2691:
2712:
2769:
898:
Shoenfield states the theorem in the form for a new function name, and constants are the same as functions of zero arguments. In formal systems that admit ordered tuples, extension by multiple constants as shown here can be accomplished by addition of a new constant tuple and the new constant names
121:
523:
775:
610:
311:
247:
28:
is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be done safely without uniqueness.
873:
802:
637:
420:
393:
342:
174:
893:
846:
826:
697:
677:
657:
543:
440:
366:
194:
147:
2844:
1070:
41:
24:
can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction.
1745:
459:
702:
969:
2810:
2142:
2300:
1088:
2753:
2155:
1478:
2160:
2150:
1887:
1740:
1093:
548:
2784:
2727:
1084:
2834:
2296:
1638:
2393:
2137:
962:
261:
1698:
1391:
1132:
2654:
2356:
2119:
2114:
1939:
1360:
1044:
2649:
2432:
2349:
2062:
1993:
1870:
1112:
2803:
2574:
2400:
2086:
1720:
1319:
2839:
2452:
2447:
2057:
1796:
1725:
1054:
955:
206:
21:
2381:
1971:
1365:
1333:
1024:
2671:
2620:
2517:
2015:
1976:
1453:
1098:
1127:
2746:
2512:
2442:
1981:
1833:
1816:
1539:
1019:
25:
2796:
2344:
2321:
2282:
2168:
2109:
1755:
1675:
1519:
1463:
1076:
913:
2634:
2361:
2339:
2306:
2199:
2045:
2030:
2003:
1954:
1838:
1773:
1598:
1564:
1559:
1433:
1264:
1241:
908:
805:
345:
2849:
2564:
2417:
2209:
1927:
1663:
1569:
1428:
1413:
1294:
1269:
2829:
2537:
2499:
2376:
2180:
2020:
1944:
1922:
1750:
1708:
1607:
1574:
1438:
1226:
1137:
851:
780:
615:
446:
398:
371:
320:
152:
8:
2739:
2666:
2557:
2542:
2522:
2479:
2366:
2316:
2242:
2187:
2124:
1917:
1912:
1860:
1628:
1617:
1289:
1189:
1117:
1108:
1104:
1039:
1034:
2776:
2695:
2464:
2427:
2412:
2405:
2388:
2192:
2174:
2040:
1966:
1949:
1902:
1715:
1624:
1458:
1443:
1403:
1355:
1340:
1328:
1284:
1259:
1029:
978:
878:
831:
811:
682:
662:
642:
528:
425:
351:
179:
132:
127:
17:
1648:
2690:
2630:
2437:
2247:
2237:
2129:
2010:
1845:
1821:
1602:
1586:
1491:
1468:
1345:
1314:
1279:
1174:
1009:
933:
2644:
2639:
2532:
2489:
2311:
2272:
2267:
2252:
2078:
2035:
1932:
1730:
1680:
1254:
1216:
2625:
2615:
2569:
2552:
2507:
2469:
2371:
2291:
2098:
2025:
1998:
1986:
1892:
1806:
1780:
1735:
1703:
1504:
1306:
1249:
1199:
1164:
1122:
197:
116:{\displaystyle \exists x_{1}\ldots \exists x_{m}\,\varphi (x_{1},\ldots ,x_{m})}
2780:
2723:
2610:
2589:
2547:
2527:
2422:
2277:
1875:
1865:
1855:
1850:
1784:
1658:
1534:
1423:
1418:
1396:
997:
395:
has the same set of theorems in the original language (i.e., without constants
2823:
2584:
2262:
1769:
1554:
1544:
1514:
1499:
1169:
2484:
2331:
2232:
2224:
2104:
2052:
1961:
1897:
1880:
1811:
1670:
1529:
1231:
1014:
2594:
2474:
1653:
1643:
1590:
1274:
1194:
1179:
1059:
1004:
518:{\displaystyle \forall {\vec {x}}\,\exists y\,\!\,\varphi (y,{\vec {x}})}
1524:
1379:
1350:
1156:
445:
Such a theory can also be conservatively extended by introducing a new
770:{\displaystyle \forall {\vec {x}}\,\varphi (f({\vec {x}}),{\vec {x}})}
2676:
2579:
1632:
1549:
1509:
1473:
1409:
1221:
1211:
1184:
947:
2661:
2459:
1907:
1612:
1206:
2257:
1049:
2711:
2768:
2719:
1801:
1147:
992:
253:
875:
prove the same theorems not involving the functional symbol
659:
by extending its language with a new functional symbol
881:
854:
834:
814:
783:
705:
685:
665:
645:
618:
551:
531:
462:
428:
401:
374:
354:
323:
264:
209:
182:
155:
135:
44:
887:
867:
840:
820:
796:
769:
691:
671:
651:
631:
604:
537:
517:
434:
414:
387:
360:
336:
305:
241:
188:
168:
141:
115:
605:{\displaystyle {\vec {x}}:=(x_{1},\ldots ,x_{n})}
486:
2821:
2804:
2747:
963:
899:having the values of elements of the tuple.
306:{\displaystyle \varphi (a_{1},\ldots ,a_{m})}
2845:Theorems in the foundations of mathematics
2811:
2797:
2754:
2740:
1155:
970:
956:
932:
721:
487:
485:
478:
74:
2822:
977:
951:
525:is a theorem of a first-order theory
2763:
2706:
242:{\displaystyle a_{1},\ldots ,a_{m}}
13:
706:
479:
463:
61:
45:
14:
2861:
940:. Addison-Wesley. pp. 55â56.
2767:
2710:
2689:
926:
764:
758:
746:
740:
731:
725:
715:
599:
567:
558:
512:
506:
491:
472:
368:, which means that the theory
300:
268:
110:
78:
1:
2650:History of mathematical logic
919:
2783:. You can help Knowledge by
2726:. You can help Knowledge by
2575:Primitive recursive function
7:
902:
10:
2866:
2762:
2705:
1639:SchröderâBernstein theorem
1366:Monadic predicate calculus
1025:Foundations of mathematics
639:be a theory obtained from
176:be a theory obtained from
2685:
2672:Philosophy of mathematics
2621:Automated theorem proving
2603:
2498:
2330:
2223:
2075:
1792:
1768:
1746:Von NeumannâBernaysâGödel
1691:
1585:
1489:
1387:
1378:
1305:
1240:
1146:
1068:
985:
699:) and adding a new axiom
2835:Mathematical logic stubs
26:Extension by definitions
2322:Self-verifying theories
2143:Tarski's axiomatization
1094:Tarski's undefinability
1089:incompleteness theorems
914:Extension by definition
2779:-related article is a
2722:-related article is a
2696:Mathematics portal
2307:Proof of impossibility
1955:propositional variable
1265:Propositional calculus
909:Conservative extension
889:
869:
842:
822:
806:conservative extension
798:
771:
693:
673:
653:
633:
606:
539:
519:
436:
416:
389:
362:
346:conservative extension
338:
307:
243:
190:
170:
143:
117:
2565:Kolmogorov complexity
2518:Computably enumerable
2418:Model complete theory
2210:Principia Mathematica
1270:Propositional formula
1099:BanachâTarski paradox
890:
870:
868:{\displaystyle T_{1}}
843:
823:
799:
797:{\displaystyle T_{1}}
772:
694:
674:
654:
634:
632:{\displaystyle T_{1}}
607:
540:
520:
437:
417:
415:{\displaystyle a_{i}}
390:
388:{\displaystyle T_{1}}
363:
339:
337:{\displaystyle T_{1}}
308:
244:
191:
171:
169:{\displaystyle T_{1}}
144:
118:
2513:ChurchâTuring thesis
2500:Computability theory
1709:continuum hypothesis
1227:Square of opposition
1085:Gödel's completeness
879:
852:
832:
828:, i.e. the theories
812:
781:
703:
683:
663:
643:
616:
549:
529:
460:
426:
399:
372:
352:
321:
262:
207:
200:with new constants
180:
153:
133:
42:
2667:Mathematical object
2558:P versus NP problem
2523:Computable function
2317:Reverse mathematics
2243:Logical consequence
2120:primitive recursive
2115:elementary function
1888:Free/bound variable
1741:TarskiâGrothendieck
1260:Logical connectives
1190:Logical equivalence
1040:Logical consequence
2840:Mathematical logic
2777:mathematical logic
2465:Transfer principle
2428:Semantics of logic
2413:Categorical theory
2389:Non-standard model
1903:Logical connective
1030:Information theory
979:Mathematical logic
938:Mathematical Logic
934:Shoenfield, Joseph
885:
865:
838:
818:
794:
767:
689:
669:
649:
629:
602:
545:, where we denote
535:
515:
432:
412:
385:
358:
334:
303:
239:
186:
166:
139:
128:first-order theory
126:is a theorem of a
113:
18:mathematical logic
2792:
2791:
2735:
2734:
2703:
2702:
2635:Abstract category
2438:Theories of truth
2248:Rule of inference
2238:Natural deduction
2219:
2218:
1764:
1763:
1469:Cartesian product
1374:
1373:
1280:Many-valued logic
1255:Boolean functions
1138:Russell's paradox
1113:diagonal argument
1010:First-order logic
888:{\displaystyle f}
841:{\displaystyle T}
821:{\displaystyle T}
761:
743:
718:
692:{\displaystyle n}
672:{\displaystyle f}
652:{\displaystyle T}
561:
538:{\displaystyle T}
509:
475:
447:functional symbol
435:{\displaystyle T}
361:{\displaystyle T}
252:and adding a new
196:by extending its
189:{\displaystyle T}
142:{\displaystyle T}
2857:
2813:
2806:
2799:
2771:
2764:
2756:
2749:
2742:
2714:
2707:
2694:
2693:
2645:History of logic
2640:Category of sets
2533:Decision problem
2312:Ordinal analysis
2253:Sequent calculus
2151:Boolean algebras
2091:
2090:
2065:
2036:logical/constant
1790:
1789:
1776:
1699:ZermeloâFraenkel
1450:Set operations:
1385:
1384:
1322:
1153:
1152:
1133:LöwenheimâSkolem
1020:Formal semantics
972:
965:
958:
949:
948:
942:
941:
930:
894:
892:
891:
886:
874:
872:
871:
866:
864:
863:
847:
845:
844:
839:
827:
825:
824:
819:
803:
801:
800:
795:
793:
792:
776:
774:
773:
768:
763:
762:
754:
745:
744:
736:
720:
719:
711:
698:
696:
695:
690:
678:
676:
675:
670:
658:
656:
655:
650:
638:
636:
635:
630:
628:
627:
611:
609:
608:
603:
598:
597:
579:
578:
563:
562:
554:
544:
542:
541:
536:
524:
522:
521:
516:
511:
510:
502:
477:
476:
468:
441:
439:
438:
433:
422:) as the theory
421:
419:
418:
413:
411:
410:
394:
392:
391:
386:
384:
383:
367:
365:
364:
359:
343:
341:
340:
335:
333:
332:
312:
310:
309:
304:
299:
298:
280:
279:
248:
246:
245:
240:
238:
237:
219:
218:
195:
193:
192:
187:
175:
173:
172:
167:
165:
164:
148:
146:
145:
140:
122:
120:
119:
114:
109:
108:
90:
89:
73:
72:
57:
56:
2865:
2864:
2860:
2859:
2858:
2856:
2855:
2854:
2820:
2819:
2818:
2817:
2761:
2760:
2704:
2699:
2688:
2681:
2626:Category theory
2616:Algebraic logic
2599:
2570:Lambda calculus
2508:Church encoding
2494:
2470:Truth predicate
2326:
2292:Complete theory
2215:
2084:
2080:
2076:
2071:
2063:
1783: and
1779:
1774:
1760:
1736:New Foundations
1704:axiom of choice
1687:
1649:Gödel numbering
1589: and
1581:
1485:
1370:
1320:
1301:
1250:Boolean algebra
1236:
1200:Equiconsistency
1165:Classical logic
1142:
1123:Halting problem
1111: and
1087: and
1075: and
1074:
1069:Theorems (
1064:
981:
976:
946:
945:
931:
927:
922:
905:
880:
877:
876:
859:
855:
853:
850:
849:
833:
830:
829:
813:
810:
809:
788:
784:
782:
779:
778:
753:
752:
735:
734:
710:
709:
704:
701:
700:
684:
681:
680:
664:
661:
660:
644:
641:
640:
623:
619:
617:
614:
613:
593:
589:
574:
570:
553:
552:
550:
547:
546:
530:
527:
526:
501:
500:
467:
466:
461:
458:
457:
452:Suppose that a
427:
424:
423:
406:
402:
400:
397:
396:
379:
375:
373:
370:
369:
353:
350:
349:
328:
324:
322:
319:
318:
294:
290:
275:
271:
263:
260:
259:
233:
229:
214:
210:
208:
205:
204:
181:
178:
177:
160:
156:
154:
151:
150:
134:
131:
130:
104:
100:
85:
81:
68:
64:
52:
48:
43:
40:
39:
31:Suppose that a
12:
11:
5:
2863:
2853:
2852:
2847:
2842:
2837:
2832:
2816:
2815:
2808:
2801:
2793:
2790:
2789:
2772:
2759:
2758:
2751:
2744:
2736:
2733:
2732:
2715:
2701:
2700:
2686:
2683:
2682:
2680:
2679:
2674:
2669:
2664:
2659:
2658:
2657:
2647:
2642:
2637:
2628:
2623:
2618:
2613:
2611:Abstract logic
2607:
2605:
2601:
2600:
2598:
2597:
2592:
2590:Turing machine
2587:
2582:
2577:
2572:
2567:
2562:
2561:
2560:
2555:
2550:
2545:
2540:
2530:
2528:Computable set
2525:
2520:
2515:
2510:
2504:
2502:
2496:
2495:
2493:
2492:
2487:
2482:
2477:
2472:
2467:
2462:
2457:
2456:
2455:
2450:
2445:
2435:
2430:
2425:
2423:Satisfiability
2420:
2415:
2410:
2409:
2408:
2398:
2397:
2396:
2386:
2385:
2384:
2379:
2374:
2369:
2364:
2354:
2353:
2352:
2347:
2340:Interpretation
2336:
2334:
2328:
2327:
2325:
2324:
2319:
2314:
2309:
2304:
2294:
2289:
2288:
2287:
2286:
2285:
2275:
2270:
2260:
2255:
2250:
2245:
2240:
2235:
2229:
2227:
2221:
2220:
2217:
2216:
2214:
2213:
2205:
2204:
2203:
2202:
2197:
2196:
2195:
2190:
2185:
2165:
2164:
2163:
2161:minimal axioms
2158:
2147:
2146:
2145:
2134:
2133:
2132:
2127:
2122:
2117:
2112:
2107:
2094:
2092:
2073:
2072:
2070:
2069:
2068:
2067:
2055:
2050:
2049:
2048:
2043:
2038:
2033:
2023:
2018:
2013:
2008:
2007:
2006:
2001:
1991:
1990:
1989:
1984:
1979:
1974:
1964:
1959:
1958:
1957:
1952:
1947:
1937:
1936:
1935:
1930:
1925:
1920:
1915:
1910:
1900:
1895:
1890:
1885:
1884:
1883:
1878:
1873:
1868:
1858:
1853:
1851:Formation rule
1848:
1843:
1842:
1841:
1836:
1826:
1825:
1824:
1814:
1809:
1804:
1799:
1793:
1787:
1770:Formal systems
1766:
1765:
1762:
1761:
1759:
1758:
1753:
1748:
1743:
1738:
1733:
1728:
1723:
1718:
1713:
1712:
1711:
1706:
1695:
1693:
1689:
1688:
1686:
1685:
1684:
1683:
1673:
1668:
1667:
1666:
1659:Large cardinal
1656:
1651:
1646:
1641:
1636:
1622:
1621:
1620:
1615:
1610:
1595:
1593:
1583:
1582:
1580:
1579:
1578:
1577:
1572:
1567:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1507:
1502:
1496:
1494:
1487:
1486:
1484:
1483:
1482:
1481:
1476:
1471:
1466:
1461:
1456:
1448:
1447:
1446:
1441:
1431:
1426:
1424:Extensionality
1421:
1419:Ordinal number
1416:
1406:
1401:
1400:
1399:
1388:
1382:
1376:
1375:
1372:
1371:
1369:
1368:
1363:
1358:
1353:
1348:
1343:
1338:
1337:
1336:
1326:
1325:
1324:
1311:
1309:
1303:
1302:
1300:
1299:
1298:
1297:
1292:
1287:
1277:
1272:
1267:
1262:
1257:
1252:
1246:
1244:
1238:
1237:
1235:
1234:
1229:
1224:
1219:
1214:
1209:
1204:
1203:
1202:
1192:
1187:
1182:
1177:
1172:
1167:
1161:
1159:
1150:
1144:
1143:
1141:
1140:
1135:
1130:
1125:
1120:
1115:
1103:Cantor's
1101:
1096:
1091:
1081:
1079:
1066:
1065:
1063:
1062:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1007:
1002:
1001:
1000:
989:
987:
983:
982:
975:
974:
967:
960:
952:
944:
943:
924:
923:
921:
918:
917:
916:
911:
904:
901:
884:
862:
858:
837:
817:
791:
787:
766:
760:
757:
751:
748:
742:
739:
733:
730:
727:
724:
717:
714:
708:
688:
668:
648:
626:
622:
601:
596:
592:
588:
585:
582:
577:
573:
569:
566:
560:
557:
534:
514:
508:
505:
499:
496:
493:
490:
484:
481:
474:
471:
465:
431:
409:
405:
382:
378:
357:
331:
327:
315:
314:
302:
297:
293:
289:
286:
283:
278:
274:
270:
267:
250:
249:
236:
232:
228:
225:
222:
217:
213:
185:
163:
159:
138:
124:
123:
112:
107:
103:
99:
96:
93:
88:
84:
80:
77:
71:
67:
63:
60:
55:
51:
47:
9:
6:
4:
3:
2:
2862:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2833:
2831:
2828:
2827:
2825:
2814:
2809:
2807:
2802:
2800:
2795:
2794:
2788:
2786:
2782:
2778:
2773:
2770:
2766:
2765:
2757:
2752:
2750:
2745:
2743:
2738:
2737:
2731:
2729:
2725:
2721:
2716:
2713:
2709:
2708:
2698:
2697:
2692:
2684:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2656:
2653:
2652:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2632:
2629:
2627:
2624:
2622:
2619:
2617:
2614:
2612:
2609:
2608:
2606:
2602:
2596:
2593:
2591:
2588:
2586:
2585:Recursive set
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2536:
2535:
2534:
2531:
2529:
2526:
2524:
2521:
2519:
2516:
2514:
2511:
2509:
2506:
2505:
2503:
2501:
2497:
2491:
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2454:
2451:
2449:
2446:
2444:
2441:
2440:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2407:
2404:
2403:
2402:
2399:
2395:
2394:of arithmetic
2392:
2391:
2390:
2387:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2359:
2358:
2355:
2351:
2348:
2346:
2343:
2342:
2341:
2338:
2337:
2335:
2333:
2329:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2302:
2301:from ZFC
2298:
2295:
2293:
2290:
2284:
2281:
2280:
2279:
2276:
2274:
2271:
2269:
2266:
2265:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2230:
2228:
2226:
2222:
2212:
2211:
2207:
2206:
2201:
2200:non-Euclidean
2198:
2194:
2191:
2189:
2186:
2184:
2183:
2179:
2178:
2176:
2173:
2172:
2170:
2166:
2162:
2159:
2157:
2154:
2153:
2152:
2148:
2144:
2141:
2140:
2139:
2135:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2106:
2103:
2102:
2100:
2096:
2095:
2093:
2088:
2082:
2077:Example
2074:
2066:
2061:
2060:
2059:
2056:
2054:
2051:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2028:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2005:
2002:
2000:
1997:
1996:
1995:
1992:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1969:
1968:
1965:
1963:
1960:
1956:
1953:
1951:
1948:
1946:
1943:
1942:
1941:
1938:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1905:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1863:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1840:
1837:
1835:
1834:by definition
1832:
1831:
1830:
1827:
1823:
1820:
1819:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1798:
1795:
1794:
1791:
1788:
1786:
1782:
1777:
1771:
1767:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1721:KripkeâPlatek
1719:
1717:
1714:
1710:
1707:
1705:
1702:
1701:
1700:
1697:
1696:
1694:
1690:
1682:
1679:
1678:
1677:
1674:
1672:
1669:
1665:
1662:
1661:
1660:
1657:
1655:
1652:
1650:
1647:
1645:
1642:
1640:
1637:
1634:
1630:
1626:
1623:
1619:
1616:
1614:
1611:
1609:
1606:
1605:
1604:
1600:
1597:
1596:
1594:
1592:
1588:
1584:
1576:
1573:
1571:
1568:
1566:
1565:constructible
1563:
1562:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1497:
1495:
1493:
1488:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1451:
1449:
1445:
1442:
1440:
1437:
1436:
1435:
1432:
1430:
1427:
1425:
1422:
1420:
1417:
1415:
1411:
1407:
1405:
1402:
1398:
1395:
1394:
1393:
1390:
1389:
1386:
1383:
1381:
1377:
1367:
1364:
1362:
1359:
1357:
1354:
1352:
1349:
1347:
1344:
1342:
1339:
1335:
1332:
1331:
1330:
1327:
1323:
1318:
1317:
1316:
1313:
1312:
1310:
1308:
1304:
1296:
1293:
1291:
1288:
1286:
1283:
1282:
1281:
1278:
1276:
1273:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1247:
1245:
1243:
1242:Propositional
1239:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1201:
1198:
1197:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1171:
1170:Logical truth
1168:
1166:
1163:
1162:
1160:
1158:
1154:
1151:
1149:
1145:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1110:
1106:
1102:
1100:
1097:
1095:
1092:
1090:
1086:
1083:
1082:
1080:
1078:
1072:
1067:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
999:
996:
995:
994:
991:
990:
988:
984:
980:
973:
968:
966:
961:
959:
954:
953:
950:
939:
935:
929:
925:
915:
912:
910:
907:
906:
900:
896:
882:
860:
856:
835:
815:
807:
789:
785:
755:
749:
737:
728:
722:
712:
686:
666:
646:
624:
620:
594:
590:
586:
583:
580:
575:
571:
564:
555:
532:
503:
497:
494:
488:
482:
469:
455:
450:
448:
443:
429:
407:
403:
380:
376:
355:
347:
329:
325:
295:
291:
287:
284:
281:
276:
272:
265:
258:
257:
256:
255:
234:
230:
226:
223:
220:
215:
211:
203:
202:
201:
199:
183:
161:
157:
136:
129:
105:
101:
97:
94:
91:
86:
82:
75:
69:
65:
58:
53:
49:
38:
37:
36:
34:
29:
27:
23:
19:
2850:Proof theory
2785:expanding it
2774:
2728:expanding it
2717:
2687:
2485:Ultraproduct
2332:Model theory
2297:Independence
2233:Formal proof
2225:Proof theory
2208:
2181:
2138:real numbers
2110:second-order
2021:Substitution
1898:Metalanguage
1839:conservative
1828:
1812:Axiom schema
1756:Constructive
1726:MorseâKelley
1692:Set theories
1671:Aleph number
1664:inaccessible
1570:Grothendieck
1454:intersection
1341:Higher-order
1329:Second-order
1275:Truth tables
1232:Venn diagram
1015:Formal proof
937:
928:
897:
453:
451:
444:
316:
251:
125:
32:
30:
15:
2830:Logic stubs
2595:Type theory
2543:undecidable
2475:Truth value
2362:equivalence
2041:non-logical
1654:Enumeration
1644:Isomorphism
1591:cardinality
1575:Von Neumann
1540:Ultrafilter
1505:Uncountable
1439:equivalence
1356:Quantifiers
1346:Fixed-point
1315:First-order
1195:Consistency
1180:Proposition
1157:Traditional
1128:Lindström's
1118:Compactness
1060:Type theory
1005:Cardinality
2824:Categories
2406:elementary
2099:arithmetic
1967:Quantifier
1945:functional
1817:Expression
1535:Transitive
1479:identities
1464:complement
1397:hereditary
1380:Set theory
920:References
679:(of arity
2677:Supertask
2580:Recursion
2538:decidable
2372:saturated
2350:of models
2273:deductive
2268:axiomatic
2188:Hilbert's
2175:Euclidean
2156:canonical
2079:axiomatic
2011:Signature
1940:Predicate
1829:Extension
1751:Ackermann
1676:Operation
1555:Universal
1545:Recursive
1520:Singleton
1515:Inhabited
1500:Countable
1490:Types of
1474:power set
1444:partition
1361:Predicate
1307:Predicate
1222:Syllogism
1212:Soundness
1185:Inference
1175:Tautology
1077:paradoxes
759:→
741:→
723:φ
716:→
707:∀
584:…
559:→
507:→
489:φ
480:∃
473:→
464:∀
285:…
266:φ
224:…
95:…
76:φ
62:∃
59:…
46:∃
35:formula
2662:Logicism
2655:timeline
2631:Concrete
2490:Validity
2460:T-schema
2453:Kripke's
2448:Tarski's
2443:semantic
2433:Strength
2382:submodel
2377:spectrum
2345:function
2193:Tarski's
2182:Elements
2169:geometry
2125:Robinson
2046:variable
2031:function
2004:spectrum
1994:Sentence
1950:variable
1893:Language
1846:Relation
1807:Automata
1797:Alphabet
1781:language
1635:-jection
1613:codomain
1599:Function
1560:Universe
1530:Infinite
1434:Relation
1217:Validity
1207:Argument
1105:theorem,
936:(1967).
903:See also
456:formula
198:language
2604:Related
2401:Diagram
2299: (
2278:Hilbert
2263:Systems
2258:Theorem
2136:of the
2081:systems
1861:Formula
1856:Grammar
1772: (
1716:General
1429:Forcing
1414:Element
1334:Monadic
1109:paradox
1050:Theorem
986:General
777:. Then
2367:finite
2130:Skolem
2083:
2058:Theory
2026:Symbol
2016:String
1999:atomic
1876:ground
1871:closed
1866:atomic
1822:ground
1785:syntax
1681:binary
1608:domain
1525:Finite
1290:finite
1148:Logics
1107:
1055:Theory
612:. Let
454:closed
149:. Let
33:closed
22:theory
2775:This
2720:logic
2718:This
2357:Model
2105:Peano
1962:Proof
1802:Arity
1731:Naive
1618:image
1550:Fuzzy
1510:Empty
1459:union
1404:Class
1045:Model
1035:Lemma
993:Axiom
804:is a
344:is a
317:Then
254:axiom
2781:stub
2724:stub
2480:Type
2283:list
2087:list
2064:list
2053:Term
1987:rank
1881:open
1775:list
1587:Maps
1492:sets
1351:Free
1321:list
1071:list
998:list
848:and
20:, a
2167:of
2149:of
2097:of
1629:Sur
1603:Map
1410:Ur-
1392:Set
895:).
808:of
348:of
16:In
2826::
2553:NP
2177::
2171::
2101::
1778:),
1633:Bi
1625:In
565::=
449::
442:.
2812:e
2805:t
2798:v
2787:.
2755:e
2748:t
2741:v
2730:.
2633:/
2548:P
2303:)
2089:)
2085:(
1982:â
1977:!
1972:â
1933:=
1928:â
1923:â
1918:â§
1913:âš
1908:ÂŹ
1631:/
1627:/
1601:/
1412:)
1408:(
1295:â
1285:3
1073:)
971:e
964:t
957:v
883:f
861:1
857:T
836:T
816:T
790:1
786:T
765:)
756:x
750:,
747:)
738:x
732:(
729:f
726:(
713:x
687:n
667:f
647:T
625:1
621:T
600:)
595:n
591:x
587:,
581:,
576:1
572:x
568:(
556:x
533:T
513:)
504:x
498:,
495:y
492:(
483:y
470:x
430:T
408:i
404:a
381:1
377:T
356:T
330:1
326:T
313:.
301:)
296:m
292:a
288:,
282:,
277:1
273:a
269:(
235:m
231:a
227:,
221:,
216:1
212:a
184:T
162:1
158:T
137:T
111:)
106:m
102:x
98:,
92:,
87:1
83:x
79:(
70:m
66:x
54:1
50:x
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