17604:
15256:
536:
6292:
5527:
13408:
4515:
345:
7855:
Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets that are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded.
6977:
12568:
336:
By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model.
6420:
11786:
5833:
5346:
4059:
13204:
1046:
6118:
5351:
10110:; if there are multiple sequences of parameters that define one of the sets, we choose the least one under this ordering. It being understood that each parameter's possible values are ordered according to the restriction of the ordering of
11220:
4349:
13546:
Actually, even this complex formula has been simplified from what the instructions given in the first paragraph would yield. But the point remains, there is a formula of set theory that is true only for the desired constructible set
1408:
13542:
14040:
4991:
747:
6853:
531:{\displaystyle \operatorname {Def} (X):={\Bigl \{}\{y\mid y\in X{\text{ and }}(X,\in )\models \Phi (y,z_{1},\ldots ,z_{n})\}~{\Big |}~\Phi {\text{ is a first-order formula and }}z_{1},\ldots ,z_{n}\in X{\Bigr \}}.}
12415:
877:
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14103:
8225:
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2312:
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12853:
6546:
7500:
2791:
11641:
1892:
8262:
5711:
4896:
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159:
of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.
10901:
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14374:
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600:
12634:
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9782:
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773:
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13595:, but that includes or is influenced by a set that is not constructible. This gives rise to the concept of relative constructibility, of which there are two flavors, denoted by
12410:
10104:
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2128:
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10671:
9203:
941:
13891:
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4243:
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3897:
14254:
10052:
14653:
12917:
9868:
9554:
9229:
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13142:
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11893:
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10271:
9042:
7533:
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2663:
2615:
2562:
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2429:
2396:
2239:
1726:
1693:
1660:
1106:
249:
10605:
10200:-tuples of parameters are well-ordered by the product ordering. The formulas with parameters are well-ordered by the ordered sum (by Gödel numbers) of well-orderings. And
9095:
6287:{\displaystyle \{y\mid y\in L\;\mathrm {and} \;\mathrm {there} \;\mathrm {exists} \;x\in S\;\mathrm {such} \;\mathrm {that} \;P(x,y)\;\mathrm {holds} \;\mathrm {in} \;L\}}
5522:{\displaystyle \{x\mid x\in L_{\alpha }\;\mathrm {and} \;x\in S\;\mathrm {and} \;P(x,z_{1},\ldots ,z_{n})\;\mathrm {holds} \;\mathrm {in} \;L_{\alpha }\}\in L_{\alpha +1}}
4807:
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12003:
11840:
11506:
11351:
10155:
10000:
9908:
9650:
9580:
9322:
8711:
8616:
8562:
8535:
8468:
8151:
8078:
8051:
8004:
7840:
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7396:
6848:
5176:
5063:
4681:
4344:
4297:
4270:
3261:
2509:
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2175:
2068:
1994:
1966:
1859:
1832:
1753:
1579:
1552:
1316:
1137:
1073:
936:
280:
129:
14501:
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11813:
11559:
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10846:
10799:
3687:
14463:
13966:
13797:
3209:
3177:
2898:
815:
684:
135:
in his 1938 paper "The
Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an
13109:
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2582:
2449:
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2014:
1912:
1339:
793:
14576:
13445:
13403:{\displaystyle \forall y(y\in s\iff (y\in L_{\omega +1}\land (\forall a(a\in y\iff a\in L_{5}\land Ord(a))\lor \forall b(b\in y\iff b\in L_{\omega }\land Ord(b)))))}
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11459:
11432:
11378:
11269:
11242:
9716:
9603:
9486:
9466:
9426:
9386:
4510:{\displaystyle y=\{s\mid s\in L_{\alpha }\;\mathrm {and} \;\mathrm {there} \;\mathrm {exists} \;z\in x\;\mathrm {such} \;\mathrm {that} \;s\in z\}\in L_{\alpha +1}}
3658:
1485:
1458:
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2529:
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1938:
1805:
1238:
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6821:
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6758:
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2148:
1627:
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1188:
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909:
557:
300:
208:
184:
99:
71:
2630:
17:
15983:
8856:
in his 1972 paper entitled "The fine structure of the constructible hierarchy". Instead of explaining the fine structure, we will give an outline of how
13971:
16658:
15720:
6972:{\displaystyle \{z\mid z\in L_{\alpha }\;\mathrm {and} \;z\;\mathrm {is} \;\mathrm {a} \;\mathrm {subset} \;\mathrm {of} \;x\}\in L_{\alpha +1}}
4901:
12563:{\displaystyle S=\{y\mid y\in L_{\alpha }\;\mathrm {and} \;\Phi (y,z_{1},\ldots ,z_{n})\;\mathrm {holds} \;\mathrm {in} \;(L_{\alpha },\in )\}}
693:
16741:
15882:
10463:
It is well known that the axiom of choice is equivalent to the ability to well-order every set. Being able to well-order the proper class
10180:
The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induction. The values of
822:
606:
12008:
17055:
148:
9237:
12100:
3557:
17213:
15161:
16001:
17068:
16391:
15409:
15222:
14047:
6415:{\displaystyle \{y\mid \mathrm {there} \;\mathrm {exists} \;\mathrm {x} \in S\;\mathrm {such} \;\mathrm {that} \;Q(x,y)\}}
14871:
are usually not actually constructible, and the properties of these models may be quite different from the properties of
8190:
17073:
17063:
16800:
16653:
16006:
15737:
15997:
17209:
15191:
15079:
15060:
15038:
8417:
2820:
2715:
2271:
16551:
13896:
12791:
17306:
17050:
15875:
12269:
8377:
11781:{\displaystyle T=\{x\in L_{\beta }:x\in S\wedge \Phi (x,z_{i})\}=\{x\in L_{\gamma }:x\in S\wedge \Phi (x,w_{i})\}}
6505:
17628:
16611:
16304:
15715:
15309:
10400:(some other standard model of ZF with the same ordinals) and we will suppose that the formula is false if either
7461:
5828:{\displaystyle \{y\mid \;\mathrm {there} \;\mathrm {exists} \;x\in S\;\mathrm {such} \;\mathrm {that} \;P(x,y)\}}
3049:
2759:
1586:
140:
17633:
16045:
15595:
11299:
7849:
17567:
17269:
17032:
17027:
16852:
16273:
15957:
1864:
8230:
5341:{\displaystyle \{x\mid x\in S\;\mathrm {and} \;P(x,z_{1},\ldots ,z_{n})\;\mathrm {holds} \;\mathrm {in} \;L\}}
4840:
4747:
4054:{\displaystyle \{x,y\}=\{s\mid s\in L_{\alpha }\;\mathrm {and} \;(s=x\;\mathrm {or} \;s=y)\}\in L_{\alpha +1}}
17562:
17345:
17262:
16975:
16906:
16783:
16025:
15489:
15368:
10851:
10676:
795:
17487:
17313:
16999:
16633:
16232:
15732:
14335:
12942:
311:
1041:{\displaystyle z=\{y\in L_{\alpha }\ {\text{and}}\ y\in z\}\in {\textrm {Def}}(L_{\alpha })=L_{\alpha +1}}
569:
17365:
17360:
16970:
16709:
16638:
15967:
15868:
15725:
15363:
15326:
12593:
12178:
10731:
9741:
9655:
5088:
2669:
752:
14658:
12376:
10057:
9913:
7116:
by their definitions and by the rank they appear at. So one chooses the least element of each member of
2100:
1758:
17294:
16884:
16278:
16246:
15937:
14379:
12988:
11054:
10650:
9182:
7844:
1941:
15380:
14974:(1984), ch. 2, "The Constructible Universe, p.58. Perspectives in Mathematical Logic, Springer-Verlag.
13854:
12858:
10961:
4215:
3902:
3869:
17584:
17533:
17430:
16928:
16889:
16366:
16011:
15414:
15299:
15287:
15282:
14900:
14228:
10005:
2794:
2584:, such as the set of (natural numbers coding) true arithmetical statements (this can be defined from
1211:
16040:
14630:
12893:
9833:
9533:
9208:
17425:
17355:
16894:
16746:
16729:
16452:
15932:
15215:
13820:
13150:
13114:
12732:
12303:
11865:
11568:
11303:
10243:
9014:
7505:
4714:
2635:
2587:
2534:
2454:
2401:
2368:
2211:
1698:
1665:
1632:
1078:
221:
10584:
9067:
4786:
4574:
4064:
3689:. Since the element relation is the same and no new elements were added, this is the empty set of
17257:
17234:
17195:
17081:
17022:
16668:
16588:
16432:
16376:
15989:
15834:
15752:
15627:
15579:
15393:
15316:
14332:
may be a (presumably non-constructible) set or a proper class. The definition of this class uses
12639:
12244:
11981:
11818:
11484:
11329:
10133:
9965:
9873:
9628:
9559:
9300:
8689:
8594:
8540:
8513:
8446:
8129:
8056:
8029:
7982:
7818:
7751:
7374:
6826:
5154:
5041:
4654:
4322:
4275:
4248:
3374:
3234:
2487:
2244:
2153:
2046:
1972:
1944:
1837:
1810:
1731:
1582:
1557:
1530:
1294:
1115:
1051:
914:
258:
107:
14468:
11791:
11537:
11383:
11274:
10906:
10824:
10777:
17547:
17274:
17252:
17219:
17112:
16958:
16943:
16916:
16867:
16751:
16686:
16511:
16472:
16346:
16177:
16154:
15786:
15667:
15292:
14257:
10504:
10107:
3663:
2322:
14436:
13951:
13766:
3182:
3150:
2871:
800:
669:
17477:
17330:
17122:
16840:
16576:
16482:
16341:
16326:
16207:
16182:
15702:
15672:
15616:
15536:
15516:
15494:
14256:, the smallest model that contains all the real numbers, which is used extensively in modern
13757:
13094:
12356:
12224:
11961:
11921:
11621:
11601:
10630:
10223:
10160:
9120:
8401:
7711:
7441:
6485:
5021:
4694:
4688:
3849:
3760:
2925:
2800:
2695:
2675:
2567:
2434:
2348:
2328:
2073:
2022:
1999:
1897:
1324:
778:
560:
15156:. Annals of Mathematics Studies. Vol. 3. Princeton, N. J.: Princeton University Press.
14555:
13415:
12765:
12686:
11034:
9807:
9100:
8968:
8942:
5843:
5650:
5615:
5580:
4520:
3823:
3797:
3286:
17450:
17412:
17289:
17093:
16933:
16857:
16835:
16663:
16621:
16520:
16487:
16351:
16139:
16050:
15776:
15766:
15600:
15531:
15484:
15424:
15304:
15171:
15102:
14915:
14910:
14905:
14416:
14141:
13826:
12573:
11437:
11410:
11356:
11247:
11227:
11215:{\displaystyle T=\{x\in L_{\beta }:x\in S\wedge \Phi (x,z_{i})\}=\{x\in S:\Phi (x,z_{i})\}}
10555:
10527:
9701:
9588:
9471:
9451:
9411:
9371:
8738:
8227:, properties of ordinals that depend on the absence of a function or other structure (i.e.
5554:
5219:
3636:
1463:
1436:
187:
14816:
14526:
14196:
14167:
14112:
13730:
13659:
13598:
7638:
is the smallest class containing all the ordinals that is a standard model of ZF. Indeed,
2514:
1403:{\displaystyle L_{\alpha }=\bigcup _{\beta <\alpha }\operatorname {Def} (L_{\beta })\!}
8:
17579:
17470:
17455:
17435:
17392:
17279:
17229:
17155:
17100:
17037:
16830:
16825:
16773:
16541:
16530:
16202:
16102:
16030:
16021:
16017:
15952:
15947:
15771:
15682:
15590:
15585:
15399:
15341:
15272:
15208:
11511:
10523:
9364:. If one discounts (for the moment) the parameters, the formulas can be given a standard
8565:
8156:
8083:
7815:. If there is a set that is a standard model of ZF, then the smallest such set is such a
5685:
4814:
3060:
2182:
1917:
1784:
1217:
15106:
17608:
17377:
17340:
17325:
17318:
17301:
17105:
17087:
16953:
16879:
16862:
16815:
16628:
16537:
16371:
16356:
16316:
16268:
16253:
16241:
16197:
16172:
15942:
15891:
15694:
15689:
15474:
15429:
15336:
15133:
15120:
15090:
15027:
14874:
14845:
14793:
14764:
14744:
14724:
14695:
14610:
14581:
14506:
14315:
14295:
14266:
13802:
13708:
13688:
13627:
13578:
13550:
13182:
13074:
13054:
13034:
13014:
12922:
12712:
12666:
12336:
12275:
12204:
12158:
12080:
12060:
11941:
11901:
11845:
11464:
11309:
11014:
10994:
10933:
10804:
10610:
10561:
10537:
10486:
10466:
10443:
10423:
10403:
10383:
10363:
10343:
10319:
10299:
10279:
10203:
10183:
10113:
9787:
9721:
9608:
9512:
9492:
9431:
9391:
9347:
9327:
9161:
9141:
9047:
8994:
8922:
8902:
8882:
8859:
8834:
8812:
8784:
8764:
8744:
8716:
8669:
8649:
8621:
8571:
8493:
8473:
8423:
8383:
8359:
8335:
8311:
8287:
8267:
8109:
8009:
7962:
7942:
7922:
7902:
7882:
7862:
7798:
7778:
7731:
7687:
7667:
7641:
7621:
7601:
7581:
7561:
7541:
7421:
7401:
7354:
7334:
7314:
7294:
7274:
7254:
7226:
7202:
7182:
7159:
7139:
7119:
7099:
7070:
7050:
7030:
7010:
6982:
6806:
6786:
6763:
6743:
6723:
6703:
6683:
6658:
6638:
6618:
6598:
6592:
6571:
6551:
6465:
6445:
6425:
6098:
6078:
6058:
6038:
6018:
5998:
5978:
5958:
5938:
5918:
5898:
5878:
5560:
5532:
5201:
5181:
5134:
5068:
5001:
4820:
4634:
4614:
4594:
4554:
4302:
4190:
4170:
4150:
4122:
4102:
3740:
3720:
3692:
3547:
3525:
3505:
3485:
3465:
3445:
3425:
3405:
3385:
3352:
3332:
3312:
3266:
3214:
3126:
3106:
3086:
3066:
3031:
3011:
2991:
2971:
2951:
2931:
2903:
2188:
2133:
1612:
1592:
1510:
1490:
1416:
1263:
1243:
1193:
1173:
1164:
1146:
894:
884:
542:
329:
285:
193:
169:
84:
74:
56:
16561:
13705:
is the intersection of all classes that are standard models of set theory and contain
9365:
8830:
2619:
2315:
17603:
17543:
17350:
17160:
17150:
17042:
16923:
16758:
16734:
16515:
16499:
16404:
16381:
16258:
16227:
16192:
16087:
15922:
15551:
15388:
15351:
15321:
15245:
15187:
15157:
15138:
15075:
15056:
15034:
14930:
8537:
cease to have those large cardinal properties, but retain the properties weaker than
4548:
215:
78:
15091:"The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"
17557:
17552:
17445:
17402:
17224:
17185:
17180:
17165:
16991:
16948:
16845:
16643:
16593:
16167:
16129:
15839:
15829:
15814:
15809:
15677:
15331:
15128:
15110:
13537:{\displaystyle \forall c\in a(\forall d\in c(d\in a\land \forall e\in d(e\in c))).}
3714:
2921:
17538:
17528:
17482:
17465:
17420:
17382:
17284:
17204:
17011:
16938:
16911:
16899:
16805:
16719:
16693:
16648:
16616:
16417:
16219:
16162:
16112:
16077:
16035:
15708:
15646:
15464:
15277:
15167:
10531:
10508:
8329:
8305:
7004:
315:
144:
27:
Particular class of sets which can be described entirely in terms of simpler sets
31:
17523:
17502:
17460:
17440:
17335:
17190:
16788:
16778:
16768:
16763:
16697:
16571:
16447:
16336:
16331:
16309:
15910:
15844:
15641:
15622:
15526:
15511:
15468:
15404:
15346:
15095:
Proceedings of the
National Academy of Sciences of the United States of America
14984:
14958:
14920:
8734:
8409:
8405:
8353:
4144:
2178:
1140:
211:
15151:
10296:
itself by a formula of set theory with no parameters, only the free-variables
151:
are true in the constructible universe. This shows that both propositions are
17622:
17497:
17175:
16682:
16467:
16457:
16427:
16412:
16082:
15849:
15651:
15565:
15560:
8853:
8643:
687:
322:
15819:
14035:{\displaystyle L_{\lambda }(A)=\bigcup _{\alpha <\lambda }L_{\alpha }(A)}
13575:
Sometimes it is desirable to find a model of set theory that is narrow like
186:
can be thought of as being built in "stages" resembling the construction of
132:
17397:
17244:
17145:
17137:
17017:
16965:
16874:
16810:
16793:
16724:
16583:
16442:
16144:
15927:
15799:
15794:
15612:
15541:
15499:
15358:
15255:
15142:
15115:
15048:
6760:. What we need here is to show that the intersection of the power set with
2091:
15186:. Springer Monographs in Mathematics (3rd millennium ed.). Springer.
4986:{\displaystyle \{x\mid x\in S\;\mathrm {and} \;P(x,z_{1},\ldots ,z_{n})\}}
3369:
because we are using the same element relation and no new sets were added.
17507:
17387:
16566:
16556:
16503:
16187:
16107:
16092:
15972:
15917:
15824:
15459:
15179:
15022:
10337:
9428:
is the formula with the smallest Gödel number that can be used to define
9388:
is the formula with the smallest Gödel number that can be used to define
7852:, one can show that the minimal model (if it exists) is a countable set.
7705:
136:
38:
12373:. Using this we can expand the definition of each constructible set. If
8733:. Furthermore, any strictly increasing class function from the class of
742:{\displaystyle L_{\lambda }:=\bigcup _{\alpha <\lambda }L_{\alpha }.}
16437:
16292:
16263:
16069:
15804:
15575:
15231:
8413:
152:
42:
17589:
17492:
16545:
16462:
16422:
16386:
16322:
16134:
16124:
16097:
15860:
15607:
15570:
15521:
15419:
15124:
13071:'s outside and one gets a formula that defines the constructible set
12295:
6823:
to show that there is an α such that the intersection is a subset of
2040:
1109:
14985:
An
Introduction to the Fine Structure of the Constructible Hierarchy
14959:
An introduction to the fine structure of the constructible hierarchy
2924:
and the interpretation uses the real element relationship, so it is
17574:
17372:
16820:
16525:
16119:
8639:
872:{\displaystyle L:=\bigcup _{\alpha \in \mathbf {Ord} }L_{\alpha }.}
10953:
17170:
15962:
657:{\displaystyle L_{\alpha +1}:=\operatorname {Def} (L_{\alpha }).}
15632:
15454:
2948:
is an inner model, i.e. it contains all the ordinal numbers of
14996:
Barwise 1975, page 60 (comment following proof of theorem 5.9)
12300:
There is a formula of set theory that expresses the idea that
1585:. Equality beyond this point does not hold. Even in models of
16714:
16060:
15905:
15504:
15264:
15200:
15009:, pp.427. Studies in Logic and the Foundations of Mathematics
8876:
could be well-ordered using only the definition given above.
4187:
whose elements are precisely the elements of the elements of
3053:
156:
12050:{\displaystyle L\cap {\mathcal {P}}(S)\subseteq L_{\delta }}
14925:
8686:, they have all the large cardinal properties weaker than
2314:
under a collection of nine explicit functions, similar to
14292:
is the class of sets whose construction is influenced by
10511:
because it also covers proper classes of non-empty sets.
9290:{\displaystyle L_{\alpha +1}=\mathrm {Def} (L_{\alpha })}
7271:
is any standard model of ZF sharing the same ordinals as
81:
that can be described entirely in terms of simpler sets.
12148:{\displaystyle L\cap {\mathcal {P}}(S)\in L_{\delta +1}}
7092:
One can show that there is a definable well-ordering of
3626:{\displaystyle \{\}=L_{0}=\{y\mid y\in L_{0}\land y=y\}}
2090:, and the bijection is constructible. So these sets are
10157:, so this definition involves transfinite recursion on
8666:. While some of these are not even initial ordinals in
3377:: Two sets are the same if they have the same elements.
2622:
14138:
contains a well-ordering of the transitive closure of
10821:
symbols (counting a constant symbol for an element of
2318:. This definition makes no reference to definability.
14877:
14848:
14819:
14796:
14767:
14747:
14727:
14698:
14661:
14633:
14613:
14584:
14558:
14529:
14523:
is a unary predicate. The intended interpretation of
14509:
14471:
14439:
14419:
14382:
14338:
14318:
14298:
14269:
14231:
14199:
14170:
14144:
14115:
14098:{\displaystyle L(A)=\bigcup _{\alpha }L_{\alpha }(A)}
14050:
13974:
13954:
13899:
13857:
13829:
13805:
13769:
13733:
13711:
13691:
13662:
13630:
13601:
13581:
13553:
13455:
13418:
13207:
13185:
13153:
13117:
13097:
13077:
13057:
13037:
13017:
12991:
12945:
12925:
12896:
12861:
12794:
12768:
12735:
12715:
12689:
12669:
12642:
12596:
12576:
12418:
12379:
12359:
12339:
12306:
12278:
12247:
12227:
12207:
12181:
12161:
12103:
12083:
12063:
12011:
11984:
11964:
11944:
11924:
11904:
11868:
11848:
11821:
11794:
11644:
11624:
11604:
11571:
11540:
11514:
11487:
11467:
11440:
11413:
11386:
11359:
11332:
11312:
11277:
11250:
11230:
11097:
11057:
11037:
11017:
10997:
10964:
10936:
10909:
10854:
10827:
10807:
10780:
10734:
10679:
10653:
10633:
10613:
10587:
10564:
10540:
10489:
10469:
10446:
10426:
10406:
10386:
10366:
10346:
10322:
10302:
10282:
10276:
Notice that this well-ordering can be defined within
10246:
10226:
10206:
10186:
10163:
10136:
10116:
10060:
10008:
9968:
9916:
9876:
9836:
9810:
9790:
9744:
9724:
9704:
9658:
9631:
9611:
9591:
9562:
9536:
9515:
9495:
9474:
9454:
9434:
9414:
9394:
9374:
9350:
9330:
9303:
9240:
9211:
9185:
9164:
9144:
9123:
9103:
9070:
9050:
9017:
8997:
8971:
8945:
8925:
8905:
8885:
8862:
8837:
8815:
8787:
8767:
8747:
8719:
8692:
8672:
8652:
8624:
8597:
8574:
8543:
8516:
8510:. So all the large cardinals whose existence implies
8496:
8476:
8449:
8426:
8386:
8362:
8338:
8314:
8290:
8270:
8233:
8193:
8159:
8132:
8112:
8086:
8059:
8032:
8012:
7985:
7965:
7945:
7925:
7905:
7885:
7865:
7821:
7801:
7781:
7754:
7734:
7714:
7690:
7670:
7644:
7624:
7604:
7584:
7564:
7544:
7508:
7464:
7444:
7424:
7404:
7377:
7357:
7337:
7317:
7297:
7277:
7257:
7229:
7205:
7185:
7162:
7142:
7122:
7102:
7073:
7067:) containing exactly one element from each member of
7053:
7033:
7013:
6985:
6856:
6829:
6809:
6789:
6766:
6746:
6726:
6706:
6686:
6661:
6641:
6621:
6601:
6574:
6554:
6508:
6488:
6468:
6448:
6428:
6300:
6121:
6101:
6081:
6061:
6041:
6021:
6001:
5981:
5961:
5941:
5921:
5901:
5881:
5846:
5714:
5688:
5653:
5618:
5583:
5563:
5535:
5354:
5228:
5204:
5184:
5157:
5137:
5091:
5071:
5044:
5024:
5004:
4904:
4843:
4823:
4789:
4750:
4717:
4697:
4657:
4637:
4617:
4597:
4577:
4557:
4523:
4352:
4325:
4305:
4278:
4251:
4218:
4193:
4173:
4153:
4125:
4105:
4067:
3938:
3905:
3872:
3852:
3826:
3800:
3763:
3743:
3723:
3695:
3666:
3639:
3560:
3528:
3508:
3488:
3468:
3448:
3428:
3408:
3388:
3355:
3335:
3315:
3289:
3269:
3237:
3217:
3185:
3153:
3129:
3109:
3089:
3069:
3034:
3014:
2994:
2974:
2954:
2934:
2906:
2874:
2864:
2823:
2803:
2762:
2718:
2698:
2678:
2638:
2590:
2570:
2564:
already contains certain non-arithmetical subsets of
2537:
2517:
2490:
2457:
2437:
2404:
2371:
2351:
2331:
2274:
2247:
2214:
2208:
Another definition, due to Gödel, characterizes each
2191:
2156:
2136:
2103:
2076:
2049:
2025:
2002:
1975:
1947:
1920:
1900:
1867:
1840:
1813:
1787:
1761:
1734:
1701:
1668:
1635:
1615:
1595:
1560:
1533:
1513:
1493:
1466:
1439:
1419:
1347:
1327:
1297:
1283:
1266:
1246:
1220:
1196:
1176:
1149:
1118:
1081:
1054:
944:
917:
897:
825:
803:
781:
755:
696:
672:
609:
572:
545:
348:
288:
261:
224:
196:
172:
147:
excluded), and also that the axiom of choice and the
110:
87:
59:
11898:
So all the constructible subsets of an infinite set
9698:
is the sequence of parameters that can be used with
9556:
in the Gödel numbering. Henceforth, we suppose that
2531:
is definable, i.e., arithmetic). On the other hand,
14413:except instead of evaluating the truth of formulas
7027:of mutually disjoint nonempty sets, there is a set
5577:and any mapping (formally defined as a proposition
2511:can be coded by natural numbers in such a way that
15026:
14883:
14863:
14834:
14802:
14782:
14753:
14733:
14721:is always a model of the axiom of choice. Even if
14713:
14682:
14647:
14619:
14599:
14570:
14544:
14515:
14495:
14457:
14425:
14405:
14368:
14324:
14304:
14284:
14248:
14214:
14185:
14164:, then this can be extended to a well-ordering of
14156:
14130:
14097:
14034:
13960:
13938:
13885:
13841:
13811:
13791:
13748:
13717:
13697:
13677:
13645:
13616:
13587:
13559:
13536:
13439:
13402:
13191:
13171:
13136:
13103:
13083:
13063:
13043:
13023:
13003:
12977:
12931:
12911:
12882:
12847:
12780:
12754:
12721:
12701:
12675:
12655:
12628:
12582:
12562:
12404:
12365:
12345:
12325:
12296:Constructible sets are definable from the ordinals
12284:
12260:
12241:, the "power set" must then have cardinal exactly
12233:
12213:
12193:
12167:
12147:
12089:
12069:
12049:
11997:
11970:
11950:
11930:
11910:
11887:
11854:
11834:
11807:
11780:
11630:
11610:
11590:
11553:
11526:
11500:
11473:
11453:
11426:
11399:
11372:
11345:
11318:
11290:
11263:
11236:
11214:
11082:
11043:
11023:
11003:
10983:
10942:
10922:
10895:
10840:
10813:
10793:
10766:
10720:
10665:
10639:
10619:
10599:
10570:
10546:
10495:
10475:
10452:
10432:
10412:
10392:
10372:
10352:
10328:
10308:
10288:
10265:
10232:
10212:
10192:
10169:
10149:
10122:
10098:
10046:
9994:
9954:
9902:
9862:
9822:
9796:
9776:
9730:
9710:
9690:
9644:
9617:
9597:
9574:
9548:
9521:
9501:
9480:
9460:
9440:
9420:
9400:
9380:
9356:
9336:
9316:
9289:
9223:
9197:
9170:
9150:
9129:
9109:
9089:
9056:
9036:
9003:
8983:
8957:
8931:
8911:
8891:
8868:
8843:
8821:
8793:
8773:
8753:
8725:
8705:
8678:
8658:
8630:
8610:
8580:
8556:
8529:
8502:
8482:
8462:
8432:
8392:
8368:
8344:
8320:
8296:
8276:
8256:
8220:{\displaystyle \mathrm {Ord} \subset L\subseteq V}
8219:
8171:
8145:
8118:
8098:
8072:
8045:
8018:
7998:
7971:
7951:
7931:
7911:
7891:
7871:
7834:
7807:
7787:
7767:
7740:
7720:
7696:
7676:
7650:
7630:
7610:
7590:
7570:
7550:
7527:
7494:
7450:
7430:
7410:
7390:
7363:
7343:
7323:
7303:
7283:
7263:
7235:
7211:
7191:
7168:
7148:
7128:
7108:
7079:
7059:
7039:
7019:
6991:
6971:
6842:
6815:
6795:
6772:
6752:
6732:
6712:
6692:
6667:
6647:
6627:
6607:
6580:
6560:
6540:
6494:
6474:
6454:
6434:
6414:
6286:
6107:
6087:
6067:
6047:
6027:
6007:
5987:
5967:
5947:
5927:
5907:
5887:
5867:
5827:
5700:
5674:
5639:
5604:
5569:
5541:
5521:
5340:
5210:
5190:
5170:
5143:
5123:
5077:
5057:
5030:
5010:
4985:
4890:
4829:
4801:
4775:
4736:
4703:
4675:
4643:
4623:
4603:
4583:
4563:
4535:
4509:
4338:
4311:
4291:
4264:
4237:
4199:
4179:
4159:
4131:
4111:
4091:
4053:
3924:
3891:
3858:
3838:
3812:
3781:
3749:
3729:
3701:
3681:
3652:
3625:
3534:
3514:
3494:
3474:
3454:
3434:
3414:
3394:
3361:
3341:
3321:
3301:
3275:
3255:
3223:
3203:
3171:
3135:
3115:
3095:
3075:
3040:
3020:
3000:
2980:
2960:
2940:
2912:
2892:
2853:
2809:
2785:
2748:
2704:
2684:
2657:
2624:
2609:
2576:
2556:
2523:
2503:
2476:
2443:
2423:
2390:
2357:
2337:
2306:
2260:
2233:
2197:
2169:
2142:
2122:
2082:
2062:
2031:
2008:
1988:
1960:
1932:
1906:
1886:
1853:
1826:
1799:
1773:
1747:
1720:
1687:
1654:
1621:
1601:
1573:
1546:
1519:
1499:
1479:
1452:
1425:
1402:
1333:
1310:
1272:
1252:
1232:
1202:
1182:
1155:
1131:
1100:
1067:
1040:
930:
903:
871:
809:
787:
767:
741:
678:
656:
594:
551:
530:
294:
274:
243:
202:
178:
123:
93:
65:
15074:. Lecture Notes in Mathematics. Springer-Verlag.
12155:. And this in turn means that the "power set" of
8179:does not hold in any other standard model of ZF.
3482:'s transitivity, they have the same elements (in
1399:
520:
467:
369:
282:. By contrast, in Gödel's constructible universe
17620:
13567:and that contains parameters only for ordinals.
12890:is the result of restricting each quantifier in
10554:requires (at least as shown above) the use of a
8737:to itself can be extended in a unique way to an
2398:(because the arithmetic definition gives one in
10954:The generalized continuum hypothesis holds in L
10514:
10220:is well-ordered by the ordered sum (indexed by
8408:become strongly Mahlo. And more generally, any
2854:{\displaystyle L_{\omega _{1}^{\mathrm {CK} }}}
2749:{\displaystyle L_{\omega _{1}^{\mathrm {CK} }}}
2307:{\displaystyle L_{\alpha }\cup \{L_{\alpha }\}}
2181:, i.e., formulas of set theory containing only
2094:in any model of set theory that includes them.
14193:. Otherwise, the axiom of choice will fail in
13939:{\displaystyle \mathrm {Def} (L_{\alpha }(A))}
12848:{\displaystyle \Psi (X,y,z_{1},\ldots ,z_{n})}
7096:, in particular based on ordering all sets in
6548:. Then one can use the axiom of separation in
5995:, but it is still a finite formula, and since
3052:, which means that it satisfies the following
306:those subsets of the previous stage that are:
15876:
15216:
15101:(12). National Academy of Sciences: 556–557.
12663:. This is equivalent to saying that: for all
8264:formulas) are preserved when going down from
1210:itself is the "constructible universe". The "
332:interpreted to range over the previous stage.
14151:
14145:
13836:
13830:
13570:
13166:
13154:
12557:
12425:
12188:
12182:
11918:have ranks with (at most) the same cardinal
11775:
11716:
11710:
11651:
11380:, and having the same first-order theory as
11209:
11169:
11163:
11104:
7246:
7156:using the axioms of union and separation in
6947:
6857:
6541:{\displaystyle L_{\alpha }\in L_{\alpha +1}}
6409:
6301:
6281:
6122:
5822:
5715:
5497:
5355:
5335:
5229:
4980:
4905:
4670:
4664:
4485:
4359:
4080:
4068:
4029:
3957:
3951:
3939:
3776:
3764:
3670:
3667:
3620:
3583:
3564:
3561:
2301:
2288:
990:
951:
459:
374:
15153:The Consistency of the Continuum Hypothesis
13051:and apply existential quantifiers over the
10507:, which is more powerful than the ordinary
8568:cease to be measurable but remain Mahlo in
7495:{\displaystyle \mathrm {Def} (L_{\alpha })}
6568:to finish showing that it is an element of
2968:and it has no "extra" sets beyond those in
2786:{\displaystyle \omega _{1}^{\mathrm {CK} }}
16068:
15883:
15869:
15223:
15209:
13350:
13346:
13284:
13280:
13230:
13226:
12534:
12525:
12507:
12462:
12450:
7458:. And the same formulas and parameters in
6943:
6934:
6913:
6907:
6898:
6894:
6882:
6462:. Again using the axiom of replacement in
6390:
6375:
6360:
6348:
6327:
6277:
6268:
6250:
6231:
6216:
6201:
6191:
6170:
6152:
6140:
5803:
5788:
5773:
5763:
5742:
5724:
5486:
5477:
5459:
5414:
5402:
5392:
5380:
5331:
5322:
5304:
5259:
5247:
4935:
4923:
4475:
4460:
4445:
4435:
4414:
4396:
4384:
4016:
4007:
3994:
3982:
1139:. Consequently, this is a tower of nested
15132:
15114:
14239:
13799:= the smallest transitive set containing
10340:regardless of whether it is evaluated in
7658:is the intersection of all such classes.
7223:assume that the axiom of choice holds in
1887:{\displaystyle \alpha =\omega _{\alpha }}
8809:There are various ways of well-ordering
8804:
8801:a nice structure of repeating segments.
8257:{\displaystyle \Pi _{1}^{\mathrm {ZF} }}
8182:
4891:{\displaystyle P(x,z_{1},\ldots ,z_{n})}
4776:{\displaystyle \omega \in L_{\omega +1}}
2241:as the intersection of the power set of
480: is a first-order formula and
15069:
15021:
13179:is constructible. It is the unique set
7502:produce the same constructible sets in
2039:is an infinite ordinal then there is a
1728:is a proper subset of the power set of
14:
17621:
15890:
15047:
14761:is not necessarily itself a member of
10896:{\displaystyle P(z_{1},\ldots ,z_{k})}
10721:{\displaystyle P(z_{1},\ldots ,z_{k})}
8564:which they also possess. For example,
15864:
15204:
15149:
15088:
14369:{\displaystyle \mathrm {Def} _{A}(X)}
12978:{\displaystyle z_{k}\in L_{\beta +1}}
10578:. Here we describe such a principle.
8404:become strongly inaccessible. Weakly
7728:is the set of ordinals that occur in
7199:is a model of ZFC only requires that
6680:In general, some subsets of a set in
2484:is arithmetical (because elements of
1190:are called "constructible" sets; and
15178:
14961:" (1974). Accessed 20 February 2023.
11306:, there must be some transitive set
6482:, we can show that there must be an
5975:is a much more complex formula than
595:{\displaystyle L_{0}:=\varnothing .}
12629:{\displaystyle z_{1},\ldots ,z_{n}}
12194:{\displaystyle \vert \delta \vert }
10767:{\displaystyle z_{1},\ldots ,z_{k}}
10106:), etc. This is called the reverse
9777:{\displaystyle w_{1},\ldots ,w_{n}}
9691:{\displaystyle z_{1},\ldots ,z_{n}}
9297:uses formulas with parameters from
6720:So the whole power set of a set in
6075:; thus we can apply replacement in
5124:{\displaystyle z_{1},\ldots ,z_{n}}
3442:and they have the same elements in
3231:is well founded. In particular, if
768:{\displaystyle \alpha <\lambda }
24:
18:Gödel's constructible universe
14683:{\displaystyle \mathrm {Def} _{A}}
14670:
14667:
14664:
14641:
14638:
14635:
14420:
14390:
14387:
14384:
14347:
14344:
14341:
13907:
13904:
13901:
13498:
13471:
13456:
13328:
13262:
13208:
12897:
12862:
12795:
12577:
12530:
12527:
12521:
12518:
12515:
12512:
12509:
12463:
12458:
12455:
12452:
12405:{\displaystyle S\in L_{\alpha +1}}
12112:
12020:
11750:
11685:
11231:
11184:
11138:
10336:. And this formula gives the same
10099:{\displaystyle z_{n-2}<w_{n-2}}
9955:{\displaystyle z_{n-1}<w_{n-1}}
9705:
9592:
9569:
9563:
9543:
9537:
9475:
9455:
9415:
9375:
9267:
9264:
9261:
8698:
8603:
8549:
8522:
8455:
8248:
8245:
8235:
8201:
8198:
8195:
7472:
7469:
7466:
6939:
6936:
6930:
6927:
6924:
6921:
6918:
6915:
6909:
6903:
6900:
6890:
6887:
6884:
6502:such that this set is a subset of
6386:
6383:
6380:
6377:
6371:
6368:
6365:
6362:
6350:
6344:
6341:
6338:
6335:
6332:
6329:
6323:
6320:
6317:
6314:
6311:
6273:
6270:
6264:
6261:
6258:
6255:
6252:
6227:
6224:
6221:
6218:
6212:
6209:
6206:
6203:
6187:
6184:
6181:
6178:
6175:
6172:
6166:
6163:
6160:
6157:
6154:
6148:
6145:
6142:
5799:
5796:
5793:
5790:
5784:
5781:
5778:
5775:
5759:
5756:
5753:
5750:
5747:
5744:
5738:
5735:
5732:
5729:
5726:
5482:
5479:
5473:
5470:
5467:
5464:
5461:
5410:
5407:
5404:
5388:
5385:
5382:
5327:
5324:
5318:
5315:
5312:
5309:
5306:
5255:
5252:
5249:
4931:
4928:
4925:
4471:
4468:
4465:
4462:
4456:
4453:
4450:
4447:
4431:
4428:
4425:
4422:
4419:
4416:
4410:
4407:
4404:
4401:
4398:
4392:
4389:
4386:
4012:
4009:
3990:
3987:
3984:
2865:L is a standard inner model of ZFC
2843:
2840:
2777:
2774:
2738:
2735:
2158:
2123:{\displaystyle {\textrm {Def}}(X)}
1774:{\displaystyle \alpha >\omega }
475:
415:
214:. In von Neumann's universe, at a
25:
17645:
14406:{\displaystyle \mathrm {Def} (X)}
13004:{\displaystyle \beta <\alpha }
12333:. It has only free variables for
11083:{\displaystyle T\in L_{\beta +1}}
10666:{\displaystyle \beta >\alpha }
9198:{\displaystyle \alpha <\beta }
8939:and we wish to determine whether
8418:list of large cardinal properties
8356:become strong limit cardinals in
4691:can be used to show each ordinal
3349:, then it is still disjoint from
1914:is inaccessible. More generally,
1581:: their elements are exactly the
1284:Additional facts about the sets L
586:
17602:
15254:
13886:{\displaystyle L_{\alpha +1}(A)}
13111:that appear in expressions like
12883:{\displaystyle \Psi (X,\ldots )}
12270:generalized continuum hypothesis
10984:{\displaystyle S\in L_{\alpha }}
8378:generalized continuum hypothesis
8153:that is a model of ZF. However,
5875:be the formula that relativizes
5018:, one can show that there is an
4238:{\displaystyle x\in L_{\alpha }}
4099:and it has the same meaning for
3925:{\displaystyle y\in L_{\alpha }}
3892:{\displaystyle x\in L_{\alpha }}
3008:might be strictly a subclass of
2797:), and conversely any subset of
850:
847:
844:
149:generalized continuum hypothesis
14249:{\displaystyle L(\mathbb {R} )}
11481:will have the same cardinal as
11011:be any constructible subset of
10047:{\displaystyle z_{n-1}=w_{n-1}}
8852:, which was first described by
8308:of cardinals remain initial in
4998:By induction on subformulas of
4272:and their elements are also in
255:subsets of the previous stage,
15230:
15029:Admissible Sets and Structures
14999:
14990:
14977:
14964:
14951:
14942:
14858:
14852:
14829:
14823:
14777:
14771:
14708:
14702:
14648:{\displaystyle \mathrm {Def} }
14594:
14588:
14539:
14533:
14490:
14472:
14452:
14440:
14400:
14394:
14363:
14357:
14279:
14273:
14243:
14235:
14209:
14203:
14180:
14174:
14125:
14119:
14092:
14086:
14060:
14054:
14029:
14023:
13991:
13985:
13933:
13930:
13924:
13911:
13880:
13874:
13786:
13780:
13743:
13737:
13672:
13666:
13640:
13634:
13611:
13605:
13528:
13525:
13522:
13510:
13483:
13468:
13434:
13428:
13397:
13394:
13391:
13388:
13385:
13379:
13347:
13334:
13322:
13319:
13313:
13281:
13268:
13259:
13231:
13227:
13214:
12912:{\displaystyle \Phi (\ldots )}
12906:
12900:
12877:
12865:
12842:
12798:
12554:
12535:
12504:
12466:
12123:
12117:
12031:
12025:
11772:
11753:
11707:
11688:
11206:
11187:
11160:
11141:
10890:
10858:
10715:
10683:
10627:to prove that for any ordinal
9863:{\displaystyle z_{n}<w_{n}}
9549:{\displaystyle \Phi <\Psi }
9284:
9271:
9224:{\displaystyle \beta =\alpha }
9205:. Henceforth, we suppose that
7489:
7476:
6979:. Thus the required set is in
6406:
6394:
6247:
6235:
5862:
5850:
5819:
5807:
5669:
5657:
5634:
5622:
5599:
5587:
5456:
5418:
5301:
5263:
4977:
4939:
4885:
4847:
4026:
3995:
3263:, then by the transitivity of
3198:
3186:
3166:
3154:
2887:
2875:
2185:) that use as parameters only
2177:formulas (with respect to the
2117:
2111:
1396:
1383:
1016:
1003:
648:
635:
456:
418:
409:
397:
361:
355:
139:of ZF set theory (that is, of
51:Gödel's constructible universe
13:
1:
17563:History of mathematical logic
15015:
13172:{\displaystyle \{5,\omega \}}
13137:{\displaystyle x=L_{\alpha }}
12755:{\displaystyle X=L_{\alpha }}
12709:if and only if [there exists
12326:{\displaystyle X=L_{\alpha }}
12057:serves as the "power set" of
11888:{\displaystyle L_{\gamma +1}}
11591:{\displaystyle K=L_{\gamma }}
10266:{\displaystyle L_{\alpha +1}}
9037:{\displaystyle L_{\alpha +1}}
7528:{\displaystyle L_{\alpha +1}}
7219:be a model of ZF, i.e. we do
6655:are precisely the subsets of
5218:), the latter is called the "
4737:{\displaystyle L_{\alpha +1}}
3846:, then there is some ordinal
2658:{\displaystyle L_{\omega +2}}
2610:{\displaystyle L_{\omega +1}}
2557:{\displaystyle L_{\omega +2}}
2477:{\displaystyle L_{\omega +1}}
2431:). Conversely, any subset of
2424:{\displaystyle L_{\omega +1}}
2391:{\displaystyle L_{\omega +1}}
2234:{\displaystyle L_{\alpha +1}}
1721:{\displaystyle L_{\alpha +1}}
1688:{\displaystyle V_{\omega +1}}
1655:{\displaystyle L_{\omega +1}}
1291:An equivalent definition for
1101:{\displaystyle L_{\alpha +1}}
244:{\displaystyle V_{\alpha +1}}
17488:Primitive recursive function
13685:for a non-constructible set
13199:that satisfies the formula:
12268:. But this is precisely the
11618:having the same cardinal as
10600:{\displaystyle n<\omega }
9090:{\displaystyle L_{\beta +1}}
8829:. Some of these involve the
6635:, such that the elements of
4802:{\displaystyle \omega \in L}
4584:{\displaystyle \varnothing }
4092:{\displaystyle \{x,y\}\in L}
3211:, which is well founded, so
1260:) is constructible, i.e. in
325:from the previous stage and,
210:. The stages are indexed by
162:
7:
15055:. Berlin: Springer-Verlag.
15033:. Berlin: Springer-Verlag.
14987:(p.2). Accessed 2021-05-12.
14894:
14790:, although it always is if
13011:. Combine formulas for the
12656:{\displaystyle L_{\alpha }}
12261:{\displaystyle \kappa ^{+}}
11998:{\displaystyle \kappa ^{+}}
11978:is the initial ordinal for
11835:{\displaystyle L_{\gamma }}
11501:{\displaystyle L_{\alpha }}
11346:{\displaystyle L_{\alpha }}
10930:if and only if it holds in
10848:as one symbol) we get that
10673:such that for any sentence
10483:(as we have done here with
10150:{\displaystyle L_{\alpha }}
9995:{\displaystyle z_{n}=w_{n}}
9903:{\displaystyle z_{n}=w_{n}}
9645:{\displaystyle L_{\alpha }}
9575:{\displaystyle \Psi =\Phi }
9368:by the natural numbers. If
9317:{\displaystyle L_{\alpha }}
8706:{\displaystyle 0^{\sharp }}
8611:{\displaystyle 0^{\sharp }}
8557:{\displaystyle 0^{\sharp }}
8530:{\displaystyle 0^{\sharp }}
8463:{\displaystyle 0^{\sharp }}
8146:{\displaystyle L_{\kappa }}
8073:{\displaystyle L_{\kappa }}
8046:{\displaystyle L_{\kappa }}
7999:{\displaystyle L_{\kappa }}
7848:of ZFC. Using the downward
7835:{\displaystyle L_{\kappa }}
7768:{\displaystyle L_{\kappa }}
7391:{\displaystyle L_{\alpha }}
7179:Notice that the proof that
6850:. Then the intersection is
6843:{\displaystyle L_{\alpha }}
5171:{\displaystyle L_{\alpha }}
5058:{\displaystyle L_{\alpha }}
4676:{\displaystyle y\cup \{y\}}
4339:{\displaystyle L_{\alpha }}
4292:{\displaystyle L_{\alpha }}
4265:{\displaystyle L_{\alpha }}
4245:, then its elements are in
3256:{\displaystyle y\in x\in L}
2504:{\displaystyle L_{\omega }}
2261:{\displaystyle L_{\alpha }}
2170:{\displaystyle \Delta _{0}}
2063:{\displaystyle L_{\alpha }}
1996:for all infinite cardinals
1989:{\displaystyle L_{\alpha }}
1961:{\displaystyle H_{\alpha }}
1854:{\displaystyle L_{\alpha }}
1827:{\displaystyle V_{\alpha }}
1748:{\displaystyle L_{\alpha }}
1574:{\displaystyle V_{\omega }}
1547:{\displaystyle L_{\omega }}
1311:{\displaystyle L_{\alpha }}
1240:", says that every set (of
1132:{\displaystyle L_{\alpha }}
1108:, which is a subset of the
1068:{\displaystyle L_{\alpha }}
931:{\displaystyle L_{\alpha }}
275:{\displaystyle V_{\alpha }}
141:Zermelo–Fraenkel set theory
124:{\displaystyle L_{\alpha }}
10:
17650:
16552:Schröder–Bernstein theorem
16279:Monadic predicate calculus
15938:Foundations of mathematics
15721:von Neumann–Bernays–Gödel
15007:Classical Recursion Theory
14496:{\displaystyle (X,\in ,A)}
11808:{\displaystyle L_{\beta }}
11554:{\displaystyle L_{\beta }}
11400:{\displaystyle L_{\beta }}
11291:{\displaystyle L_{\beta }}
10923:{\displaystyle L_{\beta }}
10841:{\displaystyle L_{\beta }}
10801:and containing fewer than
10794:{\displaystyle L_{\beta }}
10518:has a reflection principle
8919:are two different sets in
5915:, i.e. all quantifiers in
2900:is a standard model, i.e.
29:
17598:
17585:Philosophy of mathematics
17534:Automated theorem proving
17516:
17411:
17243:
17136:
16988:
16705:
16681:
16659:Von Neumann–Bernays–Gödel
16604:
16498:
16402:
16300:
16291:
16218:
16153:
16059:
15981:
15898:
15785:
15748:
15660:
15550:
15522:One-to-one correspondence
15438:
15379:
15263:
15252:
15238:
14901:Axiom of constructibility
14578:. Then the definition of
13968:is a limit ordinal, then
13571:Relative constructibility
11842:have the same theory. So
7842:. This set is called the
7247:L is absolute and minimal
3682:{\displaystyle \{\}\in L}
3502:). So they are equal (in
2130:is the set of subsets of
1212:axiom of constructibility
340:Define the Def operator:
15070:Felgner, Ulrich (1971).
14936:
14458:{\displaystyle (X,\in )}
13961:{\displaystyle \lambda }
13819:as an element, i.e. the
13792:{\displaystyle L_{0}(A)}
13091:using only the ordinals
13031:'s with the formula for
11300:Löwenheim–Skolem theorem
7850:Löwenheim–Skolem theorem
5529:. Thus the subset is in
3204:{\displaystyle (V,\in )}
3172:{\displaystyle (L,\in )}
2893:{\displaystyle (L,\in )}
1583:hereditarily finite sets
810:{\displaystyle \lambda }
679:{\displaystyle \lambda }
30:Not to be confused with
17235:Self-verifying theories
17056:Tarski's axiomatization
16007:Tarski's undefinability
16002:incompleteness theorems
15072:Models of ZF-Set Theory
14376:, which is the same as
13104:{\displaystyle \alpha }
12366:{\displaystyle \alpha }
12234:{\displaystyle \kappa }
11971:{\displaystyle \delta }
11931:{\displaystyle \kappa }
11631:{\displaystyle \alpha }
11611:{\displaystyle \gamma }
10640:{\displaystyle \alpha }
10503:) is equivalent to the
10233:{\displaystyle \alpha }
10170:{\displaystyle \alpha }
9130:{\displaystyle \alpha }
7721:{\displaystyle \kappa }
7708:of ZF, and the ordinal
7451:{\displaystyle \alpha }
6740:will usually not be in
6495:{\displaystyle \alpha }
6055:must be a mapping over
5031:{\displaystyle \alpha }
4704:{\displaystyle \alpha }
3859:{\displaystyle \alpha }
3782:{\displaystyle \{x,y\}}
3375:Axiom of extensionality
2810:{\displaystyle \omega }
2705:{\displaystyle \omega }
2685:{\displaystyle \omega }
2577:{\displaystyle \omega }
2444:{\displaystyle \omega }
2358:{\displaystyle \omega }
2338:{\displaystyle \omega }
2083:{\displaystyle \alpha }
2032:{\displaystyle \alpha }
2009:{\displaystyle \alpha }
1907:{\displaystyle \alpha }
1413:For any finite ordinal
1334:{\displaystyle \alpha }
788:{\displaystyle \alpha }
131:. It was introduced by
103:constructible hierarchy
17629:Constructible universe
17609:Mathematics portal
17220:Proof of impossibility
16868:propositional variable
16178:Propositional calculus
15480:Constructible universe
15300:Constructibility (V=L)
15116:10.1073/pnas.24.12.556
14885:
14865:
14836:
14810:is a set of ordinals.
14804:
14784:
14755:
14735:
14715:
14684:
14649:
14621:
14601:
14572:
14571:{\displaystyle y\in A}
14546:
14517:
14497:
14459:
14427:
14407:
14370:
14326:
14306:
14286:
14258:descriptive set theory
14250:
14216:
14187:
14158:
14132:
14099:
14036:
13962:
13940:
13887:
13843:
13813:
13793:
13750:
13725:and all the ordinals.
13719:
13699:
13679:
13647:
13618:
13589:
13561:
13538:
13441:
13440:{\displaystyle Ord(a)}
13404:
13193:
13173:
13138:
13105:
13085:
13065:
13045:
13025:
13005:
12979:
12933:
12913:
12884:
12849:
12782:
12781:{\displaystyle y\in X}
12756:
12723:
12703:
12702:{\displaystyle y\in S}
12677:
12657:
12630:
12584:
12564:
12406:
12367:
12347:
12327:
12286:
12262:
12235:
12215:
12195:
12169:
12149:
12097:Thus this "power set"
12091:
12071:
12051:
11999:
11972:
11952:
11932:
11912:
11889:
11856:
11836:
11809:
11782:
11632:
11612:
11592:
11555:
11528:
11502:
11475:
11455:
11428:
11401:
11374:
11347:
11320:
11292:
11265:
11238:
11216:
11084:
11045:
11044:{\displaystyle \beta }
11025:
11005:
10985:
10944:
10924:
10897:
10842:
10815:
10795:
10768:
10722:
10667:
10647:, there is an ordinal
10641:
10621:
10601:
10572:
10548:
10505:axiom of global choice
10497:
10477:
10454:
10434:
10414:
10394:
10374:
10354:
10330:
10310:
10290:
10267:
10240:) of the orderings on
10234:
10214:
10194:
10171:
10151:
10124:
10108:lexicographic ordering
10100:
10048:
9996:
9956:
9904:
9864:
9830:if and only if either
9824:
9823:{\displaystyle x<y}
9798:
9778:
9732:
9712:
9692:
9646:
9619:
9599:
9576:
9550:
9523:
9503:
9482:
9462:
9442:
9422:
9402:
9382:
9358:
9338:
9318:
9291:
9225:
9199:
9172:
9152:
9131:
9111:
9110:{\displaystyle \beta }
9091:
9058:
9038:
9005:
8985:
8984:{\displaystyle x>y}
8959:
8958:{\displaystyle x<y}
8933:
8913:
8893:
8870:
8845:
8823:
8795:
8775:
8755:
8727:
8707:
8680:
8660:
8640:closed unbounded class
8632:
8612:
8582:
8558:
8531:
8504:
8484:
8464:
8434:
8420:) will be retained in
8402:inaccessible cardinals
8394:
8370:
8346:
8322:
8298:
8278:
8258:
8221:
8173:
8147:
8120:
8100:
8074:
8047:
8020:
8000:
7973:
7953:
7933:
7913:
7893:
7873:
7836:
7809:
7789:
7769:
7742:
7722:
7698:
7678:
7652:
7632:
7612:
7592:
7572:
7552:
7529:
7496:
7452:
7432:
7412:
7392:
7365:
7345:
7325:
7305:
7285:
7265:
7237:
7213:
7193:
7170:
7150:
7130:
7110:
7081:
7061:
7041:
7021:
6993:
6973:
6844:
6817:
6797:
6774:
6754:
6734:
6714:
6694:
6669:
6649:
6629:
6609:
6582:
6562:
6542:
6496:
6476:
6456:
6436:
6416:
6288:
6109:
6089:
6069:
6049:
6029:
6009:
5989:
5969:
5949:
5929:
5909:
5889:
5869:
5868:{\displaystyle Q(x,y)}
5829:
5702:
5676:
5675:{\displaystyle P(x,z)}
5641:
5640:{\displaystyle P(x,y)}
5606:
5605:{\displaystyle P(x,y)}
5571:
5543:
5523:
5342:
5212:
5192:
5172:
5145:
5125:
5079:
5059:
5032:
5012:
4987:
4892:
4831:
4803:
4777:
4738:
4705:
4677:
4645:
4625:
4605:
4585:
4565:
4537:
4536:{\displaystyle y\in L}
4511:
4340:
4313:
4293:
4266:
4239:
4201:
4181:
4161:
4133:
4113:
4093:
4055:
3926:
3893:
3860:
3840:
3839:{\displaystyle y\in L}
3814:
3813:{\displaystyle x\in L}
3783:
3751:
3731:
3703:
3683:
3654:
3627:
3536:
3516:
3496:
3476:
3456:
3436:
3416:
3396:
3363:
3343:
3323:
3309:. If we use this same
3303:
3302:{\displaystyle y\in L}
3277:
3257:
3225:
3205:
3173:
3137:
3117:
3097:
3083:contains some element
3077:
3063:: Every non-empty set
3042:
3022:
3002:
2982:
2962:
2942:
2914:
2894:
2861:is hyperarithmetical.
2855:
2811:
2787:
2750:
2706:
2686:
2659:
2626:
2611:
2578:
2558:
2525:
2505:
2478:
2445:
2425:
2392:
2359:
2339:
2308:
2262:
2235:
2199:
2171:
2144:
2124:
2084:
2064:
2033:
2010:
1990:
1962:
1934:
1908:
1888:
1855:
1828:
1801:
1775:
1749:
1722:
1689:
1662:is a proper subset of
1656:
1623:
1603:
1575:
1548:
1521:
1501:
1487:are the same (whether
1481:
1454:
1427:
1404:
1335:
1312:
1274:
1254:
1234:
1204:
1184:
1157:
1133:
1102:
1069:
1042:
932:
905:
873:
811:
789:
769:
743:
680:
658:
596:
553:
532:
296:
276:
245:
204:
180:
125:
95:
67:
47:constructible universe
17478:Kolmogorov complexity
17431:Computably enumerable
17331:Model complete theory
17123:Principia Mathematica
16183:Propositional formula
16012:Banach–Tarski paradox
15703:Principia Mathematica
15537:Transfinite induction
15396:(i.e. set difference)
14886:
14866:
14837:
14805:
14785:
14756:
14736:
14716:
14685:
14650:
14622:
14602:
14573:
14547:
14518:
14498:
14465:, one uses the model
14460:
14428:
14426:{\displaystyle \Phi }
14408:
14371:
14327:
14307:
14287:
14251:
14217:
14188:
14159:
14157:{\displaystyle \{A\}}
14133:
14100:
14037:
13963:
13941:
13888:
13844:
13842:{\displaystyle \{A\}}
13814:
13794:
13758:transfinite recursion
13751:
13720:
13700:
13680:
13648:
13619:
13590:
13562:
13539:
13442:
13405:
13194:
13174:
13139:
13106:
13086:
13066:
13046:
13026:
13006:
12980:
12934:
12914:
12885:
12850:
12783:
12757:
12724:
12704:
12678:
12658:
12631:
12585:
12583:{\displaystyle \Phi }
12565:
12407:
12368:
12348:
12328:
12287:
12263:
12236:
12216:
12196:
12175:has cardinal at most
12170:
12150:
12092:
12072:
12052:
12000:
11973:
11958:; it follows that if
11953:
11933:
11913:
11890:
11857:
11837:
11810:
11783:
11633:
11613:
11593:
11561:, it is also true in
11556:
11529:
11503:
11476:
11456:
11454:{\displaystyle z_{i}}
11429:
11427:{\displaystyle w_{i}}
11402:
11375:
11373:{\displaystyle w_{i}}
11348:
11321:
11293:
11266:
11264:{\displaystyle z_{i}}
11239:
11237:{\displaystyle \Phi }
11217:
11085:
11046:
11031:. Then there is some
11026:
11006:
10986:
10945:
10925:
10898:
10843:
10816:
10796:
10769:
10723:
10668:
10642:
10622:
10602:
10573:
10549:
10498:
10478:
10455:
10435:
10415:
10395:
10375:
10355:
10331:
10311:
10291:
10268:
10235:
10215:
10195:
10172:
10152:
10125:
10101:
10049:
9997:
9957:
9905:
9865:
9825:
9799:
9779:
9733:
9713:
9711:{\displaystyle \Phi }
9693:
9647:
9620:
9600:
9598:{\displaystyle \Phi }
9577:
9551:
9524:
9504:
9483:
9481:{\displaystyle \Phi }
9463:
9461:{\displaystyle \Psi }
9443:
9423:
9421:{\displaystyle \Psi }
9403:
9383:
9381:{\displaystyle \Phi }
9359:
9339:
9319:
9292:
9226:
9200:
9173:
9153:
9132:
9112:
9092:
9059:
9039:
9006:
8986:
8960:
8934:
8914:
8894:
8871:
8846:
8824:
8805:L can be well-ordered
8796:
8776:
8756:
8728:
8708:
8681:
8661:
8642:of ordinals that are
8633:
8613:
8583:
8559:
8532:
8505:
8485:
8465:
8435:
8412:property weaker than
8395:
8371:
8347:
8323:
8299:
8279:
8259:
8222:
8183:L and large cardinals
8174:
8148:
8121:
8101:
8075:
8048:
8021:
8001:
7974:
7954:
7939:" result in the real
7934:
7914:
7894:
7874:
7837:
7810:
7790:
7770:
7743:
7723:
7699:
7679:
7653:
7633:
7613:
7593:
7573:
7553:
7530:
7497:
7453:
7433:
7413:
7393:
7366:
7346:
7326:
7306:
7286:
7266:
7238:
7214:
7194:
7171:
7151:
7131:
7111:
7082:
7062:
7042:
7022:
6994:
6974:
6845:
6818:
6803:. Use replacement in
6798:
6775:
6755:
6735:
6715:
6695:
6670:
6650:
6630:
6610:
6583:
6563:
6543:
6497:
6477:
6457:
6437:
6417:
6289:
6110:
6090:
6070:
6050:
6030:
6010:
5990:
5970:
5950:
5930:
5910:
5890:
5870:
5830:
5703:
5677:
5642:
5607:
5572:
5544:
5524:
5343:
5213:
5193:
5173:
5146:
5126:
5080:
5060:
5033:
5013:
4988:
4893:
4832:
4804:
4778:
4739:
4706:
4689:Transfinite induction
4678:
4646:
4626:
4606:
4586:
4566:
4551:: There exists a set
4538:
4512:
4341:
4314:
4294:
4267:
4240:
4202:
4182:
4162:
4134:
4114:
4094:
4056:
3927:
3894:
3861:
3841:
3815:
3784:
3752:
3732:
3704:
3684:
3655:
3653:{\displaystyle L_{1}}
3628:
3537:
3517:
3497:
3477:
3457:
3437:
3417:
3397:
3364:
3344:
3324:
3304:
3278:
3258:
3226:
3206:
3179:is a substructure of
3174:
3138:
3118:
3098:
3078:
3043:
3023:
3003:
2983:
2963:
2943:
2915:
2895:
2856:
2812:
2795:Church–Kleene ordinal
2788:
2751:
2707:
2687:
2660:
2627:
2612:
2579:
2559:
2526:
2506:
2479:
2446:
2426:
2393:
2360:
2340:
2309:
2263:
2236:
2200:
2172:
2145:
2125:
2085:
2065:
2034:
2011:
1991:
1963:
1935:
1909:
1889:
1856:
1829:
1802:
1781:. On the other hand,
1776:
1750:
1723:
1690:
1657:
1624:
1604:
1576:
1549:
1522:
1502:
1482:
1480:{\displaystyle V_{n}}
1455:
1453:{\displaystyle L_{n}}
1428:
1405:
1336:
1313:
1275:
1255:
1235:
1205:
1185:
1158:
1134:
1103:
1070:
1043:
933:
906:
874:
812:
790:
770:
744:
681:
659:
597:
561:transfinite recursion
554:
533:
297:
277:
246:
205:
181:
126:
96:
68:
17426:Church–Turing thesis
17413:Computability theory
16622:continuum hypothesis
16140:Square of opposition
15998:Gödel's completeness
15777:Burali-Forti paradox
15532:Set-builder notation
15485:Continuum hypothesis
15425:Symmetric difference
15150:Gödel, Kurt (1940).
15089:Gödel, Kurt (1938).
14916:Axiomatic set theory
14911:Reflection principle
14906:Statements true in L
14875:
14846:
14835:{\displaystyle L(A)}
14817:
14794:
14765:
14745:
14725:
14696:
14659:
14631:
14611:
14582:
14556:
14545:{\displaystyle A(y)}
14527:
14507:
14469:
14437:
14417:
14380:
14336:
14316:
14296:
14267:
14229:
14225:A common example is
14215:{\displaystyle L(A)}
14197:
14186:{\displaystyle L(A)}
14168:
14142:
14131:{\displaystyle L(A)}
14113:
14048:
13972:
13952:
13897:
13855:
13827:
13803:
13767:
13749:{\displaystyle L(A)}
13731:
13709:
13689:
13678:{\displaystyle L(A)}
13660:
13628:
13617:{\displaystyle L(A)}
13599:
13579:
13551:
13453:
13416:
13205:
13183:
13151:
13115:
13095:
13075:
13055:
13035:
13015:
12989:
12943:
12923:
12894:
12859:
12792:
12766:
12733:
12713:
12687:
12667:
12640:
12594:
12574:
12416:
12377:
12357:
12337:
12304:
12276:
12245:
12225:
12221:itself has cardinal
12205:
12179:
12159:
12101:
12081:
12061:
12009:
11982:
11962:
11942:
11922:
11902:
11866:
11846:
11819:
11792:
11642:
11622:
11602:
11569:
11538:
11512:
11485:
11465:
11438:
11434:substituted for the
11411:
11384:
11357:
11330:
11310:
11275:
11248:
11228:
11095:
11055:
11035:
11015:
10995:
10962:
10934:
10907:
10852:
10825:
10805:
10778:
10732:
10677:
10651:
10631:
10611:
10585:
10562:
10556:reflection principle
10538:
10528:axiom of replacement
10487:
10467:
10444:
10424:
10404:
10384:
10364:
10344:
10320:
10300:
10280:
10244:
10224:
10204:
10184:
10161:
10134:
10114:
10058:
10006:
9966:
9914:
9874:
9834:
9808:
9788:
9742:
9722:
9702:
9656:
9629:
9609:
9589:
9560:
9534:
9513:
9493:
9472:
9452:
9432:
9412:
9392:
9372:
9348:
9328:
9301:
9238:
9209:
9183:
9162:
9142:
9121:
9101:
9068:
9048:
9015:
8995:
8969:
8943:
8923:
8903:
8883:
8860:
8835:
8831:"fine structure" of
8813:
8785:
8765:
8745:
8739:elementary embedding
8717:
8690:
8670:
8650:
8622:
8595:
8572:
8566:measurable cardinals
8541:
8514:
8494:
8474:
8447:
8424:
8384:
8360:
8336:
8312:
8288:
8268:
8231:
8191:
8157:
8130:
8110:
8084:
8057:
8030:
8010:
7983:
7963:
7943:
7923:
7903:
7883:
7863:
7819:
7799:
7779:
7752:
7732:
7712:
7688:
7668:
7642:
7622:
7602:
7582:
7562:
7542:
7506:
7462:
7442:
7422:
7402:
7375:
7355:
7335:
7315:
7295:
7275:
7255:
7227:
7203:
7183:
7160:
7140:
7120:
7100:
7071:
7051:
7031:
7011:
6983:
6854:
6827:
6807:
6787:
6764:
6744:
6724:
6704:
6684:
6659:
6639:
6619:
6599:
6572:
6552:
6506:
6486:
6466:
6446:
6426:
6298:
6119:
6099:
6079:
6059:
6039:
6019:
5999:
5979:
5959:
5939:
5919:
5899:
5879:
5844:
5712:
5686:
5651:
5616:
5581:
5561:
5555:Axiom of replacement
5533:
5352:
5226:
5220:reflection principle
5202:
5182:
5155:
5135:
5089:
5069:
5042:
5022:
5002:
4902:
4841:
4837:and any proposition
4821:
4787:
4748:
4715:
4695:
4655:
4635:
4615:
4595:
4575:
4555:
4521:
4350:
4323:
4303:
4276:
4249:
4216:
4191:
4171:
4151:
4123:
4103:
4065:
3936:
3903:
3870:
3850:
3824:
3798:
3761:
3741:
3721:
3693:
3664:
3637:
3558:
3526:
3506:
3486:
3466:
3446:
3426:
3406:
3386:
3353:
3333:
3313:
3287:
3267:
3235:
3215:
3183:
3151:
3127:
3107:
3087:
3067:
3032:
3012:
2992:
2972:
2952:
2932:
2904:
2872:
2821:
2801:
2760:
2716:
2696:
2676:
2636:
2620:
2588:
2568:
2535:
2524:{\displaystyle \in }
2515:
2488:
2455:
2435:
2402:
2369:
2349:
2329:
2272:
2268:with the closure of
2245:
2212:
2189:
2154:
2134:
2101:
2074:
2047:
2023:
2000:
1973:
1945:
1918:
1898:
1865:
1838:
1811:
1785:
1759:
1732:
1699:
1666:
1633:
1613:
1593:
1558:
1531:
1511:
1491:
1464:
1437:
1417:
1345:
1325:
1295:
1264:
1244:
1218:
1194:
1174:
1147:
1116:
1079:
1052:
942:
915:
895:
823:
801:
779:
753:
694:
670:
607:
570:
543:
346:
286:
259:
222:
194:
188:von Neumann universe
170:
108:
101:is the union of the
85:
57:
17634:Works by Kurt Gödel
17580:Mathematical object
17471:P versus NP problem
17436:Computable function
17230:Reverse mathematics
17156:Logical consequence
17033:primitive recursive
17028:elementary function
16801:Free/bound variable
16654:Tarski–Grothendieck
16173:Logical connectives
16103:Logical equivalence
15953:Logical consequence
15738:Tarski–Grothendieck
15107:1938PNAS...24..556G
14607:is exactly that of
12939:. Notice that each
11527:{\displaystyle V=L}
10607:, we can use ZF in
10524:axiom of separation
9324:to define the sets
8854:Ronald Bjorn Jensen
8253:
8172:{\displaystyle V=L}
8099:{\displaystyle V=L}
7919:constructed within
7879:constructed within
7538:Furthermore, since
7331:is the same as the
6615:there exists a set
6015:was a mapping over
5701:{\displaystyle y=z}
4815:Axiom of separation
3061:Axiom of regularity
2848:
2782:
2743:
2183:bounded quantifiers
1933:{\displaystyle V=L}
1800:{\displaystyle V=L}
1233:{\displaystyle V=L}
17378:Transfer principle
17341:Semantics of logic
17326:Categorical theory
17302:Non-standard model
16816:Logical connective
15943:Information theory
15892:Mathematical logic
15327:Limitation of size
14881:
14861:
14832:
14800:
14780:
14751:
14731:
14711:
14680:
14645:
14617:
14597:
14568:
14542:
14513:
14493:
14455:
14423:
14403:
14366:
14322:
14302:
14282:
14246:
14212:
14183:
14154:
14128:
14095:
14075:
14032:
14012:
13958:
13936:
13883:
13839:
13821:transitive closure
13809:
13789:
13746:
13715:
13695:
13675:
13643:
13614:
13585:
13557:
13534:
13437:
13400:
13189:
13169:
13134:
13101:
13081:
13061:
13041:
13021:
13001:
12975:
12929:
12909:
12880:
12845:
12778:
12752:
12719:
12699:
12673:
12653:
12626:
12580:
12560:
12402:
12363:
12343:
12323:
12282:
12258:
12231:
12211:
12191:
12165:
12145:
12087:
12067:
12047:
11995:
11968:
11948:
11928:
11908:
11885:
11852:
11832:
11805:
11778:
11628:
11608:
11588:
11551:
11524:
11498:
11471:
11451:
11424:
11397:
11370:
11343:
11316:
11304:Mostowski collapse
11298:. By the downward
11288:
11261:
11234:
11212:
11080:
11041:
11021:
11001:
10981:
10940:
10920:
10893:
10838:
10811:
10791:
10764:
10718:
10663:
10637:
10617:
10597:
10568:
10544:
10493:
10473:
10450:
10430:
10410:
10390:
10370:
10350:
10326:
10306:
10286:
10263:
10230:
10210:
10190:
10167:
10147:
10120:
10096:
10044:
9992:
9952:
9900:
9860:
9820:
9794:
9784:does the same for
9774:
9728:
9708:
9688:
9642:
9615:
9595:
9572:
9546:
9519:
9499:
9478:
9468:is different from
9458:
9438:
9418:
9398:
9378:
9354:
9334:
9314:
9287:
9221:
9195:
9168:
9148:
9127:
9117:is different from
9107:
9087:
9054:
9034:
9001:
8981:
8955:
8929:
8909:
8889:
8866:
8841:
8819:
8791:
8771:
8751:
8723:
8703:
8676:
8656:
8638:, then there is a
8628:
8608:
8578:
8554:
8527:
8500:
8480:
8460:
8430:
8390:
8366:
8342:
8332:remain regular in
8318:
8294:
8274:
8254:
8234:
8217:
8169:
8143:
8116:
8096:
8070:
8043:
8016:
7996:
7969:
7949:
7929:
7909:
7889:
7869:
7832:
7805:
7785:
7765:
7738:
7718:
7694:
7674:
7648:
7628:
7608:
7588:
7568:
7548:
7525:
7492:
7448:
7438:, for any ordinal
7428:
7408:
7388:
7361:
7341:
7321:
7301:
7281:
7261:
7233:
7209:
7189:
7166:
7146:
7126:
7106:
7077:
7057:
7047:(a choice set for
7037:
7017:
6989:
6969:
6840:
6813:
6793:
6770:
6750:
6730:
6710:
6690:
6665:
6645:
6625:
6605:
6593:Axiom of power set
6578:
6558:
6538:
6492:
6472:
6452:
6442:and a subclass of
6432:
6412:
6284:
6105:
6085:
6065:
6045:
6025:
6005:
5985:
5965:
5945:
5935:are restricted to
5925:
5905:
5885:
5865:
5825:
5698:
5672:
5637:
5602:
5567:
5539:
5519:
5338:
5208:
5188:
5168:
5141:
5121:
5075:
5055:
5028:
5008:
4983:
4888:
4827:
4799:
4773:
4734:
4701:
4673:
4651:, so is the union
4641:
4621:
4601:
4581:
4561:
4533:
4507:
4336:
4309:
4289:
4262:
4235:
4197:
4177:
4157:
4129:
4109:
4089:
4051:
3922:
3889:
3856:
3836:
3810:
3779:
3747:
3727:
3699:
3679:
3650:
3623:
3548:Axiom of empty set
3532:
3512:
3492:
3472:
3452:
3432:
3412:
3392:
3359:
3339:
3319:
3299:
3273:
3253:
3221:
3201:
3169:
3143:are disjoint sets.
3133:
3113:
3093:
3073:
3038:
3018:
2998:
2978:
2958:
2938:
2910:
2890:
2851:
2829:
2807:
2783:
2763:
2746:
2724:
2702:
2682:
2655:
2607:
2574:
2554:
2521:
2501:
2474:
2441:
2421:
2388:
2355:
2335:
2304:
2258:
2231:
2205:and its elements.
2195:
2167:
2140:
2120:
2097:As defined above,
2080:
2060:
2029:
2006:
1986:
1958:
1930:
1904:
1884:
1851:
1824:
1797:
1771:
1745:
1718:
1685:
1652:
1619:
1599:
1571:
1544:
1527:or not), and thus
1517:
1497:
1477:
1450:
1423:
1400:
1376:
1331:
1308:
1270:
1250:
1230:
1200:
1180:
1153:
1129:
1098:
1065:
1038:
928:
901:
869:
855:
807:
785:
765:
739:
725:
676:
654:
592:
549:
528:
292:
272:
241:
200:
176:
121:
91:
73:, is a particular
63:
17616:
17615:
17548:Abstract category
17351:Theories of truth
17161:Rule of inference
17151:Natural deduction
17132:
17131:
16677:
16676:
16382:Cartesian product
16287:
16286:
16193:Many-valued logic
16168:Boolean functions
16051:Russell's paradox
16026:diagonal argument
15923:First-order logic
15858:
15857:
15767:Russell's paradox
15716:Zermelo–Fraenkel
15617:Dedekind-infinite
15490:Diagonal argument
15389:Cartesian product
15246:Set (mathematics)
15163:978-0-691-07927-1
14931:Ordinal definable
14884:{\displaystyle L}
14864:{\displaystyle L}
14803:{\displaystyle A}
14783:{\displaystyle L}
14754:{\displaystyle A}
14734:{\displaystyle A}
14714:{\displaystyle L}
14620:{\displaystyle L}
14600:{\displaystyle L}
14516:{\displaystyle A}
14325:{\displaystyle A}
14305:{\displaystyle A}
14285:{\displaystyle L}
14066:
13997:
13812:{\displaystyle A}
13718:{\displaystyle A}
13698:{\displaystyle A}
13646:{\displaystyle L}
13588:{\displaystyle L}
13560:{\displaystyle S}
13192:{\displaystyle s}
13147:Example: The set
13084:{\displaystyle S}
13064:{\displaystyle z}
13044:{\displaystyle S}
13024:{\displaystyle z}
12932:{\displaystyle X}
12722:{\displaystyle X}
12676:{\displaystyle y}
12570:for some formula
12346:{\displaystyle X}
12285:{\displaystyle L}
12214:{\displaystyle S}
12168:{\displaystyle S}
12090:{\displaystyle L}
12070:{\displaystyle S}
11951:{\displaystyle S}
11911:{\displaystyle S}
11855:{\displaystyle T}
11474:{\displaystyle K}
11319:{\displaystyle K}
11224:for some formula
11024:{\displaystyle S}
11004:{\displaystyle T}
10943:{\displaystyle L}
10814:{\displaystyle n}
10620:{\displaystyle V}
10571:{\displaystyle L}
10547:{\displaystyle L}
10522:Proving that the
10496:{\displaystyle L}
10476:{\displaystyle V}
10453:{\displaystyle L}
10433:{\displaystyle y}
10413:{\displaystyle x}
10393:{\displaystyle W}
10373:{\displaystyle V}
10353:{\displaystyle L}
10329:{\displaystyle y}
10309:{\displaystyle x}
10289:{\displaystyle L}
10213:{\displaystyle L}
10193:{\displaystyle n}
10123:{\displaystyle L}
9797:{\displaystyle y}
9731:{\displaystyle x}
9618:{\displaystyle n}
9522:{\displaystyle y}
9502:{\displaystyle x}
9441:{\displaystyle y}
9401:{\displaystyle x}
9357:{\displaystyle y}
9337:{\displaystyle x}
9171:{\displaystyle y}
9151:{\displaystyle x}
9064:first appears in
9057:{\displaystyle y}
9011:first appears in
9004:{\displaystyle x}
8932:{\displaystyle L}
8912:{\displaystyle y}
8892:{\displaystyle x}
8869:{\displaystyle L}
8844:{\displaystyle L}
8822:{\displaystyle L}
8794:{\displaystyle L}
8774:{\displaystyle L}
8754:{\displaystyle L}
8726:{\displaystyle L}
8679:{\displaystyle V}
8659:{\displaystyle L}
8631:{\displaystyle V}
8581:{\displaystyle L}
8503:{\displaystyle V}
8483:{\displaystyle L}
8433:{\displaystyle L}
8393:{\displaystyle L}
8369:{\displaystyle L}
8345:{\displaystyle L}
8321:{\displaystyle L}
8297:{\displaystyle L}
8277:{\displaystyle V}
8119:{\displaystyle L}
8019:{\displaystyle V}
7972:{\displaystyle L}
7952:{\displaystyle L}
7932:{\displaystyle L}
7912:{\displaystyle V}
7892:{\displaystyle L}
7872:{\displaystyle L}
7808:{\displaystyle W}
7788:{\displaystyle L}
7741:{\displaystyle W}
7697:{\displaystyle V}
7677:{\displaystyle W}
7651:{\displaystyle L}
7631:{\displaystyle L}
7611:{\displaystyle W}
7598:is a subclass of
7591:{\displaystyle L}
7571:{\displaystyle V}
7558:is a subclass of
7551:{\displaystyle L}
7431:{\displaystyle V}
7411:{\displaystyle W}
7371:. In particular,
7364:{\displaystyle V}
7344:{\displaystyle L}
7324:{\displaystyle W}
7304:{\displaystyle L}
7284:{\displaystyle V}
7264:{\displaystyle W}
7236:{\displaystyle V}
7212:{\displaystyle V}
7192:{\displaystyle L}
7169:{\displaystyle L}
7149:{\displaystyle y}
7129:{\displaystyle x}
7109:{\displaystyle L}
7080:{\displaystyle x}
7060:{\displaystyle x}
7040:{\displaystyle y}
7020:{\displaystyle x}
6992:{\displaystyle L}
6816:{\displaystyle V}
6796:{\displaystyle L}
6773:{\displaystyle L}
6753:{\displaystyle L}
6733:{\displaystyle L}
6713:{\displaystyle L}
6693:{\displaystyle L}
6668:{\displaystyle x}
6648:{\displaystyle y}
6628:{\displaystyle y}
6608:{\displaystyle x}
6581:{\displaystyle L}
6561:{\displaystyle L}
6475:{\displaystyle V}
6455:{\displaystyle L}
6435:{\displaystyle V}
6108:{\displaystyle Q}
6088:{\displaystyle V}
6068:{\displaystyle V}
6048:{\displaystyle Q}
6028:{\displaystyle L}
6008:{\displaystyle P}
5988:{\displaystyle Q}
5968:{\displaystyle Q}
5948:{\displaystyle L}
5928:{\displaystyle P}
5908:{\displaystyle L}
5888:{\displaystyle P}
5570:{\displaystyle S}
5542:{\displaystyle L}
5211:{\displaystyle L}
5191:{\displaystyle P}
5144:{\displaystyle P}
5078:{\displaystyle S}
5011:{\displaystyle P}
4830:{\displaystyle S}
4744:. In particular,
4644:{\displaystyle x}
4624:{\displaystyle y}
4604:{\displaystyle x}
4564:{\displaystyle x}
4549:Axiom of infinity
4312:{\displaystyle y}
4200:{\displaystyle x}
4180:{\displaystyle y}
4160:{\displaystyle x}
4132:{\displaystyle V}
4112:{\displaystyle L}
3750:{\displaystyle y}
3730:{\displaystyle x}
3702:{\displaystyle L}
3535:{\displaystyle L}
3515:{\displaystyle V}
3495:{\displaystyle V}
3475:{\displaystyle L}
3455:{\displaystyle L}
3435:{\displaystyle L}
3415:{\displaystyle y}
3395:{\displaystyle x}
3362:{\displaystyle x}
3342:{\displaystyle V}
3322:{\displaystyle y}
3276:{\displaystyle L}
3224:{\displaystyle L}
3136:{\displaystyle y}
3116:{\displaystyle x}
3096:{\displaystyle y}
3076:{\displaystyle x}
3041:{\displaystyle L}
3021:{\displaystyle V}
3001:{\displaystyle L}
2981:{\displaystyle V}
2961:{\displaystyle V}
2941:{\displaystyle L}
2913:{\displaystyle L}
2692:and relations on
2670:hyperarithmetical
2345:and relations on
2198:{\displaystyle X}
2143:{\displaystyle X}
2108:
1894:, for example if
1695:, and thereafter
1622:{\displaystyle L}
1602:{\displaystyle V}
1520:{\displaystyle L}
1500:{\displaystyle V}
1426:{\displaystyle n}
1361:
1273:{\displaystyle L}
1253:{\displaystyle V}
1203:{\displaystyle L}
1183:{\displaystyle L}
1156:{\displaystyle L}
1000:
980:
976:
972:
911:is an element of
904:{\displaystyle z}
832:
710:
552:{\displaystyle L}
481:
474:
464:
395:
295:{\displaystyle L}
251:to be the set of
218:stage, one takes
203:{\displaystyle V}
179:{\displaystyle L}
94:{\displaystyle L}
66:{\displaystyle L}
16:(Redirected from
17641:
17607:
17606:
17558:History of logic
17553:Category of sets
17446:Decision problem
17225:Ordinal analysis
17166:Sequent calculus
17064:Boolean algebras
17004:
17003:
16978:
16949:logical/constant
16703:
16702:
16689:
16612:Zermelo–Fraenkel
16363:Set operations:
16298:
16297:
16235:
16066:
16065:
16046:Löwenheim–Skolem
15933:Formal semantics
15885:
15878:
15871:
15862:
15861:
15840:Bertrand Russell
15830:John von Neumann
15815:Abraham Fraenkel
15810:Richard Dedekind
15772:Suslin's problem
15683:Cantor's theorem
15400:De Morgan's laws
15258:
15225:
15218:
15211:
15202:
15201:
15197:
15175:
15146:
15136:
15118:
15085:
15066:
15053:Constructibility
15049:Devlin, Keith J.
15044:
15032:
15010:
15003:
14997:
14994:
14988:
14983:K. Devlin 1975,
14981:
14975:
14972:Constructibility
14968:
14962:
14955:
14949:
14946:
14890:
14888:
14887:
14882:
14870:
14868:
14867:
14862:
14841:
14839:
14838:
14833:
14809:
14807:
14806:
14801:
14789:
14787:
14786:
14781:
14760:
14758:
14757:
14752:
14740:
14738:
14737:
14732:
14720:
14718:
14717:
14712:
14689:
14687:
14686:
14681:
14679:
14678:
14673:
14654:
14652:
14651:
14646:
14644:
14626:
14624:
14623:
14618:
14606:
14604:
14603:
14598:
14577:
14575:
14574:
14569:
14551:
14549:
14548:
14543:
14522:
14520:
14519:
14514:
14502:
14500:
14499:
14494:
14464:
14462:
14461:
14456:
14432:
14430:
14429:
14424:
14412:
14410:
14409:
14404:
14393:
14375:
14373:
14372:
14367:
14356:
14355:
14350:
14331:
14329:
14328:
14323:
14311:
14309:
14308:
14303:
14291:
14289:
14288:
14283:
14255:
14253:
14252:
14247:
14242:
14221:
14219:
14218:
14213:
14192:
14190:
14189:
14184:
14163:
14161:
14160:
14155:
14137:
14135:
14134:
14129:
14104:
14102:
14101:
14096:
14085:
14084:
14074:
14041:
14039:
14038:
14033:
14022:
14021:
14011:
13984:
13983:
13967:
13965:
13964:
13959:
13945:
13943:
13942:
13937:
13923:
13922:
13910:
13892:
13890:
13889:
13884:
13873:
13872:
13848:
13846:
13845:
13840:
13818:
13816:
13815:
13810:
13798:
13796:
13795:
13790:
13779:
13778:
13755:
13753:
13752:
13747:
13724:
13722:
13721:
13716:
13704:
13702:
13701:
13696:
13684:
13682:
13681:
13676:
13652:
13650:
13649:
13644:
13623:
13621:
13620:
13615:
13594:
13592:
13591:
13586:
13566:
13564:
13563:
13558:
13543:
13541:
13540:
13535:
13446:
13444:
13443:
13438:
13409:
13407:
13406:
13401:
13366:
13365:
13300:
13299:
13255:
13254:
13198:
13196:
13195:
13190:
13178:
13176:
13175:
13170:
13143:
13141:
13140:
13135:
13133:
13132:
13110:
13108:
13107:
13102:
13090:
13088:
13087:
13082:
13070:
13068:
13067:
13062:
13050:
13048:
13047:
13042:
13030:
13028:
13027:
13022:
13010:
13008:
13007:
13002:
12984:
12982:
12981:
12976:
12974:
12973:
12955:
12954:
12938:
12936:
12935:
12930:
12918:
12916:
12915:
12910:
12889:
12887:
12886:
12881:
12854:
12852:
12851:
12846:
12841:
12840:
12822:
12821:
12787:
12785:
12784:
12779:
12761:
12759:
12758:
12753:
12751:
12750:
12728:
12726:
12725:
12720:
12708:
12706:
12705:
12700:
12682:
12680:
12679:
12674:
12662:
12660:
12659:
12654:
12652:
12651:
12635:
12633:
12632:
12627:
12625:
12624:
12606:
12605:
12589:
12587:
12586:
12581:
12569:
12567:
12566:
12561:
12547:
12546:
12533:
12524:
12503:
12502:
12484:
12483:
12461:
12449:
12448:
12411:
12409:
12408:
12403:
12401:
12400:
12372:
12370:
12369:
12364:
12352:
12350:
12349:
12344:
12332:
12330:
12329:
12324:
12322:
12321:
12291:
12289:
12288:
12283:
12267:
12265:
12264:
12259:
12257:
12256:
12240:
12238:
12237:
12232:
12220:
12218:
12217:
12212:
12200:
12198:
12197:
12192:
12174:
12172:
12171:
12166:
12154:
12152:
12151:
12146:
12144:
12143:
12116:
12115:
12096:
12094:
12093:
12088:
12076:
12074:
12073:
12068:
12056:
12054:
12053:
12048:
12046:
12045:
12024:
12023:
12004:
12002:
12001:
11996:
11994:
11993:
11977:
11975:
11974:
11969:
11957:
11955:
11954:
11949:
11937:
11935:
11934:
11929:
11917:
11915:
11914:
11909:
11894:
11892:
11891:
11886:
11884:
11883:
11861:
11859:
11858:
11853:
11841:
11839:
11838:
11833:
11831:
11830:
11814:
11812:
11811:
11806:
11804:
11803:
11787:
11785:
11784:
11779:
11771:
11770:
11734:
11733:
11706:
11705:
11669:
11668:
11637:
11635:
11634:
11629:
11617:
11615:
11614:
11609:
11597:
11595:
11594:
11589:
11587:
11586:
11560:
11558:
11557:
11552:
11550:
11549:
11533:
11531:
11530:
11525:
11507:
11505:
11504:
11499:
11497:
11496:
11480:
11478:
11477:
11472:
11460:
11458:
11457:
11452:
11450:
11449:
11433:
11431:
11430:
11425:
11423:
11422:
11406:
11404:
11403:
11398:
11396:
11395:
11379:
11377:
11376:
11371:
11369:
11368:
11352:
11350:
11349:
11344:
11342:
11341:
11325:
11323:
11322:
11317:
11297:
11295:
11294:
11289:
11287:
11286:
11270:
11268:
11267:
11262:
11260:
11259:
11243:
11241:
11240:
11235:
11223:
11221:
11219:
11218:
11213:
11205:
11204:
11159:
11158:
11122:
11121:
11089:
11087:
11086:
11081:
11079:
11078:
11050:
11048:
11047:
11042:
11030:
11028:
11027:
11022:
11010:
11008:
11007:
11002:
10990:
10988:
10987:
10982:
10980:
10979:
10949:
10947:
10946:
10941:
10929:
10927:
10926:
10921:
10919:
10918:
10902:
10900:
10899:
10894:
10889:
10888:
10870:
10869:
10847:
10845:
10844:
10839:
10837:
10836:
10820:
10818:
10817:
10812:
10800:
10798:
10797:
10792:
10790:
10789:
10773:
10771:
10770:
10765:
10763:
10762:
10744:
10743:
10727:
10725:
10724:
10719:
10714:
10713:
10695:
10694:
10672:
10670:
10669:
10664:
10646:
10644:
10643:
10638:
10626:
10624:
10623:
10618:
10606:
10604:
10603:
10598:
10581:By induction on
10577:
10575:
10574:
10569:
10553:
10551:
10550:
10545:
10502:
10500:
10499:
10494:
10482:
10480:
10479:
10474:
10459:
10457:
10456:
10451:
10439:
10437:
10436:
10431:
10419:
10417:
10416:
10411:
10399:
10397:
10396:
10391:
10379:
10377:
10376:
10371:
10359:
10357:
10356:
10351:
10335:
10333:
10332:
10327:
10315:
10313:
10312:
10307:
10295:
10293:
10292:
10287:
10272:
10270:
10269:
10264:
10262:
10261:
10239:
10237:
10236:
10231:
10219:
10217:
10216:
10211:
10199:
10197:
10196:
10191:
10176:
10174:
10173:
10168:
10156:
10154:
10153:
10148:
10146:
10145:
10129:
10127:
10126:
10121:
10105:
10103:
10102:
10097:
10095:
10094:
10076:
10075:
10053:
10051:
10050:
10045:
10043:
10042:
10024:
10023:
10001:
9999:
9998:
9993:
9991:
9990:
9978:
9977:
9961:
9959:
9958:
9953:
9951:
9950:
9932:
9931:
9909:
9907:
9906:
9901:
9899:
9898:
9886:
9885:
9869:
9867:
9866:
9861:
9859:
9858:
9846:
9845:
9829:
9827:
9826:
9821:
9803:
9801:
9800:
9795:
9783:
9781:
9780:
9775:
9773:
9772:
9754:
9753:
9737:
9735:
9734:
9729:
9717:
9715:
9714:
9709:
9697:
9695:
9694:
9689:
9687:
9686:
9668:
9667:
9651:
9649:
9648:
9643:
9641:
9640:
9625:parameters from
9624:
9622:
9621:
9616:
9604:
9602:
9601:
9596:
9581:
9579:
9578:
9573:
9555:
9553:
9552:
9547:
9529:
9528:
9526:
9525:
9520:
9508:
9506:
9505:
9500:
9487:
9485:
9484:
9479:
9467:
9465:
9464:
9459:
9447:
9445:
9444:
9439:
9427:
9425:
9424:
9419:
9407:
9405:
9404:
9399:
9387:
9385:
9384:
9379:
9363:
9361:
9360:
9355:
9343:
9341:
9340:
9335:
9323:
9321:
9320:
9315:
9313:
9312:
9296:
9294:
9293:
9288:
9283:
9282:
9270:
9256:
9255:
9230:
9228:
9227:
9222:
9204:
9202:
9201:
9196:
9178:
9177:
9175:
9174:
9169:
9157:
9155:
9154:
9149:
9136:
9134:
9133:
9128:
9116:
9114:
9113:
9108:
9096:
9094:
9093:
9088:
9086:
9085:
9063:
9061:
9060:
9055:
9043:
9041:
9040:
9035:
9033:
9032:
9010:
9008:
9007:
9002:
8990:
8988:
8987:
8982:
8964:
8962:
8961:
8956:
8938:
8936:
8935:
8930:
8918:
8916:
8915:
8910:
8898:
8896:
8895:
8890:
8875:
8873:
8872:
8867:
8850:
8848:
8847:
8842:
8828:
8826:
8825:
8820:
8800:
8798:
8797:
8792:
8780:
8778:
8777:
8772:
8760:
8758:
8757:
8752:
8732:
8730:
8729:
8724:
8712:
8710:
8709:
8704:
8702:
8701:
8685:
8683:
8682:
8677:
8665:
8663:
8662:
8657:
8637:
8635:
8634:
8629:
8617:
8615:
8614:
8609:
8607:
8606:
8587:
8585:
8584:
8579:
8563:
8561:
8560:
8555:
8553:
8552:
8536:
8534:
8533:
8528:
8526:
8525:
8509:
8507:
8506:
8501:
8490:even if true in
8489:
8487:
8486:
8481:
8469:
8467:
8466:
8461:
8459:
8458:
8439:
8437:
8436:
8431:
8399:
8397:
8396:
8391:
8375:
8373:
8372:
8367:
8351:
8349:
8348:
8343:
8330:Regular ordinals
8327:
8325:
8324:
8319:
8306:initial ordinals
8303:
8301:
8300:
8295:
8283:
8281:
8280:
8275:
8263:
8261:
8260:
8255:
8252:
8251:
8242:
8226:
8224:
8223:
8218:
8204:
8178:
8176:
8175:
8170:
8152:
8150:
8149:
8144:
8142:
8141:
8125:
8123:
8122:
8117:
8105:
8103:
8102:
8097:
8079:
8077:
8076:
8071:
8069:
8068:
8052:
8050:
8049:
8044:
8042:
8041:
8025:
8023:
8022:
8017:
8005:
8003:
8002:
7997:
7995:
7994:
7978:
7976:
7975:
7970:
7958:
7956:
7955:
7950:
7938:
7936:
7935:
7930:
7918:
7916:
7915:
7910:
7898:
7896:
7895:
7890:
7878:
7876:
7875:
7870:
7841:
7839:
7838:
7833:
7831:
7830:
7814:
7812:
7811:
7806:
7794:
7792:
7791:
7786:
7774:
7772:
7771:
7766:
7764:
7763:
7747:
7745:
7744:
7739:
7727:
7725:
7724:
7719:
7703:
7701:
7700:
7695:
7683:
7681:
7680:
7675:
7657:
7655:
7654:
7649:
7637:
7635:
7634:
7629:
7617:
7615:
7614:
7609:
7597:
7595:
7594:
7589:
7578:and, similarly,
7577:
7575:
7574:
7569:
7557:
7555:
7554:
7549:
7534:
7532:
7531:
7526:
7524:
7523:
7501:
7499:
7498:
7493:
7488:
7487:
7475:
7457:
7455:
7454:
7449:
7437:
7435:
7434:
7429:
7417:
7415:
7414:
7409:
7397:
7395:
7394:
7389:
7387:
7386:
7370:
7368:
7367:
7362:
7350:
7348:
7347:
7342:
7330:
7328:
7327:
7322:
7310:
7308:
7307:
7302:
7290:
7288:
7287:
7282:
7270:
7268:
7267:
7262:
7242:
7240:
7239:
7234:
7218:
7216:
7215:
7210:
7198:
7196:
7195:
7190:
7175:
7173:
7172:
7167:
7155:
7153:
7152:
7147:
7135:
7133:
7132:
7127:
7115:
7113:
7112:
7107:
7086:
7084:
7083:
7078:
7066:
7064:
7063:
7058:
7046:
7044:
7043:
7038:
7026:
7024:
7023:
7018:
6998:
6996:
6995:
6990:
6978:
6976:
6975:
6970:
6968:
6967:
6942:
6933:
6912:
6906:
6893:
6881:
6880:
6849:
6847:
6846:
6841:
6839:
6838:
6822:
6820:
6819:
6814:
6802:
6800:
6799:
6794:
6779:
6777:
6776:
6771:
6759:
6757:
6756:
6751:
6739:
6737:
6736:
6731:
6719:
6717:
6716:
6711:
6699:
6697:
6696:
6691:
6674:
6672:
6671:
6666:
6654:
6652:
6651:
6646:
6634:
6632:
6631:
6626:
6614:
6612:
6611:
6606:
6587:
6585:
6584:
6579:
6567:
6565:
6564:
6559:
6547:
6545:
6544:
6539:
6537:
6536:
6518:
6517:
6501:
6499:
6498:
6493:
6481:
6479:
6478:
6473:
6461:
6459:
6458:
6453:
6441:
6439:
6438:
6433:
6421:
6419:
6418:
6413:
6389:
6374:
6353:
6347:
6326:
6293:
6291:
6290:
6285:
6276:
6267:
6230:
6215:
6190:
6169:
6151:
6114:
6112:
6111:
6106:
6094:
6092:
6091:
6086:
6074:
6072:
6071:
6066:
6054:
6052:
6051:
6046:
6034:
6032:
6031:
6026:
6014:
6012:
6011:
6006:
5994:
5992:
5991:
5986:
5974:
5972:
5971:
5966:
5954:
5952:
5951:
5946:
5934:
5932:
5931:
5926:
5914:
5912:
5911:
5906:
5894:
5892:
5891:
5886:
5874:
5872:
5871:
5866:
5834:
5832:
5831:
5826:
5802:
5787:
5762:
5741:
5707:
5705:
5704:
5699:
5681:
5679:
5678:
5673:
5646:
5644:
5643:
5638:
5611:
5609:
5608:
5603:
5576:
5574:
5573:
5568:
5557:: Given any set
5548:
5546:
5545:
5540:
5528:
5526:
5525:
5520:
5518:
5517:
5496:
5495:
5485:
5476:
5455:
5454:
5436:
5435:
5413:
5391:
5379:
5378:
5347:
5345:
5344:
5339:
5330:
5321:
5300:
5299:
5281:
5280:
5258:
5217:
5215:
5214:
5209:
5197:
5195:
5194:
5189:
5177:
5175:
5174:
5169:
5167:
5166:
5150:
5148:
5147:
5142:
5130:
5128:
5127:
5122:
5120:
5119:
5101:
5100:
5084:
5082:
5081:
5076:
5064:
5062:
5061:
5056:
5054:
5053:
5037:
5035:
5034:
5029:
5017:
5015:
5014:
5009:
4992:
4990:
4989:
4984:
4976:
4975:
4957:
4956:
4934:
4897:
4895:
4894:
4889:
4884:
4883:
4865:
4864:
4836:
4834:
4833:
4828:
4817:: Given any set
4808:
4806:
4805:
4800:
4782:
4780:
4779:
4774:
4772:
4771:
4743:
4741:
4740:
4735:
4733:
4732:
4710:
4708:
4707:
4702:
4682:
4680:
4679:
4674:
4650:
4648:
4647:
4642:
4630:
4628:
4627:
4622:
4610:
4608:
4607:
4602:
4590:
4588:
4587:
4582:
4570:
4568:
4567:
4562:
4542:
4540:
4539:
4534:
4516:
4514:
4513:
4508:
4506:
4505:
4474:
4459:
4434:
4413:
4395:
4383:
4382:
4345:
4343:
4342:
4337:
4335:
4334:
4318:
4316:
4315:
4310:
4298:
4296:
4295:
4290:
4288:
4287:
4271:
4269:
4268:
4263:
4261:
4260:
4244:
4242:
4241:
4236:
4234:
4233:
4206:
4204:
4203:
4198:
4186:
4184:
4183:
4178:
4166:
4164:
4163:
4158:
4138:
4136:
4135:
4130:
4118:
4116:
4115:
4110:
4098:
4096:
4095:
4090:
4060:
4058:
4057:
4052:
4050:
4049:
4015:
3993:
3981:
3980:
3931:
3929:
3928:
3923:
3921:
3920:
3898:
3896:
3895:
3890:
3888:
3887:
3865:
3863:
3862:
3857:
3845:
3843:
3842:
3837:
3819:
3817:
3816:
3811:
3788:
3786:
3785:
3780:
3756:
3754:
3753:
3748:
3736:
3734:
3733:
3728:
3715:Axiom of pairing
3708:
3706:
3705:
3700:
3688:
3686:
3685:
3680:
3659:
3657:
3656:
3651:
3649:
3648:
3632:
3630:
3629:
3624:
3607:
3606:
3579:
3578:
3541:
3539:
3538:
3533:
3521:
3519:
3518:
3513:
3501:
3499:
3498:
3493:
3481:
3479:
3478:
3473:
3461:
3459:
3458:
3453:
3441:
3439:
3438:
3433:
3421:
3419:
3418:
3413:
3401:
3399:
3398:
3393:
3368:
3366:
3365:
3360:
3348:
3346:
3345:
3340:
3328:
3326:
3325:
3320:
3308:
3306:
3305:
3300:
3282:
3280:
3279:
3274:
3262:
3260:
3259:
3254:
3230:
3228:
3227:
3222:
3210:
3208:
3207:
3202:
3178:
3176:
3175:
3170:
3142:
3140:
3139:
3134:
3122:
3120:
3119:
3114:
3102:
3100:
3099:
3094:
3082:
3080:
3079:
3074:
3047:
3045:
3044:
3039:
3027:
3025:
3024:
3019:
3007:
3005:
3004:
2999:
2987:
2985:
2984:
2979:
2967:
2965:
2964:
2959:
2947:
2945:
2944:
2939:
2922:transitive class
2919:
2917:
2916:
2911:
2899:
2897:
2896:
2891:
2860:
2858:
2857:
2852:
2850:
2849:
2847:
2846:
2837:
2817:that belongs to
2816:
2814:
2813:
2808:
2792:
2790:
2789:
2784:
2781:
2780:
2771:
2755:
2753:
2752:
2747:
2745:
2744:
2742:
2741:
2732:
2711:
2709:
2708:
2703:
2691:
2689:
2688:
2683:
2664:
2662:
2661:
2656:
2654:
2653:
2631:
2629:
2628:
2625:{\displaystyle }
2623:
2616:
2614:
2613:
2608:
2606:
2605:
2583:
2581:
2580:
2575:
2563:
2561:
2560:
2555:
2553:
2552:
2530:
2528:
2527:
2522:
2510:
2508:
2507:
2502:
2500:
2499:
2483:
2481:
2480:
2475:
2473:
2472:
2450:
2448:
2447:
2442:
2430:
2428:
2427:
2422:
2420:
2419:
2397:
2395:
2394:
2389:
2387:
2386:
2364:
2362:
2361:
2356:
2344:
2342:
2341:
2336:
2316:Gödel operations
2313:
2311:
2310:
2305:
2300:
2299:
2284:
2283:
2267:
2265:
2264:
2259:
2257:
2256:
2240:
2238:
2237:
2232:
2230:
2229:
2204:
2202:
2201:
2196:
2176:
2174:
2173:
2168:
2166:
2165:
2149:
2147:
2146:
2141:
2129:
2127:
2126:
2121:
2110:
2109:
2106:
2089:
2087:
2086:
2081:
2069:
2067:
2066:
2061:
2059:
2058:
2038:
2036:
2035:
2030:
2015:
2013:
2012:
2007:
1995:
1993:
1992:
1987:
1985:
1984:
1967:
1965:
1964:
1959:
1957:
1956:
1939:
1937:
1936:
1931:
1913:
1911:
1910:
1905:
1893:
1891:
1890:
1885:
1883:
1882:
1860:
1858:
1857:
1852:
1850:
1849:
1833:
1831:
1830:
1825:
1823:
1822:
1807:does imply that
1806:
1804:
1803:
1798:
1780:
1778:
1777:
1772:
1754:
1752:
1751:
1746:
1744:
1743:
1727:
1725:
1724:
1719:
1717:
1716:
1694:
1692:
1691:
1686:
1684:
1683:
1661:
1659:
1658:
1653:
1651:
1650:
1628:
1626:
1625:
1620:
1608:
1606:
1605:
1600:
1580:
1578:
1577:
1572:
1570:
1569:
1553:
1551:
1550:
1545:
1543:
1542:
1526:
1524:
1523:
1518:
1506:
1504:
1503:
1498:
1486:
1484:
1483:
1478:
1476:
1475:
1459:
1457:
1456:
1451:
1449:
1448:
1432:
1430:
1429:
1424:
1409:
1407:
1406:
1401:
1395:
1394:
1375:
1357:
1356:
1340:
1338:
1337:
1332:
1321:For any ordinal
1317:
1315:
1314:
1309:
1307:
1306:
1279:
1277:
1276:
1271:
1259:
1257:
1256:
1251:
1239:
1237:
1236:
1231:
1209:
1207:
1206:
1201:
1189:
1187:
1186:
1181:
1170:The elements of
1162:
1160:
1159:
1154:
1138:
1136:
1135:
1130:
1128:
1127:
1107:
1105:
1104:
1099:
1097:
1096:
1074:
1072:
1071:
1066:
1064:
1063:
1047:
1045:
1044:
1039:
1037:
1036:
1015:
1014:
1002:
1001:
998:
978:
977:
974:
970:
969:
968:
937:
935:
934:
929:
927:
926:
910:
908:
907:
902:
887:of all ordinals.
878:
876:
875:
870:
865:
864:
854:
853:
816:
814:
813:
808:
794:
792:
791:
786:
774:
772:
771:
766:
748:
746:
745:
740:
735:
734:
724:
706:
705:
685:
683:
682:
677:
663:
661:
660:
655:
647:
646:
625:
624:
601:
599:
598:
593:
582:
581:
558:
556:
555:
550:
537:
535:
534:
529:
524:
523:
511:
510:
492:
491:
482:
479:
472:
471:
470:
462:
455:
454:
436:
435:
396:
393:
373:
372:
301:
299:
298:
293:
281:
279:
278:
273:
271:
270:
250:
248:
247:
242:
240:
239:
209:
207:
206:
201:
185:
183:
182:
177:
130:
128:
127:
122:
120:
119:
100:
98:
97:
92:
72:
70:
69:
64:
21:
17649:
17648:
17644:
17643:
17642:
17640:
17639:
17638:
17619:
17618:
17617:
17612:
17601:
17594:
17539:Category theory
17529:Algebraic logic
17512:
17483:Lambda calculus
17421:Church encoding
17407:
17383:Truth predicate
17239:
17205:Complete theory
17128:
16997:
16993:
16989:
16984:
16976:
16696: and
16692:
16687:
16673:
16649:New Foundations
16617:axiom of choice
16600:
16562:Gödel numbering
16502: and
16494:
16398:
16283:
16233:
16214:
16163:Boolean algebra
16149:
16113:Equiconsistency
16078:Classical logic
16055:
16036:Halting problem
16024: and
16000: and
15988: and
15987:
15982:Theorems (
15977:
15894:
15889:
15859:
15854:
15781:
15760:
15744:
15709:New Foundations
15656:
15546:
15465:Cardinal number
15448:
15434:
15375:
15259:
15250:
15234:
15229:
15194:
15164:
15082:
15063:
15041:
15018:
15013:
15004:
15000:
14995:
14991:
14982:
14978:
14969:
14965:
14957:K. J. Devlin, "
14956:
14952:
14947:
14943:
14939:
14897:
14876:
14873:
14872:
14847:
14844:
14843:
14818:
14815:
14814:
14795:
14792:
14791:
14766:
14763:
14762:
14746:
14743:
14742:
14726:
14723:
14722:
14697:
14694:
14693:
14674:
14663:
14662:
14660:
14657:
14656:
14634:
14632:
14629:
14628:
14612:
14609:
14608:
14583:
14580:
14579:
14557:
14554:
14553:
14528:
14525:
14524:
14508:
14505:
14504:
14470:
14467:
14466:
14438:
14435:
14434:
14418:
14415:
14414:
14383:
14381:
14378:
14377:
14351:
14340:
14339:
14337:
14334:
14333:
14317:
14314:
14313:
14297:
14294:
14293:
14268:
14265:
14264:
14238:
14230:
14227:
14226:
14198:
14195:
14194:
14169:
14166:
14165:
14143:
14140:
14139:
14114:
14111:
14110:
14080:
14076:
14070:
14049:
14046:
14045:
14017:
14013:
14001:
13979:
13975:
13973:
13970:
13969:
13953:
13950:
13949:
13918:
13914:
13900:
13898:
13895:
13894:
13862:
13858:
13856:
13853:
13852:
13828:
13825:
13824:
13804:
13801:
13800:
13774:
13770:
13768:
13765:
13764:
13732:
13729:
13728:
13710:
13707:
13706:
13690:
13687:
13686:
13661:
13658:
13657:
13629:
13626:
13625:
13600:
13597:
13596:
13580:
13577:
13576:
13573:
13552:
13549:
13548:
13454:
13451:
13450:
13417:
13414:
13413:
13361:
13357:
13295:
13291:
13244:
13240:
13206:
13203:
13202:
13184:
13181:
13180:
13152:
13149:
13148:
13144:as parameters.
13128:
13124:
13116:
13113:
13112:
13096:
13093:
13092:
13076:
13073:
13072:
13056:
13053:
13052:
13036:
13033:
13032:
13016:
13013:
13012:
12990:
12987:
12986:
12963:
12959:
12950:
12946:
12944:
12941:
12940:
12924:
12921:
12920:
12895:
12892:
12891:
12860:
12857:
12856:
12836:
12832:
12817:
12813:
12793:
12790:
12789:
12767:
12764:
12763:
12746:
12742:
12734:
12731:
12730:
12714:
12711:
12710:
12688:
12685:
12684:
12668:
12665:
12664:
12647:
12643:
12641:
12638:
12637:
12620:
12616:
12601:
12597:
12595:
12592:
12591:
12575:
12572:
12571:
12542:
12538:
12526:
12508:
12498:
12494:
12479:
12475:
12451:
12444:
12440:
12417:
12414:
12413:
12390:
12386:
12378:
12375:
12374:
12358:
12355:
12354:
12338:
12335:
12334:
12317:
12313:
12305:
12302:
12301:
12298:
12277:
12274:
12273:
12272:relativized to
12252:
12248:
12246:
12243:
12242:
12226:
12223:
12222:
12206:
12203:
12202:
12180:
12177:
12176:
12160:
12157:
12156:
12133:
12129:
12111:
12110:
12102:
12099:
12098:
12082:
12079:
12078:
12062:
12059:
12058:
12041:
12037:
12019:
12018:
12010:
12007:
12006:
11989:
11985:
11983:
11980:
11979:
11963:
11960:
11959:
11943:
11940:
11939:
11938:as the rank of
11923:
11920:
11919:
11903:
11900:
11899:
11873:
11869:
11867:
11864:
11863:
11847:
11844:
11843:
11826:
11822:
11820:
11817:
11816:
11799:
11795:
11793:
11790:
11789:
11766:
11762:
11729:
11725:
11701:
11697:
11664:
11660:
11643:
11640:
11639:
11623:
11620:
11619:
11603:
11600:
11599:
11582:
11578:
11570:
11567:
11566:
11564:
11545:
11541:
11539:
11536:
11535:
11513:
11510:
11509:
11492:
11488:
11486:
11483:
11482:
11466:
11463:
11462:
11445:
11441:
11439:
11436:
11435:
11418:
11414:
11412:
11409:
11408:
11391:
11387:
11385:
11382:
11381:
11364:
11360:
11358:
11355:
11354:
11337:
11333:
11331:
11328:
11327:
11311:
11308:
11307:
11282:
11278:
11276:
11273:
11272:
11255:
11251:
11249:
11246:
11245:
11229:
11226:
11225:
11200:
11196:
11154:
11150:
11117:
11113:
11096:
11093:
11092:
11091:
11068:
11064:
11056:
11053:
11052:
11036:
11033:
11032:
11016:
11013:
11012:
10996:
10993:
10992:
10975:
10971:
10963:
10960:
10959:
10956:
10935:
10932:
10931:
10914:
10910:
10908:
10905:
10904:
10884:
10880:
10865:
10861:
10853:
10850:
10849:
10832:
10828:
10826:
10823:
10822:
10806:
10803:
10802:
10785:
10781:
10779:
10776:
10775:
10758:
10754:
10739:
10735:
10733:
10730:
10729:
10709:
10705:
10690:
10686:
10678:
10675:
10674:
10652:
10649:
10648:
10632:
10629:
10628:
10612:
10609:
10608:
10586:
10583:
10582:
10563:
10560:
10559:
10539:
10536:
10535:
10532:axiom of choice
10520:
10517:
10509:axiom of choice
10488:
10485:
10484:
10468:
10465:
10464:
10445:
10442:
10441:
10425:
10422:
10421:
10405:
10402:
10401:
10385:
10382:
10381:
10365:
10362:
10361:
10345:
10342:
10341:
10321:
10318:
10317:
10301:
10298:
10297:
10281:
10278:
10277:
10251:
10247:
10245:
10242:
10241:
10225:
10222:
10221:
10205:
10202:
10201:
10185:
10182:
10181:
10162:
10159:
10158:
10141:
10137:
10135:
10132:
10131:
10115:
10112:
10111:
10084:
10080:
10065:
10061:
10059:
10056:
10055:
10032:
10028:
10013:
10009:
10007:
10004:
10003:
9986:
9982:
9973:
9969:
9967:
9964:
9963:
9940:
9936:
9921:
9917:
9915:
9912:
9911:
9894:
9890:
9881:
9877:
9875:
9872:
9871:
9854:
9850:
9841:
9837:
9835:
9832:
9831:
9809:
9806:
9805:
9789:
9786:
9785:
9768:
9764:
9749:
9745:
9743:
9740:
9739:
9723:
9720:
9719:
9703:
9700:
9699:
9682:
9678:
9663:
9659:
9657:
9654:
9653:
9636:
9632:
9630:
9627:
9626:
9610:
9607:
9606:
9590:
9587:
9586:
9561:
9558:
9557:
9535:
9532:
9531:
9530:if and only if
9514:
9511:
9510:
9494:
9491:
9490:
9489:
9473:
9470:
9469:
9453:
9450:
9449:
9433:
9430:
9429:
9413:
9410:
9409:
9393:
9390:
9389:
9373:
9370:
9369:
9366:Gödel numbering
9349:
9346:
9345:
9329:
9326:
9325:
9308:
9304:
9302:
9299:
9298:
9278:
9274:
9260:
9245:
9241:
9239:
9236:
9235:
9210:
9207:
9206:
9184:
9181:
9180:
9179:if and only if
9163:
9160:
9159:
9143:
9140:
9139:
9138:
9122:
9119:
9118:
9102:
9099:
9098:
9075:
9071:
9069:
9066:
9065:
9049:
9046:
9045:
9022:
9018:
9016:
9013:
9012:
8996:
8993:
8992:
8970:
8967:
8966:
8944:
8941:
8940:
8924:
8921:
8920:
8904:
8901:
8900:
8884:
8881:
8880:
8861:
8858:
8857:
8836:
8833:
8832:
8814:
8811:
8810:
8807:
8786:
8783:
8782:
8766:
8763:
8762:
8746:
8743:
8742:
8718:
8715:
8714:
8697:
8693:
8691:
8688:
8687:
8671:
8668:
8667:
8651:
8648:
8647:
8623:
8620:
8619:
8602:
8598:
8596:
8593:
8592:
8573:
8570:
8569:
8548:
8544:
8542:
8539:
8538:
8521:
8517:
8515:
8512:
8511:
8495:
8492:
8491:
8475:
8472:
8471:
8454:
8450:
8448:
8445:
8444:
8425:
8422:
8421:
8406:Mahlo cardinals
8385:
8382:
8381:
8361:
8358:
8357:
8354:limit cardinals
8337:
8334:
8333:
8313:
8310:
8309:
8289:
8286:
8285:
8269:
8266:
8265:
8244:
8243:
8238:
8232:
8229:
8228:
8194:
8192:
8189:
8188:
8185:
8158:
8155:
8154:
8137:
8133:
8131:
8128:
8127:
8111:
8108:
8107:
8085:
8082:
8081:
8064:
8060:
8058:
8055:
8054:
8037:
8033:
8031:
8028:
8027:
8011:
8008:
8007:
7990:
7986:
7984:
7981:
7980:
7964:
7961:
7960:
7959:, and both the
7944:
7941:
7940:
7924:
7921:
7920:
7904:
7901:
7900:
7884:
7881:
7880:
7864:
7861:
7860:
7826:
7822:
7820:
7817:
7816:
7800:
7797:
7796:
7780:
7777:
7776:
7759:
7755:
7753:
7750:
7749:
7733:
7730:
7729:
7713:
7710:
7709:
7689:
7686:
7685:
7669:
7666:
7665:
7643:
7640:
7639:
7623:
7620:
7619:
7603:
7600:
7599:
7583:
7580:
7579:
7563:
7560:
7559:
7543:
7540:
7539:
7513:
7509:
7507:
7504:
7503:
7483:
7479:
7465:
7463:
7460:
7459:
7443:
7440:
7439:
7423:
7420:
7419:
7403:
7400:
7399:
7398:is the same in
7382:
7378:
7376:
7373:
7372:
7356:
7353:
7352:
7336:
7333:
7332:
7316:
7313:
7312:
7296:
7293:
7292:
7276:
7273:
7272:
7256:
7253:
7252:
7249:
7228:
7225:
7224:
7204:
7201:
7200:
7184:
7181:
7180:
7161:
7158:
7157:
7141:
7138:
7137:
7121:
7118:
7117:
7101:
7098:
7097:
7095:
7072:
7069:
7068:
7052:
7049:
7048:
7032:
7029:
7028:
7012:
7009:
7008:
7005:Axiom of choice
6984:
6981:
6980:
6957:
6953:
6935:
6914:
6908:
6899:
6883:
6876:
6872:
6855:
6852:
6851:
6834:
6830:
6828:
6825:
6824:
6808:
6805:
6804:
6788:
6785:
6784:
6765:
6762:
6761:
6745:
6742:
6741:
6725:
6722:
6721:
6705:
6702:
6701:
6700:will not be in
6685:
6682:
6681:
6660:
6657:
6656:
6640:
6637:
6636:
6620:
6617:
6616:
6600:
6597:
6596:
6573:
6570:
6569:
6553:
6550:
6549:
6526:
6522:
6513:
6509:
6507:
6504:
6503:
6487:
6484:
6483:
6467:
6464:
6463:
6447:
6444:
6443:
6427:
6424:
6423:
6376:
6361:
6349:
6328:
6310:
6299:
6296:
6295:
6269:
6251:
6217:
6202:
6171:
6153:
6141:
6120:
6117:
6116:
6100:
6097:
6096:
6080:
6077:
6076:
6060:
6057:
6056:
6040:
6037:
6036:
6020:
6017:
6016:
6000:
5997:
5996:
5980:
5977:
5976:
5960:
5957:
5956:
5940:
5937:
5936:
5920:
5917:
5916:
5900:
5897:
5896:
5880:
5877:
5876:
5845:
5842:
5841:
5789:
5774:
5743:
5725:
5713:
5710:
5709:
5687:
5684:
5683:
5652:
5649:
5648:
5617:
5614:
5613:
5582:
5579:
5578:
5562:
5559:
5558:
5534:
5531:
5530:
5507:
5503:
5491:
5487:
5478:
5460:
5450:
5446:
5431:
5427:
5403:
5381:
5374:
5370:
5353:
5350:
5349:
5323:
5305:
5295:
5291:
5276:
5272:
5248:
5227:
5224:
5223:
5203:
5200:
5199:
5183:
5180:
5179:
5178:if and only if
5162:
5158:
5156:
5153:
5152:
5136:
5133:
5132:
5115:
5111:
5096:
5092:
5090:
5087:
5086:
5070:
5067:
5066:
5049:
5045:
5043:
5040:
5039:
5023:
5020:
5019:
5003:
5000:
4999:
4971:
4967:
4952:
4948:
4924:
4903:
4900:
4899:
4879:
4875:
4860:
4856:
4842:
4839:
4838:
4822:
4819:
4818:
4788:
4785:
4784:
4761:
4757:
4749:
4746:
4745:
4722:
4718:
4716:
4713:
4712:
4696:
4693:
4692:
4656:
4653:
4652:
4636:
4633:
4632:
4616:
4613:
4612:
4596:
4593:
4592:
4576:
4573:
4572:
4556:
4553:
4552:
4522:
4519:
4518:
4495:
4491:
4461:
4446:
4415:
4397:
4385:
4378:
4374:
4351:
4348:
4347:
4330:
4326:
4324:
4321:
4320:
4319:is a subset of
4304:
4301:
4300:
4283:
4279:
4277:
4274:
4273:
4256:
4252:
4250:
4247:
4246:
4229:
4225:
4217:
4214:
4213:
4192:
4189:
4188:
4172:
4169:
4168:
4167:there is a set
4152:
4149:
4148:
4124:
4121:
4120:
4104:
4101:
4100:
4066:
4063:
4062:
4039:
4035:
4008:
3983:
3976:
3972:
3937:
3934:
3933:
3916:
3912:
3904:
3901:
3900:
3883:
3879:
3871:
3868:
3867:
3851:
3848:
3847:
3825:
3822:
3821:
3799:
3796:
3795:
3762:
3759:
3758:
3757:are sets, then
3742:
3739:
3738:
3722:
3719:
3718:
3694:
3691:
3690:
3665:
3662:
3661:
3644:
3640:
3638:
3635:
3634:
3602:
3598:
3574:
3570:
3559:
3556:
3555:
3527:
3524:
3523:
3507:
3504:
3503:
3487:
3484:
3483:
3467:
3464:
3463:
3447:
3444:
3443:
3427:
3424:
3423:
3407:
3404:
3403:
3387:
3384:
3383:
3354:
3351:
3350:
3334:
3331:
3330:
3314:
3311:
3310:
3288:
3285:
3284:
3268:
3265:
3264:
3236:
3233:
3232:
3216:
3213:
3212:
3184:
3181:
3180:
3152:
3149:
3148:
3128:
3125:
3124:
3108:
3105:
3104:
3088:
3085:
3084:
3068:
3065:
3064:
3033:
3030:
3029:
3013:
3010:
3009:
2993:
2990:
2989:
2973:
2970:
2969:
2953:
2950:
2949:
2933:
2930:
2929:
2905:
2902:
2901:
2873:
2870:
2869:
2867:
2839:
2838:
2833:
2828:
2824:
2822:
2819:
2818:
2802:
2799:
2798:
2793:stands for the
2773:
2772:
2767:
2761:
2758:
2757:
2734:
2733:
2728:
2723:
2719:
2717:
2714:
2713:
2697:
2694:
2693:
2677:
2674:
2673:
2643:
2639:
2637:
2634:
2633:
2621:
2618:
2617:
2595:
2591:
2589:
2586:
2585:
2569:
2566:
2565:
2542:
2538:
2536:
2533:
2532:
2516:
2513:
2512:
2495:
2491:
2489:
2486:
2485:
2462:
2458:
2456:
2453:
2452:
2436:
2433:
2432:
2409:
2405:
2403:
2400:
2399:
2376:
2372:
2370:
2367:
2366:
2350:
2347:
2346:
2330:
2327:
2326:
2295:
2291:
2279:
2275:
2273:
2270:
2269:
2252:
2248:
2246:
2243:
2242:
2219:
2215:
2213:
2210:
2209:
2190:
2187:
2186:
2161:
2157:
2155:
2152:
2151:
2135:
2132:
2131:
2105:
2104:
2102:
2099:
2098:
2075:
2072:
2071:
2054:
2050:
2048:
2045:
2044:
2024:
2021:
2020:
2001:
1998:
1997:
1980:
1976:
1974:
1971:
1970:
1952:
1948:
1946:
1943:
1942:
1919:
1916:
1915:
1899:
1896:
1895:
1878:
1874:
1866:
1863:
1862:
1845:
1841:
1839:
1836:
1835:
1818:
1814:
1812:
1809:
1808:
1786:
1783:
1782:
1760:
1757:
1756:
1739:
1735:
1733:
1730:
1729:
1706:
1702:
1700:
1697:
1696:
1673:
1669:
1667:
1664:
1663:
1640:
1636:
1634:
1631:
1630:
1614:
1611:
1610:
1594:
1591:
1590:
1565:
1561:
1559:
1556:
1555:
1538:
1534:
1532:
1529:
1528:
1512:
1509:
1508:
1492:
1489:
1488:
1471:
1467:
1465:
1462:
1461:
1444:
1440:
1438:
1435:
1434:
1418:
1415:
1414:
1411:
1390:
1386:
1365:
1352:
1348:
1346:
1343:
1342:
1326:
1323:
1322:
1302:
1298:
1296:
1293:
1292:
1289:
1287:
1265:
1262:
1261:
1245:
1242:
1241:
1219:
1216:
1215:
1195:
1192:
1191:
1175:
1172:
1171:
1148:
1145:
1144:
1141:transitive sets
1123:
1119:
1117:
1114:
1113:
1086:
1082:
1080:
1077:
1076:
1075:is a subset of
1059:
1055:
1053:
1050:
1049:
1026:
1022:
1010:
1006:
997:
996:
973:
964:
960:
943:
940:
939:
922:
918:
916:
913:
912:
896:
893:
892:
860:
856:
843:
836:
824:
821:
820:
802:
799:
798:
780:
777:
776:
754:
751:
750:
730:
726:
714:
701:
697:
695:
692:
691:
671:
668:
667:
642:
638:
614:
610:
608:
605:
604:
577:
573:
571:
568:
567:
544:
541:
540:
519:
518:
506:
502:
487:
483:
478:
466:
465:
450:
446:
431:
427:
394: and
392:
368:
367:
347:
344:
343:
316:formal language
310:definable by a
287:
284:
283:
266:
262:
260:
257:
256:
229:
225:
223:
220:
219:
195:
192:
191:
171:
168:
167:
165:
155:with the basic
145:axiom of choice
115:
111:
109:
106:
105:
86:
83:
82:
58:
55:
54:
35:
28:
23:
22:
15:
12:
11:
5:
17647:
17637:
17636:
17631:
17614:
17613:
17599:
17596:
17595:
17593:
17592:
17587:
17582:
17577:
17572:
17571:
17570:
17560:
17555:
17550:
17541:
17536:
17531:
17526:
17524:Abstract logic
17520:
17518:
17514:
17513:
17511:
17510:
17505:
17503:Turing machine
17500:
17495:
17490:
17485:
17480:
17475:
17474:
17473:
17468:
17463:
17458:
17453:
17443:
17441:Computable set
17438:
17433:
17428:
17423:
17417:
17415:
17409:
17408:
17406:
17405:
17400:
17395:
17390:
17385:
17380:
17375:
17370:
17369:
17368:
17363:
17358:
17348:
17343:
17338:
17336:Satisfiability
17333:
17328:
17323:
17322:
17321:
17311:
17310:
17309:
17299:
17298:
17297:
17292:
17287:
17282:
17277:
17267:
17266:
17265:
17260:
17253:Interpretation
17249:
17247:
17241:
17240:
17238:
17237:
17232:
17227:
17222:
17217:
17207:
17202:
17201:
17200:
17199:
17198:
17188:
17183:
17173:
17168:
17163:
17158:
17153:
17148:
17142:
17140:
17134:
17133:
17130:
17129:
17127:
17126:
17118:
17117:
17116:
17115:
17110:
17109:
17108:
17103:
17098:
17078:
17077:
17076:
17074:minimal axioms
17071:
17060:
17059:
17058:
17047:
17046:
17045:
17040:
17035:
17030:
17025:
17020:
17007:
17005:
16986:
16985:
16983:
16982:
16981:
16980:
16968:
16963:
16962:
16961:
16956:
16951:
16946:
16936:
16931:
16926:
16921:
16920:
16919:
16914:
16904:
16903:
16902:
16897:
16892:
16887:
16877:
16872:
16871:
16870:
16865:
16860:
16850:
16849:
16848:
16843:
16838:
16833:
16828:
16823:
16813:
16808:
16803:
16798:
16797:
16796:
16791:
16786:
16781:
16771:
16766:
16764:Formation rule
16761:
16756:
16755:
16754:
16749:
16739:
16738:
16737:
16727:
16722:
16717:
16712:
16706:
16700:
16683:Formal systems
16679:
16678:
16675:
16674:
16672:
16671:
16666:
16661:
16656:
16651:
16646:
16641:
16636:
16631:
16626:
16625:
16624:
16619:
16608:
16606:
16602:
16601:
16599:
16598:
16597:
16596:
16586:
16581:
16580:
16579:
16572:Large cardinal
16569:
16564:
16559:
16554:
16549:
16535:
16534:
16533:
16528:
16523:
16508:
16506:
16496:
16495:
16493:
16492:
16491:
16490:
16485:
16480:
16470:
16465:
16460:
16455:
16450:
16445:
16440:
16435:
16430:
16425:
16420:
16415:
16409:
16407:
16400:
16399:
16397:
16396:
16395:
16394:
16389:
16384:
16379:
16374:
16369:
16361:
16360:
16359:
16354:
16344:
16339:
16337:Extensionality
16334:
16332:Ordinal number
16329:
16319:
16314:
16313:
16312:
16301:
16295:
16289:
16288:
16285:
16284:
16282:
16281:
16276:
16271:
16266:
16261:
16256:
16251:
16250:
16249:
16239:
16238:
16237:
16224:
16222:
16216:
16215:
16213:
16212:
16211:
16210:
16205:
16200:
16190:
16185:
16180:
16175:
16170:
16165:
16159:
16157:
16151:
16150:
16148:
16147:
16142:
16137:
16132:
16127:
16122:
16117:
16116:
16115:
16105:
16100:
16095:
16090:
16085:
16080:
16074:
16072:
16063:
16057:
16056:
16054:
16053:
16048:
16043:
16038:
16033:
16028:
16016:Cantor's
16014:
16009:
16004:
15994:
15992:
15979:
15978:
15976:
15975:
15970:
15965:
15960:
15955:
15950:
15945:
15940:
15935:
15930:
15925:
15920:
15915:
15914:
15913:
15902:
15900:
15896:
15895:
15888:
15887:
15880:
15873:
15865:
15856:
15855:
15853:
15852:
15847:
15845:Thoralf Skolem
15842:
15837:
15832:
15827:
15822:
15817:
15812:
15807:
15802:
15797:
15791:
15789:
15783:
15782:
15780:
15779:
15774:
15769:
15763:
15761:
15759:
15758:
15755:
15749:
15746:
15745:
15743:
15742:
15741:
15740:
15735:
15730:
15729:
15728:
15713:
15712:
15711:
15699:
15698:
15697:
15686:
15685:
15680:
15675:
15670:
15664:
15662:
15658:
15657:
15655:
15654:
15649:
15644:
15639:
15630:
15625:
15620:
15610:
15605:
15604:
15603:
15598:
15593:
15583:
15573:
15568:
15563:
15557:
15555:
15548:
15547:
15545:
15544:
15539:
15534:
15529:
15527:Ordinal number
15524:
15519:
15514:
15509:
15508:
15507:
15502:
15492:
15487:
15482:
15477:
15472:
15462:
15457:
15451:
15449:
15447:
15446:
15443:
15439:
15436:
15435:
15433:
15432:
15427:
15422:
15417:
15412:
15407:
15405:Disjoint union
15402:
15397:
15391:
15385:
15383:
15377:
15376:
15374:
15373:
15372:
15371:
15366:
15355:
15354:
15352:Martin's axiom
15349:
15344:
15339:
15334:
15329:
15324:
15319:
15317:Extensionality
15314:
15313:
15312:
15302:
15297:
15296:
15295:
15290:
15285:
15275:
15269:
15267:
15261:
15260:
15253:
15251:
15249:
15248:
15242:
15240:
15236:
15235:
15228:
15227:
15220:
15213:
15205:
15199:
15198:
15192:
15176:
15162:
15147:
15086:
15080:
15067:
15061:
15045:
15039:
15017:
15014:
15012:
15011:
15005:P. Odifreddi,
14998:
14989:
14976:
14970:K. J. Devlin,
14963:
14950:
14940:
14938:
14935:
14934:
14933:
14928:
14923:
14921:Transitive set
14918:
14913:
14908:
14903:
14896:
14893:
14880:
14860:
14857:
14854:
14851:
14831:
14828:
14825:
14822:
14799:
14779:
14776:
14773:
14770:
14750:
14730:
14710:
14707:
14704:
14701:
14677:
14672:
14669:
14666:
14643:
14640:
14637:
14616:
14596:
14593:
14590:
14587:
14567:
14564:
14561:
14541:
14538:
14535:
14532:
14512:
14492:
14489:
14486:
14483:
14480:
14477:
14474:
14454:
14451:
14448:
14445:
14442:
14422:
14402:
14399:
14396:
14392:
14389:
14386:
14365:
14362:
14359:
14354:
14349:
14346:
14343:
14321:
14301:
14281:
14278:
14275:
14272:
14245:
14241:
14237:
14234:
14211:
14208:
14205:
14202:
14182:
14179:
14176:
14173:
14153:
14150:
14147:
14127:
14124:
14121:
14118:
14107:
14106:
14094:
14091:
14088:
14083:
14079:
14073:
14069:
14065:
14062:
14059:
14056:
14053:
14043:
14031:
14028:
14025:
14020:
14016:
14010:
14007:
14004:
14000:
13996:
13993:
13990:
13987:
13982:
13978:
13957:
13946:
13935:
13932:
13929:
13926:
13921:
13917:
13913:
13909:
13906:
13903:
13882:
13879:
13876:
13871:
13868:
13865:
13861:
13850:
13838:
13835:
13832:
13808:
13788:
13785:
13782:
13777:
13773:
13756:is defined by
13745:
13742:
13739:
13736:
13714:
13694:
13674:
13671:
13668:
13665:
13642:
13639:
13636:
13633:
13613:
13610:
13607:
13604:
13584:
13572:
13569:
13556:
13533:
13530:
13527:
13524:
13521:
13518:
13515:
13512:
13509:
13506:
13503:
13500:
13497:
13494:
13491:
13488:
13485:
13482:
13479:
13476:
13473:
13470:
13467:
13464:
13461:
13458:
13447:is short for:
13436:
13433:
13430:
13427:
13424:
13421:
13399:
13396:
13393:
13390:
13387:
13384:
13381:
13378:
13375:
13372:
13369:
13364:
13360:
13356:
13353:
13349:
13345:
13342:
13339:
13336:
13333:
13330:
13327:
13324:
13321:
13318:
13315:
13312:
13309:
13306:
13303:
13298:
13294:
13290:
13287:
13283:
13279:
13276:
13273:
13270:
13267:
13264:
13261:
13258:
13253:
13250:
13247:
13243:
13239:
13236:
13233:
13229:
13225:
13222:
13219:
13216:
13213:
13210:
13188:
13168:
13165:
13162:
13159:
13156:
13131:
13127:
13123:
13120:
13100:
13080:
13060:
13040:
13020:
13000:
12997:
12994:
12972:
12969:
12966:
12962:
12958:
12953:
12949:
12928:
12908:
12905:
12902:
12899:
12879:
12876:
12873:
12870:
12867:
12864:
12844:
12839:
12835:
12831:
12828:
12825:
12820:
12816:
12812:
12809:
12806:
12803:
12800:
12797:
12777:
12774:
12771:
12749:
12745:
12741:
12738:
12718:
12698:
12695:
12692:
12672:
12650:
12646:
12623:
12619:
12615:
12612:
12609:
12604:
12600:
12579:
12559:
12556:
12553:
12550:
12545:
12541:
12537:
12532:
12529:
12523:
12520:
12517:
12514:
12511:
12506:
12501:
12497:
12493:
12490:
12487:
12482:
12478:
12474:
12471:
12468:
12465:
12460:
12457:
12454:
12447:
12443:
12439:
12436:
12433:
12430:
12427:
12424:
12421:
12399:
12396:
12393:
12389:
12385:
12382:
12362:
12342:
12320:
12316:
12312:
12309:
12297:
12294:
12281:
12255:
12251:
12230:
12210:
12190:
12187:
12184:
12164:
12142:
12139:
12136:
12132:
12128:
12125:
12122:
12119:
12114:
12109:
12106:
12086:
12066:
12044:
12040:
12036:
12033:
12030:
12027:
12022:
12017:
12014:
11992:
11988:
11967:
11947:
11927:
11907:
11882:
11879:
11876:
11872:
11862:is in fact in
11851:
11829:
11825:
11802:
11798:
11777:
11774:
11769:
11765:
11761:
11758:
11755:
11752:
11749:
11746:
11743:
11740:
11737:
11732:
11728:
11724:
11721:
11718:
11715:
11712:
11709:
11704:
11700:
11696:
11693:
11690:
11687:
11684:
11681:
11678:
11675:
11672:
11667:
11663:
11659:
11656:
11653:
11650:
11647:
11627:
11607:
11585:
11581:
11577:
11574:
11562:
11548:
11544:
11523:
11520:
11517:
11495:
11491:
11470:
11448:
11444:
11421:
11417:
11394:
11390:
11367:
11363:
11340:
11336:
11315:
11285:
11281:
11258:
11254:
11233:
11211:
11208:
11203:
11199:
11195:
11192:
11189:
11186:
11183:
11180:
11177:
11174:
11171:
11168:
11165:
11162:
11157:
11153:
11149:
11146:
11143:
11140:
11137:
11134:
11131:
11128:
11125:
11120:
11116:
11112:
11109:
11106:
11103:
11100:
11077:
11074:
11071:
11067:
11063:
11060:
11040:
11020:
11000:
10978:
10974:
10970:
10967:
10955:
10952:
10939:
10917:
10913:
10892:
10887:
10883:
10879:
10876:
10873:
10868:
10864:
10860:
10857:
10835:
10831:
10810:
10788:
10784:
10761:
10757:
10753:
10750:
10747:
10742:
10738:
10717:
10712:
10708:
10704:
10701:
10698:
10693:
10689:
10685:
10682:
10662:
10659:
10656:
10636:
10616:
10596:
10593:
10590:
10567:
10543:
10519:
10515:
10513:
10492:
10472:
10449:
10429:
10409:
10389:
10369:
10349:
10325:
10305:
10285:
10260:
10257:
10254:
10250:
10229:
10209:
10189:
10166:
10144:
10140:
10119:
10093:
10090:
10087:
10083:
10079:
10074:
10071:
10068:
10064:
10041:
10038:
10035:
10031:
10027:
10022:
10019:
10016:
10012:
9989:
9985:
9981:
9976:
9972:
9949:
9946:
9943:
9939:
9935:
9930:
9927:
9924:
9920:
9897:
9893:
9889:
9884:
9880:
9857:
9853:
9849:
9844:
9840:
9819:
9816:
9813:
9793:
9771:
9767:
9763:
9760:
9757:
9752:
9748:
9727:
9707:
9685:
9681:
9677:
9674:
9671:
9666:
9662:
9639:
9635:
9614:
9594:
9571:
9568:
9565:
9545:
9542:
9539:
9518:
9498:
9477:
9457:
9437:
9417:
9397:
9377:
9353:
9333:
9311:
9307:
9286:
9281:
9277:
9273:
9269:
9266:
9263:
9259:
9254:
9251:
9248:
9244:
9220:
9217:
9214:
9194:
9191:
9188:
9167:
9147:
9126:
9106:
9084:
9081:
9078:
9074:
9053:
9031:
9028:
9025:
9021:
9000:
8980:
8977:
8974:
8954:
8951:
8948:
8928:
8908:
8888:
8865:
8840:
8818:
8806:
8803:
8790:
8770:
8750:
8735:indiscernibles
8722:
8700:
8696:
8675:
8655:
8627:
8605:
8601:
8577:
8551:
8547:
8524:
8520:
8499:
8479:
8457:
8453:
8429:
8410:large cardinal
8389:
8365:
8341:
8317:
8293:
8273:
8250:
8247:
8241:
8237:
8216:
8213:
8210:
8207:
8203:
8200:
8197:
8184:
8181:
8168:
8165:
8162:
8140:
8136:
8115:
8095:
8092:
8089:
8080:, we get that
8067:
8063:
8040:
8036:
8015:
7993:
7989:
7968:
7948:
7928:
7908:
7888:
7868:
7859:Because both "
7829:
7825:
7804:
7784:
7762:
7758:
7737:
7717:
7706:standard model
7693:
7673:
7661:If there is a
7647:
7627:
7607:
7587:
7567:
7547:
7522:
7519:
7516:
7512:
7491:
7486:
7482:
7478:
7474:
7471:
7468:
7447:
7427:
7407:
7385:
7381:
7360:
7340:
7320:
7300:
7280:
7260:
7248:
7245:
7232:
7208:
7188:
7177:
7176:
7165:
7145:
7125:
7105:
7093:
7089:
7088:
7076:
7056:
7036:
7016:
7007:: Given a set
7001:
7000:
6988:
6966:
6963:
6960:
6956:
6952:
6949:
6946:
6941:
6938:
6932:
6929:
6926:
6923:
6920:
6917:
6911:
6905:
6902:
6897:
6892:
6889:
6886:
6879:
6875:
6871:
6868:
6865:
6862:
6859:
6837:
6833:
6812:
6792:
6769:
6749:
6729:
6709:
6689:
6677:
6676:
6664:
6644:
6624:
6604:
6595:: For any set
6589:
6588:
6577:
6557:
6535:
6532:
6529:
6525:
6521:
6516:
6512:
6491:
6471:
6451:
6431:
6411:
6408:
6405:
6402:
6399:
6396:
6393:
6388:
6385:
6382:
6379:
6373:
6370:
6367:
6364:
6359:
6356:
6352:
6346:
6343:
6340:
6337:
6334:
6331:
6325:
6322:
6319:
6316:
6313:
6309:
6306:
6303:
6283:
6280:
6275:
6272:
6266:
6263:
6260:
6257:
6254:
6249:
6246:
6243:
6240:
6237:
6234:
6229:
6226:
6223:
6220:
6214:
6211:
6208:
6205:
6200:
6197:
6194:
6189:
6186:
6183:
6180:
6177:
6174:
6168:
6165:
6162:
6159:
6156:
6150:
6147:
6144:
6139:
6136:
6133:
6130:
6127:
6124:
6104:
6084:
6064:
6044:
6024:
6004:
5984:
5964:
5944:
5924:
5904:
5884:
5864:
5861:
5858:
5855:
5852:
5849:
5837:
5836:
5824:
5821:
5818:
5815:
5812:
5809:
5806:
5801:
5798:
5795:
5792:
5786:
5783:
5780:
5777:
5772:
5769:
5766:
5761:
5758:
5755:
5752:
5749:
5746:
5740:
5737:
5734:
5731:
5728:
5723:
5720:
5717:
5697:
5694:
5691:
5671:
5668:
5665:
5662:
5659:
5656:
5636:
5633:
5630:
5627:
5624:
5621:
5601:
5598:
5595:
5592:
5589:
5586:
5566:
5551:
5550:
5538:
5516:
5513:
5510:
5506:
5502:
5499:
5494:
5490:
5484:
5481:
5475:
5472:
5469:
5466:
5463:
5458:
5453:
5449:
5445:
5442:
5439:
5434:
5430:
5426:
5423:
5420:
5417:
5412:
5409:
5406:
5401:
5398:
5395:
5390:
5387:
5384:
5377:
5373:
5369:
5366:
5363:
5360:
5357:
5337:
5334:
5329:
5326:
5320:
5317:
5314:
5311:
5308:
5303:
5298:
5294:
5290:
5287:
5284:
5279:
5275:
5271:
5268:
5265:
5262:
5257:
5254:
5251:
5246:
5243:
5240:
5237:
5234:
5231:
5207:
5187:
5165:
5161:
5140:
5118:
5114:
5110:
5107:
5104:
5099:
5095:
5074:
5052:
5048:
5027:
5007:
4995:
4994:
4982:
4979:
4974:
4970:
4966:
4963:
4960:
4955:
4951:
4947:
4944:
4941:
4938:
4933:
4930:
4927:
4922:
4919:
4916:
4913:
4910:
4907:
4887:
4882:
4878:
4874:
4871:
4868:
4863:
4859:
4855:
4852:
4849:
4846:
4826:
4811:
4810:
4798:
4795:
4792:
4770:
4767:
4764:
4760:
4756:
4753:
4731:
4728:
4725:
4721:
4700:
4685:
4684:
4672:
4669:
4666:
4663:
4660:
4640:
4620:
4600:
4580:
4560:
4545:
4544:
4532:
4529:
4526:
4504:
4501:
4498:
4494:
4490:
4487:
4484:
4481:
4478:
4473:
4470:
4467:
4464:
4458:
4455:
4452:
4449:
4444:
4441:
4438:
4433:
4430:
4427:
4424:
4421:
4418:
4412:
4409:
4406:
4403:
4400:
4394:
4391:
4388:
4381:
4377:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4333:
4329:
4308:
4286:
4282:
4259:
4255:
4232:
4228:
4224:
4221:
4209:
4208:
4196:
4176:
4156:
4147:: For any set
4145:Axiom of union
4141:
4140:
4128:
4108:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4048:
4045:
4042:
4038:
4034:
4031:
4028:
4025:
4022:
4019:
4014:
4011:
4006:
4003:
4000:
3997:
3992:
3989:
3986:
3979:
3975:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3919:
3915:
3911:
3908:
3886:
3882:
3878:
3875:
3855:
3835:
3832:
3829:
3809:
3806:
3803:
3791:
3790:
3778:
3775:
3772:
3769:
3766:
3746:
3726:
3711:
3710:
3698:
3678:
3675:
3672:
3669:
3647:
3643:
3633:, which is in
3622:
3619:
3616:
3613:
3610:
3605:
3601:
3597:
3594:
3591:
3588:
3585:
3582:
3577:
3573:
3569:
3566:
3563:
3552:
3551:
3550:: {} is a set.
3544:
3543:
3531:
3511:
3491:
3471:
3451:
3431:
3411:
3391:
3379:
3378:
3371:
3370:
3358:
3338:
3318:
3298:
3295:
3292:
3272:
3252:
3249:
3246:
3243:
3240:
3220:
3200:
3197:
3194:
3191:
3188:
3168:
3165:
3162:
3159:
3156:
3145:
3144:
3132:
3112:
3092:
3072:
3048:is a model of
3037:
3017:
2997:
2977:
2957:
2937:
2909:
2889:
2886:
2883:
2880:
2877:
2866:
2863:
2845:
2842:
2836:
2832:
2827:
2806:
2779:
2776:
2770:
2766:
2740:
2737:
2731:
2727:
2722:
2701:
2681:
2652:
2649:
2646:
2642:
2604:
2601:
2598:
2594:
2573:
2551:
2548:
2545:
2541:
2520:
2498:
2494:
2471:
2468:
2465:
2461:
2440:
2418:
2415:
2412:
2408:
2385:
2382:
2379:
2375:
2354:
2334:
2303:
2298:
2294:
2290:
2287:
2282:
2278:
2255:
2251:
2228:
2225:
2222:
2218:
2194:
2179:Levy hierarchy
2164:
2160:
2139:
2119:
2116:
2113:
2079:
2057:
2053:
2028:
2005:
1983:
1979:
1955:
1951:
1929:
1926:
1923:
1903:
1881:
1877:
1873:
1870:
1848:
1844:
1821:
1817:
1796:
1793:
1790:
1770:
1767:
1764:
1742:
1738:
1715:
1712:
1709:
1705:
1682:
1679:
1676:
1672:
1649:
1646:
1643:
1639:
1618:
1598:
1568:
1564:
1541:
1537:
1516:
1496:
1474:
1470:
1447:
1443:
1422:
1398:
1393:
1389:
1385:
1382:
1379:
1374:
1371:
1368:
1364:
1360:
1355:
1351:
1330:
1320:
1305:
1301:
1288:
1285:
1282:
1269:
1249:
1229:
1226:
1223:
1199:
1179:
1152:
1126:
1122:
1095:
1092:
1089:
1085:
1062:
1058:
1035:
1032:
1029:
1025:
1021:
1018:
1013:
1009:
1005:
995:
992:
989:
986:
983:
967:
963:
959:
956:
953:
950:
947:
925:
921:
900:
889:
888:
868:
863:
859:
852:
849:
846:
842:
839:
835:
831:
828:
818:
806:
784:
764:
761:
758:
738:
733:
729:
723:
720:
717:
713:
709:
704:
700:
675:
664:
653:
650:
645:
641:
637:
634:
631:
628:
623:
620:
617:
613:
602:
591:
588:
585:
580:
576:
559:is defined by
548:
527:
522:
517:
514:
509:
505:
501:
498:
495:
490:
486:
477:
469:
461:
458:
453:
449:
445:
442:
439:
434:
430:
426:
423:
420:
417:
414:
411:
408:
405:
402:
399:
391:
388:
385:
382:
379:
376:
371:
366:
363:
360:
357:
354:
351:
334:
333:
326:
319:
318:of set theory,
291:
269:
265:
238:
235:
232:
228:
199:
175:
164:
161:
118:
114:
90:
62:
53:), denoted by
26:
9:
6:
4:
3:
2:
17646:
17635:
17632:
17630:
17627:
17626:
17624:
17611:
17610:
17605:
17597:
17591:
17588:
17586:
17583:
17581:
17578:
17576:
17573:
17569:
17566:
17565:
17564:
17561:
17559:
17556:
17554:
17551:
17549:
17545:
17542:
17540:
17537:
17535:
17532:
17530:
17527:
17525:
17522:
17521:
17519:
17515:
17509:
17506:
17504:
17501:
17499:
17498:Recursive set
17496:
17494:
17491:
17489:
17486:
17484:
17481:
17479:
17476:
17472:
17469:
17467:
17464:
17462:
17459:
17457:
17454:
17452:
17449:
17448:
17447:
17444:
17442:
17439:
17437:
17434:
17432:
17429:
17427:
17424:
17422:
17419:
17418:
17416:
17414:
17410:
17404:
17401:
17399:
17396:
17394:
17391:
17389:
17386:
17384:
17381:
17379:
17376:
17374:
17371:
17367:
17364:
17362:
17359:
17357:
17354:
17353:
17352:
17349:
17347:
17344:
17342:
17339:
17337:
17334:
17332:
17329:
17327:
17324:
17320:
17317:
17316:
17315:
17312:
17308:
17307:of arithmetic
17305:
17304:
17303:
17300:
17296:
17293:
17291:
17288:
17286:
17283:
17281:
17278:
17276:
17273:
17272:
17271:
17268:
17264:
17261:
17259:
17256:
17255:
17254:
17251:
17250:
17248:
17246:
17242:
17236:
17233:
17231:
17228:
17226:
17223:
17221:
17218:
17215:
17214:from ZFC
17211:
17208:
17206:
17203:
17197:
17194:
17193:
17192:
17189:
17187:
17184:
17182:
17179:
17178:
17177:
17174:
17172:
17169:
17167:
17164:
17162:
17159:
17157:
17154:
17152:
17149:
17147:
17144:
17143:
17141:
17139:
17135:
17125:
17124:
17120:
17119:
17114:
17113:non-Euclidean
17111:
17107:
17104:
17102:
17099:
17097:
17096:
17092:
17091:
17089:
17086:
17085:
17083:
17079:
17075:
17072:
17070:
17067:
17066:
17065:
17061:
17057:
17054:
17053:
17052:
17048:
17044:
17041:
17039:
17036:
17034:
17031:
17029:
17026:
17024:
17021:
17019:
17016:
17015:
17013:
17009:
17008:
17006:
17001:
16995:
16990:Example
16987:
16979:
16974:
16973:
16972:
16969:
16967:
16964:
16960:
16957:
16955:
16952:
16950:
16947:
16945:
16942:
16941:
16940:
16937:
16935:
16932:
16930:
16927:
16925:
16922:
16918:
16915:
16913:
16910:
16909:
16908:
16905:
16901:
16898:
16896:
16893:
16891:
16888:
16886:
16883:
16882:
16881:
16878:
16876:
16873:
16869:
16866:
16864:
16861:
16859:
16856:
16855:
16854:
16851:
16847:
16844:
16842:
16839:
16837:
16834:
16832:
16829:
16827:
16824:
16822:
16819:
16818:
16817:
16814:
16812:
16809:
16807:
16804:
16802:
16799:
16795:
16792:
16790:
16787:
16785:
16782:
16780:
16777:
16776:
16775:
16772:
16770:
16767:
16765:
16762:
16760:
16757:
16753:
16750:
16748:
16747:by definition
16745:
16744:
16743:
16740:
16736:
16733:
16732:
16731:
16728:
16726:
16723:
16721:
16718:
16716:
16713:
16711:
16708:
16707:
16704:
16701:
16699:
16695:
16690:
16684:
16680:
16670:
16667:
16665:
16662:
16660:
16657:
16655:
16652:
16650:
16647:
16645:
16642:
16640:
16637:
16635:
16634:Kripke–Platek
16632:
16630:
16627:
16623:
16620:
16618:
16615:
16614:
16613:
16610:
16609:
16607:
16603:
16595:
16592:
16591:
16590:
16587:
16585:
16582:
16578:
16575:
16574:
16573:
16570:
16568:
16565:
16563:
16560:
16558:
16555:
16553:
16550:
16547:
16543:
16539:
16536:
16532:
16529:
16527:
16524:
16522:
16519:
16518:
16517:
16513:
16510:
16509:
16507:
16505:
16501:
16497:
16489:
16486:
16484:
16481:
16479:
16478:constructible
16476:
16475:
16474:
16471:
16469:
16466:
16464:
16461:
16459:
16456:
16454:
16451:
16449:
16446:
16444:
16441:
16439:
16436:
16434:
16431:
16429:
16426:
16424:
16421:
16419:
16416:
16414:
16411:
16410:
16408:
16406:
16401:
16393:
16390:
16388:
16385:
16383:
16380:
16378:
16375:
16373:
16370:
16368:
16365:
16364:
16362:
16358:
16355:
16353:
16350:
16349:
16348:
16345:
16343:
16340:
16338:
16335:
16333:
16330:
16328:
16324:
16320:
16318:
16315:
16311:
16308:
16307:
16306:
16303:
16302:
16299:
16296:
16294:
16290:
16280:
16277:
16275:
16272:
16270:
16267:
16265:
16262:
16260:
16257:
16255:
16252:
16248:
16245:
16244:
16243:
16240:
16236:
16231:
16230:
16229:
16226:
16225:
16223:
16221:
16217:
16209:
16206:
16204:
16201:
16199:
16196:
16195:
16194:
16191:
16189:
16186:
16184:
16181:
16179:
16176:
16174:
16171:
16169:
16166:
16164:
16161:
16160:
16158:
16156:
16155:Propositional
16152:
16146:
16143:
16141:
16138:
16136:
16133:
16131:
16128:
16126:
16123:
16121:
16118:
16114:
16111:
16110:
16109:
16106:
16104:
16101:
16099:
16096:
16094:
16091:
16089:
16086:
16084:
16083:Logical truth
16081:
16079:
16076:
16075:
16073:
16071:
16067:
16064:
16062:
16058:
16052:
16049:
16047:
16044:
16042:
16039:
16037:
16034:
16032:
16029:
16027:
16023:
16019:
16015:
16013:
16010:
16008:
16005:
16003:
15999:
15996:
15995:
15993:
15991:
15985:
15980:
15974:
15971:
15969:
15966:
15964:
15961:
15959:
15956:
15954:
15951:
15949:
15946:
15944:
15941:
15939:
15936:
15934:
15931:
15929:
15926:
15924:
15921:
15919:
15916:
15912:
15909:
15908:
15907:
15904:
15903:
15901:
15897:
15893:
15886:
15881:
15879:
15874:
15872:
15867:
15866:
15863:
15851:
15850:Ernst Zermelo
15848:
15846:
15843:
15841:
15838:
15836:
15835:Willard Quine
15833:
15831:
15828:
15826:
15823:
15821:
15818:
15816:
15813:
15811:
15808:
15806:
15803:
15801:
15798:
15796:
15793:
15792:
15790:
15788:
15787:Set theorists
15784:
15778:
15775:
15773:
15770:
15768:
15765:
15764:
15762:
15756:
15754:
15751:
15750:
15747:
15739:
15736:
15734:
15733:Kripke–Platek
15731:
15727:
15724:
15723:
15722:
15719:
15718:
15717:
15714:
15710:
15707:
15706:
15705:
15704:
15700:
15696:
15693:
15692:
15691:
15688:
15687:
15684:
15681:
15679:
15676:
15674:
15671:
15669:
15666:
15665:
15663:
15659:
15653:
15650:
15648:
15645:
15643:
15640:
15638:
15636:
15631:
15629:
15626:
15624:
15621:
15618:
15614:
15611:
15609:
15606:
15602:
15599:
15597:
15594:
15592:
15589:
15588:
15587:
15584:
15581:
15577:
15574:
15572:
15569:
15567:
15564:
15562:
15559:
15558:
15556:
15553:
15549:
15543:
15540:
15538:
15535:
15533:
15530:
15528:
15525:
15523:
15520:
15518:
15515:
15513:
15510:
15506:
15503:
15501:
15498:
15497:
15496:
15493:
15491:
15488:
15486:
15483:
15481:
15478:
15476:
15473:
15470:
15466:
15463:
15461:
15458:
15456:
15453:
15452:
15450:
15444:
15441:
15440:
15437:
15431:
15428:
15426:
15423:
15421:
15418:
15416:
15413:
15411:
15408:
15406:
15403:
15401:
15398:
15395:
15392:
15390:
15387:
15386:
15384:
15382:
15378:
15370:
15369:specification
15367:
15365:
15362:
15361:
15360:
15357:
15356:
15353:
15350:
15348:
15345:
15343:
15340:
15338:
15335:
15333:
15330:
15328:
15325:
15323:
15320:
15318:
15315:
15311:
15308:
15307:
15306:
15303:
15301:
15298:
15294:
15291:
15289:
15286:
15284:
15281:
15280:
15279:
15276:
15274:
15271:
15270:
15268:
15266:
15262:
15257:
15247:
15244:
15243:
15241:
15237:
15233:
15226:
15221:
15219:
15214:
15212:
15207:
15206:
15203:
15195:
15193:3-540-44085-2
15189:
15185:
15181:
15177:
15173:
15169:
15165:
15159:
15155:
15154:
15148:
15144:
15140:
15135:
15130:
15126:
15122:
15117:
15112:
15108:
15104:
15100:
15096:
15092:
15087:
15083:
15081:3-540-05591-6
15077:
15073:
15068:
15064:
15062:0-387-13258-9
15058:
15054:
15050:
15046:
15042:
15040:0-387-07451-1
15036:
15031:
15030:
15024:
15020:
15019:
15008:
15002:
14993:
14986:
14980:
14973:
14967:
14960:
14954:
14945:
14941:
14932:
14929:
14927:
14924:
14922:
14919:
14917:
14914:
14912:
14909:
14907:
14904:
14902:
14899:
14898:
14892:
14878:
14855:
14849:
14826:
14820:
14811:
14797:
14774:
14768:
14748:
14728:
14705:
14699:
14691:
14675:
14614:
14591:
14585:
14565:
14562:
14559:
14536:
14530:
14510:
14487:
14484:
14481:
14478:
14475:
14449:
14446:
14443:
14433:in the model
14397:
14360:
14352:
14319:
14299:
14276:
14270:
14261:
14259:
14232:
14223:
14206:
14200:
14177:
14171:
14148:
14122:
14116:
14089:
14081:
14077:
14071:
14067:
14063:
14057:
14051:
14044:
14026:
14018:
14014:
14008:
14005:
14002:
13998:
13994:
13988:
13980:
13976:
13955:
13947:
13927:
13919:
13915:
13877:
13869:
13866:
13863:
13859:
13851:
13833:
13822:
13806:
13783:
13775:
13771:
13763:
13762:
13761:
13759:
13740:
13734:
13726:
13712:
13692:
13669:
13663:
13654:
13637:
13631:
13608:
13602:
13582:
13568:
13554:
13544:
13531:
13519:
13516:
13513:
13507:
13504:
13501:
13495:
13492:
13489:
13486:
13480:
13477:
13474:
13465:
13462:
13459:
13448:
13431:
13425:
13422:
13419:
13410:
13382:
13376:
13373:
13370:
13367:
13362:
13358:
13354:
13351:
13343:
13340:
13337:
13331:
13325:
13316:
13310:
13307:
13304:
13301:
13296:
13292:
13288:
13285:
13277:
13274:
13271:
13265:
13256:
13251:
13248:
13245:
13241:
13237:
13234:
13223:
13220:
13217:
13211:
13200:
13186:
13163:
13160:
13157:
13145:
13129:
13125:
13121:
13118:
13098:
13078:
13058:
13038:
13018:
12998:
12995:
12992:
12970:
12967:
12964:
12960:
12956:
12951:
12947:
12926:
12903:
12874:
12871:
12868:
12837:
12833:
12829:
12826:
12823:
12818:
12814:
12810:
12807:
12804:
12801:
12775:
12772:
12769:
12747:
12743:
12739:
12736:
12716:
12696:
12693:
12690:
12670:
12648:
12644:
12621:
12617:
12613:
12610:
12607:
12602:
12598:
12551:
12548:
12543:
12539:
12499:
12495:
12491:
12488:
12485:
12480:
12476:
12472:
12469:
12445:
12441:
12437:
12434:
12431:
12428:
12422:
12419:
12397:
12394:
12391:
12387:
12383:
12380:
12360:
12340:
12318:
12314:
12310:
12307:
12293:
12279:
12271:
12253:
12249:
12228:
12208:
12185:
12162:
12140:
12137:
12134:
12130:
12126:
12120:
12107:
12104:
12084:
12064:
12042:
12038:
12034:
12028:
12015:
12012:
11990:
11986:
11965:
11945:
11925:
11905:
11896:
11880:
11877:
11874:
11870:
11849:
11827:
11823:
11800:
11796:
11767:
11763:
11759:
11756:
11747:
11744:
11741:
11738:
11735:
11730:
11726:
11722:
11719:
11713:
11702:
11698:
11694:
11691:
11682:
11679:
11676:
11673:
11670:
11665:
11661:
11657:
11654:
11648:
11645:
11625:
11605:
11583:
11579:
11575:
11572:
11546:
11542:
11521:
11518:
11515:
11493:
11489:
11468:
11446:
11442:
11419:
11415:
11392:
11388:
11365:
11361:
11338:
11334:
11313:
11305:
11301:
11283:
11279:
11256:
11252:
11201:
11197:
11193:
11190:
11181:
11178:
11175:
11172:
11166:
11155:
11151:
11147:
11144:
11135:
11132:
11129:
11126:
11123:
11118:
11114:
11110:
11107:
11101:
11098:
11075:
11072:
11069:
11065:
11061:
11058:
11038:
11018:
10998:
10976:
10972:
10968:
10965:
10951:
10937:
10915:
10911:
10885:
10881:
10877:
10874:
10871:
10866:
10862:
10855:
10833:
10829:
10808:
10786:
10782:
10759:
10755:
10751:
10748:
10745:
10740:
10736:
10710:
10706:
10702:
10699:
10696:
10691:
10687:
10680:
10660:
10657:
10654:
10634:
10614:
10594:
10591:
10588:
10579:
10565:
10557:
10541:
10533:
10529:
10525:
10512:
10510:
10506:
10490:
10470:
10461:
10447:
10427:
10407:
10387:
10367:
10347:
10339:
10323:
10303:
10283:
10274:
10258:
10255:
10252:
10248:
10227:
10207:
10187:
10178:
10164:
10142:
10138:
10117:
10109:
10091:
10088:
10085:
10081:
10077:
10072:
10069:
10066:
10062:
10039:
10036:
10033:
10029:
10025:
10020:
10017:
10014:
10010:
9987:
9983:
9979:
9974:
9970:
9947:
9944:
9941:
9937:
9933:
9928:
9925:
9922:
9918:
9895:
9891:
9887:
9882:
9878:
9855:
9851:
9847:
9842:
9838:
9817:
9814:
9811:
9791:
9769:
9765:
9761:
9758:
9755:
9750:
9746:
9725:
9683:
9679:
9675:
9672:
9669:
9664:
9660:
9637:
9633:
9612:
9585:Suppose that
9583:
9566:
9540:
9516:
9496:
9435:
9395:
9367:
9351:
9331:
9309:
9305:
9279:
9275:
9257:
9252:
9249:
9246:
9242:
9232:
9218:
9215:
9212:
9192:
9189:
9186:
9165:
9145:
9124:
9104:
9082:
9079:
9076:
9072:
9051:
9029:
9026:
9023:
9019:
8998:
8978:
8975:
8972:
8952:
8949:
8946:
8926:
8906:
8886:
8877:
8863:
8855:
8851:
8838:
8816:
8802:
8788:
8781:. This gives
8768:
8748:
8740:
8736:
8720:
8694:
8673:
8653:
8645:
8644:indiscernible
8641:
8625:
8599:
8589:
8575:
8567:
8545:
8518:
8497:
8477:
8451:
8441:
8427:
8419:
8415:
8411:
8407:
8403:
8387:
8379:
8363:
8355:
8339:
8331:
8315:
8307:
8291:
8271:
8239:
8214:
8211:
8208:
8205:
8180:
8166:
8163:
8160:
8138:
8134:
8113:
8093:
8090:
8087:
8065:
8061:
8053:are the real
8038:
8034:
8013:
7991:
7987:
7966:
7946:
7926:
7906:
7886:
7866:
7857:
7853:
7851:
7847:
7846:
7845:minimal model
7827:
7823:
7802:
7782:
7760:
7756:
7735:
7715:
7707:
7691:
7671:
7664:
7659:
7645:
7625:
7605:
7585:
7565:
7545:
7536:
7520:
7517:
7514:
7510:
7484:
7480:
7445:
7425:
7405:
7383:
7379:
7358:
7338:
7318:
7298:
7278:
7258:
7244:
7230:
7222:
7206:
7186:
7163:
7143:
7123:
7103:
7091:
7090:
7074:
7054:
7034:
7014:
7006:
7003:
7002:
6986:
6964:
6961:
6958:
6954:
6950:
6944:
6895:
6877:
6873:
6869:
6866:
6863:
6860:
6835:
6831:
6810:
6790:
6782:
6767:
6747:
6727:
6707:
6687:
6679:
6678:
6662:
6642:
6622:
6602:
6594:
6591:
6590:
6575:
6555:
6533:
6530:
6527:
6523:
6519:
6514:
6510:
6489:
6469:
6449:
6429:
6403:
6400:
6397:
6391:
6357:
6354:
6307:
6304:
6278:
6244:
6241:
6238:
6232:
6198:
6195:
6192:
6137:
6134:
6131:
6128:
6125:
6102:
6082:
6062:
6042:
6022:
6002:
5982:
5962:
5942:
5922:
5902:
5882:
5859:
5856:
5853:
5847:
5839:
5838:
5816:
5813:
5810:
5804:
5770:
5767:
5764:
5721:
5718:
5695:
5692:
5689:
5666:
5663:
5660:
5654:
5631:
5628:
5625:
5619:
5596:
5593:
5590:
5584:
5564:
5556:
5553:
5552:
5536:
5514:
5511:
5508:
5504:
5500:
5492:
5488:
5451:
5447:
5443:
5440:
5437:
5432:
5428:
5424:
5421:
5415:
5399:
5396:
5393:
5375:
5371:
5367:
5364:
5361:
5358:
5332:
5296:
5292:
5288:
5285:
5282:
5277:
5273:
5269:
5266:
5260:
5244:
5241:
5238:
5235:
5232:
5221:
5205:
5185:
5163:
5159:
5138:
5116:
5112:
5108:
5105:
5102:
5097:
5093:
5072:
5050:
5046:
5025:
5005:
4997:
4996:
4972:
4968:
4964:
4961:
4958:
4953:
4949:
4945:
4942:
4936:
4920:
4917:
4914:
4911:
4908:
4880:
4876:
4872:
4869:
4866:
4861:
4857:
4853:
4850:
4844:
4824:
4816:
4813:
4812:
4796:
4793:
4790:
4768:
4765:
4762:
4758:
4754:
4751:
4729:
4726:
4723:
4719:
4698:
4690:
4687:
4686:
4667:
4661:
4658:
4638:
4618:
4611:and whenever
4598:
4578:
4558:
4550:
4547:
4546:
4530:
4527:
4524:
4502:
4499:
4496:
4492:
4488:
4482:
4479:
4476:
4442:
4439:
4436:
4379:
4375:
4371:
4368:
4365:
4362:
4356:
4353:
4331:
4327:
4306:
4284:
4280:
4257:
4253:
4230:
4226:
4222:
4219:
4211:
4210:
4194:
4174:
4154:
4146:
4143:
4142:
4126:
4106:
4086:
4083:
4077:
4074:
4071:
4046:
4043:
4040:
4036:
4032:
4023:
4020:
4017:
4004:
4001:
3998:
3977:
3973:
3969:
3966:
3963:
3960:
3954:
3948:
3945:
3942:
3917:
3913:
3909:
3906:
3884:
3880:
3876:
3873:
3853:
3833:
3830:
3827:
3807:
3804:
3801:
3793:
3792:
3773:
3770:
3767:
3744:
3724:
3716:
3713:
3712:
3696:
3676:
3673:
3645:
3641:
3617:
3614:
3611:
3608:
3603:
3599:
3595:
3592:
3589:
3586:
3580:
3575:
3571:
3567:
3554:
3553:
3549:
3546:
3545:
3529:
3509:
3489:
3469:
3449:
3429:
3409:
3389:
3381:
3380:
3376:
3373:
3372:
3356:
3336:
3316:
3296:
3293:
3290:
3270:
3250:
3247:
3244:
3241:
3238:
3218:
3195:
3192:
3189:
3163:
3160:
3157:
3147:
3146:
3130:
3110:
3090:
3070:
3062:
3059:
3058:
3057:
3055:
3051:
3035:
3015:
2995:
2975:
2955:
2935:
2927:
2923:
2907:
2884:
2881:
2878:
2862:
2834:
2830:
2825:
2804:
2796:
2768:
2764:
2729:
2725:
2720:
2699:
2679:
2671:
2666:
2650:
2647:
2644:
2640:
2602:
2599:
2596:
2592:
2571:
2549:
2546:
2543:
2539:
2518:
2496:
2492:
2469:
2466:
2463:
2459:
2451:belonging to
2438:
2416:
2413:
2410:
2406:
2383:
2380:
2377:
2373:
2352:
2332:
2324:
2319:
2317:
2296:
2292:
2285:
2280:
2276:
2253:
2249:
2226:
2223:
2220:
2216:
2206:
2192:
2184:
2180:
2162:
2137:
2114:
2095:
2093:
2077:
2055:
2051:
2042:
2026:
2017:
2003:
1981:
1977:
1968:
1953:
1949:
1927:
1924:
1921:
1901:
1879:
1875:
1871:
1868:
1846:
1842:
1819:
1815:
1794:
1791:
1788:
1768:
1765:
1762:
1740:
1736:
1713:
1710:
1707:
1703:
1680:
1677:
1674:
1670:
1647:
1644:
1641:
1637:
1616:
1596:
1588:
1584:
1566:
1562:
1539:
1535:
1514:
1494:
1472:
1468:
1445:
1441:
1420:
1391:
1387:
1380:
1377:
1372:
1369:
1366:
1362:
1358:
1353:
1349:
1328:
1319:
1303:
1299:
1281:
1267:
1247:
1227:
1224:
1221:
1213:
1197:
1177:
1168:
1166:
1150:
1142:
1124:
1120:
1111:
1093:
1090:
1087:
1083:
1060:
1056:
1033:
1030:
1027:
1023:
1019:
1011:
1007:
993:
987:
984:
981:
965:
961:
957:
954:
948:
945:
923:
919:
898:
886:
882:
866:
861:
857:
840:
837:
833:
829:
826:
819:
804:
797:
782:
762:
759:
756:
736:
731:
727:
721:
718:
715:
711:
707:
702:
698:
689:
688:limit ordinal
673:
665:
651:
643:
639:
632:
629:
626:
621:
618:
615:
611:
603:
589:
583:
578:
574:
566:
565:
564:
562:
546:
538:
525:
515:
512:
507:
503:
499:
496:
493:
488:
484:
451:
447:
443:
440:
437:
432:
428:
424:
421:
412:
406:
403:
400:
389:
386:
383:
380:
377:
364:
358:
352:
349:
341:
338:
331:
327:
324:
320:
317:
313:
309:
308:
307:
305:
289:
267:
263:
254:
236:
233:
230:
226:
217:
213:
197:
189:
173:
160:
158:
154:
150:
146:
142:
138:
134:
116:
112:
104:
88:
80:
76:
60:
52:
48:
44:
40:
33:
19:
17600:
17398:Ultraproduct
17245:Model theory
17210:Independence
17146:Formal proof
17138:Proof theory
17121:
17094:
17051:real numbers
17023:second-order
16934:Substitution
16811:Metalanguage
16752:conservative
16725:Axiom schema
16669:Constructive
16639:Morse–Kelley
16605:Set theories
16584:Aleph number
16577:inaccessible
16483:Grothendieck
16477:
16367:intersection
16254:Higher-order
16242:Second-order
16188:Truth tables
16145:Venn diagram
15928:Formal proof
15800:Georg Cantor
15795:Paul Bernays
15726:Morse–Kelley
15701:
15634:
15633:Subset
15580:hereditarily
15542:Venn diagram
15500:ordered pair
15479:
15415:Intersection
15359:Axiom schema
15183:
15180:Jech, Thomas
15152:
15098:
15094:
15071:
15052:
15028:
15023:Barwise, Jon
15006:
15001:
14992:
14979:
14971:
14966:
14953:
14944:
14813:The sets in
14812:
14692:
14655:replaced by
14262:
14224:
14108:
13760:as follows:
13727:
13655:
13574:
13545:
13449:
13411:
13201:
13146:
12299:
11897:
10957:
10580:
10521:
10462:
10275:
10179:
9584:
9233:
8878:
8808:
8590:
8470:is false in
8442:
8376:because the
8186:
7858:
7854:
7843:
7662:
7660:
7537:
7250:
7220:
7178:
6780:
6422:is a set in
3522:and thus in
2926:well-founded
2868:
2667:
2632:so it is in
2323:arithmetical
2320:
2207:
2096:
2092:equinumerous
2018:
1412:
1290:
1169:
1165:proper class
1163:itself is a
890:
883:denotes the
880:
563:as follows:
539:
342:
339:
335:
303:
252:
166:
102:
50:
46:
36:
32:Gödel metric
17508:Type theory
17456:undecidable
17388:Truth value
17275:equivalence
16954:non-logical
16567:Enumeration
16557:Isomorphism
16504:cardinality
16488:Von Neumann
16453:Ultrafilter
16418:Uncountable
16352:equivalence
16269:Quantifiers
16259:Fixed-point
16228:First-order
16108:Consistency
16093:Proposition
16070:Traditional
16041:Lindström's
16031:Compactness
15973:Type theory
15918:Cardinality
15825:Thomas Jech
15668:Alternative
15647:Uncountable
15601:Ultrafilter
15460:Cardinality
15364:replacement
15305:Determinacy
14948:Gödel 1938.
12201:. Assuming
11534:is true in
11461:; and this
11326:containing
11271:drawn from
10338:truth value
9804:. Then let
9488:, then let
9137:, then let
8126:and in any
8106:is true in
7351:defined in
7311:defined in
7291:, then the
5198:is true in
5151:is true in
2672:subsets of
2325:subsets of
2150:defined by
1433:, the sets
330:quantifiers
302:, one uses
137:inner model
39:mathematics
17623:Categories
17319:elementary
17012:arithmetic
16880:Quantifier
16858:functional
16730:Expression
16448:Transitive
16392:identities
16377:complement
16310:hereditary
16293:Set theory
15820:Kurt Gödel
15805:Paul Cohen
15642:Transitive
15410:Identities
15394:Complement
15381:Operations
15342:Regularity
15310:projective
15273:Adjunction
15232:Set theory
15184:Set Theory
15016:References
14741:is a set,
14627:only with
14263:The class
13656:The class
12729:such that
10991:, and let
10440:is not in
9718:to define
9652:. Suppose
9234:The stage
7704:that is a
5038:such that
4571:such that
3866:such that
3462:, then by
3103:such that
2988:. However
2712:belong to
2365:belong to
323:parameters
153:consistent
133:Kurt Gödel
43:set theory
17590:Supertask
17493:Recursion
17451:decidable
17285:saturated
17263:of models
17186:deductive
17181:axiomatic
17101:Hilbert's
17088:Euclidean
17069:canonical
16992:axiomatic
16924:Signature
16853:Predicate
16742:Extension
16664:Ackermann
16589:Operation
16468:Universal
16458:Recursive
16433:Singleton
16428:Inhabited
16413:Countable
16403:Types of
16387:power set
16357:partition
16274:Predicate
16220:Predicate
16135:Syllogism
16125:Soundness
16098:Inference
16088:Tautology
15990:paradoxes
15753:Paradoxes
15673:Axiomatic
15652:Universal
15628:Singleton
15623:Recursive
15566:Countable
15561:Amorphous
15420:Power set
15337:Power set
15288:dependent
15283:countable
14563:∈
14482:∈
14450:∈
14421:Φ
14082:α
14072:α
14068:⋃
14019:α
14009:λ
14003:α
13999:⋃
13981:λ
13956:λ
13920:α
13864:α
13517:∈
13505:∈
13499:∀
13496:∧
13490:∈
13478:∈
13472:∀
13463:∈
13457:∀
13368:∧
13363:ω
13355:∈
13348:⟺
13341:∈
13329:∀
13326:∨
13302:∧
13289:∈
13282:⟺
13275:∈
13263:∀
13257:∧
13246:ω
13238:∈
13228:⟺
13221:∈
13209:∀
13164:ω
13130:α
13099:α
12999:α
12993:β
12985:for some
12965:β
12957:∈
12904:…
12898:Φ
12875:…
12863:Ψ
12827:…
12796:Ψ
12773:∈
12748:α
12694:∈
12649:α
12611:…
12590:and some
12578:Φ
12552:∈
12544:α
12489:…
12464:Φ
12446:α
12438:∈
12432:∣
12392:α
12384:∈
12361:α
12319:α
12250:κ
12229:κ
12186:δ
12135:δ
12127:∈
12108:∩
12043:δ
12035:⊆
12016:∩
11987:κ
11966:δ
11926:κ
11875:γ
11828:γ
11801:β
11751:Φ
11748:∧
11742:∈
11731:γ
11723:∈
11686:Φ
11683:∧
11677:∈
11666:β
11658:∈
11626:α
11606:γ
11598:for some
11584:γ
11547:β
11494:α
11407:with the
11393:β
11353:and some
11339:α
11284:β
11244:and some
11232:Φ
11185:Φ
11176:∈
11139:Φ
11136:∧
11130:∈
11119:β
11111:∈
11070:β
11062:∈
11039:β
10977:α
10969:∈
10916:β
10903:holds in
10875:…
10834:β
10787:β
10749:…
10700:…
10661:α
10655:β
10635:α
10595:ω
10253:α
10228:α
10165:α
10143:α
10089:−
10070:−
10037:−
10018:−
9945:−
9926:−
9759:…
9706:Φ
9673:…
9638:α
9593:Φ
9570:Φ
9564:Ψ
9544:Ψ
9538:Φ
9476:Φ
9456:Ψ
9416:Ψ
9376:Φ
9310:α
9280:α
9247:α
9219:α
9213:β
9193:β
9187:α
9125:α
9105:β
9077:β
9024:α
8699:♯
8618:holds in
8604:♯
8550:♯
8523:♯
8456:♯
8443:However,
8416:(see the
8400:. Weakly
8380:holds in
8236:Π
8212:⊆
8206:⊂
8139:κ
8066:κ
8039:κ
7992:κ
7828:κ
7761:κ
7716:κ
7515:α
7485:α
7446:α
7384:α
6959:α
6951:∈
6878:α
6870:∈
6864:∣
6836:α
6528:α
6520:∈
6515:α
6490:α
6355:∈
6308:∣
6196:∈
6135:∈
6129:∣
5835:is a set.
5768:∈
5722:∣
5509:α
5501:∈
5493:α
5441:…
5397:∈
5376:α
5368:∈
5362:∣
5286:…
5242:∈
5236:∣
5164:α
5106:…
5065:contains
5051:α
5026:α
4993:is a set.
4962:…
4918:∈
4912:∣
4870:…
4794:∈
4791:ω
4783:and thus
4763:ω
4755:∈
4752:ω
4724:α
4699:α
4662:∪
4579:∅
4528:∈
4497:α
4489:∈
4480:∈
4440:∈
4380:α
4372:∈
4366:∣
4332:α
4285:α
4258:α
4231:α
4223:∈
4084:∈
4041:α
4033:∈
3978:α
3970:∈
3964:∣
3918:α
3910:∈
3885:α
3877:∈
3854:α
3831:∈
3805:∈
3789:is a set.
3674:∈
3609:∧
3596:∈
3590:∣
3294:∈
3248:∈
3242:∈
3196:∈
3164:∈
2885:∈
2831:ω
2805:ω
2765:ω
2726:ω
2700:ω
2680:ω
2645:ω
2597:ω
2572:ω
2544:ω
2519:∈
2497:ω
2464:ω
2439:ω
2411:ω
2378:ω
2353:ω
2333:ω
2297:α
2286:∪
2281:α
2254:α
2221:α
2159:Δ
2078:α
2056:α
2041:bijection
2027:α
2004:α
1982:α
1954:α
1902:α
1880:α
1876:ω
1869:α
1847:α
1820:α
1769:ω
1763:α
1741:α
1708:α
1675:ω
1642:ω
1589:in which
1567:ω
1540:ω
1392:β
1381:
1373:α
1367:β
1363:⋃
1354:α
1329:α
1304:α
1125:α
1110:power set
1088:α
1061:α
1028:α
1012:α
994:∈
985:∈
966:α
958:∈
924:α
862:α
841:∈
838:α
834:⋃
805:λ
783:α
763:λ
757:α
732:α
722:λ
716:α
712:⋃
703:λ
674:λ
644:α
633:
616:α
587:∅
513:∈
497:…
476:Φ
441:…
416:Φ
413:⊨
407:∈
387:∈
381:∣
353:
328:with the
268:α
231:α
216:successor
163:What L is
143:with the
117:α
17575:Logicism
17568:timeline
17544:Concrete
17403:Validity
17373:T-schema
17366:Kripke's
17361:Tarski's
17356:semantic
17346:Strength
17295:submodel
17290:spectrum
17258:function
17106:Tarski's
17095:Elements
17082:geometry
17038:Robinson
16959:variable
16944:function
16917:spectrum
16907:Sentence
16863:variable
16806:Language
16759:Relation
16720:Automata
16710:Alphabet
16694:language
16548:-jection
16526:codomain
16512:Function
16473:Universe
16443:Infinite
16347:Relation
16130:Validity
16120:Argument
16018:theorem,
15757:Problems
15661:Theories
15637:Superset
15613:Infinite
15442:Concepts
15322:Infinity
15239:Overview
15182:(2002).
15143:16577857
15051:(1984).
15025:(1975).
14895:See also
14891:itself.
14312:, where
12855:] where
11788:because
11508:. Since
10534:hold in
8879:Suppose
8304:. Hence
8006:and the
7136:to form
5682:implies
2043:between
1940:implies
1755:for all
1214:", aka "
938:, then
796:precedes
212:ordinals
17517:Related
17314:Diagram
17212: (
17191:Hilbert
17176:Systems
17171:Theorem
17049:of the
16994:systems
16774:Formula
16769:Grammar
16685: (
16629:General
16342:Forcing
16327:Element
16247:Monadic
16022:paradox
15963:Theorem
15899:General
15695:General
15690:Zermelo
15596:subbase
15578: (
15517:Forcing
15495:Element
15467: (
15445:Methods
15332:Pairing
15172:0002514
15134:1077160
15103:Bibcode
12412:, then
12077:within
12005:, then
8352:. Weak
7899:" and "
7775:is the
7748:, then
5222:"). So
4517:. Thus
4346:. Then
4119:as for
4061:. Thus
3932:. Then
3422:are in
2756:(where
1834:equals
1609:equals
1507:equals
690:, then
314:in the
312:formula
17280:finite
17043:Skolem
16996:
16971:Theory
16939:Symbol
16929:String
16912:atomic
16789:ground
16784:closed
16779:atomic
16735:ground
16698:syntax
16594:binary
16521:domain
16438:Finite
16203:finite
16061:Logics
16020:
15968:Theory
15586:Filter
15576:Finite
15512:Family
15455:Almost
15293:global
15278:Choice
15265:Axioms
15190:
15170:
15160:
15141:
15131:
15123:
15078:
15059:
15037:
14503:where
13412:where
11638:. And
10530:, and
9962:) or (
9738:, and
9448:, and
9408:, and
8187:Since
5612:where
4711:is in
4631:is in
4591:is in
3329:as in
3054:axioms
1143:. But
979:
971:
775:means
473:
463:
157:axioms
45:, the
17270:Model
17018:Peano
16875:Proof
16715:Arity
16644:Naive
16531:image
16463:Fuzzy
16423:Empty
16372:union
16317:Class
15958:Model
15948:Lemma
15906:Axiom
15678:Naive
15608:Fuzzy
15571:Empty
15554:types
15505:tuple
15475:Class
15469:large
15430:Union
15347:Union
15125:87239
15121:JSTOR
14937:Notes
11565:, so
11090:, so
11051:with
10728:with
10380:, or
9605:uses
9509:<
9158:<
8991:. If
8761:into
6115:. So
5131:and (
4299:. So
3717:: If
3660:. So
2920:is a
1048:. So
885:class
879:Here
749:Here
686:is a
321:with
75:class
41:, in
17393:Type
17196:list
17000:list
16977:list
16966:Term
16900:rank
16794:open
16688:list
16500:Maps
16405:sets
16264:Free
16234:list
15984:list
15911:list
15591:base
15188:ISBN
15158:ISBN
15139:PMID
15076:ISBN
15057:ISBN
15035:ISBN
14926:L(R)
14006:<
13624:and
12996:<
12788:and
12762:and
12353:and
11815:and
11302:and
10958:Let
10658:>
10592:<
10558:for
10316:and
10078:<
10054:and
10002:and
9934:<
9910:and
9870:or (
9848:<
9815:<
9541:<
9344:and
9190:<
9097:and
9044:and
8976:>
8950:<
8899:and
7418:and
5840:Let
5647:and
5085:and
3899:and
3820:and
3402:and
3123:and
2668:All
2321:All
2070:and
1766:>
1460:and
1370:<
1318:is:
760:<
719:<
304:only
79:sets
49:(or
17080:of
17062:of
17010:of
16542:Sur
16516:Map
16323:Ur-
16305:Set
15552:Set
15129:PMC
15111:doi
14842:or
14552:is
14109:If
13948:If
13823:of
12919:to
12636:in
10774:in
10420:or
10130:to
8965:or
8741:of
8713:in
8646:in
8591:If
8284:to
8026:of
7979:of
7795:of
7684:in
7663:set
7251:If
7221:not
6783:in
6095:to
5895:to
5708:),
4212:If
3794:If
3382:If
3050:ZFC
2665:).
2107:Def
2019:If
1861:if
1587:ZFC
1378:Def
1112:of
999:Def
975:and
891:If
881:Ord
666:If
630:Def
350:Def
253:all
77:of
37:In
17625::
17466:NP
17090::
17084::
17014::
16691:),
16546:Bi
16538:In
15168:MR
15166:.
15137:.
15127:.
15119:.
15109:.
15099:24
15097:.
15093:.
14690:.
14260:.
14222:.
13893:=
13653:.
12683:,
12292:.
11895:.
10950:.
10526:,
10460:.
10360:,
10273:.
10177:.
9582:.
9231:.
8588:.
8440:.
8328:.
7618:,
7535:.
7243:.
6781:is
6294:=
6035:,
5955:.
5348:=
4898:,
3737:,
3542:).
3283:,
3056::
3028:.
2928:.
2016:.
1969:=
1629:,
1554:=
1341:,
1280:.
1167:.
830::=
708::=
627::=
584::=
365::=
190:,
17546:/
17461:P
17216:)
17002:)
16998:(
16895:∀
16890:!
16885:∃
16846:=
16841:↔
16836:→
16831:∧
16826:∨
16821:¬
16544:/
16540:/
16514:/
16325:)
16321:(
16208:∞
16198:3
15986:)
15884:e
15877:t
15870:v
15635:·
15619:)
15615:(
15582:)
15471:)
15224:e
15217:t
15210:v
15196:.
15174:.
15145:.
15113::
15105::
15084:.
15065:.
15043:.
14879:L
14859:]
14856:A
14853:[
14850:L
14830:)
14827:A
14824:(
14821:L
14798:A
14778:]
14775:A
14772:[
14769:L
14749:A
14729:A
14709:]
14706:A
14703:[
14700:L
14676:A
14671:f
14668:e
14665:D
14642:f
14639:e
14636:D
14615:L
14595:]
14592:A
14589:[
14586:L
14566:A
14560:y
14540:)
14537:y
14534:(
14531:A
14511:A
14491:)
14488:A
14485:,
14479:,
14476:X
14473:(
14453:)
14447:,
14444:X
14441:(
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4220:x
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4069:{
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3961:s
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3668:{
3646:1
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3584:{
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2112:(
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1950:H
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1473:n
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198:V
174:L
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34:.
20:)
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