[{"data":1,"prerenderedAt":16946},["ShallowReactive",2],{"blog-\u002Fblog\u002F004-completeness-theorem-proofs":3,"blog-all":16922},{"id":4,"title":5,"body":6,"date":16907,"description":16908,"draft":16909,"extension":16910,"image":16911,"meta":16912,"navigation":16913,"no":16914,"path":16915,"pinned":16909,"seo":16916,"stem":16917,"tags":16918,"updated":16911,"__hash__":16921},"blog\u002Fblog\u002F004-completeness-theorem-proofs.md","完全性定理の2つの証明 ― ゲーデル流とヘンキン流",{"type":7,"value":8,"toc":16877},"minimark",[9,17,20,52,55,62,67,72,219,342,718,722,729,1017,1020,1304,1308,1472,1529,1536,1570,2236,2240,2247,2250,2342,2353,2584,2588,2597,3188,3396,3400,3758,3963,4516,4816,5037,5041,5044,5119,5470,5570,5726,5808,5883,6058,6333,6417,6421,6482,7310,7601,7760,7795,7942,7946,7991,7995,8009,8012,8342,8403,8407,8544,8695,8797,9043,9625,9659,10096,10391,10395,10497,12403,12435,13111,13143,13147,13207,14026,14114,14310,15490,15631,15733,15737,15862,15866,16265,16272,16348,16352,16355,16392,16399,16402,16640,16643,16649,16653,16760,16764,16868],[10,11,12,16],"p",{},[13,14,15],"strong",{},"完全性定理"," は、一階述語論理において「正しい（すべてのモデルで成り立つ）ものは、必ず証明できる」ことを保証する定理です。意味（モデル）と構文（証明）という、まったく別の方法で定義された2つの「正しさ」が一致することを示しています。",[10,18,19],{},"この定理には、歴史的に重要な2つの証明があります。",[21,22,23,41],"ul",{},[24,25,26,29,30,35,36,40],"li",{},[13,27,28],{},"ゲーデル流","：1929年の博士論文と1930年の論文による、元の証明（文献 ",[31,32,34],"a",{"href":33},"#ref-1","[1]","、",[31,37,39],{"href":38},"#ref-9","[9]","）。",[24,42,43,46,47,51],{},[13,44,45],{},"ヘンキン流","：1949年にヘンキンが与えた証明（文献 ",[31,48,50],{"href":49},"#ref-3","[3]","）。今日の教科書ではこちらが標準です。",[10,53,54],{},"この記事では、2つの証明の道筋を、細部を省きつつも骨組みが分かるように追いかけます。",[10,56,57],{},[58,59],"img",{"alt":60,"src":61},"2つの証明の流れ。ゲーデル流は命題論理への帰着、ヘンキン流は極大無矛盾集合を足場にする","\u002Fimages\u002Fblog\u002Fcompleteness\u002Ftwo-proofs.svg",[63,64,66],"h2",{"id":65},"_1-準備意味と証明","1. 準備：意味と証明",[68,69,71],"h3",{"id":70},"_11-意味モデルと充足","1.1 意味：モデルと充足",[10,73,74,75,123,124,152,153,156,157,187,188,218],{},"一階の言語 ",[76,77,80,104],"span",{"className":78},[79],"katex",[76,81,84],{"className":82},[83],"katex-mathml",[85,86,88],"math",{"xmlns":87},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[89,90,91,99],"semantics",{},[92,93,94],"mrow",{},[95,96,98],"mi",{"mathvariant":97},"script","L",[100,101,103],"annotation",{"encoding":102},"application\u002Fx-tex","\\mathcal{L}",[76,105,109],{"className":106,"ariaHidden":108},[107],"katex-html","true",[76,110,113,118],{"className":111},[112],"katex-base",[76,114],{"className":115,"style":117},[116],"katex-strut","height:0.6833em;",[76,119,98],{"className":120},[121,122],"mord","mathcal","（述語記号、関数記号、定数記号の集まり）を固定します。",[76,125,127,140],{"className":126},[79],[76,128,130],{"className":129},[83],[85,131,132],{"xmlns":87},[89,133,134,138],{},[92,135,136],{},[95,137,98],{"mathvariant":97},[100,139,103],{"encoding":102},[76,141,143],{"className":142,"ariaHidden":108},[107],[76,144,146,149],{"className":145},[112],[76,147],{"className":148,"style":117},[116],[76,150,98],{"className":151},[121,122]," の ",[13,154,155],{},"構造"," ",[76,158,160,175],{"className":159},[79],[76,161,163],{"className":162},[83],[85,164,165],{"xmlns":87},[89,166,167,172],{},[92,168,169],{},[95,170,171],{"mathvariant":97},"M",[100,173,174],{"encoding":102},"\\mathcal{M}",[76,176,178],{"className":177,"ariaHidden":108},[107],[76,179,181,184],{"className":180},[112],[76,182],{"className":183,"style":117},[116],[76,185,171],{"className":186},[121,122]," は、空でない領域 ",[76,189,191,204],{"className":190},[79],[76,192,194],{"className":193},[83],[85,195,196],{"xmlns":87},[89,197,198,202],{},[92,199,200],{},[95,201,171],{},[100,203,171],{"encoding":102},[76,205,207],{"className":206,"ariaHidden":108},[107],[76,208,210,213],{"className":209},[112],[76,211],{"className":212,"style":117},[116],[76,214,171],{"className":215,"style":217},[121,216],"mathnormal","margin-right:0.109em;"," と、各記号の解釈からなります。",[10,220,221,222,254,255,283,284,341],{},"文 ",[76,223,225,240],{"className":224},[79],[76,226,228],{"className":227},[83],[85,229,230],{"xmlns":87},[89,231,232,237],{},[92,233,234],{},[95,235,236],{},"σ",[100,238,239],{"encoding":102},"\\sigma",[76,241,243],{"className":242,"ariaHidden":108},[107],[76,244,246,250],{"className":245},[112],[76,247],{"className":248,"style":249},[116],"height:0.4306em;",[76,251,236],{"className":252,"style":253},[121,216],"margin-right:0.0359em;"," が ",[76,256,258,271],{"className":257},[79],[76,259,261],{"className":260},[83],[85,262,263],{"xmlns":87},[89,264,265,269],{},[92,266,267],{},[95,268,171],{"mathvariant":97},[100,270,174],{"encoding":102},[76,272,274],{"className":273,"ariaHidden":108},[107],[76,275,277,280],{"className":276},[112],[76,278],{"className":279,"style":117},[116],[76,281,171],{"className":282},[121,122]," で成り立つことを ",[76,285,287,307],{"className":286},[79],[76,288,290],{"className":289},[83],[85,291,292],{"xmlns":87},[89,293,294,304],{},[92,295,296,298,302],{},[95,297,171],{"mathvariant":97},[299,300,301],"mo",{},"⊨",[95,303,236],{},[100,305,306],{"encoding":102},"\\mathcal{M} \\models \\sigma",[76,308,310,332],{"className":309,"ariaHidden":108},[107],[76,311,313,317,320,325,329],{"className":312},[112],[76,314],{"className":315,"style":316},[116],"height:0.999em;vertical-align:-0.249em;",[76,318,171],{"className":319},[121,122],[76,321],{"className":322,"style":324},[323],"mspace","margin-right:0.2778em;",[76,326,301],{"className":327},[328],"mrel",[76,330],{"className":331,"style":324},[323],[76,333,335,338],{"className":334},[112],[76,336],{"className":337,"style":249},[116],[76,339,236],{"className":340,"style":253},[121,216]," と書きます。この「成り立つ」の定義は、タルスキによる帰納的な定義です。",[343,344,346],"definition",{"title":345},"意味論的帰結",[10,347,348,349,380,381,152,409,412,413,441,442,473,474,502,503,152,531,533,534,585,586,641,642,685,686,502,714,717],{},"文の集合 ",[76,350,352,368],{"className":351},[79],[76,353,355],{"className":354},[83],[85,356,357],{"xmlns":87},[89,358,359,365],{},[92,360,361],{},[95,362,364],{"mathvariant":363},"normal","Γ",[100,366,367],{"encoding":102},"\\Gamma",[76,369,371],{"className":370,"ariaHidden":108},[107],[76,372,374,377],{"className":373},[112],[76,375],{"className":376,"style":117},[116],[76,378,364],{"className":379},[121]," のすべての文が成り立つ構造を、",[76,382,384,397],{"className":383},[79],[76,385,387],{"className":386},[83],[85,388,389],{"xmlns":87},[89,390,391,395],{},[92,392,393],{},[95,394,364],{"mathvariant":363},[100,396,367],{"encoding":102},[76,398,400],{"className":399,"ariaHidden":108},[107],[76,401,403,406],{"className":402},[112],[76,404],{"className":405,"style":117},[116],[76,407,364],{"className":408},[121],[13,410,411],{},"モデル"," と呼ぶ。",[76,414,416,429],{"className":415},[79],[76,417,419],{"className":418},[83],[85,420,421],{"xmlns":87},[89,422,423,427],{},[92,424,425],{},[95,426,364],{"mathvariant":363},[100,428,367],{"encoding":102},[76,430,432],{"className":431,"ariaHidden":108},[107],[76,433,435,438],{"className":434},[112],[76,436],{"className":437,"style":117},[116],[76,439,364],{"className":440},[121]," のすべてのモデルで文 ",[76,443,445,460],{"className":444},[79],[76,446,448],{"className":447},[83],[85,449,450],{"xmlns":87},[89,451,452,457],{},[92,453,454],{},[95,455,456],{},"φ",[100,458,459],{"encoding":102},"\\varphi",[76,461,463],{"className":462,"ariaHidden":108},[107],[76,464,466,470],{"className":465},[112],[76,467],{"className":468,"style":469},[116],"height:0.625em;vertical-align:-0.1944em;",[76,471,456],{"className":472},[121,216]," が成り立つとき、",[76,475,477,490],{"className":476},[79],[76,478,480],{"className":479},[83],[85,481,482],{"xmlns":87},[89,483,484,488],{},[92,485,486],{},[95,487,456],{},[100,489,459],{"encoding":102},[76,491,493],{"className":492,"ariaHidden":108},[107],[76,494,496,499],{"className":495},[112],[76,497],{"className":498,"style":469},[116],[76,500,456],{"className":501},[121,216]," は ",[76,504,506,519],{"className":505},[79],[76,507,509],{"className":508},[83],[85,510,511],{"xmlns":87},[89,512,513,517],{},[92,514,515],{},[95,516,364],{"mathvariant":363},[100,518,367],{"encoding":102},[76,520,522],{"className":521,"ariaHidden":108},[107],[76,523,525,528],{"className":524},[112],[76,526],{"className":527,"style":117},[116],[76,529,364],{"className":530},[121],[13,532,345],{}," であるといい、",[76,535,537,555],{"className":536},[79],[76,538,540],{"className":539},[83],[85,541,542],{"xmlns":87},[89,543,544,552],{},[92,545,546,548,550],{},[95,547,364],{"mathvariant":363},[299,549,301],{},[95,551,456],{},[100,553,554],{"encoding":102},"\\Gamma \\models \\varphi",[76,556,558,576],{"className":557,"ariaHidden":108},[107],[76,559,561,564,567,570,573],{"className":560},[112],[76,562],{"className":563,"style":316},[116],[76,565,364],{"className":566},[121],[76,568],{"className":569,"style":324},[323],[76,571,301],{"className":572},[328],[76,574],{"className":575,"style":324},[323],[76,577,579,582],{"className":578},[112],[76,580],{"className":581,"style":469},[116],[76,583,456],{"className":584},[121,216]," と書く。特に ",[76,587,589,609],{"className":588},[79],[76,590,592],{"className":591},[83],[85,593,594],{"xmlns":87},[89,595,596,606],{},[92,597,598,600,603],{},[95,599,364],{"mathvariant":363},[299,601,602],{},"=",[95,604,605],{"mathvariant":363},"∅",[100,607,608],{"encoding":102},"\\Gamma = \\varnothing",[76,610,612,630],{"className":611,"ariaHidden":108},[107],[76,613,615,618,621,624,627],{"className":614},[112],[76,616],{"className":617,"style":117},[116],[76,619,364],{"className":620},[121],[76,622],{"className":623,"style":324},[323],[76,625,602],{"className":626},[328],[76,628],{"className":629,"style":324},[323],[76,631,633,637],{"className":632},[112],[76,634],{"className":635,"style":636},[116],"height:0.6633em;vertical-align:-0.0817em;",[76,638,605],{"className":639},[121,640],"amsrm"," のとき、",[76,643,645,661],{"className":644},[79],[76,646,648],{"className":647},[83],[85,649,650],{"xmlns":87},[89,651,652,658],{},[92,653,654,656],{},[299,655,301],{},[95,657,456],{},[100,659,660],{"encoding":102},"\\models \\varphi",[76,662,664,676],{"className":663,"ariaHidden":108},[107],[76,665,667,670,673],{"className":666},[112],[76,668],{"className":669,"style":316},[116],[76,671,301],{"className":672},[328],[76,674],{"className":675,"style":324},[323],[76,677,679,682],{"className":678},[112],[76,680],{"className":681,"style":469},[116],[76,683,456],{"className":684},[121,216]," と書き、",[76,687,689,702],{"className":688},[79],[76,690,692],{"className":691},[83],[85,693,694],{"xmlns":87},[89,695,696,700],{},[92,697,698],{},[95,699,456],{},[100,701,459],{"encoding":102},[76,703,705],{"className":704,"ariaHidden":108},[107],[76,706,708,711],{"className":707},[112],[76,709],{"className":710,"style":469},[116],[76,712,456],{"className":713},[121,216],[13,715,716],{},"妥当"," であるという。",[68,719,721],{"id":720},"_12-証明形式的な導出","1.2 証明：形式的な導出",[10,723,724,725,728],{},"一方、",[13,726,727],{},"証明"," は、公理と推論規則だけを使って、記号列を機械的に書き換えていく操作です。ここでは、命題論理の公理図式、量化子の公理、モーダスポネンス、汎化の規則からなる、ヒルベルト流の体系を考えます。",[10,730,731,759,760,788,789,842,843,871,872,922,923,980,981,502,1009,1012,1013,1016],{},[76,732,734,747],{"className":733},[79],[76,735,737],{"className":736},[83],[85,738,739],{"xmlns":87},[89,740,741,745],{},[92,742,743],{},[95,744,364],{"mathvariant":363},[100,746,367],{"encoding":102},[76,748,750],{"className":749,"ariaHidden":108},[107],[76,751,753,756],{"className":752},[112],[76,754],{"className":755,"style":117},[116],[76,757,364],{"className":758},[121]," から ",[76,761,763,776],{"className":762},[79],[76,764,766],{"className":765},[83],[85,767,768],{"xmlns":87},[89,769,770,774],{},[92,771,772],{},[95,773,456],{},[100,775,459],{"encoding":102},[76,777,779],{"className":778,"ariaHidden":108},[107],[76,780,782,785],{"className":781},[112],[76,783],{"className":784,"style":469},[116],[76,786,456],{"className":787},[121,216]," が導出できることを ",[76,790,792,811],{"className":791},[79],[76,793,795],{"className":794},[83],[85,796,797],{"xmlns":87},[89,798,799,808],{},[92,800,801,803,806],{},[95,802,364],{"mathvariant":363},[299,804,805],{},"⊢",[95,807,456],{},[100,809,810],{"encoding":102},"\\Gamma \\vdash \\varphi",[76,812,814,833],{"className":813,"ariaHidden":108},[107],[76,815,817,821,824,827,830],{"className":816},[112],[76,818],{"className":819,"style":820},[116],"height:0.6944em;",[76,822,364],{"className":823},[121],[76,825],{"className":826,"style":324},[323],[76,828,805],{"className":829},[328],[76,831],{"className":832,"style":324},[323],[76,834,836,839],{"className":835},[112],[76,837],{"className":838,"style":469},[116],[76,840,456],{"className":841},[121,216]," と書きます。また、ある ",[76,844,846,859],{"className":845},[79],[76,847,849],{"className":848},[83],[85,850,851],{"xmlns":87},[89,852,853,857],{},[92,854,855],{},[95,856,456],{},[100,858,459],{"encoding":102},[76,860,862],{"className":861,"ariaHidden":108},[107],[76,863,865,868],{"className":864},[112],[76,866],{"className":867,"style":469},[116],[76,869,456],{"className":870},[121,216]," について ",[76,873,875,892],{"className":874},[79],[76,876,878],{"className":877},[83],[85,879,880],{"xmlns":87},[89,881,882,890],{},[92,883,884,886,888],{},[95,885,364],{"mathvariant":363},[299,887,805],{},[95,889,456],{},[100,891,810],{"encoding":102},[76,893,895,913],{"className":894,"ariaHidden":108},[107],[76,896,898,901,904,907,910],{"className":897},[112],[76,899],{"className":900,"style":820},[116],[76,902,364],{"className":903},[121],[76,905],{"className":906,"style":324},[323],[76,908,805],{"className":909},[328],[76,911],{"className":912,"style":324},[323],[76,914,916,919],{"className":915},[112],[76,917],{"className":918,"style":469},[116],[76,920,456],{"className":921},[121,216]," と ",[76,924,926,947],{"className":925},[79],[76,927,929],{"className":928},[83],[85,930,931],{"xmlns":87},[89,932,933,944],{},[92,934,935,937,939,942],{},[95,936,364],{"mathvariant":363},[299,938,805],{},[95,940,941],{"mathvariant":363},"¬",[95,943,456],{},[100,945,946],{"encoding":102},"\\Gamma \\vdash \\neg\\varphi",[76,948,950,968],{"className":949,"ariaHidden":108},[107],[76,951,953,956,959,962,965],{"className":952},[112],[76,954],{"className":955,"style":820},[116],[76,957,364],{"className":958},[121],[76,960],{"className":961,"style":324},[323],[76,963,805],{"className":964},[328],[76,966],{"className":967,"style":324},[323],[76,969,971,974,977],{"className":970},[112],[76,972],{"className":973,"style":469},[116],[76,975,941],{"className":976},[121],[76,978,456],{"className":979},[121,216]," がどちらも成り立つとき、",[76,982,984,997],{"className":983},[79],[76,985,987],{"className":986},[83],[85,988,989],{"xmlns":87},[89,990,991,995],{},[92,992,993],{},[95,994,364],{"mathvariant":363},[100,996,367],{"encoding":102},[76,998,1000],{"className":999,"ariaHidden":108},[107],[76,1001,1003,1006],{"className":1002},[112],[76,1004],{"className":1005,"style":117},[116],[76,1007,364],{"className":1008},[121],[13,1010,1011],{},"矛盾している"," といい、そうでないとき ",[13,1014,1015],{},"無矛盾"," であるといいます。",[10,1018,1019],{},"この体系について、次の2つの性質は比較的簡単に示せます。証明の中で何度も使います。",[21,1021,1022,1130],{},[24,1023,1024,1027,1028,1078,1079,1129],{},[13,1025,1026],{},"健全性","：",[76,1029,1031,1048],{"className":1030},[79],[76,1032,1034],{"className":1033},[83],[85,1035,1036],{"xmlns":87},[89,1037,1038,1046],{},[92,1039,1040,1042,1044],{},[95,1041,364],{"mathvariant":363},[299,1043,805],{},[95,1045,456],{},[100,1047,810],{"encoding":102},[76,1049,1051,1069],{"className":1050,"ariaHidden":108},[107],[76,1052,1054,1057,1060,1063,1066],{"className":1053},[112],[76,1055],{"className":1056,"style":820},[116],[76,1058,364],{"className":1059},[121],[76,1061],{"className":1062,"style":324},[323],[76,1064,805],{"className":1065},[328],[76,1067],{"className":1068,"style":324},[323],[76,1070,1072,1075],{"className":1071},[112],[76,1073],{"className":1074,"style":469},[116],[76,1076,456],{"className":1077},[121,216]," ならば ",[76,1080,1082,1099],{"className":1081},[79],[76,1083,1085],{"className":1084},[83],[85,1086,1087],{"xmlns":87},[89,1088,1089,1097],{},[92,1090,1091,1093,1095],{},[95,1092,364],{"mathvariant":363},[299,1094,301],{},[95,1096,456],{},[100,1098,554],{"encoding":102},[76,1100,1102,1120],{"className":1101,"ariaHidden":108},[107],[76,1103,1105,1108,1111,1114,1117],{"className":1104},[112],[76,1106],{"className":1107,"style":316},[116],[76,1109,364],{"className":1110},[121],[76,1112],{"className":1113,"style":324},[323],[76,1115,301],{"className":1116},[328],[76,1118],{"className":1119,"style":324},[323],[76,1121,1123,1126],{"className":1122},[112],[76,1124],{"className":1125,"style":469},[116],[76,1127,456],{"className":1128},[121,216],"（証明できるものは正しい）。",[24,1131,1132,1027,1135,1078,1228,1303],{},[13,1133,1134],{},"演繹定理",[76,1136,1138,1169],{"className":1137},[79],[76,1139,1141],{"className":1140},[83],[85,1142,1143],{"xmlns":87},[89,1144,1145,1166],{},[92,1146,1147,1149,1152,1156,1159,1162,1164],{},[95,1148,364],{"mathvariant":363},[299,1150,1151],{},"∪",[299,1153,1155],{"stretchy":1154},"false","{",[95,1157,1158],{},"ψ",[299,1160,1161],{"stretchy":1154},"}",[299,1163,805],{},[95,1165,456],{},[100,1167,1168],{"encoding":102},"\\Gamma \\cup \\{\\psi\\} \\vdash \\varphi",[76,1170,1172,1192,1219],{"className":1171,"ariaHidden":108},[107],[76,1173,1175,1178,1181,1185,1189],{"className":1174},[112],[76,1176],{"className":1177,"style":117},[116],[76,1179,364],{"className":1180},[121],[76,1182],{"className":1183,"style":1184},[323],"margin-right:0.2222em;",[76,1186,1151],{"className":1187},[1188],"mbin",[76,1190],{"className":1191,"style":1184},[323],[76,1193,1195,1199,1203,1206,1210,1213,1216],{"className":1194},[112],[76,1196],{"className":1197,"style":1198},[116],"height:1em;vertical-align:-0.25em;",[76,1200,1155],{"className":1201},[1202],"mopen",[76,1204,1158],{"className":1205,"style":253},[121,216],[76,1207,1161],{"className":1208},[1209],"mclose",[76,1211],{"className":1212,"style":324},[323],[76,1214,805],{"className":1215},[328],[76,1217],{"className":1218,"style":324},[323],[76,1220,1222,1225],{"className":1221},[112],[76,1223],{"className":1224,"style":469},[116],[76,1226,456],{"className":1227},[121,216],[76,1229,1231,1254],{"className":1230},[79],[76,1232,1234],{"className":1233},[83],[85,1235,1236],{"xmlns":87},[89,1237,1238,1251],{},[92,1239,1240,1242,1244,1246,1249],{},[95,1241,364],{"mathvariant":363},[299,1243,805],{},[95,1245,1158],{},[299,1247,1248],{},"→",[95,1250,456],{},[100,1252,1253],{"encoding":102},"\\Gamma \\vdash \\psi \\to \\varphi",[76,1255,1257,1275,1294],{"className":1256,"ariaHidden":108},[107],[76,1258,1260,1263,1266,1269,1272],{"className":1259},[112],[76,1261],{"className":1262,"style":820},[116],[76,1264,364],{"className":1265},[121],[76,1267],{"className":1268,"style":324},[323],[76,1270,805],{"className":1271},[328],[76,1273],{"className":1274,"style":324},[323],[76,1276,1278,1282,1285,1288,1291],{"className":1277},[112],[76,1279],{"className":1280,"style":1281},[116],"height:0.8889em;vertical-align:-0.1944em;",[76,1283,1158],{"className":1284,"style":253},[121,216],[76,1286],{"className":1287,"style":324},[323],[76,1289,1248],{"className":1290},[328],[76,1292],{"className":1293,"style":324},[323],[76,1295,1297,1300],{"className":1296},[112],[76,1298],{"className":1299,"style":469},[116],[76,1301,456],{"className":1302},[121,216],"。",[68,1305,1307],{"id":1306},"_13-完全性定理の2つの形","1.3 完全性定理の2つの形",[1309,1310,1312],"theorem",{"title":1311},"完全性定理（強い形）",[10,1313,348,1314,1342,1343,1371,1372,1078,1422,1303],{},[76,1315,1317,1330],{"className":1316},[79],[76,1318,1320],{"className":1319},[83],[85,1321,1322],{"xmlns":87},[89,1323,1324,1328],{},[92,1325,1326],{},[95,1327,364],{"mathvariant":363},[100,1329,367],{"encoding":102},[76,1331,1333],{"className":1332,"ariaHidden":108},[107],[76,1334,1336,1339],{"className":1335},[112],[76,1337],{"className":1338,"style":117},[116],[76,1340,364],{"className":1341},[121]," と文 ",[76,1344,1346,1359],{"className":1345},[79],[76,1347,1349],{"className":1348},[83],[85,1350,1351],{"xmlns":87},[89,1352,1353,1357],{},[92,1354,1355],{},[95,1356,456],{},[100,1358,459],{"encoding":102},[76,1360,1362],{"className":1361,"ariaHidden":108},[107],[76,1363,1365,1368],{"className":1364},[112],[76,1366],{"className":1367,"style":469},[116],[76,1369,456],{"className":1370},[121,216]," について、",[76,1373,1375,1392],{"className":1374},[79],[76,1376,1378],{"className":1377},[83],[85,1379,1380],{"xmlns":87},[89,1381,1382,1390],{},[92,1383,1384,1386,1388],{},[95,1385,364],{"mathvariant":363},[299,1387,301],{},[95,1389,456],{},[100,1391,554],{"encoding":102},[76,1393,1395,1413],{"className":1394,"ariaHidden":108},[107],[76,1396,1398,1401,1404,1407,1410],{"className":1397},[112],[76,1399],{"className":1400,"style":316},[116],[76,1402,364],{"className":1403},[121],[76,1405],{"className":1406,"style":324},[323],[76,1408,301],{"className":1409},[328],[76,1411],{"className":1412,"style":324},[323],[76,1414,1416,1419],{"className":1415},[112],[76,1417],{"className":1418,"style":469},[116],[76,1420,456],{"className":1421},[121,216],[76,1423,1425,1442],{"className":1424},[79],[76,1426,1428],{"className":1427},[83],[85,1429,1430],{"xmlns":87},[89,1431,1432,1440],{},[92,1433,1434,1436,1438],{},[95,1435,364],{"mathvariant":363},[299,1437,805],{},[95,1439,456],{},[100,1441,810],{"encoding":102},[76,1443,1445,1463],{"className":1444,"ariaHidden":108},[107],[76,1446,1448,1451,1454,1457,1460],{"className":1447},[112],[76,1449],{"className":1450,"style":820},[116],[76,1452,364],{"className":1453},[121],[76,1455],{"className":1456,"style":324},[323],[76,1458,805],{"className":1459},[328],[76,1461],{"className":1462,"style":324},[323],[76,1464,1466,1469],{"className":1465},[112],[76,1467],{"className":1468,"style":469},[116],[76,1470,456],{"className":1471},[121,216],[10,1473,1474,1524,1525,1528],{},[76,1475,1477,1494],{"className":1476},[79],[76,1478,1480],{"className":1479},[83],[85,1481,1482],{"xmlns":87},[89,1483,1484,1492],{},[92,1485,1486,1488,1490],{},[95,1487,364],{"mathvariant":363},[299,1489,602],{},[95,1491,605],{"mathvariant":363},[100,1493,608],{"encoding":102},[76,1495,1497,1515],{"className":1496,"ariaHidden":108},[107],[76,1498,1500,1503,1506,1509,1512],{"className":1499},[112],[76,1501],{"className":1502,"style":117},[116],[76,1504,364],{"className":1505},[121],[76,1507],{"className":1508,"style":324},[323],[76,1510,602],{"className":1511},[328],[76,1513],{"className":1514,"style":324},[323],[76,1516,1518,1521],{"className":1517},[112],[76,1519],{"className":1520,"style":636},[116],[76,1522,605],{"className":1523},[121,640]," の場合、つまり「妥当な文は証明できる」が ",[13,1526,1527],{},"弱い形"," です。ゲーデルが1929年に証明したのは主にこちらです。",[10,1530,1531,1532,1535],{},"強い形は、次の ",[13,1533,1534],{},"モデル存在定理"," と同値です。ヘンキンの証明はこの形を直接示します。",[1309,1537,1538],{"title":1534},[10,1539,1540,1541,1569],{},"無矛盾な文の集合 ",[76,1542,1544,1557],{"className":1543},[79],[76,1545,1547],{"className":1546},[83],[85,1548,1549],{"xmlns":87},[89,1550,1551,1555],{},[92,1552,1553],{},[95,1554,364],{"mathvariant":363},[100,1556,367],{"encoding":102},[76,1558,1560],{"className":1559,"ariaHidden":108},[107],[76,1561,1563,1566],{"className":1562},[112],[76,1564],{"className":1565,"style":117},[116],[76,1567,364],{"className":1568},[121]," は、モデルを持つ。",[1571,1572,1573,1837],"proof",{},[10,1574,1575,1576,1629,1630,1696,1697,1785,1786,1836],{},"モデル存在定理を仮定し、",[76,1577,1579,1598],{"className":1578},[79],[76,1580,1582],{"className":1581},[83],[85,1583,1584],{"xmlns":87},[89,1585,1586,1595],{},[92,1587,1588,1590,1593],{},[95,1589,364],{"mathvariant":363},[299,1591,1592],{},"⊬",[95,1594,456],{},[100,1596,1597],{"encoding":102},"\\Gamma \\nvdash \\varphi",[76,1599,1601,1620],{"className":1600,"ariaHidden":108},[107],[76,1602,1604,1608,1611,1614,1617],{"className":1603},[112],[76,1605],{"className":1606,"style":1607},[116],"height:0.6889em;",[76,1609,364],{"className":1610},[121],[76,1612],{"className":1613,"style":324},[323],[76,1615,1592],{"className":1616},[328,640],[76,1618],{"className":1619,"style":324},[323],[76,1621,1623,1626],{"className":1622},[112],[76,1624],{"className":1625,"style":469},[116],[76,1627,456],{"className":1628},[121,216]," とする。このとき ",[76,1631,1633,1657],{"className":1632},[79],[76,1634,1636],{"className":1635},[83],[85,1637,1638],{"xmlns":87},[89,1639,1640,1654],{},[92,1641,1642,1644,1646,1648,1650,1652],{},[95,1643,364],{"mathvariant":363},[299,1645,1151],{},[299,1647,1155],{"stretchy":1154},[95,1649,941],{"mathvariant":363},[95,1651,456],{},[299,1653,1161],{"stretchy":1154},[100,1655,1656],{"encoding":102},"\\Gamma \\cup \\{\\neg\\varphi\\}",[76,1658,1660,1678],{"className":1659,"ariaHidden":108},[107],[76,1661,1663,1666,1669,1672,1675],{"className":1662},[112],[76,1664],{"className":1665,"style":117},[116],[76,1667,364],{"className":1668},[121],[76,1670],{"className":1671,"style":1184},[323],[76,1673,1151],{"className":1674},[1188],[76,1676],{"className":1677,"style":1184},[323],[76,1679,1681,1684,1687,1690,1693],{"className":1680},[112],[76,1682],{"className":1683,"style":1198},[116],[76,1685,1155],{"className":1686},[1202],[76,1688,941],{"className":1689},[121],[76,1691,456],{"className":1692},[121,216],[76,1694,1161],{"className":1695},[1209]," は無矛盾である。もし矛盾していれば、",[76,1698,1700,1728],{"className":1699},[79],[76,1701,1703],{"className":1702},[83],[85,1704,1705],{"xmlns":87},[89,1706,1707,1725],{},[92,1708,1709,1711,1713,1715,1717,1719,1721,1723],{},[95,1710,364],{"mathvariant":363},[299,1712,1151],{},[299,1714,1155],{"stretchy":1154},[95,1716,941],{"mathvariant":363},[95,1718,456],{},[299,1720,1161],{"stretchy":1154},[299,1722,805],{},[95,1724,456],{},[100,1726,1727],{"encoding":102},"\\Gamma \\cup \\{\\neg\\varphi\\} \\vdash \\varphi",[76,1729,1731,1749,1776],{"className":1730,"ariaHidden":108},[107],[76,1732,1734,1737,1740,1743,1746],{"className":1733},[112],[76,1735],{"className":1736,"style":117},[116],[76,1738,364],{"className":1739},[121],[76,1741],{"className":1742,"style":1184},[323],[76,1744,1151],{"className":1745},[1188],[76,1747],{"className":1748,"style":1184},[323],[76,1750,1752,1755,1758,1761,1764,1767,1770,1773],{"className":1751},[112],[76,1753],{"className":1754,"style":1198},[116],[76,1756,1155],{"className":1757},[1202],[76,1759,941],{"className":1760},[121],[76,1762,456],{"className":1763},[121,216],[76,1765,1161],{"className":1766},[1209],[76,1768],{"className":1769,"style":324},[323],[76,1771,805],{"className":1772},[328],[76,1774],{"className":1775,"style":324},[323],[76,1777,1779,1782],{"className":1778},[112],[76,1780],{"className":1781,"style":469},[116],[76,1783,456],{"className":1784},[121,216]," となり、演繹定理と命題論理から ",[76,1787,1789,1806],{"className":1788},[79],[76,1790,1792],{"className":1791},[83],[85,1793,1794],{"xmlns":87},[89,1795,1796,1804],{},[92,1797,1798,1800,1802],{},[95,1799,364],{"mathvariant":363},[299,1801,805],{},[95,1803,456],{},[100,1805,810],{"encoding":102},[76,1807,1809,1827],{"className":1808,"ariaHidden":108},[107],[76,1810,1812,1815,1818,1821,1824],{"className":1811},[112],[76,1813],{"className":1814,"style":820},[116],[76,1816,364],{"className":1817},[121],[76,1819],{"className":1820,"style":324},[323],[76,1822,805],{"className":1823},[328],[76,1825],{"className":1826,"style":324},[323],[76,1828,1830,1833],{"className":1829},[112],[76,1831],{"className":1832,"style":469},[116],[76,1834,456],{"className":1835},[121,216]," が導けてしまうからである。",[10,1838,1839,1840,1905,1906,1934,1935,502,1963,1991,1992,2043,2044,2072,2073,2125,2126,2154,2155,2183,2184,2235],{},"よって、",[76,1841,1843,1866],{"className":1842},[79],[76,1844,1846],{"className":1845},[83],[85,1847,1848],{"xmlns":87},[89,1849,1850,1864],{},[92,1851,1852,1854,1856,1858,1860,1862],{},[95,1853,364],{"mathvariant":363},[299,1855,1151],{},[299,1857,1155],{"stretchy":1154},[95,1859,941],{"mathvariant":363},[95,1861,456],{},[299,1863,1161],{"stretchy":1154},[100,1865,1656],{"encoding":102},[76,1867,1869,1887],{"className":1868,"ariaHidden":108},[107],[76,1870,1872,1875,1878,1881,1884],{"className":1871},[112],[76,1873],{"className":1874,"style":117},[116],[76,1876,364],{"className":1877},[121],[76,1879],{"className":1880,"style":1184},[323],[76,1882,1151],{"className":1883},[1188],[76,1885],{"className":1886,"style":1184},[323],[76,1888,1890,1893,1896,1899,1902],{"className":1889},[112],[76,1891],{"className":1892,"style":1198},[116],[76,1894,1155],{"className":1895},[1202],[76,1897,941],{"className":1898},[121],[76,1900,456],{"className":1901},[121,216],[76,1903,1161],{"className":1904},[1209]," はモデル ",[76,1907,1909,1922],{"className":1908},[79],[76,1910,1912],{"className":1911},[83],[85,1913,1914],{"xmlns":87},[89,1915,1916,1920],{},[92,1917,1918],{},[95,1919,171],{"mathvariant":97},[100,1921,174],{"encoding":102},[76,1923,1925],{"className":1924,"ariaHidden":108},[107],[76,1926,1928,1931],{"className":1927},[112],[76,1929],{"className":1930,"style":117},[116],[76,1932,171],{"className":1933},[121,122]," を持つ。この ",[76,1936,1938,1951],{"className":1937},[79],[76,1939,1941],{"className":1940},[83],[85,1942,1943],{"xmlns":87},[89,1944,1945,1949],{},[92,1946,1947],{},[95,1948,171],{"mathvariant":97},[100,1950,174],{"encoding":102},[76,1952,1954],{"className":1953,"ariaHidden":108},[107],[76,1955,1957,1960],{"className":1956},[112],[76,1958],{"className":1959,"style":117},[116],[76,1961,171],{"className":1962},[121,122],[76,1964,1966,1979],{"className":1965},[79],[76,1967,1969],{"className":1968},[83],[85,1970,1971],{"xmlns":87},[89,1972,1973,1977],{},[92,1974,1975],{},[95,1976,364],{"mathvariant":363},[100,1978,367],{"encoding":102},[76,1980,1982],{"className":1981,"ariaHidden":108},[107],[76,1983,1985,1988],{"className":1984},[112],[76,1986],{"className":1987,"style":117},[116],[76,1989,364],{"className":1990},[121]," のモデルであり（",[76,1993,1995,2013],{"className":1994},[79],[76,1996,1998],{"className":1997},[83],[85,1999,2000],{"xmlns":87},[89,2001,2002,2010],{},[92,2003,2004,2006,2008],{},[95,2005,171],{"mathvariant":97},[299,2007,301],{},[95,2009,364],{"mathvariant":363},[100,2011,2012],{"encoding":102},"\\mathcal{M} \\models \\Gamma",[76,2014,2016,2034],{"className":2015,"ariaHidden":108},[107],[76,2017,2019,2022,2025,2028,2031],{"className":2018},[112],[76,2020],{"className":2021,"style":316},[116],[76,2023,171],{"className":2024},[121,122],[76,2026],{"className":2027,"style":324},[323],[76,2029,301],{"className":2030},[328],[76,2032],{"className":2033,"style":324},[323],[76,2035,2037,2040],{"className":2036},[112],[76,2038],{"className":2039,"style":117},[116],[76,2041,364],{"className":2042},[121],"）、しかも ",[76,2045,2047,2060],{"className":2046},[79],[76,2048,2050],{"className":2049},[83],[85,2051,2052],{"xmlns":87},[89,2053,2054,2058],{},[92,2055,2056],{},[95,2057,456],{},[100,2059,459],{"encoding":102},[76,2061,2063],{"className":2062,"ariaHidden":108},[107],[76,2064,2066,2069],{"className":2065},[112],[76,2067],{"className":2068,"style":469},[116],[76,2070,456],{"className":2071},[121,216]," を充足しない（",[76,2074,2076,2095],{"className":2075},[79],[76,2077,2079],{"className":2078},[83],[85,2080,2081],{"xmlns":87},[89,2082,2083,2092],{},[92,2084,2085,2087,2090],{},[95,2086,171],{"mathvariant":97},[299,2088,2089],{},"⊭",[95,2091,456],{},[100,2093,2094],{"encoding":102},"\\mathcal{M} \\nvDash \\varphi",[76,2096,2098,2116],{"className":2097,"ariaHidden":108},[107],[76,2099,2101,2104,2107,2110,2113],{"className":2100},[112],[76,2102],{"className":2103,"style":1607},[116],[76,2105,171],{"className":2106},[121,122],[76,2108],{"className":2109,"style":324},[323],[76,2111,2089],{"className":2112},[328,640],[76,2114],{"className":2115,"style":324},[323],[76,2117,2119,2122],{"className":2118},[112],[76,2120],{"className":2121,"style":469},[116],[76,2123,456],{"className":2124},[121,216],"）。つまり、",[76,2127,2129,2142],{"className":2128},[79],[76,2130,2132],{"className":2131},[83],[85,2133,2134],{"xmlns":87},[89,2135,2136,2140],{},[92,2137,2138],{},[95,2139,456],{},[100,2141,459],{"encoding":102},[76,2143,2145],{"className":2144,"ariaHidden":108},[107],[76,2146,2148,2151],{"className":2147},[112],[76,2149],{"className":2150,"style":469},[116],[76,2152,456],{"className":2153},[121,216]," が成り立たない ",[76,2156,2158,2171],{"className":2157},[79],[76,2159,2161],{"className":2160},[83],[85,2162,2163],{"xmlns":87},[89,2164,2165,2169],{},[92,2166,2167],{},[95,2168,364],{"mathvariant":363},[100,2170,367],{"encoding":102},[76,2172,2174],{"className":2173,"ariaHidden":108},[107],[76,2175,2177,2180],{"className":2176},[112],[76,2178],{"className":2179,"style":117},[116],[76,2181,364],{"className":2182},[121]," のモデルが存在するので、",[76,2185,2187,2205],{"className":2186},[79],[76,2188,2190],{"className":2189},[83],[85,2191,2192],{"xmlns":87},[89,2193,2194,2202],{},[92,2195,2196,2198,2200],{},[95,2197,364],{"mathvariant":363},[299,2199,2089],{},[95,2201,456],{},[100,2203,2204],{"encoding":102},"\\Gamma \\nvDash \\varphi",[76,2206,2208,2226],{"className":2207,"ariaHidden":108},[107],[76,2209,2211,2214,2217,2220,2223],{"className":2210},[112],[76,2212],{"className":2213,"style":1607},[116],[76,2215,364],{"className":2216},[121],[76,2218],{"className":2219,"style":324},[323],[76,2221,2089],{"className":2222},[328,640],[76,2224],{"className":2225,"style":324},[323],[76,2227,2229,2232],{"className":2228},[112],[76,2230],{"className":2231,"style":469},[116],[76,2233,456],{"className":2234},[121,216]," である。対偶をとれば強い形が得られる。逆向きも同様に示せる。",[63,2237,2239],{"id":2238},"_2-ゲーデル流の証明","2. ゲーデル流の証明",[10,2241,2242,2243,2246],{},"ゲーデルの証明の核心は、",[13,2244,2245],{},"一階述語論理の問題を、無限個の命題論理の問題に帰着する"," ことです。ゲーデルが扱った言語は、関数記号を含まない（述語記号と等号のみの）言語でした。以下でも、まず等号のない場合を考えます。",[10,2248,2249],{},"ゲーデルが実際に示したのは、次の形の定理です。",[1309,2251,2253],{"title":2252},"ゲーデル（1930年）",[10,2254,2255,2256,2284,2285,2288,2289,2337,2338,2341],{},"一階の論理式 ",[76,2257,2259,2272],{"className":2258},[79],[76,2260,2262],{"className":2261},[83],[85,2263,2264],{"xmlns":87},[89,2265,2266,2270],{},[92,2267,2268],{},[95,2269,456],{},[100,2271,459],{"encoding":102},[76,2273,2275],{"className":2274,"ariaHidden":108},[107],[76,2276,2278,2281],{"className":2277},[112],[76,2279],{"className":2280,"style":469},[116],[76,2282,456],{"className":2283},[121,216]," は、",[13,2286,2287],{},"反駁可能","（",[76,2290,2292,2310],{"className":2291},[79],[76,2293,2295],{"className":2294},[83],[85,2296,2297],{"xmlns":87},[89,2298,2299,2307],{},[92,2300,2301,2303,2305],{},[299,2302,805],{},[95,2304,941],{"mathvariant":363},[95,2306,456],{},[100,2308,2309],{"encoding":102},"\\vdash \\neg\\varphi",[76,2311,2313,2325],{"className":2312,"ariaHidden":108},[107],[76,2314,2316,2319,2322],{"className":2315},[112],[76,2317],{"className":2318,"style":820},[116],[76,2320,805],{"className":2321},[328],[76,2323],{"className":2324,"style":324},[323],[76,2326,2328,2331,2334],{"className":2327},[112],[76,2329],{"className":2330,"style":469},[116],[76,2332,941],{"className":2333},[121],[76,2335,456],{"className":2336},[121,216],"）であるか、",[13,2339,2340],{},"充足可能"," であるかのどちらかである。",[10,2343,2344,2345,2348,2349,40],{},"以下の証明で作るモデルの領域は自然数（変数全体）なので、充足可能な場合は ",[13,2346,2347],{},"可算な領域で"," 充足できることも同時に分かります（文献 ",[31,2350,2352],{"href":2351},"#ref-10","[10]",[10,2354,2355,2356,2390,2391,2419,2420,2453,2454,2487,2488,2539,2540,2583],{},"この定理を ",[76,2357,2359,2375],{"className":2358},[79],[76,2360,2362],{"className":2361},[83],[85,2363,2364],{"xmlns":87},[89,2365,2366,2372],{},[92,2367,2368,2370],{},[95,2369,941],{"mathvariant":363},[95,2371,456],{},[100,2373,2374],{"encoding":102},"\\neg\\varphi",[76,2376,2378],{"className":2377,"ariaHidden":108},[107],[76,2379,2381,2384,2387],{"className":2380},[112],[76,2382],{"className":2383,"style":469},[116],[76,2385,941],{"className":2386},[121],[76,2388,456],{"className":2389},[121,216]," に当てはめると、弱い形の完全性定理が出ます。",[76,2392,2394,2407],{"className":2393},[79],[76,2395,2397],{"className":2396},[83],[85,2398,2399],{"xmlns":87},[89,2400,2401,2405],{},[92,2402,2403],{},[95,2404,456],{},[100,2406,459],{"encoding":102},[76,2408,2410],{"className":2409,"ariaHidden":108},[107],[76,2411,2413,2416],{"className":2412},[112],[76,2414],{"className":2415,"style":469},[116],[76,2417,456],{"className":2418},[121,216]," が妥当なら ",[76,2421,2423,2438],{"className":2422},[79],[76,2424,2426],{"className":2425},[83],[85,2427,2428],{"xmlns":87},[89,2429,2430,2436],{},[92,2431,2432,2434],{},[95,2433,941],{"mathvariant":363},[95,2435,456],{},[100,2437,2374],{"encoding":102},[76,2439,2441],{"className":2440,"ariaHidden":108},[107],[76,2442,2444,2447,2450],{"className":2443},[112],[76,2445],{"className":2446,"style":469},[116],[76,2448,941],{"className":2449},[121],[76,2451,456],{"className":2452},[121,216]," は充足不可能なので、",[76,2455,2457,2472],{"className":2456},[79],[76,2458,2460],{"className":2459},[83],[85,2461,2462],{"xmlns":87},[89,2463,2464,2470],{},[92,2465,2466,2468],{},[95,2467,941],{"mathvariant":363},[95,2469,456],{},[100,2471,2374],{"encoding":102},[76,2473,2475],{"className":2474,"ariaHidden":108},[107],[76,2476,2478,2481,2484],{"className":2477},[112],[76,2479],{"className":2480,"style":469},[116],[76,2482,941],{"className":2483},[121],[76,2485,456],{"className":2486},[121,216]," は反駁可能、つまり ",[76,2489,2491,2511],{"className":2490},[79],[76,2492,2494],{"className":2493},[83],[85,2495,2496],{"xmlns":87},[89,2497,2498,2508],{},[92,2499,2500,2502,2504,2506],{},[299,2501,805],{},[95,2503,941],{"mathvariant":363},[95,2505,941],{"mathvariant":363},[95,2507,456],{},[100,2509,2510],{"encoding":102},"\\vdash \\neg\\neg\\varphi",[76,2512,2514,2526],{"className":2513,"ariaHidden":108},[107],[76,2515,2517,2520,2523],{"className":2516},[112],[76,2518],{"className":2519,"style":820},[116],[76,2521,805],{"className":2522},[328],[76,2524],{"className":2525,"style":324},[323],[76,2527,2529,2532,2536],{"className":2528},[112],[76,2530],{"className":2531,"style":469},[116],[76,2533,2535],{"className":2534},[121],"¬¬",[76,2537,456],{"className":2538},[121,216]," です。したがって ",[76,2541,2543,2559],{"className":2542},[79],[76,2544,2546],{"className":2545},[83],[85,2547,2548],{"xmlns":87},[89,2549,2550,2556],{},[92,2551,2552,2554],{},[299,2553,805],{},[95,2555,456],{},[100,2557,2558],{"encoding":102},"\\vdash \\varphi",[76,2560,2562,2574],{"className":2561,"ariaHidden":108},[107],[76,2563,2565,2568,2571],{"className":2564},[112],[76,2566],{"className":2567,"style":820},[116],[76,2569,805],{"className":2570},[328],[76,2572],{"className":2573,"style":324},[323],[76,2575,2577,2580],{"className":2576},[112],[76,2578],{"className":2579,"style":469},[116],[76,2581,456],{"className":2582},[121,216]," となります。",[68,2585,2587],{"id":2586},"_21-ステップ1標準形への帰着","2.1 ステップ1：標準形への帰着",[10,2589,2590,2591,2594,2595,40],{},"まず、論理式を ",[13,2592,2593],{},"冠頭標準形","（量化子をすべて先頭に出した形）にします。ゲーデルは、スコーレムがレーヴェンハイム＝スコーレムの定理の証明で使った方法を応用し、新しい述語記号を導入することで、さらに次の形の論理式だけを考えれば十分であることを示しました（文献 ",[31,2596,2352],{"href":2351},[76,2598,2601],{"className":2599},[2600],"katex-display",[76,2602,2604,2725],{"className":2603},[79],[76,2605,2607],{"className":2606},[83],[85,2608,2610],{"xmlns":87,"display":2609},"block",[89,2611,2612,2722],{},[92,2613,2614,2616,2620,2622,2624,2627,2637,2640,2642,2649,2651,2654,2661,2663,2665,2672,2674,2676,2679,2685,2688,2691,2693,2699,2701,2707,2709,2711,2713,2719],{},[95,2615,456],{},[2617,2618,2619],"mtext",{},"  ",[299,2621,602],{},[2617,2623,2619],{},[95,2625,2626],{"mathvariant":363},"∀",[2628,2629,2630,2633],"msub",{},[95,2631,2632],{},"x",[2634,2635,2636],"mn",{},"1",[299,2638,2639],{},"⋯",[95,2641,2626],{"mathvariant":363},[2628,2643,2644,2646],{},[95,2645,2632],{},[95,2647,2648],{},"k",[2617,2650,2619],{},[95,2652,2653],{"mathvariant":363},"∃",[2628,2655,2656,2659],{},[95,2657,2658],{},"y",[2634,2660,2636],{},[299,2662,2639],{},[95,2664,2653],{"mathvariant":363},[2628,2666,2667,2669],{},[95,2668,2658],{},[95,2670,2671],{},"m",[2617,2673,2619],{},[95,2675,1158],{},[299,2677,2678],{"stretchy":1154},"(",[2628,2680,2681,2683],{},[95,2682,2632],{},[2634,2684,2636],{},[299,2686,2687],{"separator":108},",",[299,2689,2690],{},"…",[299,2692,2687],{"separator":108},[2628,2694,2695,2697],{},[95,2696,2632],{},[95,2698,2648],{},[299,2700,2687],{"separator":108},[2628,2702,2703,2705],{},[95,2704,2658],{},[2634,2706,2636],{},[299,2708,2687],{"separator":108},[299,2710,2690],{},[299,2712,2687],{"separator":108},[2628,2714,2715,2717],{},[95,2716,2658],{},[95,2718,2671],{},[299,2720,2721],{"stretchy":1154},")",[100,2723,2724],{"encoding":102},"\\varphi \\;=\\; \\forall x_1 \\cdots \\forall x_k \\; \\exists y_1 \\cdots \\exists y_m \\; \\psi(x_1, \\dots, x_k, y_1, \\dots, y_m)",[76,2726,2728,2752],{"className":2727,"ariaHidden":108},[107],[76,2729,2731,2734,2737,2740,2743,2746,2749],{"className":2730},[112],[76,2732],{"className":2733,"style":469},[116],[76,2735,456],{"className":2736},[121,216],[76,2738],{"className":2739,"style":324},[323],[76,2741],{"className":2742,"style":324},[323],[76,2744,602],{"className":2745},[328],[76,2747],{"className":2748,"style":324},[323],[76,2750],{"className":2751,"style":324},[323],[76,2753,2755,2758,2761,2817,2821,2825,2828,2831,2873,2876,2879,2920,2923,2926,2929,2932,2973,2976,2979,2982,3022,3026,3029,3032,3035,3038,3041,3081,3084,3087,3127,3130,3133,3136,3139,3142,3145,3185],{"className":2754},[112],[76,2756],{"className":2757,"style":1198},[116],[76,2759,2626],{"className":2760},[121],[76,2762,2764,2767],{"className":2763},[121],[76,2765,2632],{"className":2766},[121,216],[76,2768,2771],{"className":2769},[2770],"msupsub",[76,2772,2776,2808],{"className":2773},[2774,2775],"vlist-t","vlist-t2",[76,2777,2780,2803],{"className":2778},[2779],"vlist-r",[76,2781,2785],{"className":2782,"style":2784},[2783],"vlist","height:0.3011em;",[76,2786,2788,2793],{"style":2787},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[76,2789],{"className":2790,"style":2792},[2791],"pstrut","height:2.7em;",[76,2794,2800],{"className":2795},[2796,2797,2798,2799],"katex-sizing","reset-size6","size3","mtight",[76,2801,2636],{"className":2802},[121,2799],[76,2804,2807],{"className":2805},[2806],"vlist-s","​",[76,2809,2811],{"className":2810},[2779],[76,2812,2815],{"className":2813,"style":2814},[2783],"height:0.15em;",[76,2816],{},[76,2818],{"className":2819,"style":2820},[323],"margin-right:0.1667em;",[76,2822,2639],{"className":2823},[2824],"minner",[76,2826],{"className":2827,"style":2820},[323],[76,2829,2626],{"className":2830},[121],[76,2832,2834,2837],{"className":2833},[121],[76,2835,2632],{"className":2836},[121,216],[76,2838,2840],{"className":2839},[2770],[76,2841,2843,2865],{"className":2842},[2774,2775],[76,2844,2846,2862],{"className":2845},[2779],[76,2847,2850],{"className":2848,"style":2849},[2783],"height:0.3361em;",[76,2851,2852,2855],{"style":2787},[76,2853],{"className":2854,"style":2792},[2791],[76,2856,2858],{"className":2857},[2796,2797,2798,2799],[76,2859,2648],{"className":2860,"style":2861},[121,216,2799],"margin-right:0.0315em;",[76,2863,2807],{"className":2864},[2806],[76,2866,2868],{"className":2867},[2779],[76,2869,2871],{"className":2870,"style":2814},[2783],[76,2872],{},[76,2874],{"className":2875,"style":324},[323],[76,2877,2653],{"className":2878},[121],[76,2880,2882,2885],{"className":2881},[121],[76,2883,2658],{"className":2884,"style":253},[121,216],[76,2886,2888],{"className":2887},[2770],[76,2889,2891,2912],{"className":2890},[2774,2775],[76,2892,2894,2909],{"className":2893},[2779],[76,2895,2897],{"className":2896,"style":2784},[2783],[76,2898,2900,2903],{"style":2899},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[76,2901],{"className":2902,"style":2792},[2791],[76,2904,2906],{"className":2905},[2796,2797,2798,2799],[76,2907,2636],{"className":2908},[121,2799],[76,2910,2807],{"className":2911},[2806],[76,2913,2915],{"className":2914},[2779],[76,2916,2918],{"className":2917,"style":2814},[2783],[76,2919],{},[76,2921],{"className":2922,"style":2820},[323],[76,2924,2639],{"className":2925},[2824],[76,2927],{"className":2928,"style":2820},[323],[76,2930,2653],{"className":2931},[121],[76,2933,2935,2938],{"className":2934},[121],[76,2936,2658],{"className":2937,"style":253},[121,216],[76,2939,2941],{"className":2940},[2770],[76,2942,2944,2965],{"className":2943},[2774,2775],[76,2945,2947,2962],{"className":2946},[2779],[76,2948,2951],{"className":2949,"style":2950},[2783],"height:0.1514em;",[76,2952,2953,2956],{"style":2899},[76,2954],{"className":2955,"style":2792},[2791],[76,2957,2959],{"className":2958},[2796,2797,2798,2799],[76,2960,2671],{"className":2961},[121,216,2799],[76,2963,2807],{"className":2964},[2806],[76,2966,2968],{"className":2967},[2779],[76,2969,2971],{"className":2970,"style":2814},[2783],[76,2972],{},[76,2974],{"className":2975,"style":324},[323],[76,2977,1158],{"className":2978,"style":253},[121,216],[76,2980,2678],{"className":2981},[1202],[76,2983,2985,2988],{"className":2984},[121],[76,2986,2632],{"className":2987},[121,216],[76,2989,2991],{"className":2990},[2770],[76,2992,2994,3014],{"className":2993},[2774,2775],[76,2995,2997,3011],{"className":2996},[2779],[76,2998,3000],{"className":2999,"style":2784},[2783],[76,3001,3002,3005],{"style":2787},[76,3003],{"className":3004,"style":2792},[2791],[76,3006,3008],{"className":3007},[2796,2797,2798,2799],[76,3009,2636],{"className":3010},[121,2799],[76,3012,2807],{"className":3013},[2806],[76,3015,3017],{"className":3016},[2779],[76,3018,3020],{"className":3019,"style":2814},[2783],[76,3021],{},[76,3023,2687],{"className":3024},[3025],"mpunct",[76,3027],{"className":3028,"style":2820},[323],[76,3030,2690],{"className":3031},[2824],[76,3033],{"className":3034,"style":2820},[323],[76,3036,2687],{"className":3037},[3025],[76,3039],{"className":3040,"style":2820},[323],[76,3042,3044,3047],{"className":3043},[121],[76,3045,2632],{"className":3046},[121,216],[76,3048,3050],{"className":3049},[2770],[76,3051,3053,3073],{"className":3052},[2774,2775],[76,3054,3056,3070],{"className":3055},[2779],[76,3057,3059],{"className":3058,"style":2849},[2783],[76,3060,3061,3064],{"style":2787},[76,3062],{"className":3063,"style":2792},[2791],[76,3065,3067],{"className":3066},[2796,2797,2798,2799],[76,3068,2648],{"className":3069,"style":2861},[121,216,2799],[76,3071,2807],{"className":3072},[2806],[76,3074,3076],{"className":3075},[2779],[76,3077,3079],{"className":3078,"style":2814},[2783],[76,3080],{},[76,3082,2687],{"className":3083},[3025],[76,3085],{"className":3086,"style":2820},[323],[76,3088,3090,3093],{"className":3089},[121],[76,3091,2658],{"className":3092,"style":253},[121,216],[76,3094,3096],{"className":3095},[2770],[76,3097,3099,3119],{"className":3098},[2774,2775],[76,3100,3102,3116],{"className":3101},[2779],[76,3103,3105],{"className":3104,"style":2784},[2783],[76,3106,3107,3110],{"style":2899},[76,3108],{"className":3109,"style":2792},[2791],[76,3111,3113],{"className":3112},[2796,2797,2798,2799],[76,3114,2636],{"className":3115},[121,2799],[76,3117,2807],{"className":3118},[2806],[76,3120,3122],{"className":3121},[2779],[76,3123,3125],{"className":3124,"style":2814},[2783],[76,3126],{},[76,3128,2687],{"className":3129},[3025],[76,3131],{"className":3132,"style":2820},[323],[76,3134,2690],{"className":3135},[2824],[76,3137],{"className":3138,"style":2820},[323],[76,3140,2687],{"className":3141},[3025],[76,3143],{"className":3144,"style":2820},[323],[76,3146,3148,3151],{"className":3147},[121],[76,3149,2658],{"className":3150,"style":253},[121,216],[76,3152,3154],{"className":3153},[2770],[76,3155,3157,3177],{"className":3156},[2774,2775],[76,3158,3160,3174],{"className":3159},[2779],[76,3161,3163],{"className":3162,"style":2950},[2783],[76,3164,3165,3168],{"style":2899},[76,3166],{"className":3167,"style":2792},[2791],[76,3169,3171],{"className":3170},[2796,2797,2798,2799],[76,3172,2671],{"className":3173},[121,216,2799],[76,3175,2807],{"className":3176},[2806],[76,3178,3180],{"className":3179},[2779],[76,3181,3183],{"className":3182,"style":2814},[2783],[76,3184],{},[76,3186,2721],{"className":3187},[1209],[10,3189,3190,3191,3220,3221,3290,3291,3366,3367,3395],{},"ここで ",[76,3192,3194,3208],{"className":3193},[79],[76,3195,3197],{"className":3196},[83],[85,3198,3199],{"xmlns":87},[89,3200,3201,3205],{},[92,3202,3203],{},[95,3204,1158],{},[100,3206,3207],{"encoding":102},"\\psi",[76,3209,3211],{"className":3210,"ariaHidden":108},[107],[76,3212,3214,3217],{"className":3213},[112],[76,3215],{"className":3216,"style":1281},[116],[76,3218,1158],{"className":3219,"style":253},[121,216]," は量化子を含まない論理式です。「すべての ",[76,3222,3224,3244],{"className":3223},[79],[76,3225,3227],{"className":3226},[83],[85,3228,3229],{"xmlns":87},[89,3230,3231,3241],{},[92,3232,3233],{},[3234,3235,3236,3238],"mover",{"accent":108},[95,3237,2632],{},[299,3239,3240],{},"ˉ",[100,3242,3243],{"encoding":102},"\\bar{x}",[76,3245,3247],{"className":3246,"ariaHidden":108},[107],[76,3248,3250,3254],{"className":3249},[112],[76,3251],{"className":3252,"style":3253},[116],"height:0.5678em;",[76,3255,3258],{"className":3256},[121,3257],"katex-accent",[76,3259,3261],{"className":3260},[2774],[76,3262,3264],{"className":3263},[2779],[76,3265,3267,3277],{"className":3266,"style":3253},[2783],[76,3268,3270,3274],{"style":3269},"top:-3em;",[76,3271],{"className":3272,"style":3273},[2791],"height:3em;",[76,3275,2632],{"className":3276},[121,216],[76,3278,3279,3282],{"style":3269},[76,3280],{"className":3281,"style":3273},[2791],[76,3283,3287],{"className":3284,"style":3286},[3285],"accent-body","left:-0.2222em;",[76,3288,3240],{"className":3289},[121]," に対して、ある ",[76,3292,3294,3312],{"className":3293},[79],[76,3295,3297],{"className":3296},[83],[85,3298,3299],{"xmlns":87},[89,3300,3301,3309],{},[92,3302,3303],{},[3234,3304,3305,3307],{"accent":108},[95,3306,2658],{},[299,3308,3240],{},[100,3310,3311],{"encoding":102},"\\bar{y}",[76,3313,3315],{"className":3314,"ariaHidden":108},[107],[76,3316,3318,3322],{"className":3317},[112],[76,3319],{"className":3320,"style":3321},[116],"height:0.7622em;vertical-align:-0.1944em;",[76,3323,3325],{"className":3324},[121,3257],[76,3326,3328,3357],{"className":3327},[2774,2775],[76,3329,3331,3354],{"className":3330},[2779],[76,3332,3334,3342],{"className":3333,"style":3253},[2783],[76,3335,3336,3339],{"style":3269},[76,3337],{"className":3338,"style":3273},[2791],[76,3340,2658],{"className":3341,"style":253},[121,216],[76,3343,3344,3347],{"style":3269},[76,3345],{"className":3346,"style":3273},[2791],[76,3348,3351],{"className":3349,"style":3350},[3285],"left:-0.1944em;",[76,3352,3240],{"className":3353},[121],[76,3355,2807],{"className":3356},[2806],[76,3358,3360],{"className":3359},[2779],[76,3361,3364],{"className":3362,"style":3363},[2783],"height:0.1944em;",[76,3365],{}," が存在して、",[76,3368,3370,3383],{"className":3369},[79],[76,3371,3373],{"className":3372},[83],[85,3374,3375],{"xmlns":87},[89,3376,3377,3381],{},[92,3378,3379],{},[95,3380,1158],{},[100,3382,3207],{"encoding":102},[76,3384,3386],{"className":3385,"ariaHidden":108},[107],[76,3387,3389,3392],{"className":3388},[112],[76,3390],{"className":3391,"style":1281},[116],[76,3393,1158],{"className":3394,"style":253},[121,216]," が成り立つ」という形です。",[68,3397,3399],{"id":3398},"_22-ステップ2命題論理式の列を作る","2.2 ステップ2：命題論理式の列を作る",[10,3401,3402,3403,3597,3598,152,3626,922,3686,3757],{},"変数を ",[76,3404,3406,3447],{"className":3405},[79],[76,3407,3409],{"className":3408},[83],[85,3410,3411],{"xmlns":87},[89,3412,3413,3444],{},[92,3414,3415,3423,3425,3431,3433,3440,3442],{},[2628,3416,3417,3420],{},[95,3418,3419],{},"v",[2634,3421,3422],{},"0",[299,3424,2687],{"separator":108},[2628,3426,3427,3429],{},[95,3428,3419],{},[2634,3430,2636],{},[299,3432,2687],{"separator":108},[2628,3434,3435,3437],{},[95,3436,3419],{},[2634,3438,3439],{},"2",[299,3441,2687],{"separator":108},[299,3443,2690],{},[100,3445,3446],{"encoding":102},"v_0, v_1, v_2, \\dots",[76,3448,3450],{"className":3449,"ariaHidden":108},[107],[76,3451,3453,3456,3496,3499,3502,3542,3545,3548,3588,3591,3594],{"className":3452},[112],[76,3454],{"className":3455,"style":469},[116],[76,3457,3459,3462],{"className":3458},[121],[76,3460,3419],{"className":3461,"style":253},[121,216],[76,3463,3465],{"className":3464},[2770],[76,3466,3468,3488],{"className":3467},[2774,2775],[76,3469,3471,3485],{"className":3470},[2779],[76,3472,3474],{"className":3473,"style":2784},[2783],[76,3475,3476,3479],{"style":2899},[76,3477],{"className":3478,"style":2792},[2791],[76,3480,3482],{"className":3481},[2796,2797,2798,2799],[76,3483,3422],{"className":3484},[121,2799],[76,3486,2807],{"className":3487},[2806],[76,3489,3491],{"className":3490},[2779],[76,3492,3494],{"className":3493,"style":2814},[2783],[76,3495],{},[76,3497,2687],{"className":3498},[3025],[76,3500],{"className":3501,"style":2820},[323],[76,3503,3505,3508],{"className":3504},[121],[76,3506,3419],{"className":3507,"style":253},[121,216],[76,3509,3511],{"className":3510},[2770],[76,3512,3514,3534],{"className":3513},[2774,2775],[76,3515,3517,3531],{"className":3516},[2779],[76,3518,3520],{"className":3519,"style":2784},[2783],[76,3521,3522,3525],{"style":2899},[76,3523],{"className":3524,"style":2792},[2791],[76,3526,3528],{"className":3527},[2796,2797,2798,2799],[76,3529,2636],{"className":3530},[121,2799],[76,3532,2807],{"className":3533},[2806],[76,3535,3537],{"className":3536},[2779],[76,3538,3540],{"className":3539,"style":2814},[2783],[76,3541],{},[76,3543,2687],{"className":3544},[3025],[76,3546],{"className":3547,"style":2820},[323],[76,3549,3551,3554],{"className":3550},[121],[76,3552,3419],{"className":3553,"style":253},[121,216],[76,3555,3557],{"className":3556},[2770],[76,3558,3560,3580],{"className":3559},[2774,2775],[76,3561,3563,3577],{"className":3562},[2779],[76,3564,3566],{"className":3565,"style":2784},[2783],[76,3567,3568,3571],{"style":2899},[76,3569],{"className":3570,"style":2792},[2791],[76,3572,3574],{"className":3573},[2796,2797,2798,2799],[76,3575,3439],{"className":3576},[121,2799],[76,3578,2807],{"className":3579},[2806],[76,3581,3583],{"className":3582},[2779],[76,3584,3586],{"className":3585,"style":2814},[2783],[76,3587],{},[76,3589,2687],{"className":3590},[3025],[76,3592],{"className":3593,"style":2820},[323],[76,3595,2690],{"className":3596},[2824]," と並べます。",[76,3599,3601,3614],{"className":3600},[79],[76,3602,3604],{"className":3603},[83],[85,3605,3606],{"xmlns":87},[89,3607,3608,3612],{},[92,3609,3610],{},[95,3611,1158],{},[100,3613,3207],{"encoding":102},[76,3615,3617],{"className":3616,"ariaHidden":108},[107],[76,3618,3620,3623],{"className":3619},[112],[76,3621],{"className":3622,"style":1281},[116],[76,3624,1158],{"className":3625,"style":253},[121,216],[76,3627,3629,3646],{"className":3628},[79],[76,3630,3632],{"className":3631},[83],[85,3633,3634],{"xmlns":87},[89,3635,3636,3644],{},[92,3637,3638],{},[3234,3639,3640,3642],{"accent":108},[95,3641,2632],{},[299,3643,3240],{},[100,3645,3243],{"encoding":102},[76,3647,3649],{"className":3648,"ariaHidden":108},[107],[76,3650,3652,3655],{"className":3651},[112],[76,3653],{"className":3654,"style":3253},[116],[76,3656,3658],{"className":3657},[121,3257],[76,3659,3661],{"className":3660},[2774],[76,3662,3664],{"className":3663},[2779],[76,3665,3667,3675],{"className":3666,"style":3253},[2783],[76,3668,3669,3672],{"style":3269},[76,3670],{"className":3671,"style":3273},[2791],[76,3673,2632],{"className":3674},[121,216],[76,3676,3677,3680],{"style":3269},[76,3678],{"className":3679,"style":3273},[2791],[76,3681,3683],{"className":3682,"style":3286},[3285],[76,3684,3240],{"className":3685},[121],[76,3687,3689,3706],{"className":3688},[79],[76,3690,3692],{"className":3691},[83],[85,3693,3694],{"xmlns":87},[89,3695,3696,3704],{},[92,3697,3698],{},[3234,3699,3700,3702],{"accent":108},[95,3701,2658],{},[299,3703,3240],{},[100,3705,3311],{"encoding":102},[76,3707,3709],{"className":3708,"ariaHidden":108},[107],[76,3710,3712,3715],{"className":3711},[112],[76,3713],{"className":3714,"style":3321},[116],[76,3716,3718],{"className":3717},[121,3257],[76,3719,3721,3749],{"className":3720},[2774,2775],[76,3722,3724,3746],{"className":3723},[2779],[76,3725,3727,3735],{"className":3726,"style":3253},[2783],[76,3728,3729,3732],{"style":3269},[76,3730],{"className":3731,"style":3273},[2791],[76,3733,2658],{"className":3734,"style":253},[121,216],[76,3736,3737,3740],{"style":3269},[76,3738],{"className":3739,"style":3273},[2791],[76,3741,3743],{"className":3742,"style":3350},[3285],[76,3744,3240],{"className":3745},[121],[76,3747,2807],{"className":3748},[2806],[76,3750,3752],{"className":3751},[2779],[76,3753,3755],{"className":3754,"style":3363},[2783],[76,3756],{}," に、次の規則で変数を代入していきます。",[21,3759,3760,3856],{},[24,3761,3762,3822,3823,3855],{},[76,3763,3765,3782],{"className":3764},[79],[76,3766,3768],{"className":3767},[83],[85,3769,3770],{"xmlns":87},[89,3771,3772,3780],{},[92,3773,3774],{},[3234,3775,3776,3778],{"accent":108},[95,3777,2632],{},[299,3779,3240],{},[100,3781,3243],{"encoding":102},[76,3783,3785],{"className":3784,"ariaHidden":108},[107],[76,3786,3788,3791],{"className":3787},[112],[76,3789],{"className":3790,"style":3253},[116],[76,3792,3794],{"className":3793},[121,3257],[76,3795,3797],{"className":3796},[2774],[76,3798,3800],{"className":3799},[2779],[76,3801,3803,3811],{"className":3802,"style":3253},[2783],[76,3804,3805,3808],{"style":3269},[76,3806],{"className":3807,"style":3273},[2791],[76,3809,2632],{"className":3810},[121,216],[76,3812,3813,3816],{"style":3269},[76,3814],{"className":3815,"style":3273},[2791],[76,3817,3819],{"className":3818,"style":3286},[3285],[76,3820,3240],{"className":3821},[121]," には、",[13,3824,3825,3826,3854],{},"それまでに登場した変数の ",[76,3827,3829,3842],{"className":3828},[79],[76,3830,3832],{"className":3831},[83],[85,3833,3834],{"xmlns":87},[89,3835,3836,3840],{},[92,3837,3838],{},[95,3839,2648],{},[100,3841,2648],{"encoding":102},[76,3843,3845],{"className":3844,"ariaHidden":108},[107],[76,3846,3848,3851],{"className":3847},[112],[76,3849],{"className":3850,"style":820},[116],[76,3852,2648],{"className":3853,"style":2861},[121,216]," 個組"," を、もれなく順番に代入する。",[24,3857,3858,3929,3930,3933,3934,3962],{},[76,3859,3861,3878],{"className":3860},[79],[76,3862,3864],{"className":3863},[83],[85,3865,3866],{"xmlns":87},[89,3867,3868,3876],{},[92,3869,3870],{},[3234,3871,3872,3874],{"accent":108},[95,3873,2658],{},[299,3875,3240],{},[100,3877,3311],{"encoding":102},[76,3879,3881],{"className":3880,"ariaHidden":108},[107],[76,3882,3884,3887],{"className":3883},[112],[76,3885],{"className":3886,"style":3321},[116],[76,3888,3890],{"className":3889},[121,3257],[76,3891,3893,3921],{"className":3892},[2774,2775],[76,3894,3896,3918],{"className":3895},[2779],[76,3897,3899,3907],{"className":3898,"style":3253},[2783],[76,3900,3901,3904],{"style":3269},[76,3902],{"className":3903,"style":3273},[2791],[76,3905,2658],{"className":3906,"style":253},[121,216],[76,3908,3909,3912],{"style":3269},[76,3910],{"className":3911,"style":3273},[2791],[76,3913,3915],{"className":3914,"style":3350},[3285],[76,3916,3240],{"className":3917},[121],[76,3919,2807],{"className":3920},[2806],[76,3922,3924],{"className":3923},[2779],[76,3925,3927],{"className":3926,"style":3363},[2783],[76,3928],{}," には、毎回 ",[13,3931,3932],{},"まだ使っていない新しい変数"," を ",[76,3935,3937,3950],{"className":3936},[79],[76,3938,3940],{"className":3939},[83],[85,3941,3942],{"xmlns":87},[89,3943,3944,3948],{},[92,3945,3946],{},[95,3947,2671],{},[100,3949,2671],{"encoding":102},[76,3951,3953],{"className":3952,"ariaHidden":108},[107],[76,3954,3956,3959],{"className":3955},[112],[76,3957],{"className":3958,"style":249},[116],[76,3960,2671],{"className":3961},[121,216]," 個代入する。",[10,3964,3965,3966,35,4170,4385,4386,4415,4416,4486,4487,4515],{},"最初は ",[76,3967,3969,4011],{"className":3968},[79],[76,3970,3972],{"className":3971},[83],[85,3973,3974],{"xmlns":87},[89,3975,3976,4008],{},[92,3977,3978,3984,3986,3988,3994,3996,3998,4000,4006],{},[3234,3979,3980,3982],{"accent":108},[95,3981,2632],{},[299,3983,3240],{},[299,3985,602],{},[299,3987,2678],{"stretchy":1154},[2628,3989,3990,3992],{},[95,3991,3419],{},[2634,3993,3422],{},[299,3995,2687],{"separator":108},[299,3997,2690],{},[299,3999,2687],{"separator":108},[2628,4001,4002,4004],{},[95,4003,3419],{},[2634,4005,3422],{},[299,4007,2721],{"stretchy":1154},[100,4009,4010],{"encoding":102},"\\bar{x} = (v_0, \\dots, v_0)",[76,4012,4014,4060],{"className":4013,"ariaHidden":108},[107],[76,4015,4017,4020,4051,4054,4057],{"className":4016},[112],[76,4018],{"className":4019,"style":3253},[116],[76,4021,4023],{"className":4022},[121,3257],[76,4024,4026],{"className":4025},[2774],[76,4027,4029],{"className":4028},[2779],[76,4030,4032,4040],{"className":4031,"style":3253},[2783],[76,4033,4034,4037],{"style":3269},[76,4035],{"className":4036,"style":3273},[2791],[76,4038,2632],{"className":4039},[121,216],[76,4041,4042,4045],{"style":3269},[76,4043],{"className":4044,"style":3273},[2791],[76,4046,4048],{"className":4047,"style":3286},[3285],[76,4049,3240],{"className":4050},[121],[76,4052],{"className":4053,"style":324},[323],[76,4055,602],{"className":4056},[328],[76,4058],{"className":4059,"style":324},[323],[76,4061,4063,4066,4069,4109,4112,4115,4118,4121,4124,4127,4167],{"className":4062},[112],[76,4064],{"className":4065,"style":1198},[116],[76,4067,2678],{"className":4068},[1202],[76,4070,4072,4075],{"className":4071},[121],[76,4073,3419],{"className":4074,"style":253},[121,216],[76,4076,4078],{"className":4077},[2770],[76,4079,4081,4101],{"className":4080},[2774,2775],[76,4082,4084,4098],{"className":4083},[2779],[76,4085,4087],{"className":4086,"style":2784},[2783],[76,4088,4089,4092],{"style":2899},[76,4090],{"className":4091,"style":2792},[2791],[76,4093,4095],{"className":4094},[2796,2797,2798,2799],[76,4096,3422],{"className":4097},[121,2799],[76,4099,2807],{"className":4100},[2806],[76,4102,4104],{"className":4103},[2779],[76,4105,4107],{"className":4106,"style":2814},[2783],[76,4108],{},[76,4110,2687],{"className":4111},[3025],[76,4113],{"className":4114,"style":2820},[323],[76,4116,2690],{"className":4117},[2824],[76,4119],{"className":4120,"style":2820},[323],[76,4122,2687],{"className":4123},[3025],[76,4125],{"className":4126,"style":2820},[323],[76,4128,4130,4133],{"className":4129},[121],[76,4131,3419],{"className":4132,"style":253},[121,216],[76,4134,4136],{"className":4135},[2770],[76,4137,4139,4159],{"className":4138},[2774,2775],[76,4140,4142,4156],{"className":4141},[2779],[76,4143,4145],{"className":4144,"style":2784},[2783],[76,4146,4147,4150],{"style":2899},[76,4148],{"className":4149,"style":2792},[2791],[76,4151,4153],{"className":4152},[2796,2797,2798,2799],[76,4154,3422],{"className":4155},[121,2799],[76,4157,2807],{"className":4158},[2806],[76,4160,4162],{"className":4161},[2779],[76,4163,4165],{"className":4164,"style":2814},[2783],[76,4166],{},[76,4168,2721],{"className":4169},[1209],[76,4171,4173,4215],{"className":4172},[79],[76,4174,4176],{"className":4175},[83],[85,4177,4178],{"xmlns":87},[89,4179,4180,4212],{},[92,4181,4182,4188,4190,4192,4198,4200,4202,4204,4210],{},[3234,4183,4184,4186],{"accent":108},[95,4185,2658],{},[299,4187,3240],{},[299,4189,602],{},[299,4191,2678],{"stretchy":1154},[2628,4193,4194,4196],{},[95,4195,3419],{},[2634,4197,2636],{},[299,4199,2687],{"separator":108},[299,4201,2690],{},[299,4203,2687],{"separator":108},[2628,4205,4206,4208],{},[95,4207,3419],{},[95,4209,2671],{},[299,4211,2721],{"stretchy":1154},[100,4213,4214],{"encoding":102},"\\bar{y} = (v_1, \\dots, v_m)",[76,4216,4218,4275],{"className":4217,"ariaHidden":108},[107],[76,4219,4221,4224,4266,4269,4272],{"className":4220},[112],[76,4222],{"className":4223,"style":3321},[116],[76,4225,4227],{"className":4226},[121,3257],[76,4228,4230,4258],{"className":4229},[2774,2775],[76,4231,4233,4255],{"className":4232},[2779],[76,4234,4236,4244],{"className":4235,"style":3253},[2783],[76,4237,4238,4241],{"style":3269},[76,4239],{"className":4240,"style":3273},[2791],[76,4242,2658],{"className":4243,"style":253},[121,216],[76,4245,4246,4249],{"style":3269},[76,4247],{"className":4248,"style":3273},[2791],[76,4250,4252],{"className":4251,"style":3350},[3285],[76,4253,3240],{"className":4254},[121],[76,4256,2807],{"className":4257},[2806],[76,4259,4261],{"className":4260},[2779],[76,4262,4264],{"className":4263,"style":3363},[2783],[76,4265],{},[76,4267],{"className":4268,"style":324},[323],[76,4270,602],{"className":4271},[328],[76,4273],{"className":4274,"style":324},[323],[76,4276,4278,4281,4284,4324,4327,4330,4333,4336,4339,4342,4382],{"className":4277},[112],[76,4279],{"className":4280,"style":1198},[116],[76,4282,2678],{"className":4283},[1202],[76,4285,4287,4290],{"className":4286},[121],[76,4288,3419],{"className":4289,"style":253},[121,216],[76,4291,4293],{"className":4292},[2770],[76,4294,4296,4316],{"className":4295},[2774,2775],[76,4297,4299,4313],{"className":4298},[2779],[76,4300,4302],{"className":4301,"style":2784},[2783],[76,4303,4304,4307],{"style":2899},[76,4305],{"className":4306,"style":2792},[2791],[76,4308,4310],{"className":4309},[2796,2797,2798,2799],[76,4311,2636],{"className":4312},[121,2799],[76,4314,2807],{"className":4315},[2806],[76,4317,4319],{"className":4318},[2779],[76,4320,4322],{"className":4321,"style":2814},[2783],[76,4323],{},[76,4325,2687],{"className":4326},[3025],[76,4328],{"className":4329,"style":2820},[323],[76,4331,2690],{"className":4332},[2824],[76,4334],{"className":4335,"style":2820},[323],[76,4337,2687],{"className":4338},[3025],[76,4340],{"className":4341,"style":2820},[323],[76,4343,4345,4348],{"className":4344},[121],[76,4346,3419],{"className":4347,"style":253},[121,216],[76,4349,4351],{"className":4350},[2770],[76,4352,4354,4374],{"className":4353},[2774,2775],[76,4355,4357,4371],{"className":4356},[2779],[76,4358,4360],{"className":4359,"style":2950},[2783],[76,4361,4362,4365],{"style":2899},[76,4363],{"className":4364,"style":2792},[2791],[76,4366,4368],{"className":4367},[2796,2797,2798,2799],[76,4369,2671],{"className":4370},[121,216,2799],[76,4372,2807],{"className":4373},[2806],[76,4375,4377],{"className":4376},[2779],[76,4378,4380],{"className":4379,"style":2814},[2783],[76,4381],{},[76,4383,2721],{"className":4384},[1209]," とします。こうして得られる ",[76,4387,4389,4403],{"className":4388},[79],[76,4390,4392],{"className":4391},[83],[85,4393,4394],{"xmlns":87},[89,4395,4396,4401],{},[92,4397,4398],{},[95,4399,4400],{},"n",[100,4402,4400],{"encoding":102},[76,4404,4406],{"className":4405,"ariaHidden":108},[107],[76,4407,4409,4412],{"className":4408},[112],[76,4410],{"className":4411,"style":249},[116],[76,4413,4400],{"className":4414},[121,216]," 番目の代入例を ",[76,4417,4419,4437],{"className":4418},[79],[76,4420,4422],{"className":4421},[83],[85,4423,4424],{"xmlns":87},[89,4425,4426,4434],{},[92,4427,4428],{},[2628,4429,4430,4432],{},[95,4431,1158],{},[95,4433,4400],{},[100,4435,4436],{"encoding":102},"\\psi_n",[76,4438,4440],{"className":4439,"ariaHidden":108},[107],[76,4441,4443,4446],{"className":4442},[112],[76,4444],{"className":4445,"style":1281},[116],[76,4447,4449,4452],{"className":4448},[121],[76,4450,1158],{"className":4451,"style":253},[121,216],[76,4453,4455],{"className":4454},[2770],[76,4456,4458,4478],{"className":4457},[2774,2775],[76,4459,4461,4475],{"className":4460},[2779],[76,4462,4464],{"className":4463,"style":2950},[2783],[76,4465,4466,4469],{"style":2899},[76,4467],{"className":4468,"style":2792},[2791],[76,4470,4472],{"className":4471},[2796,2797,2798,2799],[76,4473,4400],{"className":4474},[121,216,2799],[76,4476,2807],{"className":4477},[2806],[76,4479,4481],{"className":4480},[2779],[76,4482,4484],{"className":4483,"style":2814},[2783],[76,4485],{}," とし、最初の ",[76,4488,4490,4503],{"className":4489},[79],[76,4491,4493],{"className":4492},[83],[85,4494,4495],{"xmlns":87},[89,4496,4497,4501],{},[92,4498,4499],{},[95,4500,4400],{},[100,4502,4400],{"encoding":102},[76,4504,4506],{"className":4505,"ariaHidden":108},[107],[76,4507,4509,4512],{"className":4508},[112],[76,4510],{"className":4511,"style":249},[116],[76,4513,4400],{"className":4514},[121,216]," 個の連言を",[76,4517,4519],{"className":4518},[2600],[76,4520,4522,4574],{"className":4521},[79],[76,4523,4525],{"className":4524},[83],[85,4526,4527],{"xmlns":87,"display":2609},[89,4528,4529,4571],{},[92,4530,4531,4538,4540,4542,4544,4550,4553,4559,4561,4563,4565],{},[2628,4532,4533,4536],{},[95,4534,4535],{},"B",[95,4537,4400],{},[2617,4539,2619],{},[299,4541,602],{},[2617,4543,2619],{},[2628,4545,4546,4548],{},[95,4547,1158],{},[2634,4549,2636],{},[299,4551,4552],{},"∧",[2628,4554,4555,4557],{},[95,4556,1158],{},[2634,4558,3439],{},[299,4560,4552],{},[299,4562,2639],{},[299,4564,4552],{},[2628,4566,4567,4569],{},[95,4568,1158],{},[95,4570,4400],{},[100,4572,4573],{"encoding":102},"B_n \\;=\\; \\psi_1 \\wedge \\psi_2 \\wedge \\cdots \\wedge \\psi_n",[76,4575,4577,4641,4696,4751,4770],{"className":4576,"ariaHidden":108},[107],[76,4578,4580,4584,4626,4629,4632,4635,4638],{"className":4579},[112],[76,4581],{"className":4582,"style":4583},[116],"height:0.8333em;vertical-align:-0.15em;",[76,4585,4587,4591],{"className":4586},[121],[76,4588,4535],{"className":4589,"style":4590},[121,216],"margin-right:0.0502em;",[76,4592,4594],{"className":4593},[2770],[76,4595,4597,4618],{"className":4596},[2774,2775],[76,4598,4600,4615],{"className":4599},[2779],[76,4601,4603],{"className":4602,"style":2950},[2783],[76,4604,4606,4609],{"style":4605},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[76,4607],{"className":4608,"style":2792},[2791],[76,4610,4612],{"className":4611},[2796,2797,2798,2799],[76,4613,4400],{"className":4614},[121,216,2799],[76,4616,2807],{"className":4617},[2806],[76,4619,4621],{"className":4620},[2779],[76,4622,4624],{"className":4623,"style":2814},[2783],[76,4625],{},[76,4627],{"className":4628,"style":324},[323],[76,4630],{"className":4631,"style":324},[323],[76,4633,602],{"className":4634},[328],[76,4636],{"className":4637,"style":324},[323],[76,4639],{"className":4640,"style":324},[323],[76,4642,4644,4647,4687,4690,4693],{"className":4643},[112],[76,4645],{"className":4646,"style":1281},[116],[76,4648,4650,4653],{"className":4649},[121],[76,4651,1158],{"className":4652,"style":253},[121,216],[76,4654,4656],{"className":4655},[2770],[76,4657,4659,4679],{"className":4658},[2774,2775],[76,4660,4662,4676],{"className":4661},[2779],[76,4663,4665],{"className":4664,"style":2784},[2783],[76,4666,4667,4670],{"style":2899},[76,4668],{"className":4669,"style":2792},[2791],[76,4671,4673],{"className":4672},[2796,2797,2798,2799],[76,4674,2636],{"className":4675},[121,2799],[76,4677,2807],{"className":4678},[2806],[76,4680,4682],{"className":4681},[2779],[76,4683,4685],{"className":4684,"style":2814},[2783],[76,4686],{},[76,4688],{"className":4689,"style":1184},[323],[76,4691,4552],{"className":4692},[1188],[76,4694],{"className":4695,"style":1184},[323],[76,4697,4699,4702,4742,4745,4748],{"className":4698},[112],[76,4700],{"className":4701,"style":1281},[116],[76,4703,4705,4708],{"className":4704},[121],[76,4706,1158],{"className":4707,"style":253},[121,216],[76,4709,4711],{"className":4710},[2770],[76,4712,4714,4734],{"className":4713},[2774,2775],[76,4715,4717,4731],{"className":4716},[2779],[76,4718,4720],{"className":4719,"style":2784},[2783],[76,4721,4722,4725],{"style":2899},[76,4723],{"className":4724,"style":2792},[2791],[76,4726,4728],{"className":4727},[2796,2797,2798,2799],[76,4729,3439],{"className":4730},[121,2799],[76,4732,2807],{"className":4733},[2806],[76,4735,4737],{"className":4736},[2779],[76,4738,4740],{"className":4739,"style":2814},[2783],[76,4741],{},[76,4743],{"className":4744,"style":1184},[323],[76,4746,4552],{"className":4747},[1188],[76,4749],{"className":4750,"style":1184},[323],[76,4752,4754,4758,4761,4764,4767],{"className":4753},[112],[76,4755],{"className":4756,"style":4757},[116],"height:0.5556em;",[76,4759,2639],{"className":4760},[2824],[76,4762],{"className":4763,"style":1184},[323],[76,4765,4552],{"className":4766},[1188],[76,4768],{"className":4769,"style":1184},[323],[76,4771,4773,4776],{"className":4772},[112],[76,4774],{"className":4775,"style":1281},[116],[76,4777,4779,4782],{"className":4778},[121],[76,4780,1158],{"className":4781,"style":253},[121,216],[76,4783,4785],{"className":4784},[2770],[76,4786,4788,4808],{"className":4787},[2774,2775],[76,4789,4791,4805],{"className":4790},[2779],[76,4792,4794],{"className":4793,"style":2950},[2783],[76,4795,4796,4799],{"style":2899},[76,4797],{"className":4798,"style":2792},[2791],[76,4800,4802],{"className":4801},[2796,2797,2798,2799],[76,4803,4400],{"className":4804},[121,216,2799],[76,4806,2807],{"className":4807},[2806],[76,4809,4811],{"className":4810},[2779],[76,4812,4814],{"className":4813,"style":2814},[2783],[76,4815],{},[10,4817,4818,4819,4889,4890,5032,5033,5036],{},"とおきます。",[76,4820,4822,4840],{"className":4821},[79],[76,4823,4825],{"className":4824},[83],[85,4826,4827],{"xmlns":87},[89,4828,4829,4837],{},[92,4830,4831],{},[2628,4832,4833,4835],{},[95,4834,4535],{},[95,4836,4400],{},[100,4838,4839],{"encoding":102},"B_n",[76,4841,4843],{"className":4842,"ariaHidden":108},[107],[76,4844,4846,4849],{"className":4845},[112],[76,4847],{"className":4848,"style":4583},[116],[76,4850,4852,4855],{"className":4851},[121],[76,4853,4535],{"className":4854,"style":4590},[121,216],[76,4856,4858],{"className":4857},[2770],[76,4859,4861,4881],{"className":4860},[2774,2775],[76,4862,4864,4878],{"className":4863},[2779],[76,4865,4867],{"className":4866,"style":2950},[2783],[76,4868,4869,4872],{"style":4605},[76,4870],{"className":4871,"style":2792},[2791],[76,4873,4875],{"className":4874},[2796,2797,2798,2799],[76,4876,4400],{"className":4877},[121,216,2799],[76,4879,2807],{"className":4880},[2806],[76,4882,4884],{"className":4883},[2779],[76,4885,4887],{"className":4886,"style":2814},[2783],[76,4888],{}," は量化子を含まないので、",[76,4891,4893,4927],{"className":4892},[79],[76,4894,4896],{"className":4895},[83],[85,4897,4898],{"xmlns":87},[89,4899,4900,4924],{},[92,4901,4902,4905,4907,4913,4915,4922],{},[95,4903,4904],{},"R",[299,4906,2678],{"stretchy":1154},[2628,4908,4909,4911],{},[95,4910,3419],{},[2634,4912,3439],{},[299,4914,2687],{"separator":108},[2628,4916,4917,4919],{},[95,4918,3419],{},[2634,4920,4921],{},"5",[299,4923,2721],{"stretchy":1154},[100,4925,4926],{"encoding":102},"R(v_2, v_5)",[76,4928,4930],{"className":4929,"ariaHidden":108},[107],[76,4931,4933,4936,4940,4943,4983,4986,4989,5029],{"className":4932},[112],[76,4934],{"className":4935,"style":1198},[116],[76,4937,4904],{"className":4938,"style":4939},[121,216],"margin-right:0.0077em;",[76,4941,2678],{"className":4942},[1202],[76,4944,4946,4949],{"className":4945},[121],[76,4947,3419],{"className":4948,"style":253},[121,216],[76,4950,4952],{"className":4951},[2770],[76,4953,4955,4975],{"className":4954},[2774,2775],[76,4956,4958,4972],{"className":4957},[2779],[76,4959,4961],{"className":4960,"style":2784},[2783],[76,4962,4963,4966],{"style":2899},[76,4964],{"className":4965,"style":2792},[2791],[76,4967,4969],{"className":4968},[2796,2797,2798,2799],[76,4970,3439],{"className":4971},[121,2799],[76,4973,2807],{"className":4974},[2806],[76,4976,4978],{"className":4977},[2779],[76,4979,4981],{"className":4980,"style":2814},[2783],[76,4982],{},[76,4984,2687],{"className":4985},[3025],[76,4987],{"className":4988,"style":2820},[323],[76,4990,4992,4995],{"className":4991},[121],[76,4993,3419],{"className":4994,"style":253},[121,216],[76,4996,4998],{"className":4997},[2770],[76,4999,5001,5021],{"className":5000},[2774,2775],[76,5002,5004,5018],{"className":5003},[2779],[76,5005,5007],{"className":5006,"style":2784},[2783],[76,5008,5009,5012],{"style":2899},[76,5010],{"className":5011,"style":2792},[2791],[76,5013,5015],{"className":5014},[2796,2797,2798,2799],[76,5016,4921],{"className":5017},[121,2799],[76,5019,2807],{"className":5020},[2806],[76,5022,5024],{"className":5023},[2779],[76,5025,5027],{"className":5026,"style":2814},[2783],[76,5028],{},[76,5030,2721],{"className":5031},[1209]," のような原子論理式の1つ1つを命題変数とみなせば、",[13,5034,5035],{},"命題論理の論理式"," として扱えます。",[68,5038,5040],{"id":5039},"_23-ステップ3二者択一","2.3 ステップ3：二者択一",[10,5042,5043],{},"ここで、次の2つの場合に分かれます。",[10,5045,5046],{},[13,5047,5048,5049,5118],{},"(a) どれかの ",[76,5050,5052,5069],{"className":5051},[79],[76,5053,5055],{"className":5054},[83],[85,5056,5057],{"xmlns":87},[89,5058,5059,5067],{},[92,5060,5061],{},[2628,5062,5063,5065],{},[95,5064,4535],{},[95,5066,4400],{},[100,5068,4839],{"encoding":102},[76,5070,5072],{"className":5071,"ariaHidden":108},[107],[76,5073,5075,5078],{"className":5074},[112],[76,5076],{"className":5077,"style":4583},[116],[76,5079,5081,5084],{"className":5080},[121],[76,5082,4535],{"className":5083,"style":4590},[121,216],[76,5085,5087],{"className":5086},[2770],[76,5088,5090,5110],{"className":5089},[2774,2775],[76,5091,5093,5107],{"className":5092},[2779],[76,5094,5096],{"className":5095,"style":2950},[2783],[76,5097,5098,5101],{"style":4605},[76,5099],{"className":5100,"style":2792},[2791],[76,5102,5104],{"className":5103},[2796,2797,2798,2799],[76,5105,4400],{"className":5106},[121,216,2799],[76,5108,2807],{"className":5109},[2806],[76,5111,5113],{"className":5112},[2779],[76,5114,5116],{"className":5115,"style":2814},[2783],[76,5117],{}," が、命題論理の論理式として充足不可能な場合",[10,5120,5121,5196,5197,5333,5334,502,5400,5469],{},[76,5122,5124,5144],{"className":5123},[79],[76,5125,5127],{"className":5126},[83],[85,5128,5129],{"xmlns":87},[89,5130,5131,5141],{},[92,5132,5133,5135],{},[95,5134,941],{"mathvariant":363},[2628,5136,5137,5139],{},[95,5138,4535],{},[95,5140,4400],{},[100,5142,5143],{"encoding":102},"\\neg B_n",[76,5145,5147],{"className":5146,"ariaHidden":108},[107],[76,5148,5150,5153,5156],{"className":5149},[112],[76,5151],{"className":5152,"style":4583},[116],[76,5154,941],{"className":5155},[121],[76,5157,5159,5162],{"className":5158},[121],[76,5160,4535],{"className":5161,"style":4590},[121,216],[76,5163,5165],{"className":5164},[2770],[76,5166,5168,5188],{"className":5167},[2774,2775],[76,5169,5171,5185],{"className":5170},[2779],[76,5172,5174],{"className":5173,"style":2950},[2783],[76,5175,5176,5179],{"style":4605},[76,5177],{"className":5178,"style":2792},[2791],[76,5180,5182],{"className":5181},[2796,2797,2798,2799],[76,5183,4400],{"className":5184},[121,216,2799],[76,5186,2807],{"className":5187},[2806],[76,5189,5191],{"className":5190},[2779],[76,5192,5194],{"className":5193,"style":2814},[2783],[76,5195],{}," は命題論理のトートロジーなので、述語論理でも証明でき、汎化により ",[76,5198,5200,5233],{"className":5199},[79],[76,5201,5203],{"className":5202},[83],[85,5204,5205],{"xmlns":87},[89,5206,5207,5230],{},[92,5208,5209,5211,5213,5215,5221,5224],{},[299,5210,805],{},[95,5212,941],{"mathvariant":363},[95,5214,2653],{"mathvariant":363},[3234,5216,5217,5219],{"accent":108},[95,5218,3419],{},[299,5220,3240],{},[2617,5222,5223],{}," ",[2628,5225,5226,5228],{},[95,5227,4535],{},[95,5229,4400],{},[100,5231,5232],{"encoding":102},"\\vdash \\neg \\exists \\bar{v}\\, B_n",[76,5234,5236,5248],{"className":5235,"ariaHidden":108},[107],[76,5237,5239,5242,5245],{"className":5238},[112],[76,5240],{"className":5241,"style":820},[116],[76,5243,805],{"className":5244},[328],[76,5246],{"className":5247,"style":324},[323],[76,5249,5251,5255,5259,5290,5293],{"className":5250},[112],[76,5252],{"className":5253,"style":5254},[116],"height:0.8444em;vertical-align:-0.15em;",[76,5256,5258],{"className":5257},[121],"¬∃",[76,5260,5262],{"className":5261},[121,3257],[76,5263,5265],{"className":5264},[2774],[76,5266,5268],{"className":5267},[2779],[76,5269,5271,5279],{"className":5270,"style":3253},[2783],[76,5272,5273,5276],{"style":3269},[76,5274],{"className":5275,"style":3273},[2791],[76,5277,3419],{"className":5278,"style":253},[121,216],[76,5280,5281,5284],{"style":3269},[76,5282],{"className":5283,"style":3273},[2791],[76,5285,5287],{"className":5286,"style":3286},[3285],[76,5288,3240],{"className":5289},[121],[76,5291],{"className":5292,"style":2820},[323],[76,5294,5296,5299],{"className":5295},[121],[76,5297,4535],{"className":5298,"style":4590},[121,216],[76,5300,5302],{"className":5301},[2770],[76,5303,5305,5325],{"className":5304},[2774,2775],[76,5306,5308,5322],{"className":5307},[2779],[76,5309,5311],{"className":5310,"style":2950},[2783],[76,5312,5313,5316],{"style":4605},[76,5314],{"className":5315,"style":2792},[2791],[76,5317,5319],{"className":5318},[2796,2797,2798,2799],[76,5320,4400],{"className":5321},[121,216,2799],[76,5323,2807],{"className":5324},[2806],[76,5326,5328],{"className":5327},[2779],[76,5329,5331],{"className":5330,"style":2814},[2783],[76,5332],{}," が得られます（",[76,5335,5337,5357],{"className":5336},[79],[76,5338,5340],{"className":5339},[83],[85,5341,5342],{"xmlns":87},[89,5343,5344,5354],{},[92,5345,5346,5348],{},[95,5347,2653],{"mathvariant":363},[3234,5349,5350,5352],{"accent":108},[95,5351,3419],{},[299,5353,3240],{},[100,5355,5356],{"encoding":102},"\\exists \\bar{v}",[76,5358,5360],{"className":5359,"ariaHidden":108},[107],[76,5361,5363,5366,5369],{"className":5362},[112],[76,5364],{"className":5365,"style":820},[116],[76,5367,2653],{"className":5368},[121],[76,5370,5372],{"className":5371},[121,3257],[76,5373,5375],{"className":5374},[2774],[76,5376,5378],{"className":5377},[2779],[76,5379,5381,5389],{"className":5380,"style":3253},[2783],[76,5382,5383,5386],{"style":3269},[76,5384],{"className":5385,"style":3273},[2791],[76,5387,3419],{"className":5388,"style":253},[121,216],[76,5390,5391,5394],{"style":3269},[76,5392],{"className":5393,"style":3273},[2791],[76,5395,5397],{"className":5396,"style":3286},[3285],[76,5398,3240],{"className":5399},[121],[76,5401,5403,5420],{"className":5402},[79],[76,5404,5406],{"className":5405},[83],[85,5407,5408],{"xmlns":87},[89,5409,5410,5418],{},[92,5411,5412],{},[2628,5413,5414,5416],{},[95,5415,4535],{},[95,5417,4400],{},[100,5419,4839],{"encoding":102},[76,5421,5423],{"className":5422,"ariaHidden":108},[107],[76,5424,5426,5429],{"className":5425},[112],[76,5427],{"className":5428,"style":4583},[116],[76,5430,5432,5435],{"className":5431},[121],[76,5433,4535],{"className":5434,"style":4590},[121,216],[76,5436,5438],{"className":5437},[2770],[76,5439,5441,5461],{"className":5440},[2774,2775],[76,5442,5444,5458],{"className":5443},[2779],[76,5445,5447],{"className":5446,"style":2950},[2783],[76,5448,5449,5452],{"style":4605},[76,5450],{"className":5451,"style":2792},[2791],[76,5453,5455],{"className":5454},[2796,2797,2798,2799],[76,5456,4400],{"className":5457},[121,216,2799],[76,5459,2807],{"className":5460},[2806],[76,5462,5464],{"className":5463},[2779],[76,5465,5467],{"className":5466,"style":2814},[2783],[76,5468],{}," に現れる変数すべてについての存在量化）。",[10,5471,724,5472,759,5500,5569],{},[76,5473,5475,5488],{"className":5474},[79],[76,5476,5478],{"className":5477},[83],[85,5479,5480],{"xmlns":87},[89,5481,5482,5486],{},[92,5483,5484],{},[95,5485,456],{},[100,5487,459],{"encoding":102},[76,5489,5491],{"className":5490,"ariaHidden":108},[107],[76,5492,5494,5497],{"className":5493},[112],[76,5495],{"className":5496,"style":469},[116],[76,5498,456],{"className":5499},[121,216],[76,5501,5503,5520],{"className":5502},[79],[76,5504,5506],{"className":5505},[83],[85,5507,5508],{"xmlns":87},[89,5509,5510,5518],{},[92,5511,5512],{},[2628,5513,5514,5516],{},[95,5515,4535],{},[95,5517,4400],{},[100,5519,4839],{"encoding":102},[76,5521,5523],{"className":5522,"ariaHidden":108},[107],[76,5524,5526,5529],{"className":5525},[112],[76,5527],{"className":5528,"style":4583},[116],[76,5530,5532,5535],{"className":5531},[121],[76,5533,4535],{"className":5534,"style":4590},[121,216],[76,5536,5538],{"className":5537},[2770],[76,5539,5541,5561],{"className":5540},[2774,2775],[76,5542,5544,5558],{"className":5543},[2779],[76,5545,5547],{"className":5546,"style":2950},[2783],[76,5548,5549,5552],{"style":4605},[76,5550],{"className":5551,"style":2792},[2791],[76,5553,5555],{"className":5554},[2796,2797,2798,2799],[76,5556,4400],{"className":5557},[121,216,2799],[76,5559,2807],{"className":5560},[2806],[76,5562,5564],{"className":5563},[2779],[76,5565,5567],{"className":5566,"style":2814},[2783],[76,5568],{}," の存在閉包が証明できることが、代入例の作り方から帰納法で示せます。",[76,5571,5573],{"className":5572},[2600],[76,5574,5576,5610],{"className":5575},[79],[76,5577,5579],{"className":5578},[83],[85,5580,5581],{"xmlns":87,"display":2609},[89,5582,5583,5607],{},[92,5584,5585,5587,5589,5591,5593,5599,5601],{},[299,5586,805],{},[95,5588,456],{},[299,5590,1248],{},[95,5592,2653],{"mathvariant":363},[3234,5594,5595,5597],{"accent":108},[95,5596,3419],{},[299,5598,3240],{},[2617,5600,5223],{},[2628,5602,5603,5605],{},[95,5604,4535],{},[95,5606,4400],{},[100,5608,5609],{"encoding":102},"\\vdash \\varphi \\to \\exists \\bar{v}\\, B_n",[76,5611,5613,5625,5643],{"className":5612,"ariaHidden":108},[107],[76,5614,5616,5619,5622],{"className":5615},[112],[76,5617],{"className":5618,"style":820},[116],[76,5620,805],{"className":5621},[328],[76,5623],{"className":5624,"style":324},[323],[76,5626,5628,5631,5634,5637,5640],{"className":5627},[112],[76,5629],{"className":5630,"style":469},[116],[76,5632,456],{"className":5633},[121,216],[76,5635],{"className":5636,"style":324},[323],[76,5638,1248],{"className":5639},[328],[76,5641],{"className":5642,"style":324},[323],[76,5644,5646,5649,5652,5683,5686],{"className":5645},[112],[76,5647],{"className":5648,"style":5254},[116],[76,5650,2653],{"className":5651},[121],[76,5653,5655],{"className":5654},[121,3257],[76,5656,5658],{"className":5657},[2774],[76,5659,5661],{"className":5660},[2779],[76,5662,5664,5672],{"className":5663,"style":3253},[2783],[76,5665,5666,5669],{"style":3269},[76,5667],{"className":5668,"style":3273},[2791],[76,5670,3419],{"className":5671,"style":253},[121,216],[76,5673,5674,5677],{"style":3269},[76,5675],{"className":5676,"style":3273},[2791],[76,5678,5680],{"className":5679,"style":3286},[3285],[76,5681,3240],{"className":5682},[121],[76,5684],{"className":5685,"style":2820},[323],[76,5687,5689,5692],{"className":5688},[121],[76,5690,4535],{"className":5691,"style":4590},[121,216],[76,5693,5695],{"className":5694},[2770],[76,5696,5698,5718],{"className":5697},[2774,2775],[76,5699,5701,5715],{"className":5700},[2779],[76,5702,5704],{"className":5703,"style":2950},[2783],[76,5705,5706,5709],{"style":4605},[76,5707],{"className":5708,"style":2792},[2791],[76,5710,5712],{"className":5711},[2796,2797,2798,2799],[76,5713,4400],{"className":5714},[121,216,2799],[76,5716,2807],{"className":5717},[2806],[76,5719,5721],{"className":5720},[2779],[76,5722,5724],{"className":5723,"style":2814},[2783],[76,5725],{},[10,5727,5728,5729,5776,5777,502,5805,5807],{},"この2つを合わせると ",[76,5730,5732,5749],{"className":5731},[79],[76,5733,5735],{"className":5734},[83],[85,5736,5737],{"xmlns":87},[89,5738,5739,5747],{},[92,5740,5741,5743,5745],{},[299,5742,805],{},[95,5744,941],{"mathvariant":363},[95,5746,456],{},[100,5748,2309],{"encoding":102},[76,5750,5752,5764],{"className":5751,"ariaHidden":108},[107],[76,5753,5755,5758,5761],{"className":5754},[112],[76,5756],{"className":5757,"style":820},[116],[76,5759,805],{"className":5760},[328],[76,5762],{"className":5763,"style":324},[323],[76,5765,5767,5770,5773],{"className":5766},[112],[76,5768],{"className":5769,"style":469},[116],[76,5771,941],{"className":5772},[121],[76,5774,456],{"className":5775},[121,216],"、つまり ",[76,5778,5780,5793],{"className":5779},[79],[76,5781,5783],{"className":5782},[83],[85,5784,5785],{"xmlns":87},[89,5786,5787,5791],{},[92,5788,5789],{},[95,5790,456],{},[100,5792,459],{"encoding":102},[76,5794,5796],{"className":5795,"ariaHidden":108},[107],[76,5797,5799,5802],{"className":5798},[112],[76,5800],{"className":5801,"style":469},[116],[76,5803,456],{"className":5804},[121,216],[13,5806,2287],{}," です。",[10,5809,5810],{},[13,5811,5812,5813,5882],{},"(b) すべての ",[76,5814,5816,5833],{"className":5815},[79],[76,5817,5819],{"className":5818},[83],[85,5820,5821],{"xmlns":87},[89,5822,5823,5831],{},[92,5824,5825],{},[2628,5826,5827,5829],{},[95,5828,4535],{},[95,5830,4400],{},[100,5832,4839],{"encoding":102},[76,5834,5836],{"className":5835,"ariaHidden":108},[107],[76,5837,5839,5842],{"className":5838},[112],[76,5840],{"className":5841,"style":4583},[116],[76,5843,5845,5848],{"className":5844},[121],[76,5846,4535],{"className":5847,"style":4590},[121,216],[76,5849,5851],{"className":5850},[2770],[76,5852,5854,5874],{"className":5853},[2774,2775],[76,5855,5857,5871],{"className":5856},[2779],[76,5858,5860],{"className":5859,"style":2950},[2783],[76,5861,5862,5865],{"style":4605},[76,5863],{"className":5864,"style":2792},[2791],[76,5866,5868],{"className":5867},[2796,2797,2798,2799],[76,5869,4400],{"className":5870},[121,216,2799],[76,5872,2807],{"className":5873},[2806],[76,5875,5877],{"className":5876},[2779],[76,5878,5880],{"className":5879,"style":2814},[2783],[76,5881],{}," が、命題論理の論理式として充足可能な場合",[10,5884,5885,5886,5955,5956,5984,5985,5807],{},"各 ",[76,5887,5889,5906],{"className":5888},[79],[76,5890,5892],{"className":5891},[83],[85,5893,5894],{"xmlns":87},[89,5895,5896,5904],{},[92,5897,5898],{},[2628,5899,5900,5902],{},[95,5901,4535],{},[95,5903,4400],{},[100,5905,4839],{"encoding":102},[76,5907,5909],{"className":5908,"ariaHidden":108},[107],[76,5910,5912,5915],{"className":5911},[112],[76,5913],{"className":5914,"style":4583},[116],[76,5916,5918,5921],{"className":5917},[121],[76,5919,4535],{"className":5920,"style":4590},[121,216],[76,5922,5924],{"className":5923},[2770],[76,5925,5927,5947],{"className":5926},[2774,2775],[76,5928,5930,5944],{"className":5929},[2779],[76,5931,5933],{"className":5932,"style":2950},[2783],[76,5934,5935,5938],{"style":4605},[76,5936],{"className":5937,"style":2792},[2791],[76,5939,5941],{"className":5940},[2796,2797,2798,2799],[76,5942,4400],{"className":5943},[121,216,2799],[76,5945,2807],{"className":5946},[2806],[76,5948,5950],{"className":5949},[2779],[76,5951,5953],{"className":5952,"style":2814},[2783],[76,5954],{}," には、それを真にする真理値の割り当てがあります。ただし、",[76,5957,5959,5972],{"className":5958},[79],[76,5960,5962],{"className":5961},[83],[85,5963,5964],{"xmlns":87},[89,5965,5966,5970],{},[92,5967,5968],{},[95,5969,4400],{},[100,5971,4400],{"encoding":102},[76,5973,5975],{"className":5974,"ariaHidden":108},[107],[76,5976,5978,5981],{"className":5977},[112],[76,5979],{"className":5980,"style":249},[116],[76,5982,4400],{"className":5983},[121,216]," ごとに割り当てがばらばらでは困ります。欲しいのは、",[13,5986,5987,5988,6057],{},"すべての ",[76,5989,5991,6008],{"className":5990},[79],[76,5992,5994],{"className":5993},[83],[85,5995,5996],{"xmlns":87},[89,5997,5998,6006],{},[92,5999,6000],{},[2628,6001,6002,6004],{},[95,6003,4535],{},[95,6005,4400],{},[100,6007,4839],{"encoding":102},[76,6009,6011],{"className":6010,"ariaHidden":108},[107],[76,6012,6014,6017],{"className":6013},[112],[76,6015],{"className":6016,"style":4583},[116],[76,6018,6020,6023],{"className":6019},[121],[76,6021,4535],{"className":6022,"style":4590},[121,216],[76,6024,6026],{"className":6025},[2770],[76,6027,6029,6049],{"className":6028},[2774,2775],[76,6030,6032,6046],{"className":6031},[2779],[76,6033,6035],{"className":6034,"style":2950},[2783],[76,6036,6037,6040],{"style":4605},[76,6038],{"className":6039,"style":2792},[2791],[76,6041,6043],{"className":6042},[2796,2797,2798,2799],[76,6044,4400],{"className":6045},[121,216,2799],[76,6047,2807],{"className":6048},[2806],[76,6050,6052],{"className":6051},[2779],[76,6053,6055],{"className":6054,"style":2814},[2783],[76,6056],{}," を同時に真にする1つの割り当て",[10,6059,6060,6129,6130,6199,6200,6228,6229,6232,6233,6302,6303,6332],{},[76,6061,6063,6080],{"className":6062},[79],[76,6064,6066],{"className":6065},[83],[85,6067,6068],{"xmlns":87},[89,6069,6070,6078],{},[92,6071,6072],{},[2628,6073,6074,6076],{},[95,6075,4535],{},[95,6077,4400],{},[100,6079,4839],{"encoding":102},[76,6081,6083],{"className":6082,"ariaHidden":108},[107],[76,6084,6086,6089],{"className":6085},[112],[76,6087],{"className":6088,"style":4583},[116],[76,6090,6092,6095],{"className":6091},[121],[76,6093,4535],{"className":6094,"style":4590},[121,216],[76,6096,6098],{"className":6097},[2770],[76,6099,6101,6121],{"className":6100},[2774,2775],[76,6102,6104,6118],{"className":6103},[2779],[76,6105,6107],{"className":6106,"style":2950},[2783],[76,6108,6109,6112],{"style":4605},[76,6110],{"className":6111,"style":2792},[2791],[76,6113,6115],{"className":6114},[2796,2797,2798,2799],[76,6116,4400],{"className":6117},[121,216,2799],[76,6119,2807],{"className":6120},[2806],[76,6122,6124],{"className":6123},[2779],[76,6125,6127],{"className":6126,"style":2814},[2783],[76,6128],{}," に現れる原子論理式は有限個なので、「",[76,6131,6133,6150],{"className":6132},[79],[76,6134,6136],{"className":6135},[83],[85,6137,6138],{"xmlns":87},[89,6139,6140,6148],{},[92,6141,6142],{},[2628,6143,6144,6146],{},[95,6145,4535],{},[95,6147,4400],{},[100,6149,4839],{"encoding":102},[76,6151,6153],{"className":6152,"ariaHidden":108},[107],[76,6154,6156,6159],{"className":6155},[112],[76,6157],{"className":6158,"style":4583},[116],[76,6160,6162,6165],{"className":6161},[121],[76,6163,4535],{"className":6164,"style":4590},[121,216],[76,6166,6168],{"className":6167},[2770],[76,6169,6171,6191],{"className":6170},[2774,2775],[76,6172,6174,6188],{"className":6173},[2779],[76,6175,6177],{"className":6176,"style":2950},[2783],[76,6178,6179,6182],{"style":4605},[76,6180],{"className":6181,"style":2792},[2791],[76,6183,6185],{"className":6184},[2796,2797,2798,2799],[76,6186,4400],{"className":6187},[121,216,2799],[76,6189,2807],{"className":6190},[2806],[76,6192,6194],{"className":6193},[2779],[76,6195,6197],{"className":6196,"style":2814},[2783],[76,6198],{}," を真にする割り当て」を ",[76,6201,6203,6216],{"className":6202},[79],[76,6204,6206],{"className":6205},[83],[85,6207,6208],{"xmlns":87},[89,6209,6210,6214],{},[92,6211,6212],{},[95,6213,4400],{},[100,6215,4400],{"encoding":102},[76,6217,6219],{"className":6218,"ariaHidden":108},[107],[76,6220,6222,6225],{"className":6221},[112],[76,6223],{"className":6224,"style":249},[116],[76,6226,4400],{"className":6227},[121,216]," の順に延長していく木は、各段で枝分かれが有限個の無限の木になります。今日の言葉でいえば ",[13,6230,6231],{},"ケーニヒの補題"," により、この木には無限に続く枝があります。その枝が、すべての ",[76,6234,6236,6253],{"className":6235},[79],[76,6237,6239],{"className":6238},[83],[85,6240,6241],{"xmlns":87},[89,6242,6243,6251],{},[92,6244,6245],{},[2628,6246,6247,6249],{},[95,6248,4535],{},[95,6250,4400],{},[100,6252,4839],{"encoding":102},[76,6254,6256],{"className":6255,"ariaHidden":108},[107],[76,6257,6259,6262],{"className":6258},[112],[76,6260],{"className":6261,"style":4583},[116],[76,6263,6265,6268],{"className":6264},[121],[76,6266,4535],{"className":6267,"style":4590},[121,216],[76,6269,6271],{"className":6270},[2770],[76,6272,6274,6294],{"className":6273},[2774,2775],[76,6275,6277,6291],{"className":6276},[2779],[76,6278,6280],{"className":6279,"style":2950},[2783],[76,6281,6282,6285],{"style":4605},[76,6283],{"className":6284,"style":2792},[2791],[76,6286,6288],{"className":6287},[2796,2797,2798,2799],[76,6289,4400],{"className":6290},[121,216,2799],[76,6292,2807],{"className":6293},[2806],[76,6295,6297],{"className":6296},[2779],[76,6298,6300],{"className":6299,"style":2814},[2783],[76,6301],{}," を同時に真にする割り当て ",[76,6304,6306,6320],{"className":6305},[79],[76,6307,6309],{"className":6308},[83],[85,6310,6311],{"xmlns":87},[89,6312,6313,6318],{},[92,6314,6315],{},[95,6316,6317],{},"V",[100,6319,6317],{"encoding":102},[76,6321,6323],{"className":6322,"ariaHidden":108},[107],[76,6324,6326,6329],{"className":6325},[112],[76,6327],{"className":6328,"style":117},[116],[76,6330,6317],{"className":6331,"style":1184},[121,216]," を与えます。",[6334,6335,6337],"note",{"title":6336},"ゲーデル自身の議論",[10,6338,6339,6340,6409,6410,6412,6413,40],{},"ゲーデルの論文はケーニヒの補題という名前を使っていません。原子論理式の真理値を1つずつ、「無限に多くの ",[76,6341,6343,6360],{"className":6342},[79],[76,6344,6346],{"className":6345},[83],[85,6347,6348],{"xmlns":87},[89,6349,6350,6358],{},[92,6351,6352],{},[2628,6353,6354,6356],{},[95,6355,4535],{},[95,6357,4400],{},[100,6359,4839],{"encoding":102},[76,6361,6363],{"className":6362,"ariaHidden":108},[107],[76,6364,6366,6369],{"className":6365},[112],[76,6367],{"className":6368,"style":4583},[116],[76,6370,6372,6375],{"className":6371},[121],[76,6373,4535],{"className":6374,"style":4590},[121,216],[76,6376,6378],{"className":6377},[2770],[76,6379,6381,6401],{"className":6380},[2774,2775],[76,6382,6384,6398],{"className":6383},[2779],[76,6385,6387],{"className":6386,"style":2950},[2783],[76,6388,6389,6392],{"style":4605},[76,6390],{"className":6391,"style":2792},[2791],[76,6393,6395],{"className":6394},[2796,2797,2798,2799],[76,6396,4400],{"className":6397},[121,216,2799],[76,6399,2807],{"className":6400},[2806],[76,6402,6404],{"className":6403},[2779],[76,6405,6407],{"className":6406,"style":2814},[2783],[76,6408],{}," の割り当てと両立する方」を選んで決めていく、多数決のような議論をしています（文献 ",[31,6411,2352],{"href":2351},"）。これは実質的にケーニヒの補題と同じ議論です。なお、可算な言語に対する完全性定理は、逆数学では弱ケーニヒの補題と同値であることが知られています（文献 ",[31,6414,6416],{"href":6415},"#ref-13","[13]",[68,6418,6420],{"id":6419},"_24-ステップ4変数からモデルを作る","2.4 ステップ4：変数からモデルを作る",[10,6422,6423,6424,6452,6453,6481],{},"割り当て ",[76,6425,6427,6440],{"className":6426},[79],[76,6428,6430],{"className":6429},[83],[85,6431,6432],{"xmlns":87},[89,6433,6434,6438],{},[92,6435,6436],{},[95,6437,6317],{},[100,6439,6317],{"encoding":102},[76,6441,6443],{"className":6442,"ariaHidden":108},[107],[76,6444,6446,6449],{"className":6445},[112],[76,6447],{"className":6448,"style":117},[116],[76,6450,6317],{"className":6451,"style":1184},[121,216]," から、次のように構造 ",[76,6454,6456,6469],{"className":6455},[79],[76,6457,6459],{"className":6458},[83],[85,6460,6461],{"xmlns":87},[89,6462,6463,6467],{},[92,6464,6465],{},[95,6466,171],{"mathvariant":97},[100,6468,174],{"encoding":102},[76,6470,6472],{"className":6471,"ariaHidden":108},[107],[76,6473,6475,6478],{"className":6474},[112],[76,6476],{"className":6477,"style":117},[116],[76,6479,171],{"className":6480},[121,122]," を作ります。",[21,6483,6484,6697],{},[24,6485,6486,6489,6490,6696],{},[13,6487,6488],{},"領域","：変数全体 ",[76,6491,6493,6537],{"className":6492},[79],[76,6494,6496],{"className":6495},[83],[85,6497,6498],{"xmlns":87},[89,6499,6500,6534],{},[92,6501,6502,6504,6510,6512,6518,6520,6526,6528,6530,6532],{},[299,6503,1155],{"stretchy":1154},[2628,6505,6506,6508],{},[95,6507,3419],{},[2634,6509,3422],{},[299,6511,2687],{"separator":108},[2628,6513,6514,6516],{},[95,6515,3419],{},[2634,6517,2636],{},[299,6519,2687],{"separator":108},[2628,6521,6522,6524],{},[95,6523,3419],{},[2634,6525,3439],{},[299,6527,2687],{"separator":108},[299,6529,2690],{},[2617,6531,5223],{},[299,6533,1161],{"stretchy":1154},[100,6535,6536],{"encoding":102},"\\{v_0, v_1, v_2, \\dots\\}",[76,6538,6540],{"className":6539,"ariaHidden":108},[107],[76,6541,6543,6546,6549,6589,6592,6595,6635,6638,6641,6681,6684,6687,6690,6693],{"className":6542},[112],[76,6544],{"className":6545,"style":1198},[116],[76,6547,1155],{"className":6548},[1202],[76,6550,6552,6555],{"className":6551},[121],[76,6553,3419],{"className":6554,"style":253},[121,216],[76,6556,6558],{"className":6557},[2770],[76,6559,6561,6581],{"className":6560},[2774,2775],[76,6562,6564,6578],{"className":6563},[2779],[76,6565,6567],{"className":6566,"style":2784},[2783],[76,6568,6569,6572],{"style":2899},[76,6570],{"className":6571,"style":2792},[2791],[76,6573,6575],{"className":6574},[2796,2797,2798,2799],[76,6576,3422],{"className":6577},[121,2799],[76,6579,2807],{"className":6580},[2806],[76,6582,6584],{"className":6583},[2779],[76,6585,6587],{"className":6586,"style":2814},[2783],[76,6588],{},[76,6590,2687],{"className":6591},[3025],[76,6593],{"className":6594,"style":2820},[323],[76,6596,6598,6601],{"className":6597},[121],[76,6599,3419],{"className":6600,"style":253},[121,216],[76,6602,6604],{"className":6603},[2770],[76,6605,6607,6627],{"className":6606},[2774,2775],[76,6608,6610,6624],{"className":6609},[2779],[76,6611,6613],{"className":6612,"style":2784},[2783],[76,6614,6615,6618],{"style":2899},[76,6616],{"className":6617,"style":2792},[2791],[76,6619,6621],{"className":6620},[2796,2797,2798,2799],[76,6622,2636],{"className":6623},[121,2799],[76,6625,2807],{"className":6626},[2806],[76,6628,6630],{"className":6629},[2779],[76,6631,6633],{"className":6632,"style":2814},[2783],[76,6634],{},[76,6636,2687],{"className":6637},[3025],[76,6639],{"className":6640,"style":2820},[323],[76,6642,6644,6647],{"className":6643},[121],[76,6645,3419],{"className":6646,"style":253},[121,216],[76,6648,6650],{"className":6649},[2770],[76,6651,6653,6673],{"className":6652},[2774,2775],[76,6654,6656,6670],{"className":6655},[2779],[76,6657,6659],{"className":6658,"style":2784},[2783],[76,6660,6661,6664],{"style":2899},[76,6662],{"className":6663,"style":2792},[2791],[76,6665,6667],{"className":6666},[2796,2797,2798,2799],[76,6668,3439],{"className":6669},[121,2799],[76,6671,2807],{"className":6672},[2806],[76,6674,6676],{"className":6675},[2779],[76,6677,6679],{"className":6678,"style":2814},[2783],[76,6680],{},[76,6682,2687],{"className":6683},[3025],[76,6685],{"className":6686,"style":2820},[323],[76,6688,2690],{"className":6689},[2824],[76,6691],{"className":6692,"style":2820},[323],[76,6694,1161],{"className":6695},[1209],"（自然数と同一視できる可算集合）。",[24,6698,6699,1027,6702,6994,6995,156,7036,7064,7065,7309],{},[13,6700,6701],{},"述語の解釈",[76,6703,6705,6756],{"className":6704},[79],[76,6706,6708],{"className":6707},[83],[85,6709,6710],{"xmlns":87},[89,6711,6712,6753],{},[92,6713,6714,6721,6723,6734,6736,6738,6740,6751],{},[6715,6716,6717,6719],"msup",{},[95,6718,4904],{},[95,6720,171],{"mathvariant":97},[299,6722,2678],{"stretchy":1154},[2628,6724,6725,6727],{},[95,6726,3419],{},[2628,6728,6729,6732],{},[95,6730,6731],{},"i",[2634,6733,2636],{},[299,6735,2687],{"separator":108},[299,6737,2690],{},[299,6739,2687],{"separator":108},[2628,6741,6742,6744],{},[95,6743,3419],{},[2628,6745,6746,6748],{},[95,6747,6731],{},[95,6749,6750],{},"r",[299,6752,2721],{"stretchy":1154},[100,6754,6755],{"encoding":102},"R^{\\mathcal{M}}(v_{i_1}, \\dots, v_{i_r})",[76,6757,6759],{"className":6758,"ariaHidden":108},[107],[76,6760,6762,6766,6800,6803,6891,6894,6897,6900,6903,6906,6909,6991],{"className":6761},[112],[76,6763],{"className":6764,"style":6765},[116],"height:1.0914em;vertical-align:-0.2501em;",[76,6767,6769,6772],{"className":6768},[121],[76,6770,4904],{"className":6771,"style":4939},[121,216],[76,6773,6775],{"className":6774},[2770],[76,6776,6778],{"className":6777},[2774],[76,6779,6781],{"className":6780},[2779],[76,6782,6785],{"className":6783,"style":6784},[2783],"height:0.8413em;",[76,6786,6788,6791],{"style":6787},"top:-3.063em;margin-right:0.05em;",[76,6789],{"className":6790,"style":2792},[2791],[76,6792,6794],{"className":6793},[2796,2797,2798,2799],[76,6795,6797],{"className":6796},[121,2799],[76,6798,171],{"className":6799},[121,122,2799],[76,6801,2678],{"className":6802},[1202],[76,6804,6806,6809],{"className":6805},[121],[76,6807,3419],{"className":6808,"style":253},[121,216],[76,6810,6812],{"className":6811},[2770],[76,6813,6815,6882],{"className":6814},[2774,2775],[76,6816,6818,6879],{"className":6817},[2779],[76,6819,6822],{"className":6820,"style":6821},[2783],"height:0.3117em;",[76,6823,6824,6827],{"style":2899},[76,6825],{"className":6826,"style":2792},[2791],[76,6828,6830],{"className":6829},[2796,2797,2798,2799],[76,6831,6833],{"className":6832},[121,2799],[76,6834,6836,6839],{"className":6835},[121,2799],[76,6837,6731],{"className":6838},[121,216,2799],[76,6840,6842],{"className":6841},[2770],[76,6843,6845,6870],{"className":6844},[2774,2775],[76,6846,6848,6867],{"className":6847},[2779],[76,6849,6852],{"className":6850,"style":6851},[2783],"height:0.3173em;",[76,6853,6855,6859],{"style":6854},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[76,6856],{"className":6857,"style":6858},[2791],"height:2.5em;",[76,6860,6864],{"className":6861},[2796,6862,6863,2799],"reset-size3","size1",[76,6865,2636],{"className":6866},[121,2799],[76,6868,2807],{"className":6869},[2806],[76,6871,6873],{"className":6872},[2779],[76,6874,6877],{"className":6875,"style":6876},[2783],"height:0.143em;",[76,6878],{},[76,6880,2807],{"className":6881},[2806],[76,6883,6885],{"className":6884},[2779],[76,6886,6889],{"className":6887,"style":6888},[2783],"height:0.2501em;",[76,6890],{},[76,6892,2687],{"className":6893},[3025],[76,6895],{"className":6896,"style":2820},[323],[76,6898,2690],{"className":6899},[2824],[76,6901],{"className":6902,"style":2820},[323],[76,6904,2687],{"className":6905},[3025],[76,6907],{"className":6908,"style":2820},[323],[76,6910,6912,6915],{"className":6911},[121],[76,6913,3419],{"className":6914,"style":253},[121,216],[76,6916,6918],{"className":6917},[2770],[76,6919,6921,6983],{"className":6920},[2774,2775],[76,6922,6924,6980],{"className":6923},[2779],[76,6925,6927],{"className":6926,"style":6821},[2783],[76,6928,6929,6932],{"style":2899},[76,6930],{"className":6931,"style":2792},[2791],[76,6933,6935],{"className":6934},[2796,2797,2798,2799],[76,6936,6938],{"className":6937},[121,2799],[76,6939,6941,6944],{"className":6940},[121,2799],[76,6942,6731],{"className":6943},[121,216,2799],[76,6945,6947],{"className":6946},[2770],[76,6948,6950,6972],{"className":6949},[2774,2775],[76,6951,6953,6969],{"className":6952},[2779],[76,6954,6957],{"className":6955,"style":6956},[2783],"height:0.1645em;",[76,6958,6959,6962],{"style":6854},[76,6960],{"className":6961,"style":6858},[2791],[76,6963,6965],{"className":6964},[2796,6862,6863,2799],[76,6966,6750],{"className":6967,"style":6968},[121,216,2799],"margin-right:0.0278em;",[76,6970,2807],{"className":6971},[2806],[76,6973,6975],{"className":6974},[2779],[76,6976,6978],{"className":6977,"style":6876},[2783],[76,6979],{},[76,6981,2807],{"className":6982},[2806],[76,6984,6986],{"className":6985},[2779],[76,6987,6989],{"className":6988,"style":6888},[2783],[76,6990],{},[76,6992,2721],{"className":6993},[1209]," が成り立つ ",[76,6996,6998,7017],{"className":6997},[79],[76,6999,7001],{"className":7000},[83],[85,7002,7003],{"xmlns":87},[89,7004,7005,7014],{},[92,7006,7007,7009,7012],{},[2617,7008,2619],{},[299,7010,7011],{},"⟺",[2617,7013,2619],{},[100,7015,7016],{"encoding":102},"\\iff",[76,7018,7020],{"className":7019,"ariaHidden":108},[107],[76,7021,7023,7027,7030,7033],{"className":7022},[112],[76,7024],{"className":7025,"style":7026},[116],"height:0.549em;vertical-align:-0.024em;",[76,7028],{"className":7029,"style":324},[323],[76,7031,7011],{"className":7032},[328],[76,7034],{"className":7035,"style":324},[323],[76,7037,7039,7052],{"className":7038},[79],[76,7040,7042],{"className":7041},[83],[85,7043,7044],{"xmlns":87},[89,7045,7046,7050],{},[92,7047,7048],{},[95,7049,6317],{},[100,7051,6317],{"encoding":102},[76,7053,7055],{"className":7054,"ariaHidden":108},[107],[76,7056,7058,7061],{"className":7057},[112],[76,7059],{"className":7060,"style":117},[116],[76,7062,6317],{"className":7063,"style":1184},[121,216]," が原子論理式 ",[76,7066,7068,7112],{"className":7067},[79],[76,7069,7071],{"className":7070},[83],[85,7072,7073],{"xmlns":87},[89,7074,7075,7109],{},[92,7076,7077,7079,7081,7091,7093,7095,7097,7107],{},[95,7078,4904],{},[299,7080,2678],{"stretchy":1154},[2628,7082,7083,7085],{},[95,7084,3419],{},[2628,7086,7087,7089],{},[95,7088,6731],{},[2634,7090,2636],{},[299,7092,2687],{"separator":108},[299,7094,2690],{},[299,7096,2687],{"separator":108},[2628,7098,7099,7101],{},[95,7100,3419],{},[2628,7102,7103,7105],{},[95,7104,6731],{},[95,7106,6750],{},[299,7108,2721],{"stretchy":1154},[100,7110,7111],{"encoding":102},"R(v_{i_1}, \\dots, v_{i_r})",[76,7113,7115],{"className":7114,"ariaHidden":108},[107],[76,7116,7118,7122,7125,7128,7208,7211,7214,7217,7220,7223,7226,7306],{"className":7117},[112],[76,7119],{"className":7120,"style":7121},[116],"height:1.0001em;vertical-align:-0.2501em;",[76,7123,4904],{"className":7124,"style":4939},[121,216],[76,7126,2678],{"className":7127},[1202],[76,7129,7131,7134],{"className":7130},[121],[76,7132,3419],{"className":7133,"style":253},[121,216],[76,7135,7137],{"className":7136},[2770],[76,7138,7140,7200],{"className":7139},[2774,2775],[76,7141,7143,7197],{"className":7142},[2779],[76,7144,7146],{"className":7145,"style":6821},[2783],[76,7147,7148,7151],{"style":2899},[76,7149],{"className":7150,"style":2792},[2791],[76,7152,7154],{"className":7153},[2796,2797,2798,2799],[76,7155,7157],{"className":7156},[121,2799],[76,7158,7160,7163],{"className":7159},[121,2799],[76,7161,6731],{"className":7162},[121,216,2799],[76,7164,7166],{"className":7165},[2770],[76,7167,7169,7189],{"className":7168},[2774,2775],[76,7170,7172,7186],{"className":7171},[2779],[76,7173,7175],{"className":7174,"style":6851},[2783],[76,7176,7177,7180],{"style":6854},[76,7178],{"className":7179,"style":6858},[2791],[76,7181,7183],{"className":7182},[2796,6862,6863,2799],[76,7184,2636],{"className":7185},[121,2799],[76,7187,2807],{"className":7188},[2806],[76,7190,7192],{"className":7191},[2779],[76,7193,7195],{"className":7194,"style":6876},[2783],[76,7196],{},[76,7198,2807],{"className":7199},[2806],[76,7201,7203],{"className":7202},[2779],[76,7204,7206],{"className":7205,"style":6888},[2783],[76,7207],{},[76,7209,2687],{"className":7210},[3025],[76,7212],{"className":7213,"style":2820},[323],[76,7215,2690],{"className":7216},[2824],[76,7218],{"className":7219,"style":2820},[323],[76,7221,2687],{"className":7222},[3025],[76,7224],{"className":7225,"style":2820},[323],[76,7227,7229,7232],{"className":7228},[121],[76,7230,3419],{"className":7231,"style":253},[121,216],[76,7233,7235],{"className":7234},[2770],[76,7236,7238,7298],{"className":7237},[2774,2775],[76,7239,7241,7295],{"className":7240},[2779],[76,7242,7244],{"className":7243,"style":6821},[2783],[76,7245,7246,7249],{"style":2899},[76,7247],{"className":7248,"style":2792},[2791],[76,7250,7252],{"className":7251},[2796,2797,2798,2799],[76,7253,7255],{"className":7254},[121,2799],[76,7256,7258,7261],{"className":7257},[121,2799],[76,7259,6731],{"className":7260},[121,216,2799],[76,7262,7264],{"className":7263},[2770],[76,7265,7267,7287],{"className":7266},[2774,2775],[76,7268,7270,7284],{"className":7269},[2779],[76,7271,7273],{"className":7272,"style":6956},[2783],[76,7274,7275,7278],{"style":6854},[76,7276],{"className":7277,"style":6858},[2791],[76,7279,7281],{"className":7280},[2796,6862,6863,2799],[76,7282,6750],{"className":7283,"style":6968},[121,216,2799],[76,7285,2807],{"className":7286},[2806],[76,7288,7290],{"className":7289},[2779],[76,7291,7293],{"className":7292,"style":6876},[2783],[76,7294],{},[76,7296,2807],{"className":7297},[2806],[76,7299,7301],{"className":7300},[2779],[76,7302,7304],{"className":7303,"style":6888},[2783],[76,7305],{},[76,7307,2721],{"className":7308},[1209]," を真とする。",[10,7311,5987,7312,254,7381,7409,7410,7438,7439,7499,7500,7571,7572,7600],{},[76,7313,7315,7332],{"className":7314},[79],[76,7316,7318],{"className":7317},[83],[85,7319,7320],{"xmlns":87},[89,7321,7322,7330],{},[92,7323,7324],{},[2628,7325,7326,7328],{},[95,7327,1158],{},[95,7329,4400],{},[100,7331,4436],{"encoding":102},[76,7333,7335],{"className":7334,"ariaHidden":108},[107],[76,7336,7338,7341],{"className":7337},[112],[76,7339],{"className":7340,"style":1281},[116],[76,7342,7344,7347],{"className":7343},[121],[76,7345,1158],{"className":7346,"style":253},[121,216],[76,7348,7350],{"className":7349},[2770],[76,7351,7353,7373],{"className":7352},[2774,2775],[76,7354,7356,7370],{"className":7355},[2779],[76,7357,7359],{"className":7358,"style":2950},[2783],[76,7360,7361,7364],{"style":2899},[76,7362],{"className":7363,"style":2792},[2791],[76,7365,7367],{"className":7366},[2796,2797,2798,2799],[76,7368,4400],{"className":7369},[121,216,2799],[76,7371,2807],{"className":7372},[2806],[76,7374,7376],{"className":7375},[2779],[76,7377,7379],{"className":7378,"style":2814},[2783],[76,7380],{},[76,7382,7384,7397],{"className":7383},[79],[76,7385,7387],{"className":7386},[83],[85,7388,7389],{"xmlns":87},[89,7390,7391,7395],{},[92,7392,7393],{},[95,7394,171],{"mathvariant":97},[100,7396,174],{"encoding":102},[76,7398,7400],{"className":7399,"ariaHidden":108},[107],[76,7401,7403,7406],{"className":7402},[112],[76,7404],{"className":7405,"style":117},[116],[76,7407,171],{"className":7408},[121,122]," で成り立ちます。領域の任意の ",[76,7411,7413,7426],{"className":7412},[79],[76,7414,7416],{"className":7415},[83],[85,7417,7418],{"xmlns":87},[89,7419,7420,7424],{},[92,7421,7422],{},[95,7423,2648],{},[100,7425,2648],{"encoding":102},[76,7427,7429],{"className":7428,"ariaHidden":108},[107],[76,7430,7432,7435],{"className":7431},[112],[76,7433],{"className":7434,"style":820},[116],[76,7436,2648],{"className":7437,"style":2861},[121,216]," 個組は、いつか ",[76,7440,7442,7459],{"className":7441},[79],[76,7443,7445],{"className":7444},[83],[85,7446,7447],{"xmlns":87},[89,7448,7449,7457],{},[92,7450,7451],{},[3234,7452,7453,7455],{"accent":108},[95,7454,2632],{},[299,7456,3240],{},[100,7458,3243],{"encoding":102},[76,7460,7462],{"className":7461,"ariaHidden":108},[107],[76,7463,7465,7468],{"className":7464},[112],[76,7466],{"className":7467,"style":3253},[116],[76,7469,7471],{"className":7470},[121,3257],[76,7472,7474],{"className":7473},[2774],[76,7475,7477],{"className":7476},[2779],[76,7478,7480,7488],{"className":7479,"style":3253},[2783],[76,7481,7482,7485],{"style":3269},[76,7483],{"className":7484,"style":3273},[2791],[76,7486,2632],{"className":7487},[121,216],[76,7489,7490,7493],{"style":3269},[76,7491],{"className":7492,"style":3273},[2791],[76,7494,7496],{"className":7495,"style":3286},[3285],[76,7497,3240],{"className":7498},[121]," に代入されています。そのとき ",[76,7501,7503,7520],{"className":7502},[79],[76,7504,7506],{"className":7505},[83],[85,7507,7508],{"xmlns":87},[89,7509,7510,7518],{},[92,7511,7512],{},[3234,7513,7514,7516],{"accent":108},[95,7515,2658],{},[299,7517,3240],{},[100,7519,3311],{"encoding":102},[76,7521,7523],{"className":7522,"ariaHidden":108},[107],[76,7524,7526,7529],{"className":7525},[112],[76,7527],{"className":7528,"style":3321},[116],[76,7530,7532],{"className":7531},[121,3257],[76,7533,7535,7563],{"className":7534},[2774,2775],[76,7536,7538,7560],{"className":7537},[2779],[76,7539,7541,7549],{"className":7540,"style":3253},[2783],[76,7542,7543,7546],{"style":3269},[76,7544],{"className":7545,"style":3273},[2791],[76,7547,2658],{"className":7548,"style":253},[121,216],[76,7550,7551,7554],{"style":3269},[76,7552],{"className":7553,"style":3273},[2791],[76,7555,7557],{"className":7556,"style":3350},[3285],[76,7558,3240],{"className":7559},[121],[76,7561,2807],{"className":7562},[2806],[76,7564,7566],{"className":7565},[2779],[76,7567,7569],{"className":7568,"style":3363},[2783],[76,7570],{}," に代入された新しい変数が、",[76,7573,7575,7588],{"className":7574},[79],[76,7576,7578],{"className":7577},[83],[85,7579,7580],{"xmlns":87},[89,7581,7582,7586],{},[92,7583,7584],{},[95,7585,1158],{},[100,7587,3207],{"encoding":102},[76,7589,7591],{"className":7590,"ariaHidden":108},[107],[76,7592,7594,7597],{"className":7593},[112],[76,7595],{"className":7596,"style":1281},[116],[76,7598,1158],{"className":7599,"style":253},[121,216]," を満たす「証人」になっています。したがって",[76,7602,7604],{"className":7603},[2600],[76,7605,7607,7645],{"className":7606},[79],[76,7608,7610],{"className":7609},[83],[85,7611,7612],{"xmlns":87,"display":2609},[89,7613,7614,7642],{},[92,7615,7616,7618,7620,7622,7628,7630,7632,7638,7640],{},[95,7617,171],{"mathvariant":97},[299,7619,301],{},[95,7621,2626],{"mathvariant":363},[3234,7623,7624,7626],{"accent":108},[95,7625,2632],{},[299,7627,3240],{},[2617,7629,5223],{},[95,7631,2653],{"mathvariant":363},[3234,7633,7634,7636],{"accent":108},[95,7635,2658],{},[299,7637,3240],{},[2617,7639,5223],{},[95,7641,1158],{},[100,7643,7644],{"encoding":102},"\\mathcal{M} \\models \\forall \\bar{x}\\, \\exists \\bar{y}\\, \\psi",[76,7646,7648,7666],{"className":7647,"ariaHidden":108},[107],[76,7649,7651,7654,7657,7660,7663],{"className":7650},[112],[76,7652],{"className":7653,"style":316},[116],[76,7655,171],{"className":7656},[121,122],[76,7658],{"className":7659,"style":324},[323],[76,7661,301],{"className":7662},[328],[76,7664],{"className":7665,"style":324},[323],[76,7667,7669,7672,7675,7706,7709,7712,7754,7757],{"className":7668},[112],[76,7670],{"className":7671,"style":1281},[116],[76,7673,2626],{"className":7674},[121],[76,7676,7678],{"className":7677},[121,3257],[76,7679,7681],{"className":7680},[2774],[76,7682,7684],{"className":7683},[2779],[76,7685,7687,7695],{"className":7686,"style":3253},[2783],[76,7688,7689,7692],{"style":3269},[76,7690],{"className":7691,"style":3273},[2791],[76,7693,2632],{"className":7694},[121,216],[76,7696,7697,7700],{"style":3269},[76,7698],{"className":7699,"style":3273},[2791],[76,7701,7703],{"className":7702,"style":3286},[3285],[76,7704,3240],{"className":7705},[121],[76,7707],{"className":7708,"style":2820},[323],[76,7710,2653],{"className":7711},[121],[76,7713,7715],{"className":7714},[121,3257],[76,7716,7718,7746],{"className":7717},[2774,2775],[76,7719,7721,7743],{"className":7720},[2779],[76,7722,7724,7732],{"className":7723,"style":3253},[2783],[76,7725,7726,7729],{"style":3269},[76,7727],{"className":7728,"style":3273},[2791],[76,7730,2658],{"className":7731,"style":253},[121,216],[76,7733,7734,7737],{"style":3269},[76,7735],{"className":7736,"style":3273},[2791],[76,7738,7740],{"className":7739,"style":3350},[3285],[76,7741,3240],{"className":7742},[121],[76,7744,2807],{"className":7745},[2806],[76,7747,7749],{"className":7748},[2779],[76,7750,7752],{"className":7751,"style":3363},[2783],[76,7753],{},[76,7755],{"className":7756,"style":2820},[323],[76,7758,1158],{"className":7759,"style":253},[121,216],[10,7761,7762,7763,502,7791,7794],{},"となり、",[76,7764,7766,7779],{"className":7765},[79],[76,7767,7769],{"className":7768},[83],[85,7770,7771],{"xmlns":87},[89,7772,7773,7777],{},[92,7774,7775],{},[95,7776,456],{},[100,7778,459],{"encoding":102},[76,7780,7782],{"className":7781,"ariaHidden":108},[107],[76,7783,7785,7788],{"className":7784},[112],[76,7786],{"className":7787,"style":469},[116],[76,7789,456],{"className":7790},[121,216],[13,7792,7793],{},"可算な領域で充足可能"," です。これで二者択一が示されました。",[6334,7796,7798],{"title":7797},"等号の扱い",[10,7799,7800,7801,7939,7940,40],{},"等号を含む言語では、上で作ったモデルを「",[76,7802,7804,7831],{"className":7803},[79],[76,7805,7807],{"className":7806},[83],[85,7808,7809],{"xmlns":87},[89,7810,7811,7828],{},[92,7812,7813,7819,7821],{},[2628,7814,7815,7817],{},[95,7816,3419],{},[95,7818,6731],{},[299,7820,602],{},[2628,7822,7823,7825],{},[95,7824,3419],{},[95,7826,7827],{},"j",[100,7829,7830],{"encoding":102},"v_i = v_j",[76,7832,7834,7890],{"className":7833,"ariaHidden":108},[107],[76,7835,7837,7841,7881,7884,7887],{"className":7836},[112],[76,7838],{"className":7839,"style":7840},[116],"height:0.5806em;vertical-align:-0.15em;",[76,7842,7844,7847],{"className":7843},[121],[76,7845,3419],{"className":7846,"style":253},[121,216],[76,7848,7850],{"className":7849},[2770],[76,7851,7853,7873],{"className":7852},[2774,2775],[76,7854,7856,7870],{"className":7855},[2779],[76,7857,7859],{"className":7858,"style":6821},[2783],[76,7860,7861,7864],{"style":2899},[76,7862],{"className":7863,"style":2792},[2791],[76,7865,7867],{"className":7866},[2796,2797,2798,2799],[76,7868,6731],{"className":7869},[121,216,2799],[76,7871,2807],{"className":7872},[2806],[76,7874,7876],{"className":7875},[2779],[76,7877,7879],{"className":7878,"style":2814},[2783],[76,7880],{},[76,7882],{"className":7883,"style":324},[323],[76,7885,602],{"className":7886},[328],[76,7888],{"className":7889,"style":324},[323],[76,7891,7893,7897],{"className":7892},[112],[76,7894],{"className":7895,"style":7896},[116],"height:0.7167em;vertical-align:-0.2861em;",[76,7898,7900,7903],{"className":7899},[121],[76,7901,3419],{"className":7902,"style":253},[121,216],[76,7904,7906],{"className":7905},[2770],[76,7907,7909,7930],{"className":7908},[2774,2775],[76,7910,7912,7927],{"className":7911},[2779],[76,7913,7915],{"className":7914,"style":6821},[2783],[76,7916,7917,7920],{"style":2899},[76,7918],{"className":7919,"style":2792},[2791],[76,7921,7923],{"className":7922},[2796,2797,2798,2799],[76,7924,7827],{"className":7925,"style":7926},[121,216,2799],"margin-right:0.0572em;",[76,7928,2807],{"className":7929},[2806],[76,7931,7933],{"className":7932},[2779],[76,7934,7937],{"className":7935,"style":7936},[2783],"height:0.2861em;",[76,7938],{}," が真とされる変数どうしを同一視する」ことで割った構造（商構造）を考えます。ゲーデルは論文の後半で、等号を新しい述語記号と公理（反射律・対称律・推移律・代入律）で置き換え、得られたモデルをその関係で割る方法で、等号を含む場合を扱っています（文献 ",[31,7941,2352],{"href":2351},[68,7943,7945],{"id":7944},"_25-ゲーデル流の特徴","2.5 ゲーデル流の特徴",[21,7947,7948,7954,7971],{},[24,7949,7950,7953],{},[13,7951,7952],{},"命題論理の完全性を足場にしている","：述語論理の問題を、無限個の命題論理の問題と、それらを貼り合わせる「コンパクト性」の議論に分解しています。",[24,7955,7956,7959,7960,7963,7964,7966,7967,7970],{},[13,7957,7958],{},"副産物が多い","：同じ論文で、ゲーデルは可算個の論理式の集合にも議論を広げ、",[13,7961,7962],{},"コンパクト性定理","（可算個の論理式の集合は、その有限部分集合がすべて充足可能なら、全体も充足可能）にあたる結果も得ています（文献 ",[31,7965,2352],{"href":2351},"）。モデルが可算になることから、",[13,7968,7969],{},"レーヴェンハイム＝スコーレムの定理"," も同時に得られます。",[24,7972,7973,7976,7977,7981,7982,7986,7987,7990],{},[13,7974,7975],{},"スコーレムとの関係","：一階述語論理の完全性は、スコーレムが1920年代前半に証明し1928年の論文で論じた結果から、系として導けるものでした。しかしスコーレム自身は、それを完全性定理として述べてはいません（文献 ",[31,7978,7980],{"href":7979},"#ref-11","[11]","）。ゲーデルは後年、スコーレムの研究に暗に解答が含まれていたことを、当時は知らなかったと認めています（1964年のファン・ハイエノールト宛ての手紙。文献 ",[31,7983,7985],{"href":7984},"#ref-12","[12]","）。ゲーデルの貢献は、モデルの構成を ",[13,7988,7989],{},"形式的な証明可能性"," と結びつけ、完全性という形で明確に述べた点にあります。",[63,7992,7994],{"id":7993},"_3-ヘンキン流の証明","3. ヘンキン流の証明",[10,7996,7997,7998,8001,8002,8004,8005,8008],{},"ヘンキンは、標準形への変換を使わずに、",[13,7999,8000],{},"無矛盾な理論から直接モデルを作る"," 方法を与えました（文献 ",[31,8003,50],{"href":49},"）。モデルの要素は、",[13,8006,8007],{},"閉項","（変数を含まない項）そのものです。ヘンキンは論文の冒頭で、それまでに発表された一階述語論理の完全性の証明は、ゲーデルの元の証明と、ヒルベルトとベルナイスによるその変形だけだったと述べています。",[10,8010,8011],{},"方針は次の3段階です。",[8013,8014,8016],"key-equation",{"title":8015},"ヘンキンの証明の3段階",[8017,8018,8019,8159,8199],"ol",{},[24,8020,8021,8024,8025,1078,8084,8129,8130,8158],{},[13,8022,8023],{},"証人の追加","：「",[76,8026,8028,8054],{"className":8027},[79],[76,8029,8031],{"className":8030},[83],[85,8032,8033],{"xmlns":87},[89,8034,8035,8051],{},[92,8036,8037,8039,8041,8043,8045,8047,8049],{},[95,8038,2653],{"mathvariant":363},[95,8040,2632],{},[2617,8042,5223],{},[95,8044,456],{},[299,8046,2678],{"stretchy":1154},[95,8048,2632],{},[299,8050,2721],{"stretchy":1154},[100,8052,8053],{"encoding":102},"\\exists x\\, \\varphi(x)",[76,8055,8057],{"className":8056,"ariaHidden":108},[107],[76,8058,8060,8063,8066,8069,8072,8075,8078,8081],{"className":8059},[112],[76,8061],{"className":8062,"style":1198},[116],[76,8064,2653],{"className":8065},[121],[76,8067,2632],{"className":8068},[121,216],[76,8070],{"className":8071,"style":2820},[323],[76,8073,456],{"className":8074},[121,216],[76,8076,2678],{"className":8077},[1202],[76,8079,2632],{"className":8080},[121,216],[76,8082,2721],{"className":8083},[1209],[76,8085,8087,8108],{"className":8086},[79],[76,8088,8090],{"className":8089},[83],[85,8091,8092],{"xmlns":87},[89,8093,8094,8105],{},[92,8095,8096,8098,8100,8103],{},[95,8097,456],{},[299,8099,2678],{"stretchy":1154},[95,8101,8102],{},"c",[299,8104,2721],{"stretchy":1154},[100,8106,8107],{"encoding":102},"\\varphi(c)",[76,8109,8111],{"className":8110,"ariaHidden":108},[107],[76,8112,8114,8117,8120,8123,8126],{"className":8113},[112],[76,8115],{"className":8116,"style":1198},[116],[76,8118,456],{"className":8119},[121,216],[76,8121,2678],{"className":8122},[1202],[76,8124,8102],{"className":8125},[121,216],[76,8127,2721],{"className":8128},[1209],"」という公理を加え、存在文ごとに証人となる定数 ",[76,8131,8133,8146],{"className":8132},[79],[76,8134,8136],{"className":8135},[83],[85,8137,8138],{"xmlns":87},[89,8139,8140,8144],{},[92,8141,8142],{},[95,8143,8102],{},[100,8145,8102],{"encoding":102},[76,8147,8149],{"className":8148,"ariaHidden":108},[107],[76,8150,8152,8155],{"className":8151},[112],[76,8153],{"className":8154,"style":249},[116],[76,8156,8102],{"className":8157},[121,216]," を用意する。",[24,8160,8161,8164,8165,156,8168,8198],{},[13,8162,8163],{},"極大化","：理論を、どの文についてもそれ自身か否定のどちらかを含む ",[13,8166,8167],{},"極大無矛盾集合",[76,8169,8171,8186],{"className":8170},[79],[76,8172,8174],{"className":8173},[83],[85,8175,8176],{"xmlns":87},[89,8177,8178,8183],{},[92,8179,8180],{},[95,8181,8182],{"mathvariant":363},"Δ",[100,8184,8185],{"encoding":102},"\\Delta",[76,8187,8189],{"className":8188,"ariaHidden":108},[107],[76,8190,8192,8195],{"className":8191},[112],[76,8193],{"className":8194,"style":117},[116],[76,8196,8182],{"className":8197},[121]," に拡大する。",[24,8200,8201,8204,8205,8233,8234,8341],{},[13,8202,8203],{},"項モデル","：閉項を要素とする構造 ",[76,8206,8208,8221],{"className":8207},[79],[76,8209,8211],{"className":8210},[83],[85,8212,8213],{"xmlns":87},[89,8214,8215,8219],{},[92,8216,8217],{},[95,8218,171],{"mathvariant":97},[100,8220,174],{"encoding":102},[76,8222,8224],{"className":8223,"ariaHidden":108},[107],[76,8225,8227,8230],{"className":8226},[112],[76,8228],{"className":8229,"style":117},[116],[76,8231,171],{"className":8232},[121,122]," を作り、",[76,8235,8237,8268],{"className":8236},[79],[76,8238,8240],{"className":8239},[83],[85,8241,8242],{"xmlns":87},[89,8243,8244,8265],{},[92,8245,8246,8248,8250,8252,8254,8256,8258,8260,8263],{},[95,8247,171],{"mathvariant":97},[299,8249,301],{},[95,8251,236],{},[2617,8253,2619],{},[299,8255,7011],{},[2617,8257,2619],{},[95,8259,236],{},[299,8261,8262],{},"∈",[95,8264,8182],{"mathvariant":363},[100,8266,8267],{"encoding":102},"\\mathcal{M} \\models \\sigma \\iff \\sigma \\in \\Delta",[76,8269,8271,8289,8313,8332],{"className":8270,"ariaHidden":108},[107],[76,8272,8274,8277,8280,8283,8286],{"className":8273},[112],[76,8275],{"className":8276,"style":316},[116],[76,8278,171],{"className":8279},[121,122],[76,8281],{"className":8282,"style":324},[323],[76,8284,301],{"className":8285},[328],[76,8287],{"className":8288,"style":324},[323],[76,8290,8292,8295,8298,8301,8304,8307,8310],{"className":8291},[112],[76,8293],{"className":8294,"style":7026},[116],[76,8296,236],{"className":8297,"style":253},[121,216],[76,8299],{"className":8300,"style":324},[323],[76,8302],{"className":8303,"style":324},[323],[76,8305,7011],{"className":8306},[328],[76,8308],{"className":8309,"style":324},[323],[76,8311],{"className":8312,"style":324},[323],[76,8314,8316,8320,8323,8326,8329],{"className":8315},[112],[76,8317],{"className":8318,"style":8319},[116],"height:0.5782em;vertical-align:-0.0391em;",[76,8321,236],{"className":8322,"style":253},[121,216],[76,8324],{"className":8325,"style":324},[323],[76,8327,8262],{"className":8328},[328],[76,8330],{"className":8331,"style":324},[323],[76,8333,8335,8338],{"className":8334},[112],[76,8336],{"className":8337,"style":117},[116],[76,8339,8182],{"className":8340},[121]," を示す。",[10,8343,8344,8345,8373,8374,8402],{},"以下、言語 ",[76,8346,8348,8361],{"className":8347},[79],[76,8349,8351],{"className":8350},[83],[85,8352,8353],{"xmlns":87},[89,8354,8355,8359],{},[92,8356,8357],{},[95,8358,98],{"mathvariant":97},[100,8360,103],{"encoding":102},[76,8362,8364],{"className":8363,"ariaHidden":108},[107],[76,8365,8367,8370],{"className":8366},[112],[76,8368],{"className":8369,"style":117},[116],[76,8371,98],{"className":8372},[121,122]," は可算であるとし、",[76,8375,8377,8390],{"className":8376},[79],[76,8378,8380],{"className":8379},[83],[85,8381,8382],{"xmlns":87},[89,8383,8384,8388],{},[92,8385,8386],{},[95,8387,364],{"mathvariant":363},[100,8389,367],{"encoding":102},[76,8391,8393],{"className":8392,"ariaHidden":108},[107],[76,8394,8396,8399],{"className":8395},[112],[76,8397],{"className":8398,"style":117},[116],[76,8400,364],{"className":8401},[121]," を無矛盾な文の集合とします。",[68,8404,8406],{"id":8405},"_31-ステップ1証人を追加する","3.1 ステップ1：証人を追加する",[10,8408,8409,8410,8468,8469,8539,8540,8543],{},"存在文 ",[76,8411,8413,8438],{"className":8412},[79],[76,8414,8416],{"className":8415},[83],[85,8417,8418],{"xmlns":87},[89,8419,8420,8436],{},[92,8421,8422,8424,8426,8428,8430,8432,8434],{},[95,8423,2653],{"mathvariant":363},[95,8425,2632],{},[2617,8427,5223],{},[95,8429,456],{},[299,8431,2678],{"stretchy":1154},[95,8433,2632],{},[299,8435,2721],{"stretchy":1154},[100,8437,8053],{"encoding":102},[76,8439,8441],{"className":8440,"ariaHidden":108},[107],[76,8442,8444,8447,8450,8453,8456,8459,8462,8465],{"className":8443},[112],[76,8445],{"className":8446,"style":1198},[116],[76,8448,2653],{"className":8449},[121],[76,8451,2632],{"className":8452},[121,216],[76,8454],{"className":8455,"style":2820},[323],[76,8457,456],{"className":8458},[121,216],[76,8460,2678],{"className":8461},[1202],[76,8463,2632],{"className":8464},[121,216],[76,8466,2721],{"className":8467},[1209]," ごとに新しい定数 ",[76,8470,8472,8490],{"className":8471},[79],[76,8473,8475],{"className":8474},[83],[85,8476,8477],{"xmlns":87},[89,8478,8479,8487],{},[92,8480,8481],{},[2628,8482,8483,8485],{},[95,8484,8102],{},[95,8486,456],{},[100,8488,8489],{"encoding":102},"c_\\varphi",[76,8491,8493],{"className":8492,"ariaHidden":108},[107],[76,8494,8496,8499],{"className":8495},[112],[76,8497],{"className":8498,"style":7896},[116],[76,8500,8502,8505],{"className":8501},[121],[76,8503,8102],{"className":8504},[121,216],[76,8506,8508],{"className":8507},[2770],[76,8509,8511,8531],{"className":8510},[2774,2775],[76,8512,8514,8528],{"className":8513},[2779],[76,8515,8517],{"className":8516,"style":2950},[2783],[76,8518,8519,8522],{"style":2787},[76,8520],{"className":8521,"style":2792},[2791],[76,8523,8525],{"className":8524},[2796,2797,2798,2799],[76,8526,456],{"className":8527},[121,216,2799],[76,8529,2807],{"className":8530},[2806],[76,8532,8534],{"className":8533},[2779],[76,8535,8537],{"className":8536,"style":7936},[2783],[76,8538],{}," を用意し、次の ",[13,8541,8542],{},"ヘンキン公理"," を加えます。",[76,8545,8547],{"className":8546},[2600],[76,8548,8550,8594],{"className":8549},[79],[76,8551,8553],{"className":8552},[83],[85,8554,8555],{"xmlns":87,"display":2609},[89,8556,8557,8591],{},[92,8558,8559,8561,8563,8565,8567,8569,8571,8573,8575,8577,8579,8581,8583,8589],{},[95,8560,2653],{"mathvariant":363},[95,8562,2632],{},[2617,8564,5223],{},[95,8566,456],{},[299,8568,2678],{"stretchy":1154},[95,8570,2632],{},[299,8572,2721],{"stretchy":1154},[2617,8574,2619],{},[299,8576,1248],{},[2617,8578,2619],{},[95,8580,456],{},[299,8582,2678],{"stretchy":1154},[2628,8584,8585,8587],{},[95,8586,8102],{},[95,8588,456],{},[299,8590,2721],{"stretchy":1154},[100,8592,8593],{"encoding":102},"\\exists x\\, \\varphi(x) \\;\\to\\; \\varphi(c_\\varphi)",[76,8595,8597,8639],{"className":8596,"ariaHidden":108},[107],[76,8598,8600,8603,8606,8609,8612,8615,8618,8621,8624,8627,8630,8633,8636],{"className":8599},[112],[76,8601],{"className":8602,"style":1198},[116],[76,8604,2653],{"className":8605},[121],[76,8607,2632],{"className":8608},[121,216],[76,8610],{"className":8611,"style":2820},[323],[76,8613,456],{"className":8614},[121,216],[76,8616,2678],{"className":8617},[1202],[76,8619,2632],{"className":8620},[121,216],[76,8622,2721],{"className":8623},[1209],[76,8625],{"className":8626,"style":324},[323],[76,8628],{"className":8629,"style":324},[323],[76,8631,1248],{"className":8632},[328],[76,8634],{"className":8635,"style":324},[323],[76,8637],{"className":8638,"style":324},[323],[76,8640,8642,8646,8649,8652,8692],{"className":8641},[112],[76,8643],{"className":8644,"style":8645},[116],"height:1.0361em;vertical-align:-0.2861em;",[76,8647,456],{"className":8648},[121,216],[76,8650,2678],{"className":8651},[1202],[76,8653,8655,8658],{"className":8654},[121],[76,8656,8102],{"className":8657},[121,216],[76,8659,8661],{"className":8660},[2770],[76,8662,8664,8684],{"className":8663},[2774,2775],[76,8665,8667,8681],{"className":8666},[2779],[76,8668,8670],{"className":8669,"style":2950},[2783],[76,8671,8672,8675],{"style":2787},[76,8673],{"className":8674,"style":2792},[2791],[76,8676,8678],{"className":8677},[2796,2797,2798,2799],[76,8679,456],{"className":8680},[121,216,2799],[76,8682,2807],{"className":8683},[2806],[76,8685,8687],{"className":8686},[2779],[76,8688,8690],{"className":8689,"style":7936},[2783],[76,8691],{},[76,8693,2721],{"className":8694},[1209],[10,8696,8697,8698,8726,8727,8796],{},"「何かが ",[76,8699,8701,8714],{"className":8700},[79],[76,8702,8704],{"className":8703},[83],[85,8705,8706],{"xmlns":87},[89,8707,8708,8712],{},[92,8709,8710],{},[95,8711,456],{},[100,8713,459],{"encoding":102},[76,8715,8717],{"className":8716,"ariaHidden":108},[107],[76,8718,8720,8723],{"className":8719},[112],[76,8721],{"className":8722,"style":469},[116],[76,8724,456],{"className":8725},[121,216]," を満たすなら、",[76,8728,8730,8747],{"className":8729},[79],[76,8731,8733],{"className":8732},[83],[85,8734,8735],{"xmlns":87},[89,8736,8737,8745],{},[92,8738,8739],{},[2628,8740,8741,8743],{},[95,8742,8102],{},[95,8744,456],{},[100,8746,8489],{"encoding":102},[76,8748,8750],{"className":8749,"ariaHidden":108},[107],[76,8751,8753,8756],{"className":8752},[112],[76,8754],{"className":8755,"style":7896},[116],[76,8757,8759,8762],{"className":8758},[121],[76,8760,8102],{"className":8761},[121,216],[76,8763,8765],{"className":8764},[2770],[76,8766,8768,8788],{"className":8767},[2774,2775],[76,8769,8771,8785],{"className":8770},[2779],[76,8772,8774],{"className":8773,"style":2950},[2783],[76,8775,8776,8779],{"style":2787},[76,8777],{"className":8778,"style":2792},[2791],[76,8780,8782],{"className":8781},[2796,2797,2798,2799],[76,8783,456],{"className":8784},[121,216,2799],[76,8786,2807],{"className":8787},[2806],[76,8789,8791],{"className":8790},[2779],[76,8792,8794],{"className":8793,"style":7936},[2783],[76,8795],{}," がその1つだ」という意味です。",[1309,8798,8800],{"title":8799},"補題 1：証人を加えても無矛盾",[10,8801,8802,8830,8831,254,8859,922,8887,8915,8916,9042],{},[76,8803,8805,8818],{"className":8804},[79],[76,8806,8808],{"className":8807},[83],[85,8809,8810],{"xmlns":87},[89,8811,8812,8816],{},[92,8813,8814],{},[95,8815,364],{"mathvariant":363},[100,8817,367],{"encoding":102},[76,8819,8821],{"className":8820,"ariaHidden":108},[107],[76,8822,8824,8827],{"className":8823},[112],[76,8825],{"className":8826,"style":117},[116],[76,8828,364],{"className":8829},[121]," が無矛盾で、定数 ",[76,8832,8834,8847],{"className":8833},[79],[76,8835,8837],{"className":8836},[83],[85,8838,8839],{"xmlns":87},[89,8840,8841,8845],{},[92,8842,8843],{},[95,8844,8102],{},[100,8846,8102],{"encoding":102},[76,8848,8850],{"className":8849,"ariaHidden":108},[107],[76,8851,8853,8856],{"className":8852},[112],[76,8854],{"className":8855,"style":249},[116],[76,8857,8102],{"className":8858},[121,216],[76,8860,8862,8875],{"className":8861},[79],[76,8863,8865],{"className":8864},[83],[85,8866,8867],{"xmlns":87},[89,8868,8869,8873],{},[92,8870,8871],{},[95,8872,364],{"mathvariant":363},[100,8874,367],{"encoding":102},[76,8876,8878],{"className":8877,"ariaHidden":108},[107],[76,8879,8881,8884],{"className":8880},[112],[76,8882],{"className":8883,"style":117},[116],[76,8885,364],{"className":8886},[121],[76,8888,8890,8903],{"className":8889},[79],[76,8891,8893],{"className":8892},[83],[85,8894,8895],{"xmlns":87},[89,8896,8897,8901],{},[92,8898,8899],{},[95,8900,456],{},[100,8902,459],{"encoding":102},[76,8904,8906],{"className":8905,"ariaHidden":108},[107],[76,8907,8909,8912],{"className":8908},[112],[76,8910],{"className":8911,"style":469},[116],[76,8913,456],{"className":8914},[121,216]," に現れないならば、",[76,8917,8919,8963],{"className":8918},[79],[76,8920,8922],{"className":8921},[83],[85,8923,8924],{"xmlns":87},[89,8925,8926,8960],{},[92,8927,8928,8930,8932,8934,8936,8938,8940,8942,8944,8946,8948,8950,8952,8954,8956,8958],{},[95,8929,364],{"mathvariant":363},[299,8931,1151],{},[299,8933,1155],{"stretchy":1154},[95,8935,2653],{"mathvariant":363},[95,8937,2632],{},[2617,8939,5223],{},[95,8941,456],{},[299,8943,2678],{"stretchy":1154},[95,8945,2632],{},[299,8947,2721],{"stretchy":1154},[299,8949,1248],{},[95,8951,456],{},[299,8953,2678],{"stretchy":1154},[95,8955,8102],{},[299,8957,2721],{"stretchy":1154},[299,8959,1161],{"stretchy":1154},[100,8961,8962],{"encoding":102},"\\Gamma \\cup \\{\\exists x\\, \\varphi(x) \\to \\varphi(c)\\}",[76,8964,8966,8984,9023],{"className":8965,"ariaHidden":108},[107],[76,8967,8969,8972,8975,8978,8981],{"className":8968},[112],[76,8970],{"className":8971,"style":117},[116],[76,8973,364],{"className":8974},[121],[76,8976],{"className":8977,"style":1184},[323],[76,8979,1151],{"className":8980},[1188],[76,8982],{"className":8983,"style":1184},[323],[76,8985,8987,8990,8993,8996,8999,9002,9005,9008,9011,9014,9017,9020],{"className":8986},[112],[76,8988],{"className":8989,"style":1198},[116],[76,8991,1155],{"className":8992},[1202],[76,8994,2653],{"className":8995},[121],[76,8997,2632],{"className":8998},[121,216],[76,9000],{"className":9001,"style":2820},[323],[76,9003,456],{"className":9004},[121,216],[76,9006,2678],{"className":9007},[1202],[76,9009,2632],{"className":9010},[121,216],[76,9012,2721],{"className":9013},[1209],[76,9015],{"className":9016,"style":324},[323],[76,9018,1248],{"className":9019},[328],[76,9021],{"className":9022,"style":324},[323],[76,9024,9026,9029,9032,9035,9038],{"className":9025},[112],[76,9027],{"className":9028,"style":1198},[116],[76,9030,456],{"className":9031},[121,216],[76,9033,2678],{"className":9034},[1202],[76,9036,8102],{"className":9037},[121,216],[76,9039,9041],{"className":9040},[1209],")}"," も無矛盾である。",[1571,9044,9045,9334],{},[10,9046,9047,9048,9179,9180,9261,9262,9333],{},"矛盾していると仮定する。演繹定理と命題論理により、",[76,9049,9051,9097],{"className":9050},[79],[76,9052,9054],{"className":9053},[83],[85,9055,9056],{"xmlns":87},[89,9057,9058,9094],{},[92,9059,9060,9062,9064,9066,9068,9070,9072,9074,9076,9078,9080,9082,9084,9086,9088,9090,9092],{},[95,9061,364],{"mathvariant":363},[299,9063,805],{},[95,9065,941],{"mathvariant":363},[299,9067,2678],{"stretchy":1154},[95,9069,2653],{"mathvariant":363},[95,9071,2632],{},[2617,9073,5223],{},[95,9075,456],{},[299,9077,2678],{"stretchy":1154},[95,9079,2632],{},[299,9081,2721],{"stretchy":1154},[299,9083,1248],{},[95,9085,456],{},[299,9087,2678],{"stretchy":1154},[95,9089,8102],{},[299,9091,2721],{"stretchy":1154},[299,9093,2721],{"stretchy":1154},[100,9095,9096],{"encoding":102},"\\Gamma \\vdash \\neg(\\exists x\\, \\varphi(x) \\to \\varphi(c))",[76,9098,9100,9118,9160],{"className":9099,"ariaHidden":108},[107],[76,9101,9103,9106,9109,9112,9115],{"className":9102},[112],[76,9104],{"className":9105,"style":820},[116],[76,9107,364],{"className":9108},[121],[76,9110],{"className":9111,"style":324},[323],[76,9113,805],{"className":9114},[328],[76,9116],{"className":9117,"style":324},[323],[76,9119,9121,9124,9127,9130,9133,9136,9139,9142,9145,9148,9151,9154,9157],{"className":9120},[112],[76,9122],{"className":9123,"style":1198},[116],[76,9125,941],{"className":9126},[121],[76,9128,2678],{"className":9129},[1202],[76,9131,2653],{"className":9132},[121],[76,9134,2632],{"className":9135},[121,216],[76,9137],{"className":9138,"style":2820},[323],[76,9140,456],{"className":9141},[121,216],[76,9143,2678],{"className":9144},[1202],[76,9146,2632],{"className":9147},[121,216],[76,9149,2721],{"className":9150},[1209],[76,9152],{"className":9153,"style":324},[323],[76,9155,1248],{"className":9156},[328],[76,9158],{"className":9159,"style":324},[323],[76,9161,9163,9166,9169,9172,9175],{"className":9162},[112],[76,9164],{"className":9165,"style":1198},[116],[76,9167,456],{"className":9168},[121,216],[76,9170,2678],{"className":9171},[1202],[76,9173,8102],{"className":9174},[121,216],[76,9176,9178],{"className":9177},[1209],"))","、したがって ",[76,9181,9183,9213],{"className":9182},[79],[76,9184,9186],{"className":9185},[83],[85,9187,9188],{"xmlns":87},[89,9189,9190,9210],{},[92,9191,9192,9194,9196,9198,9200,9202,9204,9206,9208],{},[95,9193,364],{"mathvariant":363},[299,9195,805],{},[95,9197,2653],{"mathvariant":363},[95,9199,2632],{},[2617,9201,5223],{},[95,9203,456],{},[299,9205,2678],{"stretchy":1154},[95,9207,2632],{},[299,9209,2721],{"stretchy":1154},[100,9211,9212],{"encoding":102},"\\Gamma \\vdash \\exists x\\, \\varphi(x)",[76,9214,9216,9234],{"className":9215,"ariaHidden":108},[107],[76,9217,9219,9222,9225,9228,9231],{"className":9218},[112],[76,9220],{"className":9221,"style":820},[116],[76,9223,364],{"className":9224},[121],[76,9226],{"className":9227,"style":324},[323],[76,9229,805],{"className":9230},[328],[76,9232],{"className":9233,"style":324},[323],[76,9235,9237,9240,9243,9246,9249,9252,9255,9258],{"className":9236},[112],[76,9238],{"className":9239,"style":1198},[116],[76,9241,2653],{"className":9242},[121],[76,9244,2632],{"className":9245},[121,216],[76,9247],{"className":9248,"style":2820},[323],[76,9250,456],{"className":9251},[121,216],[76,9253,2678],{"className":9254},[1202],[76,9256,2632],{"className":9257},[121,216],[76,9259,2721],{"className":9260},[1209]," かつ ",[76,9263,9265,9291],{"className":9264},[79],[76,9266,9268],{"className":9267},[83],[85,9269,9270],{"xmlns":87},[89,9271,9272,9288],{},[92,9273,9274,9276,9278,9280,9282,9284,9286],{},[95,9275,364],{"mathvariant":363},[299,9277,805],{},[95,9279,941],{"mathvariant":363},[95,9281,456],{},[299,9283,2678],{"stretchy":1154},[95,9285,8102],{},[299,9287,2721],{"stretchy":1154},[100,9289,9290],{"encoding":102},"\\Gamma \\vdash \\neg\\varphi(c)",[76,9292,9294,9312],{"className":9293,"ariaHidden":108},[107],[76,9295,9297,9300,9303,9306,9309],{"className":9296},[112],[76,9298],{"className":9299,"style":820},[116],[76,9301,364],{"className":9302},[121],[76,9304],{"className":9305,"style":324},[323],[76,9307,805],{"className":9308},[328],[76,9310],{"className":9311,"style":324},[323],[76,9313,9315,9318,9321,9324,9327,9330],{"className":9314},[112],[76,9316],{"className":9317,"style":1198},[116],[76,9319,941],{"className":9320},[121],[76,9322,456],{"className":9323},[121,216],[76,9325,2678],{"className":9326},[1202],[76,9328,8102],{"className":9329},[121,216],[76,9331,2721],{"className":9332},[1209]," となる。",[10,9335,9336,502,9364,9392,9393,9421,9422,9425,9426,5776,9512,9595,9596,9624],{},[76,9337,9339,9352],{"className":9338},[79],[76,9340,9342],{"className":9341},[83],[85,9343,9344],{"xmlns":87},[89,9345,9346,9350],{},[92,9347,9348],{},[95,9349,8102],{},[100,9351,8102],{"encoding":102},[76,9353,9355],{"className":9354,"ariaHidden":108},[107],[76,9356,9358,9361],{"className":9357},[112],[76,9359],{"className":9360,"style":249},[116],[76,9362,8102],{"className":9363},[121,216],[76,9365,9367,9380],{"className":9366},[79],[76,9368,9370],{"className":9369},[83],[85,9371,9372],{"xmlns":87},[89,9373,9374,9378],{},[92,9375,9376],{},[95,9377,364],{"mathvariant":363},[100,9379,367],{"encoding":102},[76,9381,9383],{"className":9382,"ariaHidden":108},[107],[76,9384,9386,9389],{"className":9385},[112],[76,9387],{"className":9388,"style":117},[116],[76,9390,364],{"className":9391},[121]," に現れないので、",[76,9394,9396,9409],{"className":9395},[79],[76,9397,9399],{"className":9398},[83],[85,9400,9401],{"xmlns":87},[89,9402,9403,9407],{},[92,9404,9405],{},[95,9406,8102],{},[100,9408,8102],{"encoding":102},[76,9410,9412],{"className":9411,"ariaHidden":108},[107],[76,9413,9415,9418],{"className":9414},[112],[76,9416],{"className":9417,"style":249},[116],[76,9419,8102],{"className":9420},[121,216]," を変数に置き換えても導出は成り立つ（",[13,9423,9424],{},"定数についての汎化","）。よって ",[76,9427,9429,9461],{"className":9428},[79],[76,9430,9432],{"className":9431},[83],[85,9433,9434],{"xmlns":87},[89,9435,9436,9458],{},[92,9437,9438,9440,9442,9444,9446,9448,9450,9452,9454,9456],{},[95,9439,364],{"mathvariant":363},[299,9441,805],{},[95,9443,2626],{"mathvariant":363},[95,9445,2632],{},[2617,9447,5223],{},[95,9449,941],{"mathvariant":363},[95,9451,456],{},[299,9453,2678],{"stretchy":1154},[95,9455,2632],{},[299,9457,2721],{"stretchy":1154},[100,9459,9460],{"encoding":102},"\\Gamma \\vdash \\forall x\\, \\neg\\varphi(x)",[76,9462,9464,9482],{"className":9463,"ariaHidden":108},[107],[76,9465,9467,9470,9473,9476,9479],{"className":9466},[112],[76,9468],{"className":9469,"style":820},[116],[76,9471,364],{"className":9472},[121],[76,9474],{"className":9475,"style":324},[323],[76,9477,805],{"className":9478},[328],[76,9480],{"className":9481,"style":324},[323],[76,9483,9485,9488,9491,9494,9497,9500,9503,9506,9509],{"className":9484},[112],[76,9486],{"className":9487,"style":1198},[116],[76,9489,2626],{"className":9490},[121],[76,9492,2632],{"className":9493},[121,216],[76,9495],{"className":9496,"style":2820},[323],[76,9498,941],{"className":9499},[121],[76,9501,456],{"className":9502},[121,216],[76,9504,2678],{"className":9505},[1202],[76,9507,2632],{"className":9508},[121,216],[76,9510,2721],{"className":9511},[1209],[76,9513,9515,9547],{"className":9514},[79],[76,9516,9518],{"className":9517},[83],[85,9519,9520],{"xmlns":87},[89,9521,9522,9544],{},[92,9523,9524,9526,9528,9530,9532,9534,9536,9538,9540,9542],{},[95,9525,364],{"mathvariant":363},[299,9527,805],{},[95,9529,941],{"mathvariant":363},[95,9531,2653],{"mathvariant":363},[95,9533,2632],{},[2617,9535,5223],{},[95,9537,456],{},[299,9539,2678],{"stretchy":1154},[95,9541,2632],{},[299,9543,2721],{"stretchy":1154},[100,9545,9546],{"encoding":102},"\\Gamma \\vdash \\neg\\exists x\\, \\varphi(x)",[76,9548,9550,9568],{"className":9549,"ariaHidden":108},[107],[76,9551,9553,9556,9559,9562,9565],{"className":9552},[112],[76,9554],{"className":9555,"style":820},[116],[76,9557,364],{"className":9558},[121],[76,9560],{"className":9561,"style":324},[323],[76,9563,805],{"className":9564},[328],[76,9566],{"className":9567,"style":324},[323],[76,9569,9571,9574,9577,9580,9583,9586,9589,9592],{"className":9570},[112],[76,9572],{"className":9573,"style":1198},[116],[76,9575,5258],{"className":9576},[121],[76,9578,2632],{"className":9579},[121,216],[76,9581],{"className":9582,"style":2820},[323],[76,9584,456],{"className":9585},[121,216],[76,9587,2678],{"className":9588},[1202],[76,9590,2632],{"className":9591},[121,216],[76,9593,2721],{"className":9594},[1209]," となり、",[76,9597,9599,9612],{"className":9598},[79],[76,9600,9602],{"className":9601},[83],[85,9603,9604],{"xmlns":87},[89,9605,9606,9610],{},[92,9607,9608],{},[95,9609,364],{"mathvariant":363},[100,9611,367],{"encoding":102},[76,9613,9615],{"className":9614,"ariaHidden":108},[107],[76,9616,9618,9621],{"className":9617},[112],[76,9619],{"className":9620,"style":117},[116],[76,9622,364],{"className":9623},[121]," が矛盾してしまう。",[10,9626,9627,9628,9658],{},"新しく加えた定数を含む存在文にも、また証人が必要です。そこで、言語の拡大を ",[76,9629,9631,9646],{"className":9630},[79],[76,9632,9634],{"className":9633},[83],[85,9635,9636],{"xmlns":87},[89,9637,9638,9643],{},[92,9639,9640],{},[95,9641,9642],{},"ω",[100,9644,9645],{"encoding":102},"\\omega",[76,9647,9649],{"className":9648,"ariaHidden":108},[107],[76,9650,9652,9655],{"className":9651},[112],[76,9653],{"className":9654,"style":249},[116],[76,9656,9642],{"className":9657,"style":253},[121,216]," 回繰り返します。",[76,9660,9662],{"className":9661},[2600],[76,9663,9665,9738],{"className":9664},[79],[76,9666,9668],{"className":9667},[83],[85,9669,9670],{"xmlns":87,"display":2609},[89,9671,9672,9735],{},[92,9673,9674,9676,9678,9684,9687,9693,9695,9701,9703,9705,9707,9709,9712,9719,9721,9729],{},[95,9675,98],{"mathvariant":97},[299,9677,602],{},[2628,9679,9680,9682],{},[95,9681,98],{"mathvariant":97},[2634,9683,3422],{},[299,9685,9686],{},"⊆",[2628,9688,9689,9691],{},[95,9690,98],{"mathvariant":97},[2634,9692,2636],{},[299,9694,9686],{},[2628,9696,9697,9699],{},[95,9698,98],{"mathvariant":97},[2634,9700,3439],{},[299,9702,9686],{},[299,9704,2639],{},[2617,9706,5223],{},[299,9708,2687],{"separator":108},[323,9710],{"width":9711},"2em",[6715,9713,9714,9716],{},[95,9715,98],{"mathvariant":97},[299,9717,9718],{},"∗",[299,9720,602],{},[9722,9723,9724,9727],"munder",{},[299,9725,9726],{},"⋃",[95,9728,2648],{},[2628,9730,9731,9733],{},[95,9732,98],{"mathvariant":97},[95,9734,2648],{},[100,9736,9737],{"encoding":102},"\\mathcal{L} = \\mathcal{L}_0 \\subseteq \\mathcal{L}_1 \\subseteq \\mathcal{L}_2 \\subseteq \\cdots, \\qquad \\mathcal{L}^* = \\bigcup_{k} \\mathcal{L}_k",[76,9739,9741,9759,9814,9869,9924,9990],{"className":9740,"ariaHidden":108},[107],[76,9742,9744,9747,9750,9753,9756],{"className":9743},[112],[76,9745],{"className":9746,"style":117},[116],[76,9748,98],{"className":9749},[121,122],[76,9751],{"className":9752,"style":324},[323],[76,9754,602],{"className":9755},[328],[76,9757],{"className":9758,"style":324},[323],[76,9760,9762,9765,9805,9808,9811],{"className":9761},[112],[76,9763],{"className":9764,"style":4583},[116],[76,9766,9768,9771],{"className":9767},[121],[76,9769,98],{"className":9770},[121,122],[76,9772,9774],{"className":9773},[2770],[76,9775,9777,9797],{"className":9776},[2774,2775],[76,9778,9780,9794],{"className":9779},[2779],[76,9781,9783],{"className":9782,"style":2784},[2783],[76,9784,9785,9788],{"style":2787},[76,9786],{"className":9787,"style":2792},[2791],[76,9789,9791],{"className":9790},[2796,2797,2798,2799],[76,9792,3422],{"className":9793},[121,2799],[76,9795,2807],{"className":9796},[2806],[76,9798,9800],{"className":9799},[2779],[76,9801,9803],{"className":9802,"style":2814},[2783],[76,9804],{},[76,9806],{"className":9807,"style":324},[323],[76,9809,9686],{"className":9810},[328],[76,9812],{"className":9813,"style":324},[323],[76,9815,9817,9820,9860,9863,9866],{"className":9816},[112],[76,9818],{"className":9819,"style":4583},[116],[76,9821,9823,9826],{"className":9822},[121],[76,9824,98],{"className":9825},[121,122],[76,9827,9829],{"className":9828},[2770],[76,9830,9832,9852],{"className":9831},[2774,2775],[76,9833,9835,9849],{"className":9834},[2779],[76,9836,9838],{"className":9837,"style":2784},[2783],[76,9839,9840,9843],{"style":2787},[76,9841],{"className":9842,"style":2792},[2791],[76,9844,9846],{"className":9845},[2796,2797,2798,2799],[76,9847,2636],{"className":9848},[121,2799],[76,9850,2807],{"className":9851},[2806],[76,9853,9855],{"className":9854},[2779],[76,9856,9858],{"className":9857,"style":2814},[2783],[76,9859],{},[76,9861],{"className":9862,"style":324},[323],[76,9864,9686],{"className":9865},[328],[76,9867],{"className":9868,"style":324},[323],[76,9870,9872,9875,9915,9918,9921],{"className":9871},[112],[76,9873],{"className":9874,"style":4583},[116],[76,9876,9878,9881],{"className":9877},[121],[76,9879,98],{"className":9880},[121,122],[76,9882,9884],{"className":9883},[2770],[76,9885,9887,9907],{"className":9886},[2774,2775],[76,9888,9890,9904],{"className":9889},[2779],[76,9891,9893],{"className":9892,"style":2784},[2783],[76,9894,9895,9898],{"style":2787},[76,9896],{"className":9897,"style":2792},[2791],[76,9899,9901],{"className":9900},[2796,2797,2798,2799],[76,9902,3439],{"className":9903},[121,2799],[76,9905,2807],{"className":9906},[2806],[76,9908,9910],{"className":9909},[2779],[76,9911,9913],{"className":9912,"style":2814},[2783],[76,9914],{},[76,9916],{"className":9917,"style":324},[323],[76,9919,9686],{"className":9920},[328],[76,9922],{"className":9923,"style":324},[323],[76,9925,9927,9931,9934,9937,9940,9943,9947,9950,9981,9984,9987],{"className":9926},[112],[76,9928],{"className":9929,"style":9930},[116],"height:0.9331em;vertical-align:-0.1944em;",[76,9932,2639],{"className":9933},[2824],[76,9935],{"className":9936,"style":2820},[323],[76,9938],{"className":9939,"style":2820},[323],[76,9941,2687],{"className":9942},[3025],[76,9944],{"className":9945,"style":9946},[323],"margin-right:2em;",[76,9948],{"className":9949,"style":2820},[323],[76,9951,9953,9956],{"className":9952},[121],[76,9954,98],{"className":9955},[121,122],[76,9957,9959],{"className":9958},[2770],[76,9960,9962],{"className":9961},[2774],[76,9963,9965],{"className":9964},[2779],[76,9966,9969],{"className":9967,"style":9968},[2783],"height:0.7387em;",[76,9970,9972,9975],{"style":9971},"top:-3.113em;margin-right:0.05em;",[76,9973],{"className":9974,"style":2792},[2791],[76,9976,9978],{"className":9977},[2796,2797,2798,2799],[76,9979,9718],{"className":9980},[1188,2799],[76,9982],{"className":9983,"style":324},[323],[76,9985,602],{"className":9986},[328],[76,9988],{"className":9989,"style":324},[323],[76,9991,9993,9997,10053,10056],{"className":9992},[112],[76,9994],{"className":9995,"style":9996},[116],"height:2.3521em;vertical-align:-1.3021em;",[76,9998,10002],{"className":9999},[10000,10001],"mop","op-limits",[76,10003,10005,10044],{"className":10004},[2774,2775],[76,10006,10008,10041],{"className":10007},[2779],[76,10009,10012,10028],{"className":10010,"style":10011},[2783],"height:1.05em;",[76,10013,10015,10019],{"style":10014},"top:-1.8479em;margin-left:0em;",[76,10016],{"className":10017,"style":10018},[2791],"height:3.05em;",[76,10020,10022],{"className":10021},[2796,2797,2798,2799],[76,10023,10025],{"className":10024},[121,2799],[76,10026,2648],{"className":10027,"style":2861},[121,216,2799],[76,10029,10031,10034],{"style":10030},"top:-3.05em;",[76,10032],{"className":10033,"style":10018},[2791],[76,10035,10036],{},[76,10037,9726],{"className":10038},[10000,10039,10040],"op-symbol","large-op",[76,10042,2807],{"className":10043},[2806],[76,10045,10047],{"className":10046},[2779],[76,10048,10051],{"className":10049,"style":10050},[2783],"height:1.3021em;",[76,10052],{},[76,10054],{"className":10055,"style":2820},[323],[76,10057,10059,10062],{"className":10058},[121],[76,10060,98],{"className":10061},[121,122],[76,10063,10065],{"className":10064},[2770],[76,10066,10068,10088],{"className":10067},[2774,2775],[76,10069,10071,10085],{"className":10070},[2779],[76,10072,10074],{"className":10073,"style":2849},[2783],[76,10075,10076,10079],{"style":2787},[76,10077],{"className":10078,"style":2792},[2791],[76,10080,10082],{"className":10081},[2796,2797,2798,2799],[76,10083,2648],{"className":10084,"style":2861},[121,216,2799],[76,10086,2807],{"className":10087},[2806],[76,10089,10091],{"className":10090},[2779],[76,10092,10094],{"className":10093,"style":2814},[2783],[76,10095],{},[10,10097,10098,2284,10186,10256,10257,10287,10288,10339,10340,10390],{},[76,10099,10101,10126],{"className":10100},[79],[76,10102,10104],{"className":10103},[83],[85,10105,10106],{"xmlns":87},[89,10107,10108,10123],{},[92,10109,10110],{},[2628,10111,10112,10114],{},[95,10113,98],{"mathvariant":97},[92,10115,10116,10118,10121],{},[95,10117,2648],{},[299,10119,10120],{},"+",[2634,10122,2636],{},[100,10124,10125],{"encoding":102},"\\mathcal{L}_{k+1}",[76,10127,10129],{"className":10128,"ariaHidden":108},[107],[76,10130,10132,10136],{"className":10131},[112],[76,10133],{"className":10134,"style":10135},[116],"height:0.8917em;vertical-align:-0.2083em;",[76,10137,10139,10142],{"className":10138},[121],[76,10140,98],{"className":10141},[121,122],[76,10143,10145],{"className":10144},[2770],[76,10146,10148,10177],{"className":10147},[2774,2775],[76,10149,10151,10174],{"className":10150},[2779],[76,10152,10154],{"className":10153,"style":2849},[2783],[76,10155,10156,10159],{"style":2787},[76,10157],{"className":10158,"style":2792},[2791],[76,10160,10162],{"className":10161},[2796,2797,2798,2799],[76,10163,10165,10168,10171],{"className":10164},[121,2799],[76,10166,2648],{"className":10167,"style":2861},[121,216,2799],[76,10169,10120],{"className":10170},[1188,2799],[76,10172,2636],{"className":10173},[121,2799],[76,10175,2807],{"className":10176},[2806],[76,10178,10180],{"className":10179},[2779],[76,10181,10184],{"className":10182,"style":10183},[2783],"height:0.2083em;",[76,10185],{},[76,10187,10189,10207],{"className":10188},[79],[76,10190,10192],{"className":10191},[83],[85,10193,10194],{"xmlns":87},[89,10195,10196,10204],{},[92,10197,10198],{},[2628,10199,10200,10202],{},[95,10201,98],{"mathvariant":97},[95,10203,2648],{},[100,10205,10206],{"encoding":102},"\\mathcal{L}_k",[76,10208,10210],{"className":10209,"ariaHidden":108},[107],[76,10211,10213,10216],{"className":10212},[112],[76,10214],{"className":10215,"style":4583},[116],[76,10217,10219,10222],{"className":10218},[121],[76,10220,98],{"className":10221},[121,122],[76,10223,10225],{"className":10224},[2770],[76,10226,10228,10248],{"className":10227},[2774,2775],[76,10229,10231,10245],{"className":10230},[2779],[76,10232,10234],{"className":10233,"style":2849},[2783],[76,10235,10236,10239],{"style":2787},[76,10237],{"className":10238,"style":2792},[2791],[76,10240,10242],{"className":10241},[2796,2797,2798,2799],[76,10243,2648],{"className":10244,"style":2861},[121,216,2799],[76,10246,2807],{"className":10247},[2806],[76,10249,10251],{"className":10250},[2779],[76,10252,10254],{"className":10253,"style":2814},[2783],[76,10255],{}," の存在文ごとに新しい定数を加えた言語です。加えたヘンキン公理全体を ",[76,10258,10260,10274],{"className":10259},[79],[76,10261,10263],{"className":10262},[83],[85,10264,10265],{"xmlns":87},[89,10266,10267,10272],{},[92,10268,10269],{},[95,10270,10271],{},"H",[100,10273,10271],{"encoding":102},[76,10275,10277],{"className":10276,"ariaHidden":108},[107],[76,10278,10280,10283],{"className":10279},[112],[76,10281],{"className":10282,"style":117},[116],[76,10284,10271],{"className":10285,"style":10286},[121,216],"margin-right:0.0813em;"," とします。",[76,10289,10291,10309],{"className":10290},[79],[76,10292,10294],{"className":10293},[83],[85,10295,10296],{"xmlns":87},[89,10297,10298,10306],{},[92,10299,10300,10302,10304],{},[95,10301,364],{"mathvariant":363},[299,10303,1151],{},[95,10305,10271],{},[100,10307,10308],{"encoding":102},"\\Gamma \\cup H",[76,10310,10312,10330],{"className":10311,"ariaHidden":108},[107],[76,10313,10315,10318,10321,10324,10327],{"className":10314},[112],[76,10316],{"className":10317,"style":117},[116],[76,10319,364],{"className":10320},[121],[76,10322],{"className":10323,"style":1184},[323],[76,10325,1151],{"className":10326},[1188],[76,10328],{"className":10329,"style":1184},[323],[76,10331,10333,10336],{"className":10332},[112],[76,10334],{"className":10335,"style":117},[116],[76,10337,10271],{"className":10338,"style":10286},[121,216]," の有限部分には有限個のヘンキン公理しか含まれないので、補題1を1つずつ（定数が導入された順に）適用すれば、",[76,10341,10343,10360],{"className":10342},[79],[76,10344,10346],{"className":10345},[83],[85,10347,10348],{"xmlns":87},[89,10349,10350,10358],{},[92,10351,10352,10354,10356],{},[95,10353,364],{"mathvariant":363},[299,10355,1151],{},[95,10357,10271],{},[100,10359,10308],{"encoding":102},[76,10361,10363,10381],{"className":10362,"ariaHidden":108},[107],[76,10364,10366,10369,10372,10375,10378],{"className":10365},[112],[76,10367],{"className":10368,"style":117},[116],[76,10370,364],{"className":10371},[121],[76,10373],{"className":10374,"style":1184},[323],[76,10376,1151],{"className":10377},[1188],[76,10379],{"className":10380,"style":1184},[323],[76,10382,10384,10387],{"className":10383},[112],[76,10385],{"className":10386,"style":117},[116],[76,10388,10271],{"className":10389,"style":10286},[121,216]," は無矛盾です。",[68,10392,10394],{"id":10393},"_32-ステップ2極大無矛盾集合に拡大する","3.2 ステップ2：極大無矛盾集合に拡大する",[1309,10396,10398],{"title":10397},"補題 2：リンデンバウムの補題",[10,10399,10400,10401,10404,10405,10433,10434,922,10462,10496],{},"無矛盾な集合は、極大無矛盾集合に拡大できる。ここで ",[13,10402,10403],{},"極大"," とは、どの文 ",[76,10406,10408,10421],{"className":10407},[79],[76,10409,10411],{"className":10410},[83],[85,10412,10413],{"xmlns":87},[89,10414,10415,10419],{},[92,10416,10417],{},[95,10418,236],{},[100,10420,239],{"encoding":102},[76,10422,10424],{"className":10423,"ariaHidden":108},[107],[76,10425,10427,10430],{"className":10426},[112],[76,10428],{"className":10429,"style":249},[116],[76,10431,236],{"className":10432,"style":253},[121,216]," についても、",[76,10435,10437,10450],{"className":10436},[79],[76,10438,10440],{"className":10439},[83],[85,10441,10442],{"xmlns":87},[89,10443,10444,10448],{},[92,10445,10446],{},[95,10447,236],{},[100,10449,239],{"encoding":102},[76,10451,10453],{"className":10452,"ariaHidden":108},[107],[76,10454,10456,10459],{"className":10455},[112],[76,10457],{"className":10458,"style":249},[116],[76,10460,236],{"className":10461,"style":253},[121,216],[76,10463,10465,10481],{"className":10464},[79],[76,10466,10468],{"className":10467},[83],[85,10469,10470],{"xmlns":87},[89,10471,10472,10478],{},[92,10473,10474,10476],{},[95,10475,941],{"mathvariant":363},[95,10477,236],{},[100,10479,10480],{"encoding":102},"\\neg\\sigma",[76,10482,10484],{"className":10483,"ariaHidden":108},[107],[76,10485,10487,10490,10493],{"className":10486},[112],[76,10488],{"className":10489,"style":249},[116],[76,10491,941],{"className":10492},[121],[76,10494,236],{"className":10495,"style":253},[121,216]," のどちらか一方を必ず含むことをいう。",[1571,10498,10499,10869,11517,12224],{},[10,10500,10501,10561,10562,10753,10754,10868],{},[76,10502,10504,10522],{"className":10503},[79],[76,10505,10507],{"className":10506},[83],[85,10508,10509],{"xmlns":87},[89,10510,10511,10519],{},[92,10512,10513],{},[6715,10514,10515,10517],{},[95,10516,98],{"mathvariant":97},[299,10518,9718],{},[100,10520,10521],{"encoding":102},"\\mathcal{L}^*",[76,10523,10525],{"className":10524,"ariaHidden":108},[107],[76,10526,10528,10532],{"className":10527},[112],[76,10529],{"className":10530,"style":10531},[116],"height:0.6887em;",[76,10533,10535,10538],{"className":10534},[121],[76,10536,98],{"className":10537},[121,122],[76,10539,10541],{"className":10540},[2770],[76,10542,10544],{"className":10543},[2774],[76,10545,10547],{"className":10546},[2779],[76,10548,10550],{"className":10549,"style":10531},[2783],[76,10551,10552,10555],{"style":6787},[76,10553],{"className":10554,"style":2792},[2791],[76,10556,10558],{"className":10557},[2796,2797,2798,2799],[76,10559,9718],{"className":10560},[1188,2799]," は可算なので、その文を ",[76,10563,10565,10603],{"className":10564},[79],[76,10566,10568],{"className":10567},[83],[85,10569,10570],{"xmlns":87},[89,10571,10572,10600],{},[92,10573,10574,10580,10582,10588,10590,10596,10598],{},[2628,10575,10576,10578],{},[95,10577,236],{},[2634,10579,3422],{},[299,10581,2687],{"separator":108},[2628,10583,10584,10586],{},[95,10585,236],{},[2634,10587,2636],{},[299,10589,2687],{"separator":108},[2628,10591,10592,10594],{},[95,10593,236],{},[2634,10595,3439],{},[299,10597,2687],{"separator":108},[299,10599,2690],{},[100,10601,10602],{"encoding":102},"\\sigma_0, \\sigma_1, \\sigma_2, \\dots",[76,10604,10606],{"className":10605,"ariaHidden":108},[107],[76,10607,10609,10612,10652,10655,10658,10698,10701,10704,10744,10747,10750],{"className":10608},[112],[76,10610],{"className":10611,"style":469},[116],[76,10613,10615,10618],{"className":10614},[121],[76,10616,236],{"className":10617,"style":253},[121,216],[76,10619,10621],{"className":10620},[2770],[76,10622,10624,10644],{"className":10623},[2774,2775],[76,10625,10627,10641],{"className":10626},[2779],[76,10628,10630],{"className":10629,"style":2784},[2783],[76,10631,10632,10635],{"style":2899},[76,10633],{"className":10634,"style":2792},[2791],[76,10636,10638],{"className":10637},[2796,2797,2798,2799],[76,10639,3422],{"className":10640},[121,2799],[76,10642,2807],{"className":10643},[2806],[76,10645,10647],{"className":10646},[2779],[76,10648,10650],{"className":10649,"style":2814},[2783],[76,10651],{},[76,10653,2687],{"className":10654},[3025],[76,10656],{"className":10657,"style":2820},[323],[76,10659,10661,10664],{"className":10660},[121],[76,10662,236],{"className":10663,"style":253},[121,216],[76,10665,10667],{"className":10666},[2770],[76,10668,10670,10690],{"className":10669},[2774,2775],[76,10671,10673,10687],{"className":10672},[2779],[76,10674,10676],{"className":10675,"style":2784},[2783],[76,10677,10678,10681],{"style":2899},[76,10679],{"className":10680,"style":2792},[2791],[76,10682,10684],{"className":10683},[2796,2797,2798,2799],[76,10685,2636],{"className":10686},[121,2799],[76,10688,2807],{"className":10689},[2806],[76,10691,10693],{"className":10692},[2779],[76,10694,10696],{"className":10695,"style":2814},[2783],[76,10697],{},[76,10699,2687],{"className":10700},[3025],[76,10702],{"className":10703,"style":2820},[323],[76,10705,10707,10710],{"className":10706},[121],[76,10708,236],{"className":10709,"style":253},[121,216],[76,10711,10713],{"className":10712},[2770],[76,10714,10716,10736],{"className":10715},[2774,2775],[76,10717,10719,10733],{"className":10718},[2779],[76,10720,10722],{"className":10721,"style":2784},[2783],[76,10723,10724,10727],{"style":2899},[76,10725],{"className":10726,"style":2792},[2791],[76,10728,10730],{"className":10729},[2796,2797,2798,2799],[76,10731,3439],{"className":10732},[121,2799],[76,10734,2807],{"className":10735},[2806],[76,10737,10739],{"className":10738},[2779],[76,10740,10742],{"className":10741,"style":2814},[2783],[76,10743],{},[76,10745,2687],{"className":10746},[3025],[76,10748],{"className":10749,"style":2820},[323],[76,10751,2690],{"className":10752},[2824]," と並べられる。",[76,10755,10757,10783],{"className":10756},[79],[76,10758,10760],{"className":10759},[83],[85,10761,10762],{"xmlns":87},[89,10763,10764,10780],{},[92,10765,10766,10772,10774,10776,10778],{},[2628,10767,10768,10770],{},[95,10769,8182],{"mathvariant":363},[2634,10771,3422],{},[299,10773,602],{},[95,10775,364],{"mathvariant":363},[299,10777,1151],{},[95,10779,10271],{},[100,10781,10782],{"encoding":102},"\\Delta_0 = \\Gamma \\cup H",[76,10784,10786,10841,10859],{"className":10785,"ariaHidden":108},[107],[76,10787,10789,10792,10832,10835,10838],{"className":10788},[112],[76,10790],{"className":10791,"style":4583},[116],[76,10793,10795,10798],{"className":10794},[121],[76,10796,8182],{"className":10797},[121],[76,10799,10801],{"className":10800},[2770],[76,10802,10804,10824],{"className":10803},[2774,2775],[76,10805,10807,10821],{"className":10806},[2779],[76,10808,10810],{"className":10809,"style":2784},[2783],[76,10811,10812,10815],{"style":2787},[76,10813],{"className":10814,"style":2792},[2791],[76,10816,10818],{"className":10817},[2796,2797,2798,2799],[76,10819,3422],{"className":10820},[121,2799],[76,10822,2807],{"className":10823},[2806],[76,10825,10827],{"className":10826},[2779],[76,10828,10830],{"className":10829,"style":2814},[2783],[76,10831],{},[76,10833],{"className":10834,"style":324},[323],[76,10836,602],{"className":10837},[328],[76,10839],{"className":10840,"style":324},[323],[76,10842,10844,10847,10850,10853,10856],{"className":10843},[112],[76,10845],{"className":10846,"style":117},[116],[76,10848,364],{"className":10849},[121],[76,10851],{"className":10852,"style":1184},[323],[76,10854,1151],{"className":10855},[1188],[76,10857],{"className":10858,"style":1184},[323],[76,10860,10862,10865],{"className":10861},[112],[76,10863],{"className":10864,"style":117},[116],[76,10866,10271],{"className":10867,"style":10286},[121,216]," から始めて、",[76,10870,10872],{"className":10871},[2600],[76,10873,10875,11012],{"className":10874},[79],[76,10876,10878],{"className":10877},[83],[85,10879,10880],{"xmlns":87,"display":2609},[89,10881,10882,11009],{},[92,10883,10884,10896,10898],{},[2628,10885,10886,10888],{},[95,10887,8182],{"mathvariant":363},[92,10889,10890,10892,10894],{},[95,10891,4400],{},[299,10893,10120],{},[2634,10895,2636],{},[299,10897,602],{},[92,10899,10900,10902],{},[299,10901,1155],{"fence":108},[10903,10904,10908,10968],"mtable",{"rowspacing":10905,"columnalign":10906,"columnspacing":10907},"0.36em","left left","1em",[10909,10910,10911,10937],"mtr",{},[10912,10913,10914],"mtd",{},[10915,10916,10917],"mstyle",{"scriptlevel":3422,"displaystyle":1154},[92,10918,10919,10925,10927,10929,10935],{},[2628,10920,10921,10923],{},[95,10922,8182],{"mathvariant":363},[95,10924,4400],{},[299,10926,1151],{},[299,10928,1155],{"stretchy":1154},[2628,10930,10931,10933],{},[95,10932,236],{},[95,10934,4400],{},[299,10936,1161],{"stretchy":1154},[10912,10938,10939],{},[10915,10940,10941],{"scriptlevel":3422,"displaystyle":1154},[92,10942,10943,10945,10951,10953,10955,10961,10963,10966],{},[299,10944,2678],{"stretchy":1154},[2628,10946,10947,10949],{},[95,10948,8182],{"mathvariant":363},[95,10950,4400],{},[299,10952,1151],{},[299,10954,1155],{"stretchy":1154},[2628,10956,10957,10959],{},[95,10958,236],{},[95,10960,4400],{},[299,10962,1161],{"stretchy":1154},[2617,10964,10965],{}," が無矛盾のとき",[299,10967,2721],{"stretchy":1154},[10909,10969,10970,10996],{},[10912,10971,10972],{},[10915,10973,10974],{"scriptlevel":3422,"displaystyle":1154},[92,10975,10976,10982,10984,10986,10988,10994],{},[2628,10977,10978,10980],{},[95,10979,8182],{"mathvariant":363},[95,10981,4400],{},[299,10983,1151],{},[299,10985,1155],{"stretchy":1154},[95,10987,941],{"mathvariant":363},[2628,10989,10990,10992],{},[95,10991,236],{},[95,10993,4400],{},[299,10995,1161],{"stretchy":1154},[10912,10997,10998],{},[10915,10999,11000],{"scriptlevel":3422,"displaystyle":1154},[92,11001,11002,11004,11007],{},[299,11003,2678],{"stretchy":1154},[2617,11005,11006],{},"そうでないとき",[299,11008,2721],{"stretchy":1154},[100,11010,11011],{"encoding":102},"\\Delta_{n+1} =\n\\begin{cases}\n\\Delta_n \\cup \\{\\sigma_n\\} & (\\Delta_n \\cup \\{\\sigma_n\\} \\text{ が無矛盾のとき}) \\\\\n\\Delta_n \\cup \\{\\neg\\sigma_n\\} & (\\text{そうでないとき})\n\\end{cases}",[76,11013,11015,11079],{"className":11014,"ariaHidden":108},[107],[76,11016,11018,11021,11070,11073,11076],{"className":11017},[112],[76,11019],{"className":11020,"style":10135},[116],[76,11022,11024,11027],{"className":11023},[121],[76,11025,8182],{"className":11026},[121],[76,11028,11030],{"className":11029},[2770],[76,11031,11033,11062],{"className":11032},[2774,2775],[76,11034,11036,11059],{"className":11035},[2779],[76,11037,11039],{"className":11038,"style":2784},[2783],[76,11040,11041,11044],{"style":2787},[76,11042],{"className":11043,"style":2792},[2791],[76,11045,11047],{"className":11046},[2796,2797,2798,2799],[76,11048,11050,11053,11056],{"className":11049},[121,2799],[76,11051,4400],{"className":11052},[121,216,2799],[76,11054,10120],{"className":11055},[1188,2799],[76,11057,2636],{"className":11058},[121,2799],[76,11060,2807],{"className":11061},[2806],[76,11063,11065],{"className":11064},[2779],[76,11066,11068],{"className":11067,"style":10183},[2783],[76,11069],{},[76,11071],{"className":11072,"style":324},[323],[76,11074,602],{"className":11075},[328],[76,11077],{"className":11078,"style":324},[323],[76,11080,11082,11086],{"className":11081},[112],[76,11083],{"className":11084,"style":11085},[116],"height:3em;vertical-align:-1.25em;",[76,11087,11089,11099,11513],{"className":11088},[2824],[76,11090,11094],{"className":11091,"style":11093},[1202,11092],"delimcenter","top:0em;",[76,11095,1155],{"className":11096},[11097,11098],"delimsizing","size4",[76,11100,11102],{"className":11101},[121],[76,11103,11105,11343,11348],{"className":11104},[10903],[76,11106,11109],{"className":11107},[11108],"col-align-l",[76,11110,11112,11334],{"className":11111},[2774,2775],[76,11113,11115,11331],{"className":11114},[2779],[76,11116,11119,11224],{"className":11117,"style":11118},[2783],"height:1.69em;",[76,11120,11122,11126],{"style":11121},"top:-3.69em;",[76,11123],{"className":11124,"style":11125},[2791],"height:3.008em;",[76,11127,11129,11169,11172,11175,11178,11181,11221],{"className":11128},[121],[76,11130,11132,11135],{"className":11131},[121],[76,11133,8182],{"className":11134},[121],[76,11136,11138],{"className":11137},[2770],[76,11139,11141,11161],{"className":11140},[2774,2775],[76,11142,11144,11158],{"className":11143},[2779],[76,11145,11147],{"className":11146,"style":2950},[2783],[76,11148,11149,11152],{"style":2787},[76,11150],{"className":11151,"style":2792},[2791],[76,11153,11155],{"className":11154},[2796,2797,2798,2799],[76,11156,4400],{"className":11157},[121,216,2799],[76,11159,2807],{"className":11160},[2806],[76,11162,11164],{"className":11163},[2779],[76,11165,11167],{"className":11166,"style":2814},[2783],[76,11168],{},[76,11170],{"className":11171,"style":1184},[323],[76,11173,1151],{"className":11174},[1188],[76,11176],{"className":11177,"style":1184},[323],[76,11179,1155],{"className":11180},[1202],[76,11182,11184,11187],{"className":11183},[121],[76,11185,236],{"className":11186,"style":253},[121,216],[76,11188,11190],{"className":11189},[2770],[76,11191,11193,11213],{"className":11192},[2774,2775],[76,11194,11196,11210],{"className":11195},[2779],[76,11197,11199],{"className":11198,"style":2950},[2783],[76,11200,11201,11204],{"style":2899},[76,11202],{"className":11203,"style":2792},[2791],[76,11205,11207],{"className":11206},[2796,2797,2798,2799],[76,11208,4400],{"className":11209},[121,216,2799],[76,11211,2807],{"className":11212},[2806],[76,11214,11216],{"className":11215},[2779],[76,11217,11219],{"className":11218,"style":2814},[2783],[76,11220],{},[76,11222,1161],{"className":11223},[1209],[76,11225,11227,11230],{"style":11226},"top:-2.25em;",[76,11228],{"className":11229,"style":11125},[2791],[76,11231,11233,11273,11276,11279,11282,11285,11288,11328],{"className":11232},[121],[76,11234,11236,11239],{"className":11235},[121],[76,11237,8182],{"className":11238},[121],[76,11240,11242],{"className":11241},[2770],[76,11243,11245,11265],{"className":11244},[2774,2775],[76,11246,11248,11262],{"className":11247},[2779],[76,11249,11251],{"className":11250,"style":2950},[2783],[76,11252,11253,11256],{"style":2787},[76,11254],{"className":11255,"style":2792},[2791],[76,11257,11259],{"className":11258},[2796,2797,2798,2799],[76,11260,4400],{"className":11261},[121,216,2799],[76,11263,2807],{"className":11264},[2806],[76,11266,11268],{"className":11267},[2779],[76,11269,11271],{"className":11270,"style":2814},[2783],[76,11272],{},[76,11274],{"className":11275,"style":1184},[323],[76,11277,1151],{"className":11278},[1188],[76,11280],{"className":11281,"style":1184},[323],[76,11283,1155],{"className":11284},[1202],[76,11286,941],{"className":11287},[121],[76,11289,11291,11294],{"className":11290},[121],[76,11292,236],{"className":11293,"style":253},[121,216],[76,11295,11297],{"className":11296},[2770],[76,11298,11300,11320],{"className":11299},[2774,2775],[76,11301,11303,11317],{"className":11302},[2779],[76,11304,11306],{"className":11305,"style":2950},[2783],[76,11307,11308,11311],{"style":2899},[76,11309],{"className":11310,"style":2792},[2791],[76,11312,11314],{"className":11313},[2796,2797,2798,2799],[76,11315,4400],{"className":11316},[121,216,2799],[76,11318,2807],{"className":11319},[2806],[76,11321,11323],{"className":11322},[2779],[76,11324,11326],{"className":11325,"style":2814},[2783],[76,11327],{},[76,11329,1161],{"className":11330},[1209],[76,11332,2807],{"className":11333},[2806],[76,11335,11337],{"className":11336},[2779],[76,11338,11341],{"className":11339,"style":11340},[2783],"height:1.19em;",[76,11342],{},[76,11344],{"className":11345,"style":11347},[11346],"arraycolsep","width:1em;",[76,11349,11351],{"className":11350},[11108],[76,11352,11354,11505],{"className":11353},[2774,2775],[76,11355,11357,11502],{"className":11356},[2779],[76,11358,11360,11482],{"className":11359,"style":11118},[2783],[76,11361,11362,11365],{"style":11121},[76,11363],{"className":11364,"style":11125},[2791],[76,11366,11368,11371,11411,11414,11417,11420,11423,11463,11466,11479],{"className":11367},[121],[76,11369,2678],{"className":11370},[1202],[76,11372,11374,11377],{"className":11373},[121],[76,11375,8182],{"className":11376},[121],[76,11378,11380],{"className":11379},[2770],[76,11381,11383,11403],{"className":11382},[2774,2775],[76,11384,11386,11400],{"className":11385},[2779],[76,11387,11389],{"className":11388,"style":2950},[2783],[76,11390,11391,11394],{"style":2787},[76,11392],{"className":11393,"style":2792},[2791],[76,11395,11397],{"className":11396},[2796,2797,2798,2799],[76,11398,4400],{"className":11399},[121,216,2799],[76,11401,2807],{"className":11402},[2806],[76,11404,11406],{"className":11405},[2779],[76,11407,11409],{"className":11408,"style":2814},[2783],[76,11410],{},[76,11412],{"className":11413,"style":1184},[323],[76,11415,1151],{"className":11416},[1188],[76,11418],{"className":11419,"style":1184},[323],[76,11421,1155],{"className":11422},[1202],[76,11424,11426,11429],{"className":11425},[121],[76,11427,236],{"className":11428,"style":253},[121,216],[76,11430,11432],{"className":11431},[2770],[76,11433,11435,11455],{"className":11434},[2774,2775],[76,11436,11438,11452],{"className":11437},[2779],[76,11439,11441],{"className":11440,"style":2950},[2783],[76,11442,11443,11446],{"style":2899},[76,11444],{"className":11445,"style":2792},[2791],[76,11447,11449],{"className":11448},[2796,2797,2798,2799],[76,11450,4400],{"className":11451},[121,216,2799],[76,11453,2807],{"className":11454},[2806],[76,11456,11458],{"className":11457},[2779],[76,11459,11461],{"className":11460,"style":2814},[2783],[76,11462],{},[76,11464,1161],{"className":11465},[1209],[76,11467,11470,11474],{"className":11468},[121,11469],"text",[76,11471,11473],{"className":11472},[121]," ",[76,11475,11478],{"className":11476},[121,11477],"cjk_fallback","が無矛盾のとき",[76,11480,2721],{"className":11481},[1209],[76,11483,11484,11487],{"style":11226},[76,11485],{"className":11486,"style":11125},[2791],[76,11488,11490,11493,11499],{"className":11489},[121],[76,11491,2678],{"className":11492},[1202],[76,11494,11496],{"className":11495},[121,11469],[76,11497,11006],{"className":11498},[121,11477],[76,11500,2721],{"className":11501},[1209],[76,11503,2807],{"className":11504},[2806],[76,11506,11508],{"className":11507},[2779],[76,11509,11511],{"className":11510,"style":11340},[2783],[76,11512],{},[76,11514],{"className":11515},[1209,11516],"nulldelimiter",[10,11518,11519,11520,922,11663,11811,11812,9261,11950,9595,12083,12153,12154,12223],{},"と定める。",[76,11521,11523,11553],{"className":11522},[79],[76,11524,11526],{"className":11525},[83],[85,11527,11528],{"xmlns":87},[89,11529,11530,11550],{},[92,11531,11532,11538,11540,11542,11548],{},[2628,11533,11534,11536],{},[95,11535,8182],{"mathvariant":363},[95,11537,4400],{},[299,11539,1151],{},[299,11541,1155],{"stretchy":1154},[2628,11543,11544,11546],{},[95,11545,236],{},[95,11547,4400],{},[299,11549,1161],{"stretchy":1154},[100,11551,11552],{"encoding":102},"\\Delta_n \\cup \\{\\sigma_n\\}",[76,11554,11556,11611],{"className":11555,"ariaHidden":108},[107],[76,11557,11559,11562,11602,11605,11608],{"className":11558},[112],[76,11560],{"className":11561,"style":4583},[116],[76,11563,11565,11568],{"className":11564},[121],[76,11566,8182],{"className":11567},[121],[76,11569,11571],{"className":11570},[2770],[76,11572,11574,11594],{"className":11573},[2774,2775],[76,11575,11577,11591],{"className":11576},[2779],[76,11578,11580],{"className":11579,"style":2950},[2783],[76,11581,11582,11585],{"style":2787},[76,11583],{"className":11584,"style":2792},[2791],[76,11586,11588],{"className":11587},[2796,2797,2798,2799],[76,11589,4400],{"className":11590},[121,216,2799],[76,11592,2807],{"className":11593},[2806],[76,11595,11597],{"className":11596},[2779],[76,11598,11600],{"className":11599,"style":2814},[2783],[76,11601],{},[76,11603],{"className":11604,"style":1184},[323],[76,11606,1151],{"className":11607},[1188],[76,11609],{"className":11610,"style":1184},[323],[76,11612,11614,11617,11620,11660],{"className":11613},[112],[76,11615],{"className":11616,"style":1198},[116],[76,11618,1155],{"className":11619},[1202],[76,11621,11623,11626],{"className":11622},[121],[76,11624,236],{"className":11625,"style":253},[121,216],[76,11627,11629],{"className":11628},[2770],[76,11630,11632,11652],{"className":11631},[2774,2775],[76,11633,11635,11649],{"className":11634},[2779],[76,11636,11638],{"className":11637,"style":2950},[2783],[76,11639,11640,11643],{"style":2899},[76,11641],{"className":11642,"style":2792},[2791],[76,11644,11646],{"className":11645},[2796,2797,2798,2799],[76,11647,4400],{"className":11648},[121,216,2799],[76,11650,2807],{"className":11651},[2806],[76,11653,11655],{"className":11654},[2779],[76,11656,11658],{"className":11657,"style":2814},[2783],[76,11659],{},[76,11661,1161],{"className":11662},[1209],[76,11664,11666,11698],{"className":11665},[79],[76,11667,11669],{"className":11668},[83],[85,11670,11671],{"xmlns":87},[89,11672,11673,11695],{},[92,11674,11675,11681,11683,11685,11687,11693],{},[2628,11676,11677,11679],{},[95,11678,8182],{"mathvariant":363},[95,11680,4400],{},[299,11682,1151],{},[299,11684,1155],{"stretchy":1154},[95,11686,941],{"mathvariant":363},[2628,11688,11689,11691],{},[95,11690,236],{},[95,11692,4400],{},[299,11694,1161],{"stretchy":1154},[100,11696,11697],{"encoding":102},"\\Delta_n \\cup \\{\\neg\\sigma_n\\}",[76,11699,11701,11756],{"className":11700,"ariaHidden":108},[107],[76,11702,11704,11707,11747,11750,11753],{"className":11703},[112],[76,11705],{"className":11706,"style":4583},[116],[76,11708,11710,11713],{"className":11709},[121],[76,11711,8182],{"className":11712},[121],[76,11714,11716],{"className":11715},[2770],[76,11717,11719,11739],{"className":11718},[2774,2775],[76,11720,11722,11736],{"className":11721},[2779],[76,11723,11725],{"className":11724,"style":2950},[2783],[76,11726,11727,11730],{"style":2787},[76,11728],{"className":11729,"style":2792},[2791],[76,11731,11733],{"className":11732},[2796,2797,2798,2799],[76,11734,4400],{"className":11735},[121,216,2799],[76,11737,2807],{"className":11738},[2806],[76,11740,11742],{"className":11741},[2779],[76,11743,11745],{"className":11744,"style":2814},[2783],[76,11746],{},[76,11748],{"className":11749,"style":1184},[323],[76,11751,1151],{"className":11752},[1188],[76,11754],{"className":11755,"style":1184},[323],[76,11757,11759,11762,11765,11768,11808],{"className":11758},[112],[76,11760],{"className":11761,"style":1198},[116],[76,11763,1155],{"className":11764},[1202],[76,11766,941],{"className":11767},[121],[76,11769,11771,11774],{"className":11770},[121],[76,11772,236],{"className":11773,"style":253},[121,216],[76,11775,11777],{"className":11776},[2770],[76,11778,11780,11800],{"className":11779},[2774,2775],[76,11781,11783,11797],{"className":11782},[2779],[76,11784,11786],{"className":11785,"style":2950},[2783],[76,11787,11788,11791],{"style":2899},[76,11789],{"className":11790,"style":2792},[2791],[76,11792,11794],{"className":11793},[2796,2797,2798,2799],[76,11795,4400],{"className":11796},[121,216,2799],[76,11798,2807],{"className":11799},[2806],[76,11801,11803],{"className":11802},[2779],[76,11804,11806],{"className":11805,"style":2814},[2783],[76,11807],{},[76,11809,1161],{"className":11810},[1209]," がどちらも矛盾するなら、演繹定理により ",[76,11813,11815,11843],{"className":11814},[79],[76,11816,11818],{"className":11817},[83],[85,11819,11820],{"xmlns":87},[89,11821,11822,11840],{},[92,11823,11824,11830,11832,11834],{},[2628,11825,11826,11828],{},[95,11827,8182],{"mathvariant":363},[95,11829,4400],{},[299,11831,805],{},[95,11833,941],{"mathvariant":363},[2628,11835,11836,11838],{},[95,11837,236],{},[95,11839,4400],{},[100,11841,11842],{"encoding":102},"\\Delta_n \\vdash \\neg\\sigma_n",[76,11844,11846,11901],{"className":11845,"ariaHidden":108},[107],[76,11847,11849,11852,11892,11895,11898],{"className":11848},[112],[76,11850],{"className":11851,"style":5254},[116],[76,11853,11855,11858],{"className":11854},[121],[76,11856,8182],{"className":11857},[121],[76,11859,11861],{"className":11860},[2770],[76,11862,11864,11884],{"className":11863},[2774,2775],[76,11865,11867,11881],{"className":11866},[2779],[76,11868,11870],{"className":11869,"style":2950},[2783],[76,11871,11872,11875],{"style":2787},[76,11873],{"className":11874,"style":2792},[2791],[76,11876,11878],{"className":11877},[2796,2797,2798,2799],[76,11879,4400],{"className":11880},[121,216,2799],[76,11882,2807],{"className":11883},[2806],[76,11885,11887],{"className":11886},[2779],[76,11888,11890],{"className":11889,"style":2814},[2783],[76,11891],{},[76,11893],{"className":11894,"style":324},[323],[76,11896,805],{"className":11897},[328],[76,11899],{"className":11900,"style":324},[323],[76,11902,11904,11907,11910],{"className":11903},[112],[76,11905],{"className":11906,"style":7840},[116],[76,11908,941],{"className":11909},[121],[76,11911,11913,11916],{"className":11912},[121],[76,11914,236],{"className":11915,"style":253},[121,216],[76,11917,11919],{"className":11918},[2770],[76,11920,11922,11942],{"className":11921},[2774,2775],[76,11923,11925,11939],{"className":11924},[2779],[76,11926,11928],{"className":11927,"style":2950},[2783],[76,11929,11930,11933],{"style":2899},[76,11931],{"className":11932,"style":2792},[2791],[76,11934,11936],{"className":11935},[2796,2797,2798,2799],[76,11937,4400],{"className":11938},[121,216,2799],[76,11940,2807],{"className":11941},[2806],[76,11943,11945],{"className":11944},[2779],[76,11946,11948],{"className":11947,"style":2814},[2783],[76,11949],{},[76,11951,11953,11979],{"className":11952},[79],[76,11954,11956],{"className":11955},[83],[85,11957,11958],{"xmlns":87},[89,11959,11960,11976],{},[92,11961,11962,11968,11970],{},[2628,11963,11964,11966],{},[95,11965,8182],{"mathvariant":363},[95,11967,4400],{},[299,11969,805],{},[2628,11971,11972,11974],{},[95,11973,236],{},[95,11975,4400],{},[100,11977,11978],{"encoding":102},"\\Delta_n \\vdash \\sigma_n",[76,11980,11982,12037],{"className":11981,"ariaHidden":108},[107],[76,11983,11985,11988,12028,12031,12034],{"className":11984},[112],[76,11986],{"className":11987,"style":5254},[116],[76,11989,11991,11994],{"className":11990},[121],[76,11992,8182],{"className":11993},[121],[76,11995,11997],{"className":11996},[2770],[76,11998,12000,12020],{"className":11999},[2774,2775],[76,12001,12003,12017],{"className":12002},[2779],[76,12004,12006],{"className":12005,"style":2950},[2783],[76,12007,12008,12011],{"style":2787},[76,12009],{"className":12010,"style":2792},[2791],[76,12012,12014],{"className":12013},[2796,2797,2798,2799],[76,12015,4400],{"className":12016},[121,216,2799],[76,12018,2807],{"className":12019},[2806],[76,12021,12023],{"className":12022},[2779],[76,12024,12026],{"className":12025,"style":2814},[2783],[76,12027],{},[76,12029],{"className":12030,"style":324},[323],[76,12032,805],{"className":12033},[328],[76,12035],{"className":12036,"style":324},[323],[76,12038,12040,12043],{"className":12039},[112],[76,12041],{"className":12042,"style":7840},[116],[76,12044,12046,12049],{"className":12045},[121],[76,12047,236],{"className":12048,"style":253},[121,216],[76,12050,12052],{"className":12051},[2770],[76,12053,12055,12075],{"className":12054},[2774,2775],[76,12056,12058,12072],{"className":12057},[2779],[76,12059,12061],{"className":12060,"style":2950},[2783],[76,12062,12063,12066],{"style":2899},[76,12064],{"className":12065,"style":2792},[2791],[76,12067,12069],{"className":12068},[2796,2797,2798,2799],[76,12070,4400],{"className":12071},[121,216,2799],[76,12073,2807],{"className":12074},[2806],[76,12076,12078],{"className":12077},[2779],[76,12079,12081],{"className":12080,"style":2814},[2783],[76,12082],{},[76,12084,12086,12104],{"className":12085},[79],[76,12087,12089],{"className":12088},[83],[85,12090,12091],{"xmlns":87},[89,12092,12093,12101],{},[92,12094,12095],{},[2628,12096,12097,12099],{},[95,12098,8182],{"mathvariant":363},[95,12100,4400],{},[100,12102,12103],{"encoding":102},"\\Delta_n",[76,12105,12107],{"className":12106,"ariaHidden":108},[107],[76,12108,12110,12113],{"className":12109},[112],[76,12111],{"className":12112,"style":4583},[116],[76,12114,12116,12119],{"className":12115},[121],[76,12117,8182],{"className":12118},[121],[76,12120,12122],{"className":12121},[2770],[76,12123,12125,12145],{"className":12124},[2774,2775],[76,12126,12128,12142],{"className":12127},[2779],[76,12129,12131],{"className":12130,"style":2950},[2783],[76,12132,12133,12136],{"style":2787},[76,12134],{"className":12135,"style":2792},[2791],[76,12137,12139],{"className":12138},[2796,2797,2798,2799],[76,12140,4400],{"className":12141},[121,216,2799],[76,12143,2807],{"className":12144},[2806],[76,12146,12148],{"className":12147},[2779],[76,12149,12151],{"className":12150,"style":2814},[2783],[76,12152],{}," 自身が矛盾する。よって各 ",[76,12155,12157,12174],{"className":12156},[79],[76,12158,12160],{"className":12159},[83],[85,12161,12162],{"xmlns":87},[89,12163,12164,12172],{},[92,12165,12166],{},[2628,12167,12168,12170],{},[95,12169,8182],{"mathvariant":363},[95,12171,4400],{},[100,12173,12103],{"encoding":102},[76,12175,12177],{"className":12176,"ariaHidden":108},[107],[76,12178,12180,12183],{"className":12179},[112],[76,12181],{"className":12182,"style":4583},[116],[76,12184,12186,12189],{"className":12185},[121],[76,12187,8182],{"className":12188},[121],[76,12190,12192],{"className":12191},[2770],[76,12193,12195,12215],{"className":12194},[2774,2775],[76,12196,12198,12212],{"className":12197},[2779],[76,12199,12201],{"className":12200,"style":2950},[2783],[76,12202,12203,12206],{"style":2787},[76,12204],{"className":12205,"style":2792},[2791],[76,12207,12209],{"className":12208},[2796,2797,2798,2799],[76,12210,4400],{"className":12211},[121,216,2799],[76,12213,2807],{"className":12214},[2806],[76,12216,12218],{"className":12217},[2779],[76,12219,12221],{"className":12220,"style":2814},[2783],[76,12222],{}," は無矛盾である。",[10,12225,12226,12373,12374,12402],{},[76,12227,12229,12257],{"className":12228},[79],[76,12230,12232],{"className":12231},[83],[85,12233,12234],{"xmlns":87},[89,12235,12236,12254],{},[92,12237,12238,12240,12242,12248],{},[95,12239,8182],{"mathvariant":363},[299,12241,602],{},[2628,12243,12244,12246],{},[299,12245,9726],{},[95,12247,4400],{},[2628,12249,12250,12252],{},[95,12251,8182],{"mathvariant":363},[95,12253,4400],{},[100,12255,12256],{"encoding":102},"\\Delta = \\bigcup_n \\Delta_n",[76,12258,12260,12278],{"className":12259,"ariaHidden":108},[107],[76,12261,12263,12266,12269,12272,12275],{"className":12262},[112],[76,12264],{"className":12265,"style":117},[116],[76,12267,8182],{"className":12268},[121],[76,12270],{"className":12271,"style":324},[323],[76,12273,602],{"className":12274},[328],[76,12276],{"className":12277,"style":324},[323],[76,12279,12281,12285,12330,12333],{"className":12280},[112],[76,12282],{"className":12283,"style":12284},[116],"height:1.0497em;vertical-align:-0.2997em;",[76,12286,12288,12293],{"className":12287},[10000],[76,12289,9726],{"className":12290,"style":12292},[10000,10039,12291],"small-op","position:relative;top:0em;",[76,12294,12296],{"className":12295},[2770],[76,12297,12299,12321],{"className":12298},[2774,2775],[76,12300,12302,12318],{"className":12301},[2779],[76,12303,12306],{"className":12304,"style":12305},[2783],"height:0.0017em;",[76,12307,12309,12312],{"style":12308},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[76,12310],{"className":12311,"style":2792},[2791],[76,12313,12315],{"className":12314},[2796,2797,2798,2799],[76,12316,4400],{"className":12317},[121,216,2799],[76,12319,2807],{"className":12320},[2806],[76,12322,12324],{"className":12323},[2779],[76,12325,12328],{"className":12326,"style":12327},[2783],"height:0.2997em;",[76,12329],{},[76,12331],{"className":12332,"style":2820},[323],[76,12334,12336,12339],{"className":12335},[121],[76,12337,8182],{"className":12338},[121],[76,12340,12342],{"className":12341},[2770],[76,12343,12345,12365],{"className":12344},[2774,2775],[76,12346,12348,12362],{"className":12347},[2779],[76,12349,12351],{"className":12350,"style":2950},[2783],[76,12352,12353,12356],{"style":2787},[76,12354],{"className":12355,"style":2792},[2791],[76,12357,12359],{"className":12358},[2796,2797,2798,2799],[76,12360,4400],{"className":12361},[121,216,2799],[76,12363,2807],{"className":12364},[2806],[76,12366,12368],{"className":12367},[2779],[76,12369,12371],{"className":12370,"style":2814},[2783],[76,12372],{}," とおくと、導出は有限個の前提しか使わないので ",[76,12375,12377,12390],{"className":12376},[79],[76,12378,12380],{"className":12379},[83],[85,12381,12382],{"xmlns":87},[89,12383,12384,12388],{},[92,12385,12386],{},[95,12387,8182],{"mathvariant":363},[100,12389,8185],{"encoding":102},[76,12391,12393],{"className":12392,"ariaHidden":108},[107],[76,12394,12396,12399],{"className":12395},[112],[76,12397],{"className":12398,"style":117},[116],[76,12400,8182],{"className":12401},[121]," も無矛盾で、作り方から極大である。",[10,12404,12405,12406,12434],{},"こうして得た ",[76,12407,12409,12422],{"className":12408},[79],[76,12410,12412],{"className":12411},[83],[85,12413,12414],{"xmlns":87},[89,12415,12416,12420],{},[92,12417,12418],{},[95,12419,8182],{"mathvariant":363},[100,12421,8185],{"encoding":102},[76,12423,12425],{"className":12424,"ariaHidden":108},[107],[76,12426,12428,12431],{"className":12427},[112],[76,12429],{"className":12430,"style":117},[116],[76,12432,8182],{"className":12433},[121]," は、次の性質を持ちます。",[21,12436,12437,12544,12701,12930],{},[24,12438,12439,1027,12442,1078,12493,1303],{},[13,12440,12441],{},"演繹で閉じている",[76,12443,12445,12463],{"className":12444},[79],[76,12446,12448],{"className":12447},[83],[85,12449,12450],{"xmlns":87},[89,12451,12452,12460],{},[92,12453,12454,12456,12458],{},[95,12455,8182],{"mathvariant":363},[299,12457,805],{},[95,12459,236],{},[100,12461,12462],{"encoding":102},"\\Delta \\vdash \\sigma",[76,12464,12466,12484],{"className":12465,"ariaHidden":108},[107],[76,12467,12469,12472,12475,12478,12481],{"className":12468},[112],[76,12470],{"className":12471,"style":820},[116],[76,12473,8182],{"className":12474},[121],[76,12476],{"className":12477,"style":324},[323],[76,12479,805],{"className":12480},[328],[76,12482],{"className":12483,"style":324},[323],[76,12485,12487,12490],{"className":12486},[112],[76,12488],{"className":12489,"style":249},[116],[76,12491,236],{"className":12492,"style":253},[121,216],[76,12494,12496,12514],{"className":12495},[79],[76,12497,12499],{"className":12498},[83],[85,12500,12501],{"xmlns":87},[89,12502,12503,12511],{},[92,12504,12505,12507,12509],{},[95,12506,236],{},[299,12508,8262],{},[95,12510,8182],{"mathvariant":363},[100,12512,12513],{"encoding":102},"\\sigma \\in \\Delta",[76,12515,12517,12535],{"className":12516,"ariaHidden":108},[107],[76,12518,12520,12523,12526,12529,12532],{"className":12519},[112],[76,12521],{"className":12522,"style":8319},[116],[76,12524,236],{"className":12525,"style":253},[121,216],[76,12527],{"className":12528,"style":324},[323],[76,12530,8262],{"className":12531},[328],[76,12533],{"className":12534,"style":324},[323],[76,12536,12538,12541],{"className":12537},[112],[76,12539],{"className":12540,"style":117},[116],[76,12542,8182],{"className":12543},[121],[24,12545,12546,1027,12549,1303],{},[13,12547,12548],{},"否定",[76,12550,12552,12585],{"className":12551},[79],[76,12553,12555],{"className":12554},[83],[85,12556,12557],{"xmlns":87},[89,12558,12559,12582],{},[92,12560,12561,12563,12565,12567,12569,12571,12573,12575,12577,12580],{},[95,12562,941],{"mathvariant":363},[95,12564,236],{},[299,12566,8262],{},[95,12568,8182],{"mathvariant":363},[2617,12570,2619],{},[299,12572,7011],{},[2617,12574,2619],{},[95,12576,236],{},[299,12578,12579],{"mathvariant":363},"∉",[95,12581,8182],{"mathvariant":363},[100,12583,12584],{"encoding":102},"\\neg\\sigma \\in \\Delta \\iff \\sigma \\notin \\Delta",[76,12586,12588,12609,12634,12692],{"className":12587,"ariaHidden":108},[107],[76,12589,12591,12594,12597,12600,12603,12606],{"className":12590},[112],[76,12592],{"className":12593,"style":8319},[116],[76,12595,941],{"className":12596},[121],[76,12598,236],{"className":12599,"style":253},[121,216],[76,12601],{"className":12602,"style":324},[323],[76,12604,8262],{"className":12605},[328],[76,12607],{"className":12608,"style":324},[323],[76,12610,12612,12616,12619,12622,12625,12628,12631],{"className":12611},[112],[76,12613],{"className":12614,"style":12615},[116],"height:0.7073em;vertical-align:-0.024em;",[76,12617,8182],{"className":12618},[121],[76,12620],{"className":12621,"style":324},[323],[76,12623],{"className":12624,"style":324},[323],[76,12626,7011],{"className":12627},[328],[76,12629],{"className":12630,"style":324},[323],[76,12632],{"className":12633,"style":324},[323],[76,12635,12637,12640,12643,12646,12689],{"className":12636},[112],[76,12638],{"className":12639,"style":1198},[116],[76,12641,236],{"className":12642,"style":253},[121,216],[76,12644],{"className":12645,"style":324},[323],[76,12647,12649,12655],{"className":12648},[328],[76,12650,12652],{"className":12651},[121],[76,12653,8262],{"className":12654},[328],[76,12656,12659],{"className":12657},[121,12658],"katex-vbox",[76,12660,12663],{"className":12661},[12662],"katex-thinbox",[76,12664,12667,12670,12685],{"className":12665},[12666],"llap",[76,12668],{"className":12669,"style":1198},[116],[76,12671,12674],{"className":12672},[12673],"katex-inner",[76,12675,12677,12681],{"className":12676},[121],[76,12678,12680],{"className":12679},[121],"\u002F",[76,12682],{"className":12683,"style":12684},[323],"margin-right:0.0556em;",[76,12686],{"className":12687},[12688],"katex-fix",[76,12690],{"className":12691,"style":324},[323],[76,12693,12695,12698],{"className":12694},[112],[76,12696],{"className":12697,"style":117},[116],[76,12699,8182],{"className":12700},[121],[24,12702,12703,1027,12706,12878,12879,1303],{},[13,12704,12705],{},"含意",[76,12707,12709,12748],{"className":12708},[79],[76,12710,12712],{"className":12711},[83],[85,12713,12714],{"xmlns":87},[89,12715,12716,12745],{},[92,12717,12718,12720,12722,12724,12727,12729,12731,12733,12735,12737,12739,12741,12743],{},[299,12719,2678],{"stretchy":1154},[95,12721,236],{},[299,12723,1248],{},[95,12725,12726],{},"τ",[299,12728,2721],{"stretchy":1154},[299,12730,8262],{},[95,12732,8182],{"mathvariant":363},[2617,12734,2619],{},[299,12736,7011],{},[2617,12738,2619],{},[95,12740,236],{},[299,12742,12579],{"mathvariant":363},[95,12744,8182],{"mathvariant":363},[100,12746,12747],{"encoding":102},"(\\sigma \\to \\tau) \\in \\Delta \\iff \\sigma \\notin \\Delta",[76,12749,12751,12772,12794,12818,12869],{"className":12750,"ariaHidden":108},[107],[76,12752,12754,12757,12760,12763,12766,12769],{"className":12753},[112],[76,12755],{"className":12756,"style":1198},[116],[76,12758,2678],{"className":12759},[1202],[76,12761,236],{"className":12762,"style":253},[121,216],[76,12764],{"className":12765,"style":324},[323],[76,12767,1248],{"className":12768},[328],[76,12770],{"className":12771,"style":324},[323],[76,12773,12775,12778,12782,12785,12788,12791],{"className":12774},[112],[76,12776],{"className":12777,"style":1198},[116],[76,12779,12726],{"className":12780,"style":12781},[121,216],"margin-right:0.1132em;",[76,12783,2721],{"className":12784},[1209],[76,12786],{"className":12787,"style":324},[323],[76,12789,8262],{"className":12790},[328],[76,12792],{"className":12793,"style":324},[323],[76,12795,12797,12800,12803,12806,12809,12812,12815],{"className":12796},[112],[76,12798],{"className":12799,"style":12615},[116],[76,12801,8182],{"className":12802},[121],[76,12804],{"className":12805,"style":324},[323],[76,12807],{"className":12808,"style":324},[323],[76,12810,7011],{"className":12811},[328],[76,12813],{"className":12814,"style":324},[323],[76,12816],{"className":12817,"style":324},[323],[76,12819,12821,12824,12827,12830,12866],{"className":12820},[112],[76,12822],{"className":12823,"style":1198},[116],[76,12825,236],{"className":12826,"style":253},[121,216],[76,12828],{"className":12829,"style":324},[323],[76,12831,12833,12839],{"className":12832},[328],[76,12834,12836],{"className":12835},[121],[76,12837,8262],{"className":12838},[328],[76,12840,12842],{"className":12841},[121,12658],[76,12843,12845],{"className":12844},[12662],[76,12846,12848,12851,12863],{"className":12847},[12666],[76,12849],{"className":12850,"style":1198},[116],[76,12852,12854],{"className":12853},[12673],[76,12855,12857,12860],{"className":12856},[121],[76,12858,12680],{"className":12859},[121],[76,12861],{"className":12862,"style":12684},[323],[76,12864],{"className":12865},[12688],[76,12867],{"className":12868,"style":324},[323],[76,12870,12872,12875],{"className":12871},[112],[76,12873],{"className":12874,"style":117},[116],[76,12876,8182],{"className":12877},[121]," または ",[76,12880,12882,12900],{"className":12881},[79],[76,12883,12885],{"className":12884},[83],[85,12886,12887],{"xmlns":87},[89,12888,12889,12897],{},[92,12890,12891,12893,12895],{},[95,12892,12726],{},[299,12894,8262],{},[95,12896,8182],{"mathvariant":363},[100,12898,12899],{"encoding":102},"\\tau \\in \\Delta",[76,12901,12903,12921],{"className":12902,"ariaHidden":108},[107],[76,12904,12906,12909,12912,12915,12918],{"className":12905},[112],[76,12907],{"className":12908,"style":8319},[116],[76,12910,12726],{"className":12911,"style":12781},[121,216],[76,12913],{"className":12914,"style":324},[323],[76,12916,8262],{"className":12917},[328],[76,12919],{"className":12920,"style":324},[323],[76,12922,12924,12927],{"className":12923},[112],[76,12925],{"className":12926,"style":117},[116],[76,12928,8182],{"className":12929},[121],[24,12931,12932,1027,12935,13016,13017,871,13045,1303],{},[13,12933,12934],{},"証人の性質",[76,12936,12938,12968],{"className":12937},[79],[76,12939,12941],{"className":12940},[83],[85,12942,12943],{"xmlns":87},[89,12944,12945,12965],{},[92,12946,12947,12949,12951,12953,12955,12957,12959,12961,12963],{},[95,12948,2653],{"mathvariant":363},[95,12950,2632],{},[2617,12952,5223],{},[95,12954,456],{},[299,12956,2678],{"stretchy":1154},[95,12958,2632],{},[299,12960,2721],{"stretchy":1154},[299,12962,8262],{},[95,12964,8182],{"mathvariant":363},[100,12966,12967],{"encoding":102},"\\exists x\\, \\varphi(x) \\in \\Delta",[76,12969,12971,13007],{"className":12970,"ariaHidden":108},[107],[76,12972,12974,12977,12980,12983,12986,12989,12992,12995,12998,13001,13004],{"className":12973},[112],[76,12975],{"className":12976,"style":1198},[116],[76,12978,2653],{"className":12979},[121],[76,12981,2632],{"className":12982},[121,216],[76,12984],{"className":12985,"style":2820},[323],[76,12987,456],{"className":12988},[121,216],[76,12990,2678],{"className":12991},[1202],[76,12993,2632],{"className":12994},[121,216],[76,12996,2721],{"className":12997},[1209],[76,12999],{"className":13000,"style":324},[323],[76,13002,8262],{"className":13003},[328],[76,13005],{"className":13006,"style":324},[323],[76,13008,13010,13013],{"className":13009},[112],[76,13011],{"className":13012,"style":117},[116],[76,13014,8182],{"className":13015},[121]," ならば、ある定数 ",[76,13018,13020,13033],{"className":13019},[79],[76,13021,13023],{"className":13022},[83],[85,13024,13025],{"xmlns":87},[89,13026,13027,13031],{},[92,13028,13029],{},[95,13030,8102],{},[100,13032,8102],{"encoding":102},[76,13034,13036],{"className":13035,"ariaHidden":108},[107],[76,13037,13039,13042],{"className":13038},[112],[76,13040],{"className":13041,"style":249},[116],[76,13043,8102],{"className":13044},[121,216],[76,13046,13048,13072],{"className":13047},[79],[76,13049,13051],{"className":13050},[83],[85,13052,13053],{"xmlns":87},[89,13054,13055,13069],{},[92,13056,13057,13059,13061,13063,13065,13067],{},[95,13058,456],{},[299,13060,2678],{"stretchy":1154},[95,13062,8102],{},[299,13064,2721],{"stretchy":1154},[299,13066,8262],{},[95,13068,8182],{"mathvariant":363},[100,13070,13071],{"encoding":102},"\\varphi(c) \\in \\Delta",[76,13073,13075,13102],{"className":13074,"ariaHidden":108},[107],[76,13076,13078,13081,13084,13087,13090,13093,13096,13099],{"className":13077},[112],[76,13079],{"className":13080,"style":1198},[116],[76,13082,456],{"className":13083},[121,216],[76,13085,2678],{"className":13086},[1202],[76,13088,8102],{"className":13089},[121,216],[76,13091,2721],{"className":13092},[1209],[76,13094],{"className":13095,"style":324},[323],[76,13097,8262],{"className":13098},[328],[76,13100],{"className":13101,"style":324},[323],[76,13103,13105,13108],{"className":13104},[112],[76,13106],{"className":13107,"style":117},[116],[76,13109,8182],{"className":13110},[121],[10,13112,13113,13114,13142],{},"最後の性質は、",[76,13115,13117,13130],{"className":13116},[79],[76,13118,13120],{"className":13119},[83],[85,13121,13122],{"xmlns":87},[89,13123,13124,13128],{},[92,13125,13126],{},[95,13127,8182],{"mathvariant":363},[100,13129,8185],{"encoding":102},[76,13131,13133],{"className":13132,"ariaHidden":108},[107],[76,13134,13136,13139],{"className":13135},[112],[76,13137],{"className":13138,"style":117},[116],[76,13140,8182],{"className":13141},[121]," がヘンキン公理を含み、演繹で閉じていることから従います。",[68,13144,13146],{"id":13145},"_33-ステップ3閉項からモデルを作る","3.3 ステップ3：閉項からモデルを作る",[10,13148,13149,13177,13178,13206],{},[76,13150,13152,13165],{"className":13151},[79],[76,13153,13155],{"className":13154},[83],[85,13156,13157],{"xmlns":87},[89,13158,13159,13163],{},[92,13160,13161],{},[95,13162,8182],{"mathvariant":363},[100,13164,8185],{"encoding":102},[76,13166,13168],{"className":13167,"ariaHidden":108},[107],[76,13169,13171,13174],{"className":13170},[112],[76,13172],{"className":13173,"style":117},[116],[76,13175,8182],{"className":13176},[121]," から、構造 ",[76,13179,13181,13194],{"className":13180},[79],[76,13182,13184],{"className":13183},[83],[85,13185,13186],{"xmlns":87},[89,13187,13188,13192],{},[92,13189,13190],{},[95,13191,171],{"mathvariant":97},[100,13193,174],{"encoding":102},[76,13195,13197],{"className":13196,"ariaHidden":108},[107],[76,13198,13200,13203],{"className":13199},[112],[76,13201],{"className":13202,"style":117},[116],[76,13204,171],{"className":13205},[121,122]," を次のように作ります。",[21,13208,13209,13272,13618],{},[24,13210,13211,1027,13213,13271],{},[13,13212,6488],{},[76,13214,13216,13233],{"className":13215},[79],[76,13217,13219],{"className":13218},[83],[85,13220,13221],{"xmlns":87},[89,13222,13223,13231],{},[92,13224,13225],{},[6715,13226,13227,13229],{},[95,13228,98],{"mathvariant":97},[299,13230,9718],{},[100,13232,10521],{"encoding":102},[76,13234,13236],{"className":13235,"ariaHidden":108},[107],[76,13237,13239,13242],{"className":13238},[112],[76,13240],{"className":13241,"style":10531},[116],[76,13243,13245,13248],{"className":13244},[121],[76,13246,98],{"className":13247},[121,122],[76,13249,13251],{"className":13250},[2770],[76,13252,13254],{"className":13253},[2774],[76,13255,13257],{"className":13256},[2779],[76,13258,13260],{"className":13259,"style":10531},[2783],[76,13261,13262,13265],{"style":6787},[76,13263],{"className":13264,"style":2792},[2791],[76,13266,13268],{"className":13267},[2796,2797,2798,2799],[76,13269,9718],{"className":13270},[1188,2799]," の閉項全体。",[24,13273,13274,1027,13277,13617],{},[13,13275,13276],{},"関数記号",[76,13278,13280,13348],{"className":13279},[79],[76,13281,13283],{"className":13282},[83],[85,13284,13285],{"xmlns":87},[89,13286,13287,13345],{},[92,13288,13289,13296,13298,13305,13307,13309,13311,13317,13319,13321,13323,13325,13331,13333,13335,13337,13343],{},[6715,13290,13291,13294],{},[95,13292,13293],{},"f",[95,13295,171],{"mathvariant":97},[299,13297,2678],{"stretchy":1154},[2628,13299,13300,13303],{},[95,13301,13302],{},"t",[2634,13304,2636],{},[299,13306,2687],{"separator":108},[299,13308,2690],{},[299,13310,2687],{"separator":108},[2628,13312,13313,13315],{},[95,13314,13302],{},[95,13316,4400],{},[299,13318,2721],{"stretchy":1154},[299,13320,602],{},[95,13322,13293],{},[299,13324,2678],{"stretchy":1154},[2628,13326,13327,13329],{},[95,13328,13302],{},[2634,13330,2636],{},[299,13332,2687],{"separator":108},[299,13334,2690],{},[299,13336,2687],{"separator":108},[2628,13338,13339,13341],{},[95,13340,13302],{},[95,13342,4400],{},[299,13344,2721],{"stretchy":1154},[100,13346,13347],{"encoding":102},"f^{\\mathcal{M}}(t_1, \\dots, t_n) = f(t_1, \\dots, t_n)",[76,13349,13351,13504],{"className":13350,"ariaHidden":108},[107],[76,13352,13354,13358,13391,13394,13434,13437,13440,13443,13446,13449,13452,13492,13495,13498,13501],{"className":13353},[112],[76,13355],{"className":13356,"style":13357},[116],"height:1.0913em;vertical-align:-0.25em;",[76,13359,13361,13365],{"className":13360},[121],[76,13362,13293],{"className":13363,"style":13364},[121,216],"margin-right:0.1076em;",[76,13366,13368],{"className":13367},[2770],[76,13369,13371],{"className":13370},[2774],[76,13372,13374],{"className":13373},[2779],[76,13375,13377],{"className":13376,"style":6784},[2783],[76,13378,13379,13382],{"style":6787},[76,13380],{"className":13381,"style":2792},[2791],[76,13383,13385],{"className":13384},[2796,2797,2798,2799],[76,13386,13388],{"className":13387},[121,2799],[76,13389,171],{"className":13390},[121,122,2799],[76,13392,2678],{"className":13393},[1202],[76,13395,13397,13400],{"className":13396},[121],[76,13398,13302],{"className":13399},[121,216],[76,13401,13403],{"className":13402},[2770],[76,13404,13406,13426],{"className":13405},[2774,2775],[76,13407,13409,13423],{"className":13408},[2779],[76,13410,13412],{"className":13411,"style":2784},[2783],[76,13413,13414,13417],{"style":2787},[76,13415],{"className":13416,"style":2792},[2791],[76,13418,13420],{"className":13419},[2796,2797,2798,2799],[76,13421,2636],{"className":13422},[121,2799],[76,13424,2807],{"className":13425},[2806],[76,13427,13429],{"className":13428},[2779],[76,13430,13432],{"className":13431,"style":2814},[2783],[76,13433],{},[76,13435,2687],{"className":13436},[3025],[76,13438],{"className":13439,"style":2820},[323],[76,13441,2690],{"className":13442},[2824],[76,13444],{"className":13445,"style":2820},[323],[76,13447,2687],{"className":13448},[3025],[76,13450],{"className":13451,"style":2820},[323],[76,13453,13455,13458],{"className":13454},[121],[76,13456,13302],{"className":13457},[121,216],[76,13459,13461],{"className":13460},[2770],[76,13462,13464,13484],{"className":13463},[2774,2775],[76,13465,13467,13481],{"className":13466},[2779],[76,13468,13470],{"className":13469,"style":2950},[2783],[76,13471,13472,13475],{"style":2787},[76,13473],{"className":13474,"style":2792},[2791],[76,13476,13478],{"className":13477},[2796,2797,2798,2799],[76,13479,4400],{"className":13480},[121,216,2799],[76,13482,2807],{"className":13483},[2806],[76,13485,13487],{"className":13486},[2779],[76,13488,13490],{"className":13489,"style":2814},[2783],[76,13491],{},[76,13493,2721],{"className":13494},[1209],[76,13496],{"className":13497,"style":324},[323],[76,13499,602],{"className":13500},[328],[76,13502],{"className":13503,"style":324},[323],[76,13505,13507,13510,13513,13516,13556,13559,13562,13565,13568,13571,13574,13614],{"className":13506},[112],[76,13508],{"className":13509,"style":1198},[116],[76,13511,13293],{"className":13512,"style":13364},[121,216],[76,13514,2678],{"className":13515},[1202],[76,13517,13519,13522],{"className":13518},[121],[76,13520,13302],{"className":13521},[121,216],[76,13523,13525],{"className":13524},[2770],[76,13526,13528,13548],{"className":13527},[2774,2775],[76,13529,13531,13545],{"className":13530},[2779],[76,13532,13534],{"className":13533,"style":2784},[2783],[76,13535,13536,13539],{"style":2787},[76,13537],{"className":13538,"style":2792},[2791],[76,13540,13542],{"className":13541},[2796,2797,2798,2799],[76,13543,2636],{"className":13544},[121,2799],[76,13546,2807],{"className":13547},[2806],[76,13549,13551],{"className":13550},[2779],[76,13552,13554],{"className":13553,"style":2814},[2783],[76,13555],{},[76,13557,2687],{"className":13558},[3025],[76,13560],{"className":13561,"style":2820},[323],[76,13563,2690],{"className":13564},[2824],[76,13566],{"className":13567,"style":2820},[323],[76,13569,2687],{"className":13570},[3025],[76,13572],{"className":13573,"style":2820},[323],[76,13575,13577,13580],{"className":13576},[121],[76,13578,13302],{"className":13579},[121,216],[76,13581,13583],{"className":13582},[2770],[76,13584,13586,13606],{"className":13585},[2774,2775],[76,13587,13589,13603],{"className":13588},[2779],[76,13590,13592],{"className":13591,"style":2950},[2783],[76,13593,13594,13597],{"style":2787},[76,13595],{"className":13596,"style":2792},[2791],[76,13598,13600],{"className":13599},[2796,2797,2798,2799],[76,13601,4400],{"className":13602},[121,216,2799],[76,13604,2807],{"className":13605},[2806],[76,13607,13609],{"className":13608},[2779],[76,13610,13612],{"className":13611,"style":2814},[2783],[76,13613],{},[76,13615,2721],{"className":13616},[1209],"（項をそのまま組み立てる）。",[24,13619,13620,1027,13623,6994,13811,156,13849,1303],{},[13,13621,13622],{},"述語記号",[76,13624,13626,13666],{"className":13625},[79],[76,13627,13629],{"className":13628},[83],[85,13630,13631],{"xmlns":87},[89,13632,13633,13663],{},[92,13634,13635,13641,13643,13649,13651,13653,13655,13661],{},[6715,13636,13637,13639],{},[95,13638,4904],{},[95,13640,171],{"mathvariant":97},[299,13642,2678],{"stretchy":1154},[2628,13644,13645,13647],{},[95,13646,13302],{},[2634,13648,2636],{},[299,13650,2687],{"separator":108},[299,13652,2690],{},[299,13654,2687],{"separator":108},[2628,13656,13657,13659],{},[95,13658,13302],{},[95,13660,4400],{},[299,13662,2721],{"stretchy":1154},[100,13664,13665],{"encoding":102},"R^{\\mathcal{M}}(t_1, \\dots, t_n)",[76,13667,13669],{"className":13668,"ariaHidden":108},[107],[76,13670,13672,13675,13707,13710,13750,13753,13756,13759,13762,13765,13768,13808],{"className":13671},[112],[76,13673],{"className":13674,"style":13357},[116],[76,13676,13678,13681],{"className":13677},[121],[76,13679,4904],{"className":13680,"style":4939},[121,216],[76,13682,13684],{"className":13683},[2770],[76,13685,13687],{"className":13686},[2774],[76,13688,13690],{"className":13689},[2779],[76,13691,13693],{"className":13692,"style":6784},[2783],[76,13694,13695,13698],{"style":6787},[76,13696],{"className":13697,"style":2792},[2791],[76,13699,13701],{"className":13700},[2796,2797,2798,2799],[76,13702,13704],{"className":13703},[121,2799],[76,13705,171],{"className":13706},[121,122,2799],[76,13708,2678],{"className":13709},[1202],[76,13711,13713,13716],{"className":13712},[121],[76,13714,13302],{"className":13715},[121,216],[76,13717,13719],{"className":13718},[2770],[76,13720,13722,13742],{"className":13721},[2774,2775],[76,13723,13725,13739],{"className":13724},[2779],[76,13726,13728],{"className":13727,"style":2784},[2783],[76,13729,13730,13733],{"style":2787},[76,13731],{"className":13732,"style":2792},[2791],[76,13734,13736],{"className":13735},[2796,2797,2798,2799],[76,13737,2636],{"className":13738},[121,2799],[76,13740,2807],{"className":13741},[2806],[76,13743,13745],{"className":13744},[2779],[76,13746,13748],{"className":13747,"style":2814},[2783],[76,13749],{},[76,13751,2687],{"className":13752},[3025],[76,13754],{"className":13755,"style":2820},[323],[76,13757,2690],{"className":13758},[2824],[76,13760],{"className":13761,"style":2820},[323],[76,13763,2687],{"className":13764},[3025],[76,13766],{"className":13767,"style":2820},[323],[76,13769,13771,13774],{"className":13770},[121],[76,13772,13302],{"className":13773},[121,216],[76,13775,13777],{"className":13776},[2770],[76,13778,13780,13800],{"className":13779},[2774,2775],[76,13781,13783,13797],{"className":13782},[2779],[76,13784,13786],{"className":13785,"style":2950},[2783],[76,13787,13788,13791],{"style":2787},[76,13789],{"className":13790,"style":2792},[2791],[76,13792,13794],{"className":13793},[2796,2797,2798,2799],[76,13795,4400],{"className":13796},[121,216,2799],[76,13798,2807],{"className":13799},[2806],[76,13801,13803],{"className":13802},[2779],[76,13804,13806],{"className":13805,"style":2814},[2783],[76,13807],{},[76,13809,2721],{"className":13810},[1209],[76,13812,13814,13831],{"className":13813},[79],[76,13815,13817],{"className":13816},[83],[85,13818,13819],{"xmlns":87},[89,13820,13821,13829],{},[92,13822,13823,13825,13827],{},[2617,13824,2619],{},[299,13826,7011],{},[2617,13828,2619],{},[100,13830,7016],{"encoding":102},[76,13832,13834],{"className":13833,"ariaHidden":108},[107],[76,13835,13837,13840,13843,13846],{"className":13836},[112],[76,13838],{"className":13839,"style":7026},[116],[76,13841],{"className":13842,"style":324},[323],[76,13844,7011],{"className":13845},[328],[76,13847],{"className":13848,"style":324},[323],[76,13850,13852,13892],{"className":13851},[79],[76,13853,13855],{"className":13854},[83],[85,13856,13857],{"xmlns":87},[89,13858,13859,13889],{},[92,13860,13861,13863,13865,13871,13873,13875,13877,13883,13885,13887],{},[95,13862,4904],{},[299,13864,2678],{"stretchy":1154},[2628,13866,13867,13869],{},[95,13868,13302],{},[2634,13870,2636],{},[299,13872,2687],{"separator":108},[299,13874,2690],{},[299,13876,2687],{"separator":108},[2628,13878,13879,13881],{},[95,13880,13302],{},[95,13882,4400],{},[299,13884,2721],{"stretchy":1154},[299,13886,8262],{},[95,13888,8182],{"mathvariant":363},[100,13890,13891],{"encoding":102},"R(t_1, \\dots, t_n) \\in \\Delta",[76,13893,13895,14017],{"className":13894,"ariaHidden":108},[107],[76,13896,13898,13901,13904,13907,13947,13950,13953,13956,13959,13962,13965,14005,14008,14011,14014],{"className":13897},[112],[76,13899],{"className":13900,"style":1198},[116],[76,13902,4904],{"className":13903,"style":4939},[121,216],[76,13905,2678],{"className":13906},[1202],[76,13908,13910,13913],{"className":13909},[121],[76,13911,13302],{"className":13912},[121,216],[76,13914,13916],{"className":13915},[2770],[76,13917,13919,13939],{"className":13918},[2774,2775],[76,13920,13922,13936],{"className":13921},[2779],[76,13923,13925],{"className":13924,"style":2784},[2783],[76,13926,13927,13930],{"style":2787},[76,13928],{"className":13929,"style":2792},[2791],[76,13931,13933],{"className":13932},[2796,2797,2798,2799],[76,13934,2636],{"className":13935},[121,2799],[76,13937,2807],{"className":13938},[2806],[76,13940,13942],{"className":13941},[2779],[76,13943,13945],{"className":13944,"style":2814},[2783],[76,13946],{},[76,13948,2687],{"className":13949},[3025],[76,13951],{"className":13952,"style":2820},[323],[76,13954,2690],{"className":13955},[2824],[76,13957],{"className":13958,"style":2820},[323],[76,13960,2687],{"className":13961},[3025],[76,13963],{"className":13964,"style":2820},[323],[76,13966,13968,13971],{"className":13967},[121],[76,13969,13302],{"className":13970},[121,216],[76,13972,13974],{"className":13973},[2770],[76,13975,13977,13997],{"className":13976},[2774,2775],[76,13978,13980,13994],{"className":13979},[2779],[76,13981,13983],{"className":13982,"style":2950},[2783],[76,13984,13985,13988],{"style":2787},[76,13986],{"className":13987,"style":2792},[2791],[76,13989,13991],{"className":13990},[2796,2797,2798,2799],[76,13992,4400],{"className":13993},[121,216,2799],[76,13995,2807],{"className":13996},[2806],[76,13998,14000],{"className":13999},[2779],[76,14001,14003],{"className":14002,"style":2814},[2783],[76,14004],{},[76,14006,2721],{"className":14007},[1209],[76,14009],{"className":14010,"style":324},[323],[76,14012,8262],{"className":14013},[328],[76,14015],{"className":14016,"style":324},[323],[76,14018,14020,14023],{"className":14019},[112],[76,14021],{"className":14022,"style":117},[116],[76,14024,8182],{"className":14025},[121],[10,14027,14028,14029,14113],{},"等号を含む言語では、「",[76,14030,14032,14059],{"className":14031},[79],[76,14033,14035],{"className":14034},[83],[85,14036,14037],{"xmlns":87},[89,14038,14039,14056],{},[92,14040,14041,14043,14045,14047,14050,14052,14054],{},[299,14042,2678],{"stretchy":1154},[95,14044,13302],{},[299,14046,602],{},[95,14048,14049],{},"s",[299,14051,2721],{"stretchy":1154},[299,14053,8262],{},[95,14055,8182],{"mathvariant":363},[100,14057,14058],{"encoding":102},"(t = s) \\in \\Delta",[76,14060,14062,14083,14104],{"className":14061,"ariaHidden":108},[107],[76,14063,14065,14068,14071,14074,14077,14080],{"className":14064},[112],[76,14066],{"className":14067,"style":1198},[116],[76,14069,2678],{"className":14070},[1202],[76,14072,13302],{"className":14073},[121,216],[76,14075],{"className":14076,"style":324},[323],[76,14078,602],{"className":14079},[328],[76,14081],{"className":14082,"style":324},[323],[76,14084,14086,14089,14092,14095,14098,14101],{"className":14085},[112],[76,14087],{"className":14088,"style":1198},[116],[76,14090,14049],{"className":14091},[121,216],[76,14093,2721],{"className":14094},[1209],[76,14096],{"className":14097,"style":324},[323],[76,14099,8262],{"className":14100},[328],[76,14102],{"className":14103,"style":324},[323],[76,14105,14107,14110],{"className":14106},[112],[76,14108],{"className":14109,"style":117},[116],[76,14111,8182],{"className":14112},[121],"」で定まる同値関係で閉項を割った商を領域にします。等号の公理により、この同値関係は関数や述語と両立します。",[1309,14115,14117],{"title":14116},"補題 3：真理補題",[10,14118,14119,14177,14178,1371,14206,1303],{},[76,14120,14122,14139],{"className":14121},[79],[76,14123,14125],{"className":14124},[83],[85,14126,14127],{"xmlns":87},[89,14128,14129,14137],{},[92,14130,14131],{},[6715,14132,14133,14135],{},[95,14134,98],{"mathvariant":97},[299,14136,9718],{},[100,14138,10521],{"encoding":102},[76,14140,14142],{"className":14141,"ariaHidden":108},[107],[76,14143,14145,14148],{"className":14144},[112],[76,14146],{"className":14147,"style":10531},[116],[76,14149,14151,14154],{"className":14150},[121],[76,14152,98],{"className":14153},[121,122],[76,14155,14157],{"className":14156},[2770],[76,14158,14160],{"className":14159},[2774],[76,14161,14163],{"className":14162},[2779],[76,14164,14166],{"className":14165,"style":10531},[2783],[76,14167,14168,14171],{"style":6787},[76,14169],{"className":14170,"style":2792},[2791],[76,14172,14174],{"className":14173},[2796,2797,2798,2799],[76,14175,9718],{"className":14176},[1188,2799]," のすべての文 ",[76,14179,14181,14194],{"className":14180},[79],[76,14182,14184],{"className":14183},[83],[85,14185,14186],{"xmlns":87},[89,14187,14188,14192],{},[92,14189,14190],{},[95,14191,236],{},[100,14193,239],{"encoding":102},[76,14195,14197],{"className":14196,"ariaHidden":108},[107],[76,14198,14200,14203],{"className":14199},[112],[76,14201],{"className":14202,"style":249},[116],[76,14204,236],{"className":14205,"style":253},[121,216],[76,14207,14209,14238],{"className":14208},[79],[76,14210,14212],{"className":14211},[83],[85,14213,14214],{"xmlns":87},[89,14215,14216,14236],{},[92,14217,14218,14220,14222,14224,14226,14228,14230,14232,14234],{},[95,14219,171],{"mathvariant":97},[299,14221,301],{},[95,14223,236],{},[2617,14225,2619],{},[299,14227,7011],{},[2617,14229,2619],{},[95,14231,236],{},[299,14233,8262],{},[95,14235,8182],{"mathvariant":363},[100,14237,8267],{"encoding":102},[76,14239,14241,14259,14283,14301],{"className":14240,"ariaHidden":108},[107],[76,14242,14244,14247,14250,14253,14256],{"className":14243},[112],[76,14245],{"className":14246,"style":316},[116],[76,14248,171],{"className":14249},[121,122],[76,14251],{"className":14252,"style":324},[323],[76,14254,301],{"className":14255},[328],[76,14257],{"className":14258,"style":324},[323],[76,14260,14262,14265,14268,14271,14274,14277,14280],{"className":14261},[112],[76,14263],{"className":14264,"style":7026},[116],[76,14266,236],{"className":14267,"style":253},[121,216],[76,14269],{"className":14270,"style":324},[323],[76,14272],{"className":14273,"style":324},[323],[76,14275,7011],{"className":14276},[328],[76,14278],{"className":14279,"style":324},[323],[76,14281],{"className":14282,"style":324},[323],[76,14284,14286,14289,14292,14295,14298],{"className":14285},[112],[76,14287],{"className":14288,"style":8319},[116],[76,14290,236],{"className":14291,"style":253},[121,216],[76,14293],{"className":14294,"style":324},[323],[76,14296,8262],{"className":14297},[328],[76,14299],{"className":14300,"style":324},[323],[76,14302,14304,14307],{"className":14303},[112],[76,14305],{"className":14306,"style":117},[116],[76,14308,8182],{"className":14309},[121],[1571,14311,14312,14432],{},[10,14313,14314,14342,14343,35,14372,35,14402,14431],{},[76,14315,14317,14330],{"className":14316},[79],[76,14318,14320],{"className":14319},[83],[85,14321,14322],{"xmlns":87},[89,14323,14324,14328],{},[92,14325,14326],{},[95,14327,236],{},[100,14329,239],{"encoding":102},[76,14331,14333],{"className":14332,"ariaHidden":108},[107],[76,14334,14336,14339],{"className":14335},[112],[76,14337],{"className":14338,"style":249},[116],[76,14340,236],{"className":14341,"style":253},[121,216]," に含まれる論理記号（",[76,14344,14346,14360],{"className":14345},[79],[76,14347,14349],{"className":14348},[83],[85,14350,14351],{"xmlns":87},[89,14352,14353,14357],{},[92,14354,14355],{},[95,14356,941],{"mathvariant":363},[100,14358,14359],{"encoding":102},"\\neg",[76,14361,14363],{"className":14362,"ariaHidden":108},[107],[76,14364,14366,14369],{"className":14365},[112],[76,14367],{"className":14368,"style":249},[116],[76,14370,941],{"className":14371},[121],[76,14373,14375,14389],{"className":14374},[79],[76,14376,14378],{"className":14377},[83],[85,14379,14380],{"xmlns":87},[89,14381,14382,14386],{},[92,14383,14384],{},[299,14385,1248],{},[100,14387,14388],{"encoding":102},"\\to",[76,14390,14392],{"className":14391,"ariaHidden":108},[107],[76,14393,14395,14399],{"className":14394},[112],[76,14396],{"className":14397,"style":14398},[116],"height:0.3669em;",[76,14400,1248],{"className":14401},[328],[76,14403,14405,14419],{"className":14404},[79],[76,14406,14408],{"className":14407},[83],[85,14409,14410],{"xmlns":87},[89,14411,14412,14416],{},[92,14413,14414],{},[95,14415,2653],{"mathvariant":363},[100,14417,14418],{"encoding":102},"\\exists",[76,14420,14422],{"className":14421,"ariaHidden":108},[107],[76,14423,14425,14428],{"className":14424},[112],[76,14426],{"className":14427,"style":820},[116],[76,14429,2653],{"className":14430},[121],"）の個数に関する帰納法で示す。",[21,14433,14434,14468,14558],{},[24,14435,14436,1027,14439,14467],{},[13,14437,14438],{},"原子文",[76,14440,14442,14455],{"className":14441},[79],[76,14443,14445],{"className":14444},[83],[85,14446,14447],{"xmlns":87},[89,14448,14449,14453],{},[92,14450,14451],{},[95,14452,171],{"mathvariant":97},[100,14454,174],{"encoding":102},[76,14456,14458],{"className":14457,"ariaHidden":108},[107],[76,14459,14461,14464],{"className":14460},[112],[76,14462],{"className":14463,"style":117},[116],[76,14465,171],{"className":14466},[121,122]," の定義そのもの。",[24,14469,14470,14528,14529,14557],{},[13,14471,14472,922,14500],{},[76,14473,14475,14488],{"className":14474},[79],[76,14476,14478],{"className":14477},[83],[85,14479,14480],{"xmlns":87},[89,14481,14482,14486],{},[92,14483,14484],{},[95,14485,941],{"mathvariant":363},[100,14487,14359],{"encoding":102},[76,14489,14491],{"className":14490,"ariaHidden":108},[107],[76,14492,14494,14497],{"className":14493},[112],[76,14495],{"className":14496,"style":249},[116],[76,14498,941],{"className":14499},[121],[76,14501,14503,14516],{"className":14502},[79],[76,14504,14506],{"className":14505},[83],[85,14507,14508],{"xmlns":87},[89,14509,14510,14514],{},[92,14511,14512],{},[299,14513,1248],{},[100,14515,14388],{"encoding":102},[76,14517,14519],{"className":14518,"ariaHidden":108},[107],[76,14520,14522,14525],{"className":14521},[112],[76,14523],{"className":14524,"style":14398},[116],[76,14526,1248],{"className":14527},[328],"：帰納法の仮定と、",[76,14530,14532,14545],{"className":14531},[79],[76,14533,14535],{"className":14534},[83],[85,14536,14537],{"xmlns":87},[89,14538,14539,14543],{},[92,14540,14541],{},[95,14542,8182],{"mathvariant":363},[100,14544,8185],{"encoding":102},[76,14546,14548],{"className":14547,"ariaHidden":108},[107],[76,14549,14551,14554],{"className":14550},[112],[76,14552],{"className":14553,"style":117},[116],[76,14555,8182],{"className":14556},[121]," の否定・含意の性質から従う。",[24,14559,14560,14620,14621],{},[13,14561,14562],{},[76,14563,14565,14590],{"className":14564},[79],[76,14566,14568],{"className":14567},[83],[85,14569,14570],{"xmlns":87},[89,14571,14572,14588],{},[92,14573,14574,14576,14578,14580,14582,14584,14586],{},[95,14575,2653],{"mathvariant":363},[95,14577,2632],{},[2617,14579,5223],{},[95,14581,456],{},[299,14583,2678],{"stretchy":1154},[95,14585,2632],{},[299,14587,2721],{"stretchy":1154},[100,14589,8053],{"encoding":102},[76,14591,14593],{"className":14592,"ariaHidden":108},[107],[76,14594,14596,14599,14602,14605,14608,14611,14614,14617],{"className":14595},[112],[76,14597],{"className":14598,"style":1198},[116],[76,14600,2653],{"className":14601},[121],[76,14603,2632],{"className":14604},[121,216],[76,14606],{"className":14607,"style":2820},[323],[76,14609,456],{"className":14610},[121,216],[76,14612,2678],{"className":14613},[1202],[76,14615,2632],{"className":14616},[121,216],[76,14618,2721],{"className":14619},[1209],"：\n",[21,14622,14623,15024],{},[24,14624,2678,14625,14655,14656,14736,14737,14802,14803,14831,14832,14875,14876,14942,14943,1303],{},[76,14626,14628,14643],{"className":14627},[79],[76,14629,14631],{"className":14630},[83],[85,14632,14633],{"xmlns":87},[89,14634,14635,14640],{},[92,14636,14637],{},[299,14638,14639],{},"⇐",[100,14641,14642],{"encoding":102},"\\Leftarrow",[76,14644,14646],{"className":14645,"ariaHidden":108},[107],[76,14647,14649,14652],{"className":14648},[112],[76,14650],{"className":14651,"style":14398},[116],[76,14653,14639],{"className":14654},[328],") ",[76,14657,14659,14688],{"className":14658},[79],[76,14660,14662],{"className":14661},[83],[85,14663,14664],{"xmlns":87},[89,14665,14666,14686],{},[92,14667,14668,14670,14672,14674,14676,14678,14680,14682,14684],{},[95,14669,2653],{"mathvariant":363},[95,14671,2632],{},[2617,14673,5223],{},[95,14675,456],{},[299,14677,2678],{"stretchy":1154},[95,14679,2632],{},[299,14681,2721],{"stretchy":1154},[299,14683,8262],{},[95,14685,8182],{"mathvariant":363},[100,14687,12967],{"encoding":102},[76,14689,14691,14727],{"className":14690,"ariaHidden":108},[107],[76,14692,14694,14697,14700,14703,14706,14709,14712,14715,14718,14721,14724],{"className":14693},[112],[76,14695],{"className":14696,"style":1198},[116],[76,14698,2653],{"className":14699},[121],[76,14701,2632],{"className":14702},[121,216],[76,14704],{"className":14705,"style":2820},[323],[76,14707,456],{"className":14708},[121,216],[76,14710,2678],{"className":14711},[1202],[76,14713,2632],{"className":14714},[121,216],[76,14716,2721],{"className":14717},[1209],[76,14719],{"className":14720,"style":324},[323],[76,14722,8262],{"className":14723},[328],[76,14725],{"className":14726,"style":324},[323],[76,14728,14730,14733],{"className":14729},[112],[76,14731],{"className":14732,"style":117},[116],[76,14734,8182],{"className":14735},[121]," なら、証人の性質により ",[76,14738,14740,14763],{"className":14739},[79],[76,14741,14743],{"className":14742},[83],[85,14744,14745],{"xmlns":87},[89,14746,14747,14761],{},[92,14748,14749,14751,14753,14755,14757,14759],{},[95,14750,456],{},[299,14752,2678],{"stretchy":1154},[95,14754,8102],{},[299,14756,2721],{"stretchy":1154},[299,14758,8262],{},[95,14760,8182],{"mathvariant":363},[100,14762,13071],{"encoding":102},[76,14764,14766,14793],{"className":14765,"ariaHidden":108},[107],[76,14767,14769,14772,14775,14778,14781,14784,14787,14790],{"className":14768},[112],[76,14770],{"className":14771,"style":1198},[116],[76,14773,456],{"className":14774},[121,216],[76,14776,2678],{"className":14777},[1202],[76,14779,8102],{"className":14780},[121,216],[76,14782,2721],{"className":14783},[1209],[76,14785],{"className":14786,"style":324},[323],[76,14788,8262],{"className":14789},[328],[76,14791],{"className":14792,"style":324},[323],[76,14794,14796,14799],{"className":14795},[112],[76,14797],{"className":14798,"style":117},[116],[76,14800,8182],{"className":14801},[121]," となる定数 ",[76,14804,14806,14819],{"className":14805},[79],[76,14807,14809],{"className":14808},[83],[85,14810,14811],{"xmlns":87},[89,14812,14813,14817],{},[92,14814,14815],{},[95,14816,8102],{},[100,14818,8102],{"encoding":102},[76,14820,14822],{"className":14821,"ariaHidden":108},[107],[76,14823,14825,14828],{"className":14824},[112],[76,14826],{"className":14827,"style":249},[116],[76,14829,8102],{"className":14830},[121,216]," がある。",[76,14833,14835,14854],{"className":14834},[79],[76,14836,14838],{"className":14837},[83],[85,14839,14840],{"xmlns":87},[89,14841,14842,14852],{},[92,14843,14844,14846,14848,14850],{},[95,14845,456],{},[299,14847,2678],{"stretchy":1154},[95,14849,8102],{},[299,14851,2721],{"stretchy":1154},[100,14853,8107],{"encoding":102},[76,14855,14857],{"className":14856,"ariaHidden":108},[107],[76,14858,14860,14863,14866,14869,14872],{"className":14859},[112],[76,14861],{"className":14862,"style":1198},[116],[76,14864,456],{"className":14865},[121,216],[76,14867,2678],{"className":14868},[1202],[76,14870,8102],{"className":14871},[121,216],[76,14873,2721],{"className":14874},[1209]," は論理記号が少ないので、帰納法の仮定より ",[76,14877,14879,14903],{"className":14878},[79],[76,14880,14882],{"className":14881},[83],[85,14883,14884],{"xmlns":87},[89,14885,14886,14900],{},[92,14887,14888,14890,14892,14894,14896,14898],{},[95,14889,171],{"mathvariant":97},[299,14891,301],{},[95,14893,456],{},[299,14895,2678],{"stretchy":1154},[95,14897,8102],{},[299,14899,2721],{"stretchy":1154},[100,14901,14902],{"encoding":102},"\\mathcal{M} \\models \\varphi(c)",[76,14904,14906,14924],{"className":14905,"ariaHidden":108},[107],[76,14907,14909,14912,14915,14918,14921],{"className":14908},[112],[76,14910],{"className":14911,"style":316},[116],[76,14913,171],{"className":14914},[121,122],[76,14916],{"className":14917,"style":324},[323],[76,14919,301],{"className":14920},[328],[76,14922],{"className":14923,"style":324},[323],[76,14925,14927,14930,14933,14936,14939],{"className":14926},[112],[76,14928],{"className":14929,"style":1198},[116],[76,14931,456],{"className":14932},[121,216],[76,14934,2678],{"className":14935},[1202],[76,14937,8102],{"className":14938},[121,216],[76,14940,2721],{"className":14941},[1209],"。よって ",[76,14944,14946,14976],{"className":14945},[79],[76,14947,14949],{"className":14948},[83],[85,14950,14951],{"xmlns":87},[89,14952,14953,14973],{},[92,14954,14955,14957,14959,14961,14963,14965,14967,14969,14971],{},[95,14956,171],{"mathvariant":97},[299,14958,301],{},[95,14960,2653],{"mathvariant":363},[95,14962,2632],{},[2617,14964,5223],{},[95,14966,456],{},[299,14968,2678],{"stretchy":1154},[95,14970,2632],{},[299,14972,2721],{"stretchy":1154},[100,14974,14975],{"encoding":102},"\\mathcal{M} \\models \\exists x\\, \\varphi(x)",[76,14977,14979,14997],{"className":14978,"ariaHidden":108},[107],[76,14980,14982,14985,14988,14991,14994],{"className":14981},[112],[76,14983],{"className":14984,"style":316},[116],[76,14986,171],{"className":14987},[121,122],[76,14989],{"className":14990,"style":324},[323],[76,14992,301],{"className":14993},[328],[76,14995],{"className":14996,"style":324},[323],[76,14998,15000,15003,15006,15009,15012,15015,15018,15021],{"className":14999},[112],[76,15001],{"className":15002,"style":1198},[116],[76,15004,2653],{"className":15005},[121],[76,15007,2632],{"className":15008},[121,216],[76,15010],{"className":15011,"style":2820},[323],[76,15013,456],{"className":15014},[121,216],[76,15016,2678],{"className":15017},[1202],[76,15019,2632],{"className":15020},[121,216],[76,15022,2721],{"className":15023},[1209],[24,15025,2678,15026,14655,15056,15136,15137,871,15166,15232,15233,1303,15299,15409,15410,1303],{},[76,15027,15029,15044],{"className":15028},[79],[76,15030,15032],{"className":15031},[83],[85,15033,15034],{"xmlns":87},[89,15035,15036,15041],{},[92,15037,15038],{},[299,15039,15040],{},"⇒",[100,15042,15043],{"encoding":102},"\\Rightarrow",[76,15045,15047],{"className":15046,"ariaHidden":108},[107],[76,15048,15050,15053],{"className":15049},[112],[76,15051],{"className":15052,"style":14398},[116],[76,15054,15040],{"className":15055},[328],[76,15057,15059,15088],{"className":15058},[79],[76,15060,15062],{"className":15061},[83],[85,15063,15064],{"xmlns":87},[89,15065,15066,15086],{},[92,15067,15068,15070,15072,15074,15076,15078,15080,15082,15084],{},[95,15069,171],{"mathvariant":97},[299,15071,301],{},[95,15073,2653],{"mathvariant":363},[95,15075,2632],{},[2617,15077,5223],{},[95,15079,456],{},[299,15081,2678],{"stretchy":1154},[95,15083,2632],{},[299,15085,2721],{"stretchy":1154},[100,15087,14975],{"encoding":102},[76,15089,15091,15109],{"className":15090,"ariaHidden":108},[107],[76,15092,15094,15097,15100,15103,15106],{"className":15093},[112],[76,15095],{"className":15096,"style":316},[116],[76,15098,171],{"className":15099},[121,122],[76,15101],{"className":15102,"style":324},[323],[76,15104,301],{"className":15105},[328],[76,15107],{"className":15108,"style":324},[323],[76,15110,15112,15115,15118,15121,15124,15127,15130,15133],{"className":15111},[112],[76,15113],{"className":15114,"style":1198},[116],[76,15116,2653],{"className":15117},[121],[76,15119,2632],{"className":15120},[121,216],[76,15122],{"className":15123,"style":2820},[323],[76,15125,456],{"className":15126},[121,216],[76,15128,2678],{"className":15129},[1202],[76,15131,2632],{"className":15132},[121,216],[76,15134,2721],{"className":15135},[1209]," なら、領域の要素、つまりある閉項 ",[76,15138,15140,15153],{"className":15139},[79],[76,15141,15143],{"className":15142},[83],[85,15144,15145],{"xmlns":87},[89,15146,15147,15151],{},[92,15148,15149],{},[95,15150,13302],{},[100,15152,13302],{"encoding":102},[76,15154,15156],{"className":15155,"ariaHidden":108},[107],[76,15157,15159,15163],{"className":15158},[112],[76,15160],{"className":15161,"style":15162},[116],"height:0.6151em;",[76,15164,13302],{"className":15165},[121,216],[76,15167,15169,15193],{"className":15168},[79],[76,15170,15172],{"className":15171},[83],[85,15173,15174],{"xmlns":87},[89,15175,15176,15190],{},[92,15177,15178,15180,15182,15184,15186,15188],{},[95,15179,171],{"mathvariant":97},[299,15181,301],{},[95,15183,456],{},[299,15185,2678],{"stretchy":1154},[95,15187,13302],{},[299,15189,2721],{"stretchy":1154},[100,15191,15192],{"encoding":102},"\\mathcal{M} \\models \\varphi(t)",[76,15194,15196,15214],{"className":15195,"ariaHidden":108},[107],[76,15197,15199,15202,15205,15208,15211],{"className":15198},[112],[76,15200],{"className":15201,"style":316},[116],[76,15203,171],{"className":15204},[121,122],[76,15206],{"className":15207,"style":324},[323],[76,15209,301],{"className":15210},[328],[76,15212],{"className":15213,"style":324},[323],[76,15215,15217,15220,15223,15226,15229],{"className":15216},[112],[76,15218],{"className":15219,"style":1198},[116],[76,15221,456],{"className":15222},[121,216],[76,15224,2678],{"className":15225},[1202],[76,15227,13302],{"className":15228},[121,216],[76,15230,2721],{"className":15231},[1209],"。帰納法の仮定より ",[76,15234,15236,15260],{"className":15235},[79],[76,15237,15239],{"className":15238},[83],[85,15240,15241],{"xmlns":87},[89,15242,15243,15257],{},[92,15244,15245,15247,15249,15251,15253,15255],{},[95,15246,456],{},[299,15248,2678],{"stretchy":1154},[95,15250,13302],{},[299,15252,2721],{"stretchy":1154},[299,15254,8262],{},[95,15256,8182],{"mathvariant":363},[100,15258,15259],{"encoding":102},"\\varphi(t) \\in \\Delta",[76,15261,15263,15290],{"className":15262,"ariaHidden":108},[107],[76,15264,15266,15269,15272,15275,15278,15281,15284,15287],{"className":15265},[112],[76,15267],{"className":15268,"style":1198},[116],[76,15270,456],{"className":15271},[121,216],[76,15273,2678],{"className":15274},[1202],[76,15276,13302],{"className":15277},[121,216],[76,15279,2721],{"className":15280},[1209],[76,15282],{"className":15283,"style":324},[323],[76,15285,8262],{"className":15286},[328],[76,15288],{"className":15289,"style":324},[323],[76,15291,15293,15296],{"className":15292},[112],[76,15294],{"className":15295,"style":117},[116],[76,15297,8182],{"className":15298},[121],[76,15300,15302,15340],{"className":15301},[79],[76,15303,15305],{"className":15304},[83],[85,15306,15307],{"xmlns":87},[89,15308,15309,15337],{},[92,15310,15311,15313,15315,15317,15319,15321,15323,15325,15327,15329,15331,15333,15335],{},[299,15312,805],{},[95,15314,456],{},[299,15316,2678],{"stretchy":1154},[95,15318,13302],{},[299,15320,2721],{"stretchy":1154},[299,15322,1248],{},[95,15324,2653],{"mathvariant":363},[95,15326,2632],{},[2617,15328,5223],{},[95,15330,456],{},[299,15332,2678],{"stretchy":1154},[95,15334,2632],{},[299,15336,2721],{"stretchy":1154},[100,15338,15339],{"encoding":102},"\\vdash \\varphi(t) \\to \\exists x\\, \\varphi(x)",[76,15341,15343,15355,15382],{"className":15342,"ariaHidden":108},[107],[76,15344,15346,15349,15352],{"className":15345},[112],[76,15347],{"className":15348,"style":820},[116],[76,15350,805],{"className":15351},[328],[76,15353],{"className":15354,"style":324},[323],[76,15356,15358,15361,15364,15367,15370,15373,15376,15379],{"className":15357},[112],[76,15359],{"className":15360,"style":1198},[116],[76,15362,456],{"className":15363},[121,216],[76,15365,2678],{"className":15366},[1202],[76,15368,13302],{"className":15369},[121,216],[76,15371,2721],{"className":15372},[1209],[76,15374],{"className":15375,"style":324},[323],[76,15377,1248],{"className":15378},[328],[76,15380],{"className":15381,"style":324},[323],[76,15383,15385,15388,15391,15394,15397,15400,15403,15406],{"className":15384},[112],[76,15386],{"className":15387,"style":1198},[116],[76,15389,2653],{"className":15390},[121],[76,15392,2632],{"className":15393},[121,216],[76,15395],{"className":15396,"style":2820},[323],[76,15398,456],{"className":15399},[121,216],[76,15401,2678],{"className":15402},[1202],[76,15404,2632],{"className":15405},[121,216],[76,15407,2721],{"className":15408},[1209]," なので、",[76,15411,15413,15442],{"className":15412},[79],[76,15414,15416],{"className":15415},[83],[85,15417,15418],{"xmlns":87},[89,15419,15420,15440],{},[92,15421,15422,15424,15426,15428,15430,15432,15434,15436,15438],{},[95,15423,2653],{"mathvariant":363},[95,15425,2632],{},[2617,15427,5223],{},[95,15429,456],{},[299,15431,2678],{"stretchy":1154},[95,15433,2632],{},[299,15435,2721],{"stretchy":1154},[299,15437,8262],{},[95,15439,8182],{"mathvariant":363},[100,15441,12967],{"encoding":102},[76,15443,15445,15481],{"className":15444,"ariaHidden":108},[107],[76,15446,15448,15451,15454,15457,15460,15463,15466,15469,15472,15475,15478],{"className":15447},[112],[76,15449],{"className":15450,"style":1198},[116],[76,15452,2653],{"className":15453},[121],[76,15455,2632],{"className":15456},[121,216],[76,15458],{"className":15459,"style":2820},[323],[76,15461,456],{"className":15462},[121,216],[76,15464,2678],{"className":15465},[1202],[76,15467,2632],{"className":15468},[121,216],[76,15470,2721],{"className":15471},[1209],[76,15473],{"className":15474,"style":324},[323],[76,15476,8262],{"className":15477},[328],[76,15479],{"className":15480,"style":324},[323],[76,15482,15484,15487],{"className":15483},[112],[76,15485],{"className":15486,"style":117},[116],[76,15488,8182],{"className":15489},[121],[10,15491,15492,15409,15544,15572,15573,15601,15602,15630],{},[76,15493,15495,15513],{"className":15494},[79],[76,15496,15498],{"className":15497},[83],[85,15499,15500],{"xmlns":87},[89,15501,15502,15510],{},[92,15503,15504,15506,15508],{},[95,15505,364],{"mathvariant":363},[299,15507,9686],{},[95,15509,8182],{"mathvariant":363},[100,15511,15512],{"encoding":102},"\\Gamma \\subseteq \\Delta",[76,15514,15516,15535],{"className":15515,"ariaHidden":108},[107],[76,15517,15519,15523,15526,15529,15532],{"className":15518},[112],[76,15520],{"className":15521,"style":15522},[116],"height:0.8193em;vertical-align:-0.136em;",[76,15524,364],{"className":15525},[121],[76,15527],{"className":15528,"style":324},[323],[76,15530,9686],{"className":15531},[328],[76,15533],{"className":15534,"style":324},[323],[76,15536,15538,15541],{"className":15537},[112],[76,15539],{"className":15540,"style":117},[116],[76,15542,8182],{"className":15543},[121],[76,15545,15547,15560],{"className":15546},[79],[76,15548,15550],{"className":15549},[83],[85,15551,15552],{"xmlns":87},[89,15553,15554,15558],{},[92,15555,15556],{},[95,15557,171],{"mathvariant":97},[100,15559,174],{"encoding":102},[76,15561,15563],{"className":15562,"ariaHidden":108},[107],[76,15564,15566,15569],{"className":15565},[112],[76,15567],{"className":15568,"style":117},[116],[76,15570,171],{"className":15571},[121,122],"（を言語 ",[76,15574,15576,15589],{"className":15575},[79],[76,15577,15579],{"className":15578},[83],[85,15580,15581],{"xmlns":87},[89,15582,15583,15587],{},[92,15584,15585],{},[95,15586,98],{"mathvariant":97},[100,15588,103],{"encoding":102},[76,15590,15592],{"className":15591,"ariaHidden":108},[107],[76,15593,15595,15598],{"className":15594},[112],[76,15596],{"className":15597,"style":117},[116],[76,15599,98],{"className":15600},[121,122]," に制限したもの）は ",[76,15603,15605,15618],{"className":15604},[79],[76,15606,15608],{"className":15607},[83],[85,15609,15610],{"xmlns":87},[89,15611,15612,15616],{},[92,15613,15614],{},[95,15615,364],{"mathvariant":363},[100,15617,367],{"encoding":102},[76,15619,15621],{"className":15620,"ariaHidden":108},[107],[76,15622,15624,15627],{"className":15623},[112],[76,15625],{"className":15626,"style":117},[116],[76,15628,364],{"className":15629},[121]," のモデルです。これでモデル存在定理、したがって強い形の完全性定理が示されました。",[15632,15633,15635],"tip",{"title":15634},"証明の要は「すべての要素に名前がある」こと",[10,15636,15637,15638,15666,15667,15695,15696,15699,15700,15728,15729,15732],{},"真理補題の ",[76,15639,15641,15654],{"className":15640},[79],[76,15642,15644],{"className":15643},[83],[85,15645,15646],{"xmlns":87},[89,15647,15648,15652],{},[92,15649,15650],{},[95,15651,2653],{"mathvariant":363},[100,15653,14418],{"encoding":102},[76,15655,15657],{"className":15656,"ariaHidden":108},[107],[76,15658,15660,15663],{"className":15659},[112],[76,15661],{"className":15662,"style":820},[116],[76,15664,2653],{"className":15665},[121]," の場合で、(",[76,15668,15670,15683],{"className":15669},[79],[76,15671,15673],{"className":15672},[83],[85,15674,15675],{"xmlns":87},[89,15676,15677,15681],{},[92,15678,15679],{},[299,15680,15040],{},[100,15682,15043],{"encoding":102},[76,15684,15686],{"className":15685,"ariaHidden":108},[107],[76,15687,15689,15692],{"className":15688},[112],[76,15690],{"className":15691,"style":14398},[116],[76,15693,15040],{"className":15694},[328],") が通るのは、モデルの要素がすべて閉項、つまり ",[13,15697,15698],{},"名前を持っている"," からです。そして (",[76,15701,15703,15716],{"className":15702},[79],[76,15704,15706],{"className":15705},[83],[85,15707,15708],{"xmlns":87},[89,15709,15710,15714],{},[92,15711,15712],{},[299,15713,14639],{},[100,15715,14642],{"encoding":102},[76,15717,15719],{"className":15718,"ariaHidden":108},[107],[76,15720,15722,15725],{"className":15721},[112],[76,15723],{"className":15724,"style":14398},[116],[76,15726,14639],{"className":15727},[328],") が通るのは、ヘンキン公理によって、存在文には必ず ",[13,15730,15731],{},"名前の付いた証人"," がいるからです。証人の追加は、この2つをかみ合わせるための工夫です。",[68,15734,15736],{"id":15735},"_34-ヘンキン流の特徴","3.4 ヘンキン流の特徴",[21,15738,15739,15745,15838,15847],{},[24,15740,15741,15744],{},[13,15742,15743],{},"標準形が不要","：論理式の形を変換せずに、理論全体を一度に扱えます。",[24,15746,15747,15750,15751,15779,15780,15808,15809,15837],{},[13,15748,15749],{},"強い形を直接示す","：無限個の文からなる理論 ",[76,15752,15754,15767],{"className":15753},[79],[76,15755,15757],{"className":15756},[83],[85,15758,15759],{"xmlns":87},[89,15760,15761,15765],{},[92,15762,15763],{},[95,15764,364],{"mathvariant":363},[100,15766,367],{"encoding":102},[76,15768,15770],{"className":15769,"ariaHidden":108},[107],[76,15771,15773,15776],{"className":15772},[112],[76,15774],{"className":15775,"style":117},[116],[76,15777,364],{"className":15778},[121]," を最初から扱えるので、コンパクト性定理も直ちに得られます。",[76,15781,15783,15796],{"className":15782},[79],[76,15784,15786],{"className":15785},[83],[85,15787,15788],{"xmlns":87},[89,15789,15790,15794],{},[92,15791,15792],{},[95,15793,364],{"mathvariant":363},[100,15795,367],{"encoding":102},[76,15797,15799],{"className":15798,"ariaHidden":108},[107],[76,15800,15802,15805],{"className":15801},[112],[76,15803],{"className":15804,"style":117},[116],[76,15806,364],{"className":15807},[121]," の有限部分がすべてモデルを持てば、",[76,15810,15812,15825],{"className":15811},[79],[76,15813,15815],{"className":15814},[83],[85,15816,15817],{"xmlns":87},[89,15818,15819,15823],{},[92,15820,15821],{},[95,15822,364],{"mathvariant":363},[100,15824,367],{"encoding":102},[76,15826,15828],{"className":15827,"ariaHidden":108},[107],[76,15829,15831,15834],{"className":15830},[112],[76,15832],{"className":15833,"style":117},[116],[76,15835,364],{"className":15836},[121]," は無矛盾だからです（導出は有限個の前提しか使わない）。",[24,15839,15840,15843,15844,15846],{},[13,15841,15842],{},"非可算の言語にも拡張しやすい","：ヘンキン自身が論文で、この方法が非可算個の記号を持つ体系にも一般化できることを指摘しています（文献 ",[31,15845,50],{"href":49},"）。文を並べる部分を、ツォルンの補題（または超限帰納法）に置き換えるだけです。モデルの大きさは言語の大きさで抑えられ、レーヴェンハイム＝スコーレムの定理の一般形も得られます。",[24,15848,15849,15852,15853,15857,15858,15861],{},[13,15850,15851],{},"他の論理にも応用できる","：同じ考え方で、ヘンキンは1950年に高階論理（型理論）の一般モデルに対する完全性を示しました（文献 ",[31,15854,15856],{"href":15855},"#ref-4","[4]","）。様相論理の ",[13,15859,15860],{},"正準モデル"," による完全性証明も、極大無矛盾集合を「世界」とする、同じ系統の方法です。",[63,15863,15865],{"id":15864},"_4-2つの証明の比較","4. 2つの証明の比較",[15867,15868,15869,15885],"table",{},[15870,15871,15872],"thead",{},[15873,15874,15875,15879,15882],"tr",{},[15876,15877,15878],"th",{},"観点",[15876,15880,15881],{},"ゲーデル流（1929–1930）",[15876,15883,15884],{},"ヘンキン流（1949）",[15886,15887,15888,15900,15911,16047,16058,16069,16254],"tbody",{},[15873,15889,15890,15894,15897],{},[15891,15892,15893],"td",{},"直接示す形",[15891,15895,15896],{},"各論理式は反駁可能か充足可能か（弱い形）",[15891,15898,15899],{},"無矛盾な集合はモデルを持つ（強い形）",[15873,15901,15902,15905,15908],{},[15891,15903,15904],{},"出発点",[15891,15906,15907],{},"1つの論理式",[15891,15909,15910],{},"文の集合（理論）全体",[15873,15912,15913,15916,16044],{},[15891,15914,15915],{},"前処理",[15891,15917,15918,15919,16043],{},"標準形 ",[76,15920,15922,15952],{"className":15921},[79],[76,15923,15925],{"className":15924},[83],[85,15926,15927],{"xmlns":87},[89,15928,15929,15949],{},[92,15930,15931,15933,15939,15941,15947],{},[95,15932,2626],{"mathvariant":363},[3234,15934,15935,15937],{"accent":108},[95,15936,2632],{},[299,15938,3240],{},[95,15940,2653],{"mathvariant":363},[3234,15942,15943,15945],{"accent":108},[95,15944,2658],{},[299,15946,3240],{},[95,15948,1158],{},[100,15950,15951],{"encoding":102},"\\forall \\bar{x} \\exists \\bar{y} \\psi",[76,15953,15955],{"className":15954,"ariaHidden":108},[107],[76,15956,15958,15961,15964,15995,15998,16040],{"className":15957},[112],[76,15959],{"className":15960,"style":1281},[116],[76,15962,2626],{"className":15963},[121],[76,15965,15967],{"className":15966},[121,3257],[76,15968,15970],{"className":15969},[2774],[76,15971,15973],{"className":15972},[2779],[76,15974,15976,15984],{"className":15975,"style":3253},[2783],[76,15977,15978,15981],{"style":3269},[76,15979],{"className":15980,"style":3273},[2791],[76,15982,2632],{"className":15983},[121,216],[76,15985,15986,15989],{"style":3269},[76,15987],{"className":15988,"style":3273},[2791],[76,15990,15992],{"className":15991,"style":3286},[3285],[76,15993,3240],{"className":15994},[121],[76,15996,2653],{"className":15997},[121],[76,15999,16001],{"className":16000},[121,3257],[76,16002,16004,16032],{"className":16003},[2774,2775],[76,16005,16007,16029],{"className":16006},[2779],[76,16008,16010,16018],{"className":16009,"style":3253},[2783],[76,16011,16012,16015],{"style":3269},[76,16013],{"className":16014,"style":3273},[2791],[76,16016,2658],{"className":16017,"style":253},[121,216],[76,16019,16020,16023],{"style":3269},[76,16021],{"className":16022,"style":3273},[2791],[76,16024,16026],{"className":16025,"style":3350},[3285],[76,16027,3240],{"className":16028},[121],[76,16030,2807],{"className":16031},[2806],[76,16033,16035],{"className":16034},[2779],[76,16036,16038],{"className":16037,"style":3363},[2783],[76,16039],{},[76,16041,1158],{"className":16042,"style":253},[121,216]," への変換",[15891,16045,16046],{},"証人となる定数とヘンキン公理の追加",[15873,16048,16049,16052,16055],{},[15891,16050,16051],{},"中心となる道具",[15891,16053,16054],{},"命題論理への帰着とケーニヒの補題",[15891,16056,16057],{},"リンデンバウムの補題（極大無矛盾集合）",[15873,16059,16060,16063,16066],{},[15891,16061,16062],{},"モデルの要素",[15891,16064,16065],{},"変数",[15891,16067,16068],{},"閉項（の同値類）",[15873,16070,16071,16074,16147],{},[15891,16072,16073],{},"鍵となる事実",[15891,16075,16076,16077,16146],{},"代入例の連言 ",[76,16078,16080,16097],{"className":16079},[79],[76,16081,16083],{"className":16082},[83],[85,16084,16085],{"xmlns":87},[89,16086,16087,16095],{},[92,16088,16089],{},[2628,16090,16091,16093],{},[95,16092,4535],{},[95,16094,4400],{},[100,16096,4839],{"encoding":102},[76,16098,16100],{"className":16099,"ariaHidden":108},[107],[76,16101,16103,16106],{"className":16102},[112],[76,16104],{"className":16105,"style":4583},[116],[76,16107,16109,16112],{"className":16108},[121],[76,16110,4535],{"className":16111,"style":4590},[121,216],[76,16113,16115],{"className":16114},[2770],[76,16116,16118,16138],{"className":16117},[2774,2775],[76,16119,16121,16135],{"className":16120},[2779],[76,16122,16124],{"className":16123,"style":2950},[2783],[76,16125,16126,16129],{"style":4605},[76,16127],{"className":16128,"style":2792},[2791],[76,16130,16132],{"className":16131},[2796,2797,2798,2799],[76,16133,4400],{"className":16134},[121,216,2799],[76,16136,2807],{"className":16137},[2806],[76,16139,16141],{"className":16140},[2779],[76,16142,16144],{"className":16143,"style":2814},[2783],[76,16145],{}," がすべて充足可能",[15891,16148,16149,16150],{},"真理補題 ",[76,16151,16153,16182],{"className":16152},[79],[76,16154,16156],{"className":16155},[83],[85,16157,16158],{"xmlns":87},[89,16159,16160,16180],{},[92,16161,16162,16164,16166,16168,16170,16172,16174,16176,16178],{},[95,16163,171],{"mathvariant":97},[299,16165,301],{},[95,16167,236],{},[2617,16169,2619],{},[299,16171,7011],{},[2617,16173,2619],{},[95,16175,236],{},[299,16177,8262],{},[95,16179,8182],{"mathvariant":363},[100,16181,8267],{"encoding":102},[76,16183,16185,16203,16227,16245],{"className":16184,"ariaHidden":108},[107],[76,16186,16188,16191,16194,16197,16200],{"className":16187},[112],[76,16189],{"className":16190,"style":316},[116],[76,16192,171],{"className":16193},[121,122],[76,16195],{"className":16196,"style":324},[323],[76,16198,301],{"className":16199},[328],[76,16201],{"className":16202,"style":324},[323],[76,16204,16206,16209,16212,16215,16218,16221,16224],{"className":16205},[112],[76,16207],{"className":16208,"style":7026},[116],[76,16210,236],{"className":16211,"style":253},[121,216],[76,16213],{"className":16214,"style":324},[323],[76,16216],{"className":16217,"style":324},[323],[76,16219,7011],{"className":16220},[328],[76,16222],{"className":16223,"style":324},[323],[76,16225],{"className":16226,"style":324},[323],[76,16228,16230,16233,16236,16239,16242],{"className":16229},[112],[76,16231],{"className":16232,"style":8319},[116],[76,16234,236],{"className":16235,"style":253},[121,216],[76,16237],{"className":16238,"style":324},[323],[76,16240,8262],{"className":16241},[328],[76,16243],{"className":16244,"style":324},[323],[76,16246,16248,16251],{"className":16247},[112],[76,16249],{"className":16250,"style":117},[116],[76,16252,8182],{"className":16253},[121],[15873,16255,16256,16259,16262],{},[15891,16257,16258],{},"一般化のしやすさ",[15891,16260,16261],{},"可算の言語が前提",[15891,16263,16264],{},"非可算の言語、高階論理、様相論理へ拡張しやすい",[10,16266,16267,16268,16271],{},"一見まったく違う証明ですが、どちらも ",[13,16269,16270],{},"記号そのもの（変数や閉項）を要素とする構造を作る"," という点では共通しています。「構文から意味を作る」ことこそが、完全性定理の証明の本質だといえます。",[6334,16273,16275],{"title":16274},"どちらも非構成的な一歩を含む",[10,16276,16277,16278,16347],{},"ゲーデル流では、各 ",[76,16279,16281,16298],{"className":16280},[79],[76,16282,16284],{"className":16283},[83],[85,16285,16286],{"xmlns":87},[89,16287,16288,16296],{},[92,16289,16290],{},[2628,16291,16292,16294],{},[95,16293,4535],{},[95,16295,4400],{},[100,16297,4839],{"encoding":102},[76,16299,16301],{"className":16300,"ariaHidden":108},[107],[76,16302,16304,16307],{"className":16303},[112],[76,16305],{"className":16306,"style":4583},[116],[76,16308,16310,16313],{"className":16309},[121],[76,16311,4535],{"className":16312,"style":4590},[121,216],[76,16314,16316],{"className":16315},[2770],[76,16317,16319,16339],{"className":16318},[2774,2775],[76,16320,16322,16336],{"className":16321},[2779],[76,16323,16325],{"className":16324,"style":2950},[2783],[76,16326,16327,16330],{"style":4605},[76,16328],{"className":16329,"style":2792},[2791],[76,16331,16333],{"className":16332},[2796,2797,2798,2799],[76,16334,4400],{"className":16335},[121,216,2799],[76,16337,2807],{"className":16338},[2806],[76,16340,16342],{"className":16341},[2779],[76,16343,16345],{"className":16344,"style":2814},[2783],[76,16346],{}," が充足可能かどうか自体は命題論理なので機械的に判定できます。しかし、ケーニヒの補題で枝を選ぶには「その先に無限に延びる枝があるか」を知る必要があり、これは一般には判定できません。ヘンキン流のリンデンバウムの補題も、「無矛盾か」という一般には判定できない性質に基づいて選択を繰り返します。そのため、どちらの証明も、モデルを具体的に計算する手順を与えるわけではありません。",[63,16349,16351],{"id":16350},"_5-その後の展開","5. その後の展開",[10,16353,16354],{},"ヘンキンの後も、完全性定理にはさまざまな証明が与えられました。",[21,16356,16357,16373],{},[24,16358,16359,16362,16363,16367,16368,16372],{},[13,16360,16361],{},"ラショーヴァとシコルスキ（1950年）","：論理式の同値類がなすブール代数を考え、その上の極大フィルター（超フィルター）からモデルを作る、代数的な証明（文献 ",[31,16364,16366],{"href":16365},"#ref-5","[5]","）。可算個の上限・下限だけを保てばよいことに注目し、ベールのカテゴリー定理という位相的な道具を使っています（文献 ",[31,16369,16371],{"href":16370},"#ref-8","[8]","）。なお、この論文は Fundamenta Mathematicae の1950年の巻に収められていますが、実際の刊行は1951年です。",[24,16374,16375,16378,16379,35,16383,16387,16388,40],{},[13,16376,16377],{},"ベートとヒンティッカ（1955年）","：証明を探す手続きそのものから、失敗した探索の跡としてモデルを読み取る、タブロー（意味論的タブロー、モデル集合）による方法（文献 ",[31,16380,16382],{"href":16381},"#ref-6","[6]",[31,16384,16386],{"href":16385},"#ref-7","[7]","）。ベートは同じ方法を1954年にはすでに考えていたとされます（文献 ",[31,16389,16391],{"href":16390},"#ref-14","[14]",[10,16393,16394,16395,40],{},"特にタブローの方法は、「証明を探す」ことと「反例となるモデルを探す」ことが表裏一体であることを示しており、自動定理証明でも使われています（文献 ",[31,16396,16398],{"href":16397},"#ref-15","[15]",[63,16400,16401],{"id":16401},"まとめ",[21,16403,16404,16411,16486,16625,16632],{},[24,16405,16406,16407,16410],{},"完全性定理は、一階述語論理で ",[13,16408,16409],{},"「正しい」と「証明できる」が一致する"," ことを述べる。",[24,16412,16413,16415,16416,16485],{},[13,16414,28],{}," は、論理式を標準形にし、命題論理式の列 ",[76,16417,16419,16436],{"className":16418},[79],[76,16420,16422],{"className":16421},[83],[85,16423,16424],{"xmlns":87},[89,16425,16426,16434],{},[92,16427,16428],{},[2628,16429,16430,16432],{},[95,16431,4535],{},[95,16433,4400],{},[100,16435,4839],{"encoding":102},[76,16437,16439],{"className":16438,"ariaHidden":108},[107],[76,16440,16442,16445],{"className":16441},[112],[76,16443],{"className":16444,"style":4583},[116],[76,16446,16448,16451],{"className":16447},[121],[76,16449,4535],{"className":16450,"style":4590},[121,216],[76,16452,16454],{"className":16453},[2770],[76,16455,16457,16477],{"className":16456},[2774,2775],[76,16458,16460,16474],{"className":16459},[2779],[76,16461,16463],{"className":16462,"style":2950},[2783],[76,16464,16465,16468],{"style":4605},[76,16466],{"className":16467,"style":2792},[2791],[76,16469,16471],{"className":16470},[2796,2797,2798,2799],[76,16472,4400],{"className":16473},[121,216,2799],[76,16475,2807],{"className":16476},[2806],[76,16478,16480],{"className":16479},[2779],[76,16481,16483],{"className":16482,"style":2814},[2783],[76,16484],{}," の二者択一（どれかが矛盾するか、すべて充足可能か）に帰着させた。後者の場合は、ケーニヒの補題で変数を要素とする可算モデルを作る。",[24,16487,16488,16490,16491,16519,16520,16624],{},[13,16489,45],{}," は、証人の追加とリンデンバウムの補題で極大無矛盾集合 ",[76,16492,16494,16507],{"className":16493},[79],[76,16495,16497],{"className":16496},[83],[85,16498,16499],{"xmlns":87},[89,16500,16501,16505],{},[92,16502,16503],{},[95,16504,8182],{"mathvariant":363},[100,16506,8185],{"encoding":102},[76,16508,16510],{"className":16509,"ariaHidden":108},[107],[76,16511,16513,16516],{"className":16512},[112],[76,16514],{"className":16515,"style":117},[116],[76,16517,8182],{"className":16518},[121]," を作り、閉項を要素とするモデルで ",[76,16521,16523,16552],{"className":16522},[79],[76,16524,16526],{"className":16525},[83],[85,16527,16528],{"xmlns":87},[89,16529,16530,16550],{},[92,16531,16532,16534,16536,16538,16540,16542,16544,16546,16548],{},[95,16533,171],{"mathvariant":97},[299,16535,301],{},[95,16537,236],{},[2617,16539,2619],{},[299,16541,7011],{},[2617,16543,2619],{},[95,16545,236],{},[299,16547,8262],{},[95,16549,8182],{"mathvariant":363},[100,16551,8267],{"encoding":102},[76,16553,16555,16573,16597,16615],{"className":16554,"ariaHidden":108},[107],[76,16556,16558,16561,16564,16567,16570],{"className":16557},[112],[76,16559],{"className":16560,"style":316},[116],[76,16562,171],{"className":16563},[121,122],[76,16565],{"className":16566,"style":324},[323],[76,16568,301],{"className":16569},[328],[76,16571],{"className":16572,"style":324},[323],[76,16574,16576,16579,16582,16585,16588,16591,16594],{"className":16575},[112],[76,16577],{"className":16578,"style":7026},[116],[76,16580,236],{"className":16581,"style":253},[121,216],[76,16583],{"className":16584,"style":324},[323],[76,16586],{"className":16587,"style":324},[323],[76,16589,7011],{"className":16590},[328],[76,16592],{"className":16593,"style":324},[323],[76,16595],{"className":16596,"style":324},[323],[76,16598,16600,16603,16606,16609,16612],{"className":16599},[112],[76,16601],{"className":16602,"style":8319},[116],[76,16604,236],{"className":16605,"style":253},[121,216],[76,16607],{"className":16608,"style":324},[323],[76,16610,8262],{"className":16611},[328],[76,16613],{"className":16614,"style":324},[323],[76,16616,16618,16621],{"className":16617},[112],[76,16619],{"className":16620,"style":117},[116],[76,16622,8182],{"className":16623},[121]," を示した。",[24,16626,16627,16628,16631],{},"どちらも ",[13,16629,16630],{},"構文から意味（モデル）を作る","。ヘンキン流は一般化しやすく、現代の論理学の標準的な道具になっている。",[24,16633,16634,16635,16639],{},"年表で20世紀の流れを見るなら、",[31,16636,16638],{"href":16637},"\u002Ftimeline\u002Fmathematical-logic-20th-century","20世紀の数理論理学の年表"," もあわせてどうぞ。",[63,16641,16642],{"id":16642},"参考文献",[10,16644,16645,16646,16648],{},"本文中の「文献 ",[76,16647,4400],{},"」は、以下の番号に対応します。書誌情報は、出版社・データベースの記録で確認しました（2026年11月14日）。",[68,16650,16652],{"id":16651},"一次文献原論文","一次文献（原論文）",[8017,16654,16655,16672,16688,16704,16717,16733,16744],{},[24,16656,16657,16661,16662,16666,16667,16671],{},[76,16658,16660],{"id":16659},"ref-1","K. Gödel",", \"Die Vollständigkeit der Axiome des logischen Funktionenkalküls\", ",[16663,16664,16665],"em",{},"Monatshefte für Mathematik und Physik",", 37 (1930), pp. 349–360.（ゲーデルの博士論文を書き直したもの。英訳は文献 ",[31,16668,16670],{"href":16669},"#ref-2","[2]"," に収録）",[24,16673,16674,16677,16678,16681,16682],{},[76,16675,16660],{"id":16676},"ref-2",", ",[16663,16679,16680],{},"Kurt Gödel: Collected Works, Volume I: Publications 1929–1936",", S. Feferman ほか編, Oxford University Press, 1986. ",[31,16683,16687],{"href":16684,"rel":16685},"https:\u002F\u002Facademic.oup.com\u002Fbook\u002F55022",[16686],"nofollow","出版社のページ",[24,16689,16690,16694,16695,16698,16699],{},[76,16691,16693],{"id":16692},"ref-3","L. Henkin",", \"The Completeness of the First-Order Functional Calculus\", ",[16663,16696,16697],{},"The Journal of Symbolic Logic",", 14(3) (1949), pp. 159–166. doi:10.2307\u002F2267044 ",[31,16700,16703],{"href":16701,"rel":16702},"https:\u002F\u002Fwww.cambridge.org\u002Fcore\u002Fjournals\u002Fjournal-of-symbolic-logic\u002Farticle\u002Fcompleteness-of-the-firstorder-functional-calculus\u002F2F7FFE04841B89975F5EA447A01481CD",[16686],"Cambridge Core",[24,16705,16706,16709,16710,16712,16713],{},[76,16707,16693],{"id":16708},"ref-4",", \"Completeness in the Theory of Types\", ",[16663,16711,16697],{},", 15(2) (1950), pp. 81–91. ",[31,16714,16703],{"href":16715,"rel":16716},"https:\u002F\u002Fwww.cambridge.org\u002Fcore\u002Fjournals\u002Fjournal-of-symbolic-logic\u002Farticle\u002Fcompleteness-in-the-theory-of-types1\u002F551D2896D3C6F197030D3D8EEA911F1E",[16686],[24,16718,16719,16723,16724,16727,16728],{},[76,16720,16722],{"id":16721},"ref-5","H. Rasiowa",", R. Sikorski, \"A proof of the completeness theorem of Gödel\", ",[16663,16725,16726],{},"Fundamenta Mathematicae",", 37 (1950, 刊行は1951), pp. 193–200. doi:10.4064\u002Ffm-37-1-193-200 ",[31,16729,16732],{"href":16730,"rel":16731},"https:\u002F\u002Feudml.org\u002Fdoc\u002F213213",[16686],"EuDML",[24,16734,16735,16739,16740,16743],{},[76,16736,16738],{"id":16737},"ref-6","E. W. Beth",", \"Semantic entailment and formal derivability\", ",[16663,16741,16742],{},"Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afd. Letterkunde",", N.R. 18(13) (1955), pp. 309–342.",[24,16745,16746,16750,16751,16754,16755],{},[76,16747,16749],{"id":16748},"ref-7","K. J. J. Hintikka",", \"Form and content in quantification theory\", ",[16663,16752,16753],{},"Two Papers on Symbolic Logic",", Acta Philosophica Fennica 8 (1955), pp. 7–55.（巻号を23とする文献もあるが、The Journal of Symbolic Logic の書評の記載に従った）",[31,16756,16759],{"href":16757,"rel":16758},"https:\u002F\u002Fcambridge.org\u002Fcore\u002Fjournals\u002Fjournal-of-symbolic-logic\u002Farticle\u002Fabs\u002Fe-w-beth-remarks-on-natural-deduction-koninklijke-nederlandse-akademie-van-wetenschappen-proceedings-series-a-vol-58-1955-pp-322325-also-indagationes-mathematicae-vol-17-1955-pp-322325-e-w-beth-semantic-entailment-and-formal-derivability-mededelingen-der-koninklijke-nederlandse-akademie-van-wetenschappen-afd-letterkunde-ns-vol-18-no-13-1955-pp-309342-k-jaakko-j-hintikka-a-new-approach-to-sentential-logic-societas-scientiarum-fennica-commentationes-physicomathematicae-vol-17-no-2-1953-14-pp-k-jaakko-j-hintikka-form-and-content-in-quantification-theory-two-papers-on-symbolic-logic-acta-philosophica-fennica-no-8-helsinki1955-pp-755-k-jaakko-j-hintikka-notes-on-quantification-theory-societas-scientiarum-fennica-commentationes-physicomathematicae-vol-17-no-12-1955-13-pp\u002F9D296A6EC1CDE819648B359BFC4EFA80",[16686],"JSL の書評（Beth と Hintikka の論文をまとめて扱う）",[68,16761,16763],{"id":16762},"二次文献解説","二次文献・解説",[8017,16765,16767,16779,16794,16808,16821,16832,16844,16855],{"start":16766},8,[24,16768,16769,16773,16774],{},[76,16770,16772],{"id":16771},"ref-8","M. Fitting",", ラショーヴァの業績に関する論考（PDF）. ",[31,16775,16778],{"href":16776,"rel":16777},"https:\u002F\u002Fid144254.securedata.net\u002Fmelvinfitting\u002Fbookspapers\u002Fpdf\u002Fpapers\u002FRasiowa.pdf",[16686],"PDF",[24,16780,16781,16677,16785,16788,16789],{},[76,16782,16784],{"id":16783},"ref-9","\"Kurt Gödel\"",[16663,16786,16787],{},"Stanford Encyclopedia of Philosophy",".（博士論文の主定理が完全性定理であること、1929年にハーンのもとで学位を取得したこと）",[31,16790,16793],{"href":16791,"rel":16792},"https:\u002F\u002Fplato.stanford.edu\u002Fentries\u002Fgoedel\u002F",[16686],"SEP",[24,16795,16796,16677,16800,16803,16804],{},[76,16797,16799],{"id":16798},"ref-10","\"Original proof of Gödel's completeness theorem\"",[16663,16801,16802],{},"Wikipedia","（英語版）.（ゲーデルの証明の構成、標準形への帰着、等号の扱い、可算集合への拡張）",[31,16805,16802],{"href":16806,"rel":16807},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FOriginal_proof_of_G%C3%B6del%27s_completeness_theorem",[16686],[24,16809,16810,16677,16814,16816,16817],{},[76,16811,16813],{"id":16812},"ref-11","\"Thoralf Skolem\"",[16663,16815,16802],{},"（英語版）.（スコーレムの1920年代の結果と完全性定理の関係）",[31,16818,16802],{"href":16819,"rel":16820},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FThoralf_Skolem",[16686],[24,16822,16823,16827,16828],{},[76,16824,16826],{"id":16825},"ref-12","J. W. Dawson",", Jr., \"The published work of Kurt Gödel: an annotated bibliography\".（ゲーデルの完全性定理の論文と、スコーレムの研究との関係についての注釈）",[31,16829,16778],{"href":16830,"rel":16831},"https:\u002F\u002Fwww.literature.wolkersdorfer.info\u002Fliterature\u002FThe%20published%20work%20of%20Kurt%20G%C3%B6del%20an%20annotated%20bibliography.pdf",[16686],[24,16833,16834,16838,16839],{},[76,16835,16837],{"id":16836},"ref-13","\"On consistency and existence in mathematics\"",", arXiv:2007.10167.（完全性定理と弱ケーニヒの補題の同値性への言及）",[31,16840,16843],{"href":16841,"rel":16842},"https:\u002F\u002Farxiv.org\u002Fpdf\u002F2007.10167",[16686],"arXiv",[24,16845,16846,16850,16851],{},[76,16847,16849],{"id":16848},"ref-14","A. S. Troelstra",", E. W. Beth の業績に関する論考（ILLC の記念論文集への寄稿）.（ベートが1954年に意味論的タブローを考案したこと）",[31,16852,16778],{"href":16853,"rel":16854},"https:\u002F\u002Ffestschriften.illc.uva.nl\u002Fj50\u002Fcontribs\u002Ftroelstra\u002Ftroelstra.pdf",[16686],[24,16856,16857,16677,16861,16863,16864],{},[76,16858,16860],{"id":16859},"ref-15","\"Method of analytic tableaux\"",[16663,16862,16802],{},"（英語版）. ",[31,16865,16802],{"href":16866,"rel":16867},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FMethod_of_analytic_tableaux",[16686],[16869,16870,16871],"blockquote",{},[10,16872,16873,16874,16876],{},"準備（1章）とヘンキン流の証明の細部（3章）の数学的な内容は、一階述語論理の標準的な教科書で扱われる事項であり、骨組みは文献 ",[31,16875,50],{"href":49}," に基づきます。",{"title":16878,"searchDepth":16879,"depth":16879,"links":16880},"",2,[16881,16887,16894,16900,16901,16902,16903],{"id":65,"depth":16879,"text":66,"children":16882},[16883,16885,16886],{"id":70,"depth":16884,"text":71},3,{"id":720,"depth":16884,"text":721},{"id":1306,"depth":16884,"text":1307},{"id":2238,"depth":16879,"text":2239,"children":16888},[16889,16890,16891,16892,16893],{"id":2586,"depth":16884,"text":2587},{"id":3398,"depth":16884,"text":3399},{"id":5039,"depth":16884,"text":5040},{"id":6419,"depth":16884,"text":6420},{"id":7944,"depth":16884,"text":7945},{"id":7993,"depth":16879,"text":7994,"children":16895},[16896,16897,16898,16899],{"id":8405,"depth":16884,"text":8406},{"id":10393,"depth":16884,"text":10394},{"id":13145,"depth":16884,"text":13146},{"id":15735,"depth":16884,"text":15736},{"id":15864,"depth":16879,"text":15865},{"id":16350,"depth":16879,"text":16351},{"id":16401,"depth":16879,"text":16401},{"id":16642,"depth":16879,"text":16642,"children":16904},[16905,16906],{"id":16651,"depth":16884,"text":16652},{"id":16762,"depth":16884,"text":16763},"2026-11-14","一階述語論理の完全性定理を、ゲーデルによる元の証明（1929–1930年）と、今日の教科書の標準であるヘンキンの証明（1949年）の2つの道筋で解説します。どちらも「記号からモデルを作る」点は同じですが、足場の置き方が大きく異なります。",false,"md",null,{},true,4,"\u002Fblog\u002F004-completeness-theorem-proofs",{"title":5,"description":16908},"blog\u002F004-completeness-theorem-proofs",[16919,85,16920],"logic","history","4pOIMk1VCxvdSztxDYqX80HPKDyPoqU9NUaxTIpBCkk",[16923,16925,16933,16939],{"path":16915,"title":5,"no":16914,"date":16907,"updated":16911,"image":16911,"description":16908,"tags":16924,"pinned":16909,"draft":16909},[16919,85,16920],{"path":16926,"title":16927,"no":16884,"date":16907,"updated":16911,"image":16911,"description":16928,"tags":16929,"pinned":16909,"draft":16909},"\u002Fblog\u002F003-design-patterns","デザインパターン再入門 ― GoF の23パターンを TypeScript で読み直す","GoF の23パターンの全体像を整理し、現代の TypeScript やフロントエンド開発でも役立つ6つのパターンを、動くコードとともに解説します。使いすぎの落とし穴と、言語やフレームワークに溶け込んだパターンの見つけ方も。",[16930,16931,16932],"design-pattern","programming","typescript",{"path":16934,"title":16935,"no":16879,"date":16907,"updated":16911,"image":16911,"description":16936,"tags":16937,"pinned":16909,"draft":16909},"\u002Fblog\u002F002-mathematical-logic-timeline","年表で見る20世紀の数理論理学 ― ゲーデルとタルスキ","不完全性定理と真理の定義不可能性定理を軸に、20世紀の数理論理学の流れを年表で整理します。",[16919,16920,16938],"timeline",{"path":16940,"title":16941,"no":16942,"date":16907,"updated":16911,"image":16911,"description":16943,"tags":16944,"pinned":16909,"draft":16909},"\u002Fblog\u002F001-backpropagation","バックプロパゲーション（誤差逆伝播法）を数式から理解する",1,"ニューラルネットワークの学習を支えるバックプロパゲーションを、連鎖律から4つの基本式を導くところまで丁寧に解説します。数値例と NumPy の実装つき。",[16945,85],"machine-learning",1794635675660]