4270:
4340:
1836:
is regular. Thus the regularity or singularity of most aleph numbers can be checked depending on whether the cardinal is a successor cardinal or a limit cardinal. Some cardinalities cannot be proven to be equal to any particular aleph, for instance the
1883:
could meaningfully be called regular or singular cardinals.Furthermore, a successor aleph would need not be regular. For instance, the union of a countable set of countable sets would not necessarily be countable. It is consistent with
2454:. Maddy cites two papers by Mirimanoff, "Les antinomies de Russell et de Burali-Forti et le problÚme fundamental de la théorie des ensembles" and "Remarques sur la théorie des ensembles et les antinomies Cantorienne", both in
1818:
1720:. They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom. Inaccessible cardinals are necessarily
525:
213:
2268:
1879:
Without the axiom of choice: there would be cardinal numbers that were not well-orderable. Moreover, the cardinal sum of an arbitrary collection could not be defined. Therefore, only the
354:
246:
604:
might fail, as in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above equivalence holds for well-orderable cardinals only.
2340:
2314:
2155:
2102:
2017:
433:
83:
2381:
2196:
1699:
1608:
1581:
1554:
1406:
1271:
691:
1668:
1124:
1098:
1072:
908:
302:
489:
390:
2044:
1944:
1913:
1874:
1507:
1480:
1453:
1379:
1352:
1325:
1298:
1242:
1211:
1184:
1153:
806:
1638:
1026:
1000:
974:
858:
2288:
2126:
1991:
1971:
1746:
1527:
1426:
1046:
948:
928:
882:
830:
779:
759:
739:
719:
660:
628:
591:
569:
545:
161:
138:
103:
53:
2220:
2067:
453:
266:
2649:
1751:
4804:
3324:
832:, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite.
3407:
2548:
3721:
1919:
be a countable union of countable sets. Furthermore, it is consistent with ZF when not including AC that every aleph bigger than
1217:(finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So
3879:
597:
Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts.
2667:
4493:
4306:
3734:
3057:
500:
169:
4821:
3739:
3729:
3466:
3319:
2672:
2663:
3875:
2521:
2502:
3217:
2225:
3972:
3716:
2541:
4799:
4393:
3277:
2970:
1885:
4679:
2711:
4233:
3935:
3698:
3693:
3518:
2939:
2623:
310:
4573:
4452:
4228:
4011:
3928:
3641:
3572:
3449:
2691:
1706:
218:
4816:
4153:
3979:
3665:
3299:
2898:
1838:
4809:
4447:
4410:
4031:
4026:
3636:
3375:
3304:
2633:
2534:
1671:
2319:
2293:
2134:
2072:
1996:
395:
62:
3960:
3550:
2944:
2912:
2603:
2451:
Early hints of the Axiom of
Replacement can be found in Cantor's letter to Dedekind and in Mirimanoff
2345:
2160:
1677:
1586:
1559:
1532:
1384:
1249:
669:
4464:
2477:
4954:
4498:
4383:
4371:
4366:
4250:
4199:
4096:
3594:
3555:
3032:
2677:
2417:
1721:
1647:
1103:
1077:
1051:
887:
271:
2706:
1728:, though not all fixed points are regular. For instance, the first fixed point is the limit of the
458:
359:
4959:
4299:
4091:
4021:
3560:
3412:
3395:
3118:
2598:
4918:
4836:
4711:
4663:
4477:
4400:
3923:
3900:
3861:
3747:
3688:
3334:
3254:
3098:
3042:
2655:
2022:
1922:
1891:
1852:
1485:
1458:
1431:
1357:
1330:
1303:
1276:
1220:
1189:
1162:
1131:
784:
120:, any cardinal number can be well-ordered, and then the following are equivalent for a cardinal
4870:
4751:
4563:
4376:
4213:
3940:
3918:
3885:
3778:
3624:
3609:
3582:
3533:
3417:
3352:
3177:
3143:
3138:
3012:
2843:
2820:
1717:
1617:
1244:
cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.
1005:
979:
953:
837:
495:
4786:
4756:
4700:
4620:
4600:
4578:
4143:
3996:
3788:
3506:
3242:
3148:
3007:
2992:
2873:
2848:
2392:
2273:
2111:
1976:
1956:
1731:
1512:
1411:
1031:
933:
913:
867:
815:
764:
744:
724:
704:
645:
613:
576:
554:
530:
146:
123:
88:
56:
38:
4860:
4850:
4684:
4615:
4568:
4508:
4388:
4116:
4078:
3955:
3759:
3599:
3523:
3501:
3329:
3287:
3186:
3153:
3017:
2805:
2716:
2446:
2047:
1846:
8:
4855:
4766:
4674:
4669:
4483:
4425:
4356:
4292:
4245:
4136:
4121:
4101:
4058:
3945:
3895:
3821:
3766:
3703:
3496:
3491:
3439:
3207:
3196:
2868:
2768:
2696:
2687:
2683:
2618:
2613:
1842:
4778:
4773:
4558:
4513:
4420:
4274:
4043:
4006:
3991:
3984:
3967:
3771:
3753:
3619:
3545:
3528:
3481:
3294:
3203:
3037:
3022:
2982:
2934:
2919:
2907:
2863:
2838:
2608:
2557:
2434:
2205:
2052:
1833:
1702:
1156:
438:
251:
3227:
4635:
4472:
4435:
4405:
4329:
4269:
4209:
4016:
3826:
3816:
3708:
3589:
3424:
3400:
3181:
3165:
3070:
3047:
2924:
2893:
2858:
2753:
2588:
2517:
2498:
2467:
T. Arai, "Bounds on provability in set theories" (2012, p.2). Accessed 4 August 2022.
861:
1841:, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see
4923:
4913:
4898:
4893:
4761:
4415:
4223:
4218:
4111:
4068:
3890:
3851:
3846:
3831:
3657:
3614:
3511:
3309:
3259:
2833:
2795:
2489:
2426:
4792:
4730:
4548:
4361:
4204:
4194:
4148:
4131:
4086:
4048:
3950:
3870:
3677:
3604:
3577:
3565:
3471:
3385:
3359:
3314:
3282:
3083:
2885:
2828:
2778:
2743:
2701:
2442:
1915:
be the limit of a countable sequence of countable ordinals as well as the set of
1829:
721:
are finite. A finite sequence of finite ordinals always has a finite maximum, so
663:
601:
117:
28:
4928:
4725:
4706:
4610:
4595:
4552:
4488:
4430:
4189:
4168:
4126:
4106:
4001:
3856:
3454:
3444:
3434:
3429:
3363:
3237:
3113:
3002:
2997:
2975:
2576:
2412:
1713:
608:
4948:
4933:
4735:
4649:
4644:
4163:
3841:
3348:
3133:
3123:
3093:
3078:
2748:
2508:
1813:{\displaystyle \aleph _{0},\aleph _{\omega },\aleph _{\omega _{\omega }},...}
1214:
635:
4903:
4883:
4878:
4696:
4625:
4583:
4442:
4339:
4063:
3910:
3811:
3803:
3683:
3631:
3540:
3476:
3459:
3390:
3249:
3108:
2810:
2593:
1880:
1725:
4908:
4543:
4173:
4053:
3232:
3222:
3169:
2853:
2773:
2758:
2638:
2583:
1916:
2480:". Annals of Pure and Applied Logic vol. 170, no. 2 (2019), pp.251--271.
113:. Finite cardinal numbers are typically not called regular or singular.
4888:
4659:
4315:
3103:
2958:
2929:
2735:
2438:
1947:
809:
639:
106:
32:
20:
4691:
4654:
4605:
4503:
4255:
4158:
3211:
3128:
3088:
3052:
2988:
2800:
2790:
2763:
2526:
638:
that is not the limit of a set of smaller ordinals that as a set has
2430:
1610:
is the first infinite cardinal that is singular (the first infinite
4240:
4038:
3486:
3191:
2785:
2105:
3836:
2628:
600:
The situation is slightly more complicated in contexts where the
548:
4716:
4538:
4588:
4348:
4284:
3380:
2726:
2571:
1849:
postulates that the cardinality of the continuum is equal to
1670:). Proving the existence of singular cardinals requires the
1674:, and in fact the inability to prove the existence of
741:
cannot be the limit of any sequence of type less than
666:, though some initial ordinals are not regular, e.g.,
2478:
Small embedding characterizations for large cardinals
2348:
2322:
2296:
2276:
2228:
2208:
2163:
2137:
2114:
2075:
2055:
2025:
1999:
1979:
1959:
1925:
1894:
1855:
1754:
1734:
1680:
1650:
1620:
1589:
1562:
1535:
1515:
1488:
1461:
1434:
1414:
1387:
1360:
1333:
1306:
1279:
1252:
1223:
1192:
1165:
1134:
1106:
1080:
1054:
1034:
1008:
982:
956:
936:
916:
890:
870:
840:
818:
812:) is a regular cardinal because its initial ordinal,
787:
767:
747:
727:
707:
672:
648:
616:
579:
557:
533:
503:
461:
441:
398:
362:
313:
274:
254:
221:
172:
149:
126:
91:
65:
41:
884:. It is singular, since it is not a limit ordinal.
2514:Set Theory, An Introduction to Independence Proofs
2375:
2334:
2308:
2282:
2262:
2214:
2190:
2149:
2120:
2096:
2061:
2038:
2011:
1985:
1965:
1938:
1907:
1868:
1812:
1740:
1693:
1662:
1632:
1602:
1575:
1548:
1521:
1501:
1474:
1447:
1420:
1400:
1373:
1346:
1319:
1292:
1265:
1236:
1205:
1178:
1147:
1118:
1092:
1066:
1040:
1020:
994:
968:
942:
922:
902:
876:
852:
824:
800:
773:
753:
733:
713:
685:
654:
622:
585:
563:
539:
520:{\displaystyle \operatorname {Set} _{<\kappa }}
519:
483:
447:
427:
384:
348:
296:
260:
240:
208:{\displaystyle \kappa =\sum _{i\in I}\lambda _{i}}
207:
155:
132:
97:
77:
47:
930:. It can be written as the limit of the sequence
4946:
1273:is the next cardinal number after the sequence
547:and all functions between them is closed under
2263:{\displaystyle j({\textrm {crit}}(j))=\kappa }
4300:
2542:
1074:is the limit of a sequence of type less than
1716:that are also regular are known as (weakly)
2290:is uncountable and regular iff there is an
55:is a regular cardinal if and only if every
4307:
4293:
2734:
2549:
2535:
1028:, and so on. This sequence has order type
109:cardinals that are not regular are called
349:{\displaystyle S=\bigcup _{i\in I}S_{i}}
241:{\displaystyle \lambda _{i}<\kappa }
4947:
2556:
1100:whose elements are ordinals less than
781:, and is therefore a regular ordinal.
761:whose elements are ordinals less than
16:Type of cardinal number in mathematics
4288:
2530:
2411:
1876:, which is regular assuming choice.
35:. More explicitly, this means that
2415:(1988), "Believing the axioms. I",
2157:, say that an elementary embedding
13:
2335:{\displaystyle \theta >\alpha }
2309:{\displaystyle \alpha >\kappa }
2150:{\displaystyle \kappa <\theta }
2097:{\displaystyle j(\alpha )=\kappa }
2027:
2012:{\displaystyle \alpha <\kappa }
1927:
1857:
1782:
1769:
1756:
1682:
1591:
1564:
1509:, and so on, which has order type
1362:
1335:
1308:
1281:
1254:
1225:
1194:
1167:
1136:
789:
428:{\displaystyle |S_{i}|<\kappa }
78:{\displaystyle C\subseteq \kappa }
14:
4971:
2376:{\displaystyle j:M\to H(\theta )}
2191:{\displaystyle j:M\to H(\theta )}
1694:{\displaystyle \aleph _{\omega }}
1603:{\displaystyle \aleph _{\omega }}
1576:{\displaystyle \aleph _{\omega }}
1549:{\displaystyle \omega _{\omega }}
1401:{\displaystyle \omega _{\omega }}
1381:, and so on. Its initial ordinal
1266:{\displaystyle \aleph _{\omega }}
686:{\displaystyle \omega _{\omega }}
662:. A regular ordinal is always an
527:of sets of cardinality less than
4338:
4268:
1946:is singular (a result proved by
1583:. Assuming the axiom of choice,
910:is the next limit ordinal after
593:is a regular ordinal (see below)
1663:{\displaystyle \omega +\omega }
1119:{\displaystyle \omega +\omega }
1093:{\displaystyle \omega +\omega }
1067:{\displaystyle \omega +\omega }
903:{\displaystyle \omega +\omega }
297:{\displaystyle |I|\geq \kappa }
4314:
2470:
2461:
2405:
2370:
2364:
2358:
2251:
2248:
2242:
2232:
2185:
2179:
2173:
2085:
2079:
484:{\displaystyle |S|<\kappa }
471:
463:
415:
400:
385:{\displaystyle |I|<\kappa }
372:
364:
284:
276:
1:
4229:History of mathematical logic
2398:
2342:, there is a small embedding
1823:
1408:is the limit of the sequence
1186:, so the cardinals less than
4154:Primitive recursive function
2019:that are critical points of
1839:cardinality of the continuum
1126:; therefore it is singular.
7:
2456:L'Enseignement Mathématique
2386:
2039:{\displaystyle \Sigma _{1}}
1939:{\displaystyle \aleph _{0}}
1908:{\displaystyle \omega _{1}}
1869:{\displaystyle \aleph _{1}}
1820:and is therefore singular.
1502:{\displaystyle \omega _{3}}
1475:{\displaystyle \omega _{2}}
1448:{\displaystyle \omega _{1}}
1374:{\displaystyle \aleph _{3}}
1347:{\displaystyle \aleph _{2}}
1320:{\displaystyle \aleph _{1}}
1293:{\displaystyle \aleph _{0}}
1237:{\displaystyle \aleph _{1}}
1206:{\displaystyle \aleph _{1}}
1179:{\displaystyle \aleph _{0}}
1148:{\displaystyle \aleph _{1}}
801:{\displaystyle \aleph _{0}}
696:
10:
4976:
4805:von NeumannâBernaysâGödel
3218:SchröderâBernstein theorem
2945:Monadic predicate calculus
2604:Foundations of mathematics
1993:is regular iff the set of
4869:
4832:
4744:
4634:
4606:One-to-one correspondence
4522:
4463:
4347:
4336:
4322:
4264:
4251:Philosophy of mathematics
4200:Automated theorem proving
4182:
4077:
3909:
3802:
3654:
3371:
3347:
3325:Von NeumannâBernaysâGödel
3270:
3164:
3068:
2966:
2957:
2884:
2819:
2725:
2647:
2564:
2418:Journal of Symbolic Logic
1709:to postulate this axiom.
1640:, and the first infinite
1633:{\displaystyle \omega +1}
1021:{\displaystyle \omega +3}
995:{\displaystyle \omega +2}
969:{\displaystyle \omega +1}
853:{\displaystyle \omega +1}
693:(see the example below).
551:of cardinality less than
31:that is equal to its own
2476:Holy, LĂŒcke, Njegomir, "
3901:Self-verifying theories
3722:Tarski's axiomatization
2673:Tarski's undefinability
2668:incompleteness theorems
2283:{\displaystyle \kappa }
2121:{\displaystyle \kappa }
1986:{\displaystyle \kappa }
1966:{\displaystyle \kappa }
1741:{\displaystyle \omega }
1556:is singular, and so is
1522:{\displaystyle \omega }
1421:{\displaystyle \omega }
1041:{\displaystyle \omega }
943:{\displaystyle \omega }
923:{\displaystyle \omega }
877:{\displaystyle \omega }
825:{\displaystyle \omega }
774:{\displaystyle \omega }
754:{\displaystyle \omega }
734:{\displaystyle \omega }
714:{\displaystyle \omega }
701:The ordinals less than
655:{\displaystyle \alpha }
623:{\displaystyle \alpha }
586:{\displaystyle \kappa }
564:{\displaystyle \kappa }
540:{\displaystyle \kappa }
156:{\displaystyle \kappa }
133:{\displaystyle \kappa }
116:In the presence of the
98:{\displaystyle \kappa }
48:{\displaystyle \kappa }
4564:Constructible universe
4384:Constructibility (V=L)
4275:Mathematics portal
3886:Proof of impossibility
3534:propositional variable
2844:Propositional calculus
2495:Elements of Set Theory
2377:
2336:
2310:
2284:
2264:
2216:
2192:
2151:
2122:
2098:
2063:
2040:
2013:
1987:
1967:
1940:
1909:
1870:
1814:
1742:
1718:inaccessible cardinals
1695:
1664:
1634:
1604:
1577:
1550:
1523:
1503:
1476:
1449:
1422:
1402:
1375:
1348:
1321:
1294:
1267:
1238:
1207:
1180:
1149:
1120:
1094:
1068:
1042:
1022:
996:
970:
944:
924:
904:
878:
854:
826:
802:
775:
755:
735:
715:
687:
656:
624:
587:
565:
541:
521:
485:
449:
429:
386:
350:
298:
262:
242:
209:
163:is a regular cardinal.
157:
134:
99:
79:
49:
4787:Principia Mathematica
4621:Transfinite induction
4480:(i.e. set difference)
4144:Kolmogorov complexity
4097:Computably enumerable
3997:Model complete theory
3789:Principia Mathematica
2849:Propositional formula
2678:BanachâTarski paradox
2393:Inaccessible cardinal
2378:
2337:
2311:
2285:
2265:
2217:
2193:
2152:
2123:
2099:
2064:
2048:elementary embeddings
2041:
2014:
1988:
1968:
1941:
1910:
1871:
1815:
1743:
1696:
1665:
1635:
1605:
1578:
1551:
1524:
1504:
1477:
1450:
1423:
1403:
1376:
1349:
1322:
1295:
1268:
1239:
1208:
1181:
1150:
1121:
1095:
1069:
1043:
1023:
997:
971:
945:
925:
905:
879:
855:
827:
803:
776:
756:
736:
716:
688:
657:
625:
588:
566:
542:
522:
486:
450:
430:
387:
351:
299:
263:
243:
210:
158:
135:
100:
80:
50:
4861:Burali-Forti paradox
4616:Set-builder notation
4569:Continuum hypothesis
4509:Symmetric difference
4092:ChurchâTuring thesis
4079:Computability theory
3288:continuum hypothesis
2806:Square of opposition
2664:Gödel's completeness
2346:
2320:
2316:such that for every
2294:
2274:
2226:
2206:
2161:
2135:
2112:
2073:
2053:
2023:
1997:
1977:
1973:is a limit ordinal,
1957:
1923:
1892:
1853:
1847:continuum hypothesis
1752:
1732:
1678:
1672:axiom of replacement
1648:
1644:that is singular is
1618:
1614:that is singular is
1587:
1560:
1533:
1513:
1486:
1459:
1432:
1412:
1385:
1358:
1331:
1304:
1277:
1250:
1221:
1190:
1163:
1157:next cardinal number
1132:
1104:
1078:
1052:
1032:
1006:
980:
954:
934:
914:
888:
868:
838:
816:
785:
765:
745:
725:
705:
670:
646:
614:
577:
555:
531:
501:
459:
439:
396:
360:
311:
272:
252:
219:
170:
147:
124:
89:
63:
39:
4822:TarskiâGrothendieck
4246:Mathematical object
4137:P versus NP problem
4102:Computable function
3896:Reverse mathematics
3822:Logical consequence
3699:primitive recursive
3694:elementary function
3467:Free/bound variable
3320:TarskiâGrothendieck
2839:Logical connectives
2769:Logical equivalence
2619:Logical consequence
2490:Herbert B. Enderton
1712:Uncountable (weak)
862:next ordinal number
4411:Limitation of size
4044:Transfer principle
4007:Semantics of logic
3992:Categorical theory
3968:Non-standard model
3482:Logical connective
2609:Information theory
2558:Mathematical logic
2373:
2332:
2306:
2280:
2260:
2222:is transitive and
2212:
2188:
2147:
2118:
2094:
2059:
2036:
2009:
1983:
1963:
1936:
1905:
1866:
1834:successor cardinal
1832:holds, then every
1810:
1738:
1703:Zermelo set theory
1691:
1660:
1630:
1600:
1573:
1546:
1519:
1499:
1472:
1445:
1418:
1398:
1371:
1344:
1317:
1290:
1263:
1234:
1203:
1176:
1145:
1116:
1090:
1064:
1038:
1018:
992:
966:
940:
920:
900:
874:
850:
822:
798:
771:
751:
731:
711:
683:
652:
620:
583:
561:
537:
517:
481:
445:
425:
382:
346:
335:
294:
258:
238:
205:
194:
153:
130:
111:singular cardinals
95:
75:
45:
4942:
4941:
4851:Russell's paradox
4800:ZermeloâFraenkel
4701:Dedekind-infinite
4574:Diagonal argument
4473:Cartesian product
4330:Set (mathematics)
4282:
4281:
4214:Abstract category
4017:Theories of truth
3827:Rule of inference
3817:Natural deduction
3798:
3797:
3343:
3342:
3048:Cartesian product
2953:
2952:
2859:Many-valued logic
2834:Boolean functions
2717:Russell's paradox
2692:diagonal argument
2589:First-order logic
2239:
2215:{\displaystyle M}
2062:{\displaystyle j}
448:{\displaystyle i}
320:
261:{\displaystyle i}
179:
4967:
4955:Cardinal numbers
4924:Bertrand Russell
4914:John von Neumann
4899:Abraham Fraenkel
4894:Richard Dedekind
4856:Suslin's problem
4767:Cantor's theorem
4484:De Morgan's laws
4342:
4309:
4302:
4295:
4286:
4285:
4273:
4272:
4224:History of logic
4219:Category of sets
4112:Decision problem
3891:Ordinal analysis
3832:Sequent calculus
3730:Boolean algebras
3670:
3669:
3644:
3615:logical/constant
3369:
3368:
3355:
3278:ZermeloâFraenkel
3029:Set operations:
2964:
2963:
2901:
2732:
2731:
2712:LöwenheimâSkolem
2599:Formal semantics
2551:
2544:
2537:
2528:
2527:
2511:
2492:
2481:
2474:
2468:
2465:
2459:
2453:
2409:
2382:
2380:
2379:
2374:
2341:
2339:
2338:
2333:
2315:
2313:
2312:
2307:
2289:
2287:
2286:
2281:
2269:
2267:
2266:
2261:
2241:
2240:
2237:
2221:
2219:
2218:
2213:
2197:
2195:
2194:
2189:
2156:
2154:
2153:
2148:
2127:
2125:
2124:
2119:
2103:
2101:
2100:
2095:
2068:
2066:
2065:
2060:
2045:
2043:
2042:
2037:
2035:
2034:
2018:
2016:
2015:
2010:
1992:
1990:
1989:
1984:
1972:
1970:
1969:
1964:
1945:
1943:
1942:
1937:
1935:
1934:
1914:
1912:
1911:
1906:
1904:
1903:
1875:
1873:
1872:
1867:
1865:
1864:
1843:Easton's theorem
1819:
1817:
1816:
1811:
1797:
1796:
1795:
1794:
1777:
1776:
1764:
1763:
1747:
1745:
1744:
1739:
1700:
1698:
1697:
1692:
1690:
1689:
1669:
1667:
1666:
1661:
1639:
1637:
1636:
1631:
1609:
1607:
1606:
1601:
1599:
1598:
1582:
1580:
1579:
1574:
1572:
1571:
1555:
1553:
1552:
1547:
1545:
1544:
1528:
1526:
1525:
1520:
1508:
1506:
1505:
1500:
1498:
1497:
1481:
1479:
1478:
1473:
1471:
1470:
1454:
1452:
1451:
1446:
1444:
1443:
1427:
1425:
1424:
1419:
1407:
1405:
1404:
1399:
1397:
1396:
1380:
1378:
1377:
1372:
1370:
1369:
1353:
1351:
1350:
1345:
1343:
1342:
1326:
1324:
1323:
1318:
1316:
1315:
1299:
1297:
1296:
1291:
1289:
1288:
1272:
1270:
1269:
1264:
1262:
1261:
1243:
1241:
1240:
1235:
1233:
1232:
1212:
1210:
1209:
1204:
1202:
1201:
1185:
1183:
1182:
1177:
1175:
1174:
1154:
1152:
1151:
1146:
1144:
1143:
1125:
1123:
1122:
1117:
1099:
1097:
1096:
1091:
1073:
1071:
1070:
1065:
1047:
1045:
1044:
1039:
1027:
1025:
1024:
1019:
1001:
999:
998:
993:
975:
973:
972:
967:
949:
947:
946:
941:
929:
927:
926:
921:
909:
907:
906:
901:
883:
881:
880:
875:
859:
857:
856:
851:
831:
829:
828:
823:
807:
805:
804:
799:
797:
796:
780:
778:
777:
772:
760:
758:
757:
752:
740:
738:
737:
732:
720:
718:
717:
712:
692:
690:
689:
684:
682:
681:
661:
659:
658:
653:
629:
627:
626:
621:
592:
590:
589:
584:
570:
568:
567:
562:
546:
544:
543:
538:
526:
524:
523:
518:
516:
515:
490:
488:
487:
482:
474:
466:
454:
452:
451:
446:
434:
432:
431:
426:
418:
413:
412:
403:
391:
389:
388:
383:
375:
367:
355:
353:
352:
347:
345:
344:
334:
303:
301:
300:
295:
287:
279:
267:
265:
264:
259:
247:
245:
244:
239:
231:
230:
214:
212:
211:
206:
204:
203:
193:
162:
160:
159:
154:
139:
137:
136:
131:
104:
102:
101:
96:
85:has cardinality
84:
82:
81:
76:
54:
52:
51:
46:
25:regular cardinal
4975:
4974:
4970:
4969:
4968:
4966:
4965:
4964:
4960:Ordinal numbers
4945:
4944:
4943:
4938:
4865:
4844:
4828:
4793:New Foundations
4740:
4630:
4549:Cardinal number
4532:
4518:
4459:
4343:
4334:
4318:
4313:
4283:
4278:
4267:
4260:
4205:Category theory
4195:Algebraic logic
4178:
4149:Lambda calculus
4087:Church encoding
4073:
4049:Truth predicate
3905:
3871:Complete theory
3794:
3663:
3659:
3655:
3650:
3642:
3362: and
3358:
3353:
3339:
3315:New Foundations
3283:axiom of choice
3266:
3228:Gödel numbering
3168: and
3160:
3064:
2949:
2899:
2880:
2829:Boolean algebra
2815:
2779:Equiconsistency
2744:Classical logic
2721:
2702:Halting problem
2690: and
2666: and
2654: and
2653:
2648:Theorems (
2643:
2560:
2555:
2507:
2488:
2485:
2484:
2475:
2471:
2466:
2462:
2431:10.2307/2274520
2413:Maddy, Penelope
2410:
2406:
2401:
2389:
2347:
2344:
2343:
2321:
2318:
2317:
2295:
2292:
2291:
2275:
2272:
2271:
2236:
2235:
2227:
2224:
2223:
2207:
2204:
2203:
2200:small embedding
2162:
2159:
2158:
2136:
2133:
2132:
2113:
2110:
2109:
2074:
2071:
2070:
2054:
2051:
2050:
2030:
2026:
2024:
2021:
2020:
1998:
1995:
1994:
1978:
1975:
1974:
1958:
1955:
1954:
1930:
1926:
1924:
1921:
1920:
1899:
1895:
1893:
1890:
1889:
1860:
1856:
1854:
1851:
1850:
1830:axiom of choice
1826:
1790:
1786:
1785:
1781:
1772:
1768:
1759:
1755:
1753:
1750:
1749:
1733:
1730:
1729:
1714:limit cardinals
1685:
1681:
1679:
1676:
1675:
1649:
1646:
1645:
1619:
1616:
1615:
1594:
1590:
1588:
1585:
1584:
1567:
1563:
1561:
1558:
1557:
1540:
1536:
1534:
1531:
1530:
1514:
1511:
1510:
1493:
1489:
1487:
1484:
1483:
1466:
1462:
1460:
1457:
1456:
1439:
1435:
1433:
1430:
1429:
1413:
1410:
1409:
1392:
1388:
1386:
1383:
1382:
1365:
1361:
1359:
1356:
1355:
1338:
1334:
1332:
1329:
1328:
1311:
1307:
1305:
1302:
1301:
1284:
1280:
1278:
1275:
1274:
1257:
1253:
1251:
1248:
1247:
1228:
1224:
1222:
1219:
1218:
1197:
1193:
1191:
1188:
1187:
1170:
1166:
1164:
1161:
1160:
1139:
1135:
1133:
1130:
1129:
1105:
1102:
1101:
1079:
1076:
1075:
1053:
1050:
1049:
1033:
1030:
1029:
1007:
1004:
1003:
981:
978:
977:
955:
952:
951:
935:
932:
931:
915:
912:
911:
889:
886:
885:
869:
866:
865:
839:
836:
835:
817:
814:
813:
792:
788:
786:
783:
782:
766:
763:
762:
746:
743:
742:
726:
723:
722:
706:
703:
702:
699:
677:
673:
671:
668:
667:
664:initial ordinal
647:
644:
643:
632:regular ordinal
615:
612:
611:
602:axiom of choice
578:
575:
574:
556:
553:
552:
532:
529:
528:
508:
504:
502:
499:
498:
470:
462:
460:
457:
456:
440:
437:
436:
414:
408:
404:
399:
397:
394:
393:
371:
363:
361:
358:
357:
340:
336:
324:
312:
309:
308:
283:
275:
273:
270:
269:
253:
250:
249:
226:
222:
220:
217:
216:
199:
195:
183:
171:
168:
167:
148:
145:
144:
125:
122:
121:
118:axiom of choice
90:
87:
86:
64:
61:
60:
40:
37:
36:
29:cardinal number
17:
12:
11:
5:
4973:
4963:
4962:
4957:
4940:
4939:
4937:
4936:
4931:
4929:Thoralf Skolem
4926:
4921:
4916:
4911:
4906:
4901:
4896:
4891:
4886:
4881:
4875:
4873:
4867:
4866:
4864:
4863:
4858:
4853:
4847:
4845:
4843:
4842:
4839:
4833:
4830:
4829:
4827:
4826:
4825:
4824:
4819:
4814:
4813:
4812:
4797:
4796:
4795:
4783:
4782:
4781:
4770:
4769:
4764:
4759:
4754:
4748:
4746:
4742:
4741:
4739:
4738:
4733:
4728:
4723:
4714:
4709:
4704:
4694:
4689:
4688:
4687:
4682:
4677:
4667:
4657:
4652:
4647:
4641:
4639:
4632:
4631:
4629:
4628:
4623:
4618:
4613:
4611:Ordinal number
4608:
4603:
4598:
4593:
4592:
4591:
4586:
4576:
4571:
4566:
4561:
4556:
4546:
4541:
4535:
4533:
4531:
4530:
4527:
4523:
4520:
4519:
4517:
4516:
4511:
4506:
4501:
4496:
4491:
4489:Disjoint union
4486:
4481:
4475:
4469:
4467:
4461:
4460:
4458:
4457:
4456:
4455:
4450:
4439:
4438:
4436:Martin's axiom
4433:
4428:
4423:
4418:
4413:
4408:
4403:
4401:Extensionality
4398:
4397:
4396:
4386:
4381:
4380:
4379:
4374:
4369:
4359:
4353:
4351:
4345:
4344:
4337:
4335:
4333:
4332:
4326:
4324:
4320:
4319:
4312:
4311:
4304:
4297:
4289:
4280:
4279:
4265:
4262:
4261:
4259:
4258:
4253:
4248:
4243:
4238:
4237:
4236:
4226:
4221:
4216:
4207:
4202:
4197:
4192:
4190:Abstract logic
4186:
4184:
4180:
4179:
4177:
4176:
4171:
4169:Turing machine
4166:
4161:
4156:
4151:
4146:
4141:
4140:
4139:
4134:
4129:
4124:
4119:
4109:
4107:Computable set
4104:
4099:
4094:
4089:
4083:
4081:
4075:
4074:
4072:
4071:
4066:
4061:
4056:
4051:
4046:
4041:
4036:
4035:
4034:
4029:
4024:
4014:
4009:
4004:
4002:Satisfiability
3999:
3994:
3989:
3988:
3987:
3977:
3976:
3975:
3965:
3964:
3963:
3958:
3953:
3948:
3943:
3933:
3932:
3931:
3926:
3919:Interpretation
3915:
3913:
3907:
3906:
3904:
3903:
3898:
3893:
3888:
3883:
3873:
3868:
3867:
3866:
3865:
3864:
3854:
3849:
3839:
3834:
3829:
3824:
3819:
3814:
3808:
3806:
3800:
3799:
3796:
3795:
3793:
3792:
3784:
3783:
3782:
3781:
3776:
3775:
3774:
3769:
3764:
3744:
3743:
3742:
3740:minimal axioms
3737:
3726:
3725:
3724:
3713:
3712:
3711:
3706:
3701:
3696:
3691:
3686:
3673:
3671:
3652:
3651:
3649:
3648:
3647:
3646:
3634:
3629:
3628:
3627:
3622:
3617:
3612:
3602:
3597:
3592:
3587:
3586:
3585:
3580:
3570:
3569:
3568:
3563:
3558:
3553:
3543:
3538:
3537:
3536:
3531:
3526:
3516:
3515:
3514:
3509:
3504:
3499:
3494:
3489:
3479:
3474:
3469:
3464:
3463:
3462:
3457:
3452:
3447:
3437:
3432:
3430:Formation rule
3427:
3422:
3421:
3420:
3415:
3405:
3404:
3403:
3393:
3388:
3383:
3378:
3372:
3366:
3349:Formal systems
3345:
3344:
3341:
3340:
3338:
3337:
3332:
3327:
3322:
3317:
3312:
3307:
3302:
3297:
3292:
3291:
3290:
3285:
3274:
3272:
3268:
3267:
3265:
3264:
3263:
3262:
3252:
3247:
3246:
3245:
3238:Large cardinal
3235:
3230:
3225:
3220:
3215:
3201:
3200:
3199:
3194:
3189:
3174:
3172:
3162:
3161:
3159:
3158:
3157:
3156:
3151:
3146:
3136:
3131:
3126:
3121:
3116:
3111:
3106:
3101:
3096:
3091:
3086:
3081:
3075:
3073:
3066:
3065:
3063:
3062:
3061:
3060:
3055:
3050:
3045:
3040:
3035:
3027:
3026:
3025:
3020:
3010:
3005:
3003:Extensionality
3000:
2998:Ordinal number
2995:
2985:
2980:
2979:
2978:
2967:
2961:
2955:
2954:
2951:
2950:
2948:
2947:
2942:
2937:
2932:
2927:
2922:
2917:
2916:
2915:
2905:
2904:
2903:
2890:
2888:
2882:
2881:
2879:
2878:
2877:
2876:
2871:
2866:
2856:
2851:
2846:
2841:
2836:
2831:
2825:
2823:
2817:
2816:
2814:
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2782:
2781:
2771:
2766:
2761:
2756:
2751:
2746:
2740:
2738:
2729:
2723:
2722:
2720:
2719:
2714:
2709:
2704:
2699:
2694:
2682:Cantor's
2680:
2675:
2670:
2660:
2658:
2645:
2644:
2642:
2641:
2636:
2631:
2626:
2621:
2616:
2611:
2606:
2601:
2596:
2591:
2586:
2581:
2580:
2579:
2568:
2566:
2562:
2561:
2554:
2553:
2546:
2539:
2531:
2525:
2524:
2505:
2483:
2482:
2469:
2460:
2425:(2): 481â511,
2403:
2402:
2400:
2397:
2396:
2395:
2388:
2385:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2331:
2328:
2325:
2305:
2302:
2299:
2279:
2259:
2256:
2253:
2250:
2247:
2244:
2234:
2231:
2211:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2146:
2143:
2140:
2131:For cardinals
2117:
2093:
2090:
2087:
2084:
2081:
2078:
2058:
2033:
2029:
2008:
2005:
2002:
1982:
1962:
1933:
1929:
1902:
1898:
1863:
1859:
1825:
1822:
1809:
1806:
1803:
1800:
1793:
1789:
1784:
1780:
1775:
1771:
1767:
1762:
1758:
1737:
1726:aleph function
1688:
1684:
1659:
1656:
1653:
1629:
1626:
1623:
1597:
1593:
1570:
1566:
1543:
1539:
1518:
1496:
1492:
1469:
1465:
1442:
1438:
1417:
1395:
1391:
1368:
1364:
1341:
1337:
1314:
1310:
1287:
1283:
1260:
1256:
1231:
1227:
1200:
1196:
1173:
1169:
1142:
1138:
1115:
1112:
1109:
1089:
1086:
1083:
1063:
1060:
1057:
1037:
1017:
1014:
1011:
991:
988:
985:
965:
962:
959:
939:
919:
899:
896:
893:
873:
849:
846:
843:
821:
795:
791:
770:
750:
730:
710:
698:
695:
680:
676:
651:
619:
595:
594:
582:
572:
560:
536:
514:
511:
507:
492:
480:
477:
473:
469:
465:
444:
424:
421:
417:
411:
407:
402:
381:
378:
374:
370:
366:
343:
339:
333:
330:
327:
323:
319:
316:
305:
293:
290:
286:
282:
278:
257:
237:
234:
229:
225:
202:
198:
192:
189:
186:
182:
178:
175:
164:
152:
129:
94:
74:
71:
68:
44:
15:
9:
6:
4:
3:
2:
4972:
4961:
4958:
4956:
4953:
4952:
4950:
4935:
4934:Ernst Zermelo
4932:
4930:
4927:
4925:
4922:
4920:
4919:Willard Quine
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4876:
4874:
4872:
4871:Set theorists
4868:
4862:
4859:
4857:
4854:
4852:
4849:
4848:
4846:
4840:
4838:
4835:
4834:
4831:
4823:
4820:
4818:
4817:KripkeâPlatek
4815:
4811:
4808:
4807:
4806:
4803:
4802:
4801:
4798:
4794:
4791:
4790:
4789:
4788:
4784:
4780:
4777:
4776:
4775:
4772:
4771:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4749:
4747:
4743:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4720:
4715:
4713:
4710:
4708:
4705:
4702:
4698:
4695:
4693:
4690:
4686:
4683:
4681:
4678:
4676:
4673:
4672:
4671:
4668:
4665:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4642:
4640:
4637:
4633:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4590:
4587:
4585:
4582:
4581:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4554:
4550:
4547:
4545:
4542:
4540:
4537:
4536:
4534:
4528:
4525:
4524:
4521:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4479:
4476:
4474:
4471:
4470:
4468:
4466:
4462:
4454:
4453:specification
4451:
4449:
4446:
4445:
4444:
4441:
4440:
4437:
4434:
4432:
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4395:
4392:
4391:
4390:
4387:
4385:
4382:
4378:
4375:
4373:
4370:
4368:
4365:
4364:
4363:
4360:
4358:
4355:
4354:
4352:
4350:
4346:
4341:
4331:
4328:
4327:
4325:
4321:
4317:
4310:
4305:
4303:
4298:
4296:
4291:
4290:
4287:
4277:
4276:
4271:
4263:
4257:
4254:
4252:
4249:
4247:
4244:
4242:
4239:
4235:
4232:
4231:
4230:
4227:
4225:
4222:
4220:
4217:
4215:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4187:
4185:
4181:
4175:
4172:
4170:
4167:
4165:
4164:Recursive set
4162:
4160:
4157:
4155:
4152:
4150:
4147:
4145:
4142:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4114:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4084:
4082:
4080:
4076:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4033:
4030:
4028:
4025:
4023:
4020:
4019:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3986:
3983:
3982:
3981:
3978:
3974:
3973:of arithmetic
3971:
3970:
3969:
3966:
3962:
3959:
3957:
3954:
3952:
3949:
3947:
3944:
3942:
3939:
3938:
3937:
3934:
3930:
3927:
3925:
3922:
3921:
3920:
3917:
3916:
3914:
3912:
3908:
3902:
3899:
3897:
3894:
3892:
3889:
3887:
3884:
3881:
3880:from ZFC
3877:
3874:
3872:
3869:
3863:
3860:
3859:
3858:
3855:
3853:
3850:
3848:
3845:
3844:
3843:
3840:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3809:
3807:
3805:
3801:
3791:
3790:
3786:
3785:
3780:
3779:non-Euclidean
3777:
3773:
3770:
3768:
3765:
3763:
3762:
3758:
3757:
3755:
3752:
3751:
3749:
3745:
3741:
3738:
3736:
3733:
3732:
3731:
3727:
3723:
3720:
3719:
3718:
3714:
3710:
3707:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3685:
3682:
3681:
3679:
3675:
3674:
3672:
3667:
3661:
3656:Example
3653:
3645:
3640:
3639:
3638:
3635:
3633:
3630:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3607:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3584:
3581:
3579:
3576:
3575:
3574:
3571:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3548:
3547:
3544:
3542:
3539:
3535:
3532:
3530:
3527:
3525:
3522:
3521:
3520:
3517:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3484:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3443:
3442:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3423:
3419:
3416:
3414:
3413:by definition
3411:
3410:
3409:
3406:
3402:
3399:
3398:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3373:
3370:
3367:
3365:
3361:
3356:
3350:
3346:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3300:KripkeâPlatek
3298:
3296:
3293:
3289:
3286:
3284:
3281:
3280:
3279:
3276:
3275:
3273:
3269:
3261:
3258:
3257:
3256:
3253:
3251:
3248:
3244:
3241:
3240:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3213:
3209:
3205:
3202:
3198:
3195:
3193:
3190:
3188:
3185:
3184:
3183:
3179:
3176:
3175:
3173:
3171:
3167:
3163:
3155:
3152:
3150:
3147:
3145:
3144:constructible
3142:
3141:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3076:
3074:
3072:
3067:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3030:
3028:
3024:
3021:
3019:
3016:
3015:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2990:
2986:
2984:
2981:
2977:
2974:
2973:
2972:
2969:
2968:
2965:
2962:
2960:
2956:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2914:
2911:
2910:
2909:
2906:
2902:
2897:
2896:
2895:
2892:
2891:
2889:
2887:
2883:
2875:
2872:
2870:
2867:
2865:
2862:
2861:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2826:
2824:
2822:
2821:Propositional
2818:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2780:
2777:
2776:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2749:Logical truth
2747:
2745:
2742:
2741:
2739:
2737:
2733:
2730:
2728:
2724:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2689:
2685:
2681:
2679:
2676:
2674:
2671:
2669:
2665:
2662:
2661:
2659:
2657:
2651:
2646:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2578:
2575:
2574:
2573:
2570:
2569:
2567:
2563:
2559:
2552:
2547:
2545:
2540:
2538:
2533:
2532:
2529:
2523:
2522:0-444-85401-0
2519:
2515:
2510:
2509:Kenneth Kunen
2506:
2504:
2503:0-12-238440-7
2500:
2496:
2491:
2487:
2486:
2479:
2473:
2464:
2457:
2452:
2448:
2444:
2440:
2436:
2432:
2428:
2424:
2420:
2419:
2414:
2408:
2404:
2394:
2391:
2390:
2384:
2367:
2361:
2355:
2352:
2349:
2329:
2326:
2323:
2303:
2300:
2297:
2277:
2270:. A cardinal
2257:
2254:
2245:
2229:
2209:
2201:
2182:
2176:
2170:
2167:
2164:
2144:
2141:
2138:
2129:
2115:
2107:
2091:
2088:
2082:
2076:
2056:
2049:
2031:
2006:
2003:
2000:
1980:
1960:
1951:
1949:
1931:
1918:
1900:
1896:
1887:
1882:
1881:aleph numbers
1877:
1861:
1848:
1844:
1840:
1835:
1831:
1821:
1807:
1804:
1801:
1798:
1791:
1787:
1778:
1773:
1765:
1760:
1735:
1727:
1723:
1719:
1715:
1710:
1708:
1704:
1686:
1673:
1657:
1654:
1651:
1643:
1642:limit ordinal
1627:
1624:
1621:
1613:
1595:
1568:
1541:
1537:
1516:
1494:
1490:
1467:
1463:
1440:
1436:
1415:
1393:
1389:
1366:
1339:
1312:
1285:
1258:
1245:
1229:
1216:
1198:
1171:
1159:greater than
1158:
1140:
1127:
1113:
1110:
1107:
1087:
1084:
1081:
1061:
1058:
1055:
1035:
1015:
1012:
1009:
989:
986:
983:
963:
960:
957:
937:
917:
897:
894:
891:
871:
864:greater than
863:
847:
844:
841:
833:
819:
811:
793:
768:
748:
728:
708:
694:
678:
674:
665:
649:
641:
637:
636:limit ordinal
633:
617:
610:
605:
603:
598:
580:
573:
558:
550:
534:
512:
509:
505:
497:
493:
478:
475:
467:
442:
422:
419:
409:
405:
379:
376:
368:
341:
337:
331:
328:
325:
321:
317:
314:
306:
291:
288:
280:
255:
235:
232:
227:
223:
200:
196:
190:
187:
184:
180:
176:
173:
165:
150:
143:
142:
141:
127:
119:
114:
112:
108:
92:
72:
69:
66:
58:
42:
34:
30:
26:
22:
4884:Georg Cantor
4879:Paul Bernays
4810:MorseâKelley
4785:
4718:
4717:Subset
4664:hereditarily
4626:Venn diagram
4584:ordered pair
4499:Intersection
4443:Axiom schema
4266:
4064:Ultraproduct
3911:Model theory
3876:Independence
3812:Formal proof
3804:Proof theory
3787:
3760:
3717:real numbers
3689:second-order
3600:Substitution
3477:Metalanguage
3418:conservative
3391:Axiom schema
3335:Constructive
3305:MorseâKelley
3271:Set theories
3250:Aleph number
3243:inaccessible
3149:Grothendieck
3033:intersection
2920:Higher-order
2908:Second-order
2854:Truth tables
2811:Venn diagram
2594:Formal proof
2513:
2494:
2472:
2463:
2455:
2450:
2422:
2416:
2407:
2199:
2130:
1952:
1917:real numbers
1878:
1827:
1722:fixed points
1711:
1705:is what led
1641:
1611:
1246:
1128:
834:
700:
631:
607:An infinite
606:
599:
596:
115:
110:
107:well-ordered
105:. Infinite
24:
18:
4909:Thomas Jech
4752:Alternative
4731:Uncountable
4685:Ultrafilter
4544:Cardinality
4448:replacement
4389:Determinacy
4174:Type theory
4122:undecidable
4054:Truth value
3941:equivalence
3620:non-logical
3233:Enumeration
3223:Isomorphism
3170:cardinality
3154:Von Neumann
3119:Ultrafilter
3084:Uncountable
3018:equivalence
2935:Quantifiers
2925:Fixed-point
2894:First-order
2774:Consistency
2759:Proposition
2736:Traditional
2707:Lindström's
2697:Compactness
2639:Type theory
2584:Cardinality
634:if it is a
4949:Categories
4904:Kurt Gödel
4889:Paul Cohen
4726:Transitive
4494:Identities
4478:Complement
4465:Operations
4426:Regularity
4394:projective
4357:Adjunction
4316:Set theory
3985:elementary
3678:arithmetic
3546:Quantifier
3524:functional
3396:Expression
3114:Transitive
3058:identities
3043:complement
2976:hereditary
2959:Set theory
2399:References
1948:Moti Gitik
1824:Properties
1748:-sequence
810:aleph-null
642:less than
640:order type
33:cofinality
21:set theory
4837:Paradoxes
4757:Axiomatic
4736:Universal
4712:Singleton
4707:Recursive
4650:Countable
4645:Amorphous
4504:Power set
4421:Power set
4372:dependent
4367:countable
4256:Supertask
4159:Recursion
4117:decidable
3951:saturated
3929:of models
3852:deductive
3847:axiomatic
3767:Hilbert's
3754:Euclidean
3735:canonical
3658:axiomatic
3590:Signature
3519:Predicate
3408:Extension
3330:Ackermann
3255:Operation
3134:Universal
3124:Recursive
3099:Singleton
3094:Inhabited
3079:Countable
3069:Types of
3053:power set
3023:partition
2940:Predicate
2886:Predicate
2801:Syllogism
2791:Soundness
2764:Inference
2754:Tautology
2656:paradoxes
2368:θ
2359:→
2330:α
2324:θ
2304:κ
2298:α
2278:κ
2258:κ
2183:θ
2174:→
2145:θ
2139:κ
2116:κ
2092:κ
2083:α
2028:Σ
2007:κ
2001:α
1981:κ
1961:κ
1928:ℵ
1897:ω
1858:ℵ
1792:ω
1788:ω
1783:ℵ
1774:ω
1770:ℵ
1757:ℵ
1736:ω
1687:ω
1683:ℵ
1658:ω
1652:ω
1622:ω
1596:ω
1592:ℵ
1569:ω
1565:ℵ
1542:ω
1538:ω
1517:ω
1491:ω
1464:ω
1437:ω
1416:ω
1394:ω
1390:ω
1363:ℵ
1336:ℵ
1309:ℵ
1282:ℵ
1259:ω
1255:ℵ
1226:ℵ
1215:countable
1195:ℵ
1168:ℵ
1137:ℵ
1114:ω
1108:ω
1088:ω
1082:ω
1062:ω
1056:ω
1036:ω
1010:ω
984:ω
958:ω
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918:ω
898:ω
892:ω
872:ω
842:ω
820:ω
790:ℵ
769:ω
749:ω
729:ω
709:ω
679:ω
675:ω
650:α
618:α
581:κ
559:κ
535:κ
513:κ
479:κ
423:κ
380:κ
356:, and if
329:∈
322:⋃
292:κ
289:≥
236:κ
224:λ
197:λ
188:∈
181:∑
174:κ
151:κ
128:κ
93:κ
73:κ
70:⊆
57:unbounded
43:κ
4841:Problems
4745:Theories
4721:Superset
4697:Infinite
4526:Concepts
4406:Infinity
4323:Overview
4241:Logicism
4234:timeline
4210:Concrete
4069:Validity
4039:T-schema
4032:Kripke's
4027:Tarski's
4022:semantic
4012:Strength
3961:submodel
3956:spectrum
3924:function
3772:Tarski's
3761:Elements
3748:geometry
3704:Robinson
3625:variable
3610:function
3583:spectrum
3573:Sentence
3529:variable
3472:Language
3425:Relation
3386:Automata
3376:Alphabet
3360:language
3214:-jection
3192:codomain
3178:Function
3139:Universe
3109:Infinite
3013:Relation
2796:Validity
2786:Argument
2684:theorem,
2387:See also
1707:Fraenkel
697:Examples
549:colimits
496:category
435:for all
248:for all
4779:General
4774:Zermelo
4680:subbase
4662: (
4601:Forcing
4579:Element
4551: (
4529:Methods
4416:Pairing
4183:Related
3980:Diagram
3878: (
3857:Hilbert
3842:Systems
3837:Theorem
3715:of the
3660:systems
3440:Formula
3435:Grammar
3351: (
3295:General
3008:Forcing
2993:Element
2913:Monadic
2688:paradox
2629:Theorem
2565:General
2458:(1917).
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1845:). The
1828:If the
1724:of the
1612:ordinal
1155:is the
860:is the
609:ordinal
455:, then
268:, then
59:subset
4670:Filter
4660:Finite
4596:Family
4539:Almost
4377:global
4362:Choice
4349:Axioms
3946:finite
3709:Skolem
3662:
3637:Theory
3605:Symbol
3595:String
3578:atomic
3455:ground
3450:closed
3445:atomic
3401:ground
3364:syntax
3260:binary
3187:domain
3104:Finite
2869:finite
2727:Logics
2686:
2634:Theory
2520:
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2437:
27:is a
4762:Naive
4692:Fuzzy
4655:Empty
4638:types
4589:tuple
4559:Class
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4431:Union
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3684:Peano
3541:Proof
3381:Arity
3310:Naive
3197:image
3129:Fuzzy
3089:Empty
3038:union
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2624:Model
2614:Lemma
2572:Axiom
2435:JSTOR
2069:with
1888:that
1529:, so
1048:, so
630:is a
4675:base
4059:Type
3862:list
3666:list
3643:list
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3566:rank
3460:open
3354:list
3166:Maps
3071:sets
2930:Free
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