6758:
4410:
321:, the existence of a bijection between two sets becomes more difficult to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed; it does not take proper account of the fact that all three sets are
4320:
2969:
2551:
970:, and it was later supported by the independence of CH from the axioms of ZFC since these axioms are enough to establish the elementary properties of sets and cardinalities. In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another. Although the
1660:(the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen's model in which CH fails is a model in which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed by Cohen to prove
977:
At least two other axioms have been proposed that have implications for the continuum hypothesis, although these axioms have not currently found wide acceptance in the mathematical community. In 1986, Chris
Freiling presented an argument against CH by showing that the negation of CH is equivalent to
251:
proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set
2798:
891:
The independence from ZFC means that proving or disproving the CH within ZFC is impossible. However, Gödel and Cohen's negative results are not universally accepted as disposing of all interest in the continuum hypothesis. The continuum hypothesis remains an active topic of research; see
2374:
2787:
1149:
approach to set theory and argues that "the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and, as a result, it can no longer be settled in the manner formerly hoped for". In a related vein,
663:, which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds, and constructs another model which contains more sets than the original, in a way that CH does not hold in the new model. Cohen was awarded the
1154:
wrote that he does "not agree with the pure
Platonic view that the interesting problems in set theory can be decided, that we just have to discover the additional axiom. My mental picture is that we have many possible set theories, all conforming to ZFC".
2631:
907:
were among the first genuinely mathematical statements shown to be independent of ZF set theory. Although the existence of some statements independent of ZFC had already been known more than two decades prior: for example, assuming
2964:{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\leq \aleph _{\alpha }^{\aleph _{\alpha }}\leq (2^{\aleph _{\alpha }})^{\aleph _{\alpha }}=2^{\aleph _{\alpha }\cdot \aleph _{\alpha }}=2^{\aleph _{\alpha }}=\aleph _{\alpha +1}}
209:
of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by
2546:{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\leq \aleph _{\beta +1}^{\aleph _{\beta }}=(2^{\aleph _{\beta }})^{\aleph _{\beta }}=2^{\aleph _{\beta }\cdot \aleph _{\beta }}=2^{\aleph _{\beta }}=\aleph _{\beta +1}}
352:). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question.
2695:
2354:
2226:
2294:
2104:
2166:
502:
1648:
2703:
3637:
1527:, AD), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than some
1742:
1385:
2035:
1329:
1460:
421:
847:
611:
148:
1790:
1936:
1988:
1853:
746:
1133:
192:
1689:
1569:
1246:
1205:
807:
355:
The continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set,
1998:
Although the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation
1033:
780:
1886:
For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsets of B. Thus for any infinite cardinals A and B,
1060:
564:
537:
1506:
325:. It turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (
1270:
2558:
1873:
1810:
1480:
1408:
1290:
867:
702:
959:
can actually be used to argue in favor of CH, because among models that have the same reals, models with "more" sets of reals have a better chance of satisfying CH.
1098:
1883:
th successor of κ. On the other hand, László Patai proved that if γ is an ordinal and for each infinite cardinal κ, 2 is the γth successor of κ, then γ is finite.
4062:
962:
Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. This viewpoint was advanced as early as 1923 by
648:
of ZF set theory, assuming only the axioms of ZF. The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are
3061:
659:
Cohen showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof. To prove his result, Cohen developed the method of
5137:
2790:
1072:
argued that CH is not a definite mathematical problem. He proposed a theory of "definiteness" using a semi-intuitionistic subsystem of ZF that accepts
3565:
3034:
4333:
1135:. He conjectured that CH is not definite according to this notion, and proposed that CH should, therefore, be considered not to have a truth value.
17:
6794:
2648:
2307:
5812:
4874:
4305:
3754:
2179:
947:
of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. Parallel arguments were made for and against the
6962:
3728:
684:. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if
2241:
1062:, thus falsifying CH. The Star axiom was bolstered by an independent May 2021 proof showing the Star axiom can be derived from a variation of
2051:
2119:
435:
5895:
5036:
1066:. However, Woodin stated in the 2010s that he now instead believes CH to be true, based on his belief in his new "ultimate L" conjecture.
920:
scheme) expressing the consistency of ZFC, that is also independent of it. The latter independence result indeed holds for many theories.
670:
The independence proof just described shows that CH is independent of ZFC. Further research has shown that CH is independent of all known
235:
Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it. It became the first on David
Hilbert's
2782:{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\geq \aleph _{\alpha }^{\operatorname {cf} (\aleph _{\alpha })}>\aleph _{\alpha }}
3829:
4324:
1578:
936:
and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though a
240:
6209:
3854:
1694:
1337:
345:
3449:
3413:
6367:
4284:
Proceedings of
Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp. 81–92.
4242:
3574:
974:
does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false.
5155:
913:
653:
6222:
5545:
4563:
4376:
2001:
6787:
1295:
1425:
6930:
6919:
6227:
6217:
5954:
5807:
5160:
4891:
4031:
5151:
681:
359:, of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into
6924:
6914:
6363:
4289:
3536:. Positions more or less like this may be found in Haskell Curry , Abraham Robinson , and Paul Cohen .
3115:
206:
87:"Any subset of the real numbers is either finite, or countably infinite, or has the cardinality of the real numbers."
5705:
3029:
374:
6894:
6460:
6204:
5029:
1990:
holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals.
997:
does not reject Woodin's argument outright but urges caution. Woodin proposed a new hypothesis that he labeled the
812:
35:
6904:
6899:
6879:
6874:
5765:
5458:
4869:
4463:
3481:
979:
929:
622:
569:
106:
92:
6967:
5199:
4749:
1755:
6952:
6909:
6889:
6884:
6780:
6721:
6423:
6186:
6181:
6006:
5427:
5111:
1889:
933:
4213:
1941:
1815:
6957:
6864:
6716:
6499:
6416:
6129:
6060:
5937:
5179:
4643:
4522:
711:
349:
1103:
157:
6869:
6844:
6641:
6467:
6153:
5787:
5386:
4886:
677:
3779:
Rittberg, Colin J. (March 2015). "How Woodin changed his mind: new thoughts on the
Continuum Hypothesis".
6849:
6829:
6819:
6519:
6514:
6124:
5863:
5792:
5121:
5022:
4879:
4517:
4480:
3000:
6859:
6854:
6839:
6834:
6824:
6814:
6448:
6038:
5432:
5400:
5091:
4230:
1667:
1534:
1218:
1177:
966:, even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as
928:
Gödel believed that CH is false, and that his proof that CH is consistent with ZFC only shows that the
785:
630:
310:
and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}.
253:
211:
4534:
652:
with ZF, provided ZF itself is consistent. The latter condition cannot be proved in ZF itself, due to
6982:
6738:
6687:
6584:
6082:
6043:
5520:
5165:
4568:
4453:
4441:
4436:
4081:
3721:
3683:
Hamkins, Joel David (January 2015). "Is the Dream
Solution of the Continuum Hypothesis Attainable?".
1657:
1004:
971:
948:
916:, which were published in 1931, establish that there is a formal statement (one for each appropriate
751:
5194:
6579:
6509:
6048:
5900:
5883:
5606:
5086:
4369:
3363:
6411:
6388:
6349:
6235:
6176:
5822:
5742:
5586:
5530:
5143:
4988:
4906:
4781:
4733:
4547:
4470:
1146:
1038:
937:
542:
515:
3107:
2626:{\displaystyle \aleph _{\beta +1}=2^{\aleph _{\beta }}\leq \aleph _{\alpha }^{\aleph _{\beta }}}
6701:
6428:
6406:
6373:
6266:
6112:
6097:
6070:
6021:
5905:
5840:
5665:
5631:
5626:
5500:
5331:
5308:
4940:
4821:
4633:
4446:
3358:
1516:
1485:
956:
944:
641:
223:
30:
This article is about the hypothesis in set theory. For the assumption in fluid mechanics, see
3995:
Moore, Gregory H. (2011). "Early history of the generalized continuum hypothesis: 1878–1938".
1255:
6803:
6631:
6484:
6276:
5994:
5730:
5636:
5495:
5480:
5361:
5336:
4856:
4826:
4770:
4690:
4670:
4648:
3818:
3055:
1858:
1795:
1465:
1393:
1275:
873:
852:
687:
660:
236:
202:
6604:
6566:
6443:
6247:
6087:
6011:
5989:
5817:
5674:
5641:
5505:
5293:
5204:
4930:
4920:
4754:
4685:
4578:
4458:
4112:
4016:
3669:
3527:
3295:
3236:
3147:
1524:
1083:
1077:
672:
244:
31:
8:
6977:
6733:
6624:
6609:
6589:
6546:
6433:
6383:
6309:
6254:
6191:
5984:
5979:
5927:
5695:
5684:
5356:
5256:
5184:
5175:
5171:
5106:
5101:
4925:
4836:
4744:
4739:
4553:
4495:
4426:
4362:
4340:
3005:
1661:
1063:
967:
4277:, 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH.
3299:
3240:
3151:
1167:(GCH) states that if an infinite set's cardinality lies between that of an infinite set
6762:
6531:
6494:
6479:
6472:
6455:
6259:
6241:
6107:
6033:
6016:
5969:
5782:
5691:
5525:
5510:
5470:
5422:
5407:
5395:
5351:
5326:
5096:
5045:
4848:
4843:
4628:
4583:
4490:
4208:
4184:
4116:
4090:
4054:
3946:
3928:
3901:
3883:
3871:
3796:
3692:
3657:
3618:
3610:
3515:
3376:
3313:
3259:
3224:
3165:
3100:
1142:
5715:
3593:
Freiling, Chris (1986). "Axioms of
Symmetry: Throwing darts at the real number line".
3326:
3283:
3178:
3135:
917:
6972:
6757:
6697:
6504:
6314:
6304:
6196:
6077:
5912:
5888:
5669:
5653:
5558:
5535:
5412:
5381:
5346:
5241:
5076:
4705:
4542:
4505:
4475:
4399:
4337:
4285:
4238:
4032:"Two classical surprises concerning the Axiom of Choice and the Continuum Hypothesis"
3847:
3800:
3570:
3331:
3264:
3209:
3183:
3111:
2980:
1752:
proved that (assuming the consistency of very large cardinals) it is consistent that
4120:
3965:
3905:
3622:
3434:
3398:
3225:"The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"
6711:
6706:
6599:
6556:
6378:
6339:
6334:
6319:
6145:
6102:
5999:
5797:
5747:
5321:
5283:
4993:
4983:
4968:
4963:
4831:
4485:
4176:
4100:
4046:
4004:
3977:
3950:
3938:
3893:
3814:
3788:
3702:
3649:
3602:
3507:
3368:
3321:
3303:
3254:
3244:
3173:
3155:
3043:
1069:
3942:
909:
640:(AC) is adopted (making ZFC). Gödel's proof shows that CH and AC both hold in the
6692:
6682:
6636:
6619:
6574:
6536:
6438:
6358:
6165:
6092:
6065:
6053:
5959:
5873:
5847:
5802:
5770:
5571:
5373:
5316:
5266:
5231:
5189:
4862:
4800:
4618:
4431:
4217:
4108:
4079:
Merimovich, Carmi (2007). "A power function with a fixed finite gap everywhere".
4012:
3665:
3523:
1745:
1520:
1509:
1073:
994:
986:. Freiling believes this axiom is "intuitively clear" but others have disagreed.
952:
904:
637:
509:
318:
294:
to have the same cardinality means that it is possible to "pair off" elements of
278:
265:
96:
656:, but is widely believed to be true and can be proved in stronger set theories.
6677:
6656:
6614:
6594:
6489:
6344:
5942:
5932:
5922:
5917:
5851:
5725:
5601:
5490:
5485:
5463:
5064:
4998:
4795:
4776:
4680:
4665:
4622:
4558:
4500:
3466:
3288:
Proceedings of the
National Academy of Sciences of the United States of America
3140:
Proceedings of the
National Academy of Sciences of the United States of America
1749:
1572:
1388:
1151:
1136:
990:
897:
893:
881:
6772:
3981:
3897:
3792:
6946:
6651:
6329:
5836:
5621:
5611:
5581:
5566:
5236:
5003:
4805:
4719:
4714:
4104:
4008:
3706:
3560:
3047:
330:
4973:
4203:
This article incorporates material from
Generalized continuum hypothesis on
3080:
1653:
626:
248:
215:
6551:
6398:
6299:
6291:
6171:
6119:
6028:
5964:
5947:
5878:
5737:
5596:
5298:
5081:
4953:
4948:
4766:
4695:
4653:
4512:
4409:
4280:
Martin, D. (1976). "Hilbert's first problem: the continuum hypothesis," in
3335:
3268:
3249:
3187:
3160:
3025:
2690:{\displaystyle \aleph _{\beta }\geq \operatorname {cf} (\aleph _{\alpha })}
2349:{\displaystyle \aleph _{\beta }\geq \operatorname {cf} (\aleph _{\alpha })}
1528:
1249:
664:
364:
322:
198:
100:
60:
3308:
2221:{\displaystyle \aleph _{\beta }<\operatorname {cf} (\aleph _{\alpha })}
1531:, and thus can be ordered. This is done by showing that n is smaller than
6661:
6541:
5720:
5710:
5657:
5341:
5261:
5246:
5126:
5071:
4978:
4613:
3638:"Throwing a dart at Freiling's argument against the continuum hypothesis"
2990:
2985:
1419:
1100:
is mathematically "definite" if the semi-intuitionistic theory can prove
983:
932:
axioms do not adequately characterize the universe of sets. Gödel was a
645:
341:
272:
151:
76:
68:
44:
3661:
3349:
Feferman, Solomon (February 1999). "Does mathematics need new axioms?".
2289:{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\alpha +1}}
1664:, which shows it is consistent with ZFC for arbitrarily large cardinals
888:
in those fields have subsequently been shown to be independent as well.
5591:
5446:
5417:
5223:
4958:
4729:
4385:
4306:"How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer"
4204:
4188:
4058:
3755:"How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer"
3614:
3519:
3380:
2233:
2099:{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\beta +1}}
885:
705:
649:
48:
3470:
3284:"The independence of the Continuum Hypothesis, [part] II"
2995:
2161:{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\alpha }}
497:{\displaystyle \nexists S\colon \aleph _{0}<|S|<2^{\aleph _{0}}}
286:(a one-to-one correspondence) between them. Intuitively, for two sets
6743:
6646:
5699:
5616:
5576:
5540:
5476:
5288:
5278:
5251:
5014:
4761:
4724:
4675:
4573:
4345:
4095:
3933:
3653:
3317:
3169:
3136:"The independence of the Continuum Hypothesis, [part I]"
1172:
923:
566:, and the continuum hypothesis is in turn equivalent to the equality
283:
4180:
4157:. Columbus, Ohio: Charles E. Merrill. p. 147, exercise 76.
4050:
3606:
3511:
3372:
1879: ≥ 1, it is consistent with ZFC that for each κ, 2 is the
6728:
6526:
5974:
5679:
5273:
3498:
Goodman, Nicolas D. (1979). "Mathematics as an objective science".
877:
368:
3888:
3697:
1993:
982:, a statement derived by arguing from particular intuitions about
872:
The continuum hypothesis is closely related to many statements in
6324:
5116:
4319:
337:
314:
201:
in 1878, and establishing its truth or falsehood is the first of
72:
1643:{\displaystyle 2^{\aleph _{0}+n}\,=\,2\cdot \,2^{\aleph _{0}+n}}
4786:
4608:
1482:. The continuum hypothesis is the special case for the ordinal
963:
748:. However, per König's theorem, it is not consistent to assume
3819:"Is the Continuum Hypothesis a definite mathematical problem?"
943:
Historically, mathematicians who favored a "rich" and "large"
636:
Gödel showed that CH cannot be disproved from ZF, even if the
5868:
5214:
5059:
4658:
4418:
4354:
4331:
4296:
4136:
Mathematische und naturwissenschaftliche Berichte aus Ungarn
3102:
Georg Cantor: His mathematics and philosophy of the infinite
676:
in the context of ZFC. Moreover, it has been shown that the
1812:. Later Woodin extended this by showing the consistency of
4282:
Mathematical Developments Arising from Hilbert's Problems,
993:
has attracted considerable attention since the year 2000.
336:
Cantor gave two proofs that the cardinality of the set of
3966:"On transfinite cardinal numbers of the exponential form"
1737:{\displaystyle 2^{\aleph _{\alpha }}=\aleph _{\alpha +1}}
1380:{\displaystyle \aleph _{\alpha +1}=2^{\aleph _{\alpha }}}
4167:
Maddy, Penelope (June 1988). "Believing the axioms, ".
621:
The independence of the continuum hypothesis (CH) from
423:, the continuum hypothesis can be restated as follows:
99:(ZFC), this is equivalent to the following equation in
4134:
Patai, L. (1930). "Untersuchungen über die א-reihe".
2801:
2706:
2651:
2561:
2377:
2310:
2244:
2182:
2122:
2054:
2004:
1944:
1892:
1861:
1818:
1798:
1758:
1697:
1670:
1581:
1537:
1488:
1468:
1428:
1396:
1340:
1298:
1278:
1258:
1221:
1180:
1106:
1086:
1080:
for unbounded ones, and suggested that a proposition
1041:
1007:
855:
815:
788:
754:
714:
690:
572:
545:
518:
438:
377:
160:
109:
3541:
2030:{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}}
1422:
provide an alternative notation for this condition:
884:. As a result of its independence, many substantial
1324:{\displaystyle \lambda <\kappa <2^{\lambda }}
1158:
1139:wrote a critical commentary on Feferman's article.
1001:, or "Star axiom". The Star axiom would imply that
4303:
4273:, reprinted in Benacerraf and Putnam's collection
4146:
3566:Set Theory: An Introduction to Independence Proofs
3099:
2963:
2781:
2689:
2625:
2545:
2348:
2288:
2220:
2160:
2098:
2029:
1982:
1930:
1867:
1847:
1804:
1784:
1736:
1683:
1642:
1563:
1500:
1474:
1455:{\displaystyle \aleph _{\alpha }=\beth _{\alpha }}
1454:
1402:
1379:
1323:
1284:
1264:
1240:
1199:
1127:
1092:
1054:
1027:
924:Arguments for and against the continuum hypothesis
861:
841:
801:
774:
740:
696:
605:
558:
531:
496:
415:
186:
142:
3342:
3060:: CS1 maint: DOI inactive as of September 2024 (
1938:. If A and B are finite, the stronger inequality
205:presented in 1900. The answer to this problem is
6944:
4209:Creative Commons Attribution/Share-Alike License
900:for an overview of the current research status.
6802:
3588:
3586:
3229:Proceedings of the National Academy of Sciences
3035:Journal für die Reine und Angewandte Mathematik
1994:Implications of GCH for cardinal exponentiation
259:
221:The name of the hypothesis comes from the term
4237:. Mineola, New York City: Dover Publications.
4175:(2). Association for Symbolic Logic: 481–511.
4152:
4072:
3601:(1). Association for Symbolic Logic: 190–200.
3428:
3426:
3091:
416:{\displaystyle |\mathbb {R} |=2^{\aleph _{0}}}
59:) is a hypothesis about the possible sizes of
6788:
5030:
4370:
3921:Bulletin of the American Mathematical Society
3839:
3635:
3569:. Amsterdam, NL: North-Holland. p. 171.
3553:
3459:
3392:
3390:
3199:
3197:
1515:Like CH, GCH is also independent of ZFC, but
1211:, then it has the same cardinality as either
989:A difficult argument against CH developed by
842:{\displaystyle \aleph _{\omega _{1}+\omega }}
708:, then there is a forcing extension in which
512:, there is a unique smallest cardinal number
4254:An Introduction to Independence for Analysts
3957:
3807:
3722:"Has the Continuum Hypothesis been settled?"
3583:
2037:in all cases. GCH implies that for ordinals
340:is strictly smaller than that of the set of
4251:
4153:Hayden, Seymour; Kennison, John F. (1968).
4023:
3864:
3746:
3713:
3676:
3491:
3423:
3129:
3127:
3085:The Consistency of the Continuum-Hypothesis
3075:
3073:
3071:
1512:. For the early history of GCH, see Moore.
606:{\displaystyle 2^{\aleph _{0}}=\aleph _{1}}
143:{\displaystyle 2^{\aleph _{0}}=\aleph _{1}}
6795:
6781:
5222:
5037:
5023:
4377:
4363:
4294:
4078:
3919:Shelah, Saharon (2003). "Logical dreams".
3912:
3387:
3194:
1785:{\displaystyle 2^{\kappa }>\kappa ^{+}}
306:is paired off with exactly one element of
4127:
4094:
3988:
3932:
3887:
3752:
3696:
3362:
3325:
3307:
3258:
3248:
3177:
3159:
3018:
1931:{\displaystyle A<B\to 2^{A}\leq 2^{B}}
1875:. Carmi Merimovich showed that, for each
1656:showed that GCH is a consequence of ZF +
1616:
1609:
1605:
384:
197:The continuum hypothesis was advanced by
34:. For the album by Epoch of Unlight, see
4260:
3963:
3874:(2012). "The set-theoretic multiverse".
3845:
3813:
3778:
3629:
3592:
3465:
3435:"The Continuum Hypothesis, Part II"
3348:
3275:
3124:
3068:
3030:"Ein Beitrag zur Mannigfaltigkeitslehre"
1983:{\displaystyle A<B\to 2^{A}<2^{B}}
1848:{\displaystyle 2^{\kappa }=\kappa ^{++}}
1523:(AC) (and therefore the negation of the
616:
329:) as the set of integers: they are both
302:in such a fashion that every element of
241:International Congress of Mathematicians
4235:Set theory and the continuum hypothesis
4029:
3870:
3826:Exploring the Frontiers of Independence
3719:
3682:
3497:
3478:Exploring the Frontiers of Independence
3399:"The Continuum Hypothesis, Part I"
3203:
3106:. Princeton University Press. pp.
741:{\displaystyle 2^{\aleph _{0}}=\kappa }
247:was at that point not yet formulated.
14:
6945:
5044:
3918:
3848:"Feferman on the indefiniteness of CH"
3432:
3396:
3097:
3024:
1128:{\displaystyle (\phi \lor \neg \phi )}
313:With infinite sets such as the set of
187:{\displaystyle \beth _{1}=\aleph _{1}}
6963:Basic concepts in infinite set theory
6776:
5018:
4358:
4332:
4229:
4166:
4133:
3994:
3781:Archive for History of Exact Sciences
3559:
3547:
3281:
3222:
3133:
3079:
940:, also tended towards rejecting CH.
680:can be any cardinal consistent with
4304:Wolchover, Natalie (15 July 2021).
4271:What is Cantor's Continuum Problem?
3753:Wolchover, Natalie (15 July 2021).
3134:Cohen, Paul J. (15 December 1963).
1650:; for the full proof, see Gillman.
951:, which implies CH. More recently,
625:(ZF) follows from combined work of
346:Cantor's first uncountability proof
270:Two sets are said to have the same
24:
4252:Dales, H.G.; Woodin, W.H. (1987).
4223:
3685:Notre Dame Journal of Formal Logic
3282:Cohen, Paul J. (15 January 1964).
2946:
2931:
2911:
2898:
2878:
2861:
2838:
2828:
2813:
2803:
2770:
2752:
2733:
2718:
2708:
2675:
2653:
2612:
2602:
2587:
2563:
2528:
2513:
2493:
2480:
2460:
2443:
2420:
2404:
2389:
2379:
2334:
2312:
2271:
2256:
2246:
2206:
2184:
2149:
2134:
2124:
2081:
2066:
2056:
2016:
2006:
1792:holds for every infinite cardinal
1719:
1704:
1672:
1623:
1588:
1544:
1430:
1366:
1342:
1224:
1183:
1116:
1043:
1014:
817:
790:
761:
721:
594:
579:
547:
520:
483:
449:
402:
175:
131:
116:
25:
6994:
4313:
3500:The American Mathematical Monthly
1684:{\displaystyle \aleph _{\alpha }}
1564:{\displaystyle 2^{\aleph _{0}+n}}
1519:proved that ZF + GCH implies the
1241:{\displaystyle {\mathcal {P}}(S)}
1200:{\displaystyle {\mathcal {P}}(S)}
1076:for bounded quantifiers but uses
903:The continuum hypothesis and the
802:{\displaystyle \aleph _{\omega }}
27:Proposition in mathematical logic
6756:
4408:
4318:
4068:from the original on 2022-10-10.
3860:from the original on 2012-03-19.
3835:from the original on 2022-10-10.
3487:from the original on 2012-01-24.
3455:from the original on 2022-10-10.
3419:from the original on 2022-10-10.
1165:generalized continuum hypothesis
1159:Generalized continuum hypothesis
849:or any cardinal with cofinality
237:list of important open questions
214:, complementing earlier work by
71:is strictly between that of the
36:The Continuum Hypothesis (album)
18:Generalized Continuum Hypothesis
3772:
3734:from the original on 2022-10-10
1028:{\displaystyle 2^{\aleph _{0}}}
914:Gödel's incompleteness theorems
775:{\displaystyle 2^{\aleph _{0}}}
654:Gödel's incompleteness theorems
4384:
4207:, which is licensed under the
3964:Jourdain, Philip E.B. (1905).
3216:
3098:Dauben, Joseph Warren (1990).
2873:
2852:
2761:
2748:
2684:
2671:
2455:
2434:
2343:
2330:
2215:
2202:
1954:
1902:
1571:which is smaller than its own
1235:
1229:
1194:
1188:
1122:
1107:
470:
462:
389:
379:
13:
1:
6717:History of mathematical logic
4039:American Mathematical Monthly
3943:10.1090/s0273-0979-03-00981-9
3351:American Mathematical Monthly
3087:. Princeton University Press.
3011:
1508:. GCH was first suggested by
704:is a cardinal of uncountable
6642:Primitive recursive function
3876:The Review of Symbolic Logic
980:Freiling's axiom of symmetry
678:cardinality of the continuum
260:Cardinality of infinite sets
7:
4155:Zermelo-Fraenkel Set Theory
3001:Second continuum hypothesis
2974:
1055:{\displaystyle \aleph _{2}}
623:Zermelo–Fraenkel set theory
559:{\displaystyle \aleph _{0}}
532:{\displaystyle \aleph _{1}}
243:in the year 1900 in Paris.
93:Zermelo–Fraenkel set theory
10:
6999:
5706:Schröder–Bernstein theorem
5433:Monadic predicate calculus
5092:Foundations of mathematics
4875:von Neumann–Bernays–Gödel
4297:"The Continuum Hypothesis"
4261:Enderton, Herbert (1977).
4196:
3997:Bulletin of Symbolic Logic
3828:. Harvard lecture series.
3636:Bagemihl, F. (1989–1990).
3532:This view is often called
3480:. Harvard lecture series.
3471:"The Continuum Hypothesis"
363:. As the real numbers are
350:Cantor's diagonal argument
263:
239:that was presented at the
230:
203:Hilbert's 23 problems
29:
6810:
6752:
6739:Philosophy of mathematics
6688:Automated theorem proving
6670:
6565:
6397:
6290:
6142:
5859:
5835:
5813:Von Neumann–Bernays–Gödel
5758:
5652:
5556:
5454:
5445:
5372:
5307:
5213:
5135:
5052:
4939:
4902:
4814:
4704:
4676:One-to-one correspondence
4592:
4533:
4417:
4406:
4392:
4275:Philosophy of Mathematics
4169:Journal of Symbolic Logic
4082:Journal of Symbolic Logic
4030:Gillman, Leonard (2002).
3982:10.1080/14786440509463254
3898:10.1017/S1755020311000359
3793:10.1007/s00407-014-0142-8
3595:Journal of Symbolic Logic
2637:The third equality (when
2360:The first equality (when
1501:{\displaystyle \alpha =1}
1412:Cantor's aleph hypothesis
972:axiom of constructibility
949:axiom of constructibility
912:and the consistency ZFC,
910:good soundness properties
3846:Koellner, Peter (2011).
3707:10.1215/00294527-2835047
3433:Woodin, W. Hugh (2001).
3397:Woodin, W. Hugh (2001).
3048:10.1515/crll.1878.84.242
1575:—this uses the equality
1331:. GCH is equivalent to:
1265:{\displaystyle \lambda }
252:– was proved in 1963 by
6389:Self-verifying theories
6210:Tarski's axiomatization
5161:Tarski's undefinability
5156:incompleteness theorems
3204:Goldrei, Derek (1996).
1868:{\displaystyle \kappa }
1805:{\displaystyle \kappa }
1475:{\displaystyle \alpha }
1403:{\displaystyle \alpha }
1285:{\displaystyle \kappa }
862:{\displaystyle \omega }
697:{\displaystyle \kappa }
667:in 1966 for his proof.
150:, or even shorter with
67:"There is no set whose
6763:Mathematics portal
6374:Proof of impossibility
6022:propositional variable
5332:Propositional calculus
4634:Constructible universe
4454:Constructibility (V=L)
4341:"Continuum Hypothesis"
4323:Quotations related to
4263:Elements of Set Theory
4105:10.2178/jsl/1185803615
4009:10.2178/bsl/1318855631
3970:Philosophical Magazine
3720:Foreman, Matt (2003).
3642:Real Analysis Exchange
3250:10.1073/pnas.24.12.556
3161:10.1073/pnas.50.6.1143
3050:(inactive 2024-09-11).
2965:
2783:
2691:
2627:
2547:
2350:
2290:
2222:
2162:
2100:
2031:
1984:
1932:
1869:
1849:
1806:
1786:
1738:
1685:
1644:
1565:
1502:
1476:
1456:
1404:
1381:
1325:
1286:
1266:
1242:
1201:
1129:
1094:
1056:
1029:
957:ontological maximalism
863:
843:
803:
776:
742:
698:
642:constructible universe
607:
560:
533:
498:
417:
371:of the integers, i.e.
227:for the real numbers.
188:
144:
89:
81:
6953:Forcing (mathematics)
6632:Kolmogorov complexity
6585:Computably enumerable
6485:Model complete theory
6277:Principia Mathematica
5337:Propositional formula
5166:Banach–Tarski paradox
4857:Principia Mathematica
4691:Transfinite induction
4550:(i.e. set difference)
3309:10.1073/pnas.51.1.105
2966:
2784:
2692:
2628:
2548:
2351:
2291:
2223:
2163:
2101:
2032:
1985:
1933:
1870:
1850:
1807:
1787:
1739:
1686:
1645:
1566:
1503:
1477:
1457:
1410:(occasionally called
1405:
1382:
1326:
1287:
1272:there is no cardinal
1267:
1243:
1202:
1130:
1095:
1093:{\displaystyle \phi }
1057:
1030:
955:has pointed out that
864:
844:
804:
777:
743:
699:
673:large cardinal axioms
617:Independence from ZFC
608:
561:
534:
499:
418:
189:
145:
85:
65:
6958:Independence results
6580:Church–Turing thesis
6567:Computability theory
5776:continuum hypothesis
5294:Square of opposition
5152:Gödel's completeness
4931:Burali-Forti paradox
4686:Set-builder notation
4639:Continuum hypothesis
4579:Symmetric difference
4325:Continuum hypothesis
3223:Gödel, Kurt (1938).
2799:
2704:
2649:
2559:
2375:
2308:
2242:
2180:
2120:
2052:
2002:
1942:
1890:
1859:
1816:
1796:
1756:
1695:
1668:
1579:
1535:
1525:axiom of determinacy
1486:
1466:
1426:
1394:
1338:
1296:
1276:
1256:
1219:
1178:
1104:
1084:
1078:intuitionistic logic
1039:
1005:
853:
813:
786:
752:
712:
688:
570:
543:
516:
436:
428:Continuum hypothesis
375:
245:Axiomatic set theory
158:
107:
53:continuum hypothesis
32:Continuum assumption
6734:Mathematical object
6625:P versus NP problem
6590:Computable function
6384:Reverse mathematics
6310:Logical consequence
6187:primitive recursive
6182:elementary function
5955:Free/bound variable
5808:Tarski–Grothendieck
5327:Logical connectives
5257:Logical equivalence
5107:Logical consequence
4892:Tarski–Grothendieck
3872:Hamkins, Joel David
3300:1964PNAS...51..105C
3241:1938PNAS...24..556G
3152:1963PNAS...50.1143C
2848:
2823:
2765:
2728:
2622:
2430:
2399:
2266:
2144:
2076:
2026:
1691:to fail to satisfy
1248:. That is, for any
431: —
6968:Hilbert's problems
6804:Hilbert's problems
6532:Transfer principle
6495:Semantics of logic
6480:Categorical theory
6456:Non-standard model
5970:Logical connective
5097:Information theory
5046:Mathematical logic
4481:Limitation of size
4338:Weisstein, Eric W.
4231:Cohen, Paul Joseph
4216:2017-02-08 at the
3442:Notices of the AMS
3406:Notices of the AMS
3210:Chapman & Hall
3206:Classic Set Theory
2961:
2827:
2802:
2779:
2732:
2707:
2687:
2623:
2601:
2543:
2403:
2378:
2368:+1) follows from:
2346:
2286:
2245:
2218:
2158:
2123:
2096:
2055:
2027:
2005:
1980:
1928:
1865:
1845:
1802:
1782:
1734:
1681:
1640:
1561:
1498:
1472:
1462:for every ordinal
1452:
1400:
1377:
1321:
1282:
1262:
1238:
1197:
1143:Joel David Hamkins
1125:
1090:
1052:
1025:
859:
839:
799:
772:
738:
694:
603:
556:
529:
494:
429:
413:
282:if there exists a
184:
140:
6940:
6939:
6770:
6769:
6702:Abstract category
6505:Theories of truth
6315:Rule of inference
6305:Natural deduction
6286:
6285:
5831:
5830:
5536:Cartesian product
5441:
5440:
5347:Many-valued logic
5322:Boolean functions
5205:Russell's paradox
5180:diagonal argument
5077:First-order logic
5012:
5011:
4921:Russell's paradox
4870:Zermelo–Fraenkel
4771:Dedekind-infinite
4644:Diagonal argument
4543:Cartesian product
4400:Set (mathematics)
4265:. Academic Press.
4244:978-0-486-46921-8
3972:. Series 6.
3815:Feferman, Solomon
3576:978-0-444-85401-8
2981:Absolute infinite
427:
298:with elements of
83:Or equivalently:
16:(Redirected from
6990:
6983:Cardinal numbers
6797:
6790:
6783:
6774:
6773:
6761:
6760:
6712:History of logic
6707:Category of sets
6600:Decision problem
6379:Ordinal analysis
6320:Sequent calculus
6218:Boolean algebras
6158:
6157:
6132:
6103:logical/constant
5857:
5856:
5843:
5766:Zermelo–Fraenkel
5517:Set operations:
5452:
5451:
5389:
5220:
5219:
5200:Löwenheim–Skolem
5087:Formal semantics
5039:
5032:
5025:
5016:
5015:
4994:Bertrand Russell
4984:John von Neumann
4969:Abraham Fraenkel
4964:Richard Dedekind
4926:Suslin's problem
4837:Cantor's theorem
4554:De Morgan's laws
4412:
4379:
4372:
4365:
4356:
4355:
4351:
4350:
4334:Szudzik, Matthew
4322:
4309:
4300:
4295:McGough, Nancy.
4266:
4257:
4248:
4192:
4159:
4158:
4150:
4144:
4143:
4131:
4125:
4124:
4098:
4076:
4070:
4069:
4067:
4036:
4027:
4021:
4020:
3992:
3986:
3985:
3961:
3955:
3954:
3936:
3916:
3910:
3909:
3891:
3868:
3862:
3861:
3859:
3852:
3843:
3837:
3836:
3834:
3823:
3811:
3805:
3804:
3776:
3770:
3769:
3767:
3765:
3750:
3744:
3743:
3741:
3739:
3733:
3726:
3717:
3711:
3710:
3700:
3680:
3674:
3673:
3654:10.2307/44152014
3633:
3627:
3626:
3590:
3581:
3580:
3557:
3551:
3545:
3539:
3538:
3495:
3489:
3488:
3486:
3475:
3463:
3457:
3456:
3454:
3439:
3430:
3421:
3420:
3418:
3403:
3394:
3385:
3384:
3366:
3346:
3340:
3339:
3329:
3311:
3279:
3273:
3272:
3262:
3252:
3220:
3214:
3213:
3201:
3192:
3191:
3181:
3163:
3146:(6): 1143–1148.
3131:
3122:
3121:
3105:
3095:
3089:
3088:
3077:
3066:
3065:
3059:
3051:
3022:
3006:Wetzel's problem
2970:
2968:
2967:
2962:
2960:
2959:
2941:
2940:
2939:
2938:
2921:
2920:
2919:
2918:
2906:
2905:
2888:
2887:
2886:
2885:
2871:
2870:
2869:
2868:
2847:
2846:
2845:
2835:
2822:
2821:
2820:
2810:
2788:
2786:
2785:
2780:
2778:
2777:
2764:
2760:
2759:
2740:
2727:
2726:
2725:
2715:
2697:) follows from:
2696:
2694:
2693:
2688:
2683:
2682:
2661:
2660:
2632:
2630:
2629:
2624:
2621:
2620:
2619:
2609:
2597:
2596:
2595:
2594:
2577:
2576:
2552:
2550:
2549:
2544:
2542:
2541:
2523:
2522:
2521:
2520:
2503:
2502:
2501:
2500:
2488:
2487:
2470:
2469:
2468:
2467:
2453:
2452:
2451:
2450:
2429:
2428:
2427:
2417:
2398:
2397:
2396:
2386:
2355:
2353:
2352:
2347:
2342:
2341:
2320:
2319:
2295:
2293:
2292:
2287:
2285:
2284:
2265:
2264:
2263:
2253:
2227:
2225:
2224:
2219:
2214:
2213:
2192:
2191:
2167:
2165:
2164:
2159:
2157:
2156:
2143:
2142:
2141:
2131:
2105:
2103:
2102:
2097:
2095:
2094:
2075:
2074:
2073:
2063:
2036:
2034:
2033:
2028:
2025:
2024:
2023:
2013:
1989:
1987:
1986:
1981:
1979:
1978:
1966:
1965:
1937:
1935:
1934:
1929:
1927:
1926:
1914:
1913:
1874:
1872:
1871:
1866:
1854:
1852:
1851:
1846:
1844:
1843:
1828:
1827:
1811:
1809:
1808:
1803:
1791:
1789:
1788:
1783:
1781:
1780:
1768:
1767:
1743:
1741:
1740:
1735:
1733:
1732:
1714:
1713:
1712:
1711:
1690:
1688:
1687:
1682:
1680:
1679:
1662:Easton's theorem
1649:
1647:
1646:
1641:
1639:
1638:
1631:
1630:
1604:
1603:
1596:
1595:
1570:
1568:
1567:
1562:
1560:
1559:
1552:
1551:
1507:
1505:
1504:
1499:
1481:
1479:
1478:
1473:
1461:
1459:
1458:
1453:
1451:
1450:
1438:
1437:
1409:
1407:
1406:
1401:
1386:
1384:
1383:
1378:
1376:
1375:
1374:
1373:
1356:
1355:
1330:
1328:
1327:
1322:
1320:
1319:
1291:
1289:
1288:
1283:
1271:
1269:
1268:
1263:
1247:
1245:
1244:
1239:
1228:
1227:
1206:
1204:
1203:
1198:
1187:
1186:
1171:and that of the
1134:
1132:
1131:
1126:
1099:
1097:
1096:
1091:
1070:Solomon Feferman
1064:Martin's maximum
1061:
1059:
1058:
1053:
1051:
1050:
1034:
1032:
1031:
1026:
1024:
1023:
1022:
1021:
1000:
968:Skolem's paradox
930:Zermelo–Fraenkel
868:
866:
865:
860:
848:
846:
845:
840:
838:
837:
830:
829:
808:
806:
805:
800:
798:
797:
781:
779:
778:
773:
771:
770:
769:
768:
747:
745:
744:
739:
731:
730:
729:
728:
703:
701:
700:
695:
612:
610:
609:
604:
602:
601:
589:
588:
587:
586:
565:
563:
562:
557:
555:
554:
538:
536:
535:
530:
528:
527:
503:
501:
500:
495:
493:
492:
491:
490:
473:
465:
457:
456:
432:
422:
420:
419:
414:
412:
411:
410:
409:
392:
387:
382:
319:rational numbers
193:
191:
190:
185:
183:
182:
170:
169:
149:
147:
146:
141:
139:
138:
126:
125:
124:
123:
21:
6998:
6997:
6993:
6992:
6991:
6989:
6988:
6987:
6943:
6942:
6941:
6936:
6806:
6801:
6771:
6766:
6755:
6748:
6693:Category theory
6683:Algebraic logic
6666:
6637:Lambda calculus
6575:Church encoding
6561:
6537:Truth predicate
6393:
6359:Complete theory
6282:
6151:
6147:
6143:
6138:
6130:
5850: and
5846:
5841:
5827:
5803:New Foundations
5771:axiom of choice
5754:
5716:Gödel numbering
5656: and
5648:
5552:
5437:
5387:
5368:
5317:Boolean algebra
5303:
5267:Equiconsistency
5232:Classical logic
5209:
5190:Halting problem
5178: and
5154: and
5142: and
5141:
5136:Theorems (
5131:
5048:
5043:
5013:
5008:
4935:
4914:
4898:
4863:New Foundations
4810:
4700:
4619:Cardinal number
4602:
4588:
4529:
4413:
4404:
4388:
4383:
4316:
4245:
4226:
4224:Further reading
4218:Wayback Machine
4199:
4181:10.2307/2274520
4163:
4162:
4151:
4147:
4132:
4128:
4077:
4073:
4065:
4051:10.2307/2695444
4034:
4028:
4024:
3993:
3989:
3962:
3958:
3917:
3913:
3869:
3865:
3857:
3850:
3844:
3840:
3832:
3821:
3812:
3808:
3777:
3773:
3763:
3761:
3759:Quanta Magazine
3751:
3747:
3737:
3735:
3731:
3724:
3718:
3714:
3681:
3677:
3634:
3630:
3607:10.2307/2273955
3591:
3584:
3577:
3558:
3554:
3546:
3542:
3512:10.2307/2320581
3496:
3492:
3484:
3473:
3467:Koellner, Peter
3464:
3460:
3452:
3437:
3431:
3424:
3416:
3401:
3395:
3388:
3373:10.2307/2589047
3347:
3343:
3280:
3276:
3235:(12): 556–557.
3221:
3217:
3202:
3195:
3132:
3125:
3118:
3096:
3092:
3078:
3069:
3053:
3052:
3042:(84): 242–258.
3023:
3019:
3014:
2977:
2949:
2945:
2934:
2930:
2929:
2925:
2914:
2910:
2901:
2897:
2896:
2892:
2881:
2877:
2876:
2872:
2864:
2860:
2859:
2855:
2841:
2837:
2836:
2831:
2816:
2812:
2811:
2806:
2800:
2797:
2796:
2791:König's theorem
2773:
2769:
2755:
2751:
2741:
2736:
2721:
2717:
2716:
2711:
2705:
2702:
2701:
2678:
2674:
2656:
2652:
2650:
2647:
2646:
2615:
2611:
2610:
2605:
2590:
2586:
2585:
2581:
2566:
2562:
2560:
2557:
2556:
2531:
2527:
2516:
2512:
2511:
2507:
2496:
2492:
2483:
2479:
2478:
2474:
2463:
2459:
2458:
2454:
2446:
2442:
2441:
2437:
2423:
2419:
2418:
2407:
2392:
2388:
2387:
2382:
2376:
2373:
2372:
2337:
2333:
2315:
2311:
2309:
2306:
2305:
2274:
2270:
2259:
2255:
2254:
2249:
2243:
2240:
2239:
2209:
2205:
2187:
2183:
2181:
2178:
2177:
2152:
2148:
2137:
2133:
2132:
2127:
2121:
2118:
2117:
2084:
2080:
2069:
2065:
2064:
2059:
2053:
2050:
2049:
2019:
2015:
2014:
2009:
2003:
2000:
1999:
1996:
1974:
1970:
1961:
1957:
1943:
1940:
1939:
1922:
1918:
1909:
1905:
1891:
1888:
1887:
1860:
1857:
1856:
1836:
1832:
1823:
1819:
1817:
1814:
1813:
1797:
1794:
1793:
1776:
1772:
1763:
1759:
1757:
1754:
1753:
1722:
1718:
1707:
1703:
1702:
1698:
1696:
1693:
1692:
1675:
1671:
1669:
1666:
1665:
1626:
1622:
1621:
1617:
1591:
1587:
1586:
1582:
1580:
1577:
1576:
1547:
1543:
1542:
1538:
1536:
1533:
1532:
1521:axiom of choice
1510:Philip Jourdain
1487:
1484:
1483:
1467:
1464:
1463:
1446:
1442:
1433:
1429:
1427:
1424:
1423:
1395:
1392:
1391:
1369:
1365:
1364:
1360:
1345:
1341:
1339:
1336:
1335:
1315:
1311:
1297:
1294:
1293:
1277:
1274:
1273:
1257:
1254:
1253:
1223:
1222:
1220:
1217:
1216:
1182:
1181:
1179:
1176:
1175:
1161:
1105:
1102:
1101:
1085:
1082:
1081:
1074:classical logic
1046:
1042:
1040:
1037:
1036:
1017:
1013:
1012:
1008:
1006:
1003:
1002:
998:
953:Matthew Foreman
926:
918:Gödel numbering
905:axiom of choice
854:
851:
850:
825:
821:
820:
816:
814:
811:
810:
793:
789:
787:
784:
783:
764:
760:
759:
755:
753:
750:
749:
724:
720:
719:
715:
713:
710:
709:
689:
686:
685:
682:König's theorem
638:axiom of choice
619:
597:
593:
582:
578:
577:
573:
571:
568:
567:
550:
546:
544:
541:
540:
523:
519:
517:
514:
513:
510:axiom of choice
506:
486:
482:
481:
477:
469:
461:
452:
448:
437:
434:
433:
430:
405:
401:
400:
396:
388:
383:
378:
376:
373:
372:
279:cardinal number
268:
266:Cardinal number
262:
233:
178:
174:
165:
161:
159:
156:
155:
134:
130:
119:
115:
114:
110:
108:
105:
104:
97:axiom of choice
47:, specifically
39:
28:
23:
22:
15:
12:
11:
5:
6996:
6986:
6985:
6980:
6975:
6970:
6965:
6960:
6955:
6938:
6937:
6935:
6934:
6927:
6922:
6917:
6912:
6907:
6902:
6897:
6892:
6887:
6882:
6877:
6872:
6867:
6862:
6857:
6852:
6847:
6842:
6837:
6832:
6827:
6822:
6817:
6811:
6808:
6807:
6800:
6799:
6792:
6785:
6777:
6768:
6767:
6753:
6750:
6749:
6747:
6746:
6741:
6736:
6731:
6726:
6725:
6724:
6714:
6709:
6704:
6695:
6690:
6685:
6680:
6678:Abstract logic
6674:
6672:
6668:
6667:
6665:
6664:
6659:
6657:Turing machine
6654:
6649:
6644:
6639:
6634:
6629:
6628:
6627:
6622:
6617:
6612:
6607:
6597:
6595:Computable set
6592:
6587:
6582:
6577:
6571:
6569:
6563:
6562:
6560:
6559:
6554:
6549:
6544:
6539:
6534:
6529:
6524:
6523:
6522:
6517:
6512:
6502:
6497:
6492:
6490:Satisfiability
6487:
6482:
6477:
6476:
6475:
6465:
6464:
6463:
6453:
6452:
6451:
6446:
6441:
6436:
6431:
6421:
6420:
6419:
6414:
6407:Interpretation
6403:
6401:
6395:
6394:
6392:
6391:
6386:
6381:
6376:
6371:
6361:
6356:
6355:
6354:
6353:
6352:
6342:
6337:
6327:
6322:
6317:
6312:
6307:
6302:
6296:
6294:
6288:
6287:
6284:
6283:
6281:
6280:
6272:
6271:
6270:
6269:
6264:
6263:
6262:
6257:
6252:
6232:
6231:
6230:
6228:minimal axioms
6225:
6214:
6213:
6212:
6201:
6200:
6199:
6194:
6189:
6184:
6179:
6174:
6161:
6159:
6140:
6139:
6137:
6136:
6135:
6134:
6122:
6117:
6116:
6115:
6110:
6105:
6100:
6090:
6085:
6080:
6075:
6074:
6073:
6068:
6058:
6057:
6056:
6051:
6046:
6041:
6031:
6026:
6025:
6024:
6019:
6014:
6004:
6003:
6002:
5997:
5992:
5987:
5982:
5977:
5967:
5962:
5957:
5952:
5951:
5950:
5945:
5940:
5935:
5925:
5920:
5918:Formation rule
5915:
5910:
5909:
5908:
5903:
5893:
5892:
5891:
5881:
5876:
5871:
5866:
5860:
5854:
5837:Formal systems
5833:
5832:
5829:
5828:
5826:
5825:
5820:
5815:
5810:
5805:
5800:
5795:
5790:
5785:
5780:
5779:
5778:
5773:
5762:
5760:
5756:
5755:
5753:
5752:
5751:
5750:
5740:
5735:
5734:
5733:
5726:Large cardinal
5723:
5718:
5713:
5708:
5703:
5689:
5688:
5687:
5682:
5677:
5662:
5660:
5650:
5649:
5647:
5646:
5645:
5644:
5639:
5634:
5624:
5619:
5614:
5609:
5604:
5599:
5594:
5589:
5584:
5579:
5574:
5569:
5563:
5561:
5554:
5553:
5551:
5550:
5549:
5548:
5543:
5538:
5533:
5528:
5523:
5515:
5514:
5513:
5508:
5498:
5493:
5491:Extensionality
5488:
5486:Ordinal number
5483:
5473:
5468:
5467:
5466:
5455:
5449:
5443:
5442:
5439:
5438:
5436:
5435:
5430:
5425:
5420:
5415:
5410:
5405:
5404:
5403:
5393:
5392:
5391:
5378:
5376:
5370:
5369:
5367:
5366:
5365:
5364:
5359:
5354:
5344:
5339:
5334:
5329:
5324:
5319:
5313:
5311:
5305:
5304:
5302:
5301:
5296:
5291:
5286:
5281:
5276:
5271:
5270:
5269:
5259:
5254:
5249:
5244:
5239:
5234:
5228:
5226:
5217:
5211:
5210:
5208:
5207:
5202:
5197:
5192:
5187:
5182:
5170:Cantor's
5168:
5163:
5158:
5148:
5146:
5133:
5132:
5130:
5129:
5124:
5119:
5114:
5109:
5104:
5099:
5094:
5089:
5084:
5079:
5074:
5069:
5068:
5067:
5056:
5054:
5050:
5049:
5042:
5041:
5034:
5027:
5019:
5010:
5009:
5007:
5006:
5001:
4999:Thoralf Skolem
4996:
4991:
4986:
4981:
4976:
4971:
4966:
4961:
4956:
4951:
4945:
4943:
4937:
4936:
4934:
4933:
4928:
4923:
4917:
4915:
4913:
4912:
4909:
4903:
4900:
4899:
4897:
4896:
4895:
4894:
4889:
4884:
4883:
4882:
4867:
4866:
4865:
4853:
4852:
4851:
4840:
4839:
4834:
4829:
4824:
4818:
4816:
4812:
4811:
4809:
4808:
4803:
4798:
4793:
4784:
4779:
4774:
4764:
4759:
4758:
4757:
4752:
4747:
4737:
4727:
4722:
4717:
4711:
4709:
4702:
4701:
4699:
4698:
4693:
4688:
4683:
4681:Ordinal number
4678:
4673:
4668:
4663:
4662:
4661:
4656:
4646:
4641:
4636:
4631:
4626:
4616:
4611:
4605:
4603:
4601:
4600:
4597:
4593:
4590:
4589:
4587:
4586:
4581:
4576:
4571:
4566:
4561:
4559:Disjoint union
4556:
4551:
4545:
4539:
4537:
4531:
4530:
4528:
4527:
4526:
4525:
4520:
4509:
4508:
4506:Martin's axiom
4503:
4498:
4493:
4488:
4483:
4478:
4473:
4471:Extensionality
4468:
4467:
4466:
4456:
4451:
4450:
4449:
4444:
4439:
4429:
4423:
4421:
4415:
4414:
4407:
4405:
4403:
4402:
4396:
4394:
4390:
4389:
4382:
4381:
4374:
4367:
4359:
4353:
4352:
4315:
4314:External links
4312:
4311:
4310:
4301:
4292:
4278:
4267:
4258:
4249:
4243:
4225:
4222:
4221:
4220:
4198:
4195:
4194:
4193:
4161:
4160:
4145:
4126:
4089:(2): 361–417.
4071:
4045:(6): 544–553.
4022:
4003:(4): 489–532.
3987:
3956:
3927:(2): 203–228.
3923:. New Series.
3911:
3882:(3): 416–449.
3863:
3838:
3806:
3787:(2): 125–151.
3771:
3745:
3712:
3675:
3648:(1): 342–345.
3628:
3582:
3575:
3561:Kunen, Kenneth
3552:
3550:, p. 500.
3540:
3506:(7): 540–551.
3490:
3458:
3448:(7): 681–690.
3422:
3412:(6): 567–576.
3386:
3341:
3294:(1): 105–110.
3274:
3215:
3193:
3123:
3116:
3090:
3067:
3016:
3015:
3013:
3010:
3009:
3008:
3003:
2998:
2993:
2988:
2983:
2976:
2973:
2972:
2971:
2958:
2955:
2952:
2948:
2944:
2937:
2933:
2928:
2924:
2917:
2913:
2909:
2904:
2900:
2895:
2891:
2884:
2880:
2875:
2867:
2863:
2858:
2854:
2851:
2844:
2840:
2834:
2830:
2826:
2819:
2815:
2809:
2805:
2794:
2776:
2772:
2768:
2763:
2758:
2754:
2750:
2747:
2744:
2739:
2735:
2731:
2724:
2720:
2714:
2710:
2686:
2681:
2677:
2673:
2670:
2667:
2664:
2659:
2655:
2635:
2634:
2618:
2614:
2608:
2604:
2600:
2593:
2589:
2584:
2580:
2575:
2572:
2569:
2565:
2554:
2540:
2537:
2534:
2530:
2526:
2519:
2515:
2510:
2506:
2499:
2495:
2491:
2486:
2482:
2477:
2473:
2466:
2462:
2457:
2449:
2445:
2440:
2436:
2433:
2426:
2422:
2416:
2413:
2410:
2406:
2402:
2395:
2391:
2385:
2381:
2358:
2357:
2345:
2340:
2336:
2332:
2329:
2326:
2323:
2318:
2314:
2283:
2280:
2277:
2273:
2269:
2262:
2258:
2252:
2248:
2237:
2236:operation; and
2217:
2212:
2208:
2204:
2201:
2198:
2195:
2190:
2186:
2155:
2151:
2147:
2140:
2136:
2130:
2126:
2115:
2093:
2090:
2087:
2083:
2079:
2072:
2068:
2062:
2058:
2022:
2018:
2012:
2008:
1995:
1992:
1977:
1973:
1969:
1964:
1960:
1956:
1953:
1950:
1947:
1925:
1921:
1917:
1912:
1908:
1904:
1901:
1898:
1895:
1864:
1842:
1839:
1835:
1831:
1826:
1822:
1801:
1779:
1775:
1771:
1766:
1762:
1744:. Much later,
1731:
1728:
1725:
1721:
1717:
1710:
1706:
1701:
1678:
1674:
1637:
1634:
1629:
1625:
1620:
1615:
1612:
1608:
1602:
1599:
1594:
1590:
1585:
1573:Hartogs number
1558:
1555:
1550:
1546:
1541:
1497:
1494:
1491:
1471:
1449:
1445:
1441:
1436:
1432:
1416:
1415:
1399:
1372:
1368:
1363:
1359:
1354:
1351:
1348:
1344:
1318:
1314:
1310:
1307:
1304:
1301:
1281:
1261:
1237:
1234:
1231:
1226:
1196:
1193:
1190:
1185:
1160:
1157:
1152:Saharon Shelah
1137:Peter Koellner
1124:
1121:
1118:
1115:
1112:
1109:
1089:
1049:
1045:
1020:
1016:
1011:
991:W. Hugh Woodin
925:
922:
898:Peter Koellner
882:measure theory
858:
836:
833:
828:
824:
819:
796:
792:
767:
763:
758:
737:
734:
727:
723:
718:
693:
618:
615:
600:
596:
592:
585:
581:
576:
553:
549:
526:
522:
489:
485:
480:
476:
472:
468:
464:
460:
455:
451:
447:
444:
441:
425:
408:
404:
399:
395:
391:
386:
381:
331:countable sets
264:Main article:
261:
258:
232:
229:
181:
177:
173:
168:
164:
137:
133:
129:
122:
118:
113:
26:
9:
6:
4:
3:
2:
6995:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6950:
6948:
6932:
6928:
6926:
6923:
6921:
6918:
6916:
6913:
6911:
6908:
6906:
6903:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6861:
6858:
6856:
6853:
6851:
6848:
6846:
6843:
6841:
6838:
6836:
6833:
6831:
6828:
6826:
6823:
6821:
6818:
6816:
6813:
6812:
6809:
6805:
6798:
6793:
6791:
6786:
6784:
6779:
6778:
6775:
6765:
6764:
6759:
6751:
6745:
6742:
6740:
6737:
6735:
6732:
6730:
6727:
6723:
6720:
6719:
6718:
6715:
6713:
6710:
6708:
6705:
6703:
6699:
6696:
6694:
6691:
6689:
6686:
6684:
6681:
6679:
6676:
6675:
6673:
6669:
6663:
6660:
6658:
6655:
6653:
6652:Recursive set
6650:
6648:
6645:
6643:
6640:
6638:
6635:
6633:
6630:
6626:
6623:
6621:
6618:
6616:
6613:
6611:
6608:
6606:
6603:
6602:
6601:
6598:
6596:
6593:
6591:
6588:
6586:
6583:
6581:
6578:
6576:
6573:
6572:
6570:
6568:
6564:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6528:
6525:
6521:
6518:
6516:
6513:
6511:
6508:
6507:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6488:
6486:
6483:
6481:
6478:
6474:
6471:
6470:
6469:
6466:
6462:
6461:of arithmetic
6459:
6458:
6457:
6454:
6450:
6447:
6445:
6442:
6440:
6437:
6435:
6432:
6430:
6427:
6426:
6425:
6422:
6418:
6415:
6413:
6410:
6409:
6408:
6405:
6404:
6402:
6400:
6396:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6372:
6369:
6368:from ZFC
6365:
6362:
6360:
6357:
6351:
6348:
6347:
6346:
6343:
6341:
6338:
6336:
6333:
6332:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6308:
6306:
6303:
6301:
6298:
6297:
6295:
6293:
6289:
6279:
6278:
6274:
6273:
6268:
6267:non-Euclidean
6265:
6261:
6258:
6256:
6253:
6251:
6250:
6246:
6245:
6243:
6240:
6239:
6237:
6233:
6229:
6226:
6224:
6221:
6220:
6219:
6215:
6211:
6208:
6207:
6206:
6202:
6198:
6195:
6193:
6190:
6188:
6185:
6183:
6180:
6178:
6175:
6173:
6170:
6169:
6167:
6163:
6162:
6160:
6155:
6149:
6144:Example
6141:
6133:
6128:
6127:
6126:
6123:
6121:
6118:
6114:
6111:
6109:
6106:
6104:
6101:
6099:
6096:
6095:
6094:
6091:
6089:
6086:
6084:
6081:
6079:
6076:
6072:
6069:
6067:
6064:
6063:
6062:
6059:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6037:
6036:
6035:
6032:
6030:
6027:
6023:
6020:
6018:
6015:
6013:
6010:
6009:
6008:
6005:
6001:
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5981:
5978:
5976:
5973:
5972:
5971:
5968:
5966:
5963:
5961:
5958:
5956:
5953:
5949:
5946:
5944:
5941:
5939:
5936:
5934:
5931:
5930:
5929:
5926:
5924:
5921:
5919:
5916:
5914:
5911:
5907:
5904:
5902:
5901:by definition
5899:
5898:
5897:
5894:
5890:
5887:
5886:
5885:
5882:
5880:
5877:
5875:
5872:
5870:
5867:
5865:
5862:
5861:
5858:
5855:
5853:
5849:
5844:
5838:
5834:
5824:
5821:
5819:
5816:
5814:
5811:
5809:
5806:
5804:
5801:
5799:
5796:
5794:
5791:
5789:
5788:Kripke–Platek
5786:
5784:
5781:
5777:
5774:
5772:
5769:
5768:
5767:
5764:
5763:
5761:
5757:
5749:
5746:
5745:
5744:
5741:
5739:
5736:
5732:
5729:
5728:
5727:
5724:
5722:
5719:
5717:
5714:
5712:
5709:
5707:
5704:
5701:
5697:
5693:
5690:
5686:
5683:
5681:
5678:
5676:
5673:
5672:
5671:
5667:
5664:
5663:
5661:
5659:
5655:
5651:
5643:
5640:
5638:
5635:
5633:
5632:constructible
5630:
5629:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5605:
5603:
5600:
5598:
5595:
5593:
5590:
5588:
5585:
5583:
5580:
5578:
5575:
5573:
5570:
5568:
5565:
5564:
5562:
5560:
5555:
5547:
5544:
5542:
5539:
5537:
5534:
5532:
5529:
5527:
5524:
5522:
5519:
5518:
5516:
5512:
5509:
5507:
5504:
5503:
5502:
5499:
5497:
5494:
5492:
5489:
5487:
5484:
5482:
5478:
5474:
5472:
5469:
5465:
5462:
5461:
5460:
5457:
5456:
5453:
5450:
5448:
5444:
5434:
5431:
5429:
5426:
5424:
5421:
5419:
5416:
5414:
5411:
5409:
5406:
5402:
5399:
5398:
5397:
5394:
5390:
5385:
5384:
5383:
5380:
5379:
5377:
5375:
5371:
5363:
5360:
5358:
5355:
5353:
5350:
5349:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5314:
5312:
5310:
5309:Propositional
5306:
5300:
5297:
5295:
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5275:
5272:
5268:
5265:
5264:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5237:Logical truth
5235:
5233:
5230:
5229:
5227:
5225:
5221:
5218:
5216:
5212:
5206:
5203:
5201:
5198:
5196:
5193:
5191:
5188:
5186:
5183:
5181:
5177:
5173:
5169:
5167:
5164:
5162:
5159:
5157:
5153:
5150:
5149:
5147:
5145:
5139:
5134:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5066:
5063:
5062:
5061:
5058:
5057:
5055:
5051:
5047:
5040:
5035:
5033:
5028:
5026:
5021:
5020:
5017:
5005:
5004:Ernst Zermelo
5002:
5000:
4997:
4995:
4992:
4990:
4989:Willard Quine
4987:
4985:
4982:
4980:
4977:
4975:
4972:
4970:
4967:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4946:
4944:
4942:
4941:Set theorists
4938:
4932:
4929:
4927:
4924:
4922:
4919:
4918:
4916:
4910:
4908:
4905:
4904:
4901:
4893:
4890:
4888:
4887:Kripke–Platek
4885:
4881:
4878:
4877:
4876:
4873:
4872:
4871:
4868:
4864:
4861:
4860:
4859:
4858:
4854:
4850:
4847:
4846:
4845:
4842:
4841:
4838:
4835:
4833:
4830:
4828:
4825:
4823:
4820:
4819:
4817:
4813:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4790:
4785:
4783:
4780:
4778:
4775:
4772:
4768:
4765:
4763:
4760:
4756:
4753:
4751:
4748:
4746:
4743:
4742:
4741:
4738:
4735:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4712:
4710:
4707:
4703:
4697:
4694:
4692:
4689:
4687:
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4660:
4657:
4655:
4652:
4651:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4630:
4627:
4624:
4620:
4617:
4615:
4612:
4610:
4607:
4606:
4604:
4598:
4595:
4594:
4591:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4555:
4552:
4549:
4546:
4544:
4541:
4540:
4538:
4536:
4532:
4524:
4523:specification
4521:
4519:
4516:
4515:
4514:
4511:
4510:
4507:
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4465:
4462:
4461:
4460:
4457:
4455:
4452:
4448:
4445:
4443:
4440:
4438:
4435:
4434:
4433:
4430:
4428:
4425:
4424:
4422:
4420:
4416:
4411:
4401:
4398:
4397:
4395:
4391:
4387:
4380:
4375:
4373:
4368:
4366:
4361:
4360:
4357:
4348:
4347:
4342:
4339:
4335:
4330:
4329:
4328:
4327:at Wikiquote
4326:
4321:
4307:
4302:
4298:
4293:
4291:
4290:0-8218-1428-1
4287:
4283:
4279:
4276:
4272:
4268:
4264:
4259:
4255:
4250:
4246:
4240:
4236:
4232:
4228:
4227:
4219:
4215:
4212:
4210:
4206:
4201:
4200:
4190:
4186:
4182:
4178:
4174:
4170:
4165:
4164:
4156:
4149:
4141:
4138:(in German).
4137:
4130:
4122:
4118:
4114:
4110:
4106:
4102:
4097:
4092:
4088:
4084:
4083:
4075:
4064:
4060:
4056:
4052:
4048:
4044:
4040:
4033:
4026:
4018:
4014:
4010:
4006:
4002:
3998:
3991:
3983:
3979:
3976:(49): 42–56.
3975:
3971:
3967:
3960:
3952:
3948:
3944:
3940:
3935:
3930:
3926:
3922:
3915:
3907:
3903:
3899:
3895:
3890:
3885:
3881:
3877:
3873:
3867:
3856:
3849:
3842:
3831:
3827:
3820:
3816:
3810:
3802:
3798:
3794:
3790:
3786:
3782:
3775:
3760:
3756:
3749:
3730:
3723:
3716:
3708:
3704:
3699:
3694:
3690:
3686:
3679:
3671:
3667:
3663:
3659:
3655:
3651:
3647:
3643:
3639:
3632:
3624:
3620:
3616:
3612:
3608:
3604:
3600:
3596:
3589:
3587:
3578:
3572:
3568:
3567:
3562:
3556:
3549:
3544:
3537:
3535:
3529:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3494:
3483:
3479:
3472:
3468:
3462:
3451:
3447:
3443:
3436:
3429:
3427:
3415:
3411:
3407:
3400:
3393:
3391:
3382:
3378:
3374:
3370:
3365:
3364:10.1.1.37.295
3360:
3357:(2): 99–111.
3356:
3352:
3345:
3337:
3333:
3328:
3323:
3319:
3315:
3310:
3305:
3301:
3297:
3293:
3289:
3285:
3278:
3270:
3266:
3261:
3256:
3251:
3246:
3242:
3238:
3234:
3230:
3226:
3219:
3211:
3207:
3200:
3198:
3189:
3185:
3180:
3175:
3171:
3167:
3162:
3157:
3153:
3149:
3145:
3141:
3137:
3130:
3128:
3119:
3117:9780691024479
3113:
3109:
3104:
3103:
3094:
3086:
3082:
3076:
3074:
3072:
3063:
3057:
3049:
3045:
3041:
3037:
3036:
3031:
3027:
3026:Cantor, Georg
3021:
3017:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2978:
2956:
2953:
2950:
2942:
2935:
2926:
2922:
2915:
2907:
2902:
2893:
2889:
2882:
2865:
2856:
2849:
2842:
2832:
2824:
2817:
2807:
2795:
2792:
2774:
2766:
2756:
2745:
2742:
2737:
2729:
2722:
2712:
2700:
2699:
2698:
2679:
2668:
2665:
2662:
2657:
2644:
2640:
2616:
2606:
2598:
2591:
2582:
2578:
2573:
2570:
2567:
2555:
2538:
2535:
2532:
2524:
2517:
2508:
2504:
2497:
2489:
2484:
2475:
2471:
2464:
2447:
2438:
2431:
2424:
2414:
2411:
2408:
2400:
2393:
2383:
2371:
2370:
2369:
2367:
2363:
2338:
2327:
2324:
2321:
2316:
2303:
2299:
2281:
2278:
2275:
2267:
2260:
2250:
2238:
2235:
2231:
2210:
2199:
2196:
2193:
2188:
2175:
2171:
2153:
2145:
2138:
2128:
2116:
2113:
2109:
2091:
2088:
2085:
2077:
2070:
2060:
2048:
2047:
2046:
2044:
2040:
2020:
2010:
1991:
1975:
1971:
1967:
1962:
1958:
1951:
1948:
1945:
1923:
1919:
1915:
1910:
1906:
1899:
1896:
1893:
1884:
1882:
1878:
1862:
1840:
1837:
1833:
1829:
1824:
1820:
1799:
1777:
1773:
1769:
1764:
1760:
1751:
1747:
1729:
1726:
1723:
1715:
1708:
1699:
1676:
1663:
1659:
1655:
1651:
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1606:
1600:
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1492:
1489:
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1390:
1370:
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1079:
1075:
1071:
1067:
1065:
1047:
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1009:
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992:
987:
985:
984:probabilities
981:
975:
973:
969:
965:
960:
958:
954:
950:
946:
941:
939:
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931:
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911:
906:
901:
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883:
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632:
628:
624:
614:
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590:
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551:
539:greater than
524:
511:
508:Assuming the
505:
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320:
316:
311:
309:
305:
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285:
281:
280:
275:
274:
267:
257:
255:
250:
246:
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238:
228:
226:
225:
224:the continuum
219:
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213:
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204:
200:
195:
179:
171:
166:
162:
153:
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127:
120:
111:
102:
101:aleph numbers
98:
94:
88:
84:
80:
78:
74:
70:
64:
63:. It states:
62:
61:infinite sets
58:
55:(abbreviated
54:
50:
46:
41:
37:
33:
19:
6754:
6552:Ultraproduct
6399:Model theory
6364:Independence
6300:Formal proof
6292:Proof theory
6275:
6248:
6205:real numbers
6177:second-order
6088:Substitution
5965:Metalanguage
5906:conservative
5879:Axiom schema
5823:Constructive
5793:Morse–Kelley
5775:
5759:Set theories
5738:Aleph number
5731:inaccessible
5637:Grothendieck
5521:intersection
5408:Higher-order
5396:Second-order
5342:Truth tables
5299:Venn diagram
5082:Formal proof
4954:Georg Cantor
4949:Paul Bernays
4880:Morse–Kelley
4855:
4788:
4787:Subset
4734:hereditarily
4696:Venn diagram
4654:ordered pair
4638:
4569:Intersection
4513:Axiom schema
4344:
4317:
4281:
4274:
4270:
4262:
4256:. Cambridge.
4253:
4234:
4202:
4172:
4168:
4154:
4148:
4139:
4135:
4129:
4096:math/0005179
4086:
4080:
4074:
4042:
4038:
4025:
4000:
3996:
3990:
3973:
3969:
3959:
3934:math/0211398
3924:
3920:
3914:
3879:
3875:
3866:
3841:
3825:
3809:
3784:
3780:
3774:
3762:. Retrieved
3758:
3748:
3736:. Retrieved
3715:
3688:
3684:
3678:
3645:
3641:
3631:
3598:
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3564:
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3543:
3533:
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3477:
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3445:
3441:
3409:
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3344:
3291:
3287:
3277:
3232:
3228:
3218:
3205:
3143:
3139:
3101:
3093:
3084:
3056:cite journal
3039:
3033:
3020:
2642:
2638:
2636:
2365:
2361:
2359:
2301:
2297:
2229:
2173:
2169:
2111:
2107:
2042:
2038:
1997:
1885:
1880:
1876:
1652:
1529:aleph number
1514:
1420:beth numbers
1417:
1411:
1212:
1208:
1168:
1164:
1162:
1141:
1068:
988:
976:
961:
942:
927:
902:
890:
876:, point set
871:
671:
669:
665:Fields Medal
658:
635:
620:
507:
426:
365:equinumerous
360:
356:
354:
342:real numbers
335:
326:
312:
307:
303:
299:
295:
291:
287:
277:
271:
269:
234:
222:
220:
199:Georg Cantor
196:
152:beth numbers
90:
86:
82:
77:real numbers
66:
56:
52:
42:
40:
6662:Type theory
6610:undecidable
6542:Truth value
6429:equivalence
6108:non-logical
5721:Enumeration
5711:Isomorphism
5658:cardinality
5642:Von Neumann
5607:Ultrafilter
5572:Uncountable
5506:equivalence
5423:Quantifiers
5413:Fixed-point
5382:First-order
5262:Consistency
5247:Proposition
5224:Traditional
5195:Lindström's
5185:Compactness
5127:Type theory
5072:Cardinality
4979:Thomas Jech
4822:Alternative
4801:Uncountable
4755:Ultrafilter
4614:Cardinality
4518:replacement
4459:Determinacy
4269:Gödel, K.:
3764:30 December
3738:25 February
3081:Gödel, Kurt
2991:Cardinality
2986:Beth number
1145:proposes a
999:"(*)-axiom"
886:conjectures
646:inner model
327:cardinality
273:cardinality
207:independent
69:cardinality
45:mathematics
6978:Hypotheses
6947:Categories
6473:elementary
6166:arithmetic
6034:Quantifier
6012:functional
5884:Expression
5602:Transitive
5546:identities
5531:complement
5464:hereditary
5447:Set theory
4974:Kurt Gödel
4959:Paul Cohen
4796:Transitive
4564:Identities
4548:Complement
4535:Operations
4496:Regularity
4464:projective
4427:Adjunction
4386:Set theory
4205:PlanetMath
4142:: 127–142.
3548:Maddy 1988
3012:References
2234:cofinality
1855:for every
1654:Kurt Gödel
1517:Sierpiński
1387:for every
1292:such that
1147:multiverse
706:cofinality
650:consistent
631:Paul Cohen
627:Kurt Gödel
254:Paul Cohen
249:Kurt Gödel
216:Kurt Gödel
212:Paul Cohen
49:set theory
6744:Supertask
6647:Recursion
6605:decidable
6439:saturated
6417:of models
6340:deductive
6335:axiomatic
6255:Hilbert's
6242:Euclidean
6223:canonical
6146:axiomatic
6078:Signature
6007:Predicate
5896:Extension
5818:Ackermann
5743:Operation
5622:Universal
5612:Recursive
5587:Singleton
5582:Inhabited
5567:Countable
5557:Types of
5541:power set
5511:partition
5428:Predicate
5374:Predicate
5289:Syllogism
5279:Soundness
5252:Inference
5242:Tautology
5144:paradoxes
4907:Paradoxes
4827:Axiomatic
4806:Universal
4782:Singleton
4777:Recursive
4720:Countable
4715:Amorphous
4574:Power set
4491:Power set
4442:dependent
4437:countable
4346:MathWorld
4233:(2008) .
3889:1108.4223
3801:122205863
3698:1203.4026
3534:formalism
3359:CiteSeerX
2951:α
2947:ℵ
2936:α
2932:ℵ
2916:α
2912:ℵ
2908:⋅
2903:α
2899:ℵ
2883:α
2879:ℵ
2866:α
2862:ℵ
2850:≤
2843:α
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2833:α
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2825:≤
2818:β
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2808:α
2804:ℵ
2775:α
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2757:α
2753:ℵ
2746:
2738:α
2734:ℵ
2730:≥
2723:β
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2713:α
2709:ℵ
2680:α
2676:ℵ
2669:
2663:≥
2658:β
2654:ℵ
2617:β
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2607:α
2603:ℵ
2599:≤
2592:β
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2568:β
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2465:β
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2409:β
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2401:≤
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2384:α
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2339:α
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2328:
2322:≥
2317:β
2313:ℵ
2276:α
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2261:β
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2251:α
2247:ℵ
2211:α
2207:ℵ
2200:
2189:β
2185:ℵ
2154:α
2150:ℵ
2139:β
2135:ℵ
2129:α
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2086:β
2082:ℵ
2071:β
2067:ℵ
2061:α
2057:ℵ
2021:β
2017:ℵ
2011:α
2007:ℵ
1955:→
1916:≤
1903:→
1863:κ
1834:κ
1825:κ
1800:κ
1774:κ
1765:κ
1724:α
1720:ℵ
1709:α
1705:ℵ
1677:α
1673:ℵ
1624:ℵ
1614:⋅
1589:ℵ
1545:ℵ
1490:α
1470:α
1448:α
1444:ℶ
1435:α
1431:ℵ
1398:α
1371:α
1367:ℵ
1347:α
1343:ℵ
1317:λ
1306:κ
1300:λ
1280:κ
1260:λ
1252:cardinal
1173:power set
1120:ϕ
1117:¬
1114:∨
1111:ϕ
1088:ϕ
1044:ℵ
1015:ℵ
938:formalist
934:platonist
857:ω
835:ω
823:ω
818:ℵ
795:ω
791:ℵ
762:ℵ
736:κ
722:ℵ
692:κ
595:ℵ
580:ℵ
548:ℵ
521:ℵ
484:ℵ
450:ℵ
446::
440:∄
403:ℵ
367:with the
284:bijection
218:in 1940.
176:ℵ
163:ℶ
132:ℵ
117:ℵ
95:with the
6973:Infinity
6729:Logicism
6722:timeline
6698:Concrete
6557:Validity
6527:T-schema
6520:Kripke's
6515:Tarski's
6510:semantic
6500:Strength
6449:submodel
6444:spectrum
6412:function
6260:Tarski's
6249:Elements
6236:geometry
6192:Robinson
6113:variable
6098:function
6071:spectrum
6061:Sentence
6017:variable
5960:Language
5913:Relation
5874:Automata
5864:Alphabet
5848:language
5702:-jection
5680:codomain
5666:Function
5627:Universe
5597:Infinite
5501:Relation
5284:Validity
5274:Argument
5172:theorem,
4911:Problems
4815:Theories
4791:Superset
4767:Infinite
4596:Concepts
4476:Infinity
4393:Overview
4214:Archived
4121:15577499
4063:Archived
3906:33807508
3855:Archived
3830:Archived
3817:(2011).
3729:Archived
3662:44152014
3623:38174418
3563:(1980).
3482:Archived
3469:(2011).
3450:Archived
3414:Archived
3336:16591132
3269:16577857
3188:16578557
3083:(1940).
3028:(1878).
2975:See also
2793:, while:
2641:+1 <
2553:, while:
2300:+1 <
2228:, where
2172:+1 <
1250:infinite
945:universe
878:topology
874:analysis
369:powerset
338:integers
323:infinite
315:integers
75:and the
73:integers
6671:Related
6468:Diagram
6366: (
6345:Hilbert
6330:Systems
6325:Theorem
6203:of the
6148:systems
5928:Formula
5923:Grammar
5839: (
5783:General
5496:Forcing
5481:Element
5401:Monadic
5176:paradox
5117:Theorem
5053:General
4849:General
4844:Zermelo
4750:subbase
4732: (
4671:Forcing
4649:Element
4621: (
4599:Methods
4486:Pairing
4197:Sources
4189:2274520
4113:2320282
4059:2695444
4017:2896574
3951:1510438
3670:1042552
3615:2273955
3528:0542765
3520:2320581
3381:2589047
3296:Bibcode
3260:1077160
3237:Bibcode
3148:Bibcode
2996:Ω-logic
2633: ;
2232:is the
1746:Foreman
1389:ordinal
995:Foreman
661:forcing
231:History
6434:finite
6197:Skolem
6150:
6125:Theory
6093:Symbol
6083:String
6066:atomic
5943:ground
5938:closed
5933:atomic
5889:ground
5852:syntax
5748:binary
5675:domain
5592:Finite
5357:finite
5215:Logics
5174:
5122:Theory
4740:Filter
4730:Finite
4666:Family
4609:Almost
4447:global
4432:Choice
4419:Axioms
4336:&
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1750:Woodin
964:Skolem
894:Woodin
644:L, an
51:, the
6424:Model
6172:Peano
6029:Proof
5869:Arity
5798:Naive
5685:image
5617:Fuzzy
5577:Empty
5526:union
5471:Class
5112:Model
5102:Lemma
5060:Axiom
4832:Naive
4762:Fuzzy
4725:Empty
4708:types
4659:tuple
4629:Class
4623:large
4584:Union
4501:Union
4185:JSTOR
4117:S2CID
4091:arXiv
4066:(PDF)
4055:JSTOR
4035:(PDF)
3947:S2CID
3929:arXiv
3902:S2CID
3884:arXiv
3858:(PDF)
3851:(PDF)
3833:(PDF)
3822:(PDF)
3797:S2CID
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2789:, by
2296:when
2168:when
2106:when
344:(see
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6350:list
6154:list
6131:list
6120:Term
6054:rank
5948:open
5842:list
5654:Maps
5559:sets
5418:Free
5388:list
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5065:list
4745:base
4286:ISBN
4239:ISBN
3766:2021
3740:2006
3571:ISBN
3332:PMID
3265:PMID
3184:PMID
3112:ISBN
3062:link
3040:1878
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2645:and
2304:and
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2176:and
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1968:<
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348:and
290:and
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6216:of
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5696:Sur
5670:Map
5477:Ur-
5459:Set
4706:Set
4177:doi
4101:doi
4047:doi
4043:109
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1959:2
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1924:B
1920:2
1911:A
1907:2
1900:B
1894:A
1881:n
1877:n
1841:+
1838:+
1830:=
1821:2
1778:+
1761:2
1730:1
1727:+
1716:=
1700:2
1636:n
1633:+
1628:0
1619:2
1611:2
1607:=
1601:n
1598:+
1593:0
1584:2
1557:n
1554:+
1549:0
1540:2
1496:1
1493:=
1440:=
1362:2
1358:=
1353:1
1350:+
1313:2
1236:)
1233:S
1230:(
1225:P
1213:S
1209:S
1195:)
1192:S
1189:(
1184:P
1169:S
1123:)
1108:(
1048:2
1019:0
1010:2
832:+
827:1
766:0
757:2
733:=
726:0
717:2
599:1
591:=
584:0
575:2
552:0
525:1
488:0
479:2
471:|
467:S
463:|
454:0
443:S
407:0
398:2
394:=
390:|
385:R
380:|
361:S
357:S
308:T
304:S
300:T
296:S
292:T
288:S
180:1
172:=
167:1
136:1
128:=
121:0
112:2
38:.
20:)
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