11469],{"className":11468},[434],[33,11470,11472],{"className":11471},[438],[33,11473,11475],{"className":11474},[443],[33,11476,11479],{"className":11477,"style":11478},[447],"height:0.8452em;",[33,11480,11482,11485],{"style":11481},"top:-3.063em;margin-right:0.05em;",[33,11483],{"className":11484,"style":456},[455],[33,11486,11488],{"className":11487},[460,461,462,463],[33,11489,11491],{"className":11490},[119,463],[33,11492,56],{"className":11493},[119,120,463]," (the set of infinite binary sequences of 0s and 1s), and ",[33,11496,11498,11523],{"className":11497,"translate":37},[36],[33,11499,11501],{"className":11500},[41],[43,11502,11503],{"xmlns":45},[47,11504,11505,11521],{},[50,11506,11507,11509,11511,11513,11515],{},[58,11508,64],{"stretchy":63},[66,11510,68],{},[58,11512,72],{"separator":71},[66,11514,75],{},[11426,11516,11517,11519],{},[58,11518,97],{"stretchy":63},[53,11520,56],{"mathvariant":55},[99,11522,11434],{"encoding":101},[33,11524,11526],{"className":11525,"ariaHidden":71},[106],[33,11527,11529,11532,11535,11538,11541,11544,11547],{"className":11528},[110],[33,11530],{"className":11531,"style":11444},[114],[33,11533,64],{"className":11534},[143],[33,11536,68],{"className":11537},[119],[33,11539,72],{"className":11540},[150],[33,11542],{"className":11543,"style":154},[124],[33,11545,75],{"className":11546},[119],[33,11548,11550,11553],{"className":11549},[192],[33,11551,97],{"className":11552},[192],[33,11554,11556],{"className":11555},[434],[33,11557,11559],{"className":11558},[438],[33,11560,11562],{"className":11561},[443],[33,11563,11565],{"className":11564,"style":11478},[447],[33,11566,11567,11570],{"style":11481},[33,11568],{"className":11569,"style":456},[455],[33,11571,11573],{"className":11572},[460,461,462,463],[33,11574,11576],{"className":11575},[119,463],[33,11577,56],{"className":11578},[119,120,463]," is, in essence, ",[33,11581,11583,11603],{"className":11582,"translate":37},[36],[33,11584,11586],{"className":11585},[41],[43,11587,11588],{"xmlns":45},[47,11589,11590,11600],{},[50,11591,11592,11594,11596,11598],{},[53,11593,8364],{"mathvariant":8363},[58,11595,389],{"stretchy":63},[53,11597,56],{"mathvariant":55},[58,11599,398],{"stretchy":63},[99,11601,11602],{"encoding":101},"\\mathcal{P}(\\mathbb{N})",[33,11604,11606],{"className":11605,"ariaHidden":71},[106],[33,11607,11609,11612,11615,11618,11621],{"className":11608},[110],[33,11610],{"className":11611,"style":139},[114],[33,11613,8364],{"className":11614,"style":8387},[119,8386],[33,11616,389],{"className":11617},[143],[33,11619,56],{"className":11620},[119,120],[33,11622,398],{"className":11623},[192]," (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 ",[33,11626,11628,11646],{"className":11627,"translate":37},[36],[33,11629,11631],{"className":11630},[41],[43,11632,11633],{"xmlns":45},[47,11634,11635,11643],{},[50,11636,11637,11639,11641],{},[53,11638,214],{},[58,11640,60],{},[53,11642,56],{"mathvariant":55},[99,11644,11645],{"encoding":101},"A = \\mathbb{N}",[33,11647,11649,11667],{"className":11648,"ariaHidden":71},[106],[33,11650,11652,11655,11658,11661,11664],{"className":11651},[110],[33,11653],{"className":11654,"style":226},[114],[33,11656,214],{"className":11657},[119,230],[33,11659],{"className":11660,"style":125},[124],[33,11662,60],{"className":11663},[129],[33,11665],{"className":11666,"style":125},[124],[33,11668,11670,11673],{"className":11669},[110],[33,11671],{"className":11672,"style":115},[114],[33,11674,56],{"className":11675},[119,120],[10,11677,11678],{},"While we cannot represent infinite sets inside a computer, we can verify the logical skeleton of the proof using a finite set:",[7819,11680,11682],{"className":7821,"code":11681,"language":7823,"meta":7824,"style":7824},"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",[7826,11683,11684,11694,11699,11709,11721,11769,11773,11791,11801,11835,11839,11844,11877,11899,11903,11932,11965,11969],{"__ignoreMap":7824},[33,11685,11686,11688,11691],{"class":7830,"line":7831},[33,11687,7835],{"class":7834},[33,11689,11690],{"class":7838}," cantor_diagonal_demo",[33,11692,11693],{"class":7842},"(A):\n",[33,11695,11696],{"class":7830,"line":7855},[33,11697,11698],{"class":7950},"    # Construct an \"arbitrary\" mapping f: A -> P(A) for demonstration\n",[33,11700,11701,11704,11706],{"class":7830,"line":7862},[33,11702,11703],{"class":7842},"    f ",[33,11705,60],{"class":7834},[33,11707,11708],{"class":7842}," {}\n",[33,11710,11711,11713,11716,11718],{"class":7830,"line":7868},[33,11712,7893],{"class":7834},[33,11714,11715],{"class":7842}," i ",[33,11717,7899],{"class":7834},[33,11719,11720],{"class":7842}," A:\n",[33,11722,11723,11726,11728,11731,11734,11737,11740,11742,11745,11748,11751,11753,11756,11758,11761,11764,11767],{"class":7830,"line":7874},[33,11724,11725],{"class":7842},"        f[i] ",[33,11727,60],{"class":7834},[33,11729,11730],{"class":7848}," frozenset",[33,11732,11733],{"class":7842},"(j ",[33,11735,11736],{"class":7834},"for",[33,11738,11739],{"class":7842}," j ",[33,11741,7899],{"class":7834},[33,11743,11744],{"class":7842}," A ",[33,11746,11747],{"class":7834},"if",[33,11749,11750],{"class":7842}," (i ",[33,11752,1607],{"class":7834},[33,11754,11755],{"class":7842}," j) ",[33,11757,7973],{"class":7834},[33,11759,11760],{"class":7848}," 2",[33,11762,11763],{"class":7834}," ==",[33,11765,11766],{"class":7848}," 0",[33,11768,7933],{"class":7842},[33,11770,11771],{"class":7830,"line":7879},[33,11772,8016],{"emptyLinePlaceholder":8015},[33,11774,11775,11778,11780,11783,11785,11788],{"class":7830,"line":7890},[33,11776,11777],{"class":7848},"    print",[33,11779,389],{"class":7842},[33,11781,11782],{"class":7858},"\"Set A =\"",[33,11784,8041],{"class":7842},[33,11786,11787],{"class":7848},"set",[33,11789,11790],{"class":7842},"(A))\n",[33,11792,11793,11795,11797,11799],{"class":7830,"line":7908},[33,11794,7893],{"class":7834},[33,11796,11715],{"class":7842},[33,11798,7899],{"class":7834},[33,11800,11720],{"class":7842},[33,11802,11803,11806,11808,11810,11813,11815,11817,11819,11822,11825,11828,11830,11833],{"class":7830,"line":7936},[33,11804,11805],{"class":7848},"        print",[33,11807,389],{"class":7842},[33,11809,277],{"class":7834},[33,11811,11812],{"class":7858},"\"  f(",[33,11814,64],{"class":7848},[33,11816,3866],{"class":7842},[33,11818,97],{"class":7848},[33,11820,11821],{"class":7858},") = ",[33,11823,11824],{"class":7848},"{set",[33,11826,11827],{"class":7842},"(f[i])",[33,11829,97],{"class":7848},[33,11831,11832],{"class":7858},"\"",[33,11834,7933],{"class":7842},[33,11836,11837],{"class":7830,"line":7954},[33,11838,8016],{"emptyLinePlaceholder":8015},[33,11840,11841],{"class":7830,"line":7982},[33,11842,11843],{"class":7950},"    # Diagonal construction: D = { i in A : i not in f(i) }\n",[33,11845,11846,11849,11851,11853,11856,11858,11860,11862,11864,11866,11868,11871,11874],{"class":7830,"line":7994},[33,11847,11848],{"class":7842},"    D ",[33,11850,60],{"class":7834},[33,11852,11730],{"class":7848},[33,11854,11855],{"class":7842},"(i ",[33,11857,11736],{"class":7834},[33,11859,11715],{"class":7842},[33,11861,7899],{"class":7834},[33,11863,11744],{"class":7842},[33,11865,11747],{"class":7834},[33,11867,11715],{"class":7842},[33,11869,11870],{"class":7834},"not",[33,11872,11873],{"class":7834}," in",[33,11875,11876],{"class":7842}," f[i])\n",[33,11878,11879,11881,11883,11885,11888,11890,11893,11895,11897],{"class":7830,"line":8012},[33,11880,11777],{"class":7848},[33,11882,389],{"class":7842},[33,11884,277],{"class":7834},[33,11886,11887],{"class":7858},"\"Diagonal-constructed set D = ",[33,11889,11824],{"class":7848},[33,11891,11892],{"class":7842},"(D)",[33,11894,97],{"class":7848},[33,11896,11832],{"class":7858},[33,11898,7933],{"class":7842},[33,11900,11901],{"class":7830,"line":8019},[33,11902,8016],{"emptyLinePlaceholder":8015},[33,11904,11905,11908,11910,11913,11915,11917,11919,11921,11923,11926,11929],{"class":7830,"line":8024},[33,11906,11907],{"class":7842},"    matches ",[33,11909,60],{"class":7834},[33,11911,11912],{"class":7842}," [i ",[33,11914,11736],{"class":7834},[33,11916,11715],{"class":7842},[33,11918,7899],{"class":7834},[33,11920,11744],{"class":7842},[33,11922,11747],{"class":7834},[33,11924,11925],{"class":7842}," f[i] ",[33,11927,11928],{"class":7834},"==",[33,11930,11931],{"class":7842}," D]\n",[33,11933,11934,11936,11938,11940,11943,11945,11948,11950,11953,11956,11959,11961,11963],{"class":7830,"line":8035},[33,11935,11777],{"class":7848},[33,11937,389],{"class":7842},[33,11939,277],{"class":7834},[33,11941,11942],{"class":7858},"\"Does there exist i such that f(i) = D? -> ",[33,11944,64],{"class":7848},[33,11946,11947],{"class":7842},"matches ",[33,11949,11747],{"class":7834},[33,11951,11952],{"class":7842}," matches ",[33,11954,11955],{"class":7834},"else",[33,11957,11958],{"class":7858}," 'Does not exist (contradiction established)'",[33,11960,97],{"class":7848},[33,11962,11832],{"class":7858},[33,11964,7933],{"class":7842},[33,11966,11967],{"class":7830,"line":8060},[33,11968,8016],{"emptyLinePlaceholder":8015},[33,11970,11971,11974,11976,11978,11980,11982,11984,11986,11988],{"class":7830,"line":8083},[33,11972,11973],{"class":7842},"cantor_diagonal_demo([",[33,11975,68],{"class":7848},[33,11977,8041],{"class":7842},[33,11979,75],{"class":7848},[33,11981,8041],{"class":7842},[33,11983,80],{"class":7848},[33,11985,8041],{"class":7842},[33,11987,85],{"class":7848},[33,11989,11990],{"class":7842},"])\n",[10,11992,11993],{},[14,11994,8134],{},[7819,11996,11999],{"className":11997,"code":11998,"language":1798},[8138],"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",[7826,12000,11998],{"__ignoreMap":7824},[10,12002,12003,12004,12032,12033,12036,12037,12065,12066,1191,12069,1364],{},"Here, the ",[33,12005,12007,12020],{"className":12006,"translate":37},[36],[33,12008,12010],{"className":12009},[41],[43,12011,12012],{"xmlns":45},[47,12013,12014,12018],{},[50,12015,12016],{},[53,12017,8959],{},[99,12019,8959],{"encoding":101},[33,12021,12023],{"className":12022,"ariaHidden":71},[106],[33,12024,12026,12029],{"className":12025},[110],[33,12027],{"className":12028,"style":226},[114],[33,12030,8959],{"className":12031,"style":9006},[119,230]," constructed happens to be the ",[14,12034,12035],{},"empty set",", and this particular mapping ",[33,12038,12040,12053],{"className":12039,"translate":37},[36],[33,12041,12043],{"className":12042},[41],[43,12044,12045],{"xmlns":45},[47,12046,12047,12051],{},[50,12048,12049],{},[53,12050,277],{},[99,12052,277],{"encoding":101},[33,12054,12056],{"className":12055,"ariaHidden":71},[106],[33,12057,12059,12062],{"className":12058},[110],[33,12060],{"className":12061,"style":300},[114],[33,12063,277],{"className":12064,"style":304},[119,230]," 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 ",[14,12067,12068],{},"every possible mapping",[33,12070,12072,12085],{"className":12071,"translate":37},[36],[33,12073,12075],{"className":12074},[41],[43,12076,12077],{"xmlns":45},[47,12078,12079,12083],{},[50,12080,12081],{},[53,12082,277],{},[99,12084,277],{"encoding":101},[33,12086,12088],{"className":12087,"ariaHidden":71},[106],[33,12089,12091,12094],{"className":12090},[110],[33,12092],{"className":12093,"style":300},[114],[33,12095,277],{"className":12096,"style":304},[119,230],[22,12098,12100],{"id":12099},"the-undecidability-of-the-halting-problem","The Undecidability of the Halting Problem",[10,12102,12103,12104,12107],{},"One of the most striking applications of the diagonal argument appears in Alan Turing's 1936 proof of the ",[14,12105,12106],{},"undecidability of the halting problem",". This result directly laid the foundation for the entire field of computability theory.",[10,12109,12110,12113,12114,12143,12144,12172,12173,12201,12202,12230],{},[14,12111,12112],{},"The Halting Problem."," Given a program ",[33,12115,12117,12130],{"className":12116,"translate":37},[36],[33,12118,12120],{"className":12119},[41],[43,12121,12122],{"xmlns":45},[47,12123,12124,12128],{},[50,12125,12126],{},[53,12127,8364],{},[99,12129,8364],{"encoding":101},[33,12131,12133],{"className":12132,"ariaHidden":71},[106],[33,12134,12136,12139],{"className":12135},[110],[33,12137],{"className":12138,"style":226},[114],[33,12140,8364],{"className":12141,"style":12142},[119,230],"margin-right:0.1389em;"," and an input ",[33,12145,12147,12160],{"className":12146,"translate":37},[36],[33,12148,12150],{"className":12149},[41],[43,12151,12152],{"xmlns":45},[47,12153,12154,12158],{},[50,12155,12156],{},[53,12157,2430],{},[99,12159,2430],{"encoding":101},[33,12161,12163],{"className":12162,"ariaHidden":71},[106],[33,12164,12166,12169],{"className":12165},[110],[33,12167],{"className":12168,"style":6054},[114],[33,12170,2430],{"className":12171},[119,230],", does there exist a universal algorithm capable of determining whether ",[33,12174,12176,12189],{"className":12175,"translate":37},[36],[33,12177,12179],{"className":12178},[41],[43,12180,12181],{"xmlns":45},[47,12182,12183,12187],{},[50,12184,12185],{},[53,12186,8364],{},[99,12188,8364],{"encoding":101},[33,12190,12192],{"className":12191,"ariaHidden":71},[106],[33,12193,12195,12198],{"className":12194},[110],[33,12196],{"className":12197,"style":226},[114],[33,12199,8364],{"className":12200,"style":12142},[119,230],", when run on input ",[33,12203,12205,12218],{"className":12204,"translate":37},[36],[33,12206,12208],{"className":12207},[41],[43,12209,12210],{"xmlns":45},[47,12211,12212,12216],{},[50,12213,12214],{},[53,12215,2430],{},[99,12217,2430],{"encoding":101},[33,12219,12221],{"className":12220,"ariaHidden":71},[106],[33,12222,12224,12227],{"className":12223},[110],[33,12225],{"className":12226,"style":6054},[114],[33,12228,2430],{"className":12229},[119,230],", will eventually halt (and return a result) or will run forever (enter an infinite loop)?",[10,12232,12233,12234,12237],{},"Turing proved: ",[14,12235,12236],{},"no such universal algorithm exists"," (a variant of the diagonal argument).",[10,12239,12240,12242],{},[14,12241,8639],{}," the existence of such a \"universal halting decider,\" which we formalize as a function:",[33,12244,12246],{"className":12245,"translate":37},[2403],[33,12247,12249,12336],{"className":12248,"translate":37},[36],[33,12250,12252],{"className":12251},[41],[43,12253,12254],{"xmlns":45,"display":2412},[47,12255,12256,12333],{},[50,12257,12258,12261,12263,12265,12267,12269,12271,12273],{},[92,12259,12260],{},"halts",[58,12262,389],{"stretchy":63},[53,12264,8364],{},[58,12266,72],{"separator":71},[53,12268,2430],{},[58,12270,398],{"stretchy":63},[58,12272,60],{},[50,12274,12275,12277],{},[58,12276,64],{"fence":71},[1557,12278,12279,12306],{"rowspacing":1559,"columnalign":1560,"columnspacing":1561},[1563,12280,12281,12287],{},[1566,12282,12283],{},[1569,12284,12285],{"scriptlevel":68,"displaystyle":63},[92,12286,71],{},[1566,12288,12289],{},[1569,12290,12291],{"scriptlevel":68,"displaystyle":63},[50,12292,12293,12295,12297,12299,12301,12303],{},[92,12294,4350],{},[53,12296,8364],{},[58,12298,389],{"stretchy":63},[53,12300,2430],{},[58,12302,398],{"stretchy":63},[92,12304,12305],{}," halts",[1563,12307,12308,12314],{},[1566,12309,12310],{},[1569,12311,12312],{"scriptlevel":68,"displaystyle":63},[92,12313,63],{},[1566,12315,12316],{},[1569,12317,12318],{"scriptlevel":68,"displaystyle":63},[50,12319,12320,12322,12324,12326,12328,12330],{},[92,12321,4350],{},[53,12323,8364],{},[58,12325,389],{"stretchy":63},[53,12327,2430],{},[58,12329,398],{"stretchy":63},[92,12331,12332],{}," does not halt",[99,12334,12335],{"encoding":101},"\\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}",[33,12337,12339,12378],{"className":12338,"ariaHidden":71},[106],[33,12340,12342,12345,12351,12354,12357,12360,12363,12366,12369,12372,12375],{"className":12341},[110],[33,12343],{"className":12344,"style":139},[114],[33,12346,12348],{"className":12347},[119,1798],[33,12349,12260],{"className":12350},[119],[33,12352,389],{"className":12353},[143],[33,12355,8364],{"className":12356,"style":12142},[119,230],[33,12358,72],{"className":12359},[150],[33,12361],{"className":12362,"style":154},[124],[33,12364,2430],{"className":12365},[119,230],[33,12367,398],{"className":12368},[192],[33,12370],{"className":12371,"style":125},[124],[33,12373,60],{"className":12374},[129],[33,12376],{"className":12377,"style":125},[124],[33,12379,12381,12384],{"className":12380},[110],[33,12382],{"className":12383,"style":1666},[114],[33,12385,12387,12393,12540],{"className":12386},[185],[33,12388,12390],{"className":12389,"style":1674},[143,1673],[33,12391,64],{"className":12392},[1678,1679],[33,12394,12396],{"className":12395},[119],[33,12397,12399,12450,12453],{"className":12398},[1557],[33,12400,12402],{"className":12401},[1689],[33,12403,12405,12442],{"className":12404},[438,439],[33,12406,12408,12439],{"className":12407},[443],[33,12409,12411,12425],{"className":12410,"style":1699},[447],[33,12412,12413,12416],{"style":1702},[33,12414],{"className":12415,"style":1706},[455],[33,12417,12419],{"className":12418},[119],[33,12420,12422],{"className":12421},[119,1798],[33,12423,71],{"className":12424},[119],[33,12426,12427,12430],{"style":1719},[33,12428],{"className":12429,"style":1706},[455],[33,12431,12433],{"className":12432},[119],[33,12434,12436],{"className":12435},[119,1798],[33,12437,63],{"className":12438},[119],[33,12440,471],{"className":12441},[470],[33,12443,12445],{"className":12444},[443],[33,12446,12448],{"className":12447,"style":1764},[447],[33,12449],{},[33,12451],{"className":12452,"style":1771},[1770],[33,12454,12456],{"className":12455},[1689],[33,12457,12459,12532],{"className":12458},[438,439],[33,12460,12462,12529],{"className":12461},[443],[33,12463,12465,12497],{"className":12464,"style":1699},[447],[33,12466,12467,12470],{"style":1702},[33,12468],{"className":12469,"style":1706},[455],[33,12471,12473,12479,12482,12485,12488,12491],{"className":12472},[119],[33,12474,12476],{"className":12475},[119,1798],[33,12477,4350],{"className":12478},[119],[33,12480,8364],{"className":12481,"style":12142},[119,230],[33,12483,389],{"className":12484},[143],[33,12486,2430],{"className":12487},[119,230],[33,12489,398],{"className":12490},[192],[33,12492,12494],{"className":12493},[119,1798],[33,12495,12305],{"className":12496},[119],[33,12498,12499,12502],{"style":1719},[33,12500],{"className":12501,"style":1706},[455],[33,12503,12505,12511,12514,12517,12520,12523],{"className":12504},[119],[33,12506,12508],{"className":12507},[119,1798],[33,12509,4350],{"className":12510},[119],[33,12512,8364],{"className":12513,"style":12142},[119,230],[33,12515,389],{"className":12516},[143],[33,12518,2430],{"className":12519},[119,230],[33,12521,398],{"className":12522},[192],[33,12524,12526],{"className":12525},[119,1798],[33,12527,12332],{"className":12528},[119],[33,12530,471],{"className":12531},[470],[33,12533,12535],{"className":12534},[443],[33,12536,12538],{"className":12537,"style":1764},[447],[33,12539],{},[33,12541],{"className":12542},[192,1833],[10,12544,12545],{},"This function itself must always return a correct answer within finite time (this is the meaning of \"universal\").",[10,12547,12548,1191,12551,12579],{},[14,12549,12550],{},"Construct a diagonal program",[33,12552,12554,12567],{"className":12553,"translate":37},[36],[33,12555,12557],{"className":12556},[41],[43,12558,12559],{"xmlns":45},[47,12560,12561,12565],{},[50,12562,12563],{},[53,12564,8959],{},[99,12566,8959],{"encoding":101},[33,12568,12570],{"className":12569,"ariaHidden":71},[106],[33,12571,12573,12576],{"className":12572},[110],[33,12574],{"className":12575,"style":226},[114],[33,12577,8959],{"className":12578,"style":9006},[119,230],", whose behavior is defined as follows:",[33,12581,12583],{"className":12582,"translate":37},[2403],[33,12584,12586,12677],{"className":12585,"translate":37},[36],[33,12587,12589],{"className":12588},[41],[43,12590,12591],{"xmlns":45,"display":2412},[47,12592,12593,12674],{},[50,12594,12595,12597,12599,12601,12603,12605],{},[53,12596,8959],{},[58,12598,389],{"stretchy":63},[53,12600,8364],{},[58,12602,398],{"stretchy":63},[58,12604,60],{},[50,12606,12607,12609],{},[58,12608,64],{"fence":71},[1557,12610,12611,12643],{"rowspacing":1559,"columnalign":1560,"columnspacing":1561},[1563,12612,12613,12620],{},[1566,12614,12615],{},[1569,12616,12617],{"scriptlevel":68,"displaystyle":63},[92,12618,12619],{},"infinite loop",[1566,12621,12622],{},[1569,12623,12624],{"scriptlevel":68,"displaystyle":63},[50,12625,12626,12629,12631,12633,12635,12637,12639,12641],{},[92,12627,12628],{},"if halts",[58,12630,389],{"stretchy":63},[53,12632,8364],{},[58,12634,72],{"separator":71},[53,12636,8364],{},[58,12638,398],{"stretchy":63},[58,12640,60],{},[92,12642,71],{},[1563,12644,12645,12652],{},[1566,12646,12647],{},[1569,12648,12649],{"scriptlevel":68,"displaystyle":63},[92,12650,12651],{},"halt immediately",[1566,12653,12654],{},[1569,12655,12656],{"scriptlevel":68,"displaystyle":63},[50,12657,12658,12660,12662,12664,12666,12668,12670,12672],{},[92,12659,12628],{},[58,12661,389],{"stretchy":63},[53,12663,8364],{},[58,12665,72],{"separator":71},[53,12667,8364],{},[58,12669,398],{"stretchy":63},[58,12671,60],{},[92,12673,63],{},[99,12675,12676],{"encoding":101},"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}",[33,12678,12680,12707],{"className":12679,"ariaHidden":71},[106],[33,12681,12683,12686,12689,12692,12695,12698,12701,12704],{"className":12682},[110],[33,12684],{"className":12685,"style":139},[114],[33,12687,8959],{"className":12688,"style":9006},[119,230],[33,12690,389],{"className":12691},[143],[33,12693,8364],{"className":12694,"style":12142},[119,230],[33,12696,398],{"className":12697},[192],[33,12699],{"className":12700,"style":125},[124],[33,12702,60],{"className":12703},[129],[33,12705],{"className":12706,"style":125},[124],[33,12708,12710,12713],{"className":12709},[110],[33,12711],{"className":12712,"style":1666},[114],[33,12714,12716,12722,12911],{"className":12715},[185],[33,12717,12719],{"className":12718,"style":1674},[143,1673],[33,12720,64],{"className":12721},[1678,1679],[33,12723,12725],{"className":12724},[119],[33,12726,12728,12779,12782],{"className":12727},[1557],[33,12729,12731],{"className":12730},[1689],[33,12732,12734,12771],{"className":12733},[438,439],[33,12735,12737,12768],{"className":12736},[443],[33,12738,12740,12754],{"className":12739,"style":1699},[447],[33,12741,12742,12745],{"style":1702},[33,12743],{"className":12744,"style":1706},[455],[33,12746,12748],{"className":12747},[119],[33,12749,12751],{"className":12750},[119,1798],[33,12752,12619],{"className":12753},[119],[33,12755,12756,12759],{"style":1719},[33,12757],{"className":12758,"style":1706},[455],[33,12760,12762],{"className":12761},[119],[33,12763,12765],{"className":12764},[119,1798],[33,12766,12651],{"className":12767},[119],[33,12769,471],{"className":12770},[470],[33,12772,12774],{"className":12773},[443],[33,12775,12777],{"className":12776,"style":1764},[447],[33,12778],{},[33,12780],{"className":12781,"style":1771},[1770],[33,12783,12785],{"className":12784},[1689],[33,12786,12788,12903],{"className":12787},[438,439],[33,12789,12791,12900],{"className":12790},[443],[33,12792,12794,12847],{"className":12793,"style":1699},[447],[33,12795,12796,12799],{"style":1702},[33,12797],{"className":12798,"style":1706},[455],[33,12800,12802,12808,12814,12817,12820,12823,12826,12829,12832,12835,12838,12841],{"className":12801},[119],[33,12803,12805],{"className":12804},[119,1798],[33,12806,4350],{"className":12807},[119],[33,12809,12811],{"className":12810},[119,1798],[33,12812,12260],{"className":12813},[119],[33,12815,389],{"className":12816},[143],[33,12818,8364],{"className":12819,"style":12142},[119,230],[33,12821,72],{"className":12822},[150],[33,12824],{"className":12825,"style":154},[124],[33,12827,8364],{"className":12828,"style":12142},[119,230],[33,12830,398],{"className":12831},[192],[33,12833],{"className":12834,"style":125},[124],[33,12836,60],{"className":12837},[129],[33,12839],{"className":12840,"style":125},[124],[33,12842,12844],{"className":12843},[119,1798],[33,12845,71],{"className":12846},[119],[33,12848,12849,12852],{"style":1719},[33,12850],{"className":12851,"style":1706},[455],[33,12853,12855,12861,12867,12870,12873,12876,12879,12882,12885,12888,12891,12894],{"className":12854},[119],[33,12856,12858],{"className":12857},[119,1798],[33,12859,4350],{"className":12860},[119],[33,12862,12864],{"className":12863},[119,1798],[33,12865,12260],{"className":12866},[119],[33,12868,389],{"className":12869},[143],[33,12871,8364],{"className":12872,"style":12142},[119,230],[33,12874,72],{"className":12875},[150],[33,12877],{"className":12878,"style":154},[124],[33,12880,8364],{"className":12881,"style":12142},[119,230],[33,12883,398],{"className":12884},[192],[33,12886],{"className":12887,"style":125},[124],[33,12889,60],{"className":12890},[129],[33,12892],{"className":12893,"style":125},[124],[33,12895,12897],{"className":12896},[119,1798],[33,12898,63],{"className":12899},[119],[33,12901,471],{"className":12902},[470],[33,12904,12906],{"className":12905},[443],[33,12907,12909],{"className":12908,"style":1764},[447],[33,12910],{},[33,12912],{"className":12913},[192,1833],[10,12915,12916,12917,12920,12921,12981,12982,13010,13011,13039],{},"Note the critical operation here: ",[14,12918,12919],{},"feeding a program to itself as its own input"," (",[33,12922,12924,12948],{"className":12923,"translate":37},[36],[33,12925,12927],{"className":12926},[41],[43,12928,12929],{"xmlns":45},[47,12930,12931,12945],{},[50,12932,12933,12935,12937,12939,12941,12943],{},[92,12934,12260],{},[58,12936,389],{"stretchy":63},[53,12938,8364],{},[58,12940,72],{"separator":71},[53,12942,8364],{},[58,12944,398],{"stretchy":63},[99,12946,12947],{"encoding":101},"\\text{halts}(P, P)",[33,12949,12951],{"className":12950,"ariaHidden":71},[106],[33,12952,12954,12957,12963,12966,12969,12972,12975,12978],{"className":12953},[110],[33,12955],{"className":12956,"style":139},[114],[33,12958,12960],{"className":12959},[119,1798],[33,12961,12260],{"className":12962},[119],[33,12964,389],{"className":12965},[143],[33,12967,8364],{"className":12968,"style":12142},[119,230],[33,12970,72],{"className":12971},[150],[33,12973],{"className":12974,"style":154},[124],[33,12976,8364],{"className":12977,"style":12142},[119,230],[33,12979,398],{"className":12980},[192],"). This is the direct analogue of the diagonal argument's operation of \"taking the ",[33,12983,12985,12998],{"className":12984,"translate":37},[36],[33,12986,12988],{"className":12987},[41],[43,12989,12990],{"xmlns":45},[47,12991,12992,12996],{},[50,12993,12994],{},[53,12995,1309],{},[99,12997,1309],{"encoding":101},[33,12999,13001],{"className":13000,"ariaHidden":71},[106],[33,13002,13004,13007],{"className":13003},[110],[33,13005],{"className":13006,"style":6054},[114],[33,13008,1309],{"className":13009},[119,230],"-th digit of the ",[33,13012,13014,13027],{"className":13013,"translate":37},[36],[33,13015,13017],{"className":13016},[41],[43,13018,13019],{"xmlns":45},[47,13020,13021,13025],{},[50,13022,13023],{},[53,13024,1309],{},[99,13026,1309],{"encoding":101},[33,13028,13030],{"className":13029,"ariaHidden":71},[106],[33,13031,13033,13036],{"className":13032},[110],[33,13034],{"className":13035,"style":6054},[114],[33,13037,1309],{"className":13038},[119,230],"-th number\" — the self-referential move of making a program \"examine\" itself.",[10,13041,13042,13045,13046,1364],{},[14,13043,13044],{},"Deriving the contradiction."," We now examine the concrete execution ",[33,13047,13049,13069],{"className":13048,"translate":37},[36],[33,13050,13052],{"className":13051},[41],[43,13053,13054],{"xmlns":45},[47,13055,13056,13066],{},[50,13057,13058,13060,13062,13064],{},[53,13059,8959],{},[58,13061,389],{"stretchy":63},[53,13063,8959],{},[58,13065,398],{"stretchy":63},[99,13067,13068],{"encoding":101},"D(D)",[33,13070,13072],{"className":13071,"ariaHidden":71},[106],[33,13073,13075,13078,13081,13084,13087],{"className":13074},[110],[33,13076],{"className":13077,"style":139},[114],[33,13079,8959],{"className":13080,"style":9006},[119,230],[33,13082,389],{"className":13083},[143],[33,13085,8959],{"className":13086,"style":9006},[119,230],[33,13088,398],{"className":13089},[192],[343,13091,13092,13301],{},[346,13093,9450,13094,13180,13181,13224,13225,8041,13253,13296,13297,13300],{},[33,13095,13097,13125],{"className":13096,"translate":37},[36],[33,13098,13100],{"className":13099},[41],[43,13101,13102],{"xmlns":45},[47,13103,13104,13122],{},[50,13105,13106,13108,13110,13112,13114,13116,13118,13120],{},[92,13107,12260],{},[58,13109,389],{"stretchy":63},[53,13111,8959],{},[58,13113,72],{"separator":71},[53,13115,8959],{},[58,13117,398],{"stretchy":63},[58,13119,60],{},[92,13121,71],{},[99,13123,13124],{"encoding":101},"\\text{halts}(D, D) = \\text{true}",[33,13126,13128,13167],{"className":13127,"ariaHidden":71},[106],[33,13129,13131,13134,13140,13143,13146,13149,13152,13155,13158,13161,13164],{"className":13130},[110],[33,13132],{"className":13133,"style":139},[114],[33,13135,13137],{"className":13136},[119,1798],[33,13138,12260],{"className":13139},[119],[33,13141,389],{"className":13142},[143],[33,13144,8959],{"className":13145,"style":9006},[119,230],[33,13147,72],{"className":13148},[150],[33,13150],{"className":13151,"style":154},[124],[33,13153,8959],{"className":13154,"style":9006},[119,230],[33,13156,398],{"className":13157},[192],[33,13159],{"className":13160,"style":125},[124],[33,13162,60],{"className":13163},[129],[33,13165],{"className":13166,"style":125},[124],[33,13168,13170,13174],{"className":13169},[110],[33,13171],{"className":13172,"style":13173},[114],"height:0.6151em;",[33,13175,13177],{"className":13176},[119,1798],[33,13178,71],{"className":13179},[119]," (i.e., it is judged that \"",[33,13182,13184,13203],{"className":13183,"translate":37},[36],[33,13185,13187],{"className":13186},[41],[43,13188,13189],{"xmlns":45},[47,13190,13191,13201],{},[50,13192,13193,13195,13197,13199],{},[53,13194,8959],{},[58,13196,389],{"stretchy":63},[53,13198,8959],{},[58,13200,398],{"stretchy":63},[99,13202,13068],{"encoding":101},[33,13204,13206],{"className":13205,"ariaHidden":71},[106],[33,13207,13209,13212,13215,13218,13221],{"className":13208},[110],[33,13210],{"className":13211,"style":139},[114],[33,13213,8959],{"className":13214,"style":9006},[119,230],[33,13216,389],{"className":13217},[143],[33,13219,8959],{"className":13220,"style":9006},[119,230],[33,13222,398],{"className":13223},[192]," halts\"), then by the definition of ",[33,13226,13228,13241],{"className":13227,"translate":37},[36],[33,13229,13231],{"className":13230},[41],[43,13232,13233],{"xmlns":45},[47,13234,13235,13239],{},[50,13236,13237],{},[53,13238,8959],{},[99,13240,8959],{"encoding":101},[33,13242,13244],{"className":13243,"ariaHidden":71},[106],[33,13245,13247,13250],{"className":13246},[110],[33,13248],{"className":13249,"style":226},[114],[33,13251,8959],{"className":13252,"style":9006},[119,230],[33,13254,13256,13275],{"className":13255,"translate":37},[36],[33,13257,13259],{"className":13258},[41],[43,13260,13261],{"xmlns":45},[47,13262,13263,13273],{},[50,13264,13265,13267,13269,13271],{},[53,13266,8959],{},[58,13268,389],{"stretchy":63},[53,13270,8959],{},[58,13272,398],{"stretchy":63},[99,13274,13068],{"encoding":101},[33,13276,13278],{"className":13277,"ariaHidden":71},[106],[33,13279,13281,13284,13287,13290,13293],{"className":13280},[110],[33,13282],{"className":13283,"style":139},[114],[33,13285,8959],{"className":13286,"style":9006},[119,230],[33,13288,389],{"className":13289},[143],[33,13291,8959],{"className":13292,"style":9006},[119,230],[33,13294,398],{"className":13295},[192]," should ",[14,13298,13299],{},"loop infinitely",". This contradicts \"it halts.\"",[346,13302,9450,13303,13180,13388,13431,13432,8041,13460,13296,13503,13506],{},[33,13304,13306,13334],{"className":13305,"translate":37},[36],[33,13307,13309],{"className":13308},[41],[43,13310,13311],{"xmlns":45},[47,13312,13313,13331],{},[50,13314,13315,13317,13319,13321,13323,13325,13327,13329],{},[92,13316,12260],{},[58,13318,389],{"stretchy":63},[53,13320,8959],{},[58,13322,72],{"separator":71},[53,13324,8959],{},[58,13326,398],{"stretchy":63},[58,13328,60],{},[92,13330,63],{},[99,13332,13333],{"encoding":101},"\\text{halts}(D, D) = \\text{false}",[33,13335,13337,13376],{"className":13336,"ariaHidden":71},[106],[33,13338,13340,13343,13349,13352,13355,13358,13361,13364,13367,13370,13373],{"className":13339},[110],[33,13341],{"className":13342,"style":139},[114],[33,13344,13346],{"className":13345},[119,1798],[33,13347,12260],{"className":13348},[119],[33,13350,389],{"className":13351},[143],[33,13353,8959],{"className":13354,"style":9006},[119,230],[33,13356,72],{"className":13357},[150],[33,13359],{"className":13360,"style":154},[124],[33,13362,8959],{"className":13363,"style":9006},[119,230],[33,13365,398],{"className":13366},[192],[33,13368],{"className":13369,"style":125},[124],[33,13371,60],{"className":13372},[129],[33,13374],{"className":13375,"style":125},[124],[33,13377,13379,13382],{"className":13378},[110],[33,13380],{"className":13381,"style":916},[114],[33,13383,13385],{"className":13384},[119,1798],[33,13386,63],{"className":13387},[119],[33,13389,13391,13410],{"className":13390,"translate":37},[36],[33,13392,13394],{"className":13393},[41],[43,13395,13396],{"xmlns":45},[47,13397,13398,13408],{},[50,13399,13400,13402,13404,13406],{},[53,13401,8959],{},[58,13403,389],{"stretchy":63},[53,13405,8959],{},[58,13407,398],{"stretchy":63},[99,13409,13068],{"encoding":101},[33,13411,13413],{"className":13412,"ariaHidden":71},[106],[33,13414,13416,13419,13422,13425,13428],{"className":13415},[110],[33,13417],{"className":13418,"style":139},[114],[33,13420,8959],{"className":13421,"style":9006},[119,230],[33,13423,389],{"className":13424},[143],[33,13426,8959],{"className":13427,"style":9006},[119,230],[33,13429,398],{"className":13430},[192]," does not halt\"), then by the definition of ",[33,13433,13435,13448],{"className":13434,"translate":37},[36],[33,13436,13438],{"className":13437},[41],[43,13439,13440],{"xmlns":45},[47,13441,13442,13446],{},[50,13443,13444],{},[53,13445,8959],{},[99,13447,8959],{"encoding":101},[33,13449,13451],{"className":13450,"ariaHidden":71},[106],[33,13452,13454,13457],{"className":13453},[110],[33,13455],{"className":13456,"style":226},[114],[33,13458,8959],{"className":13459,"style":9006},[119,230],[33,13461,13463,13482],{"className":13462,"translate":37},[36],[33,13464,13466],{"className":13465},[41],[43,13467,13468],{"xmlns":45},[47,13469,13470,13480],{},[50,13471,13472,13474,13476,13478],{},[53,13473,8959],{},[58,13475,389],{"stretchy":63},[53,13477,8959],{},[58,13479,398],{"stretchy":63},[99,13481,13068],{"encoding":101},[33,13483,13485],{"className":13484,"ariaHidden":71},[106],[33,13486,13488,13491,13494,13497,13500],{"className":13487},[110],[33,13489],{"className":13490,"style":139},[114],[33,13492,8959],{"className":13493,"style":9006},[119,230],[33,13495,389],{"className":13496},[143],[33,13498,8959],{"className":13499,"style":9006},[119,230],[33,13501,398],{"className":13502},[192],[14,13504,13505],{},"halt immediately",". This contradicts \"it does not halt.\"",[10,13508,13509,13510,13542,13543],{},"Both cases lead to self-contradiction. Hence the assumption that \"a universal halting decider ",[33,13511,13513,13527],{"className":13512,"translate":37},[36],[33,13514,13516],{"className":13515},[41],[43,13517,13518],{"xmlns":45},[47,13519,13520,13524],{},[50,13521,13522],{},[92,13523,12260],{},[99,13525,13526],{"encoding":101},"\\text{halts}",[33,13528,13530],{"className":13529,"ariaHidden":71},[106],[33,13531,13533,13536],{"className":13532},[110],[33,13534],{"className":13535,"style":916},[114],[33,13537,13539],{"className":13538},[119,1798],[33,13540,12260],{"className":13541},[119]," exists\" is false. ",[14,13544,13545],{},"The halting problem is undecidable.",[10,13547,13548,13549,13742],{},"If one enumerates all possible programs according to some rule (",[33,13550,13552,13590],{"className":13551,"translate":37},[36],[33,13553,13555],{"className":13554},[41],[43,13556,13557],{"xmlns":45},[47,13558,13559,13587],{},[50,13560,13561,13567,13569,13575,13577,13583,13585],{},[365,13562,13563,13565],{},[53,13564,8364],{},[66,13566,75],{},[58,13568,72],{"separator":71},[365,13570,13571,13573],{},[53,13572,8364],{},[66,13574,80],{},[58,13576,72],{"separator":71},[365,13578,13579,13581],{},[53,13580,8364],{},[66,13582,85],{},[58,13584,72],{"separator":71},[58,13586,90],{},[99,13588,13589],{"encoding":101},"P_1, P_2, P_3, \\dots",[33,13591,13593],{"className":13592,"ariaHidden":71},[106],[33,13594,13596,13600,13641,13644,13647,13687,13690,13693,13733,13736,13739],{"className":13595},[110],[33,13597],{"className":13598,"style":13599},[114],"height:0.8778em;vertical-align:-0.1944em;",[33,13601,13603,13606],{"className":13602},[119],[33,13604,8364],{"className":13605,"style":12142},[119,230],[33,13607,13609],{"className":13608},[434],[33,13610,13612,13633],{"className":13611},[438,439],[33,13613,13615,13630],{"className":13614},[443],[33,13616,13618],{"className":13617,"style":448},[447],[33,13619,13621,13624],{"style":13620},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[33,13622],{"className":13623,"style":456},[455],[33,13625,13627],{"className":13626},[460,461,462,463],[33,13628,75],{"className":13629},[119,463],[33,13631,471],{"className":13632},[470],[33,13634,13636],{"className":13635},[443],[33,13637,13639],{"className":13638,"style":478},[447],[33,13640],{},[33,13642,72],{"className":13643},[150],[33,13645],{"className":13646,"style":154},[124],[33,13648,13650,13653],{"className":13649},[119],[33,13651,8364],{"className":13652,"style":12142},[119,230],[33,13654,13656],{"className":13655},[434],[33,13657,13659,13679],{"className":13658},[438,439],[33,13660,13662,13676],{"className":13661},[443],[33,13663,13665],{"className":13664,"style":448},[447],[33,13666,13667,13670],{"style":13620},[33,13668],{"className":13669,"style":456},[455],[33,13671,13673],{"className":13672},[460,461,462,463],[33,13674,80],{"className":13675},[119,463],[33,13677,471],{"className":13678},[470],[33,13680,13682],{"className":13681},[443],[33,13683,13685],{"className":13684,"style":478},[447],[33,13686],{},[33,13688,72],{"className":13689},[150],[33,13691],{"className":13692,"style":154},[124],[33,13694,13696,13699],{"className":13695},[119],[33,13697,8364],{"className":13698,"style":12142},[119,230],[33,13700,13702],{"className":13701},[434],[33,13703,13705,13725],{"className":13704},[438,439],[33,13706,13708,13722],{"className":13707},[443],[33,13709,13711],{"className":13710,"style":448},[447],[33,13712,13713,13716],{"style":13620},[33,13714],{"className":13715,"style":456},[455],[33,13717,13719],{"className":13718},[460,461,462,463],[33,13720,85],{"className":13721},[119,463],[33,13723,471],{"className":13724},[470],[33,13726,13728],{"className":13727},[443],[33,13729,13731],{"className":13730,"style":478},[447],[33,13732],{},[33,13734,72],{"className":13735},[150],[33,13737],{"className":13738,"style":154},[124],[33,13740,90],{"className":13741},[185]," — this is feasible, since programs are essentially finite-length strings) and imagines an infinite table:",[33,13744,13746],{"className":13745,"translate":37},[2403],[33,13747,13749,13914],{"className":13748,"translate":37},[36],[33,13750,13752],{"className":13751},[41],[43,13753,13754],{"xmlns":45,"display":2412},[47,13755,13756,13911],{},[50,13757,13758,13760,13909],{},[58,13759,389],{"fence":71},[1557,13761,13763,13823,13883],{"rowspacing":6688,"columnalign":13762,"columnspacing":1561},"center center center",[1563,13764,13765,13791,13817],{},[1566,13766,13767],{},[1569,13768,13769],{"scriptlevel":68,"displaystyle":63},[50,13770,13771,13773,13775,13781,13783,13789],{},[92,13772,12260],{},[58,13774,389],{"stretchy":63},[365,13776,13777,13779],{},[53,13778,8364],{},[66,13780,75],{},[58,13782,72],{"separator":71},[365,13784,13785,13787],{},[53,13786,8364],{},[66,13788,75],{},[58,13790,398],{"stretchy":63},[1566,13792,13793],{},[1569,13794,13795],{"scriptlevel":68,"displaystyle":63},[50,13796,13797,13799,13801,13807,13809,13815],{},[92,13798,12260],{},[58,13800,389],{"stretchy":63},[365,13802,13803,13805],{},[53,13804,8364],{},[66,13806,75],{},[58,13808,72],{"separator":71},[365,13810,13811,13813],{},[53,13812,8364],{},[66,13814,80],{},[58,13816,398],{"stretchy":63},[1566,13818,13819],{},[1569,13820,13821],{"scriptlevel":68,"displaystyle":63},[58,13822,6738],{"lspace":2419,"rspace":2419},[1563,13824,13825,13851,13877],{},[1566,13826,13827],{},[1569,13828,13829],{"scriptlevel":68,"displaystyle":63},[50,13830,13831,13833,13835,13841,13843,13849],{},[92,13832,12260],{},[58,13834,389],{"stretchy":63},[365,13836,13837,13839],{},[53,13838,8364],{},[66,13840,80],{},[58,13842,72],{"separator":71},[365,13844,13845,13847],{},[53,13846,8364],{},[66,13848,75],{},[58,13850,398],{"stretchy":63},[1566,13852,13853],{},[1569,13854,13855],{"scriptlevel":68,"displaystyle":63},[50,13856,13857,13859,13861,13867,13869,13875],{},[92,13858,12260],{},[58,13860,389],{"stretchy":63},[365,13862,13863,13865],{},[53,13864,8364],{},[66,13866,80],{},[58,13868,72],{"separator":71},[365,13870,13871,13873],{},[53,13872,8364],{},[66,13874,80],{},[58,13876,398],{"stretchy":63},[1566,13878,13879],{},[1569,13880,13881],{"scriptlevel":68,"displaystyle":63},[58,13882,6738],{"lspace":2419,"rspace":2419},[1563,13884,13885,13897,13903],{},[1566,13886,13887],{},[1569,13888,13889],{"scriptlevel":68,"displaystyle":63},[50,13890,13891,13893],{},[53,13892,2686],{"mathvariant":374},[2688,13894,13895],{"height":2419,"voffset":2419},[124,13896],{"mathbackground":2692,"width":2419,"height":2693},[1566,13898,13899],{},[1569,13900,13901],{"scriptlevel":68,"displaystyle":63},[50,13902],{},[1566,13904,13905],{},[1569,13906,13907],{"scriptlevel":68,"displaystyle":63},[58,13908,6875],{"lspace":2419,"rspace":2419},[58,13910,398],{"fence":71},[99,13912,13913],{"encoding":101},"\\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}",[33,13915,13917],{"className":13916,"ariaHidden":71},[106],[33,13918,13920,13924],{"className":13919},[110],[33,13921],{"className":13922,"style":13923},[114],"height:4.26em;vertical-align:-1.88em;",[33,13925,13927,13972,14547],{"className":13926},[185],[33,13928,13930],{"className":13929},[143],[33,13931,13933],{"className":13932},[1678,6900],[33,13934,13936,13963],{"className":13935},[438,439],[33,13937,13939,13960],{"className":13938},[443],[33,13940,13943],{"className":13941,"style":13942},[447],"height:2.35em;",[33,13944,13946,13950],{"style":13945},"top:-4.35em;",[33,13947],{"className":13948,"style":13949},[455],"height:6.2em;",[33,13951,13953],{"style":13952},"width:0.875em;height:4.2em;",[6922,13954,13957],{"xmlns":6924,"width":6925,"height":13955,"viewBox":13956},"4.2em","0 0 875 4200",[6929,13958],{"d":13959},"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",[33,13961,471],{"className":13962},[470],[33,13964,13966],{"className":13965},[443],[33,13967,13970],{"className":13968,"style":13969},[447],"height:1.85em;",[33,13971],{},[33,13973,13975],{"className":13974},[119],[33,13976,13978,14234,14237,14240,14485,14488,14491],{"className":13977},[1557],[33,13979,13981],{"className":13980},[6953],[33,13982,13984,14225],{"className":13983},[438,439],[33,13985,13987,14222],{"className":13986},[443],[33,13988,13991,14098,14204],{"className":13989,"style":13990},[447],"height:2.38em;",[33,13992,13994,13997],{"style":13993},"top:-5.2275em;",[33,13995],{"className":13996,"style":2963},[455],[33,13998,14000,14006,14009,14049,14052,14055,14095],{"className":13999},[119],[33,14001,14003],{"className":14002},[119,1798],[33,14004,12260],{"className":14005},[119],[33,14007,389],{"className":14008},[143],[33,14010,14012,14015],{"className":14011},[119],[33,14013,8364],{"className":14014,"style":12142},[119,230],[33,14016,14018],{"className":14017},[434],[33,14019,14021,14041],{"className":14020},[438,439],[33,14022,14024,14038],{"className":14023},[443],[33,14025,14027],{"className":14026,"style":448},[447],[33,14028,14029,14032],{"style":13620},[33,14030],{"className":14031,"style":456},[455],[33,14033,14035],{"className":14034},[460,461,462,463],[33,14036,75],{"className":14037},[119,463],[33,14039,471],{"className":14040},[470],[33,14042,14044],{"className":14043},[443],[33,14045,14047],{"className":14046,"style":478},[447],[33,14048],{},[33,14050,72],{"className":14051},[150],[33,14053],{"className":14054,"style":154},[124],[33,14056,14058,14061],{"className":14057},[119],[33,14059,8364],{"className":14060,"style":12142},[119,230],[33,14062,14064],{"className":14063},[434],[33,14065,14067,14087],{"className":14066},[438,439],[33,14068,14070,14084],{"className":14069},[443],[33,14071,14073],{"className":14072,"style":448},[447],[33,14074,14075,14078],{"style":13620},[33,14076],{"className":14077,"style":456},[455],[33,14079,14081],{"className":14080},[460,461,462,463],[33,14082,75],{"className":14083},[119,463],[33,14085,471],{"className":14086},[470],[33,14088,14090],{"className":14089},[443],[33,14091,14093],{"className":14092,"style":478},[447],[33,14094],{},[33,14096,398],{"className":14097},[192],[33,14099,14100,14103],{"style":3382},[33,14101],{"className":14102,"style":2963},[455],[33,14104,14106,14112,14115,14155,14158,14161,14201],{"className":14105},[119],[33,14107,14109],{"className":14108},[119,1798],[33,14110,12260],{"className":14111},[119],[33,14113,389],{"className":14114},[143],[33,14116,14118,14121],{"className":14117},[119],[33,14119,8364],{"className":14120,"style":12142},[119,230],[33,14122,14124],{"className":14123},[434],[33,14125,14127,14147],{"className":14126},[438,439],[33,14128,14130,14144],{"className":14129},[443],[33,14131,14133],{"className":14132,"style":448},[447],[33,14134,14135,14138],{"style":13620},[33,14136],{"className":14137,"style":456},[455],[33,14139,14141],{"className":14140},[460,461,462,463],[33,14142,80],{"className":14143},[119,463],[33,14145,471],{"className":14146},[470],[33,14148,14150],{"className":14149},[443],[33,14151,14153],{"className":14152,"style":478},[447],[33,14154],{},[33,14156,72],{"className":14157},[150],[33,14159],{"className":14160,"style":154},[124],[33,14162,14164,14167],{"className":14163},[119],[33,14165,8364],{"className":14166,"style":12142},[119,230],[33,14168,14170],{"className":14169},[434],[33,14171,14173,14193],{"className":14172},[438,439],[33,14174,14176,14190],{"className":14175},[443],[33,14177,14179],{"className":14178,"style":448},[447],[33,14180,14181,14184],{"style":13620},[33,14182],{"className":14183,"style":456},[455],[33,14185,14187],{"className":14186},[460,461,462,463],[33,14188,75],{"className":14189},[119,463],[33,14191,471],{"className":14192},[470],[33,14194,14196],{"className":14195},[443],[33,14197,14199],{"className":14198,"style":478},[447],[33,14200],{},[33,14202,398],{"className":14203},[192],[33,14205,14207,14210],{"style":14206},"top:-2.1675em;",[33,14208],{"className":14209,"style":2963},[455],[33,14211,14213],{"className":14212},[119],[33,14214,14216,14219],{"className":14215},[119],[33,14217,2686],{"className":14218},[119],[33,14220],{"className":14221,"style":3831},[119,3830],[33,14223,471],{"className":14224},[470],[33,14226,14228],{"className":14227},[443],[33,14229,14232],{"className":14230,"style":14231},[447],"height:1.88em;",[33,14233],{},[33,14235],{"className":14236,"style":7207},[1770],[33,14238],{"className":14239,"style":7207},[1770],[33,14241,14243],{"className":14242},[6953],[33,14244,14246,14477],{"className":14245},[438,439],[33,14247,14249,14474],{"className":14248},[443],[33,14250,14252,14359,14465],{"className":14251,"style":13990},[447],[33,14253,14255,14258],{"style":14254},"top:-5.04em;",[33,14256],{"className":14257,"style":2733},[455],[33,14259,14261,14267,14270,14310,14313,14316,14356],{"className":14260},[119],[33,14262,14264],{"className":14263},[119,1798],[33,14265,12260],{"className":14266},[119],[33,14268,389],{"className":14269},[143],[33,14271,14273,14276],{"className":14272},[119],[33,14274,8364],{"className":14275,"style":12142},[119,230],[33,14277,14279],{"className":14278},[434],[33,14280,14282,14302],{"className":14281},[438,439],[33,14283,14285,14299],{"className":14284},[443],[33,14286,14288],{"className":14287,"style":448},[447],[33,14289,14290,14293],{"style":13620},[33,14291],{"className":14292,"style":456},[455],[33,14294,14296],{"className":14295},[460,461,462,463],[33,14297,75],{"className":14298},[119,463],[33,14300,471],{"className":14301},[470],[33,14303,14305],{"className":14304},[443],[33,14306,14308],{"className":14307,"style":478},[447],[33,14309],{},[33,14311,72],{"className":14312},[150],[33,14314],{"className":14315,"style":154},[124],[33,14317,14319,14322],{"className":14318},[119],[33,14320,8364],{"className":14321,"style":12142},[119,230],[33,14323,14325],{"className":14324},[434],[33,14326,14328,14348],{"className":14327},[438,439],[33,14329,14331,14345],{"className":14330},[443],[33,14332,14334],{"className":14333,"style":448},[447],[33,14335,14336,14339],{"style":13620},[33,14337],{"className":14338,"style":456},[455],[33,14340,14342],{"className":14341},[460,461,462,463],[33,14343,80],{"className":14344},[119,463],[33,14346,471],{"className":14347},[470],[33,14349,14351],{"className":14350},[443],[33,14352,14354],{"className":14353,"style":478},[447],[33,14355],{},[33,14357,398],{"className":14358},[192],[33,14360,14361,14364],{"style":2828},[33,14362],{"className":14363,"style":2733},[455],[33,14365,14367,14373,14376,14416,14419,14422,14462],{"className":14366},[119],[33,14368,14370],{"className":14369},[119,1798],[33,14371,12260],{"className":14372},[119],[33,14374,389],{"className":14375},[143],[33,14377,14379,14382],{"className":14378},[119],[33,14380,8364],{"className":14381,"style":12142},[119,230],[33,14383,14385],{"className":14384},[434],[33,14386,14388,14408],{"className":14387},[438,439],[33,14389,14391,14405],{"className":14390},[443],[33,14392,14394],{"className":14393,"style":448},[447],[33,14395,14396,14399],{"style":13620},[33,14397],{"className":14398,"style":456},[455],[33,14400,14402],{"className":14401},[460,461,462,463],[33,14403,80],{"className":14404},[119,463],[33,14406,471],{"className":14407},[470],[33,14409,14411],{"className":14410},[443],[33,14412,14414],{"className":14413,"style":478},[447],[33,14415],{},[33,14417,72],{"className":14418},[150],[33,14420],{"className":14421,"style":154},[124],[33,14423,14425,14428],{"className":14424},[119],[33,14426,8364],{"className":14427,"style":12142},[119,230],[33,14429,14431],{"className":14430},[434],[33,14432,14434,14454],{"className":14433},[438,439],[33,14435,14437,14451],{"className":14436},[443],[33,14438,14440],{"className":14439,"style":448},[447],[33,14441,14442,14445],{"style":13620},[33,14443],{"className":14444,"style":456},[455],[33,14446,14448],{"className":14447},[460,461,462,463],[33,14449,80],{"className":14450},[119,463],[33,14452,471],{"className":14453},[470],[33,14455,14457],{"className":14456},[443],[33,14458,14460],{"className":14459,"style":478},[447],[33,14461],{},[33,14463,398],{"className":14464},[192],[33,14466,14468,14471],{"style":14467},"top:-1.98em;",[33,14469],{"className":14470,"style":2733},[455],[33,14472],{"className":14473},[119],[33,14475,471],{"className":14476},[470],[33,14478,14480],{"className":14479},[443],[33,14481,14483],{"className":14482,"style":14231},[447],[33,14484],{},[33,14486],{"className":14487,"style":7207},[1770],[33,14489],{"className":14490,"style":7207},[1770],[33,14492,14494],{"className":14493},[6953],[33,14495,14497,14539],{"className":14496},[438,439],[33,14498,14500,14536],{"className":14499},[443],[33,14501,14503,14514,14525],{"className":14502,"style":13990},[447],[33,14504,14505,14508],{"style":14254},[33,14506],{"className":14507,"style":2733},[455],[33,14509,14511],{"className":14510},[119],[33,14512,6738],{"className":14513},[185],[33,14515,14516,14519],{"style":2828},[33,14517],{"className":14518,"style":2733},[455],[33,14520,14522],{"className":14521},[119],[33,14523,6738],{"className":14524},[185],[33,14526,14527,14530],{"style":14467},[33,14528],{"className":14529,"style":2733},[455],[33,14531,14533],{"className":14532},[119],[33,14534,6875],{"className":14535},[185],[33,14537,471],{"className":14538},[470],[33,14540,14542],{"className":14541},[443],[33,14543,14545],{"className":14544,"style":14231},[447],[33,14546],{},[33,14548,14550],{"className":14549},[192],[33,14551,14553],{"className":14552},[1678,6900],[33,14554,14556,14577],{"className":14555},[438,439],[33,14557,14559,14574],{"className":14558},[443],[33,14560,14562],{"className":14561,"style":13942},[447],[33,14563,14564,14567],{"style":13945},[33,14565],{"className":14566,"style":13949},[455],[33,14568,14569],{"style":13952},[6922,14570,14571],{"xmlns":6924,"width":6925,"height":13955,"viewBox":13956},[6929,14572],{"d":14573},"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",[33,14575,471],{"className":14576},[470],[33,14578,14580],{"className":14579},[443],[33,14581,14583],{"className":14582,"style":13969},[447],[33,14584],{},[10,14586,14587,14588,14616,14617,1191,14619,14761,14762,14765],{},"The behavior of the diagonal program ",[33,14589,14591,14604],{"className":14590,"translate":37},[36],[33,14592,14594],{"className":14593},[41],[43,14595,14596],{"xmlns":45},[47,14597,14598,14602],{},[50,14599,14600],{},[53,14601,8959],{},[99,14603,8959],{"encoding":101},[33,14605,14607],{"className":14606,"ariaHidden":71},[106],[33,14608,14610,14613],{"className":14609},[110],[33,14611],{"className":14612,"style":226},[114],[33,14614,8959],{"className":14615,"style":9006},[119,230]," depends precisely on the ",[14,14618,6667],{},[33,14620,14622,14654],{"className":14621,"translate":37},[36],[33,14623,14625],{"className":14624},[41],[43,14626,14627],{"xmlns":45},[47,14628,14629,14651],{},[50,14630,14631,14633,14635,14641,14643,14649],{},[92,14632,12260],{},[58,14634,389],{"stretchy":63},[365,14636,14637,14639],{},[53,14638,8364],{},[53,14640,3866],{},[58,14642,72],{"separator":71},[365,14644,14645,14647],{},[53,14646,8364],{},[53,14648,3866],{},[58,14650,398],{"stretchy":63},[99,14652,14653],{"encoding":101},"\\text{halts}(P_i, P_i)",[33,14655,14657],{"className":14656,"ariaHidden":71},[106],[33,14658,14660,14663,14669,14672,14712,14715,14718,14758],{"className":14659},[110],[33,14661],{"className":14662,"style":139},[114],[33,14664,14666],{"className":14665},[119,1798],[33,14667,12260],{"className":14668},[119],[33,14670,389],{"className":14671},[143],[33,14673,14675,14678],{"className":14674},[119],[33,14676,8364],{"className":14677,"style":12142},[119,230],[33,14679,14681],{"className":14680},[434],[33,14682,14684,14704],{"className":14683},[438,439],[33,14685,14687,14701],{"className":14686},[443],[33,14688,14690],{"className":14689,"style":3901},[447],[33,14691,14692,14695],{"style":13620},[33,14693],{"className":14694,"style":456},[455],[33,14696,14698],{"className":14697},[460,461,462,463],[33,14699,3866],{"className":14700},[119,230,463],[33,14702,471],{"className":14703},[470],[33,14705,14707],{"className":14706},[443],[33,14708,14710],{"className":14709,"style":478},[447],[33,14711],{},[33,14713,72],{"className":14714},[150],[33,14716],{"className":14717,"style":154},[124],[33,14719,14721,14724],{"className":14720},[119],[33,14722,8364],{"className":14723,"style":12142},[119,230],[33,14725,14727],{"className":14726},[434],[33,14728,14730,14750],{"className":14729},[438,439],[33,14731,14733,14747],{"className":14732},[443],[33,14734,14736],{"className":14735,"style":3901},[447],[33,14737,14738,14741],{"style":13620},[33,14739],{"className":14740,"style":456},[455],[33,14742,14744],{"className":14743},[460,461,462,463],[33,14745,3866],{"className":14746},[119,230,463],[33,14748,471],{"className":14749},[470],[33,14751,14753],{"className":14752},[443],[33,14754,14756],{"className":14755,"style":478},[447],[33,14757],{},[33,14759,398],{"className":14760},[192]," of this table, and it deliberately behaves in the opposite manner. This is ",[14,14763,14764],{},"structurally identical"," to Cantor's construction of a real number \"each of whose digits differs from the corresponding diagonal digit.\"",[10,14767,14768],{},"We cannot actually implement a universal halting decider (since none exists!), but we can demonstrate through code how this logical contradiction concretely arises:",[7819,14770,14772],{"className":7821,"code":14771,"language":7823,"meta":7824,"style":7824},"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",[7826,14773,14774,14782,14792,14796,14812,14817,14825,14835,14852,14862,14869,14873,14889,14904,14918,14936,14943,14956,14963,14970,14974,14995],{"__ignoreMap":7824},[33,14775,14776,14779],{"class":7830,"line":7831},[33,14777,14778],{"class":7834},"import",[33,14780,14781],{"class":7842}," sys\n",[33,14783,14784,14787,14790],{"class":7830,"line":7855},[33,14785,14786],{"class":7842},"sys.setrecursionlimit(",[33,14788,14789],{"class":7848},"50",[33,14791,7933],{"class":7842},[33,14793,14794],{"class":7830,"line":7862},[33,14795,8016],{"emptyLinePlaceholder":8015},[33,14797,14798,14800,14803,14806,14808,14810],{"class":7830,"line":7868},[33,14799,7835],{"class":7834},[33,14801,14802],{"class":7838}," hypothetical_halts",[33,14804,14805],{"class":7842},"(f, x, fuel",[33,14807,60],{"class":7834},[33,14809,7849],{"class":7848},[33,14811,7852],{"class":7842},[33,14813,14814],{"class":7830,"line":7874},[33,14815,14816],{"class":7858},"    \"\"\"Hypothetical halting decider (for paradox demonstration only; not a genuinely feasible implementation)\"\"\"\n",[33,14818,14819,14822],{"class":7830,"line":7879},[33,14820,14821],{"class":7834},"    try",[33,14823,14824],{"class":7842},":\n",[33,14826,14827,14830,14832],{"class":7830,"line":7890},[33,14828,14829],{"class":7842},"        result ",[33,14831,60],{"class":7834},[33,14833,14834],{"class":7842}," f(x, fuel)\n",[33,14836,14837,14840,14843,14846,14849],{"class":7830,"line":7908},[33,14838,14839],{"class":7834},"        return",[33,14841,14842],{"class":7842}," result ",[33,14844,14845],{"class":7834},"is",[33,14847,14848],{"class":7834}," not",[33,14850,14851],{"class":7848}," None\n",[33,14853,14854,14857,14860],{"class":7830,"line":7936},[33,14855,14856],{"class":7834},"    except",[33,14858,14859],{"class":7848}," RecursionError",[33,14861,14824],{"class":7842},[33,14863,14864,14866],{"class":7830,"line":7954},[33,14865,14839],{"class":7834},[33,14867,14868],{"class":7848}," False\n",[33,14870,14871],{"class":7830,"line":7982},[33,14872,8016],{"emptyLinePlaceholder":8015},[33,14874,14875,14877,14880,14883,14885,14887],{"class":7830,"line":7994},[33,14876,7835],{"class":7834},[33,14878,14879],{"class":7838}," D",[33,14881,14882],{"class":7842},"(x, fuel",[33,14884,60],{"class":7834},[33,14886,7849],{"class":7848},[33,14888,7852],{"class":7842},[33,14890,14891,14894,14897,14900,14902],{"class":7830,"line":8012},[33,14892,14893],{"class":7834},"    if",[33,14895,14896],{"class":7842}," fuel ",[33,14898,14899],{"class":7834},"\u003C=",[33,14901,11766],{"class":7848},[33,14903,14824],{"class":7842},[33,14905,14906,14909,14911,14913,14916],{"class":7830,"line":8019},[33,14907,14908],{"class":7834},"        raise",[33,14910,14859],{"class":7848},[33,14912,389],{"class":7842},[33,14914,14915],{"class":7858},"\"fuel exhausted, simulating infinite loop\"",[33,14917,7933],{"class":7842},[33,14919,14920,14923,14925,14928,14931,14934],{"class":7830,"line":8024},[33,14921,14922],{"class":7842},"    will_halt ",[33,14924,60],{"class":7834},[33,14926,14927],{"class":7842}," hypothetical_halts(x, x, fuel ",[33,14929,14930],{"class":7834},"-",[33,14932,14933],{"class":7848}," 1",[33,14935,7933],{"class":7842},[33,14937,14938,14940],{"class":7830,"line":8035},[33,14939,14893],{"class":7834},[33,14941,14942],{"class":7842}," will_halt:\n",[33,14944,14945,14947,14949,14951,14954],{"class":7830,"line":8060},[33,14946,14908],{"class":7834},[33,14948,14859],{"class":7848},[33,14950,389],{"class":7842},[33,14952,14953],{"class":7858},"\"simulating infinite loop: halts says it halts, so D deliberately does not halt\"",[33,14955,7933],{"class":7842},[33,14957,14958,14961],{"class":7830,"line":8083},[33,14959,14960],{"class":7834},"    else",[33,14962,14824],{"class":7842},[33,14964,14965,14967],{"class":7830,"line":8096},[33,14966,14839],{"class":7834},[33,14968,14969],{"class":7858}," \"D halted\"\n",[33,14971,14972],{"class":7830,"line":8102},[33,14973,8016],{"emptyLinePlaceholder":8015},[33,14975,14976,14979,14981,14984,14988,14990,14993],{"class":7830,"line":8107},[33,14977,14978],{"class":7842},"outcome ",[33,14980,60],{"class":7834},[33,14982,14983],{"class":7842}," D(D, ",[33,14985,14987],{"class":14986},"s9osk","fuel",[33,14989,60],{"class":7834},[33,14991,14992],{"class":7848},"8",[33,14994,7933],{"class":7842},[33,14996,14997,14999,15001,15004],{"class":7830,"line":8118},[33,14998,8121],{"class":7848},[33,15000,389],{"class":7842},[33,15002,15003],{"class":7858},"\"Execution result of D(D):\"",[33,15005,15006],{"class":7842},", outcome)\n",[10,15008,15009],{},[14,15010,8134],{},[7819,15012,15015],{"className":15013,"code":15014,"language":1798},[8138],"=== 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",[7826,15016,15014],{"__ignoreMap":7824},[10,15018,15019,15020,15023,15024,15026,15027,15030],{},"The ",[7826,15021,15022],{},"hypothetical_halts"," in this code is plainly not a genuinely \"universal\" decider — it operates only within a finite ",[7826,15025,14987],{}," (step budget), which is the sole reason we are able to execute it on a real computer. This, in fact, corroborates Turing's conclusion: ",[14,15028,15029],{},"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).",[22,15032,15034],{"id":15033},"gödels-incompleteness-theorems","Gödel's Incompleteness Theorems",[10,15036,15037],{},"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:",[2122,15039,15040],{},[10,15041,15042,15045],{},[14,15043,15044],{},"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.",[10,15047,15048],{},"Gödel's proof proceeds in three broad steps:",[15050,15051,15052,15058,15414],"ol",{},[346,15053,15054,15057],{},[14,15055,15056],{},"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.",[346,15059,15060,15063,15064,15093,15094,15169,15172,15173,15176,15177,15222,15223,15253,15254,15340,15341,15384,15385,15413],{},[14,15061,15062],{},"Constructing a self-referential proposition."," Exploiting the encoding technique, one constructs a proposition ",[33,15065,15067,15081],{"className":15066,"translate":37},[36],[33,15068,15070],{"className":15069},[41],[43,15071,15072],{"xmlns":45},[47,15073,15074,15079],{},[50,15075,15076],{},[53,15077,15078],{},"G",[99,15080,15078],{"encoding":101},[33,15082,15084],{"className":15083,"ariaHidden":71},[106],[33,15085,15087,15090],{"className":15086},[110],[33,15088],{"className":15089,"style":226},[114],[33,15091,15078],{"className":15092},[119,230]," whose essential content is:",[33,15095,15097],{"className":15096,"translate":37},[2403],[33,15098,15100,15126],{"className":15099,"translate":37},[36],[33,15101,15103],{"className":15102},[41],[43,15104,15105],{"xmlns":45,"display":2412},[47,15106,15107,15123],{},[50,15108,15109,15111,15113,15115,15118,15120],{},[53,15110,15078],{},[58,15112,280],{},[58,15114,60],{},[92,15116,15117],{},"\"Proposition ",[53,15119,15078],{},[92,15121,15122],{}," itself is unprovable within this system\"",[99,15124,15125],{"encoding":101},"G := \\text{\"Proposition } G \\text{ itself is unprovable within this system\"}",[33,15127,15129,15148],{"className":15128,"ariaHidden":71},[106],[33,15130,15132,15135,15138,15141,15145],{"className":15131},[110],[33,15133],{"className":15134,"style":226},[114],[33,15136,15078],{"className":15137},[119,230],[33,15139],{"className":15140,"style":125},[124],[33,15142,15144],{"className":15143},[129],":=",[33,15146],{"className":15147,"style":125},[124],[33,15149,15151,15154,15160,15163],{"className":15150},[110],[33,15152],{"className":15153,"style":300},[114],[33,15155,15157],{"className":15156},[119,1798],[33,15158,15117],{"className":15159},[119],[33,15161,15078],{"className":15162},[119,230],[33,15164,15166],{"className":15165},[119,1798],[33,15167,15122],{"className":15168},[119],[15170,15171],"br",{},"This is a highly self-referential construction, whose mathematical foundation is known as the ",[14,15174,15175],{},"Diagonal Lemma"," (or fixed-point lemma): for any formula ",[33,15178,15180,15201],{"className":15179,"translate":37},[36],[33,15181,15183],{"className":15182},[41],[43,15184,15185],{"xmlns":45},[47,15186,15187,15198],{},[50,15188,15189,15192,15194,15196],{},[53,15190,15191],{},"ϕ",[58,15193,389],{"stretchy":63},[53,15195,2430],{},[58,15197,398],{"stretchy":63},[99,15199,15200],{"encoding":101},"\\phi(x)",[33,15202,15204],{"className":15203,"ariaHidden":71},[106],[33,15205,15207,15210,15213,15216,15219],{"className":15206},[110],[33,15208],{"className":15209,"style":139},[114],[33,15211,15191],{"className":15212},[119,230],[33,15214,389],{"className":15215},[143],[33,15217,2430],{"className":15218},[119,230],[33,15220,398],{"className":15221},[192]," with a single free variable, one can construct a sentence ",[33,15224,15226,15241],{"className":15225,"translate":37},[36],[33,15227,15229],{"className":15228},[41],[43,15230,15231],{"xmlns":45},[47,15232,15233,15238],{},[50,15234,15235],{},[53,15236,15237],{},"ψ",[99,15239,15240],{"encoding":101},"\\psi",[33,15242,15244],{"className":15243,"ariaHidden":71},[106],[33,15245,15247,15250],{"className":15246},[110],[33,15248],{"className":15249,"style":300},[114],[33,15251,15237],{"className":15252,"style":2112},[119,230]," such that the system proves ",[33,15255,15257,15292],{"className":15256,"translate":37},[36],[33,15258,15260],{"className":15259},[41],[43,15261,15262],{"xmlns":45},[47,15263,15264,15289],{},[50,15265,15266,15268,15271,15273,15275,15280,15282,15287],{},[53,15267,15237],{},[58,15269,15270],{},"↔",[53,15272,15191],{},[58,15274,389],{"stretchy":63},[58,15276,15277],{},[53,15278,15279],{"mathvariant":374},"⌜",[53,15281,15237],{},[58,15283,15284],{},[53,15285,15286],{"mathvariant":374},"⌝",[58,15288,398],{"stretchy":63},[99,15290,15291],{"encoding":101},"\\psi \\leftrightarrow \\phi(\\ulcorner \\psi \\urcorner)",[33,15293,15295,15313],{"className":15294,"ariaHidden":71},[106],[33,15296,15298,15301,15304,15307,15310],{"className":15297},[110],[33,15299],{"className":15300,"style":300},[114],[33,15302,15237],{"className":15303,"style":2112},[119,230],[33,15305],{"className":15306,"style":125},[124],[33,15308,15270],{"className":15309},[129],[33,15311],{"className":15312,"style":125},[124],[33,15314,15316,15319,15322,15325,15330,15333,15337],{"className":15315},[110],[33,15317],{"className":15318,"style":139},[114],[33,15320,15191],{"className":15321},[119,230],[33,15323,389],{"className":15324},[143],[33,15326,15329],{"className":15327},[143,15328],"amsrm","┌",[33,15331,15237],{"className":15332,"style":2112},[119,230],[33,15334,15336],{"className":15335},[192,15328],"┐",[33,15338,398],{"className":15339},[192]," (where ",[33,15342,15344,15366],{"className":15343,"translate":37},[36],[33,15345,15347],{"className":15346},[41],[43,15348,15349],{"xmlns":45},[47,15350,15351,15363],{},[50,15352,15353,15357,15359],{},[58,15354,15355],{},[53,15356,15279],{"mathvariant":374},[53,15358,15237],{},[58,15360,15361],{},[53,15362,15286],{"mathvariant":374},[99,15364,15365],{"encoding":101},"\\ulcorner \\psi \\urcorner",[33,15367,15369],{"className":15368,"ariaHidden":71},[106],[33,15370,15372,15375,15378,15381],{"className":15371},[110],[33,15373],{"className":15374,"style":300},[114],[33,15376,15329],{"className":15377},[143,15328],[33,15379,15237],{"className":15380,"style":2112},[119,230],[33,15382,15336],{"className":15383},[192,15328]," denotes the Gödel number of ",[33,15386,15388,15401],{"className":15387,"translate":37},[36],[33,15389,15391],{"className":15390},[41],[43,15392,15393],{"xmlns":45},[47,15394,15395,15399],{},[50,15396,15397],{},[53,15398,15237],{},[99,15400,15240],{"encoding":101},[33,15402,15404],{"className":15403,"ariaHidden":71},[106],[33,15405,15407,15410],{"className":15406},[110],[33,15408],{"className":15409,"style":300},[114],[33,15411,15237],{"className":15412,"style":2112},[119,230],"). 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.",[346,15415,15416,15419],{},[14,15417,15418],{},"Deriving a contradictory dilemma.",[343,15420,15421,15511,15607],{},[346,15422,15423,15424,15452,15453,15481,15482,15510],{},"If the system can prove ",[33,15425,15427,15440],{"className":15426,"translate":37},[36],[33,15428,15430],{"className":15429},[41],[43,15431,15432],{"xmlns":45},[47,15433,15434,15438],{},[50,15435,15436],{},[53,15437,15078],{},[99,15439,15078],{"encoding":101},[33,15441,15443],{"className":15442,"ariaHidden":71},[106],[33,15444,15446,15449],{"className":15445},[110],[33,15447],{"className":15448,"style":226},[114],[33,15450,15078],{"className":15451},[119,230],", then, by the meaning of ",[33,15454,15456,15469],{"className":15455,"translate":37},[36],[33,15457,15459],{"className":15458},[41],[43,15460,15461],{"xmlns":45},[47,15462,15463,15467],{},[50,15464,15465],{},[53,15466,15078],{},[99,15468,15078],{"encoding":101},[33,15470,15472],{"className":15471,"ariaHidden":71},[106],[33,15473,15475,15478],{"className":15474},[110],[33,15476],{"className":15477,"style":226},[114],[33,15479,15078],{"className":15480},[119,230]," (\"",[33,15483,15485,15498],{"className":15484,"translate":37},[36],[33,15486,15488],{"className":15487},[41],[43,15489,15490],{"xmlns":45},[47,15491,15492,15496],{},[50,15493,15494],{},[53,15495,15078],{},[99,15497,15078],{"encoding":101},[33,15499,15501],{"className":15500,"ariaHidden":71},[106],[33,15502,15504,15507],{"className":15503},[110],[33,15505],{"className":15506,"style":226},[114],[33,15508,15078],{"className":15509},[119,230]," is unprovable\"), the system would simultaneously prove a false proposition — contradicting the consistency of the system.",[346,15512,15513,15514,15542,15543,15577,15578,15606],{},"If the system can disprove ",[33,15515,15517,15530],{"className":15516,"translate":37},[36],[33,15518,15520],{"className":15519},[41],[43,15521,15522],{"xmlns":45},[47,15523,15524,15528],{},[50,15525,15526],{},[53,15527,15078],{},[99,15529,15078],{"encoding":101},[33,15531,15533],{"className":15532,"ariaHidden":71},[106],[33,15534,15536,15539],{"className":15535},[110],[33,15537],{"className":15538,"style":226},[114],[33,15540,15078],{"className":15541},[119,230]," (i.e., prove ",[33,15544,15546,15562],{"className":15545,"translate":37},[36],[33,15547,15549],{"className":15548},[41],[43,15550,15551],{"xmlns":45},[47,15552,15553,15559],{},[50,15554,15555,15557],{},[53,15556,10006],{"mathvariant":374},[53,15558,15078],{},[99,15560,15561],{"encoding":101},"\\neg G",[33,15563,15565],{"className":15564,"ariaHidden":71},[106],[33,15566,15568,15571,15574],{"className":15567},[110],[33,15569],{"className":15570,"style":226},[114],[33,15572,10006],{"className":15573},[119],[33,15575,15078],{"className":15576},[119,230],", which amounts to proving \"",[33,15579,15581,15594],{"className":15580,"translate":37},[36],[33,15582,15584],{"className":15583},[41],[43,15585,15586],{"xmlns":45},[47,15587,15588,15592],{},[50,15589,15590],{},[53,15591,15078],{},[99,15593,15078],{"encoding":101},[33,15595,15597],{"className":15596,"ariaHidden":71},[106],[33,15598,15600,15603],{"className":15599},[110],[33,15601],{"className":15602,"style":226},[114],[33,15604,15078],{"className":15605},[119,230]," is provable\"), then, under reasonable supplementary conditions, this likewise leads to contradiction.",[346,15608,15609,15610,15638],{},"Hence ",[33,15611,15613,15626],{"className":15612,"translate":37},[36],[33,15614,15616],{"className":15615},[41],[43,15617,15618],{"xmlns":45},[47,15619,15620,15624],{},[50,15621,15622],{},[53,15623,15078],{},[99,15625,15078],{"encoding":101},[33,15627,15629],{"className":15628,"ariaHidden":71},[106],[33,15630,15632,15635],{"className":15631},[110],[33,15633],{"className":15634,"style":226},[114],[33,15636,15078],{"className":15637},[119,230]," can be neither proved nor disproved — it is an \"undecidable proposition\" within the system.",[10,15640,15641],{},"Commonality with the preceding two proofs:",[10504,15643,15644,15774],{},[10507,15645,15646],{},[10510,15647,15648,15650,15681,15712,15743],{},[10513,15649],{},[10513,15651,15652,15653],{},"Diagonal Number ",[33,15654,15656,15669],{"className":15655,"translate":37},[36],[33,15657,15659],{"className":15658},[41],[43,15660,15661],{"xmlns":45},[47,15662,15663,15667],{},[50,15664,15665],{},[53,15666,4070],{},[99,15668,4070],{"encoding":101},[33,15670,15672],{"className":15671,"ariaHidden":71},[106],[33,15673,15675,15678],{"className":15674},[110],[33,15676],{"className":15677,"style":4114},[114],[33,15679,4070],{"className":15680,"style":2112},[119,230],[10513,15682,15683,15684],{},"Diagonal Set ",[33,15685,15687,15700],{"className":15686,"translate":37},[36],[33,15688,15690],{"className":15689},[41],[43,15691,15692],{"xmlns":45},[47,15693,15694,15698],{},[50,15695,15696],{},[53,15697,8959],{},[99,15699,8959],{"encoding":101},[33,15701,15703],{"className":15702,"ariaHidden":71},[106],[33,15704,15706,15709],{"className":15705},[110],[33,15707],{"className":15708,"style":226},[114],[33,15710,8959],{"className":15711,"style":9006},[119,230],[10513,15713,15714,15715],{},"Diagonal Program ",[33,15716,15718,15731],{"className":15717,"translate":37},[36],[33,15719,15721],{"className":15720},[41],[43,15722,15723],{"xmlns":45},[47,15724,15725,15729],{},[50,15726,15727],{},[53,15728,8959],{},[99,15730,8959],{"encoding":101},[33,15732,15734],{"className":15733,"ariaHidden":71},[106],[33,15735,15737,15740],{"className":15736},[110],[33,15738],{"className":15739,"style":226},[114],[33,15741,8959],{"className":15742,"style":9006},[119,230],[10513,15744,15745,15746],{},"Gödel Sentence ",[33,15747,15749,15762],{"className":15748,"translate":37},[36],[33,15750,15752],{"className":15751},[41],[43,15753,15754],{"xmlns":45},[47,15755,15756,15760],{},[50,15757,15758],{},[53,15759,15078],{},[99,15761,15078],{"encoding":101},[33,15763,15765],{"className":15764,"ariaHidden":71},[106],[33,15766,15768,15771],{"className":15767},[110],[33,15769],{"className":15770,"style":226},[114],[33,15772,15078],{"className":15773},[119,230],[10519,15775,15776,15793,15810],{},[10510,15777,15778,15781,15784,15787,15790],{},[10524,15779,15780],{},"Self-referential object",[10524,15782,15783],{},"Each digit \"avoids\" itself",[10524,15785,15786],{},"Collects elements \"not belonging to themselves\"",[10524,15788,15789],{},"Uses its own behavior to negate the judgment about itself",[10524,15791,15792],{},"Asserts \"I am unprovable\"",[10510,15794,15795,15798,15801,15804,15807],{},[10524,15796,15797],{},"Mechanism relied upon",[10524,15799,15800],{},"Diagonal of the decimal expansion",[10524,15802,15803],{},"Self-reference in set membership",[10524,15805,15806],{},"A program fed to itself as input",[10524,15808,15809],{},"Self-reference realized via Gödel numbering",[10510,15811,15812,15815,15846,15877,15880],{},[10524,15813,15814],{},"Source of contradiction",[10524,15816,15817,15818],{},"The assumed bijection cannot cover ",[33,15819,15821,15834],{"className":15820,"translate":37},[36],[33,15822,15824],{"className":15823},[41],[43,15825,15826],{"xmlns":45},[47,15827,15828,15832],{},[50,15829,15830],{},[53,15831,4070],{},[99,15833,4070],{"encoding":101},[33,15835,15837],{"className":15836,"ariaHidden":71},[106],[33,15838,15840,15843],{"className":15839},[110],[33,15841],{"className":15842,"style":4114},[114],[33,15844,4070],{"className":15845,"style":2112},[119,230],[10524,15847,15848,15849],{},"The assumed surjection cannot cover ",[33,15850,15852,15865],{"className":15851,"translate":37},[36],[33,15853,15855],{"className":15854},[41],[43,15856,15857],{"xmlns":45},[47,15858,15859,15863],{},[50,15860,15861],{},[53,15862,8959],{},[99,15864,8959],{"encoding":101},[33,15866,15868],{"className":15867,"ariaHidden":71},[106],[33,15869,15871,15874],{"className":15870},[110],[33,15872],{"className":15873,"style":226},[114],[33,15875,8959],{"className":15876,"style":9006},[119,230],[10524,15878,15879],{},"The assumed decider yields paradox",[10524,15881,15882],{},"The assumed provability yields paradox",[10,15884,15885],{},"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.",[22,15887,15889],{"id":15888},"further-extensions","Further Extensions",[10,15891,15892],{},"Beyond the three classical applications discussed above, the diagonal argument and its underlying idea permeate numerous corners of computer science and mathematics:",[343,15894,15895,15901,15964,15970],{},[346,15896,15897,15900],{},[14,15898,15899],{},"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.",[346,15902,15903,15906,15907,4049],{},[14,15904,15905],{},"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 ",[33,15908,15910,15930],{"className":15909,"translate":37},[36],[33,15911,15913],{"className":15912},[41],[43,15914,15915],{"xmlns":45},[47,15916,15917,15927],{},[50,15918,15919,15921,15924],{},[53,15920,8364],{},[58,15922,15923],{},"⊊",[92,15925,15926],{},"EXP",[99,15928,15929],{"encoding":101},"P \\subsetneq \\text{EXP}",[33,15931,15933,15952],{"className":15932,"ariaHidden":71},[106],[33,15934,15936,15940,15943,15946,15949],{"className":15935},[110],[33,15937],{"className":15938,"style":15939},[114],"height:0.8193em;vertical-align:-0.136em;",[33,15941,8364],{"className":15942,"style":12142},[119,230],[33,15944],{"className":15945,"style":125},[124],[33,15947,15923],{"className":15948},[129,15328],[33,15950],{"className":15951,"style":125},[124],[33,15953,15955,15958],{"className":15954},[110],[33,15956],{"className":15957,"style":226},[114],[33,15959,15961],{"className":15960},[119,1798],[33,15962,15926],{"className":15963},[119],[346,15965,15966,15969],{},[14,15967,15968],{},"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.",[346,15971,15972,15975,15976,16116,16117,16266],{},[14,15973,15974],{},"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,\" ",[33,15977,15979,16010],{"className":15978,"translate":37},[36],[33,15980,15982],{"className":15981},[41],[43,15983,15984],{"xmlns":45},[47,15985,15986,16007],{},[50,15987,15988,15991,15993,15995,15997,15999,16001,16003,16005],{},[53,15989,15990],{},"R",[58,15992,60],{},[58,15994,64],{"stretchy":63},[53,15996,2430],{},[58,15998,280],{},[53,16000,2430],{},[58,16002,8978],{"mathvariant":374},[53,16004,2430],{},[58,16006,97],{"stretchy":63},[99,16008,16009],{"encoding":101},"R = \\{x : x \\notin x\\}",[33,16011,16013,16032,16053,16104],{"className":16012,"ariaHidden":71},[106],[33,16014,16016,16019,16023,16026,16029],{"className":16015},[110],[33,16017],{"className":16018,"style":226},[114],[33,16020,15990],{"className":16021,"style":16022},[119,230],"margin-right:0.0077em;",[33,16024],{"className":16025,"style":125},[124],[33,16027,60],{"className":16028},[129],[33,16030],{"className":16031,"style":125},[124],[33,16033,16035,16038,16041,16044,16047,16050],{"className":16034},[110],[33,16036],{"className":16037,"style":139},[114],[33,16039,64],{"className":16040},[143],[33,16042,2430],{"className":16043},[119,230],[33,16045],{"className":16046,"style":125},[124],[33,16048,280],{"className":16049},[129],[33,16051],{"className":16052,"style":125},[124],[33,16054,16056,16059,16062,16065,16101],{"className":16055},[110],[33,16057],{"className":16058,"style":139},[114],[33,16060,2430],{"className":16061},[119,230],[33,16063],{"className":16064,"style":125},[124],[33,16066,16068,16074],{"className":16067},[129],[33,16069,16071],{"className":16070},[119],[33,16072,762],{"className":16073},[129],[33,16075,16077],{"className":16076},[119,493],[33,16078,16080],{"className":16079},[497],[33,16081,16083,16086,16098],{"className":16082},[9088],[33,16084],{"className":16085,"style":139},[114],[33,16087,16089],{"className":16088},[508],[33,16090,16092,16095],{"className":16091},[119],[33,16093,1577],{"className":16094},[119],[33,16096],{"className":16097,"style":9104},[124],[33,16099],{"className":16100},[519],[33,16102],{"className":16103,"style":125},[124],[33,16105,16107,16110,16113],{"className":16106},[110],[33,16108],{"className":16109,"style":139},[114],[33,16111,2430],{"className":16112},[119,230],[33,16114,97],{"className":16115},[192],", shares an identical structure with the set ",[33,16118,16120,16155],{"className":16119,"translate":37},[36],[33,16121,16123],{"className":16122},[41],[43,16124,16125],{"xmlns":45},[47,16126,16127,16153],{},[50,16128,16129,16131,16133,16135,16137,16139,16141,16143,16145,16147,16149,16151],{},[53,16130,8959],{},[58,16132,60],{},[58,16134,64],{"stretchy":63},[53,16136,369],{},[58,16138,280],{},[53,16140,369],{},[58,16142,8978],{"mathvariant":374},[53,16144,277],{},[58,16146,389],{"stretchy":63},[53,16148,369],{},[58,16150,398],{"stretchy":63},[58,16152,97],{"stretchy":63},[99,16154,10886],{"encoding":101},[33,16156,16158,16176,16197,16248],{"className":16157,"ariaHidden":71},[106],[33,16159,16161,16164,16167,16170,16173],{"className":16160},[110],[33,16162],{"className":16163,"style":226},[114],[33,16165,8959],{"className":16166,"style":9006},[119,230],[33,16168],{"className":16169,"style":125},[124],[33,16171,60],{"className":16172},[129],[33,16174],{"className":16175,"style":125},[124],[33,16177,16179,16182,16185,16188,16191,16194],{"className":16178},[110],[33,16180],{"className":16181,"style":139},[114],[33,16183,64],{"className":16184},[143],[33,16186,369],{"className":16187},[119,230],[33,16189],{"className":16190,"style":125},[124],[33,16192,280],{"className":16193},[129],[33,16195],{"className":16196,"style":125},[124],[33,16198,16200,16203,16206,16209,16245],{"className":16199},[110],[33,16201],{"className":16202,"style":139},[114],[33,16204,369],{"className":16205},[119,230],[33,16207],{"className":16208,"style":125},[124],[33,16210,16212,16218],{"className":16211},[129],[33,16213,16215],{"className":16214},[119],[33,16216,762],{"className":16217},[129],[33,16219,16221],{"className":16220},[119,493],[33,16222,16224],{"className":16223},[497],[33,16225,16227,16230,16242],{"className":16226},[9088],[33,16228],{"className":16229,"style":139},[114],[33,16231,16233],{"className":16232},[508],[33,16234,16236,16239],{"className":16235},[119],[33,16237,1577],{"className":16238},[119],[33,16240],{"className":16241,"style":9104},[124],[33,16243],{"className":16244},[519],[33,16246],{"className":16247,"style":125},[124],[33,16249,16251,16254,16257,16260,16263],{"className":16250},[110],[33,16252],{"className":16253,"style":139},[114],[33,16255,277],{"className":16256,"style":304},[119,230],[33,16258,389],{"className":16259},[143],[33,16261,369],{"className":16262},[119,230],[33,16264,10998],{"className":16265},[192]," in the proof of Cantor's Theorem — both are products of \"self-reference + self-exclusion.\"",[10,16268,16269],{},"If one were to distill the diagonal argument into a reusable \"template of thought,\" the outline would run approximately as follows:",[15050,16271,16272,16278,16284,16351],{},[346,16273,16274,16277],{},[14,16275,16276],{},"Point of departure (reductio)."," Assume the existence of a \"perfect coverage\" or \"universal judgment\" — a bijection, a surjection, a decider, a proof system.",[346,16279,16280,16283],{},[14,16281,16282],{},"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).",[346,16285,16286,16289,16290,16292,16293,16321,16322,16350],{},[14,16287,16288],{},"Verify its evasive character."," Prove that this new object differs systematically from ",[14,16291,5970],{}," item within the scope of the assumed coverage (typically through the pattern of \"the ",[33,16294,16296,16309],{"className":16295,"translate":37},[36],[33,16297,16299],{"className":16298},[41],[43,16300,16301],{"xmlns":45},[47,16302,16303,16307],{},[50,16304,16305],{},[53,16306,1309],{},[99,16308,1309],{"encoding":101},[33,16310,16312],{"className":16311,"ariaHidden":71},[106],[33,16313,16315,16318],{"className":16314},[110],[33,16316],{"className":16317,"style":6054},[114],[33,16319,1309],{"className":16320},[119,230],"-th item differs at the ",[33,16323,16325,16338],{"className":16324,"translate":37},[36],[33,16326,16328],{"className":16327},[41],[43,16329,16330],{"xmlns":45},[47,16331,16332,16336],{},[50,16333,16334],{},[53,16335,1309],{},[99,16337,1309],{"encoding":101},[33,16339,16341],{"className":16340,"ariaHidden":71},[106],[33,16342,16344,16347],{"className":16343},[110],[33,16345],{"className":16346,"style":6054},[114],[33,16348,1309],{"className":16349},[119,230],"-th position\").",[346,16352,16353,16356],{},[14,16354,16355],{},"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.",[10,16358,16359],{},"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.",[16361,16362,16363],"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":7824,"searchDepth":7855,"depth":7855,"links":16365},[16366,16367,16368,16369,16372,16373,16374],{"id":24,"depth":7855,"text":25},{"id":2119,"depth":7855,"text":2120},{"id":7813,"depth":7855,"text":7814},{"id":8307,"depth":7855,"text":8308,"children":16370},[16371],{"id":8633,"depth":7862,"text":8634},{"id":12099,"depth":7855,"text":12100},{"id":15033,"depth":7855,"text":15034},{"id":15888,"depth":7855,"text":15889},null,"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.",false,"md",{},"2026-07-31","\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en",{"title":5,"description":16376},"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.",[16386,16387,16388],"mathematics","cs","computation","ojh1o2Ivk_OuHQYv0CedmmmGt5_ORitJINjxETvihPw",1785513140878]