4671:
2330:
1381:
1188:
1194:
1794:
1069:
2143:, note at end of §2.3 on page 27: "Those who do not allow relations on an empty universe consider (∃x)x=x and its consequences as theses; we, however, do not share this abhorrence, with so little logical ground, of a vacuum."
512:, choice, and sometimes infinity. Also the axioms of separation and collection here are weaker than the corresponding axioms in ZFC because the formulas φ used in these are limited to bounded quantifiers only.
2048:
1499:
1376:{\displaystyle \land \,\forall t\in p\,{\Big (}{\big (}a\in t\,\land \,\forall x\in t\,(x=a){\big )}\,\lor \,{\big (}a\in t\land b\in t\land \forall x\in t\,(x=a\,\lor \,x=b){\big )}{\Big )}.}
1075:
1973:
1887:
1673:
1567:
916:
589:
490:
3050:
169:
1923:
137:
1661:
1602:
968:
683:
654:
3725:
2787:
620:
547:
1837:
1817:
1629:
1431:
1411:
446:
2104:
3808:
2949:
1390:
What follows are two steps of collection of sets, followed by a restriction through separation. All results are also expressed using set builder notation.
988:
4122:
289:
4280:
3068:
4135:
3458:
2476:
2296:
3720:
2804:
4140:
4130:
3867:
3073:
3618:
3064:
1988:
448:
is postulated to exist, such as in the axiom of infinity, then the axiom of empty set is redundant because it is equal to the subset
4276:
2182:
2136:
4373:
4117:
2942:
3678:
3371:
3112:
2782:
4695:
4634:
4336:
4099:
4094:
3919:
3340:
3024:
2203:
492:. Furthermore, the existence of a member in the universe of discourse, i.e., ∃x(x=x), is implied in certain formulations of
4629:
4412:
4329:
4042:
3973:
3850:
3092:
2556:
2435:
1183:{\displaystyle \land \,\exists s\in p\,{\big (}a\in s\,\land \,b\in s\,\land \,\forall x\in s\,(x=a\,\lor \,x=b){\big )}}
4554:
4380:
4066:
3299:
1436:
3705:
2792:
2073:
4037:
3776:
3034:
2935:
2430:
2393:
4432:
4427:
1789:{\displaystyle \forall a\in A\,\exists p\in P\,\psi (a,b,p)\,\land \,\forall p\in P\,\exists a\in A\,\psi (a,b,p)\,.}
4361:
3951:
3345:
3313:
3004:
2099:
3078:
1928:
1842:
1518:
4651:
4600:
4497:
3995:
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3433:
2481:
2373:
2361:
2356:
2194:
4492:
3107:
4422:
3961:
3813:
3796:
3519:
2999:
2289:
2077:
519:, which amounts to applying induction to the complement of a set (the class of all sets not in the given set).
853:
4324:
4301:
4262:
4148:
4089:
3735:
3655:
3499:
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2819:
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2460:
2383:
562:
451:
413:
179:
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4010:
3983:
3934:
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2734:
2546:
2366:
2089:
417:
4544:
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3907:
3643:
3549:
3408:
3393:
3274:
3249:
2769:
2683:
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4356:
4160:
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3924:
3902:
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3206:
3117:
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142:
83:
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113:
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3897:
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3608:
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3269:
3169:
3097:
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3019:
3014:
2838:
2749:
2662:
2657:
2652:
2466:
2408:
2346:
2282:
2260:
1634:
1575:
516:
323:
193:
932:
4675:
4444:
4407:
4392:
4385:
4368:
4154:
4020:
3946:
3929:
3882:
3695:
3604:
3438:
3423:
3383:
3335:
3320:
3308:
3264:
3239:
3009:
2958:
2761:
2756:
2541:
2496:
2403:
2244:
2215:
2094:
661:
632:
625:
409:
405:
227:
107:
4172:
3628:
4610:
4417:
4227:
4217:
4109:
3990:
3825:
3801:
3582:
3566:
3471:
3448:
3325:
3294:
3259:
3154:
2989:
2618:
2455:
2447:
2418:
2388:
2319:
2178:
2132:
493:
398:
185:
172:
605:
4624:
4619:
4512:
4469:
4291:
4252:
4247:
4232:
4058:
4015:
3912:
3710:
3660:
3234:
3196:
2906:
2896:
2881:
2876:
2744:
2398:
2236:
2207:
237:
30:
2268:
4605:
4595:
4549:
4532:
4487:
4449:
4351:
4271:
4078:
4005:
3978:
3966:
3872:
3786:
3760:
3715:
3683:
3484:
3286:
3229:
3179:
3144:
3102:
2775:
2713:
2531:
2351:
2264:
2174:
509:
91:
531:
4590:
4569:
4527:
4507:
4402:
4257:
3855:
3845:
3835:
3830:
3764:
3638:
3514:
3403:
3398:
3376:
2977:
2911:
2708:
2689:
2593:
2578:
2535:
2471:
2413:
2125:
1822:
1802:
1614:
1416:
1396:
599:
556:
551:
431:
404:
KP with infinity is denoted by KPω. These axioms lead to close connections between KP,
267:
4689:
4564:
4242:
3749:
3534:
3524:
3494:
3479:
3149:
2916:
2886:
2718:
2632:
2627:
1064:{\displaystyle \exists r\in p\,{\big (}a\in r\,\land \,\forall x\in r\,(x=a){\big )}}
716:
4464:
4311:
4212:
4204:
4084:
4032:
3941:
3877:
3860:
3791:
3650:
3509:
3211:
2994:
2866:
2861:
2679:
2608:
2566:
2425:
2329:
781:
592:
317:
4574:
4454:
3633:
3623:
3570:
3254:
3174:
3159:
3039:
2984:
2891:
2526:
87:
3504:
3359:
3330:
3136:
2871:
2739:
2642:
2305:
4656:
4559:
3612:
3529:
3489:
3453:
3389:
3201:
3191:
3164:
2674:
2637:
2588:
2486:
2127:
A course in model theory: an introduction to contemporary mathematical logic
231:
4641:
4439:
3887:
3592:
3186:
4237:
3029:
2248:
2219:
2192:
Gostanian, Richard (1980). "Constructible Models of
Subsystems of ZF".
515:
The axiom of induction in the context of KP is stronger than the usual
2927:
2699:
2521:
2058:
305:
2240:
2211:
2076:. KP fails to prove some common theorems in set theory, such as the
2227:
Kripke, S. (1964), "Transfinite recursion on admissible ordinals",
182:: Two sets are the same if and only if they have the same elements.
3781:
3127:
2972:
2571:
2338:
496:, in which case the axiom of empty set follows from the axiom of Δ
689:
if it is a standard model of KP set theory without the axiom of Δ
90:
and
Richard Platek. The theory can be thought of as roughly the
2931:
2278:
2043:{\displaystyle A\times B:=\bigcup \{A\times \{b\}\mid b\in B\}}
2274:
65:
508:
As noted, the above are weaker than ZFC as they exclude the
71:
51:
42:
308:
of the original set containing precisely those elements
749:
is a standard model of KP, then the set of ordinals in
1991:
1931:
1902:
1845:
1825:
1805:
1676:
1637:
1617:
1578:
1521:
1439:
1419:
1399:
1197:
1078:
991:
935:
856:
664:
635:
608:
565:
534:
454:
434:
145:
116:
74:
48:
36:
1925:
in front of that last formula and one finds the set
68:
62:
59:
45:
39:
4583:
4478:
4310:
4203:
4055:
3748:
3671:
3565:
3469:
3358:
3285:
3220:
3135:
3126:
3048:
2965:
2852:
2815:
2727:
2617:
2505:
2446:
2337:
2312:
56:
33:
2124:
2042:
1967:
1917:
1881:
1831:
1811:
1788:
1655:
1623:
1596:
1561:
1494:{\displaystyle A\times \{b\}=\{(a,b)\mid a\in A\}}
1493:
1425:
1405:
1375:
1182:
1063:
962:
910:
677:
648:
614:
583:
541:
484:
440:
163:
131:
1365:
1217:
230:: There exists a set with no members, called the
94:part of ZFC and is considerably weaker than it.
282:are precisely the elements of the elements of
2943:
2290:
1358:
1285:
1273:
1224:
1175:
1098:
1056:
1007:
8:
2037:
2022:
2016:
2007:
1968:{\displaystyle \{A\times \{b\}\mid b\in B\}}
1962:
1947:
1941:
1932:
1882:{\displaystyle \{A\times \{b\}\mid b\in B\}}
1876:
1861:
1855:
1846:
1650:
1644:
1631:ought to stand for this collection of pairs
1591:
1585:
1562:{\displaystyle \exists a\in A\,\psi (a,b,p)}
1488:
1458:
1452:
1446:
905:
902:
890:
884:
878:
875:
578:
566:
479:
455:
110:. This means any quantification is the form
106:formula is one all of whose quantifiers are
504:Comparison with Zermelo-Fraenkel set theory
3769:
3364:
3132:
2950:
2936:
2928:
2297:
2283:
2275:
2156:(1989) p.421. North-Holland, 0-444-87295-7
1990:
1930:
1901:
1844:
1824:
1804:
1782:
1757:
1744:
1731:
1727:
1702:
1689:
1675:
1636:
1616:
1577:
1534:
1520:
1438:
1418:
1398:
1364:
1363:
1357:
1356:
1343:
1339:
1326:
1284:
1283:
1282:
1278:
1272:
1271:
1255:
1242:
1238:
1223:
1222:
1216:
1215:
1214:
1201:
1196:
1174:
1173:
1160:
1156:
1143:
1130:
1126:
1116:
1112:
1097:
1096:
1095:
1082:
1077:
1055:
1054:
1038:
1025:
1021:
1006:
1005:
1004:
990:
934:
855:
669:
663:
640:
634:
607:
564:
538:
533:
453:
433:
144:
115:
2105:Kripke–Platek set theory with urelements
911:{\displaystyle (a,b):=\{\{a\},\{a,b\}\}}
711:is an admissible ordinal if and only if
2115:
824:}, is the same as the unordered pair {
584:{\displaystyle \langle A,\in \rangle }
485:{\displaystyle \{x\in c\mid x\neq x\}}
394:Some but not all authors include an
7:
500:-separation, and is thus redundant.
1903:
1745:
1732:
1690:
1677:
1522:
1314:
1243:
1202:
1131:
1083:
1026:
992:
847:}, and then also the ordered pair
146:
117:
14:
16:System of mathematical set theory
4669:
2328:
29:
2257:Foundations of recursion theory
1819:and collecting with respect to
1413:and collecting with respect to
420:, without changing any axioms.
164:{\displaystyle \exists u\in v.}
2204:Association for Symbolic Logic
1918:{\displaystyle \exists b\in B}
1779:
1761:
1724:
1706:
1667:-formula characterizing it is
1556:
1538:
1473:
1461:
1353:
1327:
1268:
1256:
1170:
1144:
1051:
1039:
957:
939:
869:
857:
816:The singleton set with member
772:are sets, then there is a set
132:{\displaystyle \forall u\in v}
1:
4630:History of mathematical logic
2255:Platek, Richard Alan (1966),
1656:{\displaystyle A\times \{b\}}
1597:{\displaystyle A\times \{b\}}
4555:Primitive recursive function
963:{\displaystyle \psi (a,b,p)}
595:of Kripke–Platek set theory.
406:generalized recursion theory
719:and there does not exist a
678:{\displaystyle L_{\alpha }}
649:{\displaystyle L_{\alpha }}
358:) holds, then for all sets
4712:
3619:Schröder–Bernstein theorem
3346:Monadic predicate calculus
3005:Foundations of mathematics
2788:von Neumann–Bernays–Gödel
2154:Classical Recursion Theory
2100:Hereditarily countable set
982:) is given by the lengthy
753:is an admissible ordinal.
278:such that the elements of
4665:
4652:Philosophy of mathematics
4601:Automated theorem proving
3772:
3726:Von Neumann–Bernays–Gödel
3367:
2589:One-to-one correspondence
2326:
2229:Journal of Symbolic Logic
2195:Journal of Symbolic Logic
2169:Devlin, Keith J. (1984).
412:. KP can be studied as a
296:: Given any set and any Δ
204:) holds for all elements
2078:Mostowski collapse lemma
839:The singleton, the set {
757:Cartesian products exist
21:Kripke–Platek set theory
4302:Self-verifying theories
4123:Tarski's axiomatization
3074:Tarski's undefinability
3069:incompleteness theorems
2074:Bachmann–Howard ordinal
2072:of KPω is given by the
615:{\displaystyle \alpha }
414:constructive set theory
180:Axiom of extensionality
102:In its formulation, a Δ
4676:Mathematics portal
4287:Proof of impossibility
3935:propositional variable
3245:Propositional calculus
2547:Constructible universe
2374:Constructibility (V=L)
2123:Poizat, Bruno (2000).
2090:Constructible universe
2044:
1969:
1919:
1883:
1833:
1813:
1790:
1657:
1625:
1598:
1563:
1495:
1427:
1407:
1377:
1184:
1065:
964:
912:
780:which consists of all
727:for which there is a Σ
679:
650:
616:
585:
543:
486:
442:
418:law of excluded middle
248:are sets, then so is {
200:the assumption that φ(
165:
133:
4696:Systems of set theory
4545:Kolmogorov complexity
4498:Computably enumerable
4398:Model complete theory
4190:Principia Mathematica
3250:Propositional formula
3079:Banach–Tarski paradox
2770:Principia Mathematica
2604:Transfinite induction
2463:(i.e. set difference)
2045:
1982:Finally, the desired
1970:
1920:
1884:
1834:
1814:
1791:
1658:
1626:
1599:
1564:
1496:
1428:
1408:
1378:
1185:
1066:
974:stands for the pair (
965:
913:
680:
656:is an admissible set.
651:
617:
586:
544:
487:
443:
316:) holds. (This is an
264:as its only elements.
220:) holds for all sets
166:
134:
4493:Church–Turing thesis
4480:Computability theory
3689:continuum hypothesis
3207:Square of opposition
3065:Gödel's completeness
2844:Burali-Forti paradox
2599:Set-builder notation
2552:Continuum hypothesis
2492:Symmetric difference
2070:consistency strength
1989:
1929:
1900:
1843:
1823:
1803:
1674:
1635:
1615:
1576:
1519:
1437:
1417:
1397:
1195:
1076:
989:
933:
854:
662:
633:
606:
563:
532:
452:
432:
408:, and the theory of
366:such that for every
342:), if for every set
256:}, a set containing
143:
114:
84:axiomatic set theory
4647:Mathematical object
4538:P versus NP problem
4503:Computable function
4297:Reverse mathematics
4223:Logical consequence
4100:primitive recursive
4095:elementary function
3868:Free/bound variable
3721:Tarski–Grothendieck
3240:Logical connectives
3170:Logical equivalence
3020:Logical consequence
2805:Tarski–Grothendieck
2261:Stanford University
1839:, some superset of
1433:, some superset of
832:}, by the axiom of
542:{\displaystyle A\,}
523:Related definitions
517:axiom of regularity
410:admissible ordinals
362:there exists a set
346:there exists a set
4445:Transfer principle
4408:Semantics of logic
4393:Categorical theory
4369:Non-standard model
3883:Logical connective
3010:Information theory
2959:Mathematical logic
2394:Limitation of size
2095:Admissible ordinal
2040:
1965:
1915:
1879:
1829:
1809:
1786:
1653:
1621:
1594:
1559:
1491:
1423:
1403:
1373:
1180:
1061:
960:
908:
675:
646:
626:admissible ordinal
612:
581:
539:
482:
438:
228:Axiom of empty set
196:, if for all sets
186:Axiom of induction
161:
129:
4683:
4682:
4615:Abstract category
4418:Theories of truth
4228:Rule of inference
4218:Natural deduction
4199:
4198:
3744:
3743:
3449:Cartesian product
3354:
3353:
3260:Many-valued logic
3235:Boolean functions
3118:Russell's paradox
3093:diagonal argument
2990:First-order logic
2925:
2924:
2834:Russell's paradox
2783:Zermelo–Fraenkel
2684:Dedekind-infinite
2557:Diagonal argument
2456:Cartesian product
2320:Set (mathematics)
2259:, Thesis (Ph.D.)–
1975:itself exists by
1832:{\displaystyle B}
1812:{\displaystyle A}
1624:{\displaystyle P}
1604:itself exists by
1572:grants that just
1426:{\displaystyle A}
1406:{\displaystyle b}
494:first-order logic
441:{\displaystyle c}
399:Axiom of infinity
274:, there is a set
4703:
4674:
4673:
4625:History of logic
4620:Category of sets
4513:Decision problem
4292:Ordinal analysis
4233:Sequent calculus
4131:Boolean algebras
4071:
4070:
4045:
4016:logical/constant
3770:
3756:
3679:Zermelo–Fraenkel
3430:Set operations:
3365:
3302:
3133:
3113:Löwenheim–Skolem
3000:Formal semantics
2952:
2945:
2938:
2929:
2907:Bertrand Russell
2897:John von Neumann
2882:Abraham Fraenkel
2877:Richard Dedekind
2839:Suslin's problem
2750:Cantor's theorem
2467:De Morgan's laws
2332:
2299:
2292:
2285:
2276:
2271:
2251:
2223:
2188:
2171:Constructibility
2157:
2150:
2144:
2142:
2130:
2120:
2049:
2047:
2046:
2041:
1974:
1972:
1971:
1966:
1924:
1922:
1921:
1916:
1888:
1886:
1885:
1880:
1838:
1836:
1835:
1830:
1818:
1816:
1815:
1810:
1795:
1793:
1792:
1787:
1662:
1660:
1659:
1654:
1630:
1628:
1627:
1622:
1603:
1601:
1600:
1595:
1568:
1566:
1565:
1560:
1500:
1498:
1497:
1492:
1432:
1430:
1429:
1424:
1412:
1410:
1409:
1404:
1382:
1380:
1379:
1374:
1369:
1368:
1362:
1361:
1289:
1288:
1277:
1276:
1228:
1227:
1221:
1220:
1189:
1187:
1186:
1181:
1179:
1178:
1102:
1101:
1070:
1068:
1067:
1062:
1060:
1059:
1011:
1010:
970:expressing that
969:
967:
966:
961:
917:
915:
914:
909:
723: <
684:
682:
681:
676:
674:
673:
655:
653:
652:
647:
645:
644:
621:
619:
618:
613:
590:
588:
587:
582:
548:
546:
545:
540:
491:
489:
488:
483:
447:
445:
444:
439:
416:by dropping the
238:Axiom of pairing
216:) holds, then φ(
170:
168:
167:
162:
138:
136:
135:
130:
81:
80:
77:
76:
73:
70:
67:
64:
61:
58:
54:
53:
50:
47:
44:
41:
38:
35:
4711:
4710:
4706:
4705:
4704:
4702:
4701:
4700:
4686:
4685:
4684:
4679:
4668:
4661:
4606:Category theory
4596:Algebraic logic
4579:
4550:Lambda calculus
4488:Church encoding
4474:
4450:Truth predicate
4306:
4272:Complete theory
4195:
4064:
4060:
4056:
4051:
4043:
3763: and
3759:
3754:
3740:
3716:New Foundations
3684:axiom of choice
3667:
3629:Gödel numbering
3569: and
3561:
3465:
3350:
3300:
3281:
3230:Boolean algebra
3216:
3180:Equiconsistency
3145:Classical logic
3122:
3103:Halting problem
3091: and
3067: and
3055: and
3054:
3049:Theorems (
3044:
2961:
2956:
2926:
2921:
2848:
2827:
2811:
2776:New Foundations
2723:
2613:
2532:Cardinal number
2515:
2501:
2442:
2333:
2324:
2308:
2303:
2254:
2241:10.2307/2271646
2226:
2212:10.2307/2273185
2191:
2185:
2175:Springer-Verlag
2168:
2165:
2160:
2151:
2147:
2139:
2122:
2121:
2117:
2113:
2086:
2066:
1987:
1986:
1927:
1926:
1898:
1897:
1841:
1840:
1821:
1820:
1801:
1800:
1672:
1671:
1666:
1633:
1632:
1613:
1612:
1574:
1573:
1517:
1516:
1511:
1435:
1434:
1415:
1414:
1395:
1394:
1393:Firstly, given
1193:
1192:
1074:
1073:
987:
986:
931:
930:
928:
852:
851:
759:
737:) mapping from
736:
730:
705:
703:Admissible sets
700:
692:
665:
660:
659:
636:
631:
630:
604:
603:
561:
560:
530:
529:
525:
510:power set axiom
506:
499:
450:
449:
430:
429:
426:
333:
327:
299:
293:
234:and denoted {}.
212:entails that φ(
141:
140:
112:
111:
105:
100:
55:
32:
28:
17:
12:
11:
5:
4709:
4707:
4699:
4698:
4688:
4687:
4681:
4680:
4666:
4663:
4662:
4660:
4659:
4654:
4649:
4644:
4639:
4638:
4637:
4627:
4622:
4617:
4608:
4603:
4598:
4593:
4591:Abstract logic
4587:
4585:
4581:
4580:
4578:
4577:
4572:
4570:Turing machine
4567:
4562:
4557:
4552:
4547:
4542:
4541:
4540:
4535:
4530:
4525:
4520:
4510:
4508:Computable set
4505:
4500:
4495:
4490:
4484:
4482:
4476:
4475:
4473:
4472:
4467:
4462:
4457:
4452:
4447:
4442:
4437:
4436:
4435:
4430:
4425:
4415:
4410:
4405:
4403:Satisfiability
4400:
4395:
4390:
4389:
4388:
4378:
4377:
4376:
4366:
4365:
4364:
4359:
4354:
4349:
4344:
4334:
4333:
4332:
4327:
4320:Interpretation
4316:
4314:
4308:
4307:
4305:
4304:
4299:
4294:
4289:
4284:
4274:
4269:
4268:
4267:
4266:
4265:
4255:
4250:
4240:
4235:
4230:
4225:
4220:
4215:
4209:
4207:
4201:
4200:
4197:
4196:
4194:
4193:
4185:
4184:
4183:
4182:
4177:
4176:
4175:
4170:
4165:
4145:
4144:
4143:
4141:minimal axioms
4138:
4127:
4126:
4125:
4114:
4113:
4112:
4107:
4102:
4097:
4092:
4087:
4074:
4072:
4053:
4052:
4050:
4049:
4048:
4047:
4035:
4030:
4029:
4028:
4023:
4018:
4013:
4003:
3998:
3993:
3988:
3987:
3986:
3981:
3971:
3970:
3969:
3964:
3959:
3954:
3944:
3939:
3938:
3937:
3932:
3927:
3917:
3916:
3915:
3910:
3905:
3900:
3895:
3890:
3880:
3875:
3870:
3865:
3864:
3863:
3858:
3853:
3848:
3838:
3833:
3831:Formation rule
3828:
3823:
3822:
3821:
3816:
3806:
3805:
3804:
3794:
3789:
3784:
3779:
3773:
3767:
3750:Formal systems
3746:
3745:
3742:
3741:
3739:
3738:
3733:
3728:
3723:
3718:
3713:
3708:
3703:
3698:
3693:
3692:
3691:
3686:
3675:
3673:
3669:
3668:
3666:
3665:
3664:
3663:
3653:
3648:
3647:
3646:
3639:Large cardinal
3636:
3631:
3626:
3621:
3616:
3602:
3601:
3600:
3595:
3590:
3575:
3573:
3563:
3562:
3560:
3559:
3558:
3557:
3552:
3547:
3537:
3532:
3527:
3522:
3517:
3512:
3507:
3502:
3497:
3492:
3487:
3482:
3476:
3474:
3467:
3466:
3464:
3463:
3462:
3461:
3456:
3451:
3446:
3441:
3436:
3428:
3427:
3426:
3421:
3411:
3406:
3404:Extensionality
3401:
3399:Ordinal number
3396:
3386:
3381:
3380:
3379:
3368:
3362:
3356:
3355:
3352:
3351:
3349:
3348:
3343:
3338:
3333:
3328:
3323:
3318:
3317:
3316:
3306:
3305:
3304:
3291:
3289:
3283:
3282:
3280:
3279:
3278:
3277:
3272:
3267:
3257:
3252:
3247:
3242:
3237:
3232:
3226:
3224:
3218:
3217:
3215:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3183:
3182:
3172:
3167:
3162:
3157:
3152:
3147:
3141:
3139:
3130:
3124:
3123:
3121:
3120:
3115:
3110:
3105:
3100:
3095:
3083:Cantor's
3081:
3076:
3071:
3061:
3059:
3046:
3045:
3043:
3042:
3037:
3032:
3027:
3022:
3017:
3012:
3007:
3002:
2997:
2992:
2987:
2982:
2981:
2980:
2969:
2967:
2963:
2962:
2957:
2955:
2954:
2947:
2940:
2932:
2923:
2922:
2920:
2919:
2914:
2912:Thoralf Skolem
2909:
2904:
2899:
2894:
2889:
2884:
2879:
2874:
2869:
2864:
2858:
2856:
2850:
2849:
2847:
2846:
2841:
2836:
2830:
2828:
2826:
2825:
2822:
2816:
2813:
2812:
2810:
2809:
2808:
2807:
2802:
2797:
2796:
2795:
2780:
2779:
2778:
2766:
2765:
2764:
2753:
2752:
2747:
2742:
2737:
2731:
2729:
2725:
2724:
2722:
2721:
2716:
2711:
2706:
2697:
2692:
2687:
2677:
2672:
2671:
2670:
2665:
2660:
2650:
2640:
2635:
2630:
2624:
2622:
2615:
2614:
2612:
2611:
2606:
2601:
2596:
2594:Ordinal number
2591:
2586:
2581:
2576:
2575:
2574:
2569:
2559:
2554:
2549:
2544:
2539:
2529:
2524:
2518:
2516:
2514:
2513:
2510:
2506:
2503:
2502:
2500:
2499:
2494:
2489:
2484:
2479:
2474:
2472:Disjoint union
2469:
2464:
2458:
2452:
2450:
2444:
2443:
2441:
2440:
2439:
2438:
2433:
2422:
2421:
2419:Martin's axiom
2416:
2411:
2406:
2401:
2396:
2391:
2386:
2384:Extensionality
2381:
2376:
2371:
2370:
2369:
2364:
2359:
2349:
2343:
2341:
2335:
2334:
2327:
2325:
2323:
2322:
2316:
2314:
2310:
2309:
2304:
2302:
2301:
2294:
2287:
2279:
2273:
2272:
2252:
2224:
2189:
2183:
2164:
2161:
2159:
2158:
2152:P. Odifreddi,
2145:
2137:
2114:
2112:
2109:
2108:
2107:
2102:
2097:
2092:
2085:
2082:
2065:
2062:
2051:
2050:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1914:
1911:
1908:
1905:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1828:
1808:
1797:
1796:
1785:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1756:
1753:
1750:
1747:
1743:
1740:
1737:
1734:
1730:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1701:
1698:
1695:
1692:
1688:
1685:
1682:
1679:
1664:
1652:
1649:
1646:
1643:
1640:
1620:
1593:
1590:
1587:
1584:
1581:
1570:
1569:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1533:
1530:
1527:
1524:
1509:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1422:
1402:
1388:
1387:
1386:
1385:
1384:
1383:
1372:
1367:
1360:
1355:
1352:
1349:
1346:
1342:
1338:
1335:
1332:
1329:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1287:
1281:
1275:
1270:
1267:
1264:
1261:
1258:
1254:
1251:
1248:
1245:
1241:
1237:
1234:
1231:
1226:
1219:
1213:
1210:
1207:
1204:
1200:
1177:
1172:
1169:
1166:
1163:
1159:
1155:
1152:
1149:
1146:
1142:
1139:
1136:
1133:
1129:
1125:
1122:
1119:
1115:
1111:
1108:
1105:
1100:
1094:
1091:
1088:
1085:
1081:
1058:
1053:
1050:
1047:
1044:
1041:
1037:
1034:
1031:
1028:
1024:
1020:
1017:
1014:
1009:
1003:
1000:
997:
994:
959:
956:
953:
950:
947:
944:
941:
938:
926:
925:. A possible Δ
919:
918:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
834:extensionality
792:) of elements
758:
755:
732:
728:
704:
701:
699:
696:
695:
694:
690:
672:
668:
657:
643:
639:
611:
600:ordinal number
596:
580:
577:
574:
571:
568:
537:
524:
521:
505:
502:
497:
481:
478:
475:
472:
469:
466:
463:
460:
457:
437:
425:
422:
402:
401:
392:
391:
331:
325:
321:
304:), there is a
297:
291:
287:
270:: For any set
268:Axiom of union
265:
235:
225:
183:
173:Lévy hierarchy
160:
157:
154:
151:
148:
128:
125:
122:
119:
103:
99:
96:
27:), pronounced
15:
13:
10:
9:
6:
4:
3:
2:
4708:
4697:
4694:
4693:
4691:
4678:
4677:
4672:
4664:
4658:
4655:
4653:
4650:
4648:
4645:
4643:
4640:
4636:
4633:
4632:
4631:
4628:
4626:
4623:
4621:
4618:
4616:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4592:
4589:
4588:
4586:
4582:
4576:
4573:
4571:
4568:
4566:
4565:Recursive set
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4539:
4536:
4534:
4531:
4529:
4526:
4524:
4521:
4519:
4516:
4515:
4514:
4511:
4509:
4506:
4504:
4501:
4499:
4496:
4494:
4491:
4489:
4486:
4485:
4483:
4481:
4477:
4471:
4468:
4466:
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4446:
4443:
4441:
4438:
4434:
4431:
4429:
4426:
4424:
4421:
4420:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4387:
4384:
4383:
4382:
4379:
4375:
4374:of arithmetic
4372:
4371:
4370:
4367:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4339:
4338:
4335:
4331:
4328:
4326:
4323:
4322:
4321:
4318:
4317:
4315:
4313:
4309:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4282:
4281:from ZFC
4278:
4275:
4273:
4270:
4264:
4261:
4260:
4259:
4256:
4254:
4251:
4249:
4246:
4245:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4210:
4208:
4206:
4202:
4192:
4191:
4187:
4186:
4181:
4180:non-Euclidean
4178:
4174:
4171:
4169:
4166:
4164:
4163:
4159:
4158:
4156:
4153:
4152:
4150:
4146:
4142:
4139:
4137:
4134:
4133:
4132:
4128:
4124:
4121:
4120:
4119:
4115:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4091:
4088:
4086:
4083:
4082:
4080:
4076:
4075:
4073:
4068:
4062:
4057:Example
4054:
4046:
4041:
4040:
4039:
4036:
4034:
4031:
4027:
4024:
4022:
4019:
4017:
4014:
4012:
4009:
4008:
4007:
4004:
4002:
3999:
3997:
3994:
3992:
3989:
3985:
3982:
3980:
3977:
3976:
3975:
3972:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3949:
3948:
3945:
3943:
3940:
3936:
3933:
3931:
3928:
3926:
3923:
3922:
3921:
3918:
3914:
3911:
3909:
3906:
3904:
3901:
3899:
3896:
3894:
3891:
3889:
3886:
3885:
3884:
3881:
3879:
3876:
3874:
3871:
3869:
3866:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3844:
3843:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3820:
3817:
3815:
3814:by definition
3812:
3811:
3810:
3807:
3803:
3800:
3799:
3798:
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3778:
3775:
3774:
3771:
3768:
3766:
3762:
3757:
3751:
3747:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3712:
3709:
3707:
3704:
3702:
3701:Kripke–Platek
3699:
3697:
3694:
3690:
3687:
3685:
3682:
3681:
3680:
3677:
3676:
3674:
3670:
3662:
3659:
3658:
3657:
3654:
3652:
3649:
3645:
3642:
3641:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3614:
3610:
3606:
3603:
3599:
3596:
3594:
3591:
3589:
3586:
3585:
3584:
3580:
3577:
3576:
3574:
3572:
3568:
3564:
3556:
3553:
3551:
3548:
3546:
3545:constructible
3543:
3542:
3541:
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3481:
3478:
3477:
3475:
3473:
3468:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3431:
3429:
3425:
3422:
3420:
3417:
3416:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3391:
3387:
3385:
3382:
3378:
3375:
3374:
3373:
3370:
3369:
3366:
3363:
3361:
3357:
3347:
3344:
3342:
3339:
3337:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3315:
3312:
3311:
3310:
3307:
3303:
3298:
3297:
3296:
3293:
3292:
3290:
3288:
3284:
3276:
3273:
3271:
3268:
3266:
3263:
3262:
3261:
3258:
3256:
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3236:
3233:
3231:
3228:
3227:
3225:
3223:
3222:Propositional
3219:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3181:
3178:
3177:
3176:
3173:
3171:
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3150:Logical truth
3148:
3146:
3143:
3142:
3140:
3138:
3134:
3131:
3129:
3125:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3090:
3086:
3082:
3080:
3077:
3075:
3072:
3070:
3066:
3063:
3062:
3060:
3058:
3052:
3047:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2979:
2976:
2975:
2974:
2971:
2970:
2968:
2964:
2960:
2953:
2948:
2946:
2941:
2939:
2934:
2933:
2930:
2918:
2917:Ernst Zermelo
2915:
2913:
2910:
2908:
2905:
2903:
2902:Willard Quine
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2883:
2880:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2860:
2859:
2857:
2855:
2854:Set theorists
2851:
2845:
2842:
2840:
2837:
2835:
2832:
2831:
2829:
2823:
2821:
2818:
2817:
2814:
2806:
2803:
2801:
2800:Kripke–Platek
2798:
2794:
2791:
2790:
2789:
2786:
2785:
2784:
2781:
2777:
2774:
2773:
2772:
2771:
2767:
2763:
2760:
2759:
2758:
2755:
2754:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2732:
2730:
2726:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2703:
2698:
2696:
2693:
2691:
2688:
2685:
2681:
2678:
2676:
2673:
2669:
2666:
2664:
2661:
2659:
2656:
2655:
2654:
2651:
2648:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2625:
2623:
2620:
2616:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2573:
2570:
2568:
2565:
2564:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2537:
2533:
2530:
2528:
2525:
2523:
2520:
2519:
2517:
2511:
2508:
2507:
2504:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2462:
2459:
2457:
2454:
2453:
2451:
2449:
2445:
2437:
2436:specification
2434:
2432:
2429:
2428:
2427:
2424:
2423:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2368:
2365:
2363:
2360:
2358:
2355:
2354:
2353:
2350:
2348:
2345:
2344:
2342:
2340:
2336:
2331:
2321:
2318:
2317:
2315:
2311:
2307:
2300:
2295:
2293:
2288:
2286:
2281:
2280:
2277:
2270:
2266:
2262:
2258:
2253:
2250:
2246:
2242:
2238:
2234:
2230:
2225:
2221:
2217:
2213:
2209:
2205:
2201:
2197:
2196:
2190:
2186:
2184:0-387-13258-9
2180:
2176:
2172:
2167:
2166:
2162:
2155:
2149:
2146:
2140:
2138:0-387-98655-3
2134:
2129:
2128:
2119:
2116:
2110:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2087:
2083:
2081:
2079:
2075:
2071:
2063:
2061:
2060:
2056:
2034:
2031:
2028:
2025:
2019:
2013:
2010:
2004:
2001:
1998:
1995:
1992:
1985:
1984:
1983:
1980:
1978:
1959:
1956:
1953:
1950:
1944:
1938:
1935:
1912:
1909:
1906:
1894:
1892:
1873:
1870:
1867:
1864:
1858:
1852:
1849:
1826:
1806:
1783:
1776:
1773:
1770:
1767:
1764:
1758:
1754:
1751:
1748:
1741:
1738:
1735:
1728:
1721:
1718:
1715:
1712:
1709:
1703:
1699:
1696:
1693:
1686:
1683:
1680:
1670:
1669:
1668:
1647:
1641:
1638:
1618:
1609:
1607:
1588:
1582:
1579:
1553:
1550:
1547:
1544:
1541:
1535:
1531:
1528:
1525:
1515:
1514:
1513:
1506:
1504:
1485:
1482:
1479:
1476:
1470:
1467:
1464:
1455:
1449:
1443:
1440:
1420:
1400:
1391:
1370:
1350:
1347:
1344:
1340:
1336:
1333:
1330:
1323:
1320:
1317:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1279:
1265:
1262:
1259:
1252:
1249:
1246:
1239:
1235:
1232:
1229:
1211:
1208:
1205:
1198:
1191:
1190:
1167:
1164:
1161:
1157:
1153:
1150:
1147:
1140:
1137:
1134:
1127:
1123:
1120:
1117:
1113:
1109:
1106:
1103:
1092:
1089:
1086:
1079:
1072:
1071:
1048:
1045:
1042:
1035:
1032:
1029:
1022:
1018:
1015:
1012:
1001:
998:
995:
985:
984:
983:
981:
977:
973:
954:
951:
948:
945:
942:
936:
924:
921:all exist by
899:
896:
893:
887:
881:
872:
866:
863:
860:
850:
849:
848:
846:
842:
837:
835:
831:
827:
823:
819:
814:
813:
809:
807:
803:
799:
795:
791:
787:
783:
782:ordered pairs
779:
775:
771:
767:
763:
756:
754:
752:
748:
744:
740:
735:
726:
722:
718:
717:limit ordinal
714:
710:
702:
697:
688:
685:is called an
670:
666:
658:
641:
637:
628:
627:
623:is called an
622:
609:
601:
597:
594:
575:
572:
569:
558:
554:
553:
535:
527:
526:
522:
520:
518:
513:
511:
503:
501:
495:
476:
473:
470:
467:
464:
461:
458:
435:
423:
421:
419:
415:
411:
407:
400:
397:
396:
395:
389:
385:
381:
377:
373:
369:
365:
361:
357:
353:
349:
345:
341:
337:
330:: Given any Δ
329:
322:
319:
315:
311:
307:
303:
295:
288:
285:
281:
277:
273:
269:
266:
263:
259:
255:
251:
247:
243:
239:
236:
233:
229:
226:
223:
219:
215:
211:
207:
203:
199:
195:
191:
187:
184:
181:
178:
177:
176:
174:
158:
155:
152:
149:
126:
123:
120:
109:
97:
95:
93:
89:
86:developed by
85:
79:
26:
22:
4667:
4465:Ultraproduct
4312:Model theory
4277:Independence
4213:Formal proof
4205:Proof theory
4188:
4161:
4118:real numbers
4090:second-order
4001:Substitution
3878:Metalanguage
3819:conservative
3792:Axiom schema
3736:Constructive
3706:Morse–Kelley
3700:
3672:Set theories
3651:Aleph number
3644:inaccessible
3550:Grothendieck
3434:intersection
3321:Higher-order
3309:Second-order
3255:Truth tables
3212:Venn diagram
2995:Formal proof
2867:Georg Cantor
2862:Paul Bernays
2799:
2793:Morse–Kelley
2768:
2701:
2700:Subset
2647:hereditarily
2609:Venn diagram
2567:ordered pair
2482:Intersection
2426:Axiom schema
2256:
2232:
2228:
2199:
2193:
2170:
2163:Bibliography
2153:
2148:
2131:. Springer.
2126:
2118:
2067:
2054:
2052:
1981:
1976:
1895:
1890:
1798:
1610:
1605:
1571:
1507:
1502:
1392:
1389:
979:
975:
971:
922:
920:
844:
840:
838:
833:
829:
825:
821:
817:
815:
811:
810:
805:
801:
797:
793:
789:
785:
777:
773:
769:
765:
761:
760:
750:
746:
742:
738:
733:
724:
720:
712:
708:
707:The ordinal
706:
693:-collection.
687:amenable set
686:
624:
602:
550:
514:
507:
427:
403:
393:
387:
383:
382:such that φ(
379:
375:
371:
367:
363:
359:
355:
351:
350:such that φ(
347:
343:
339:
335:
318:axiom schema
313:
312:for which φ(
309:
301:
283:
279:
275:
271:
261:
257:
253:
249:
245:
241:
221:
217:
213:
209:
205:
201:
197:
189:
101:
24:
20:
18:
4575:Type theory
4523:undecidable
4455:Truth value
4342:equivalence
4021:non-logical
3634:Enumeration
3624:Isomorphism
3571:cardinality
3555:Von Neumann
3520:Ultrafilter
3485:Uncountable
3419:equivalence
3336:Quantifiers
3326:Fixed-point
3295:First-order
3175:Consistency
3160:Proposition
3137:Traditional
3108:Lindström's
3098:Compactness
3040:Type theory
2985:Cardinality
2892:Thomas Jech
2735:Alternative
2714:Uncountable
2668:Ultrafilter
2527:Cardinality
2431:replacement
2379:Determinacy
2235:: 161–162,
820:, written {
428:If any set
374:there is a
328:-collection
294:-separation
92:predicative
88:Saul Kripke
4386:elementary
4079:arithmetic
3947:Quantifier
3925:functional
3797:Expression
3515:Transitive
3459:identities
3444:complement
3377:hereditary
3360:Set theory
2887:Kurt Gödel
2872:Paul Cohen
2709:Transitive
2477:Identities
2461:Complement
2448:Operations
2409:Regularity
2347:Adjunction
2306:Set theory
2173:. Berlin:
2111:References
2053:exists by
1977:separation
1891:collection
1889:exists by
1663:, then a Δ
1606:separation
1512:-formula
1503:collection
1501:exists by
557:transitive
552:admissible
549:is called
334:formula φ(
324:Axiom of Δ
300:formula φ(
290:Axiom of Δ
192:) being a
4657:Supertask
4560:Recursion
4518:decidable
4352:saturated
4330:of models
4253:deductive
4248:axiomatic
4168:Hilbert's
4155:Euclidean
4136:canonical
4059:axiomatic
3991:Signature
3920:Predicate
3809:Extension
3731:Ackermann
3656:Operation
3535:Universal
3525:Recursive
3500:Singleton
3495:Inhabited
3480:Countable
3470:Types of
3454:power set
3424:partition
3341:Predicate
3287:Predicate
3202:Syllogism
3192:Soundness
3165:Inference
3155:Tautology
3057:paradoxes
2820:Paradoxes
2740:Axiomatic
2719:Universal
2695:Singleton
2690:Recursive
2633:Countable
2628:Amorphous
2487:Power set
2404:Power set
2362:dependent
2357:countable
2064:Metalogic
2032:∈
2026:∣
2014:×
2005:⋃
1996:×
1957:∈
1951:∣
1939:×
1910:∈
1904:∃
1871:∈
1865:∣
1853:×
1759:ψ
1752:∈
1746:∃
1739:∈
1733:∀
1729:∧
1704:ψ
1697:∈
1691:∃
1684:∈
1678:∀
1642:×
1583:×
1536:ψ
1529:∈
1523:∃
1483:∈
1477:∣
1444:×
1341:∨
1321:∈
1315:∀
1312:∧
1306:∈
1300:∧
1294:∈
1280:∨
1250:∈
1244:∀
1240:∧
1233:∈
1209:∈
1203:∀
1199:∧
1158:∨
1138:∈
1132:∀
1128:∧
1121:∈
1114:∧
1107:∈
1090:∈
1084:∃
1080:∧
1033:∈
1027:∀
1023:∧
1016:∈
999:∈
993:∃
937:ψ
929:-formula
671:α
642:α
610:α
579:⟩
576:∈
567:⟨
555:if it is
474:≠
468:∣
462:∈
424:Empty set
232:empty set
171:(See the
153:∈
147:∃
124:∈
118:∀
4690:Category
4642:Logicism
4635:timeline
4611:Concrete
4470:Validity
4440:T-schema
4433:Kripke's
4428:Tarski's
4423:semantic
4413:Strength
4362:submodel
4357:spectrum
4325:function
4173:Tarski's
4162:Elements
4149:geometry
4105:Robinson
4026:variable
4011:function
3984:spectrum
3974:Sentence
3930:variable
3873:Language
3826:Relation
3787:Automata
3777:Alphabet
3761:language
3615:-jection
3593:codomain
3579:Function
3540:Universe
3510:Infinite
3414:Relation
3197:Validity
3187:Argument
3085:theorem,
2824:Problems
2728:Theories
2704:Superset
2680:Infinite
2509:Concepts
2389:Infinity
2313:Overview
2084:See also
1896:Putting
762:Theorem:
698:Theorems
390:) holds.
82:, is an
4584:Related
4381:Diagram
4279: (
4258:Hilbert
4243:Systems
4238:Theorem
4116:of the
4061:systems
3841:Formula
3836:Grammar
3752: (
3696:General
3409:Forcing
3394:Element
3314:Monadic
3089:paradox
3030:Theorem
2966:General
2762:General
2757:Zermelo
2663:subbase
2645: (
2584:Forcing
2562:Element
2534: (
2512:Methods
2399:Pairing
2269:2615453
2249:2271646
2220:2273185
2206:: 237.
923:pairing
194:formula
108:bounded
4347:finite
4110:Skolem
4063:
4038:Theory
4006:Symbol
3996:String
3979:atomic
3856:ground
3851:closed
3846:atomic
3802:ground
3765:syntax
3661:binary
3588:domain
3505:Finite
3270:finite
3128:Logics
3087:
3035:Theory
2653:Filter
2643:Finite
2579:Family
2522:Almost
2367:global
2352:Choice
2339:Axioms
2267:
2247:
2218:
2181:
2135:
2059:Q.E.D.
1799:Given
812:Proof:
745:. If
528:A set
306:subset
98:Axioms
4337:Model
4085:Peano
3942:Proof
3782:Arity
3711:Naive
3598:image
3530:Fuzzy
3490:Empty
3439:union
3384:Class
3025:Model
3015:Lemma
2973:Axiom
2745:Naive
2675:Fuzzy
2638:Empty
2621:types
2572:tuple
2542:Class
2536:large
2497:Union
2414:Union
2245:JSTOR
2216:JSTOR
2202:(2).
2055:union
1508:The Δ
741:onto
715:is a
593:model
591:is a
240:: If
4460:Type
4263:list
4067:list
4044:list
4033:Term
3967:rank
3861:open
3755:list
3567:Maps
3472:sets
3331:Free
3301:list
3051:list
2978:list
2658:base
2179:ISBN
2133:ISBN
2068:The
800:and
768:and
559:and
260:and
188:: φ(
19:The
4147:of
4129:of
4077:of
3609:Sur
3583:Map
3390:Ur-
3372:Set
2619:Set
2237:doi
2208:doi
2080:.
1611:If
836:.
804:of
796:of
764:If
629:if
598:An
378:in
370:in
208:of
175:.)
139:or
4692::
4533:NP
4157::
4151::
4081::
3758:),
3613:Bi
3605:In
2265:MR
2263:,
2243:,
2233:29
2231:,
2214:.
2200:45
2198:.
2177:.
2057:.
2002::=
1979:.
1893:.
1608:.
1505:.
978:,
873::=
843:,
828:,
808:.
788:,
731:(L
386:,
354:,
338:,
320:.)
252:,
244:,
66:ɑː
25:KP
4613:/
4528:P
4283:)
4069:)
4065:(
3962:∀
3957:!
3952:∃
3913:=
3908:↔
3903:→
3898:∧
3893:∨
3888:¬
3611:/
3607:/
3581:/
3392:)
3388:(
3275:∞
3265:3
3053:)
2951:e
2944:t
2937:v
2702:·
2686:)
2682:(
2649:)
2538:)
2298:e
2291:t
2284:v
2239::
2222:.
2210::
2187:.
2141:.
2038:}
2035:B
2029:b
2023:}
2020:b
2017:{
2011:A
2008:{
1999:B
1993:A
1963:}
1960:B
1954:b
1948:}
1945:b
1942:{
1936:A
1933:{
1913:B
1907:b
1877:}
1874:B
1868:b
1862:}
1859:b
1856:{
1850:A
1847:{
1827:B
1807:A
1784:.
1780:)
1777:p
1774:,
1771:b
1768:,
1765:a
1762:(
1755:A
1749:a
1742:P
1736:p
1725:)
1722:p
1719:,
1716:b
1713:,
1710:a
1707:(
1700:P
1694:p
1687:A
1681:a
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