3162:
387:), large cardinal axioms "say" that we are considering all the sets we're "supposed" to be considering, whereas their negations are "restrictive" and say that we're considering only some of those sets. Moreover the consequences of large cardinal axioms seem to fall into natural patterns (see Maddy, "Believing the Axioms, II"). For these reasons, such set theorists tend to consider large cardinal axioms to have a preferred status among extensions of ZFC, one not shared by axioms of less clear motivation (such as
821:
64:
There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial
300:
The observation that large cardinal axioms are linearly ordered by consistency strength is just that, an observation, not a theorem. (Without an accepted definition of large cardinal property, it is not subject to proof in the ordinary sense.) Also, it is not known in every case which of the three
410:
would assert that standard set theory is by definition the study of the consequences of ZFC, and while they might not be opposed in principle to studying the consequences of other systems, they see no reason to single out large cardinals as preferred. There are also realists who deny that
57:, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in
320:
The order of consistency strength is not necessarily the same as the order of the size of the smallest witness to a large cardinal axiom. For example, the existence of a
700:
317:. It is also noteworthy that many combinatorial statements are exactly equiconsistent with some large cardinal rather than, say, being intermediate between them.
415:
is a proper motivation, and even believe that large cardinal axioms are false. And finally, there are some who deny that the negations of large cardinal axioms
1541:
2216:
1278:
380:, L, which does not satisfy the statement "there is a measurable cardinal" (even though it contains the measurable cardinal as an ordinal).
2299:
1440:
94:
There is no generally agreed precise definition of what a large cardinal property is, though essentially everyone agrees that those in the
294:
45:. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=Ï
482:
2613:
2771:
593:
1559:
2626:
1949:
967:
787:
396:
377:
88:
760:
80:
is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property.
2631:
2621:
2358:
2211:
1564:
1295:
1555:
2767:
559:
535:
513:
465:
432:
423:
model in L that believes there exists a measurable cardinal, even though L itself does not satisfy that proposition.
95:
2109:
2864:
2608:
1433:
2169:
1862:
1273:
383:
Thus, from a certain point of view held by many set theorists (especially those inspired by the tradition of the
17:
1603:
1153:
3125:
2827:
2590:
2585:
2410:
1831:
1515:
407:
3186:
3120:
2903:
2820:
2533:
2464:
2341:
1583:
1047:
926:
360:
can be seen in some natural way as submodels of those in which the axioms hold. For example, if there is an
3045:
2871:
2557:
2191:
1790:
1290:
506:
Set Theory: An
Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76)
83:
Most working set theorists believe that the large cardinal axioms that are currently being considered are
2923:
2918:
2528:
2267:
2196:
1525:
1426:
1283:
921:
884:
87:
with ZFC. These axioms are strong enough to imply the consistency of ZFC. This has the consequence (via
2852:
2442:
1836:
1804:
1495:
305:
has asked, "s there some theorem explaining this, or is our vision just more uniform than we realize?"
938:
3142:
3091:
2988:
2486:
2447:
1924:
1569:
972:
864:
852:
847:
1598:
3191:
2983:
2913:
2452:
2304:
2287:
2010:
1490:
780:
2815:
2792:
2753:
2639:
2580:
2226:
2146:
1990:
1934:
1547:
1392:
1310:
1185:
1137:
951:
874:
756:
3105:
2832:
2810:
2777:
2670:
2516:
2501:
2474:
2425:
2309:
2244:
2069:
2035:
2030:
1904:
1735:
1712:
1344:
1225:
1037:
857:
412:
365:
223:
These are mutually exclusive, unless one of the theories in question is actually inconsistent.
119:
580:, Lecture Notes in Mathematics, vol. 669, Springer Berlin / Heidelberg, pp. 99â275,
3035:
2888:
2680:
2398:
2134:
2040:
1899:
1884:
1765:
1740:
1260:
1230:
1174:
1094:
1074:
1052:
361:
341:
325:
145:
A remarkable observation about large cardinal axioms is that they appear to occur in strict
3008:
2970:
2847:
2651:
2491:
2415:
2393:
2221:
2179:
2078:
2045:
1909:
1697:
1608:
1334:
1324:
1158:
1089:
1042:
982:
869:
337:
150:
8:
3137:
3028:
3013:
2993:
2950:
2837:
2787:
2713:
2658:
2595:
2388:
2383:
2331:
2099:
2088:
1760:
1660:
1588:
1579:
1575:
1510:
1505:
1329:
1240:
1148:
1143:
957:
899:
837:
773:
549:
369:
91:) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent).
3166:
2935:
2898:
2883:
2876:
2859:
2663:
2645:
2511:
2437:
2420:
2373:
2186:
2095:
1929:
1914:
1874:
1826:
1811:
1799:
1755:
1730:
1500:
1449:
1252:
1247:
1032:
987:
894:
682:
666:
658:
629:
454:
384:
39:
2119:
3161:
3101:
2908:
2718:
2708:
2600:
2481:
2316:
2292:
2073:
2057:
1962:
1939:
1816:
1785:
1750:
1645:
1480:
1109:
946:
909:
879:
810:
728:
696:
589:
555:
531:
509:
461:
388:
670:
364:, then "cutting the universe off" at the height of the first such cardinal yields a
61:'s phrase, as quantifying the fact "that if you want more you have to assume more".
3115:
3110:
3003:
2960:
2782:
2743:
2738:
2723:
2549:
2506:
2403:
2201:
2151:
1725:
1687:
1397:
1387:
1372:
1367:
1235:
889:
723:
704:
650:
621:
581:
545:
153:. That is, no exception is known to the following: Given two large cardinal axioms
328:, but assuming both exist, the first huge is smaller than the first supercompact.
3096:
3086:
3040:
3023:
2978:
2940:
2842:
2762:
2569:
2496:
2469:
2457:
2363:
2277:
2251:
2206:
2174:
1975:
1777:
1720:
1670:
1635:
1593:
1266:
1204:
1022:
842:
241:
49:). The proposition that such cardinals exist cannot be proved in the most common
42:
310:
3081:
3060:
3018:
2998:
2893:
2748:
2346:
2336:
2326:
2321:
2255:
2005:
1894:
1889:
1867:
1468:
1402:
1199:
1180:
1084:
1069:
962:
904:
678:
609:
420:
306:
302:
293:
is itself consistent (provided of course that it really is). This follows from
50:
551:
The Higher
Infinite : Large Cardinals in Set Theory from Their Beginnings
110:
is that the existence of such a cardinal is not known to be inconsistent with
3180:
3055:
2733:
2240:
2025:
2015:
1985:
1970:
1640:
1407:
1209:
1123:
1118:
321:
1377:
324:
is much stronger, in terms of consistency strength, than the existence of a
2955:
2802:
2703:
2695:
2575:
2523:
2432:
2368:
2351:
2282:
2141:
2000:
1702:
1485:
1357:
1352:
1170:
1099:
1057:
916:
820:
695:
486:
353:
146:
3065:
2945:
2124:
2114:
2061:
1745:
1665:
1650:
1530:
1475:
1382:
1017:
523:
1995:
1850:
1821:
1627:
1362:
1133:
796:
662:
633:
585:
399:
in this group would state, more simply, that large cardinal axioms are
130:
84:
58:
31:
708:
570:
406:
This point of view is by no means universal among set theorists. Some
314:
3147:
3050:
2103:
2020:
1980:
1944:
1880:
1692:
1682:
1655:
1418:
1165:
1128:
1079:
977:
687:
654:
625:
3132:
2930:
2378:
2083:
1677:
345:
2728:
1520:
106:
A necessary condition for a property of cardinal numbers to be a
1190:
1012:
349:
2272:
1618:
1463:
1062:
829:
765:
737:
Woodin, W. Hugh (2001). "The continuum hypothesis, part II".
391:) or others that they consider intuitively unlikely (such as
368:
in which there is no inaccessible cardinal. Or if there is a
419:
restrictive, pointing out that (for example) there can be a
286:
is consistent, even with the additional hypothesis that ZFC+
528:
Set theory, third millennium edition (revised and expanded)
483:"Does anyone still seriously doubt the consistency of ZFC?"
456:
392:
111:
54:
571:"The evolution of large cardinal axioms in set theory"
709:"Strong axioms of infinity and elementary embeddings"
336:
Large cardinals are understood in the context of the
66:
641:
Maddy, Penelope (1988). "Believing the Axioms, II".
140:
376:powerset operation rather than the full one yields
331:
453:
118:would be an uncountable initial ordinal for which
3178:
65:point among distinct philosophical schools (see
568:
1434:
781:
739:Notices of the American Mathematical Society
522:
503:
137:imply that any such large cardinals exist.
1626:
1441:
1427:
788:
774:
727:
686:
544:
569:Kanamori, Akihiro; Magidor, M. (1978),
348:operation, which collects together all
167:, exactly one of three things happens:
14:
3179:
1448:
736:
677:
1422:
769:
640:
608:
295:Gödel's second incompleteness theorem
101:
89:Gödel's second incompleteness theorem
681:(2002). "The Future of Set Theory".
480:
451:
761:Stanford Encyclopedia of Philosophy
612:(1988). "Believing the Axioms, I".
313:, the main unsolved problem of his
24:
251:is consistency-wise stronger than
25:
3203:
757:"Large Cardinals and Determinacy"
750:
460:. Oxford University Press. viii.
433:List of large cardinal properties
309:, however, deduces this from the
178:is consistent if and only if ZFC+
141:Hierarchy of consistency strength
96:list of large cardinal properties
38:is a certain kind of property of
3160:
819:
332:Motivations and epistemic status
171:Unless ZFC is inconsistent, ZFC+
67:Motivations and epistemic status
356:in which large cardinal axioms
98:are large cardinal properties.
795:
474:
445:
378:Gödel's constructible universe
13:
1:
3121:History of mathematical logic
497:
129:is a model of ZFC. If ZFC is
30:In the mathematical field of
3046:Primitive recursive function
729:10.1016/0003-4843(78)90031-1
716:Annals of Mathematical Logic
481:Joel, Hamkins (2022-12-24).
258:(vice versa for case 3). If
7:
426:
352:of a given set. Typically,
10:
3208:
2110:SchröderâBernstein theorem
1837:Monadic predicate calculus
1496:Foundations of mathematics
1279:von NeumannâBernaysâGödel
554:(2nd ed.). Springer.
3156:
3143:Philosophy of mathematics
3092:Automated theorem proving
3074:
2969:
2801:
2694:
2546:
2263:
2239:
2217:Von NeumannâBernaysâGödel
2162:
2056:
1960:
1858:
1849:
1776:
1711:
1617:
1539:
1456:
1343:
1306:
1218:
1108:
1080:One-to-one correspondence
996:
937:
828:
817:
803:
643:Journal of Symbolic Logic
614:Journal of Symbolic Logic
244:. In case 2, we say that
114:and that such a cardinal
508:. Elsevier Science Ltd.
438:
340:V, which is built up by
2793:Self-verifying theories
2614:Tarski's axiomatization
1565:Tarski's undefinability
1560:incompleteness theorems
342:transfinitely iterating
226:In case 1, we say that
108:large cardinal property
36:large cardinal property
3167:Mathematics portal
2778:Proof of impossibility
2426:propositional variable
1736:Propositional calculus
1038:Constructible universe
865:Constructibility (V=L)
413:ontological maximalism
53:of set theory, namely
3036:Kolmogorov complexity
2989:Computably enumerable
2889:Model complete theory
2681:Principia Mathematica
1741:Propositional formula
1570:BanachâTarski paradox
1261:Principia Mathematica
1095:Transfinite induction
954:(i.e. set difference)
504:Drake, F. R. (1974).
372:, then iterating the
362:inaccessible cardinal
326:supercompact cardinal
3187:Axioms of set theory
2984:ChurchâTuring thesis
2971:Computability theory
2180:continuum hypothesis
1698:Square of opposition
1556:Gödel's completeness
1335:Burali-Forti paradox
1090:Set-builder notation
1043:Continuum hypothesis
983:Symmetric difference
701:William N. Reinhardt
452:Bell, J. L. (1985).
338:von Neumann universe
151:consistency strength
76:large cardinal axiom
3138:Mathematical object
3029:P versus NP problem
2994:Computable function
2788:Reverse mathematics
2714:Logical consequence
2591:primitive recursive
2586:elementary function
2359:Free/bound variable
2212:TarskiâGrothendieck
1731:Logical connectives
1661:Logical equivalence
1511:Logical consequence
1296:TarskiâGrothendieck
370:measurable cardinal
2936:Transfer principle
2899:Semantics of logic
2884:Categorical theory
2860:Non-standard model
2374:Logical connective
1501:Information theory
1450:Mathematical logic
885:Limitation of size
697:Solovay, Robert M.
586:10.1007/BFb0103104
102:Partial definition
27:Set theory concept
3174:
3173:
3106:Abstract category
2909:Theories of truth
2719:Rule of inference
2709:Natural deduction
2690:
2689:
2235:
2234:
1940:Cartesian product
1845:
1844:
1751:Many-valued logic
1726:Boolean functions
1609:Russell's paradox
1584:diagonal argument
1481:First-order logic
1416:
1415:
1325:Russell's paradox
1274:ZermeloâFraenkel
1175:Dedekind-infinite
1048:Diagonal argument
947:Cartesian product
811:Set (mathematics)
595:978-3-540-08926-1
578:Higher Set Theory
546:Kanamori, Akihiro
279:cannot prove ZFC+
265:is stronger than
202:is consistent; or
16:(Redirected from
3199:
3165:
3164:
3116:History of logic
3111:Category of sets
3004:Decision problem
2783:Ordinal analysis
2724:Sequent calculus
2622:Boolean algebras
2562:
2561:
2536:
2507:logical/constant
2261:
2260:
2247:
2170:ZermeloâFraenkel
1921:Set operations:
1856:
1855:
1793:
1624:
1623:
1604:LöwenheimâSkolem
1491:Formal semantics
1443:
1436:
1429:
1420:
1419:
1398:Bertrand Russell
1388:John von Neumann
1373:Abraham Fraenkel
1368:Richard Dedekind
1330:Suslin's problem
1241:Cantor's theorem
958:De Morgan's laws
823:
790:
783:
776:
767:
766:
746:
733:
731:
713:
705:Akihiro Kanamori
692:
690:
674:
637:
605:
604:
602:
575:
565:
541:
519:
491:
490:
478:
472:
471:
459:
449:
395:). The hardcore
212:proves that ZFC+
195:proves that ZFC+
133:, then ZFC does
78:
77:
43:cardinal numbers
21:
3207:
3206:
3202:
3201:
3200:
3198:
3197:
3196:
3192:Large cardinals
3177:
3176:
3175:
3170:
3159:
3152:
3097:Category theory
3087:Algebraic logic
3070:
3041:Lambda calculus
2979:Church encoding
2965:
2941:Truth predicate
2797:
2763:Complete theory
2686:
2555:
2551:
2547:
2542:
2534:
2254: and
2250:
2245:
2231:
2207:New Foundations
2175:axiom of choice
2158:
2120:Gödel numbering
2060: and
2052:
1956:
1841:
1791:
1772:
1721:Boolean algebra
1707:
1671:Equiconsistency
1636:Classical logic
1613:
1594:Halting problem
1582: and
1558: and
1546: and
1545:
1540:Theorems (
1535:
1452:
1447:
1417:
1412:
1339:
1318:
1302:
1267:New Foundations
1214:
1104:
1023:Cardinal number
1006:
992:
933:
824:
815:
799:
794:
753:
711:
679:Shelah, Saharon
655:10.2307/2274569
626:10.2307/2274520
610:Maddy, Penelope
600:
598:
596:
573:
562:
538:
516:
500:
495:
494:
479:
475:
468:
450:
446:
441:
429:
393:V = L
334:
292:
285:
278:
271:
264:
257:
250:
239:
232:
218:
211:
201:
194:
184:
177:
166:
159:
143:
127:
104:
75:
74:
48:
28:
23:
22:
18:Large cardinals
15:
12:
11:
5:
3205:
3195:
3194:
3189:
3172:
3171:
3157:
3154:
3153:
3151:
3150:
3145:
3140:
3135:
3130:
3129:
3128:
3118:
3113:
3108:
3099:
3094:
3089:
3084:
3082:Abstract logic
3078:
3076:
3072:
3071:
3069:
3068:
3063:
3061:Turing machine
3058:
3053:
3048:
3043:
3038:
3033:
3032:
3031:
3026:
3021:
3016:
3011:
3001:
2999:Computable set
2996:
2991:
2986:
2981:
2975:
2973:
2967:
2966:
2964:
2963:
2958:
2953:
2948:
2943:
2938:
2933:
2928:
2927:
2926:
2921:
2916:
2906:
2901:
2896:
2894:Satisfiability
2891:
2886:
2881:
2880:
2879:
2869:
2868:
2867:
2857:
2856:
2855:
2850:
2845:
2840:
2835:
2825:
2824:
2823:
2818:
2811:Interpretation
2807:
2805:
2799:
2798:
2796:
2795:
2790:
2785:
2780:
2775:
2765:
2760:
2759:
2758:
2757:
2756:
2746:
2741:
2731:
2726:
2721:
2716:
2711:
2706:
2700:
2698:
2692:
2691:
2688:
2687:
2685:
2684:
2676:
2675:
2674:
2673:
2668:
2667:
2666:
2661:
2656:
2636:
2635:
2634:
2632:minimal axioms
2629:
2618:
2617:
2616:
2605:
2604:
2603:
2598:
2593:
2588:
2583:
2578:
2565:
2563:
2544:
2543:
2541:
2540:
2539:
2538:
2526:
2521:
2520:
2519:
2514:
2509:
2504:
2494:
2489:
2484:
2479:
2478:
2477:
2472:
2462:
2461:
2460:
2455:
2450:
2445:
2435:
2430:
2429:
2428:
2423:
2418:
2408:
2407:
2406:
2401:
2396:
2391:
2386:
2381:
2371:
2366:
2361:
2356:
2355:
2354:
2349:
2344:
2339:
2329:
2324:
2322:Formation rule
2319:
2314:
2313:
2312:
2307:
2297:
2296:
2295:
2285:
2280:
2275:
2270:
2264:
2258:
2241:Formal systems
2237:
2236:
2233:
2232:
2230:
2229:
2224:
2219:
2214:
2209:
2204:
2199:
2194:
2189:
2184:
2183:
2182:
2177:
2166:
2164:
2160:
2159:
2157:
2156:
2155:
2154:
2144:
2139:
2138:
2137:
2130:Large cardinal
2127:
2122:
2117:
2112:
2107:
2093:
2092:
2091:
2086:
2081:
2066:
2064:
2054:
2053:
2051:
2050:
2049:
2048:
2043:
2038:
2028:
2023:
2018:
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1967:
1965:
1958:
1957:
1955:
1954:
1953:
1952:
1947:
1942:
1937:
1932:
1927:
1919:
1918:
1917:
1912:
1902:
1897:
1895:Extensionality
1892:
1890:Ordinal number
1887:
1877:
1872:
1871:
1870:
1859:
1853:
1847:
1846:
1843:
1842:
1840:
1839:
1834:
1829:
1824:
1819:
1814:
1809:
1808:
1807:
1797:
1796:
1795:
1782:
1780:
1774:
1773:
1771:
1770:
1769:
1768:
1763:
1758:
1748:
1743:
1738:
1733:
1728:
1723:
1717:
1715:
1709:
1708:
1706:
1705:
1700:
1695:
1690:
1685:
1680:
1675:
1674:
1673:
1663:
1658:
1653:
1648:
1643:
1638:
1632:
1630:
1621:
1615:
1614:
1612:
1611:
1606:
1601:
1596:
1591:
1586:
1574:Cantor's
1572:
1567:
1562:
1552:
1550:
1537:
1536:
1534:
1533:
1528:
1523:
1518:
1513:
1508:
1503:
1498:
1493:
1488:
1483:
1478:
1473:
1472:
1471:
1460:
1458:
1454:
1453:
1446:
1445:
1438:
1431:
1423:
1414:
1413:
1411:
1410:
1405:
1403:Thoralf Skolem
1400:
1395:
1390:
1385:
1380:
1375:
1370:
1365:
1360:
1355:
1349:
1347:
1341:
1340:
1338:
1337:
1332:
1327:
1321:
1319:
1317:
1316:
1313:
1307:
1304:
1303:
1301:
1300:
1299:
1298:
1293:
1288:
1287:
1286:
1271:
1270:
1269:
1257:
1256:
1255:
1244:
1243:
1238:
1233:
1228:
1222:
1220:
1216:
1215:
1213:
1212:
1207:
1202:
1197:
1188:
1183:
1178:
1168:
1163:
1162:
1161:
1156:
1151:
1141:
1131:
1126:
1121:
1115:
1113:
1106:
1105:
1103:
1102:
1097:
1092:
1087:
1085:Ordinal number
1082:
1077:
1072:
1067:
1066:
1065:
1060:
1050:
1045:
1040:
1035:
1030:
1020:
1015:
1009:
1007:
1005:
1004:
1001:
997:
994:
993:
991:
990:
985:
980:
975:
970:
965:
963:Disjoint union
960:
955:
949:
943:
941:
935:
934:
932:
931:
930:
929:
924:
913:
912:
910:Martin's axiom
907:
902:
897:
892:
887:
882:
877:
875:Extensionality
872:
867:
862:
861:
860:
855:
850:
840:
834:
832:
826:
825:
818:
816:
814:
813:
807:
805:
801:
800:
793:
792:
785:
778:
770:
764:
763:
752:
751:External links
749:
748:
747:
734:
693:
675:
649:(3): 736â764.
638:
620:(2): 481â511.
606:
594:
566:
560:
542:
536:
520:
514:
499:
496:
493:
492:
473:
466:
443:
442:
440:
437:
436:
435:
428:
425:
421:transitive set
389:Martin's axiom
333:
330:
303:Saharon Shelah
290:
283:
276:
269:
262:
255:
248:
242:equiconsistent
237:
230:
221:
220:
219:is consistent.
216:
209:
203:
199:
192:
186:
185:is consistent;
182:
175:
164:
157:
142:
139:
123:
103:
100:
51:axiomatization
46:
26:
9:
6:
4:
3:
2:
3204:
3193:
3190:
3188:
3185:
3184:
3182:
3169:
3168:
3163:
3155:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3127:
3124:
3123:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3079:
3077:
3073:
3067:
3064:
3062:
3059:
3057:
3056:Recursive set
3054:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3006:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2976:
2974:
2972:
2968:
2962:
2959:
2957:
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2925:
2922:
2920:
2917:
2915:
2912:
2911:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2878:
2875:
2874:
2873:
2870:
2866:
2865:of arithmetic
2863:
2862:
2861:
2858:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2830:
2829:
2826:
2822:
2819:
2817:
2814:
2813:
2812:
2809:
2808:
2806:
2804:
2800:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2773:
2772:from ZFC
2769:
2766:
2764:
2761:
2755:
2752:
2751:
2750:
2747:
2745:
2742:
2740:
2737:
2736:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2701:
2699:
2697:
2693:
2683:
2682:
2678:
2677:
2672:
2671:non-Euclidean
2669:
2665:
2662:
2660:
2657:
2655:
2654:
2650:
2649:
2647:
2644:
2643:
2641:
2637:
2633:
2630:
2628:
2625:
2624:
2623:
2619:
2615:
2612:
2611:
2610:
2606:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2573:
2571:
2567:
2566:
2564:
2559:
2553:
2548:Example
2545:
2537:
2532:
2531:
2530:
2527:
2525:
2522:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2499:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2476:
2473:
2471:
2468:
2467:
2466:
2463:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2440:
2439:
2436:
2434:
2431:
2427:
2424:
2422:
2419:
2417:
2414:
2413:
2412:
2409:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2376:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2334:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2311:
2308:
2306:
2305:by definition
2303:
2302:
2301:
2298:
2294:
2291:
2290:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2265:
2262:
2259:
2257:
2253:
2248:
2242:
2238:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2192:KripkeâPlatek
2190:
2188:
2185:
2181:
2178:
2176:
2173:
2172:
2171:
2168:
2167:
2165:
2161:
2153:
2150:
2149:
2148:
2145:
2143:
2140:
2136:
2133:
2132:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2108:
2105:
2101:
2097:
2094:
2090:
2087:
2085:
2082:
2080:
2077:
2076:
2075:
2071:
2068:
2067:
2065:
2063:
2059:
2055:
2047:
2044:
2042:
2039:
2037:
2036:constructible
2034:
2033:
2032:
2029:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1968:
1966:
1964:
1959:
1951:
1948:
1946:
1943:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1922:
1920:
1916:
1913:
1911:
1908:
1907:
1906:
1903:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1882:
1878:
1876:
1873:
1869:
1866:
1865:
1864:
1861:
1860:
1857:
1854:
1852:
1848:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1806:
1803:
1802:
1801:
1798:
1794:
1789:
1788:
1787:
1784:
1783:
1781:
1779:
1775:
1767:
1764:
1762:
1759:
1757:
1754:
1753:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1722:
1719:
1718:
1716:
1714:
1713:Propositional
1710:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1672:
1669:
1668:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1641:Logical truth
1639:
1637:
1634:
1633:
1631:
1629:
1625:
1622:
1620:
1616:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1581:
1577:
1573:
1571:
1568:
1566:
1563:
1561:
1557:
1554:
1553:
1551:
1549:
1543:
1538:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1470:
1467:
1466:
1465:
1462:
1461:
1459:
1455:
1451:
1444:
1439:
1437:
1432:
1430:
1425:
1424:
1421:
1409:
1408:Ernst Zermelo
1406:
1404:
1401:
1399:
1396:
1394:
1393:Willard Quine
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1350:
1348:
1346:
1345:Set theorists
1342:
1336:
1333:
1331:
1328:
1326:
1323:
1322:
1320:
1314:
1312:
1309:
1308:
1305:
1297:
1294:
1292:
1291:KripkeâPlatek
1289:
1285:
1282:
1281:
1280:
1277:
1276:
1275:
1272:
1268:
1265:
1264:
1263:
1262:
1258:
1254:
1251:
1250:
1249:
1246:
1245:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1224:
1223:
1221:
1217:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1194:
1189:
1187:
1184:
1182:
1179:
1176:
1172:
1169:
1167:
1164:
1160:
1157:
1155:
1152:
1150:
1147:
1146:
1145:
1142:
1139:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1116:
1114:
1111:
1107:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1064:
1061:
1059:
1056:
1055:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1028:
1024:
1021:
1019:
1016:
1014:
1011:
1010:
1008:
1002:
999:
998:
995:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
953:
950:
948:
945:
944:
942:
940:
936:
928:
927:specification
925:
923:
920:
919:
918:
915:
914:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
859:
856:
854:
851:
849:
846:
845:
844:
841:
839:
836:
835:
833:
831:
827:
822:
812:
809:
808:
806:
802:
798:
791:
786:
784:
779:
777:
772:
771:
768:
762:
758:
755:
754:
745:(7): 681â690.
744:
740:
735:
730:
725:
722:(1): 73â116.
721:
717:
710:
706:
702:
698:
694:
689:
684:
680:
676:
672:
668:
664:
660:
656:
652:
648:
644:
639:
635:
631:
627:
623:
619:
615:
611:
607:
601:September 25,
597:
591:
587:
583:
579:
572:
567:
563:
561:3-540-00384-3
557:
553:
552:
547:
543:
539:
537:3-540-44085-2
533:
529:
525:
521:
517:
515:0-444-10535-2
511:
507:
502:
501:
488:
484:
477:
469:
467:0-19-853241-5
463:
458:
457:
448:
444:
434:
431:
430:
424:
422:
418:
414:
409:
404:
402:
398:
394:
390:
386:
381:
379:
375:
371:
367:
363:
359:
355:
351:
347:
343:
339:
329:
327:
323:
322:huge cardinal
318:
316:
312:
308:
304:
301:cases holds.
298:
296:
289:
282:
275:
268:
261:
254:
247:
243:
236:
229:
224:
215:
208:
204:
198:
191:
187:
181:
174:
170:
169:
168:
163:
156:
152:
148:
138:
136:
132:
128:
126:
122:
117:
113:
109:
99:
97:
92:
90:
86:
81:
79:
70:
68:
62:
60:
56:
52:
44:
41:
37:
33:
19:
3158:
2956:Ultraproduct
2803:Model theory
2768:Independence
2704:Formal proof
2696:Proof theory
2679:
2652:
2609:real numbers
2581:second-order
2492:Substitution
2369:Metalanguage
2310:conservative
2283:Axiom schema
2227:Constructive
2197:MorseâKelley
2163:Set theories
2142:Aleph number
2135:inaccessible
2129:
2041:Grothendieck
1925:intersection
1812:Higher-order
1800:Second-order
1746:Truth tables
1703:Venn diagram
1486:Formal proof
1358:Georg Cantor
1353:Paul Bernays
1284:MorseâKelley
1259:
1192:
1191:Subset
1138:hereditarily
1100:Venn diagram
1058:ordered pair
1026:
973:Intersection
917:Axiom schema
742:
738:
719:
715:
688:math/0211397
646:
642:
617:
613:
599:, retrieved
577:
550:
530:. Springer.
527:
524:Jech, Thomas
505:
487:MathOverflow
476:
455:
447:
416:
405:
400:
382:
373:
357:
335:
319:
311:Ω-conjecture
299:
287:
280:
273:
266:
259:
252:
245:
234:
227:
225:
222:
213:
206:
196:
189:
179:
172:
161:
154:
147:linear order
144:
134:
124:
120:
115:
107:
105:
93:
82:
73:
71:
63:
35:
29:
3066:Type theory
3014:undecidable
2946:Truth value
2833:equivalence
2512:non-logical
2125:Enumeration
2115:Isomorphism
2062:cardinality
2046:Von Neumann
2011:Ultrafilter
1976:Uncountable
1910:equivalence
1827:Quantifiers
1817:Fixed-point
1786:First-order
1666:Consistency
1651:Proposition
1628:Traditional
1599:Lindström's
1589:Compactness
1531:Type theory
1476:Cardinality
1383:Thomas Jech
1226:Alternative
1205:Uncountable
1159:Ultrafilter
1018:Cardinality
922:replacement
870:Determinacy
272:, then ZFC+
40:transfinite
3181:Categories
2877:elementary
2570:arithmetic
2438:Quantifier
2416:functional
2288:Expression
2006:Transitive
1950:identities
1935:complement
1868:hereditary
1851:Set theory
1378:Kurt Gödel
1363:Paul Cohen
1200:Transitive
968:Identities
952:Complement
939:Operations
900:Regularity
838:Adjunction
797:Set theory
498:References
408:formalists
131:consistent
85:consistent
59:Dana Scott
32:set theory
3148:Supertask
3051:Recursion
3009:decidable
2843:saturated
2821:of models
2744:deductive
2739:axiomatic
2659:Hilbert's
2646:Euclidean
2627:canonical
2550:axiomatic
2482:Signature
2411:Predicate
2300:Extension
2222:Ackermann
2147:Operation
2026:Universal
2016:Recursive
1991:Singleton
1986:Inhabited
1971:Countable
1961:Types of
1945:power set
1915:partition
1832:Predicate
1778:Predicate
1693:Syllogism
1683:Soundness
1656:Inference
1646:Tautology
1548:paradoxes
1311:Paradoxes
1231:Axiomatic
1210:Universal
1186:Singleton
1181:Recursive
1124:Countable
1119:Amorphous
978:Power set
895:Power set
853:dependent
848:countable
374:definable
3133:Logicism
3126:timeline
3102:Concrete
2961:Validity
2931:T-schema
2924:Kripke's
2919:Tarski's
2914:semantic
2904:Strength
2853:submodel
2848:spectrum
2816:function
2664:Tarski's
2653:Elements
2640:geometry
2596:Robinson
2517:variable
2502:function
2475:spectrum
2465:Sentence
2421:variable
2364:Language
2317:Relation
2278:Automata
2268:Alphabet
2252:language
2106:-jection
2084:codomain
2070:Function
2031:Universe
2001:Infinite
1905:Relation
1688:Validity
1678:Argument
1576:theorem,
1315:Problems
1219:Theories
1195:Superset
1171:Infinite
1000:Concepts
880:Infinity
804:Overview
707:(1978).
671:16544090
548:(2003).
526:(2002).
427:See also
397:realists
366:universe
346:powerset
69:below).
3075:Related
2872:Diagram
2770: (
2749:Hilbert
2734:Systems
2729:Theorem
2607:of the
2552:systems
2332:Formula
2327:Grammar
2243: (
2187:General
1900:Forcing
1885:Element
1805:Monadic
1580:paradox
1521:Theorem
1457:General
1253:General
1248:Zermelo
1154:subbase
1136: (
1075:Forcing
1053:Element
1025: (
1003:Methods
890:Pairing
759:at the
663:2274569
634:2274520
350:subsets
315:Ω-logic
2838:finite
2601:Skolem
2554:
2529:Theory
2497:Symbol
2487:String
2470:atomic
2347:ground
2342:closed
2337:atomic
2293:ground
2256:syntax
2152:binary
2079:domain
1996:Finite
1761:finite
1619:Logics
1578:
1526:Theory
1144:Filter
1134:Finite
1070:Family
1013:Almost
858:global
843:Choice
830:Axioms
669:
661:
632:
592:
558:
534:
512:
464:
354:models
307:Woodin
2828:Model
2576:Peano
2433:Proof
2273:Arity
2202:Naive
2089:image
2021:Fuzzy
1981:Empty
1930:union
1875:Class
1516:Model
1506:Lemma
1464:Axiom
1236:Naive
1166:Fuzzy
1129:Empty
1112:types
1063:tuple
1033:Class
1027:large
988:Union
905:Union
712:(PDF)
683:arXiv
667:S2CID
659:JSTOR
630:JSTOR
574:(PDF)
439:Notes
385:Cabal
2951:Type
2754:list
2558:list
2535:list
2524:Term
2458:rank
2352:open
2246:list
2058:Maps
1963:sets
1822:Free
1792:list
1542:list
1469:list
1149:base
603:2022
590:ISBN
556:ISBN
532:ISBN
510:ISBN
462:ISBN
401:true
358:fail
344:the
240:are
233:and
205:ZFC+
188:ZFC+
160:and
34:, a
2638:of
2620:of
2568:of
2100:Sur
2074:Map
1881:Ur-
1863:Set
1110:Set
724:doi
651:doi
622:doi
582:doi
417:are
149:by
135:not
55:ZFC
3183::
3024:NP
2648::
2642::
2572::
2249:),
2104:Bi
2096:In
743:48
741:.
720:13
718:.
714:.
703:;
699:;
665:.
657:.
647:53
645:.
628:.
618:53
616:.
588:,
576:,
485:.
403:.
297:.
112:ZF
72:A
3104:/
3019:P
2774:)
2560:)
2556:(
2453:â
2448:!
2443:â
2404:=
2399:â
2394:â
2389:â§
2384:âš
2379:ÂŹ
2102:/
2098:/
2072:/
1883:)
1879:(
1766:â
1756:3
1544:)
1442:e
1435:t
1428:v
1193:·
1177:)
1173:(
1140:)
1029:)
789:e
782:t
775:v
732:.
726::
691:.
685::
673:.
653::
636:.
624::
584::
564:.
540:.
518:.
489:.
470:.
291:1
288:A
284:2
281:A
277:1
274:A
270:1
267:A
263:2
260:A
256:2
253:A
249:1
246:A
238:2
235:A
231:1
228:A
217:1
214:A
210:2
207:A
200:2
197:A
193:1
190:A
183:2
180:A
176:1
173:A
165:2
162:A
158:1
155:A
125:Î
121:L
116:Î
47:α
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.