3158:
376:), 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
810:
53:
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
289:
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
399:
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
46:, 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
309:
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
689:
306:. It is also noteworthy that many combinatorial statements are exactly equiconsistent with some large cardinal rather than, say, being intermediate between them.
404:
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
1537:
2212:
1274:
369:, L, which does not satisfy the statement "there is a measurable cardinal" (even though it contains the measurable cardinal as an ordinal).
2295:
1436:
83:
There is no generally agreed precise definition of what a large cardinal property is, though essentially everyone agrees that those in the
283:
34:. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=Ï
471:
2609:
2767:
582:
1555:
2622:
1945:
963:
776:
385:
366:
77:
749:
69:
is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property.
2627:
2617:
2354:
2207:
1560:
1291:
1551:
2763:
548:
524:
502:
454:
421:
412:
model in L that believes there exists a measurable cardinal, even though L itself does not satisfy that proposition.
84:
2105:
2860:
2604:
1429:
2165:
1858:
1269:
863:
372:
Thus, from a certain point of view held by many set theorists (especially those inspired by the tradition of the
1599:
1149:
3121:
2823:
2586:
2581:
2406:
1827:
1511:
396:
3182:
3116:
2899:
2816:
2529:
2460:
2337:
1579:
1043:
922:
349:
can be seen in some natural way as submodels of those in which the axioms hold. For example, if there is an
3041:
2867:
2553:
2187:
1786:
1286:
495:
Set Theory: An
Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76)
72:
Most working set theorists believe that the large cardinal axioms that are currently being considered are
2919:
2914:
2524:
2263:
2192:
1521:
1422:
1279:
917:
880:
76:
with ZFC. These axioms are strong enough to imply the consistency of ZFC. This has the consequence (via
2848:
2438:
1832:
1800:
1491:
294:
has asked, "s there some theorem explaining this, or is our vision just more uniform than we realize?"
934:
3138:
3087:
2984:
2482:
2443:
1920:
1565:
968:
853:
841:
836:
1594:
3187:
2979:
2909:
2448:
2300:
2283:
2006:
1486:
769:
2811:
2788:
2749:
2635:
2576:
2222:
2142:
1986:
1930:
1543:
1388:
1306:
1181:
1133:
947:
870:
745:
3101:
2828:
2806:
2773:
2666:
2512:
2497:
2470:
2421:
2305:
2240:
2065:
2031:
2026:
1900:
1731:
1708:
1340:
1221:
1033:
846:
401:
354:
212:
These are mutually exclusive, unless one of the theories in question is actually inconsistent.
108:
569:, Lecture Notes in Mathematics, vol. 669, Springer Berlin / Heidelberg, pp. 99â275,
3031:
2884:
2676:
2394:
2130:
2036:
1895:
1880:
1761:
1736:
1256:
1226:
1170:
1090:
1070:
1048:
350:
330:
314:
134:
A remarkable observation about large cardinal axioms is that they appear to occur in strict
3004:
2966:
2843:
2647:
2487:
2411:
2389:
2217:
2175:
2074:
2041:
1905:
1693:
1604:
1330:
1320:
1154:
1085:
1038:
978:
858:
326:
139:
8:
3133:
3024:
3009:
2989:
2946:
2833:
2783:
2709:
2654:
2591:
2384:
2379:
2327:
2095:
2084:
1756:
1656:
1584:
1575:
1571:
1506:
1501:
1325:
1236:
1144:
1139:
953:
895:
826:
762:
538:
358:
80:) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent).
3162:
2931:
2894:
2879:
2872:
2855:
2659:
2641:
2507:
2433:
2416:
2369:
2182:
2091:
1925:
1910:
1870:
1822:
1807:
1795:
1751:
1726:
1496:
1445:
1248:
1243:
1028:
983:
890:
671:
655:
647:
618:
443:
373:
28:
2115:
3157:
3097:
2904:
2714:
2704:
2596:
2477:
2312:
2288:
2069:
2053:
1958:
1935:
1812:
1781:
1746:
1641:
1476:
1105:
942:
905:
875:
799:
717:
685:
578:
544:
520:
498:
450:
377:
659:
353:, then "cutting the universe off" at the height of the first such cardinal yields a
50:'s phrase, as quantifying the fact "that if you want more you have to assume more".
3111:
3106:
2999:
2956:
2778:
2739:
2734:
2719:
2545:
2502:
2399:
2197:
2147:
1721:
1683:
1393:
1383:
1368:
1363:
1231:
885:
712:
693:
639:
610:
570:
534:
142:. That is, no exception is known to the following: Given two large cardinal axioms
317:, but assuming both exist, the first huge is smaller than the first supercompact.
3092:
3082:
3036:
3019:
2974:
2936:
2838:
2758:
2565:
2492:
2465:
2453:
2359:
2273:
2247:
2202:
2170:
1971:
1773:
1716:
1666:
1631:
1589:
1262:
1200:
1018:
831:
230:
38:). The proposition that such cardinals exist cannot be proved in the most common
31:
299:
3077:
3056:
3014:
2994:
2889:
2744:
2342:
2332:
2322:
2317:
2251:
2001:
1890:
1885:
1863:
1464:
1398:
1195:
1176:
1080:
1065:
958:
900:
667:
598:
409:
295:
291:
282:
is itself consistent (provided of course that it really is). This follows from
39:
540:
The Higher
Infinite : Large Cardinals in Set Theory from Their Beginnings
99:
is that the existence of such a cardinal is not known to be inconsistent with
3176:
3051:
2729:
2236:
2021:
2011:
1981:
1966:
1636:
1403:
1205:
1119:
1114:
310:
1373:
313:
is much stronger, in terms of consistency strength, than the existence of a
2951:
2798:
2699:
2691:
2571:
2519:
2428:
2364:
2347:
2278:
2137:
1996:
1698:
1481:
1353:
1348:
1166:
1095:
1053:
912:
809:
684:
475:
342:
135:
3061:
2941:
2120:
2110:
2057:
1741:
1661:
1646:
1526:
1471:
1378:
1013:
512:
1991:
1846:
1817:
1623:
1358:
1129:
785:
651:
622:
574:
388:
in this group would state, more simply, that large cardinal axioms are
119:
73:
47:
20:
697:
559:
395:
This point of view is by no means universal among set theorists. Some
303:
3143:
3046:
2099:
2016:
1976:
1940:
1876:
1688:
1678:
1651:
1414:
1161:
1124:
1075:
973:
676:
643:
614:
3128:
2926:
2374:
2079:
1673:
334:
2724:
1516:
95:
A necessary condition for a property of cardinal numbers to be a
1186:
1008:
338:
2268:
1614:
1459:
1058:
818:
754:
726:
Woodin, W. Hugh (2001). "The continuum hypothesis, part II".
380:) or others that they consider intuitively unlikely (such as
357:
in which there is no inaccessible cardinal. Or if there is a
408:
restrictive, pointing out that (for example) there can be a
275:
is consistent, even with the additional hypothesis that ZFC+
517:
Set theory, third millennium edition (revised and expanded)
472:"Does anyone still seriously doubt the consistency of ZFC?"
445:
381:
100:
43:
560:"The evolution of large cardinal axioms in set theory"
698:"Strong axioms of infinity and elementary embeddings"
325:
Large cardinals are understood in the context of the
55:
630:
Maddy, Penelope (1988). "Believing the Axioms, II".
129:
365:powerset operation rather than the full one yields
320:
442:
107:would be an uncountable initial ordinal for which
3174:
54:point among distinct philosophical schools (see
557:
1430:
770:
728:Notices of the American Mathematical Society
511:
492:
126:imply that any such large cardinals exist.
1622:
1437:
1423:
777:
763:
716:
675:
533:
558:Kanamori, Akihiro; Magidor, M. (1978),
337:operation, which collects together all
156:, exactly one of three things happens:
3175:
1444:
725:
666:
1418:
758:
629:
597:
284:Gödel's second incompleteness theorem
90:
78:Gödel's second incompleteness theorem
670:(2002). "The Future of Set Theory".
469:
440:
750:Stanford Encyclopedia of Philosophy
601:(1988). "Believing the Axioms, I".
302:, the main unsolved problem of his
13:
240:is consistency-wise stronger than
14:
3199:
746:"Large Cardinals and Determinacy"
739:
449:. Oxford University Press. viii.
422:List of large cardinal properties
298:, however, deduces this from the
167:is consistent if and only if ZFC+
130:Hierarchy of consistency strength
85:list of large cardinal properties
27:is a certain kind of property of
3156:
808:
321:Motivations and epistemic status
160:Unless ZFC is inconsistent, ZFC+
56:Motivations and epistemic status
345:in which large cardinal axioms
87:are large cardinal properties.
784:
463:
434:
367:Gödel's constructible universe
1:
3117:History of mathematical logic
486:
118:is a model of ZFC. If ZFC is
19:In the mathematical field of
3042:Primitive recursive function
718:10.1016/0003-4843(78)90031-1
705:Annals of Mathematical Logic
470:Joel, Hamkins (2022-12-24).
247:(vice versa for case 3). If
7:
415:
341:of a given set. Typically,
10:
3204:
2106:SchröderâBernstein theorem
1833:Monadic predicate calculus
1492:Foundations of mathematics
1275:von NeumannâBernaysâGödel
543:(2nd ed.). Springer.
3152:
3139:Philosophy of mathematics
3088:Automated theorem proving
3070:
2965:
2797:
2690:
2542:
2259:
2235:
2213:Von NeumannâBernaysâGödel
2158:
2052:
1956:
1854:
1845:
1772:
1707:
1613:
1535:
1452:
1339:
1302:
1214:
1104:
1076:One-to-one correspondence
992:
933:
817:
806:
792:
632:Journal of Symbolic Logic
603:Journal of Symbolic Logic
233:. In case 2, we say that
103:and that such a cardinal
497:. Elsevier Science Ltd.
427:
329:V, which is built up by
2789:Self-verifying theories
2610:Tarski's axiomatization
1561:Tarski's undefinability
1556:incompleteness theorems
331:transfinitely iterating
215:In case 1, we say that
97:large cardinal property
25:large cardinal property
3163:Mathematics portal
2774:Proof of impossibility
2422:propositional variable
1732:Propositional calculus
1034:Constructible universe
854:Constructibility (V=L)
402:ontological maximalism
42:of set theory, namely
3032:Kolmogorov complexity
2985:Computably enumerable
2885:Model complete theory
2677:Principia Mathematica
1737:Propositional formula
1566:BanachâTarski paradox
1257:Principia Mathematica
1091:Transfinite induction
950:(i.e. set difference)
493:Drake, F. R. (1974).
361:, then iterating the
351:inaccessible cardinal
315:supercompact cardinal
3183:Axioms of set theory
2980:ChurchâTuring thesis
2967:Computability theory
2176:continuum hypothesis
1694:Square of opposition
1552:Gödel's completeness
1331:Burali-Forti paradox
1086:Set-builder notation
1039:Continuum hypothesis
979:Symmetric difference
690:William N. Reinhardt
441:Bell, J. L. (1985).
327:von Neumann universe
140:consistency strength
65:large cardinal axiom
3134:Mathematical object
3025:P versus NP problem
2990:Computable function
2784:Reverse mathematics
2710:Logical consequence
2587:primitive recursive
2582:elementary function
2355:Free/bound variable
2208:TarskiâGrothendieck
1727:Logical connectives
1657:Logical equivalence
1507:Logical consequence
1292:TarskiâGrothendieck
359:measurable cardinal
2932:Transfer principle
2895:Semantics of logic
2880:Categorical theory
2856:Non-standard model
2370:Logical connective
1497:Information theory
1446:Mathematical logic
881:Limitation of size
686:Solovay, Robert M.
575:10.1007/BFb0103104
91:Partial definition
16:Set theory concept
3170:
3169:
3102:Abstract category
2905:Theories of truth
2715:Rule of inference
2705:Natural deduction
2686:
2685:
2231:
2230:
1936:Cartesian product
1841:
1840:
1747:Many-valued logic
1722:Boolean functions
1605:Russell's paradox
1580:diagonal argument
1477:First-order logic
1412:
1411:
1321:Russell's paradox
1270:ZermeloâFraenkel
1171:Dedekind-infinite
1044:Diagonal argument
943:Cartesian product
800:Set (mathematics)
584:978-3-540-08926-1
567:Higher Set Theory
535:Kanamori, Akihiro
268:cannot prove ZFC+
254:is stronger than
191:is consistent; or
3195:
3161:
3160:
3112:History of logic
3107:Category of sets
3000:Decision problem
2779:Ordinal analysis
2720:Sequent calculus
2618:Boolean algebras
2558:
2557:
2532:
2503:logical/constant
2257:
2256:
2243:
2166:ZermeloâFraenkel
1917:Set operations:
1852:
1851:
1789:
1620:
1619:
1600:LöwenheimâSkolem
1487:Formal semantics
1439:
1432:
1425:
1416:
1415:
1394:Bertrand Russell
1384:John von Neumann
1369:Abraham Fraenkel
1364:Richard Dedekind
1326:Suslin's problem
1237:Cantor's theorem
954:De Morgan's laws
812:
779:
772:
765:
756:
755:
735:
722:
720:
702:
694:Akihiro Kanamori
681:
679:
663:
626:
594:
593:
591:
564:
554:
530:
508:
480:
479:
467:
461:
460:
448:
438:
384:). The hardcore
201:proves that ZFC+
184:proves that ZFC+
122:, then ZFC does
67:
66:
32:cardinal numbers
3203:
3202:
3198:
3197:
3196:
3194:
3193:
3192:
3188:Large cardinals
3173:
3172:
3171:
3166:
3155:
3148:
3093:Category theory
3083:Algebraic logic
3066:
3037:Lambda calculus
2975:Church encoding
2961:
2937:Truth predicate
2793:
2759:Complete theory
2682:
2551:
2547:
2543:
2538:
2530:
2250: and
2246:
2241:
2227:
2203:New Foundations
2171:axiom of choice
2154:
2116:Gödel numbering
2056: and
2048:
1952:
1837:
1787:
1768:
1717:Boolean algebra
1703:
1667:Equiconsistency
1632:Classical logic
1609:
1590:Halting problem
1578: and
1554: and
1542: and
1541:
1536:Theorems (
1531:
1448:
1443:
1413:
1408:
1335:
1314:
1298:
1263:New Foundations
1210:
1100:
1019:Cardinal number
1002:
988:
929:
813:
804:
788:
783:
742:
700:
668:Shelah, Saharon
644:10.2307/2274569
615:10.2307/2274520
599:Maddy, Penelope
589:
587:
585:
562:
551:
527:
505:
489:
484:
483:
468:
464:
457:
439:
435:
430:
418:
382:V = L
323:
281:
274:
267:
260:
253:
246:
239:
228:
221:
207:
200:
190:
183:
173:
166:
155:
148:
132:
116:
93:
64:
63:
37:
17:
12:
11:
5:
3201:
3191:
3190:
3185:
3168:
3167:
3153:
3150:
3149:
3147:
3146:
3141:
3136:
3131:
3126:
3125:
3124:
3114:
3109:
3104:
3095:
3090:
3085:
3080:
3078:Abstract logic
3074:
3072:
3068:
3067:
3065:
3064:
3059:
3057:Turing machine
3054:
3049:
3044:
3039:
3034:
3029:
3028:
3027:
3022:
3017:
3012:
3007:
2997:
2995:Computable set
2992:
2987:
2982:
2977:
2971:
2969:
2963:
2962:
2960:
2959:
2954:
2949:
2944:
2939:
2934:
2929:
2924:
2923:
2922:
2917:
2912:
2902:
2897:
2892:
2890:Satisfiability
2887:
2882:
2877:
2876:
2875:
2865:
2864:
2863:
2853:
2852:
2851:
2846:
2841:
2836:
2831:
2821:
2820:
2819:
2814:
2807:Interpretation
2803:
2801:
2795:
2794:
2792:
2791:
2786:
2781:
2776:
2771:
2761:
2756:
2755:
2754:
2753:
2752:
2742:
2737:
2727:
2722:
2717:
2712:
2707:
2702:
2696:
2694:
2688:
2687:
2684:
2683:
2681:
2680:
2672:
2671:
2670:
2669:
2664:
2663:
2662:
2657:
2652:
2632:
2631:
2630:
2628:minimal axioms
2625:
2614:
2613:
2612:
2601:
2600:
2599:
2594:
2589:
2584:
2579:
2574:
2561:
2559:
2540:
2539:
2537:
2536:
2535:
2534:
2522:
2517:
2516:
2515:
2510:
2505:
2500:
2490:
2485:
2480:
2475:
2474:
2473:
2468:
2458:
2457:
2456:
2451:
2446:
2441:
2431:
2426:
2425:
2424:
2419:
2414:
2404:
2403:
2402:
2397:
2392:
2387:
2382:
2377:
2367:
2362:
2357:
2352:
2351:
2350:
2345:
2340:
2335:
2325:
2320:
2318:Formation rule
2315:
2310:
2309:
2308:
2303:
2293:
2292:
2291:
2281:
2276:
2271:
2266:
2260:
2254:
2237:Formal systems
2233:
2232:
2229:
2228:
2226:
2225:
2220:
2215:
2210:
2205:
2200:
2195:
2190:
2185:
2180:
2179:
2178:
2173:
2162:
2160:
2156:
2155:
2153:
2152:
2151:
2150:
2140:
2135:
2134:
2133:
2126:Large cardinal
2123:
2118:
2113:
2108:
2103:
2089:
2088:
2087:
2082:
2077:
2062:
2060:
2050:
2049:
2047:
2046:
2045:
2044:
2039:
2034:
2024:
2019:
2014:
2009:
2004:
1999:
1994:
1989:
1984:
1979:
1974:
1969:
1963:
1961:
1954:
1953:
1951:
1950:
1949:
1948:
1943:
1938:
1933:
1928:
1923:
1915:
1914:
1913:
1908:
1898:
1893:
1891:Extensionality
1888:
1886:Ordinal number
1883:
1873:
1868:
1867:
1866:
1855:
1849:
1843:
1842:
1839:
1838:
1836:
1835:
1830:
1825:
1820:
1815:
1810:
1805:
1804:
1803:
1793:
1792:
1791:
1778:
1776:
1770:
1769:
1767:
1766:
1765:
1764:
1759:
1754:
1744:
1739:
1734:
1729:
1724:
1719:
1713:
1711:
1705:
1704:
1702:
1701:
1696:
1691:
1686:
1681:
1676:
1671:
1670:
1669:
1659:
1654:
1649:
1644:
1639:
1634:
1628:
1626:
1617:
1611:
1610:
1608:
1607:
1602:
1597:
1592:
1587:
1582:
1570:Cantor's
1568:
1563:
1558:
1548:
1546:
1533:
1532:
1530:
1529:
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1484:
1479:
1474:
1469:
1468:
1467:
1456:
1454:
1450:
1449:
1442:
1441:
1434:
1427:
1419:
1410:
1409:
1407:
1406:
1401:
1399:Thoralf Skolem
1396:
1391:
1386:
1381:
1376:
1371:
1366:
1361:
1356:
1351:
1345:
1343:
1337:
1336:
1334:
1333:
1328:
1323:
1317:
1315:
1313:
1312:
1309:
1303:
1300:
1299:
1297:
1296:
1295:
1294:
1289:
1284:
1283:
1282:
1267:
1266:
1265:
1253:
1252:
1251:
1240:
1239:
1234:
1229:
1224:
1218:
1216:
1212:
1211:
1209:
1208:
1203:
1198:
1193:
1184:
1179:
1174:
1164:
1159:
1158:
1157:
1152:
1147:
1137:
1127:
1122:
1117:
1111:
1109:
1102:
1101:
1099:
1098:
1093:
1088:
1083:
1081:Ordinal number
1078:
1073:
1068:
1063:
1062:
1061:
1056:
1046:
1041:
1036:
1031:
1026:
1016:
1011:
1005:
1003:
1001:
1000:
997:
993:
990:
989:
987:
986:
981:
976:
971:
966:
961:
959:Disjoint union
956:
951:
945:
939:
937:
931:
930:
928:
927:
926:
925:
920:
909:
908:
906:Martin's axiom
903:
898:
893:
888:
883:
878:
873:
871:Extensionality
868:
867:
866:
856:
851:
850:
849:
844:
839:
829:
823:
821:
815:
814:
807:
805:
803:
802:
796:
794:
790:
789:
782:
781:
774:
767:
759:
753:
752:
741:
740:External links
738:
737:
736:
723:
682:
664:
638:(3): 736â764.
627:
609:(2): 481â511.
595:
583:
555:
549:
531:
525:
509:
503:
488:
485:
482:
481:
462:
455:
432:
431:
429:
426:
425:
424:
417:
414:
410:transitive set
378:Martin's axiom
322:
319:
292:Saharon Shelah
279:
272:
265:
258:
251:
244:
237:
231:equiconsistent
226:
219:
210:
209:
208:is consistent.
205:
198:
192:
188:
181:
175:
174:is consistent;
171:
164:
153:
146:
131:
128:
112:
92:
89:
40:axiomatization
35:
15:
9:
6:
4:
3:
2:
3200:
3189:
3186:
3184:
3181:
3180:
3178:
3165:
3164:
3159:
3151:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3123:
3120:
3119:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3075:
3073:
3069:
3063:
3060:
3058:
3055:
3053:
3052:Recursive set
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3002:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2972:
2970:
2968:
2964:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2921:
2918:
2916:
2913:
2911:
2908:
2907:
2906:
2903:
2901:
2898:
2896:
2893:
2891:
2888:
2886:
2883:
2881:
2878:
2874:
2871:
2870:
2869:
2866:
2862:
2861:of arithmetic
2859:
2858:
2857:
2854:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2826:
2825:
2822:
2818:
2815:
2813:
2810:
2809:
2808:
2805:
2804:
2802:
2800:
2796:
2790:
2787:
2785:
2782:
2780:
2777:
2775:
2772:
2769:
2768:from ZFC
2765:
2762:
2760:
2757:
2751:
2748:
2747:
2746:
2743:
2741:
2738:
2736:
2733:
2732:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2697:
2695:
2693:
2689:
2679:
2678:
2674:
2673:
2668:
2667:non-Euclidean
2665:
2661:
2658:
2656:
2653:
2651:
2650:
2646:
2645:
2643:
2640:
2639:
2637:
2633:
2629:
2626:
2624:
2621:
2620:
2619:
2615:
2611:
2608:
2607:
2606:
2602:
2598:
2595:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2569:
2567:
2563:
2562:
2560:
2555:
2549:
2544:Example
2541:
2533:
2528:
2527:
2526:
2523:
2521:
2518:
2514:
2511:
2509:
2506:
2504:
2501:
2499:
2496:
2495:
2494:
2491:
2489:
2486:
2484:
2481:
2479:
2476:
2472:
2469:
2467:
2464:
2463:
2462:
2459:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2436:
2435:
2432:
2430:
2427:
2423:
2420:
2418:
2415:
2413:
2410:
2409:
2408:
2405:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2383:
2381:
2378:
2376:
2373:
2372:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2330:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2307:
2304:
2302:
2301:by definition
2299:
2298:
2297:
2294:
2290:
2287:
2286:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2267:
2265:
2262:
2261:
2258:
2255:
2253:
2249:
2244:
2238:
2234:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2188:KripkeâPlatek
2186:
2184:
2181:
2177:
2174:
2172:
2169:
2168:
2167:
2164:
2163:
2161:
2157:
2149:
2146:
2145:
2144:
2141:
2139:
2136:
2132:
2129:
2128:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2101:
2097:
2093:
2090:
2086:
2083:
2081:
2078:
2076:
2073:
2072:
2071:
2067:
2064:
2063:
2061:
2059:
2055:
2051:
2043:
2040:
2038:
2035:
2033:
2032:constructible
2030:
2029:
2028:
2025:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1964:
1962:
1960:
1955:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1918:
1916:
1912:
1909:
1907:
1904:
1903:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1882:
1878:
1874:
1872:
1869:
1865:
1862:
1861:
1860:
1857:
1856:
1853:
1850:
1848:
1844:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1802:
1799:
1798:
1797:
1794:
1790:
1785:
1784:
1783:
1780:
1779:
1777:
1775:
1771:
1763:
1760:
1758:
1755:
1753:
1750:
1749:
1748:
1745:
1743:
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1723:
1720:
1718:
1715:
1714:
1712:
1710:
1709:Propositional
1706:
1700:
1697:
1695:
1692:
1690:
1687:
1685:
1682:
1680:
1677:
1675:
1672:
1668:
1665:
1664:
1663:
1660:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1637:Logical truth
1635:
1633:
1630:
1629:
1627:
1625:
1621:
1618:
1616:
1612:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1577:
1573:
1569:
1567:
1564:
1562:
1559:
1557:
1553:
1550:
1549:
1547:
1545:
1539:
1534:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1466:
1463:
1462:
1461:
1458:
1457:
1455:
1451:
1447:
1440:
1435:
1433:
1428:
1426:
1421:
1420:
1417:
1405:
1404:Ernst Zermelo
1402:
1400:
1397:
1395:
1392:
1390:
1389:Willard Quine
1387:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1346:
1344:
1342:
1341:Set theorists
1338:
1332:
1329:
1327:
1324:
1322:
1319:
1318:
1316:
1310:
1308:
1305:
1304:
1301:
1293:
1290:
1288:
1287:KripkeâPlatek
1285:
1281:
1278:
1277:
1276:
1273:
1272:
1271:
1268:
1264:
1261:
1260:
1259:
1258:
1254:
1250:
1247:
1246:
1245:
1242:
1241:
1238:
1235:
1233:
1230:
1228:
1225:
1223:
1220:
1219:
1217:
1213:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1190:
1185:
1183:
1180:
1178:
1175:
1172:
1168:
1165:
1163:
1160:
1156:
1153:
1151:
1148:
1146:
1143:
1142:
1141:
1138:
1135:
1131:
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1112:
1110:
1107:
1103:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1060:
1057:
1055:
1052:
1051:
1050:
1047:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1024:
1020:
1017:
1015:
1012:
1010:
1007:
1006:
1004:
998:
995:
994:
991:
985:
982:
980:
977:
975:
972:
970:
967:
965:
962:
960:
957:
955:
952:
949:
946:
944:
941:
940:
938:
936:
932:
924:
923:specification
921:
919:
916:
915:
914:
911:
910:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
882:
879:
877:
874:
872:
869:
865:
862:
861:
860:
857:
855:
852:
848:
845:
843:
840:
838:
835:
834:
833:
830:
828:
825:
824:
822:
820:
816:
811:
801:
798:
797:
795:
791:
787:
780:
775:
773:
768:
766:
761:
760:
757:
751:
747:
744:
743:
734:(7): 681â690.
733:
729:
724:
719:
714:
711:(1): 73â116.
710:
706:
699:
695:
691:
687:
683:
678:
673:
669:
665:
661:
657:
653:
649:
645:
641:
637:
633:
628:
624:
620:
616:
612:
608:
604:
600:
596:
590:September 25,
586:
580:
576:
572:
568:
561:
556:
552:
550:3-540-00384-3
546:
542:
541:
536:
532:
528:
526:3-540-44085-2
522:
518:
514:
510:
506:
504:0-444-10535-2
500:
496:
491:
490:
477:
473:
466:
458:
456:0-19-853241-5
452:
447:
446:
437:
433:
423:
420:
419:
413:
411:
407:
403:
398:
393:
391:
387:
383:
379:
375:
370:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
328:
318:
316:
312:
311:huge cardinal
307:
305:
301:
297:
293:
290:cases holds.
287:
285:
278:
271:
264:
257:
250:
243:
236:
232:
225:
218:
213:
204:
197:
193:
187:
180:
176:
170:
163:
159:
158:
157:
152:
145:
141:
137:
127:
125:
121:
117:
115:
111:
106:
102:
98:
88:
86:
81:
79:
75:
70:
68:
59:
57:
51:
49:
45:
41:
33:
30:
26:
22:
3154:
2952:Ultraproduct
2799:Model theory
2764:Independence
2700:Formal proof
2692:Proof theory
2675:
2648:
2605:real numbers
2577:second-order
2488:Substitution
2365:Metalanguage
2306:conservative
2279:Axiom schema
2223:Constructive
2193:MorseâKelley
2159:Set theories
2138:Aleph number
2131:inaccessible
2125:
2037:Grothendieck
1921:intersection
1808:Higher-order
1796:Second-order
1742:Truth tables
1699:Venn diagram
1482:Formal proof
1354:Georg Cantor
1349:Paul Bernays
1280:MorseâKelley
1255:
1188:
1187:Subset
1134:hereditarily
1096:Venn diagram
1054:ordered pair
1022:
969:Intersection
913:Axiom schema
731:
727:
708:
704:
677:math/0211397
635:
631:
606:
602:
588:, retrieved
566:
539:
519:. Springer.
516:
513:Jech, Thomas
494:
476:MathOverflow
465:
444:
436:
405:
394:
389:
371:
362:
346:
324:
308:
300:Ω-conjecture
288:
276:
269:
262:
255:
248:
241:
234:
223:
216:
214:
211:
202:
195:
185:
178:
168:
161:
150:
143:
136:linear order
133:
123:
113:
109:
104:
96:
94:
82:
71:
62:
60:
52:
24:
18:
3062:Type theory
3010:undecidable
2942:Truth value
2829:equivalence
2508:non-logical
2121:Enumeration
2111:Isomorphism
2058:cardinality
2042:Von Neumann
2007:Ultrafilter
1972:Uncountable
1906:equivalence
1823:Quantifiers
1813:Fixed-point
1782:First-order
1662:Consistency
1647:Proposition
1624:Traditional
1595:Lindström's
1585:Compactness
1527:Type theory
1472:Cardinality
1379:Thomas Jech
1222:Alternative
1201:Uncountable
1155:Ultrafilter
1014:Cardinality
918:replacement
859:Determinacy
261:, then ZFC+
29:transfinite
3177:Categories
2873:elementary
2566:arithmetic
2434:Quantifier
2412:functional
2284:Expression
2002:Transitive
1946:identities
1931:complement
1864:hereditary
1847:Set theory
1374:Kurt Gödel
1359:Paul Cohen
1196:Transitive
964:Identities
948:Complement
935:Operations
896:Regularity
864:projective
827:Adjunction
786:Set theory
487:References
397:formalists
120:consistent
74:consistent
48:Dana Scott
21:set theory
3144:Supertask
3047:Recursion
3005:decidable
2839:saturated
2817:of models
2740:deductive
2735:axiomatic
2655:Hilbert's
2642:Euclidean
2623:canonical
2546:axiomatic
2478:Signature
2407:Predicate
2296:Extension
2218:Ackermann
2143:Operation
2022:Universal
2012:Recursive
1987:Singleton
1982:Inhabited
1967:Countable
1957:Types of
1941:power set
1911:partition
1828:Predicate
1774:Predicate
1689:Syllogism
1679:Soundness
1652:Inference
1642:Tautology
1544:paradoxes
1307:Paradoxes
1227:Axiomatic
1206:Universal
1182:Singleton
1177:Recursive
1120:Countable
1115:Amorphous
974:Power set
891:Power set
842:dependent
837:countable
363:definable
3129:Logicism
3122:timeline
3098:Concrete
2957:Validity
2927:T-schema
2920:Kripke's
2915:Tarski's
2910:semantic
2900:Strength
2849:submodel
2844:spectrum
2812:function
2660:Tarski's
2649:Elements
2636:geometry
2592:Robinson
2513:variable
2498:function
2471:spectrum
2461:Sentence
2417:variable
2360:Language
2313:Relation
2274:Automata
2264:Alphabet
2248:language
2102:-jection
2080:codomain
2066:Function
2027:Universe
1997:Infinite
1901:Relation
1684:Validity
1674:Argument
1572:theorem,
1311:Problems
1215:Theories
1191:Superset
1167:Infinite
996:Concepts
876:Infinity
793:Overview
696:(1978).
660:16544090
537:(2003).
515:(2002).
416:See also
386:realists
355:universe
335:powerset
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3071:Related
2868:Diagram
2766: (
2745:Hilbert
2730:Systems
2725:Theorem
2603:of the
2548:systems
2328:Formula
2323:Grammar
2239: (
2183:General
1896:Forcing
1881:Element
1801:Monadic
1576:paradox
1517:Theorem
1453:General
1249:General
1244:Zermelo
1150:subbase
1132: (
1071:Forcing
1049:Element
1021: (
999:Methods
886:Pairing
748:at the
652:2274569
623:2274520
339:subsets
304:Ω-logic
2834:finite
2597:Skolem
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2525:Theory
2493:Symbol
2483:String
2466:atomic
2343:ground
2338:closed
2333:atomic
2289:ground
2252:syntax
2148:binary
2075:domain
1992:Finite
1757:finite
1615:Logics
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1140:Filter
1130:Finite
1066:Family
1009:Almost
847:global
832:Choice
819:Axioms
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343:models
296:Woodin
2824:Model
2572:Peano
2429:Proof
2269:Arity
2198:Naive
2085:image
2017:Fuzzy
1977:Empty
1926:union
1871:Class
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1502:Lemma
1460:Axiom
1232:Naive
1162:Fuzzy
1125:Empty
1108:types
1059:tuple
1029:Class
1023:large
984:Union
901:Union
701:(PDF)
672:arXiv
656:S2CID
648:JSTOR
619:JSTOR
563:(PDF)
428:Notes
374:Cabal
2947:Type
2750:list
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2520:Term
2454:rank
2348:open
2242:list
2054:Maps
1959:sets
1818:Free
1788:list
1538:list
1465:list
1145:base
592:2022
579:ISBN
545:ISBN
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499:ISBN
451:ISBN
390:true
347:fail
333:the
229:are
222:and
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