4116:
924:, which shows that if ZFC + "there is an inaccessible cardinal" is consistent, then it cannot prove its own consistency. Because ZFC + "there is an inaccessible cardinal" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + "there is an inaccessible cardinal" then this latter theory would be able to prove its own consistency, which is impossible if it is consistent.
970:. Thus, this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the
919:
The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle. The proof sketched in the previous paragraph that the consistency of ZFC implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, assuming that ZFC is
915:
is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles". This shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals.
127:. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case
951:
There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal
1001:
This is a relatively weak large cardinal axiom since it amounts to saying that â is 1-inaccessible in the language of the next section, where â denotes the least ordinal not in V, i.e. the class of all ordinals in your model.
116:
The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is
1746:
1451:
1224:
is ambiguous and has at least three incompatible meanings. Many authors use it to mean a regular limit of strongly inaccessible cardinals (1-inaccessible). Other authors use it to mean that
111:
1500:
1367:
1748:
is only required to be 'elementary' with respect to a finite set of formulas. Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation
844:
is a standard model of ZFC which contains no strong inaccessibles. Thus, the consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles". Similarly, either
1919:
1880:
1847:
1598:
1393:
78:
570:
1095:
is unbounded in Îș. In this case the 0-weakly inaccessible cardinals are the regular cardinals and the 1-weakly inaccessible cardinals are the weakly inaccessible cardinals.
427:
is weakly inaccessible. Thus, ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of
2050:
610:
1978:
1564:
2005:
1777:
1625:
913:
842:
522:
405:
351:
241:
195:
154:
2097:
4194:
1946:
1808:
1665:
1645:
1532:
866:
795:
726:
685:
472:
425:
375:
321:
297:
653:
2070:
2025:
920:
consistent, no proof that the consistency of ZFC implies the consistency of ZFC + "there is an inaccessible cardinal" can be formalized in ZFC. This follows from
1688:
886:
815:
775:
755:
706:
492:
452:
264:
are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible.
253:, every other infinite cardinal number is regular or a (weak) limit. However, only a rather large cardinal number can be both and thus weakly inaccessible.
2495:
1064:
is regular). In this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinal
986:. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with
1213:
such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of
267:
The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a
3170:
573:
3253:
2394:
921:
1102:-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles. For example, denote by
208:
Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the
1925:) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a transitive model of ZFC.
927:
There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC. One such argument, presented by
3567:
260:
is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and
209:
4166:
K. Hauser, "Indescribable cardinals and elementary embeddings". Journal of
Symbolic Logic vol. 56, iss. 2 (1991), pp.439--457.
3725:
2362:
2238:
2513:
3580:
2903:
1017:-inaccessible cardinal" is ambiguous and different authors use inequivalent definitions. One definition is that a cardinal
2353:
Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory",
1693:
1398:
378:
3585:
3575:
3312:
3165:
2518:
1780:
2509:
2171:
3721:
2261:
2158:
1787:
1214:
3063:
3818:
3562:
2387:
1299:
Hyper-hyper-inaccessible cardinals and so on can be defined in similar ways, and as usual this term is ambiguous.
3123:
2816:
525:
300:
2557:
4079:
3781:
3544:
3539:
3364:
2785:
2469:
1567:
4074:
3857:
3774:
3487:
3418:
3295:
2537:
83:
4157:
A. Enayat, "Analogues of the MacDowell-Specker_theorem for set theory" (2020), p.10. Accessed 9 March 2024.
3999:
3825:
3511:
3145:
2744:
935:
of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending
1460:
1339:
4196:
ISILC Logic
Conference: Proceedings of the International Summer Institute and Logic Colloquium, Kiel 1974
3877:
3872:
3482:
3221:
3150:
2479:
2380:
613:
1885:
1859:
1813:
1577:
1372:
57:
3806:
3396:
2790:
2758:
2449:
533:
4096:
4045:
3942:
3440:
3401:
2878:
2523:
2552:
4214:
3937:
3867:
3406:
3258:
3241:
2964:
2444:
2030:
1302:
Using "weakly inaccessible" instead of "inaccessible", similar definitions can be made for "weakly
582:
1951:
1537:
3769:
3746:
3707:
3593:
3534:
3180:
3100:
2944:
2888:
2501:
2326:
2286:
1983:
1755:
1603:
1454:
891:
820:
500:
383:
329:
219:
173:
132:
46:
if it satisfies the following three conditions: it is uncountable, it is not a sum of fewer than
4059:
3786:
3764:
3731:
3624:
3470:
3455:
3428:
3379:
3263:
3198:
3023:
2989:
2984:
2858:
2689:
2666:
2270:
2135:
991:
664:
2319:"Ăber Grenzzahlen und Mengenbereiche: neue Untersuchungen ĂŒber die Grundlagen der Mengenlehre"
2075:
868:
to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of
576:
which excludes global choice, replacing limitation of size by replacement and ordinary choice;
4179:
3989:
3842:
3634:
3352:
2994:
2853:
2838:
2719:
2694:
2176:
1931:
1793:
1650:
1630:
1517:
983:
851:
780:
711:
670:
457:
410:
360:
306:
282:
268:
623:
3962:
3924:
3801:
3605:
3445:
3369:
3347:
3175:
3133:
3032:
2999:
2863:
2651:
2562:
2130:
1333:
324:
2055:
2010:
212:
holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.
8:
4091:
3982:
3967:
3947:
3904:
3791:
3741:
3667:
3612:
3549:
3342:
3337:
3285:
3053:
3042:
2714:
2614:
2542:
2533:
2529:
2464:
2459:
2251:
124:
35:
4120:
3889:
3852:
3837:
3830:
3813:
3617:
3599:
3465:
3391:
3374:
3327:
3140:
3049:
2883:
2868:
2828:
2780:
2765:
2753:
2709:
2684:
2454:
2403:
2278:
2222:
2211:
2153:, Studies in Logic and the Foundations of Mathematics, vol. 76, Elsevier Science,
1850:
1673:
979:
871:
800:
760:
740:
691:
477:
437:
3073:
2318:
4115:
4055:
3862:
3672:
3662:
3554:
3435:
3270:
3246:
3027:
3011:
2916:
2893:
2770:
2739:
2704:
2599:
2434:
2358:
2343:
2303:
2257:
2234:
2215:
2203:
2154:
1922:
4148:", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.
2355:
From
Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics
4069:
4064:
3957:
3914:
3736:
3697:
3692:
3677:
3503:
3460:
3357:
3155:
3105:
2679:
2641:
2335:
2295:
2247:
2193:
2185:
2108:
995:
202:
121:
4175:
K. J. Devlin, "Indescribability
Properties and Small Large Cardinals" (1974). In
4050:
4040:
3994:
3977:
3932:
3894:
3796:
3716:
3523:
3450:
3423:
3411:
3317:
3231:
3205:
3160:
3128:
2929:
2731:
2674:
2624:
2589:
2547:
2167:
250:
27:
24:
4035:
4014:
3972:
3952:
3847:
3702:
3300:
3290:
3280:
3275:
3209:
3083:
2959:
2848:
2843:
2821:
2422:
2114:
1316:
are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.
1313:
1240:
428:
257:
2198:
4208:
4009:
3687:
3194:
2979:
2969:
2939:
2924:
2594:
2347:
2314:
2307:
2274:
2207:
731:
2299:
1005:
158:
is strongly inaccessible). Weakly inaccessible cardinals were introduced by
3909:
3756:
3657:
3649:
3529:
3477:
3386:
3322:
3305:
3236:
3095:
2954:
2656:
2439:
1319:
975:
354:
246:
34:
if it cannot be obtained from smaller cardinals by the usual operations of
2339:
687:
does not need to be inaccessible, or even a cardinal number, in order for
4019:
3899:
3078:
3068:
3015:
2699:
2619:
2604:
2484:
2429:
2226:
2125:
4145:
2253:
The Higher
Infinite: Large Cardinals in Set Theory from Their Beginnings
931:, p. 279), is that the class of all ordinals of a particular model
2949:
2804:
2775:
2581:
2189:
20:
4101:
4004:
3057:
2974:
2934:
2898:
2834:
2646:
2636:
2609:
2372:
1257:-inaccessible. Other authors use the definition that for any ordinal
987:
4086:
3884:
3332:
3037:
2631:
2120:
1507:
990:). This axiomatic system is useful to prove for example that every
1690:
has a somewhat weaker reflection property, where the substructure
667:). It is worth pointing out that the first claim can be weakened:
3682:
2474:
946:
3226:
2572:
2417:
2279:"Sur une propriété caractéristique des nombres inaccessibles"
1856:
In this case, by the reflection property above, there exists
1574:â„ 0. On the other hand, there is not necessarily an ordinal
377:
is strongly inaccessible. Furthermore, ZF implies that the
1306:-inaccessible", "weakly hyper-inaccessible", and "weakly
1320:
Two model-theoretic characterisations of inaccessibility
1009:-inaccessible cardinals and hyper-inaccessible cardinals
572:
is one of the intended models of
Mendelson's version of
4182:
2078:
2058:
2033:
2013:
1986:
1954:
1934:
1888:
1862:
1816:
1796:
1758:
1696:
1676:
1653:
1633:
1606:
1580:
1540:
1520:
1463:
1401:
1375:
1342:
894:
874:
854:
823:
803:
783:
763:
743:
714:
694:
673:
626:
585:
536:
503:
480:
460:
440:
413:
386:
363:
332:
309:
285:
222:
176:
135:
86:
60:
1741:{\displaystyle (V_{\alpha },\in ,U\cap V_{\alpha })}
1446:{\displaystyle (V_{\alpha },\in ,U\cap V_{\alpha })}
249:) is a regular strong limit cardinal. Assuming the
4188:
2091:
2064:
2044:
2019:
1999:
1972:
1940:
1913:
1874:
1841:
1802:
1771:
1740:
1682:
1659:
1639:
1619:
1592:
1558:
1526:
1494:
1445:
1387:
1361:
907:
880:
860:
836:
809:
789:
769:
749:
720:
700:
679:
647:
604:
564:
516:
486:
466:
446:
419:
399:
369:
345:
315:
291:
235:
189:
148:
105:
72:
170:; in the latter they were referred to along with
4206:
2269:
1239:-inaccessible.) It is occasionally used to mean
1117:inaccessible cardinal, then the fixed points of
163:
4198:, Lecture Notes in Mathematics, vol. 499 (1974)
2357:, Oxford University Press, pp. 1208â1233,
2172:"GrundzĂŒge einer Theorie der geordneten Mengen"
1253:is also ambiguous. Some authors use it to mean
1124:are the 1-inaccessible cardinals. Then letting
2151:Set Theory: An Introduction to Large Cardinals
2388:
1752:can be defined, semantic truth itself (i.e.
1276:is hyper-inaccessible and for every ordinal
1144:-inaccessible cardinal, the fixed points of
947:Existence of a proper class of inaccessibles
271:, the two ideas being intimately connected.
2221:
2027:being inaccessible (in some given model of
928:
777:contains no strong inaccessible or, taking
2580:
2395:
2381:
797:to be the smallest strong inaccessible in
2197:
2166:
848:contains no weak inaccessible or, taking
274:
159:
2246:
1179:-inaccessible is a fixed point of every
2313:
1157:+1)-inaccessible cardinals (the values
939:and preserving powerset of elements of
167:
4207:
2402:
1036:is inaccessible and for every ordinal
106:{\displaystyle 2^{\alpha }<\kappa }
2376:
2352:
2148:
922:Gödel's second incompleteness theorem
16:Type of infinite number in set theory
1495:{\displaystyle (V_{\kappa },\in ,U)}
1362:{\displaystyle U\subset V_{\kappa }}
956:, there is an inaccessible cardinal
574:Von NeumannâBernaysâGödel set theory
162:, and strongly inaccessible ones by
303:with Choice (ZFC) implies that the
13:
2233:(3rd ed.), New York: Dekker,
2080:
2038:
2035:
1956:
1914:{\displaystyle (V_{\alpha },\in )}
1875:{\displaystyle \alpha <\kappa }
1842:{\displaystyle (V_{\kappa },\in )}
1593:{\displaystyle \alpha >\kappa }
1542:
1388:{\displaystyle \alpha <\kappa }
730:to be a standard model of ZF (see
224:
178:
137:
73:{\displaystyle \alpha <\kappa }
14:
4226:
1079:is regular and for every ordinal
612:is one of the intended models of
524:is one of the intended models of
4114:
1790:can be shown, which states that
1232:-inaccessible. (It can never be
1091:-weakly inaccessibles less than
565:{\displaystyle Def(V_{\kappa })}
210:generalized continuum hypothesis
1810:is inaccessible if and only if
1328:is inaccessible if and only if
1288:-hyper-inaccessibles less than
454:is a standard model of ZFC and
4169:
4160:
4151:
4138:
2141:
1908:
1889:
1836:
1817:
1788:Zermelo's categoricity theorem
1735:
1697:
1489:
1464:
1440:
1402:
982:: every set is contained in a
642:
636:
559:
546:
205:“limit numbers”).
164:SierpiĆski & Tarski (1930)
38:. More precisely, a cardinal
1:
4131:
4075:History of mathematical logic
2045:{\displaystyle \mathrm {ZF} }
605:{\displaystyle V_{\kappa +1}}
4000:Primitive recursive function
1973:{\displaystyle \Pi _{1}^{1}}
1559:{\displaystyle \Pi _{n}^{0}}
1502:. (In fact, the set of such
7:
2102:
2000:{\displaystyle V_{\kappa }}
1772:{\displaystyle \vDash _{V}}
1620:{\displaystyle V_{\kappa }}
908:{\displaystyle L_{\kappa }}
837:{\displaystyle V_{\kappa }}
526:ZermeloâFraenkel set theory
517:{\displaystyle V_{\kappa }}
407:is a model of ZFC whenever
400:{\displaystyle L_{\kappa }}
346:{\displaystyle V_{\kappa }}
301:ZermeloâFraenkel set theory
236:{\displaystyle \aleph _{0}}
190:{\displaystyle \aleph _{0}}
149:{\displaystyle \aleph _{0}}
10:
4231:
3064:SchröderâBernstein theorem
2791:Monadic predicate calculus
2450:Foundations of mathematics
2256:(2nd ed.), Springer,
2231:Introduction to set theory
1670:It is provable in ZF that
1667:th inaccessible cardinal.
1627:, and if this holds, then
1336:property: for all subsets
960:which is strictly larger,
757:is a model of ZFC. Either
4110:
4097:Philosophy of mathematics
4046:Automated theorem proving
4028:
3923:
3755:
3648:
3500:
3217:
3193:
3171:Von NeumannâBernaysâGödel
3116:
3010:
2914:
2812:
2803:
2730:
2665:
2571:
2493:
2410:
1056:(and thus of cardinality
1048:-inaccessibles less than
2092:{\displaystyle \Pi _{1}}
1921:is a standard model of (
929:HrbĂĄÄek & Jech (1999
4189:{\displaystyle \vDash }
3747:Self-verifying theories
3568:Tarski's axiomatization
2519:Tarski's undefinability
2514:incompleteness theorems
2351:. English translation:
2327:Fundamenta Mathematicae
2300:10.4064/fm-15-1-292-300
2287:Fundamenta Mathematicae
1941:{\displaystyle \kappa }
1803:{\displaystyle \kappa }
1660:{\displaystyle \kappa }
1640:{\displaystyle \kappa }
1527:{\displaystyle \kappa }
1455:elementary substructure
1175:is a limit ordinal, an
861:{\displaystyle \kappa }
790:{\displaystyle \kappa }
721:{\displaystyle \kappa }
680:{\displaystyle \kappa }
614:MorseâKelley set theory
467:{\displaystyle \kappa }
420:{\displaystyle \kappa }
370:{\displaystyle \kappa }
316:{\displaystyle \kappa }
292:{\displaystyle \kappa }
50:cardinals smaller than
4190:
4146:Zermelo and Set Theory
4121:Mathematics portal
3732:Proof of impossibility
3380:propositional variable
2690:Propositional calculus
2136:Constructible universe
2093:
2066:
2046:
2021:
2001:
1974:
1942:
1915:
1876:
1843:
1804:
1773:
1742:
1684:
1661:
1641:
1621:
1594:
1560:
1528:
1496:
1447:
1389:
1363:
1310:-hyper-inaccessible".
1215:large cardinal numbers
909:
882:
862:
838:
811:
791:
771:
751:
722:
702:
681:
665:constructible universe
659:-definable subsets of
649:
648:{\displaystyle Def(X)}
606:
566:
518:
488:
474:is an inaccessible in
468:
448:
421:
401:
371:
347:
317:
299:is a cardinal number.
293:
275:Models and consistency
237:
191:
150:
107:
74:
4191:
3990:Kolmogorov complexity
3943:Computably enumerable
3843:Model complete theory
3635:Principia Mathematica
2695:Propositional formula
2524:BanachâTarski paradox
2340:10.4064/fm-16-1-29-47
2177:Mathematische Annalen
2149:Drake, F. R. (1974),
2094:
2067:
2047:
2022:
2002:
1975:
1943:
1916:
1877:
1844:
1805:
1774:
1743:
1685:
1662:
1642:
1622:
1595:
1561:
1529:
1497:
1448:
1390:
1364:
984:Grothendieck universe
910:
883:
863:
839:
812:
792:
772:
752:
723:
703:
682:
650:
607:
567:
519:
489:
469:
449:
422:
402:
372:
348:
318:
294:
269:Grothendieck universe
238:
192:
151:
108:
75:
44:strongly inaccessible
4180:
3938:ChurchâTuring thesis
3925:Computability theory
3134:continuum hypothesis
2652:Square of opposition
2510:Gödel's completeness
2131:Von Neumann universe
2076:
2065:{\displaystyle \pi }
2056:
2031:
2020:{\displaystyle \pi }
2011:
1984:
1952:
1932:
1886:
1860:
1814:
1794:
1786:Secondly, under ZFC
1756:
1694:
1674:
1651:
1631:
1604:
1578:
1538:
1518:
1461:
1399:
1373:
1340:
1324:Firstly, a cardinal
1073:-weakly inaccessible
892:
872:
852:
821:
801:
781:
761:
741:
712:
692:
671:
624:
583:
534:
501:
478:
458:
438:
411:
384:
361:
330:
325:Von Neumann universe
307:
283:
220:
174:
133:
84:
58:
4092:Mathematical object
3983:P versus NP problem
3948:Computable function
3742:Reverse mathematics
3668:Logical consequence
3545:primitive recursive
3540:elementary function
3313:Free/bound variable
3166:TarskiâGrothendieck
2685:Logical connectives
2615:Logical equivalence
2465:Logical consequence
2117:, a stronger notion
2007:, while a cardinal
1969:
1928:Inaccessibility of
1555:
1270:-hyper-inaccessible
1251:-hyper-inaccessible
994:has an appropriate
125:weak limit cardinal
118:weakly inaccessible
36:cardinal arithmetic
4186:
3890:Transfer principle
3853:Semantics of logic
3838:Categorical theory
3814:Non-standard model
3328:Logical connective
2455:Information theory
2404:Mathematical logic
2271:SierpiĆski, WacĆaw
2199:10338.dmlcz/100813
2190:10.1007/BF01451165
2089:
2062:
2042:
2017:
1997:
1970:
1955:
1938:
1911:
1872:
1839:
1800:
1769:
1738:
1680:
1657:
1637:
1617:
1590:
1556:
1541:
1524:
1492:
1443:
1385:
1359:
1332:has the following
1222:hyper-inaccessible
1028:, for any ordinal
905:
878:
858:
834:
807:
787:
767:
747:
718:
698:
677:
645:
602:
562:
514:
484:
464:
444:
417:
397:
367:
343:
313:
289:
233:
187:
146:
103:
70:
4128:
4127:
4060:Abstract category
3863:Theories of truth
3673:Rule of inference
3663:Natural deduction
3644:
3643:
3189:
3188:
2894:Cartesian product
2799:
2798:
2705:Many-valued logic
2680:Boolean functions
2563:Russell's paradox
2538:diagonal argument
2435:First-order logic
2364:978-0-19-853271-2
2248:Kanamori, Akihiro
2240:978-0-8247-7915-3
2111:, a weaker notion
1779:) cannot, due to
1683:{\displaystyle V}
881:{\displaystyle V}
810:{\displaystyle V}
770:{\displaystyle V}
750:{\displaystyle V}
701:{\displaystyle V}
487:{\displaystyle V}
447:{\displaystyle V}
4222:
4199:
4195:
4193:
4192:
4187:
4173:
4167:
4164:
4158:
4155:
4149:
4142:
4119:
4118:
4070:History of logic
4065:Category of sets
3958:Decision problem
3737:Ordinal analysis
3678:Sequent calculus
3576:Boolean algebras
3516:
3515:
3490:
3461:logical/constant
3215:
3214:
3201:
3124:ZermeloâFraenkel
2875:Set operations:
2810:
2809:
2747:
2578:
2577:
2558:LöwenheimâSkolem
2445:Formal semantics
2397:
2390:
2383:
2374:
2373:
2367:
2350:
2323:
2310:
2283:
2266:
2243:
2218:
2201:
2168:Hausdorff, Felix
2163:
2109:Worldly cardinal
2098:
2096:
2095:
2090:
2088:
2087:
2071:
2069:
2068:
2063:
2051:
2049:
2048:
2043:
2041:
2026:
2024:
2023:
2018:
2006:
2004:
2003:
1998:
1996:
1995:
1979:
1977:
1976:
1971:
1968:
1963:
1947:
1945:
1944:
1939:
1920:
1918:
1917:
1912:
1901:
1900:
1881:
1879:
1878:
1873:
1848:
1846:
1845:
1840:
1829:
1828:
1809:
1807:
1806:
1801:
1781:Tarski's theorem
1778:
1776:
1775:
1770:
1768:
1767:
1751:
1747:
1745:
1744:
1739:
1734:
1733:
1709:
1708:
1689:
1687:
1686:
1681:
1666:
1664:
1663:
1658:
1646:
1644:
1643:
1638:
1626:
1624:
1623:
1618:
1616:
1615:
1599:
1597:
1596:
1591:
1565:
1563:
1562:
1557:
1554:
1549:
1533:
1531:
1530:
1525:
1513:
1508:closed unbounded
1501:
1499:
1498:
1493:
1476:
1475:
1452:
1450:
1449:
1444:
1439:
1438:
1414:
1413:
1394:
1392:
1391:
1386:
1368:
1366:
1365:
1360:
1358:
1357:
1331:
1327:
1295:
1292:is unbounded in
1291:
1275:
1264:
1238:
1231:
1227:
1094:
1078:
1067:
1063:
1059:
1055:
1052:is unbounded in
1051:
1035:
1020:
996:Yoneda embedding
969:
959:
914:
912:
911:
906:
904:
903:
887:
885:
884:
879:
867:
865:
864:
859:
847:
843:
841:
840:
835:
833:
832:
816:
814:
813:
808:
796:
794:
793:
788:
776:
774:
773:
768:
756:
754:
753:
748:
729:
727:
725:
724:
719:
707:
705:
704:
699:
686:
684:
683:
678:
654:
652:
651:
646:
611:
609:
608:
603:
601:
600:
571:
569:
568:
563:
558:
557:
523:
521:
520:
515:
513:
512:
493:
491:
490:
485:
473:
471:
470:
465:
453:
451:
450:
445:
426:
424:
423:
418:
406:
404:
403:
398:
396:
395:
376:
374:
373:
368:
357:of ZFC whenever
352:
350:
349:
344:
342:
341:
323:th level of the
322:
320:
319:
314:
298:
296:
295:
290:
263:
244:
242:
240:
239:
234:
232:
231:
196:
194:
193:
188:
186:
185:
160:Hausdorff (1908)
157:
155:
153:
152:
147:
145:
144:
112:
110:
109:
104:
96:
95:
79:
77:
76:
71:
53:
49:
41:
4230:
4229:
4225:
4224:
4223:
4221:
4220:
4219:
4215:Large cardinals
4205:
4204:
4203:
4202:
4181:
4178:
4177:
4174:
4170:
4165:
4161:
4156:
4152:
4143:
4139:
4134:
4129:
4124:
4113:
4106:
4051:Category theory
4041:Algebraic logic
4024:
3995:Lambda calculus
3933:Church encoding
3919:
3895:Truth predicate
3751:
3717:Complete theory
3640:
3509:
3505:
3501:
3496:
3488:
3208: and
3204:
3199:
3185:
3161:New Foundations
3129:axiom of choice
3112:
3074:Gödel numbering
3014: and
3006:
2910:
2795:
2745:
2726:
2675:Boolean algebra
2661:
2625:Equiconsistency
2590:Classical logic
2567:
2548:Halting problem
2536: and
2512: and
2500: and
2499:
2494:Theorems (
2489:
2406:
2401:
2371:
2365:
2321:
2281:
2264:
2241:
2161:
2144:
2105:
2083:
2079:
2077:
2074:
2073:
2057:
2054:
2053:
2034:
2032:
2029:
2028:
2012:
2009:
2008:
1991:
1987:
1985:
1982:
1981:
1964:
1959:
1953:
1950:
1949:
1933:
1930:
1929:
1896:
1892:
1887:
1884:
1883:
1861:
1858:
1857:
1824:
1820:
1815:
1812:
1811:
1795:
1792:
1791:
1763:
1759:
1757:
1754:
1753:
1749:
1729:
1725:
1704:
1700:
1695:
1692:
1691:
1675:
1672:
1671:
1652:
1649:
1648:
1632:
1629:
1628:
1611:
1607:
1605:
1602:
1601:
1579:
1576:
1575:
1550:
1545:
1539:
1536:
1535:
1519:
1516:
1515:
1511:
1471:
1467:
1462:
1459:
1458:
1434:
1430:
1409:
1405:
1400:
1397:
1396:
1374:
1371:
1370:
1369:, there exists
1353:
1349:
1341:
1338:
1337:
1329:
1325:
1322:
1314:Mahlo cardinals
1293:
1289:
1273:
1272:if and only if
1262:
1236:
1233:
1229:
1225:
1204:
1187:
1166:
1152:
1132:
1123:
1108:
1092:
1076:
1065:
1061:
1057:
1053:
1049:
1033:
1018:
1011:
961:
957:
949:
899:
895:
893:
890:
889:
873:
870:
869:
853:
850:
849:
845:
828:
824:
822:
819:
818:
802:
799:
798:
782:
779:
778:
762:
759:
758:
742:
739:
738:
728:
713:
710:
709:
693:
690:
689:
688:
672:
669:
668:
658:
655:is the set of Î
625:
622:
621:
590:
586:
584:
581:
580:
553:
549:
535:
532:
531:
508:
504:
502:
499:
498:
479:
476:
475:
459:
456:
455:
439:
436:
435:
412:
409:
408:
391:
387:
385:
382:
381:
362:
359:
358:
337:
333:
331:
328:
327:
308:
305:
304:
284:
281:
280:
277:
261:
251:axiom of choice
227:
223:
221:
218:
217:
215:
181:
177:
175:
172:
171:
140:
136:
134:
131:
130:
128:
91:
87:
85:
82:
81:
59:
56:
55:
51:
47:
39:
17:
12:
11:
5:
4228:
4218:
4217:
4201:
4200:
4185:
4168:
4159:
4150:
4144:A. Kanamori, "
4136:
4135:
4133:
4130:
4126:
4125:
4111:
4108:
4107:
4105:
4104:
4099:
4094:
4089:
4084:
4083:
4082:
4072:
4067:
4062:
4053:
4048:
4043:
4038:
4036:Abstract logic
4032:
4030:
4026:
4025:
4023:
4022:
4017:
4015:Turing machine
4012:
4007:
4002:
3997:
3992:
3987:
3986:
3985:
3980:
3975:
3970:
3965:
3955:
3953:Computable set
3950:
3945:
3940:
3935:
3929:
3927:
3921:
3920:
3918:
3917:
3912:
3907:
3902:
3897:
3892:
3887:
3882:
3881:
3880:
3875:
3870:
3860:
3855:
3850:
3848:Satisfiability
3845:
3840:
3835:
3834:
3833:
3823:
3822:
3821:
3811:
3810:
3809:
3804:
3799:
3794:
3789:
3779:
3778:
3777:
3772:
3765:Interpretation
3761:
3759:
3753:
3752:
3750:
3749:
3744:
3739:
3734:
3729:
3719:
3714:
3713:
3712:
3711:
3710:
3700:
3695:
3685:
3680:
3675:
3670:
3665:
3660:
3654:
3652:
3646:
3645:
3642:
3641:
3639:
3638:
3630:
3629:
3628:
3627:
3622:
3621:
3620:
3615:
3610:
3590:
3589:
3588:
3586:minimal axioms
3583:
3572:
3571:
3570:
3559:
3558:
3557:
3552:
3547:
3542:
3537:
3532:
3519:
3517:
3498:
3497:
3495:
3494:
3493:
3492:
3480:
3475:
3474:
3473:
3468:
3463:
3458:
3448:
3443:
3438:
3433:
3432:
3431:
3426:
3416:
3415:
3414:
3409:
3404:
3399:
3389:
3384:
3383:
3382:
3377:
3372:
3362:
3361:
3360:
3355:
3350:
3345:
3340:
3335:
3325:
3320:
3315:
3310:
3309:
3308:
3303:
3298:
3293:
3283:
3278:
3276:Formation rule
3273:
3268:
3267:
3266:
3261:
3251:
3250:
3249:
3239:
3234:
3229:
3224:
3218:
3212:
3195:Formal systems
3191:
3190:
3187:
3186:
3184:
3183:
3178:
3173:
3168:
3163:
3158:
3153:
3148:
3143:
3138:
3137:
3136:
3131:
3120:
3118:
3114:
3113:
3111:
3110:
3109:
3108:
3098:
3093:
3092:
3091:
3084:Large cardinal
3081:
3076:
3071:
3066:
3061:
3047:
3046:
3045:
3040:
3035:
3020:
3018:
3008:
3007:
3005:
3004:
3003:
3002:
2997:
2992:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2947:
2942:
2937:
2932:
2927:
2921:
2919:
2912:
2911:
2909:
2908:
2907:
2906:
2901:
2896:
2891:
2886:
2881:
2873:
2872:
2871:
2866:
2856:
2851:
2849:Extensionality
2846:
2844:Ordinal number
2841:
2831:
2826:
2825:
2824:
2813:
2807:
2801:
2800:
2797:
2796:
2794:
2793:
2788:
2783:
2778:
2773:
2768:
2763:
2762:
2761:
2751:
2750:
2749:
2736:
2734:
2728:
2727:
2725:
2724:
2723:
2722:
2717:
2712:
2702:
2697:
2692:
2687:
2682:
2677:
2671:
2669:
2663:
2662:
2660:
2659:
2654:
2649:
2644:
2639:
2634:
2629:
2628:
2627:
2617:
2612:
2607:
2602:
2597:
2592:
2586:
2584:
2575:
2569:
2568:
2566:
2565:
2560:
2555:
2550:
2545:
2540:
2528:Cantor's
2526:
2521:
2516:
2506:
2504:
2491:
2490:
2488:
2487:
2482:
2477:
2472:
2467:
2462:
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2426:
2425:
2414:
2412:
2408:
2407:
2400:
2399:
2392:
2385:
2377:
2370:
2369:
2363:
2315:Zermelo, Ernst
2311:
2275:Tarski, Alfred
2267:
2262:
2244:
2239:
2223:HrbĂĄÄek, Karel
2219:
2184:(4): 435â505,
2164:
2159:
2145:
2143:
2140:
2139:
2138:
2133:
2128:
2123:
2118:
2115:Mahlo cardinal
2112:
2104:
2101:
2086:
2082:
2061:
2040:
2037:
2016:
1994:
1990:
1980:property over
1967:
1962:
1958:
1937:
1910:
1907:
1904:
1899:
1895:
1891:
1871:
1868:
1865:
1849:is a model of
1838:
1835:
1832:
1827:
1823:
1819:
1799:
1766:
1762:
1737:
1732:
1728:
1724:
1721:
1718:
1715:
1712:
1707:
1703:
1699:
1679:
1656:
1636:
1614:
1610:
1589:
1586:
1583:
1553:
1548:
1544:
1523:
1514:.) Therefore,
1491:
1488:
1485:
1482:
1479:
1474:
1470:
1466:
1442:
1437:
1433:
1429:
1426:
1423:
1420:
1417:
1412:
1408:
1404:
1384:
1381:
1378:
1356:
1352:
1348:
1345:
1321:
1318:
1241:Mahlo cardinal
1234:
1200:
1183:
1161:
1148:
1128:
1121:
1106:
1010:
1004:
972:universe axiom
948:
945:
902:
898:
877:
857:
831:
827:
806:
786:
766:
746:
717:
708:
697:
676:
656:
644:
641:
638:
635:
632:
629:
618:
617:
599:
596:
593:
589:
577:
561:
556:
552:
548:
545:
542:
539:
529:
511:
507:
483:
463:
443:
429:large cardinal
416:
394:
390:
379:Gödel universe
366:
340:
336:
312:
288:
276:
273:
230:
226:
184:
180:
168:Zermelo (1930)
143:
139:
102:
99:
94:
90:
69:
66:
63:
15:
9:
6:
4:
3:
2:
4227:
4216:
4213:
4212:
4210:
4197:
4183:
4172:
4163:
4154:
4147:
4141:
4137:
4123:
4122:
4117:
4109:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4081:
4078:
4077:
4076:
4073:
4071:
4068:
4066:
4063:
4061:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4033:
4031:
4027:
4021:
4018:
4016:
4013:
4011:
4010:Recursive set
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3960:
3959:
3956:
3954:
3951:
3949:
3946:
3944:
3941:
3939:
3936:
3934:
3931:
3930:
3928:
3926:
3922:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3879:
3876:
3874:
3871:
3869:
3866:
3865:
3864:
3861:
3859:
3856:
3854:
3851:
3849:
3846:
3844:
3841:
3839:
3836:
3832:
3829:
3828:
3827:
3824:
3820:
3819:of arithmetic
3817:
3816:
3815:
3812:
3808:
3805:
3803:
3800:
3798:
3795:
3793:
3790:
3788:
3785:
3784:
3783:
3780:
3776:
3773:
3771:
3768:
3767:
3766:
3763:
3762:
3760:
3758:
3754:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3727:
3726:from ZFC
3723:
3720:
3718:
3715:
3709:
3706:
3705:
3704:
3701:
3699:
3696:
3694:
3691:
3690:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3664:
3661:
3659:
3656:
3655:
3653:
3651:
3647:
3637:
3636:
3632:
3631:
3626:
3625:non-Euclidean
3623:
3619:
3616:
3614:
3611:
3609:
3608:
3604:
3603:
3601:
3598:
3597:
3595:
3591:
3587:
3584:
3582:
3579:
3578:
3577:
3573:
3569:
3566:
3565:
3564:
3560:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3531:
3528:
3527:
3525:
3521:
3520:
3518:
3513:
3507:
3502:Example
3499:
3491:
3486:
3485:
3484:
3481:
3479:
3476:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3454:
3453:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3430:
3427:
3425:
3422:
3421:
3420:
3417:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3394:
3393:
3390:
3388:
3385:
3381:
3378:
3376:
3373:
3371:
3368:
3367:
3366:
3363:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3330:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3288:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3265:
3262:
3260:
3259:by definition
3257:
3256:
3255:
3252:
3248:
3245:
3244:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3219:
3216:
3213:
3211:
3207:
3202:
3196:
3192:
3182:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3162:
3159:
3157:
3154:
3152:
3149:
3147:
3146:KripkeâPlatek
3144:
3142:
3139:
3135:
3132:
3130:
3127:
3126:
3125:
3122:
3121:
3119:
3115:
3107:
3104:
3103:
3102:
3099:
3097:
3094:
3090:
3087:
3086:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3059:
3055:
3051:
3048:
3044:
3041:
3039:
3036:
3034:
3031:
3030:
3029:
3025:
3022:
3021:
3019:
3017:
3013:
3009:
3001:
2998:
2996:
2993:
2991:
2990:constructible
2988:
2987:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2928:
2926:
2923:
2922:
2920:
2918:
2913:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2876:
2874:
2870:
2867:
2865:
2862:
2861:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2836:
2832:
2830:
2827:
2823:
2820:
2819:
2818:
2815:
2814:
2811:
2808:
2806:
2802:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2760:
2757:
2756:
2755:
2752:
2748:
2743:
2742:
2741:
2738:
2737:
2735:
2733:
2729:
2721:
2718:
2716:
2713:
2711:
2708:
2707:
2706:
2703:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2672:
2670:
2668:
2667:Propositional
2664:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2626:
2623:
2622:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2595:Logical truth
2593:
2591:
2588:
2587:
2585:
2583:
2579:
2576:
2574:
2570:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2535:
2531:
2527:
2525:
2522:
2520:
2517:
2515:
2511:
2508:
2507:
2505:
2503:
2497:
2492:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2424:
2421:
2420:
2419:
2416:
2415:
2413:
2409:
2405:
2398:
2393:
2391:
2386:
2384:
2379:
2378:
2375:
2366:
2360:
2356:
2349:
2345:
2341:
2337:
2333:
2329:
2328:
2320:
2316:
2312:
2309:
2305:
2301:
2297:
2293:
2289:
2288:
2280:
2276:
2272:
2268:
2265:
2263:3-540-00384-3
2259:
2255:
2254:
2249:
2245:
2242:
2236:
2232:
2228:
2224:
2220:
2217:
2213:
2209:
2205:
2200:
2195:
2191:
2187:
2183:
2179:
2178:
2173:
2169:
2165:
2162:
2160:0-444-10535-2
2156:
2152:
2147:
2146:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2116:
2113:
2110:
2107:
2106:
2100:
2084:
2059:
2014:
1992:
1988:
1965:
1960:
1935:
1926:
1924:
1905:
1902:
1897:
1893:
1869:
1866:
1863:
1854:
1852:
1833:
1830:
1825:
1821:
1797:
1789:
1784:
1782:
1764:
1760:
1730:
1726:
1722:
1719:
1716:
1713:
1710:
1705:
1701:
1677:
1668:
1654:
1634:
1612:
1608:
1587:
1584:
1581:
1573:
1569:
1568:indescribable
1551:
1546:
1521:
1509:
1505:
1486:
1483:
1480:
1477:
1472:
1468:
1456:
1435:
1431:
1427:
1424:
1421:
1418:
1415:
1410:
1406:
1382:
1379:
1376:
1354:
1350:
1346:
1343:
1335:
1317:
1315:
1311:
1309:
1305:
1300:
1297:
1287:
1284:, the set of
1283:
1279:
1271:
1269:
1261:, a cardinal
1260:
1256:
1252:
1250:
1244:
1242:
1223:
1218:
1216:
1212:
1208:
1203:
1199:
1195:
1191:
1186:
1182:
1178:
1174:
1170:
1164:
1160:
1156:
1151:
1147:
1143:
1140:
1136:
1131:
1127:
1120:
1116:
1112:
1105:
1101:
1096:
1090:
1087:, the set of
1086:
1082:
1074:
1072:
1047:
1044:, the set of
1043:
1039:
1031:
1027:
1026:-inaccessible
1025:
1016:
1008:
1003:
999:
997:
993:
989:
985:
981:
977:
973:
968:
964:
955:
944:
942:
938:
934:
930:
925:
923:
917:
900:
896:
875:
855:
829:
825:
804:
784:
764:
744:
735:
733:
715:
695:
674:
666:
662:
639:
633:
630:
627:
615:
597:
594:
591:
587:
578:
575:
554:
550:
543:
540:
537:
530:
527:
509:
505:
497:
496:
495:
481:
461:
441:
432:
430:
414:
392:
388:
380:
364:
356:
338:
334:
326:
310:
302:
286:
279:Suppose that
272:
270:
265:
259:
254:
252:
248:
228:
213:
211:
206:
204:
200:
182:
169:
165:
161:
141:
126:
123:
119:
114:
100:
97:
92:
88:
67:
64:
61:
45:
37:
33:
29:
26:
22:
4176:
4171:
4162:
4153:
4140:
4112:
3910:Ultraproduct
3757:Model theory
3722:Independence
3658:Formal proof
3650:Proof theory
3633:
3606:
3563:real numbers
3535:second-order
3446:Substitution
3323:Metalanguage
3264:conservative
3237:Axiom schema
3181:Constructive
3151:MorseâKelley
3117:Set theories
3096:Aleph number
3089:inaccessible
3088:
2995:Grothendieck
2879:intersection
2766:Higher-order
2754:Second-order
2700:Truth tables
2657:Venn diagram
2440:Formal proof
2354:
2331:
2325:
2291:
2285:
2252:
2230:
2227:Jech, Thomas
2181:
2175:
2150:
1927:
1855:
1851:second order
1785:
1669:
1647:must be the
1571:
1503:
1323:
1312:
1307:
1303:
1301:
1298:
1285:
1281:
1277:
1267:
1266:
1258:
1254:
1248:
1247:
1245:
1221:
1219:
1210:
1206:
1201:
1197:
1193:
1189:
1184:
1180:
1176:
1172:
1168:
1162:
1158:
1154:
1149:
1145:
1141:
1138:
1134:
1129:
1125:
1118:
1114:
1110:
1103:
1099:
1097:
1088:
1084:
1080:
1070:
1069:
1045:
1041:
1037:
1029:
1023:
1022:
1014:
1012:
1006:
1000:
976:Grothendieck
971:
966:
962:
953:
950:
940:
936:
932:
926:
918:
736:
660:
619:
433:
278:
266:
255:
214:
207:
198:
117:
115:
43:
32:inaccessible
31:
18:
4020:Type theory
3968:undecidable
3900:Truth value
3787:equivalence
3466:non-logical
3079:Enumeration
3069:Isomorphism
3016:cardinality
3000:Von Neumann
2965:Ultrafilter
2930:Uncountable
2864:equivalence
2781:Quantifiers
2771:Fixed-point
2740:First-order
2620:Consistency
2605:Proposition
2582:Traditional
2553:Lindström's
2543:Compactness
2485:Type theory
2430:Cardinality
2294:: 292â300,
2142:Works cited
2126:Inner model
2052:containing
1923:first order
1196:(the value
199:Grenzzahlen
120:if it is a
25:uncountable
4132:References
3831:elementary
3524:arithmetic
3392:Quantifier
3370:functional
3242:Expression
2960:Transitive
2904:identities
2889:complement
2822:hereditary
2805:Set theory
1882:such that
1600:such that
1395:such that
1334:reflection
1068:is called
1021:is called
1013:The term "
988:urelements
247:aleph-null
21:set theory
4184:⊨
4102:Supertask
4005:Recursion
3963:decidable
3797:saturated
3775:of models
3698:deductive
3693:axiomatic
3613:Hilbert's
3600:Euclidean
3581:canonical
3504:axiomatic
3436:Signature
3365:Predicate
3254:Extension
3176:Ackermann
3101:Operation
2980:Universal
2970:Recursive
2945:Singleton
2940:Inhabited
2925:Countable
2915:Types of
2899:power set
2869:partition
2786:Predicate
2732:Predicate
2647:Syllogism
2637:Soundness
2610:Inference
2600:Tautology
2502:paradoxes
2348:0016-2736
2334:: 29â47,
2308:0016-2736
2216:119648544
2208:0025-5831
2081:Π
2060:π
2015:π
1993:κ
1957:Π
1936:κ
1906:∈
1898:α
1870:κ
1864:α
1834:∈
1826:κ
1798:κ
1761:⊨
1731:α
1723:∩
1714:∈
1706:α
1655:κ
1635:κ
1613:κ
1588:κ
1582:α
1543:Π
1522:κ
1481:∈
1473:κ
1436:α
1428:∩
1419:∈
1411:α
1383:κ
1377:α
1355:κ
1347:⊂
1246:The term
1220:The term
1209:) is the
1153:are the (
1137:) be the
901:κ
856:κ
830:κ
785:κ
716:κ
675:κ
592:κ
555:κ
510:κ
462:κ
415:κ
393:κ
365:κ
339:κ
311:κ
287:κ
225:ℵ
179:ℵ
138:ℵ
101:κ
93:α
68:κ
62:α
4209:Category
4087:Logicism
4080:timeline
4056:Concrete
3915:Validity
3885:T-schema
3878:Kripke's
3873:Tarski's
3868:semantic
3858:Strength
3807:submodel
3802:spectrum
3770:function
3618:Tarski's
3607:Elements
3594:geometry
3550:Robinson
3471:variable
3456:function
3429:spectrum
3419:Sentence
3375:variable
3318:Language
3271:Relation
3232:Automata
3222:Alphabet
3206:language
3060:-jection
3038:codomain
3024:Function
2985:Universe
2955:Infinite
2859:Relation
2642:Validity
2632:Argument
2530:theorem,
2317:(1930),
2277:(1930),
2250:(2003),
2229:(1999),
2170:(1908),
2121:Club set
2103:See also
1570:for all
1060:, since
992:category
737:Suppose
80:implies
28:cardinal
4029:Related
3826:Diagram
3724: (
3703:Hilbert
3688:Systems
3683:Theorem
3561:of the
3506:systems
3286:Formula
3281:Grammar
3197: (
3141:General
2854:Forcing
2839:Element
2759:Monadic
2534:paradox
2475:Theorem
2411:General
1171:)). If
980:Verdier
888:, then
494:, then
258:ordinal
243:
216:
203:English
156:
129:
122:regular
3792:finite
3555:Skolem
3508:
3483:Theory
3451:Symbol
3441:String
3424:atomic
3301:ground
3296:closed
3291:atomic
3247:ground
3210:syntax
3106:binary
3033:domain
2950:Finite
2715:finite
2573:Logics
2532:
2480:Theory
2361:
2346:
2306:
2260:
2237:
2214:
2206:
2157:
1453:is an
1113:) the
620:Here,
262:ω
54:, and
52:κ
48:κ
40:κ
3782:Model
3530:Peano
3387:Proof
3227:Arity
3156:Naive
3043:image
2975:Fuzzy
2935:Empty
2884:union
2829:Class
2470:Model
2460:Lemma
2418:Axiom
2322:(PDF)
2282:(PDF)
2212:S2CID
2072:) is
1948:is a
1853:ZFC.
1280:<
1192:<
1083:<
1040:<
1032:, if
965:<
732:below
663:(see
355:model
353:is a
23:, an
3905:Type
3708:list
3512:list
3489:list
3478:Term
3412:rank
3306:open
3200:list
3012:Maps
2917:sets
2776:Free
2746:list
2496:list
2423:list
2359:ISBN
2344:ISSN
2304:ISSN
2258:ISBN
2235:ISBN
2204:ISSN
2155:ISBN
1867:<
1585:>
1380:<
1188:for
1098:The
978:and
579:and
166:and
98:<
65:<
3592:of
3574:of
3522:of
3054:Sur
3028:Map
2835:Ur-
2817:Set
2336:doi
2296:doi
2194:hdl
2186:doi
1534:is
1510:in
1506:is
1457:of
1265:is
1228:is
1075:if
974:of
734:).
434:If
256:An
197:as
42:is
30:is
19:In
4211::
3978:NP
3602::
3596::
3526::
3203:),
3058:Bi
3050:In
2342:,
2332:16
2330:,
2324:,
2302:,
2292:15
2290:,
2284:,
2273:;
2225:;
2210:,
2202:,
2192:,
2182:65
2180:,
2174:,
2099:.
1783:.
1296:.
1243:.
1237:+1
1217:.
1165:+1
998:.
943:.
817:,
431:.
113:.
4058:/
3973:P
3728:)
3514:)
3510:(
3407:â
3402:!
3397:â
3358:=
3353:â
3348:â
3343:â§
3338:âš
3333:ÂŹ
3056:/
3052:/
3026:/
2837:)
2833:(
2720:â
2710:3
2498:)
2396:e
2389:t
2382:v
2368:.
2338::
2298::
2196::
2188::
2085:1
2039:F
2036:Z
1989:V
1966:1
1961:1
1909:)
1903:,
1894:V
1890:(
1837:)
1831:,
1822:V
1818:(
1765:V
1750:â§
1736:)
1727:V
1720:U
1717:,
1711:,
1702:V
1698:(
1678:V
1609:V
1572:n
1566:-
1552:0
1547:n
1512:Îș
1504:α
1490:)
1487:U
1484:,
1478:,
1469:V
1465:(
1441:)
1432:V
1425:U
1422:,
1416:,
1407:V
1403:(
1351:V
1344:U
1330:Îș
1326:Îș
1308:α
1304:α
1294:Îș
1290:Îș
1286:ÎČ
1282:α
1278:ÎČ
1274:Îș
1268:α
1263:Îș
1259:α
1255:α
1249:α
1235:Îș
1230:Îș
1226:Îș
1211:λ
1207:λ
1205:(
1202:α
1198:Ï
1194:α
1190:ÎČ
1185:ÎČ
1181:Ï
1177:α
1173:α
1169:λ
1167:(
1163:ÎČ
1159:Ï
1155:ÎČ
1150:ÎČ
1146:Ï
1142:ÎČ
1139:λ
1135:λ
1133:(
1130:ÎČ
1126:Ï
1122:0
1119:Ï
1115:λ
1111:λ
1109:(
1107:0
1104:Ï
1100:α
1093:Îș
1089:ÎČ
1085:α
1081:ÎČ
1077:Îș
1071:α
1066:Îș
1062:Îș
1058:Îș
1054:Îș
1050:Îș
1046:ÎČ
1042:α
1038:ÎČ
1034:Îș
1030:α
1024:α
1019:Îș
1015:α
1007:α
967:Îș
963:Ό
958:Îș
954:Ό
941:M
937:M
933:M
897:L
876:V
846:V
826:V
805:V
765:V
745:V
696:V
661:X
657:0
643:)
640:X
637:(
634:f
631:e
628:D
616:.
598:1
595:+
588:V
560:)
551:V
547:(
544:f
541:e
538:D
528:;
506:V
482:V
442:V
389:L
335:V
245:(
229:0
201:(
183:0
142:0
89:2
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