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bent, who believe that the axiom of constructibility is either true or false, most believe that it is false. This is in part because it seems unnecessarily "restrictive", as it allows only certain subsets of a given set (for example,
151:
without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of
ZermeloâFraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the
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Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as an axiom for set theory in the same way as the ZFC axioms. Among set theorists of a
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formula known as the "analytical form of the axiom of constructibility" that has some associations to the set-theoretic axiom V=L. For example, some cases where
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can't exist), with no clear reason to believe that these are all of them. In part it is because the axiom is contradicted by sufficiently strong
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Especially from the 1950s to the 1970s, there have been some investigations into formulating an analogue of the axiom of constructibility for
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1215:"Embeddings of the universe into the constructible universe, current state of knowledge, CUNY Set Theory Seminar, March 2015"
635:{\displaystyle M\cap {\mathcal {P}}(\omega )\vDash {\textrm {Analytical}}\;{\textrm {form}}\;{\textrm {of}}\;{\textrm {V=L}}}
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1248:, Observations Concerning Elementary Extensions of Ï-models. II (1973, p.227). Accessed 2021 November 3.
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Stable sets, a characterization of ÎČâ-models of full second-order arithmetic and some related facts
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of all sets (the formula for which can be given explicitly). In particular,
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Accepting the axiom of constructibility (which asserts that every set is
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Inductive
Definitions and Reflecting Properties of Admissible Ordinals
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thereby establishing that AC and GCH are also relatively consistent.
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2014:
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430:{\displaystyle {\mathcal {P}}(\omega )\vDash {\textrm {Constr}}(X)}
251:) that endow those cardinals with their large cardinal properties.
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The axiom of constructibility implies the non-existence of those
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The major significance of the axiom of constructibility is in
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of those large cardinals (when they exist in a supermodel of
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316:. A few results stand out in the study of such analogues:
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The existence of a primitive recursive class surjection
1259:Ï-models of second-order arithmetic and admissible sets
296:. This point of view is especially associated with the
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917:) are also relatively consistent to ZF set theory.
1170:{\displaystyle F:{\textrm {Ord}}\to {\textrm {V}}}
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475:{\displaystyle X\in {\mathcal {P}}(\omega )\cap L}
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854:Gödel's proof was complemented in later years by
131:). Generalizations of this axiom are explored in
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994:Here is a list of propositions that hold in the
46:but its sources remain unclear because it lacks
101:in mathematics that asserts that every set is
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247:, it excludes the auxiliary structures (e.g.
285:{\displaystyle 0^{\sharp }\subseteq \omega }
119:, is inconsistent with the proposition that
16:Possible axiom for set theory in mathematics
962:. Unsourced material may be challenged and
862:, i.e. that the negations of these axioms (
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314:subsystems of second-order arithmetic
243:), and they are still initial ordinals in
143:The axiom of constructibility implies the
1213:Hamkins, Joel David (February 27, 2015).
982:Learn how and when to remove this message
77:Learn how and when to remove this message
1277:(pp.176--177). Accessed 2021 November 3.
1261:(1975, p.105). Accessed 2021 November 3.
841:{\displaystyle V=L\implies AC\land GCH,}
1212:
920:
748:can prove a contradiction, then so can
562:{\displaystyle M\vDash {\textrm {V=L}}}
204:, all of which are independent of ZFC.
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1379:
1303:
1294:(1974, p.23). Accessed 30 August 2022.
1185:) these propositions also hold in the
3111:
1353:
378:{\displaystyle {\textrm {Constr}}(X)}
960:adding citations to reliable sources
927:
669:Von NeumannâBernaysâGödel set theory
18:
1344:Mathematical Association of America
1227:from the original on April 23, 2024
858:'s result that both AC and GCH are
115:. The axiom, first investigated by
13:
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869:
704:is relatively consistent (i.e. if
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515:
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172:
105:. The axiom is usually written as
14:
3787:
1331:
129:list of large cardinal properties
3161:
3091:
1339:How many real numbers are there?
1008:generalized continuum hypothesis
932:
665:generalized continuum hypothesis
307:
300:, or the "California school" as
154:generalized continuum hypothesis
23:
646:
345:{\displaystyle \Sigma _{2}^{1}}
189:{\displaystyle \Delta _{2}^{1}}
138:
3137:
1280:
1251:
1239:
1206:
1157:
1061:that imply the existence of a
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735:
723:
671:. (The proof carries over to
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463:
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366:
1:
3052:History of mathematical logic
1199:
1122:The existence of a definable
2977:Primitive recursive function
532:{\displaystyle \Pi _{3}^{1}}
7:
1085:) set of reals that is not
673:ZermeloâFraenkel set theory
149:ZermeloâFraenkel set theory
10:
3792:
3621:von NeumannâBernaysâGödel
2041:SchröderâBernstein theorem
1768:Monadic predicate calculus
1427:Foundations of mathematics
160:, and the existence of an
3685:
3648:
3560:
3450:
3422:One-to-one correspondence
3338:
3279:
3170:
3159:
3145:
3087:
3074:Philosophy of mathematics
3023:Automated theorem proving
3005:
2900:
2732:
2625:
2477:
2194:
2170:
2148:Von NeumannâBernaysâGödel
2093:
1987:
1891:
1789:
1780:
1707:
1642:
1548:
1470:
1387:
1057:The non existence of all
910:{\displaystyle \lnot GCH}
741:{\displaystyle ZFC+(V=L)}
678:Namely Gödel proved that
655:'s proof of the relative
91:axiom of constructibility
881:{\displaystyle \lnot AC}
502:is a constructible real.
32:This article includes a
2724:Self-verifying theories
2545:Tarski's axiomatization
1496:Tarski's undefinability
1491:incompleteness theorems
61:more precise citations.
3776:Constructible universe
3380:Constructible universe
3207:Constructibility (V=L)
3098:Mathematics portal
2709:Proof of impossibility
2357:propositional variable
1667:Propositional calculus
1171:
1094:Whitehead's conjecture
996:constructible universe
911:
882:
842:
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765:
742:
698:
636:
563:
533:
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476:
431:
379:
346:
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190:
3603:Principia Mathematica
3437:Transfinite induction
3296:(i.e. set difference)
2967:Kolmogorov complexity
2920:Computably enumerable
2820:Model complete theory
2612:Principia Mathematica
1672:Propositional formula
1501:BanachâTarski paradox
1172:
1081:(with respect to the
1054:and as a consequence
1050:The non-existence of
1010:and as a consequence
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843:
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637:
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534:
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477:
432:
380:
347:
294:large cardinal axioms
287:
191:
125:large cardinal axioms
3771:Axioms of set theory
3677:Burali-Forti paradox
3432:Set-builder notation
3385:Continuum hypothesis
3325:Symmetric difference
2915:ChurchâTuring thesis
2902:Computability theory
2111:continuum hypothesis
1629:Square of opposition
1487:Gödel's completeness
1187:von Neumann universe
1141:
1083:analytical hierarchy
1044:The negation of the
956:improve this section
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866:
800:
775:
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511:
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389:
356:
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215:greater or equal to
213:consistency strength
168:
123:exists and stronger
3638:TarskiâGrothendieck
3069:Mathematical object
2960:P versus NP problem
2925:Computable function
2719:Reverse mathematics
2645:Logical consequence
2522:primitive recursive
2517:elementary function
2290:Free/bound variable
2143:TarskiâGrothendieck
1662:Logical connectives
1592:Logical equivalence
1442:Logical consequence
1069:The existence of a
1063:measurable cardinal
921:Statements true in
697:{\displaystyle V=L}
528:
341:
185:
158:Suslin's hypothesis
3227:Limitation of size
2867:Transfer principle
2830:Semantics of logic
2815:Categorical theory
2791:Non-standard model
2305:Logical connective
1432:Information theory
1381:Mathematical logic
1167:
1117:free abelian group
907:
878:
838:
787:{\displaystyle ZF}
784:
764:{\displaystyle ZF}
761:
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156:, the negation of
133:inner model theory
34:list of references
3758:
3757:
3667:Russell's paradox
3616:ZermeloâFraenkel
3517:Dedekind-infinite
3390:Diagonal argument
3289:Cartesian product
3153:Set (mathematics)
3105:
3104:
3037:Abstract category
2840:Theories of truth
2650:Rule of inference
2640:Natural deduction
2621:
2620:
2166:
2165:
1871:Cartesian product
1776:
1775:
1682:Many-valued logic
1657:Boolean functions
1540:Russell's paradox
1515:diagonal argument
1412:First-order logic
1164:
1154:
1046:Suslin hypothesis
1038:The existence of
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991:
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495:{\displaystyle X}
415:
363:
235:does contain the
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86:
79:
3783:
3740:Bertrand Russell
3730:John von Neumann
3715:Abraham Fraenkel
3710:Richard Dedekind
3672:Suslin's problem
3583:Cantor's theorem
3300:De Morgan's laws
3165:
3132:
3125:
3118:
3109:
3108:
3096:
3095:
3047:History of logic
3042:Category of sets
2935:Decision problem
2714:Ordinal analysis
2655:Sequent calculus
2553:Boolean algebras
2493:
2492:
2467:
2438:logical/constant
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2191:
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2101:ZermeloâFraenkel
1852:Set operations:
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1724:
1555:
1554:
1535:LöwenheimâSkolem
1422:Formal semantics
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1367:
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1351:
1350:
1342:, Keith Devlin,
1326:
1309:Constructibility
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237:initial ordinals
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57:this article by
48:inline citations
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3609:New Foundations
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3365:Cardinal number
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3028:Category theory
3018:Algebraic logic
3001:
2972:Lambda calculus
2910:Church encoding
2896:
2872:Truth predicate
2728:
2694:Complete theory
2617:
2486:
2482:
2478:
2473:
2465:
2185: and
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2162:
2138:New Foundations
2106:axiom of choice
2089:
2051:Gödel numbering
1991: and
1983:
1887:
1772:
1722:
1703:
1652:Boolean algebra
1638:
1602:Equiconsistency
1567:Classical logic
1544:
1525:Halting problem
1513: and
1489: and
1477: and
1476:
1471:Theorems (
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1313:Springer-Verlag
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1220:jdh.hamkins.org
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1059:large cardinals
1015:axiom of choice
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771:), and that in
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661:axiom of choice
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320:John Addison's
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209:large cardinals
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145:axiom of choice
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38:related reading
28:
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11:
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3745:Thoralf Skolem
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3702:
3697:
3691:
3689:
3683:
3682:
3680:
3679:
3674:
3669:
3663:
3661:
3659:
3658:
3655:
3649:
3646:
3645:
3643:
3642:
3641:
3640:
3635:
3630:
3629:
3628:
3613:
3612:
3611:
3599:
3598:
3597:
3586:
3585:
3580:
3575:
3570:
3564:
3562:
3558:
3557:
3555:
3554:
3549:
3544:
3539:
3530:
3525:
3520:
3510:
3505:
3504:
3503:
3498:
3493:
3483:
3473:
3468:
3463:
3457:
3455:
3448:
3447:
3445:
3444:
3439:
3434:
3429:
3427:Ordinal number
3424:
3419:
3414:
3409:
3408:
3407:
3402:
3392:
3387:
3382:
3377:
3372:
3362:
3357:
3351:
3349:
3347:
3346:
3343:
3339:
3336:
3335:
3333:
3332:
3327:
3322:
3317:
3312:
3307:
3305:Disjoint union
3302:
3297:
3291:
3285:
3283:
3277:
3276:
3274:
3273:
3272:
3271:
3266:
3255:
3254:
3252:Martin's axiom
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3217:Extensionality
3214:
3209:
3204:
3203:
3202:
3197:
3192:
3182:
3176:
3174:
3168:
3167:
3160:
3158:
3156:
3155:
3149:
3147:
3143:
3142:
3135:
3134:
3127:
3120:
3112:
3103:
3102:
3088:
3085:
3084:
3082:
3081:
3076:
3071:
3066:
3061:
3060:
3059:
3049:
3044:
3039:
3030:
3025:
3020:
3015:
3013:Abstract logic
3009:
3007:
3003:
3002:
3000:
2999:
2994:
2992:Turing machine
2989:
2984:
2979:
2974:
2969:
2964:
2963:
2962:
2957:
2952:
2947:
2942:
2932:
2930:Computable set
2927:
2922:
2917:
2912:
2906:
2904:
2898:
2897:
2895:
2894:
2889:
2884:
2879:
2874:
2869:
2864:
2859:
2858:
2857:
2852:
2847:
2837:
2832:
2827:
2825:Satisfiability
2822:
2817:
2812:
2811:
2810:
2800:
2799:
2798:
2788:
2787:
2786:
2781:
2776:
2771:
2766:
2756:
2755:
2754:
2749:
2742:Interpretation
2738:
2736:
2730:
2729:
2727:
2726:
2721:
2716:
2711:
2706:
2696:
2691:
2690:
2689:
2688:
2687:
2677:
2672:
2662:
2657:
2652:
2647:
2642:
2637:
2631:
2629:
2623:
2622:
2619:
2618:
2616:
2615:
2607:
2606:
2605:
2604:
2599:
2598:
2597:
2592:
2587:
2567:
2566:
2565:
2563:minimal axioms
2560:
2549:
2548:
2547:
2536:
2535:
2534:
2529:
2524:
2519:
2514:
2509:
2496:
2494:
2475:
2474:
2472:
2471:
2470:
2469:
2457:
2452:
2451:
2450:
2445:
2440:
2435:
2425:
2420:
2415:
2410:
2409:
2408:
2403:
2393:
2392:
2391:
2386:
2381:
2376:
2366:
2361:
2360:
2359:
2354:
2349:
2339:
2338:
2337:
2332:
2327:
2322:
2317:
2312:
2302:
2297:
2292:
2287:
2286:
2285:
2280:
2275:
2270:
2260:
2255:
2253:Formation rule
2250:
2245:
2244:
2243:
2238:
2228:
2227:
2226:
2216:
2211:
2206:
2201:
2195:
2189:
2172:Formal systems
2168:
2167:
2164:
2163:
2161:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2114:
2113:
2108:
2097:
2095:
2091:
2090:
2088:
2087:
2086:
2085:
2075:
2070:
2069:
2068:
2061:Large cardinal
2058:
2053:
2048:
2043:
2038:
2024:
2023:
2022:
2017:
2012:
1997:
1995:
1985:
1984:
1982:
1981:
1980:
1979:
1974:
1969:
1959:
1954:
1949:
1944:
1939:
1934:
1929:
1924:
1919:
1914:
1909:
1904:
1898:
1896:
1889:
1888:
1886:
1885:
1884:
1883:
1878:
1873:
1868:
1863:
1858:
1850:
1849:
1848:
1843:
1833:
1828:
1826:Extensionality
1823:
1821:Ordinal number
1818:
1808:
1803:
1802:
1801:
1790:
1784:
1778:
1777:
1774:
1773:
1771:
1770:
1765:
1760:
1755:
1750:
1745:
1740:
1739:
1738:
1728:
1727:
1726:
1713:
1711:
1705:
1704:
1702:
1701:
1700:
1699:
1694:
1689:
1679:
1674:
1669:
1664:
1659:
1654:
1648:
1646:
1640:
1639:
1637:
1636:
1631:
1626:
1621:
1616:
1611:
1606:
1605:
1604:
1594:
1589:
1584:
1579:
1574:
1569:
1563:
1561:
1552:
1546:
1545:
1543:
1542:
1537:
1532:
1527:
1522:
1517:
1505:Cantor's
1503:
1498:
1493:
1483:
1481:
1468:
1467:
1465:
1464:
1459:
1454:
1449:
1444:
1439:
1434:
1429:
1424:
1419:
1414:
1409:
1404:
1403:
1402:
1391:
1389:
1385:
1384:
1377:
1376:
1369:
1362:
1354:
1348:
1347:
1333:
1332:External links
1330:
1328:
1327:
1321:
1300:
1297:
1296:
1279:
1263:
1250:
1238:
1204:
1203:
1201:
1198:
1179:
1178:
1159:
1149:
1146:
1135:
1120:
1090:
1076:
1067:
1066:
1065:
1048:
1042:
1036:
1031:
1030:
1029:
1019:
1018:
1017:
990:
989:
940:
938:
931:
925:
919:
906:
903:
900:
897:
877:
874:
871:
849:
848:
837:
834:
831:
828:
825:
822:
819:
815:
811:
808:
805:
783:
780:
760:
757:
737:
734:
731:
728:
725:
722:
719:
716:
713:
693:
690:
687:
648:
645:
644:
643:
600:
597:
594:
591:
586:
581:
578:
551:
548:
526:
521:
517:
504:
503:
491:
471:
468:
465:
462:
459:
454:
449:
446:
426:
423:
420:
410:
407:
404:
401:
396:
374:
371:
368:
339:
334:
330:
309:
306:
302:Saharon Shelah
281:
278:
273:
269:
220:
198:non-measurable
183:
178:
174:
140:
137:
93:is a possible
85:
84:
42:external links
31:
29:
22:
15:
9:
6:
4:
3:
2:
3788:
3777:
3774:
3772:
3769:
3768:
3766:
3751:
3750:Ernst Zermelo
3748:
3746:
3743:
3741:
3738:
3736:
3735:Willard Quine
3733:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3698:
3696:
3693:
3692:
3690:
3688:
3687:Set theorists
3684:
3678:
3675:
3673:
3670:
3668:
3665:
3664:
3662:
3656:
3654:
3651:
3650:
3647:
3639:
3636:
3634:
3633:KripkeâPlatek
3631:
3627:
3624:
3623:
3622:
3619:
3618:
3617:
3614:
3610:
3607:
3606:
3605:
3604:
3600:
3596:
3593:
3592:
3591:
3588:
3587:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3565:
3563:
3559:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3536:
3531:
3529:
3526:
3524:
3521:
3518:
3514:
3511:
3509:
3506:
3502:
3499:
3497:
3494:
3492:
3489:
3488:
3487:
3484:
3481:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3459:
3458:
3456:
3453:
3449:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3406:
3403:
3401:
3398:
3397:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3376:
3373:
3370:
3366:
3363:
3361:
3358:
3356:
3353:
3352:
3350:
3344:
3341:
3340:
3337:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3295:
3292:
3290:
3287:
3286:
3284:
3282:
3278:
3270:
3269:specification
3267:
3265:
3262:
3261:
3260:
3257:
3256:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3201:
3198:
3196:
3193:
3191:
3188:
3187:
3186:
3183:
3181:
3178:
3177:
3175:
3173:
3169:
3164:
3154:
3151:
3150:
3148:
3144:
3140:
3133:
3128:
3126:
3121:
3119:
3114:
3113:
3110:
3100:
3099:
3094:
3086:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3058:
3055:
3054:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3014:
3011:
3010:
3008:
3004:
2998:
2995:
2993:
2990:
2988:
2987:Recursive set
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2965:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2937:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2907:
2905:
2903:
2899:
2893:
2890:
2888:
2885:
2883:
2880:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2860:
2856:
2853:
2851:
2848:
2846:
2843:
2842:
2841:
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2809:
2806:
2805:
2804:
2801:
2797:
2796:of arithmetic
2794:
2793:
2792:
2789:
2785:
2782:
2780:
2777:
2775:
2772:
2770:
2767:
2765:
2762:
2761:
2760:
2757:
2753:
2750:
2748:
2745:
2744:
2743:
2740:
2739:
2737:
2735:
2731:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2704:
2703:from ZFC
2700:
2697:
2695:
2692:
2686:
2683:
2682:
2681:
2678:
2676:
2673:
2671:
2668:
2667:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2632:
2630:
2628:
2624:
2614:
2613:
2609:
2608:
2603:
2602:non-Euclidean
2600:
2596:
2593:
2591:
2588:
2586:
2585:
2581:
2580:
2578:
2575:
2574:
2572:
2568:
2564:
2561:
2559:
2556:
2555:
2554:
2550:
2546:
2543:
2542:
2541:
2537:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2508:
2505:
2504:
2502:
2498:
2497:
2495:
2490:
2484:
2479:Example
2476:
2468:
2463:
2462:
2461:
2458:
2456:
2453:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2430:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2407:
2404:
2402:
2399:
2398:
2397:
2394:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2371:
2370:
2367:
2365:
2362:
2358:
2355:
2353:
2350:
2348:
2345:
2344:
2343:
2340:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2307:
2306:
2303:
2301:
2298:
2296:
2293:
2291:
2288:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2265:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2242:
2239:
2237:
2236:by definition
2234:
2233:
2232:
2229:
2225:
2222:
2221:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2196:
2193:
2190:
2188:
2184:
2179:
2173:
2169:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2123:KripkeâPlatek
2121:
2119:
2116:
2112:
2109:
2107:
2104:
2103:
2102:
2099:
2098:
2096:
2092:
2084:
2081:
2080:
2079:
2076:
2074:
2071:
2067:
2064:
2063:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2036:
2032:
2028:
2025:
2021:
2018:
2016:
2013:
2011:
2008:
2007:
2006:
2002:
1999:
1998:
1996:
1994:
1990:
1986:
1978:
1975:
1973:
1970:
1968:
1967:constructible
1965:
1964:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1899:
1897:
1895:
1890:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1853:
1851:
1847:
1844:
1842:
1839:
1838:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1813:
1809:
1807:
1804:
1800:
1797:
1796:
1795:
1792:
1791:
1788:
1785:
1783:
1779:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1737:
1734:
1733:
1732:
1729:
1725:
1720:
1719:
1718:
1715:
1714:
1712:
1710:
1706:
1698:
1695:
1693:
1690:
1688:
1685:
1684:
1683:
1680:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1658:
1655:
1653:
1650:
1649:
1647:
1645:
1644:Propositional
1641:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1617:
1615:
1612:
1610:
1607:
1603:
1600:
1599:
1598:
1595:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1572:Logical truth
1570:
1568:
1565:
1564:
1562:
1560:
1556:
1553:
1551:
1547:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1512:
1508:
1504:
1502:
1499:
1497:
1494:
1492:
1488:
1485:
1484:
1482:
1480:
1474:
1469:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1405:
1401:
1398:
1397:
1396:
1393:
1392:
1390:
1386:
1382:
1375:
1370:
1368:
1363:
1361:
1356:
1355:
1352:
1345:
1341:
1340:
1336:
1335:
1324:
1322:3-540-13258-9
1318:
1314:
1310:
1306:
1305:Devlin, Keith
1302:
1301:
1293:
1289:
1283:
1276:
1270:
1268:
1260:
1254:
1247:
1242:
1231:September 22,
1226:
1222:
1221:
1216:
1209:
1205:
1197:
1194:
1192:
1188:
1184:
1183:constructible
1147:
1144:
1136:
1133:
1129:
1125:
1121:
1118:
1114:
1110:
1106:
1102:
1099:
1098:abelian group
1095:
1092:The truth of
1091:
1088:
1084:
1072:
1068:
1064:
1060:
1056:
1055:
1053:
1049:
1047:
1043:
1041:
1037:
1035:
1034:Global square
1032:
1028:
1025:
1024:
1023:
1020:
1016:
1012:
1011:
1009:
1005:
1004:
1003:
1001:
997:
986:
983:
975:
972:November 2017
965:
961:
957:
951:
950:
946:
941:This section
939:
935:
930:
929:
924:
918:
904:
901:
898:
875:
872:
861:
857:
852:
835:
832:
829:
826:
823:
820:
817:
809:
806:
803:
796:
795:
794:
781:
778:
758:
755:
732:
729:
726:
720:
717:
714:
711:
691:
688:
685:
676:
674:
670:
666:
662:
658:
654:
598:
592:
579:
576:
549:
546:
524:
519:
506:
505:
489:
469:
466:
460:
447:
444:
421:
408:
402:
369:
337:
332:
319:
318:
317:
315:
308:In arithmetic
305:
303:
299:
295:
279:
276:
267:
257:
252:
250:
246:
242:
238:
234:
230:
226:
218:
214:
210:
205:
203:
199:
181:
176:
163:
159:
155:
150:
146:
136:
134:
130:
126:
122:
118:
114:
113:
109:
104:
103:constructible
100:
96:
92:
81:
78:
70:
60:
56:
50:
49:
43:
39:
35:
30:
21:
20:
3700:Georg Cantor
3695:Paul Bernays
3626:MorseâKelley
3601:
3534:
3533:Subset
3480:hereditarily
3442:Venn diagram
3400:ordered pair
3315:Intersection
3259:Axiom schema
3206:
3089:
2887:Ultraproduct
2734:Model theory
2699:Independence
2635:Formal proof
2627:Proof theory
2610:
2583:
2540:real numbers
2512:second-order
2423:Substitution
2300:Metalanguage
2241:conservative
2214:Axiom schema
2158:Constructive
2128:MorseâKelley
2094:Set theories
2073:Aleph number
2066:inaccessible
1972:Grothendieck
1856:intersection
1743:Higher-order
1731:Second-order
1677:Truth tables
1634:Venn diagram
1417:Formal proof
1338:
1308:
1286:W. Richter,
1282:
1253:
1241:
1229:. Retrieved
1218:
1208:
1195:
1180:
1127:
1112:
1108:
1100:
1070:
999:
998:(denoted by
993:
978:
969:
954:Please help
942:
922:
859:
853:
850:
677:
650:
647:Significance
311:
253:
244:
240:
232:
228:
206:
202:real numbers
147:(AC), given
142:
139:Implications
111:
107:
106:
90:
88:
73:
64:
53:Please help
45:
3725:Thomas Jech
3568:Alternative
3547:Uncountable
3501:Ultrafilter
3360:Cardinality
3264:replacement
3212:Determinacy
2997:Type theory
2945:undecidable
2877:Truth value
2764:equivalence
2443:non-logical
2056:Enumeration
2046:Isomorphism
1993:cardinality
1977:Von Neumann
1942:Ultrafilter
1907:Uncountable
1841:equivalence
1758:Quantifiers
1748:Fixed-point
1717:First-order
1597:Consistency
1582:Proposition
1559:Traditional
1530:Lindström's
1520:Compactness
1462:Type theory
1407:Cardinality
1346:, June 2001
1115:) = 0 is a
1096:that every
1022:Diamondsuit
860:independent
657:consistency
507:There is a
59:introducing
3765:Categories
3720:Kurt Gödel
3705:Paul Cohen
3542:Transitive
3310:Identities
3294:Complement
3281:Operations
3242:Regularity
3180:Adjunction
3139:Set theory
2808:elementary
2501:arithmetic
2369:Quantifier
2347:functional
2219:Expression
1937:Transitive
1881:identities
1866:complement
1799:hereditary
1782:Set theory
1273:W. Marek,
1257:W. Marek,
1200:References
1130:satisfies
1124:well-order
1087:measurable
856:Paul Cohen
653:Kurt Gödel
604:Analytical
385:such that
164:(in fact,
162:analytical
121:zero sharp
117:Kurt Gödel
99:set theory
3653:Paradoxes
3573:Axiomatic
3552:Universal
3528:Singleton
3523:Recursive
3466:Countable
3461:Amorphous
3320:Power set
3237:Power set
3195:dependent
3190:countable
3079:Supertask
2982:Recursion
2940:decidable
2774:saturated
2752:of models
2675:deductive
2670:axiomatic
2590:Hilbert's
2577:Euclidean
2558:canonical
2481:axiomatic
2413:Signature
2342:Predicate
2231:Extension
2153:Ackermann
2078:Operation
1957:Universal
1947:Recursive
1922:Singleton
1917:Inhabited
1902:Countable
1892:Types of
1876:power set
1846:partition
1763:Predicate
1709:Predicate
1624:Syllogism
1614:Soundness
1587:Inference
1577:Tautology
1479:paradoxes
1158:→
943:does not
896:¬
870:¬
824:∧
814:⟹
599:⊨
593:ω
580:∩
550:⊨
516:Π
467:∩
461:ω
448:∈
409:⊨
403:ω
329:Σ
280:ω
277:⊆
272:♯
231:. While
173:Δ
3657:Problems
3561:Theories
3537:Superset
3513:Infinite
3342:Concepts
3222:Infinity
3146:Overview
3064:Logicism
3057:timeline
3033:Concrete
2892:Validity
2862:T-schema
2855:Kripke's
2850:Tarski's
2845:semantic
2835:Strength
2784:submodel
2779:spectrum
2747:function
2595:Tarski's
2584:Elements
2571:geometry
2527:Robinson
2448:variable
2433:function
2406:spectrum
2396:Sentence
2352:variable
2295:Language
2248:Relation
2209:Automata
2199:Alphabet
2183:language
2037:-jection
2015:codomain
2001:Function
1962:Universe
1932:Infinite
1836:Relation
1619:Validity
1609:Argument
1507:theorem,
1307:(1984).
1288:P. Aczel
1246:W. Marek
1225:Archived
1191:analysis
1040:morasses
1027:Clubsuit
663:and the
352:formula
249:measures
67:May 2017
3595:General
3590:Zermelo
3496:subbase
3478: (
3417:Forcing
3395:Element
3367: (
3345:Methods
3232:Pairing
3006:Related
2803:Diagram
2701: (
2680:Hilbert
2665:Systems
2660:Theorem
2538:of the
2483:systems
2263:Formula
2258:Grammar
2174: (
2118:General
1831:Forcing
1816:Element
1736:Monadic
1511:paradox
1452:Theorem
1388:General
964:removed
949:sources
659:of the
482:, i.e.
256:realist
200:set of
55:improve
3486:Filter
3476:Finite
3412:Family
3355:Almost
3200:global
3185:Choice
3172:Axioms
2769:finite
2532:Skolem
2485:
2460:Theory
2428:Symbol
2418:String
2401:atomic
2278:ground
2273:closed
2268:atomic
2224:ground
2187:syntax
2083:binary
2010:domain
1927:Finite
1692:finite
1550:Logics
1509:
1457:Theory
1319:
414:Constr
362:Constr
3578:Naive
3508:Fuzzy
3471:Empty
3454:types
3405:tuple
3375:Class
3369:large
3330:Union
3247:Union
2759:Model
2507:Peano
2364:Proof
2204:Arity
2133:Naive
2020:image
1952:Fuzzy
1912:Empty
1861:union
1806:Class
1447:Model
1437:Lemma
1395:Axiom
1132:V=HOD
1103:with
298:Cabal
225:ErdĆs
211:with
127:(see
95:axiom
40:, or
3491:base
2882:Type
2685:list
2489:list
2466:list
2455:Term
2389:rank
2283:open
2177:list
1989:Maps
1894:sets
1753:Free
1723:list
1473:list
1400:list
1317:ISBN
1233:2024
1013:The
1006:The
947:any
945:cite
888:and
612:form
569:iff
437:iff
97:for
89:The
3452:Set
2569:of
2551:of
2499:of
2031:Sur
2005:Map
1812:Ur-
1794:Set
1153:Ord
1105:Ext
1002:):
958:by
667:to
628:V=L
555:V=L
227:in
3767::
2955:NP
2579::
2573::
2503::
2180:),
2035:Bi
2027:In
1315:.
1311:.
1290:,
1266:^
1223:.
1217:.
1193:.
1111:,
620:of
196:)
135:.
110:=
44:,
36:,
3535:·
3519:)
3515:(
3482:)
3371:)
3131:e
3124:t
3117:v
3035:/
2950:P
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2491:)
2487:(
2384:â
2379:!
2374:â
2335:=
2330:â
2325:â
2320:â§
2315:âš
2310:ÂŹ
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2029:/
2003:/
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1810:(
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1325:.
1235:.
1163:V
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1107:(
1101:A
1089:.
1077:2
1071:ÎŁ
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1000:L
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966:.
952:.
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876:C
873:A
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833:H
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818:A
810:L
807:=
804:V
782:F
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727:V
724:(
721:+
718:C
715:F
712:Z
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689:=
686:V
596:)
590:(
585:P
577:M
547:M
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458:(
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419:(
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400:(
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177:2
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