36:
4982:
160:
4990:
5438:
2352:
5020:
Zermelo and
Fraenkel had corresponded heavily in 1921; the axiom of replacement was a major topic of this exchange. Fraenkel initiated correspondence with Zermelo sometime in March 1921. However, his letters before the one dated 6 May 1921 are lost. Zermelo first admitted to a gap in his system in a
4322:
Separation is still important, however, for use in fragments of ZFC, because of historical considerations, and for comparison with alternative axiomatizations of set theory. A formulation of set theory that does not include the axiom of replacement will likely include some form of the axiom of
4318:
This result shows that it is possible to axiomatize ZFC with a single infinite axiom schema. Because at least one such infinite schema is required (ZFC is not finitely axiomatizable), this shows that the axiom schema of replacement can stand as the only infinite axiom schema in ZFC if desired.
2165:
shows that each of these theories contains a sentence, "expressing" the theory's own consistency, that is unprovable in that theory, if that theory is consistent - this result is often loosely expressed as the claim that neither of these theories can prove its own consistency, if it is
5017:. Although it is Skolem's first order version of the axiom list that we use today, he usually gets no credit since each individual axiom was developed earlier by either Zermelo or Fraenkel. The phrase “Zermelo-Fraenkel set theory” was first used in print by von Neumann in 1928.
1289:
2452:
is closely related to and frequently confused with the axiom schema of replacement. Over the remainder of the ZF axioms, it is equivalent to the axiom schema of replacement. The axiom of collection is stronger than replacement in the absence of the
1946:, which can itself be shown to be uncountable. The construction uses replacement twice; once to ensure an ordinal assignment for each well ordered set and again to replace well ordered sets by their ordinals. This is a special case of the result of
1523:
2184:
showed that a version of replacement with parameters removed, i.e. the following schema, is equivalent to the original form. In particular the equivalence holds in the presence of the axioms of extensionality, pairing, union and powerset.
5033:
turns into a set again" (parenthetical completion and translation by
Ebbinghaus). Fraenkel's 1922 publication thanked Zermelo for helpful arguments. Prior to this publication, Fraenkel publicly announced his new axiom at a meeting of the
4711:
5322:. Maddy cites two papers by Mirimanoff, "Les antinomies de Russell et de Burali-Forti et le problème fundamental de la théorie des ensembles" and "Remarques sur la théorie des ensembles et les antinomies Cantorienne", both in
3290:
3910:
3562:
2341:
4323:
separation, to ensure that its models contain a sufficiently rich collection of sets. In the study of models of set theory, it is sometimes useful to consider models of ZFC without replacement, such as the models
5107:
Zermelo himself never accepted Skolem's formulation of the axiom schema of replacement. At one point he called Skolem's approach “set theory of the impoverished”. Zermelo envisaged a system that would allow for
5042:
on 22 September 1921. Zermelo was present at this meeting; in the discussion following
Fraenkel's talk he accepted the axiom of replacement in general terms, but expressed reservations regarding its extent.
1717:
in finite types, which in turn are sufficient to formalize the bulk of mathematics. Although the axiom schema of replacement is a standard axiom in set theory today, it is often omitted from systems of
1026:
1825:
5021:
reply to
Fraenkel dated 9 May 1921. On 10 July 1921, Fraenkel completed and submitted for publication a paper (published in 1922) that described his axiom as allowing arbitrary replacements: "If
1362:
1031:
2087:
3019:
5013:
later in the same year (and published in 1923). Zermelo himself incorporated
Fraenkel's axiom in his revised system he published in 1930, which also included as a new axiom von Neumann's
4952:
4910:
4579:
978:
2737:
4293:
4199:
2018:
1729:
At any rate, the axiom schema drastically increases the strength of ZF, both in terms of the theorems it can prove - for example the sets shown to exist - and also in terms of its
4524:
762:
2159:
1940:
1905:
1357:
4348:
4428:
4072:
4043:
4001:
3972:
3806:
3717:
601:
430:
4868:
4845:
4762:
1867:
1315:
5895:
4599:
4399:
3625:
3585:
2556:
559:
533:
467:
361:
4822:
4782:
4735:
4448:
4246:
4139:
3930:
3353:
3082:
3062:
2797:
2679:
2530:
2510:
1753:
asserts the existence of an infinite set ω = {0, 1, 2, ...}. One may hope to define ω·2 as the union of the sequence {ω, ω + 1, ω + 2,...}. However, arbitrary such
1546:
1349:
1018:
914:
704:
668:
388:
4802:
4375:
4313:
4219:
4112:
4092:
3777:
3757:
3737:
3688:
3668:
3645:
3605:
3453:
3433:
3413:
3393:
3373:
3333:
3313:
3042:
2777:
2757:
2656:
2636:
2616:
2596:
2576:
2490:
2442:
2422:
2402:
2382:
2038:
1695:
1666:
1646:
1626:
1606:
1586:
1566:
998:
887:
858:
838:
809:
789:
641:
621:
507:
487:
326:
306:
278:
250:
230:
210:
190:
4604:
5046:
Thoralf Skolem made public his discovery of the gap in
Zermelo's system (the same gap that Fraenkel had found) in a talk he gave on 6 July 1922 at the 5th
2162:
3090:
2468:
While replacement can be read to say that the image of a function is a set, collection speaks about images of relations and then merely says that some
3811:
5104:." In the same year, Fraenkel wrote a review of Skolem's paper, in which Fraenkel simply stated that Skolem's considerations correspond to his own.
1773:
of two copies of ω, with the second copy greater than the first – but that this is not an ordinal since it is not totally ordered by inclusion.
2112:. A later, more careful analysis by Martin of the result showed that it only requires replacement for functions with domain an arbitrary countable
4319:
Because the axiom schema of separation is not independent, it is sometimes omitted from contemporary statements of the
Zermelo-Fraenkel axioms.
4455:
3481:
5054:; the proceedings of this congress were published in 1923. Skolem presented a resolution in terms of first-order definable replacements: "Let
5139:
2191:
1757:
of ordinals need not be sets - for example, the class of all ordinals is not a set. Replacement now allows one to replace each finite number
896:
Because it is impossible to quantify over definable functions in first-order logic, one instance of the schema is included for each formula
5584:
5404:
1953:
In light of the above, the existence of an assignment of an ordinal to every well-ordered set requires replacement as well. Similarly the
57:
2089:
has too high of a rank for its existence to be provable from set theory with just the axiom of power set, choice and without replacement.
5912:
147:
only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining
5209:
5187:
5165:
79:
1284:{\displaystyle {\begin{aligned}\forall w_{1},\ldots ,w_{n}\,\forall A\,(\ \Longrightarrow \ \exists B\,\forall y\,)\end{aligned}}}
5890:
5120:, Zermelo's disapproval of Skolem's approach marked the end of Zermelo's influence on the developments of set theory and logic.
109:
1954:
5047:
1787:
6045:
5664:
5543:
5907:
5900:
2043:
5501:
5035:
890:
50:
44:
3468:
2805:
1705:
The axiom schema of replacement is not necessary for the proofs of most theorems of ordinary mathematics. Indeed,
5589:
5481:
5469:
5464:
5285:
3739:
fulfills the relevant instance of the axiom schema of separation and one is done. Otherwise, choose such a fixed
2109:
1781:
1318:
5246:
Gitman, Victoria; Joel David
Hamkins; Johnstone, Thomas A. (2011). "What is the theory ZFC without power set?".
4915:
4873:
4529:
61:
5397:
919:
2684:
2180:
Some simplifications may be made to the axiom schema of replacement to obtain different equivalent versions.
6009:
5927:
5802:
5754:
5568:
5491:
5117:
4451:
4251:
4144:
2458:
1710:
4476:
is equivalent to the axiom of replacement, assuming the axiom of infinity. LĂ©vy's principle is as follows:
1971:
5961:
5842:
5654:
5474:
4354:
2462:
281:
143:
from that class to a second class, the axiom states that the second class is also a set. However, because
4483:
1765:, and thus guarantees that this class is a set. As a clarification, note that one can easily construct a
713:
5877:
5791:
5711:
5691:
5669:
5319:
Early hints of the Axiom of
Replacement can be found in Cantor's letter to Dedekind and in Mirimanoff
2137:
5951:
5941:
5775:
5706:
5659:
5599:
5486:
4473:
4467:
1518:{\displaystyle {\begin{aligned}\forall A\,(\ \Longrightarrow \ \exists B\,\forall y\,)\end{aligned}}}
707:
121:
5116:, which followed from Skolem's first-order axiomatization. According to the biography of Zermelo by
5946:
5857:
5770:
5765:
5760:
5574:
5516:
5454:
5390:
5113:
5014:
1828:
148:
113:
1910:
1875:
5869:
5864:
5649:
5604:
5511:
5302:
5247:
4326:
3472:
1832:
1754:
1706:
128:
4981:
4404:
4048:
4006:
3977:
3935:
3782:
3693:
564:
393:
4850:
4827:
4740:
1846:
1297:
5726:
5563:
5555:
5526:
5496:
5427:
5205:
5183:
5161:
5135:
2161:
is the first one which can be shown to exist in ZF but not in Z. For clarification, note that
1784:, can be constructed as follows – the set of countable well orders exists as a subset of
1750:
159:
136:
117:
4584:
4384:
3610:
3570:
6014:
6004:
5989:
5984:
5852:
5506:
5294:
4998:
2535:
2105:
1766:
1738:
538:
512:
437:
331:
5314:
4807:
4767:
4720:
4433:
4224:
4117:
3915:
3338:
3067:
3047:
2782:
2664:
2515:
2495:
1531:
1327:
1003:
899:
673:
646:
366:
5883:
5821:
5639:
5459:
5310:
5154:
4989:
2492:
has no minimality requirement, i.e. this variant also lacks the uniqueness requirement on
2454:
2132:
2093:
1962:
1958:
1942:). Replace each well-ordered set with its ordinal. This is the set of countable ordinals ω
1836:
1730:
5236:", pp.74--75. Bulletin of Symbolic Logic vol. 18, no. 1 (2012). Accessed 22 August 2023.
6019:
5816:
5797:
5701:
5686:
5643:
5579:
5521:
5280:
5149:
5109:
5010:
4787:
4706:{\displaystyle \phi (x_{1},\ldots ,x_{n})\iff \phi ^{V_{\alpha }}(x_{1},\ldots ,x_{n})}
4360:
4298:
4204:
4097:
4077:
3762:
3742:
3722:
3673:
3653:
3630:
3590:
3471:, the other axiom schema in ZFC, is implied by the axiom schema of replacement and the
3438:
3418:
3398:
3378:
3358:
3318:
3298:
3027:
2762:
2742:
2641:
2621:
2601:
2581:
2561:
2475:
2427:
2407:
2387:
2358:
2181:
2023:
1947:
1770:
1746:
1671:
1651:
1631:
1611:
1591:
1571:
1551:
983:
863:
843:
814:
794:
774:
626:
606:
492:
472:
432:
311:
291:
263:
235:
215:
195:
166:
840:
is also a set. This can be seen as a principle of smallness: the axiom states that if
6039:
6024:
5994:
5826:
5740:
5735:
5197:
4963:
4378:
2128:
1742:
124:
is also a set. It is necessary for the construction of certain infinite sets in ZF.
17:
5974:
5969:
5787:
5716:
5674:
5533:
5437:
5270:", p.73. Bulletin of Symbolic Logic vol. 18, no. 1 (2012). Accessed 22 August 2023.
4971:
3285:{\displaystyle \forall w_{1},\ldots ,w_{n}\,\forall A\,\exists B\,\forall x\in A\,}
285:
101:
4717:
This is a schema that consists of countably many statements, one for each formula
1769:
that is isomorphic to ω·2 without resorting to replacement – simply take the
139:
of its elements. Thus, if one class is "small enough" to be a set, and there is a
5999:
5634:
5175:
3905:{\displaystyle \phi (x,y):=(\theta (x)\land y=x)\lor (\neg \theta (x)\land y=a)}
2120:
2101:
1719:
1714:
132:
5979:
5847:
5750:
5413:
4975:
140:
93:
2096:
showed that at least some instances of replacement are required to show that
5782:
5745:
5696:
5594:
3024:
The axiom schema is sometimes stated without prior restrictions (apart from
2097:
1870:
2351:
5051:
3557:{\displaystyle \forall A\,\exists B\,\forall C\,(C\in B\Leftrightarrow )}
2469:
5267:
5233:
5306:
2336:{\displaystyle \forall A\,(\ \Longrightarrow \ \exists B\,\forall y\,)}
2113:
257:
2658:
in the original set, with no guarantee that it will contain only one.
5807:
5629:
3912:
for use with replacement. Using function notation for this predicate
2472:
of the relation's image is a set. In other words, the resulting set
1733:
consistency strength, compared to Z. Some important examples follow:
5298:
3355:. However, the axiom schema as stated requires that, if an element
5679:
5446:
5252:
4988:
4980:
2350:
1723:
158:
105:
5112:. He also objected strongly to the philosophical implications of
4450:
is functional. The axiom of separation is explicitly included in
1776:
Larger ordinals rely on replacement less directly. For example, ω
5039:
889:
is also small enough to be a set. It is implied by the stronger
5386:
3719:, or it does not. In the latter case, taking the empty set for
5180:
Set Theory: The Third
Millennium Edition, Revised and Expanded
144:
29:
5382:
1020:. In the formal language of set theory, the axiom schema is:
5009:). The axiom was independently discovered and announced by
4295:, is granted to be a set by the axiom of replacement. This
2618:
whose existence is asserted must contain at least one such
4974:'s unpublished works, and it appeared again informally in
127:
The axiom schema is motivated by the idea that whether a
5058:
be a definite proposition that holds for certain pairs (
4141:
indeed constitutes a class function. In turn, the image
916:
in the language of set theory with free variables among
3475:. Recall that the axiom schema of separation includes
1820:{\displaystyle P({\mathbb {N} }\times {\mathbb {N} })}
4918:
4876:
4853:
4830:
4810:
4790:
4770:
4743:
4723:
4607:
4587:
4532:
4486:
4436:
4407:
4387:
4363:
4329:
4301:
4254:
4227:
4207:
4147:
4120:
4100:
4080:
4051:
4009:
3980:
3938:
3918:
3814:
3785:
3765:
3745:
3725:
3696:
3676:
3656:
3633:
3613:
3593:
3573:
3484:
3441:
3421:
3401:
3381:
3361:
3341:
3321:
3301:
3093:
3070:
3050:
3030:
2808:
2785:
2765:
2745:
2687:
2667:
2644:
2624:
2604:
2584:
2564:
2538:
2518:
2498:
2478:
2430:
2410:
2390:
2361:
2194:
2140:
2046:
2026:
1974:
1913:
1878:
1849:
1790:
1674:
1654:
1634:
1614:
1594:
1574:
1554:
1534:
1360:
1330:
1300:
1029:
1006:
986:
922:
902:
866:
846:
817:
797:
777:
716:
676:
649:
629:
609:
567:
541:
515:
495:
475:
440:
396:
369:
334:
314:
294:
266:
238:
218:
198:
169:
2461:
but weaker in the framework of IZF, which lacks the
5960:
5923:
5835:
5725:
5613:
5554:
5445:
5420:
1745:greater than ω requires the replacement axiom. The
363:holds. There is a corresponding definable function
5202:Set Theory: An Introduction to Independence Proofs
5153:
4946:
4904:
4862:
4839:
4816:
4796:
4776:
4756:
4729:
4705:
4593:
4573:
4518:
4442:
4422:
4393:
4369:
4342:
4307:
4287:
4240:
4213:
4193:
4133:
4106:
4086:
4066:
4037:
3995:
3966:
3924:
3904:
3800:
3771:
3751:
3731:
3711:
3682:
3662:
3639:
3619:
3599:
3579:
3556:
3447:
3427:
3407:
3387:
3367:
3347:
3327:
3307:
3284:
3076:
3056:
3036:
3013:
2791:
2771:
2751:
2731:
2673:
2650:
2630:
2610:
2590:
2570:
2550:
2524:
2504:
2484:
2436:
2416:
2396:
2376:
2335:
2153:
2081:
2032:
2012:
1934:
1899:
1861:
1819:
1689:
1660:
1640:
1620:
1600:
1580:
1560:
1540:
1517:
1343:
1309:
1283:
1012:
992:
972:
908:
881:
852:
832:
803:
783:
756:
698:
662:
635:
615:
595:
553:
527:
501:
481:
461:
424:
382:
355:
320:
300:
272:
244:
224:
204:
184:
2082:{\displaystyle \{A^{n}\mid n\in {\mathbb {N} }\}}
5001:in 1922 is what makes modern set theory Zermelo-
4970:). Some informal approximation to it existed in
4962:The axiom schema of replacement was not part of
4074:is false. By case analysis, the possible values
3455:. The resulting axiom schema is also called the
5363:
5361:
5132:Ernst Zermelo: An Approach to His Life and Work
1950:, and the general case can be proved similarly.
2131:of Z whose existence can be proved in ZF. The
5398:
5351:
5349:
5347:
5345:
5343:
5341:
4315:precisely validates the axiom of separation.
3335:that are not associated to any other sets by
3014:{\displaystyle \forall w_{1},\ldots ,w_{n}\,}
1961:to each set requires replacement, as well as
1648:reached this way can be collected into a set
791:is a definable class function, as above, and
8:
4282:
4261:
4188:
4154:
2076:
2047:
751:
717:
1749:ω·2 = ω + ω is the first such ordinal. The
5405:
5391:
5383:
4947:{\displaystyle \forall (x\in V_{\alpha })}
4905:{\displaystyle \exists (x\in V_{\alpha })}
4650:
4646:
4574:{\displaystyle \phi (x_{1},\ldots ,x_{n})}
1968:For sets of tuples recursively defined as
5251:
5134:, Springer Science & Business Media,
4935:
4917:
4893:
4875:
4852:
4829:
4809:
4789:
4769:
4748:
4742:
4722:
4694:
4675:
4660:
4655:
4637:
4618:
4606:
4586:
4562:
4543:
4531:
4510:
4491:
4485:
4435:
4406:
4386:
4362:
4334:
4328:
4300:
4253:
4232:
4226:
4206:
4161:
4146:
4125:
4119:
4099:
4079:
4050:
4014:
4008:
3979:
3943:
3937:
3917:
3813:
3784:
3764:
3744:
3724:
3695:
3675:
3655:
3632:
3612:
3592:
3572:
3505:
3498:
3491:
3483:
3440:
3420:
3400:
3380:
3360:
3340:
3320:
3300:
3270:
3251:
3228:
3204:
3185:
3153:
3140:
3133:
3126:
3120:
3101:
3092:
3069:
3049:
3029:
2999:
2980:
2957:
2944:
2931:
2924:
2903:
2884:
2858:
2854:
2841:
2835:
2816:
2807:
2784:
2764:
2744:
2711:
2692:
2686:
2666:
2643:
2623:
2603:
2583:
2563:
2537:
2517:
2497:
2477:
2429:
2409:
2389:
2360:
2302:
2274:
2267:
2224:
2214:
2201:
2193:
2145:
2139:
2071:
2070:
2069:
2054:
2045:
2025:
1992:
1979:
1973:
1912:
1907:. A set of relations is thus a subset of
1877:
1848:
1809:
1808:
1807:
1799:
1798:
1797:
1789:
1673:
1653:
1633:
1613:
1593:
1573:
1553:
1533:
1480:
1452:
1445:
1402:
1392:
1371:
1361:
1359:
1335:
1329:
1324:For clarity, in the case of no variables
1299:
1256:
1237:
1214:
1186:
1179:
1146:
1127:
1104:
1094:
1073:
1066:
1060:
1041:
1030:
1028:
1005:
985:
973:{\displaystyle w_{1},\dotsc ,w_{n},A,x,y}
946:
927:
921:
901:
865:
845:
816:
796:
776:
724:
715:
681:
675:
654:
648:
628:
608:
572:
566:
540:
514:
494:
474:
439:
401:
395:
374:
368:
333:
313:
293:
265:
237:
217:
197:
168:
80:Learn how and when to remove this message
43:This article includes a list of general
5225:
5048:Congress of Scandinavian Mathematicians
4985:Abraham Fraenkel, between 1939 and 1949
3587:in the language of set theory in which
2732:{\displaystyle w_{1},\dotsc ,w_{n},x,y}
469:. Consider the (possibly proper) class
163:Axiom schema of replacement: the image
4966:'s 1908 axiomatisation of set theory (
4288:{\displaystyle A\cap \{x:\theta (x)\}}
4194:{\displaystyle B:=\{F_{a}(x):x\in A\}}
2532:is not required to be a function—some
2355:Axiom schema of collection: the image
4003:is true and as the constant function
2163:Gödel's second incompleteness theorem
2013:{\displaystyle A^{n}=A^{n-1}\times A}
7:
3395:is associated with at least one set
3295:In this case, there may be elements
2512:. That is, the relation defined by
5283:(1988), "Believing the axioms. I",
4519:{\displaystyle x_{1},\ldots ,x_{n}}
4430:when stipulating that the relation
2661:Suppose that the free variables of
2404:under the definable class function
1737:Using the modern definition due to
1588:correspondence, akin to a function
757:{\displaystyle \{F_{P}(x):x\in A\}}
212:under the definable class function
5097:ranges over all elements of a set
5086:ranges over the elements of a set
4919:
4877:
4854:
4831:
4353:The proof given above assumes the
3872:
3499:
3492:
3485:
3216:
3157:
3141:
3134:
3127:
3094:
2945:
2932:
2925:
2918:
2855:
2848:
2809:
2290:
2268:
2261:
2215:
2208:
2195:
2142:
1468:
1446:
1439:
1393:
1378:
1365:
1301:
1202:
1180:
1173:
1095:
1080:
1067:
1034:
860:is small enough to be a set, then
49:it lacks sufficient corresponding
25:
5130:Ebbinghaus, Heinz-Dieter (2007),
5070:; assume further, that for every
2154:{\displaystyle \aleph _{\omega }}
5436:
4784:with all quantifiers bounded to
3650:The proof is as follows: Either
1761:in ω with the corresponding ω +
489:defined such that for every set
34:
3435:will contain at least one such
2171:Relation to other axiom schemas
2119:ZF with replacement proves the
1955:von Neumann cardinal assignment
1741:, proving the existence of any
5114:countable models of set theory
4941:
4922:
4899:
4880:
4700:
4668:
4647:
4643:
4611:
4568:
4536:
4417:
4411:
4279:
4273:
4173:
4167:
4061:
4055:
4026:
4020:
3990:
3984:
3955:
3949:
3899:
3884:
3878:
3869:
3863:
3848:
3842:
3836:
3830:
3818:
3795:
3789:
3706:
3700:
3551:
3548:
3545:
3539:
3521:
3518:
3506:
3279:
3276:
3232:
3213:
3210:
3166:
3154:
3008:
3005:
2961:
2915:
2912:
2909:
2865:
2845:
2842:
2598:. In this case, the image set
2459:constructive counterpart of ZF
2371:
2365:
2330:
2327:
2324:
2306:
2287:
2275:
2255:
2249:
2246:
2228:
2205:
2202:
1929:
1917:
1894:
1882:
1814:
1794:
1684:
1678:
1508:
1505:
1502:
1484:
1465:
1453:
1433:
1427:
1424:
1406:
1375:
1372:
1274:
1271:
1268:
1218:
1199:
1187:
1167:
1161:
1158:
1108:
1077:
1074:
876:
870:
827:
821:
736:
730:
693:
687:
584:
578:
456:
444:
413:
407:
350:
338:
179:
173:
1:
5025:is a set and each element of
131:is a set depends only on the
4993:Thoralf Skolem, in the 1930s
4526:and any first-order formula
4474:reflection principle for ZFC
4350:in von Neumann's hierarchy.
2799:. Then the axiom schema is:
1935:{\displaystyle P(A\times A)}
1900:{\displaystyle P(A\times A)}
5324:L'Enseignement Mathématique
5036:German Mathematical Society
4824:but with every instance of
4343:{\displaystyle V_{\delta }}
3457:axiom schema of boundedness
1869:, and so an element of the
891:axiom of limitation of size
811:is any set, then the image
769:axiom schema of replacement
535:if and only if there is an
112:(ZF) that asserts that the
110:Zermelo–Fraenkel set theory
98:axiom schema of replacement
6062:
5896:von Neumann–Bernays–Gödel
4465:
4423:{\displaystyle \theta (x)}
4067:{\displaystyle \theta (x)}
4038:{\displaystyle F_{a}(x)=a}
3996:{\displaystyle \theta (x)}
3967:{\displaystyle F_{a}(x)=x}
3932:, it acts as the identity
3801:{\displaystyle \theta (a)}
3712:{\displaystyle \theta (a)}
3469:axiom schema of separation
2450:axiom schema of collection
1722:and foundation systems in
1709:(Z) already can interpret
596:{\displaystyle F_{P}(x)=y}
425:{\displaystyle F_{P}(x)=y}
288:) such that for every set
5697:One-to-one correspondence
5434:
5286:Journal of Symbolic Logic
5074:there exists at most one
4863:{\displaystyle \forall x}
4840:{\displaystyle \exists x}
4757:{\displaystyle \phi ^{M}}
4357:for the proposition that
2110:Borel determinacy theorem
1862:{\displaystyle A\times A}
1782:first uncountable ordinal
1319:uniqueness quantification
1310:{\displaystyle \exists !}
135:of the class, not on the
5355:Ebbinghaus, pp. 135-138.
5268:In Praise of Replacement
5234:In Praise of Replacement
5118:Heinz-Dieter Ebbinghaus
4594:{\displaystyle \alpha }
4456:bounded variant thereof
4452:constructive set theory
4394:{\displaystyle \theta }
3620:{\displaystyle \theta }
3580:{\displaystyle \theta }
2558:may correspond to many
2104:. The proven result is
1711:second-order arithmetic
623:is called the image of
64:more precise citations.
5655:Constructible universe
5482:Constructibility (V=L)
4994:
4986:
4948:
4906:
4864:
4841:
4818:
4798:
4778:
4758:
4731:
4707:
4595:
4575:
4520:
4444:
4424:
4395:
4371:
4355:law of excluded middle
4344:
4309:
4289:
4242:
4215:
4195:
4135:
4108:
4088:
4068:
4039:
3997:
3968:
3926:
3906:
3802:
3773:
3753:
3733:
3713:
3684:
3670:contains some element
3664:
3641:
3627:that does not mention
3621:
3601:
3581:
3558:
3449:
3429:
3409:
3389:
3369:
3349:
3329:
3309:
3286:
3078:
3058:
3044:not occurring free in
3038:
3015:
2793:
2773:
2753:
2733:
2675:
2652:
2632:
2612:
2592:
2572:
2552:
2551:{\displaystyle x\in A}
2526:
2506:
2486:
2463:law of excluded middle
2445:
2438:
2418:
2398:
2378:
2337:
2155:
2083:
2034:
2014:
1936:
1901:
1863:
1821:
1691:
1662:
1642:
1622:
1602:
1582:
1562:
1542:
1519:
1351:, this simplifies to:
1345:
1311:
1285:
1014:
994:
974:
910:
883:
854:
834:
805:
785:
758:
700:
664:
637:
617:
597:
555:
554:{\displaystyle x\in A}
529:
528:{\displaystyle y\in B}
503:
483:
463:
462:{\displaystyle P(x,y)}
426:
384:
357:
356:{\displaystyle P(x,y)}
322:
308:there is a unique set
302:
280:is a definable binary
274:
253:
246:
226:
206:
186:
5878:Principia Mathematica
5712:Transfinite induction
5571:(i.e. set difference)
5029:is replaced by then
4992:
4984:
4949:
4907:
4865:
4842:
4819:
4817:{\displaystyle \phi }
4799:
4779:
4777:{\displaystyle \phi }
4759:
4732:
4730:{\displaystyle \phi }
4708:
4596:
4576:
4521:
4445:
4443:{\displaystyle \phi }
4425:
4396:
4372:
4345:
4310:
4290:
4243:
4241:{\displaystyle F_{a}}
4216:
4196:
4136:
4134:{\displaystyle F_{a}}
4109:
4089:
4069:
4040:
3998:
3969:
3927:
3925:{\displaystyle \phi }
3907:
3803:
3774:
3754:
3734:
3714:
3685:
3665:
3642:
3622:
3602:
3582:
3559:
3450:
3430:
3415:, then the image set
3410:
3390:
3370:
3350:
3348:{\displaystyle \phi }
3330:
3310:
3287:
3079:
3077:{\displaystyle \phi }
3059:
3057:{\displaystyle \phi }
3039:
3016:
2794:
2792:{\displaystyle \phi }
2774:
2754:
2734:
2676:
2674:{\displaystyle \phi }
2653:
2633:
2613:
2593:
2573:
2553:
2527:
2525:{\displaystyle \phi }
2507:
2505:{\displaystyle \phi }
2487:
2439:
2419:
2399:
2379:
2354:
2338:
2156:
2084:
2035:
2015:
1937:
1902:
1864:
1822:
1692:
1663:
1643:
1623:
1603:
1583:
1563:
1543:
1541:{\displaystyle \phi }
1520:
1346:
1344:{\displaystyle w_{i}}
1312:
1286:
1015:
1013:{\displaystyle \phi }
995:
975:
911:
909:{\displaystyle \phi }
884:
855:
835:
806:
786:
759:
701:
699:{\displaystyle F_{P}}
665:
663:{\displaystyle F_{P}}
638:
618:
598:
556:
530:
504:
484:
464:
427:
385:
383:{\displaystyle F_{P}}
358:
323:
303:
275:
247:
227:
207:
187:
162:
27:Concept in set theory
6046:Axioms of set theory
5952:Burali-Forti paradox
5707:Set-builder notation
5660:Continuum hypothesis
5600:Symmetric difference
5050:, which was held in
4916:
4874:
4851:
4828:
4808:
4788:
4768:
4741:
4721:
4605:
4585:
4530:
4484:
4468:Reflection principle
4434:
4405:
4385:
4381:by a set validating
4361:
4327:
4299:
4252:
4225:
4205:
4145:
4118:
4098:
4078:
4049:
4007:
3978:
3936:
3916:
3812:
3783:
3763:
3743:
3723:
3694:
3674:
3654:
3631:
3611:
3591:
3571:
3482:
3439:
3419:
3399:
3379:
3359:
3339:
3319:
3299:
3091:
3068:
3064:) on the predicate,
3048:
3028:
2806:
2783:
2763:
2743:
2685:
2665:
2642:
2622:
2602:
2582:
2562:
2536:
2516:
2496:
2476:
2428:
2408:
2388:
2359:
2192:
2138:
2044:
2024:
1972:
1911:
1876:
1847:
1788:
1672:
1652:
1632:
1612:
1592:
1572:
1552:
1532:
1358:
1328:
1298:
1027:
1004:
984:
920:
900:
864:
844:
815:
795:
775:
714:
708:set-builder notation
674:
647:
627:
607:
565:
539:
513:
493:
473:
438:
394:
367:
332:
312:
292:
264:
236:
216:
196:
167:
120:under any definable
18:Axiom of replacement
5913:Tarski–Grothendieck
5376:Ebbinghaus, p. 184.
5367:Ebbinghaus, p. 189.
5160:, Springer-Verlag,
5015:axiom of foundation
4997:Its publication by
4094:are unique for any
2424:falls inside a set
1548:specifies a unique
1294:For the meaning of
5502:Limitation of size
5335:Ebbinghaus, p. 92.
5082:is true. Then, as
4995:
4987:
4944:
4902:
4860:
4837:
4814:
4794:
4774:
4754:
4727:
4703:
4591:
4581:, there exists an
4571:
4516:
4440:
4420:
4391:
4367:
4340:
4305:
4285:
4238:
4211:
4191:
4131:
4104:
4084:
4064:
4035:
3993:
3964:
3922:
3902:
3798:
3769:
3749:
3729:
3709:
3680:
3660:
3637:
3617:
3607:is not free, i.e.
3597:
3577:
3554:
3473:axiom of empty set
3445:
3425:
3405:
3385:
3365:
3345:
3325:
3305:
3282:
3074:
3054:
3034:
3011:
2789:
2769:
2749:
2729:
2671:
2648:
2628:
2608:
2588:
2568:
2548:
2522:
2502:
2482:
2446:
2434:
2414:
2394:
2384:of the domain set
2374:
2333:
2151:
2123:of Z, as the set V
2079:
2030:
2010:
1932:
1897:
1859:
1817:
1707:Zermelo set theory
1687:
1658:
1638:
1618:
1598:
1578:
1558:
1538:
1515:
1513:
1341:
1307:
1281:
1279:
1010:
990:
970:
906:
879:
850:
830:
801:
781:
754:
696:
660:
633:
613:
593:
551:
525:
499:
479:
459:
422:
380:
353:
318:
298:
270:
254:
242:
222:
202:
192:of the domain set
182:
6033:
6032:
5942:Russell's paradox
5891:Zermelo–Fraenkel
5792:Dedekind-infinite
5665:Diagonal argument
5564:Cartesian product
5428:Set (mathematics)
5141:978-3-540-49553-6
4797:{\displaystyle M}
4370:{\displaystyle A}
4308:{\displaystyle B}
4248:, i.e. the class
4214:{\displaystyle A}
4107:{\displaystyle x}
4087:{\displaystyle y}
3772:{\displaystyle A}
3752:{\displaystyle a}
3732:{\displaystyle B}
3683:{\displaystyle a}
3663:{\displaystyle A}
3640:{\displaystyle B}
3600:{\displaystyle B}
3567:for each formula
3448:{\displaystyle y}
3428:{\displaystyle B}
3408:{\displaystyle y}
3388:{\displaystyle A}
3368:{\displaystyle x}
3328:{\displaystyle A}
3308:{\displaystyle x}
3037:{\displaystyle B}
2772:{\displaystyle B}
2752:{\displaystyle A}
2651:{\displaystyle x}
2631:{\displaystyle y}
2611:{\displaystyle B}
2591:{\displaystyle B}
2571:{\displaystyle y}
2485:{\displaystyle B}
2437:{\displaystyle B}
2417:{\displaystyle f}
2397:{\displaystyle A}
2377:{\displaystyle f}
2260:
2254:
2033:{\displaystyle A}
1751:axiom of infinity
1690:{\displaystyle F}
1661:{\displaystyle B}
1641:{\displaystyle y}
1621:{\displaystyle A}
1601:{\displaystyle F}
1581:{\displaystyle y}
1561:{\displaystyle x}
1438:
1432:
1172:
1166:
993:{\displaystyle B}
882:{\displaystyle F}
853:{\displaystyle A}
833:{\displaystyle F}
804:{\displaystyle A}
784:{\displaystyle F}
636:{\displaystyle A}
616:{\displaystyle B}
502:{\displaystyle y}
482:{\displaystyle B}
321:{\displaystyle y}
301:{\displaystyle x}
273:{\displaystyle P}
245:{\displaystyle B}
232:is itself a set,
225:{\displaystyle F}
205:{\displaystyle A}
185:{\displaystyle F}
90:
89:
82:
16:(Redirected from
6053:
6015:Bertrand Russell
6005:John von Neumann
5990:Abraham Fraenkel
5985:Richard Dedekind
5947:Suslin's problem
5858:Cantor's theorem
5575:De Morgan's laws
5440:
5407:
5400:
5393:
5384:
5377:
5374:
5368:
5365:
5356:
5353:
5336:
5333:
5327:
5321:
5277:
5271:
5264:
5258:
5257:
5255:
5243:
5237:
5230:
5214:
5192:
5170:
5159:
5156:Naive Set Theory
5144:
5066:) in the domain
4999:Abraham Fraenkel
4953:
4951:
4950:
4945:
4940:
4939:
4911:
4909:
4908:
4903:
4898:
4897:
4869:
4867:
4866:
4861:
4846:
4844:
4843:
4838:
4823:
4821:
4820:
4815:
4803:
4801:
4800:
4795:
4783:
4781:
4780:
4775:
4763:
4761:
4760:
4755:
4753:
4752:
4736:
4734:
4733:
4728:
4712:
4710:
4709:
4704:
4699:
4698:
4680:
4679:
4667:
4666:
4665:
4664:
4642:
4641:
4623:
4622:
4600:
4598:
4597:
4592:
4580:
4578:
4577:
4572:
4567:
4566:
4548:
4547:
4525:
4523:
4522:
4517:
4515:
4514:
4496:
4495:
4449:
4447:
4446:
4441:
4429:
4427:
4426:
4421:
4400:
4398:
4397:
4392:
4376:
4374:
4373:
4368:
4349:
4347:
4346:
4341:
4339:
4338:
4314:
4312:
4311:
4306:
4294:
4292:
4291:
4286:
4247:
4245:
4244:
4239:
4237:
4236:
4220:
4218:
4217:
4212:
4200:
4198:
4197:
4192:
4166:
4165:
4140:
4138:
4137:
4132:
4130:
4129:
4113:
4111:
4110:
4105:
4093:
4091:
4090:
4085:
4073:
4071:
4070:
4065:
4044:
4042:
4041:
4036:
4019:
4018:
4002:
4000:
3999:
3994:
3973:
3971:
3970:
3965:
3948:
3947:
3931:
3929:
3928:
3923:
3911:
3909:
3908:
3903:
3807:
3805:
3804:
3799:
3778:
3776:
3775:
3770:
3758:
3756:
3755:
3750:
3738:
3736:
3735:
3730:
3718:
3716:
3715:
3710:
3689:
3687:
3686:
3681:
3669:
3667:
3666:
3661:
3646:
3644:
3643:
3638:
3626:
3624:
3623:
3618:
3606:
3604:
3603:
3598:
3586:
3584:
3583:
3578:
3563:
3561:
3560:
3555:
3454:
3452:
3451:
3446:
3434:
3432:
3431:
3426:
3414:
3412:
3411:
3406:
3394:
3392:
3391:
3386:
3374:
3372:
3371:
3366:
3354:
3352:
3351:
3346:
3334:
3332:
3331:
3326:
3314:
3312:
3311:
3306:
3291:
3289:
3288:
3283:
3275:
3274:
3256:
3255:
3209:
3208:
3190:
3189:
3125:
3124:
3106:
3105:
3083:
3081:
3080:
3075:
3063:
3061:
3060:
3055:
3043:
3041:
3040:
3035:
3020:
3018:
3017:
3012:
3004:
3003:
2985:
2984:
2908:
2907:
2889:
2888:
2840:
2839:
2821:
2820:
2798:
2796:
2795:
2790:
2778:
2776:
2775:
2770:
2758:
2756:
2755:
2750:
2738:
2736:
2735:
2730:
2716:
2715:
2697:
2696:
2680:
2678:
2677:
2672:
2657:
2655:
2654:
2649:
2637:
2635:
2634:
2629:
2617:
2615:
2614:
2609:
2597:
2595:
2594:
2589:
2577:
2575:
2574:
2569:
2557:
2555:
2554:
2549:
2531:
2529:
2528:
2523:
2511:
2509:
2508:
2503:
2491:
2489:
2488:
2483:
2443:
2441:
2440:
2435:
2423:
2421:
2420:
2415:
2403:
2401:
2400:
2395:
2383:
2381:
2380:
2375:
2342:
2340:
2339:
2334:
2258:
2252:
2160:
2158:
2157:
2152:
2150:
2149:
2106:Donald A. Martin
2088:
2086:
2085:
2080:
2075:
2074:
2059:
2058:
2039:
2037:
2036:
2031:
2019:
2017:
2016:
2011:
2003:
2002:
1984:
1983:
1957:which assigns a
1941:
1939:
1938:
1933:
1906:
1904:
1903:
1898:
1868:
1866:
1865:
1860:
1826:
1824:
1823:
1818:
1813:
1812:
1803:
1802:
1767:well-ordered set
1696:
1694:
1693:
1688:
1667:
1665:
1664:
1659:
1647:
1645:
1644:
1639:
1627:
1625:
1624:
1619:
1607:
1605:
1604:
1599:
1587:
1585:
1584:
1579:
1567:
1565:
1564:
1559:
1547:
1545:
1544:
1539:
1524:
1522:
1521:
1516:
1514:
1436:
1430:
1350:
1348:
1347:
1342:
1340:
1339:
1316:
1314:
1313:
1308:
1290:
1288:
1287:
1282:
1280:
1261:
1260:
1242:
1241:
1170:
1164:
1151:
1150:
1132:
1131:
1065:
1064:
1046:
1045:
1019:
1017:
1016:
1011:
999:
997:
996:
991:
979:
977:
976:
971:
951:
950:
932:
931:
915:
913:
912:
907:
888:
886:
885:
880:
859:
857:
856:
851:
839:
837:
836:
831:
810:
808:
807:
802:
790:
788:
787:
782:
763:
761:
760:
755:
729:
728:
705:
703:
702:
697:
686:
685:
669:
667:
666:
661:
659:
658:
642:
640:
639:
634:
622:
620:
619:
614:
602:
600:
599:
594:
577:
576:
560:
558:
557:
552:
534:
532:
531:
526:
508:
506:
505:
500:
488:
486:
485:
480:
468:
466:
465:
460:
431:
429:
428:
423:
406:
405:
389:
387:
386:
381:
379:
378:
362:
360:
359:
354:
327:
325:
324:
319:
307:
305:
304:
299:
284:(which may be a
279:
277:
276:
271:
251:
249:
248:
243:
231:
229:
228:
223:
211:
209:
208:
203:
191:
189:
188:
183:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
6061:
6060:
6056:
6055:
6054:
6052:
6051:
6050:
6036:
6035:
6034:
6029:
5956:
5935:
5919:
5884:New Foundations
5831:
5721:
5640:Cardinal number
5623:
5609:
5550:
5441:
5432:
5416:
5411:
5381:
5380:
5375:
5371:
5366:
5359:
5354:
5339:
5334:
5330:
5299:10.2307/2274520
5281:Maddy, Penelope
5279:
5278:
5274:
5265:
5261:
5245:
5244:
5240:
5231:
5227:
5222:
5212:
5196:
5190:
5174:
5168:
5150:Halmos, Paul R.
5148:
5142:
5129:
5126:
5110:large cardinals
5102:
5091:
4960:
4931:
4914:
4913:
4889:
4872:
4871:
4849:
4848:
4826:
4825:
4806:
4805:
4786:
4785:
4766:
4765:
4744:
4739:
4738:
4719:
4718:
4690:
4671:
4656:
4651:
4633:
4614:
4603:
4602:
4583:
4582:
4558:
4539:
4528:
4527:
4506:
4487:
4482:
4481:
4470:
4464:
4432:
4431:
4403:
4402:
4383:
4382:
4359:
4358:
4330:
4325:
4324:
4297:
4296:
4250:
4249:
4228:
4223:
4222:
4203:
4202:
4157:
4143:
4142:
4121:
4116:
4115:
4096:
4095:
4076:
4075:
4047:
4046:
4010:
4005:
4004:
3976:
3975:
3939:
3934:
3933:
3914:
3913:
3810:
3809:
3781:
3780:
3779:that validates
3761:
3760:
3741:
3740:
3721:
3720:
3692:
3691:
3672:
3671:
3652:
3651:
3629:
3628:
3609:
3608:
3589:
3588:
3569:
3568:
3480:
3479:
3465:
3437:
3436:
3417:
3416:
3397:
3396:
3377:
3376:
3357:
3356:
3337:
3336:
3317:
3316:
3297:
3296:
3266:
3247:
3200:
3181:
3116:
3097:
3089:
3088:
3066:
3065:
3046:
3045:
3026:
3025:
2995:
2976:
2899:
2880:
2831:
2812:
2804:
2803:
2781:
2780:
2761:
2760:
2741:
2740:
2707:
2688:
2683:
2682:
2663:
2662:
2640:
2639:
2620:
2619:
2600:
2599:
2580:
2579:
2560:
2559:
2534:
2533:
2514:
2513:
2494:
2493:
2474:
2473:
2455:power set axiom
2426:
2425:
2406:
2405:
2386:
2385:
2357:
2356:
2349:
2190:
2189:
2178:
2176:Simplifications
2173:
2141:
2136:
2135:
2133:cardinal number
2126:
2094:Harvey Friedman
2050:
2042:
2041:
2022:
2021:
1988:
1975:
1970:
1969:
1963:axiom of choice
1959:cardinal number
1945:
1909:
1908:
1874:
1873:
1845:
1844:
1843:is a subset of
1786:
1785:
1779:
1731:proof-theoretic
1703:
1670:
1669:
1650:
1649:
1630:
1629:
1610:
1609:
1590:
1589:
1570:
1569:
1550:
1549:
1530:
1529:
1512:
1511:
1390:
1356:
1355:
1331:
1326:
1325:
1296:
1295:
1278:
1277:
1252:
1233:
1142:
1123:
1092:
1056:
1037:
1025:
1024:
1002:
1001:
1000:is not free in
982:
981:
942:
923:
918:
917:
898:
897:
862:
861:
842:
841:
813:
812:
793:
792:
773:
772:
771:states that if
720:
712:
711:
677:
672:
671:
650:
645:
644:
625:
624:
605:
604:
568:
563:
562:
537:
536:
511:
510:
491:
490:
471:
470:
436:
435:
397:
392:
391:
370:
365:
364:
330:
329:
310:
309:
290:
289:
262:
261:
234:
233:
214:
213:
194:
193:
165:
164:
157:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
6059:
6057:
6049:
6048:
6038:
6037:
6031:
6030:
6028:
6027:
6022:
6020:Thoralf Skolem
6017:
6012:
6007:
6002:
5997:
5992:
5987:
5982:
5977:
5972:
5966:
5964:
5958:
5957:
5955:
5954:
5949:
5944:
5938:
5936:
5934:
5933:
5930:
5924:
5921:
5920:
5918:
5917:
5916:
5915:
5910:
5905:
5904:
5903:
5888:
5887:
5886:
5874:
5873:
5872:
5861:
5860:
5855:
5850:
5845:
5839:
5837:
5833:
5832:
5830:
5829:
5824:
5819:
5814:
5805:
5800:
5795:
5785:
5780:
5779:
5778:
5773:
5768:
5758:
5748:
5743:
5738:
5732:
5730:
5723:
5722:
5720:
5719:
5714:
5709:
5704:
5702:Ordinal number
5699:
5694:
5689:
5684:
5683:
5682:
5677:
5667:
5662:
5657:
5652:
5647:
5637:
5632:
5626:
5624:
5622:
5621:
5618:
5614:
5611:
5610:
5608:
5607:
5602:
5597:
5592:
5587:
5582:
5580:Disjoint union
5577:
5572:
5566:
5560:
5558:
5552:
5551:
5549:
5548:
5547:
5546:
5541:
5530:
5529:
5527:Martin's axiom
5524:
5519:
5514:
5509:
5504:
5499:
5494:
5492:Extensionality
5489:
5484:
5479:
5478:
5477:
5472:
5467:
5457:
5451:
5449:
5443:
5442:
5435:
5433:
5431:
5430:
5424:
5422:
5418:
5417:
5412:
5410:
5409:
5402:
5395:
5387:
5379:
5378:
5369:
5357:
5337:
5328:
5293:(2): 481–511,
5272:
5266:A. Kanamori, "
5259:
5238:
5232:A. Kanamori, "
5224:
5223:
5221:
5218:
5217:
5216:
5210:
5198:Kunen, Kenneth
5194:
5188:
5172:
5166:
5146:
5140:
5125:
5122:
5100:
5089:
5011:Thoralf Skolem
4959:
4956:
4954:respectively.
4943:
4938:
4934:
4930:
4927:
4924:
4921:
4901:
4896:
4892:
4888:
4885:
4882:
4879:
4870:replaced with
4859:
4856:
4836:
4833:
4813:
4793:
4773:
4751:
4747:
4726:
4715:
4714:
4702:
4697:
4693:
4689:
4686:
4683:
4678:
4674:
4670:
4663:
4659:
4654:
4649:
4645:
4640:
4636:
4632:
4629:
4626:
4621:
4617:
4613:
4610:
4590:
4570:
4565:
4561:
4557:
4554:
4551:
4546:
4542:
4538:
4535:
4513:
4509:
4505:
4502:
4499:
4494:
4490:
4466:Main article:
4463:
4460:
4439:
4419:
4416:
4413:
4410:
4401:, and for any
4390:
4366:
4337:
4333:
4304:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4235:
4231:
4210:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4164:
4160:
4156:
4153:
4150:
4128:
4124:
4103:
4083:
4063:
4060:
4057:
4054:
4034:
4031:
4028:
4025:
4022:
4017:
4013:
3992:
3989:
3986:
3983:
3963:
3960:
3957:
3954:
3951:
3946:
3942:
3921:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3797:
3794:
3791:
3788:
3768:
3748:
3728:
3708:
3705:
3702:
3699:
3679:
3659:
3636:
3616:
3596:
3576:
3565:
3564:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3504:
3501:
3497:
3494:
3490:
3487:
3464:
3461:
3444:
3424:
3404:
3384:
3364:
3344:
3324:
3304:
3293:
3292:
3281:
3278:
3273:
3269:
3265:
3262:
3259:
3254:
3250:
3246:
3243:
3240:
3237:
3234:
3231:
3227:
3224:
3221:
3218:
3215:
3212:
3207:
3203:
3199:
3196:
3193:
3188:
3184:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3152:
3149:
3146:
3143:
3139:
3136:
3132:
3129:
3123:
3119:
3115:
3112:
3109:
3104:
3100:
3096:
3073:
3053:
3033:
3022:
3021:
3010:
3007:
3002:
2998:
2994:
2991:
2988:
2983:
2979:
2975:
2972:
2969:
2966:
2963:
2960:
2956:
2953:
2950:
2947:
2943:
2940:
2937:
2934:
2930:
2927:
2923:
2920:
2917:
2914:
2911:
2906:
2902:
2898:
2895:
2892:
2887:
2883:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2857:
2853:
2850:
2847:
2844:
2838:
2834:
2830:
2827:
2824:
2819:
2815:
2811:
2788:
2768:
2748:
2739:; but neither
2728:
2725:
2722:
2719:
2714:
2710:
2706:
2703:
2700:
2695:
2691:
2670:
2647:
2627:
2607:
2587:
2567:
2547:
2544:
2541:
2521:
2501:
2481:
2433:
2413:
2393:
2373:
2370:
2367:
2364:
2348:
2345:
2344:
2343:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2273:
2270:
2266:
2263:
2257:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2223:
2220:
2217:
2213:
2210:
2207:
2204:
2200:
2197:
2177:
2174:
2172:
2169:
2168:
2167:
2148:
2144:
2124:
2117:
2090:
2078:
2073:
2068:
2065:
2062:
2057:
2053:
2049:
2029:
2020:and for large
2009:
2006:
2001:
1998:
1995:
1991:
1987:
1982:
1978:
1966:
1951:
1948:Hartogs number
1943:
1931:
1928:
1925:
1922:
1919:
1916:
1896:
1893:
1890:
1887:
1884:
1881:
1858:
1855:
1852:
1816:
1811:
1806:
1801:
1796:
1793:
1777:
1774:
1771:disjoint union
1747:ordinal number
1702:
1699:
1686:
1683:
1680:
1677:
1657:
1637:
1617:
1597:
1577:
1557:
1537:
1526:
1525:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1451:
1448:
1444:
1441:
1435:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1401:
1398:
1395:
1391:
1389:
1386:
1383:
1380:
1377:
1374:
1370:
1367:
1364:
1363:
1338:
1334:
1306:
1303:
1292:
1291:
1276:
1273:
1270:
1267:
1264:
1259:
1255:
1251:
1248:
1245:
1240:
1236:
1232:
1229:
1226:
1223:
1220:
1217:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1185:
1182:
1178:
1175:
1169:
1163:
1160:
1157:
1154:
1149:
1145:
1141:
1138:
1135:
1130:
1126:
1122:
1119:
1116:
1113:
1110:
1107:
1103:
1100:
1097:
1093:
1091:
1088:
1085:
1082:
1079:
1076:
1072:
1069:
1063:
1059:
1055:
1052:
1049:
1044:
1040:
1036:
1033:
1032:
1009:
989:
969:
966:
963:
960:
957:
954:
949:
945:
941:
938:
935:
930:
926:
905:
878:
875:
872:
869:
849:
829:
826:
823:
820:
800:
780:
753:
750:
747:
744:
741:
738:
735:
732:
727:
723:
719:
695:
692:
689:
684:
680:
670:, and denoted
657:
653:
632:
612:
592:
589:
586:
583:
580:
575:
571:
550:
547:
544:
524:
521:
518:
498:
478:
458:
455:
452:
449:
446:
443:
433:if and only if
421:
418:
415:
412:
409:
404:
400:
377:
373:
352:
349:
346:
343:
340:
337:
317:
297:
269:
241:
221:
201:
181:
178:
175:
172:
156:
153:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6058:
6047:
6044:
6043:
6041:
6026:
6025:Ernst Zermelo
6023:
6021:
6018:
6016:
6013:
6011:
6010:Willard Quine
6008:
6006:
6003:
6001:
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5981:
5978:
5976:
5973:
5971:
5968:
5967:
5965:
5963:
5962:Set theorists
5959:
5953:
5950:
5948:
5945:
5943:
5940:
5939:
5937:
5931:
5929:
5926:
5925:
5922:
5914:
5911:
5909:
5908:Kripke–Platek
5906:
5902:
5899:
5898:
5897:
5894:
5893:
5892:
5889:
5885:
5882:
5881:
5880:
5879:
5875:
5871:
5868:
5867:
5866:
5863:
5862:
5859:
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5840:
5838:
5834:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5811:
5806:
5804:
5801:
5799:
5796:
5793:
5789:
5786:
5784:
5781:
5777:
5774:
5772:
5769:
5767:
5764:
5763:
5762:
5759:
5756:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5733:
5731:
5728:
5724:
5718:
5715:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5693:
5690:
5688:
5685:
5681:
5678:
5676:
5673:
5672:
5671:
5668:
5666:
5663:
5661:
5658:
5656:
5653:
5651:
5648:
5645:
5641:
5638:
5636:
5633:
5631:
5628:
5627:
5625:
5619:
5616:
5615:
5612:
5606:
5603:
5601:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5581:
5578:
5576:
5573:
5570:
5567:
5565:
5562:
5561:
5559:
5557:
5553:
5545:
5544:specification
5542:
5540:
5537:
5536:
5535:
5532:
5531:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5510:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5483:
5480:
5476:
5473:
5471:
5468:
5466:
5463:
5462:
5461:
5458:
5456:
5453:
5452:
5450:
5448:
5444:
5439:
5429:
5426:
5425:
5423:
5419:
5415:
5408:
5403:
5401:
5396:
5394:
5389:
5388:
5385:
5373:
5370:
5364:
5362:
5358:
5352:
5350:
5348:
5346:
5344:
5342:
5338:
5332:
5329:
5325:
5320:
5316:
5312:
5308:
5304:
5300:
5296:
5292:
5288:
5287:
5282:
5276:
5273:
5269:
5263:
5260:
5254:
5249:
5242:
5239:
5235:
5229:
5226:
5219:
5213:
5211:0-444-86839-9
5207:
5203:
5199:
5195:
5191:
5189:3-540-44085-2
5185:
5181:
5177:
5173:
5169:
5167:0-387-90092-6
5163:
5158:
5157:
5151:
5147:
5143:
5137:
5133:
5128:
5127:
5123:
5121:
5119:
5115:
5111:
5105:
5103:
5096:
5092:
5085:
5081:
5077:
5073:
5069:
5065:
5061:
5057:
5053:
5049:
5044:
5041:
5037:
5032:
5028:
5024:
5018:
5016:
5012:
5008:
5004:
5000:
4991:
4983:
4979:
4977:
4973:
4969:
4965:
4964:Ernst Zermelo
4957:
4955:
4936:
4932:
4928:
4925:
4894:
4890:
4886:
4883:
4857:
4834:
4811:
4791:
4771:
4749:
4745:
4724:
4695:
4691:
4687:
4684:
4681:
4676:
4672:
4661:
4657:
4652:
4638:
4634:
4630:
4627:
4624:
4619:
4615:
4608:
4588:
4563:
4559:
4555:
4552:
4549:
4544:
4540:
4533:
4511:
4507:
4503:
4500:
4497:
4492:
4488:
4479:
4478:
4477:
4475:
4469:
4461:
4459:
4457:
4453:
4437:
4414:
4408:
4388:
4380:
4364:
4356:
4351:
4335:
4331:
4320:
4316:
4302:
4276:
4270:
4267:
4264:
4258:
4255:
4233:
4229:
4208:
4185:
4182:
4179:
4176:
4170:
4162:
4158:
4151:
4148:
4126:
4122:
4101:
4081:
4058:
4052:
4032:
4029:
4023:
4015:
4011:
3987:
3981:
3961:
3958:
3952:
3944:
3940:
3919:
3896:
3893:
3890:
3887:
3881:
3875:
3866:
3860:
3857:
3854:
3851:
3845:
3839:
3833:
3827:
3824:
3821:
3815:
3808:. Now define
3792:
3786:
3766:
3746:
3726:
3703:
3697:
3677:
3657:
3648:
3634:
3614:
3594:
3574:
3542:
3536:
3533:
3530:
3527:
3524:
3515:
3512:
3509:
3502:
3495:
3488:
3478:
3477:
3476:
3474:
3470:
3462:
3460:
3458:
3442:
3422:
3402:
3382:
3362:
3342:
3322:
3302:
3271:
3267:
3263:
3260:
3257:
3252:
3248:
3244:
3241:
3238:
3235:
3229:
3225:
3222:
3219:
3205:
3201:
3197:
3194:
3191:
3186:
3182:
3178:
3175:
3172:
3169:
3163:
3160:
3150:
3147:
3144:
3137:
3130:
3121:
3117:
3113:
3110:
3107:
3102:
3098:
3087:
3086:
3085:
3071:
3051:
3031:
3000:
2996:
2992:
2989:
2986:
2981:
2977:
2973:
2970:
2967:
2964:
2958:
2954:
2951:
2948:
2941:
2938:
2935:
2928:
2921:
2904:
2900:
2896:
2893:
2890:
2885:
2881:
2877:
2874:
2871:
2868:
2862:
2859:
2851:
2836:
2832:
2828:
2825:
2822:
2817:
2813:
2802:
2801:
2800:
2786:
2766:
2746:
2726:
2723:
2720:
2717:
2712:
2708:
2704:
2701:
2698:
2693:
2689:
2668:
2659:
2645:
2625:
2605:
2585:
2565:
2545:
2542:
2539:
2519:
2499:
2479:
2471:
2466:
2464:
2460:
2456:
2451:
2431:
2411:
2391:
2368:
2362:
2353:
2346:
2321:
2318:
2315:
2312:
2309:
2303:
2299:
2296:
2293:
2284:
2281:
2278:
2271:
2264:
2243:
2240:
2237:
2234:
2231:
2225:
2221:
2218:
2211:
2198:
2188:
2187:
2186:
2183:
2175:
2170:
2164:
2146:
2134:
2130:
2122:
2118:
2115:
2111:
2107:
2103:
2099:
2095:
2091:
2066:
2063:
2060:
2055:
2051:
2027:
2007:
2004:
1999:
1996:
1993:
1989:
1985:
1980:
1976:
1967:
1964:
1960:
1956:
1952:
1949:
1926:
1923:
1920:
1914:
1891:
1888:
1885:
1879:
1872:
1856:
1853:
1850:
1842:
1838:
1834:
1830:
1804:
1791:
1783:
1775:
1772:
1768:
1764:
1760:
1756:
1752:
1748:
1744:
1743:limit ordinal
1740:
1736:
1735:
1734:
1732:
1727:
1725:
1721:
1716:
1712:
1708:
1700:
1698:
1681:
1675:
1655:
1635:
1615:
1595:
1575:
1555:
1535:
1499:
1496:
1493:
1490:
1487:
1481:
1477:
1474:
1471:
1462:
1459:
1456:
1449:
1442:
1421:
1418:
1415:
1412:
1409:
1403:
1399:
1396:
1387:
1384:
1381:
1368:
1354:
1353:
1352:
1336:
1332:
1322:
1320:
1304:
1265:
1262:
1257:
1253:
1249:
1246:
1243:
1238:
1234:
1230:
1227:
1224:
1221:
1215:
1211:
1208:
1205:
1196:
1193:
1190:
1183:
1176:
1155:
1152:
1147:
1143:
1139:
1136:
1133:
1128:
1124:
1120:
1117:
1114:
1111:
1105:
1101:
1098:
1089:
1086:
1083:
1070:
1061:
1057:
1053:
1050:
1047:
1042:
1038:
1023:
1022:
1021:
1007:
987:
967:
964:
961:
958:
955:
952:
947:
943:
939:
936:
933:
928:
924:
903:
894:
892:
873:
867:
847:
824:
818:
798:
778:
770:
765:
748:
745:
742:
739:
733:
725:
721:
709:
690:
682:
678:
655:
651:
630:
610:
590:
587:
581:
573:
569:
548:
545:
542:
522:
519:
516:
496:
476:
453:
450:
447:
441:
434:
419:
416:
410:
402:
398:
375:
371:
347:
344:
341:
335:
315:
295:
287:
283:
267:
260:
259:
239:
219:
199:
176:
170:
161:
154:
152:
150:
146:
142:
138:
134:
130:
125:
123:
119:
115:
111:
107:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
5975:Georg Cantor
5970:Paul Bernays
5901:Morse–Kelley
5876:
5809:
5808:Subset
5755:hereditarily
5717:Venn diagram
5675:ordered pair
5590:Intersection
5538:
5534:Axiom schema
5372:
5331:
5323:
5318:
5290:
5284:
5275:
5262:
5241:
5228:
5204:, Elsevier,
5201:
5182:, Springer,
5179:
5176:Jech, Thomas
5155:
5131:
5106:
5098:
5094:
5087:
5083:
5079:
5075:
5071:
5067:
5063:
5059:
5055:
5045:
5030:
5026:
5022:
5019:
5006:
5005:set theory (
5002:
4996:
4967:
4961:
4716:
4471:
4352:
4321:
4317:
3649:
3566:
3466:
3456:
3294:
3023:
2660:
2467:
2449:
2447:
2179:
1840:
1762:
1758:
1728:
1713:and much of
1704:
1701:Applications
1528:So whenever
1527:
1323:
1293:
895:
768:
766:
286:proper class
256:
255:
126:
97:
91:
76:
67:
48:
6000:Thomas Jech
5843:Alternative
5822:Uncountable
5776:Ultrafilter
5635:Cardinality
5539:replacement
5487:Determinacy
3690:validating
2779:is free in
2182:Azriel LĂ©vy
2166:consistent.
2121:consistency
2098:Borel games
2092:Similarly,
1739:von Neumann
1720:type theory
1715:type theory
1628:, then all
133:cardinality
62:introducing
5995:Kurt Gödel
5980:Paul Cohen
5817:Transitive
5585:Identities
5569:Complement
5556:Operations
5517:Regularity
5455:Adjunction
5414:Set theory
5124:References
5078:such that
4976:Mirimanoff
4601:such that
4462:Reflection
4114:, meaning
3463:Separation
2681:are among
2470:superclass
2347:Collection
2102:determined
2040:, the set
1829:separation
1668:, akin to
706:or (using
328:such that
141:surjection
94:set theory
70:March 2013
45:references
5928:Paradoxes
5848:Axiomatic
5827:Universal
5803:Singleton
5798:Recursive
5741:Countable
5736:Amorphous
5595:Power set
5512:Power set
5470:dependent
5465:countable
5253:1110.2430
5220:Citations
5152:(1974) ,
4937:α
4929:∈
4920:∀
4895:α
4887:∈
4878:∃
4855:∀
4832:∃
4812:ϕ
4772:ϕ
4746:ϕ
4725:ϕ
4685:…
4662:α
4653:ϕ
4648:⟺
4628:…
4609:ϕ
4589:α
4553:…
4534:ϕ
4501:…
4438:ϕ
4409:θ
4389:θ
4379:inhabited
4336:δ
4271:θ
4259:∩
4183:∈
4053:θ
4045:wherever
3982:θ
3974:wherever
3920:ϕ
3888:∧
3876:θ
3873:¬
3867:∨
3852:∧
3840:θ
3816:ϕ
3787:θ
3698:θ
3615:θ
3575:θ
3537:θ
3534:∧
3528:∈
3519:⇔
3513:∈
3500:∀
3493:∃
3486:∀
3343:ϕ
3261:…
3230:ϕ
3223:∈
3217:∃
3214:⇒
3195:…
3164:ϕ
3158:∃
3148:∈
3142:∀
3135:∃
3128:∀
3111:…
3095:∀
3072:ϕ
3052:ϕ
2990:…
2959:ϕ
2952:∈
2946:∃
2939:∈
2933:∀
2926:∃
2919:∀
2916:⇒
2894:…
2863:ϕ
2856:∃
2849:∀
2826:…
2810:∀
2787:ϕ
2702:…
2669:ϕ
2638:for each
2543:∈
2520:ϕ
2500:ϕ
2304:ϕ
2297:∈
2291:∃
2288:⇔
2282:∈
2269:∀
2262:∃
2256:⟹
2226:ϕ
2216:∃
2209:∀
2196:∀
2147:ω
2143:ℵ
2067:∈
2061:∣
2005:×
1997:−
1924:×
1889:×
1871:power set
1854:×
1805:×
1536:ϕ
1482:ϕ
1475:∈
1469:∃
1466:⇔
1460:∈
1447:∀
1440:∃
1434:⟹
1404:ϕ
1394:∃
1385:∈
1379:∀
1366:∀
1302:∃
1247:…
1216:ϕ
1209:∈
1203:∃
1200:⇔
1194:∈
1181:∀
1174:∃
1168:⟹
1137:…
1106:ϕ
1096:∃
1087:∈
1081:∀
1068:∀
1051:…
1035:∀
1008:ϕ
937:…
904:ϕ
746:∈
546:∈
520:∈
155:Statement
6040:Category
5932:Problems
5836:Theories
5812:Superset
5788:Infinite
5617:Concepts
5497:Infinity
5421:Overview
5200:(1980),
5178:(2003),
5052:Helsinki
5038:held in
5003:Fraenkel
4978:(1917).
4737:. Here,
4480:For any
1837:relation
1833:powerset
1726:theory.
390:, where
282:relation
149:formulas
5870:General
5865:Zermelo
5771:subbase
5753: (
5692:Forcing
5670:Element
5642: (
5620:Methods
5507:Pairing
5326:(1917).
5315:0947855
5307:2274520
4958:History
4804:, i.e.
4472:LĂ©vy's
4454:, or a
2457:or its
2114:ordinal
1755:classes
258:Suppose
122:mapping
116:of any
58:improve
5761:Filter
5751:Finite
5687:Family
5630:Almost
5475:global
5460:Choice
5447:Axioms
5313:
5305:
5208:
5186:
5164:
5138:
4972:Cantor
4764:means
4221:under
2578:'s in
2259:
2253:
1780:, the
1437:
1431:
1317:, see
1171:
1165:
980:; but
643:under
106:axioms
102:schema
96:, the
47:, but
5853:Naive
5783:Fuzzy
5746:Empty
5729:types
5680:tuple
5650:Class
5644:large
5605:Union
5522:Union
5303:JSTOR
5248:arXiv
2129:model
2127:is a
1724:topos
561:with
129:class
114:image
100:is a
5766:base
5206:ISBN
5184:ISBN
5162:ISBN
5136:ISBN
5040:Jena
4912:and
4847:and
3467:The
2759:nor
2448:The
2100:are
1831:and
1568:-to-
767:The
137:rank
5727:Set
5295:doi
5007:ZFC
4377:is
4201:of
3759:in
3647:.
3375:of
3315:in
2125:ω·2
2108:'s
1839:on
1835:(a
1827:by
1608:on
145:ZFC
118:set
108:in
104:of
92:In
6042::
5360:^
5340:^
5317:,
5311:MR
5309:,
5301:,
5291:53
5289:,
5093:,
5062:,
4458:.
4152::=
3834::=
3459:.
3084::
2465:.
1697:.
1321:.
893:.
764:.
710:)
603:.
509:,
151:.
5810:·
5794:)
5790:(
5757:)
5646:)
5406:e
5399:t
5392:v
5297::
5256:.
5250::
5215:.
5193:.
5171:.
5145:.
5101:b
5099:M
5095:b
5090:a
5088:M
5084:a
5080:U
5076:b
5072:a
5068:B
5064:b
5060:a
5056:U
5031:M
5027:M
5023:M
4968:Z
4942:)
4933:V
4926:x
4923:(
4900:)
4891:V
4884:x
4881:(
4858:x
4835:x
4792:M
4750:M
4713:.
4701:)
4696:n
4692:x
4688:,
4682:,
4677:1
4673:x
4669:(
4658:V
4644:)
4639:n
4635:x
4631:,
4625:,
4620:1
4616:x
4612:(
4569:)
4564:n
4560:x
4556:,
4550:,
4545:1
4541:x
4537:(
4512:n
4508:x
4504:,
4498:,
4493:1
4489:x
4418:)
4415:x
4412:(
4365:A
4332:V
4303:B
4283:}
4280:)
4277:x
4274:(
4268::
4265:x
4262:{
4256:A
4234:a
4230:F
4209:A
4189:}
4186:A
4180:x
4177::
4174:)
4171:x
4168:(
4163:a
4159:F
4155:{
4149:B
4127:a
4123:F
4102:x
4082:y
4062:)
4059:x
4056:(
4033:a
4030:=
4027:)
4024:x
4021:(
4016:a
4012:F
3991:)
3988:x
3985:(
3962:x
3959:=
3956:)
3953:x
3950:(
3945:a
3941:F
3900:)
3897:a
3894:=
3891:y
3885:)
3882:x
3879:(
3870:(
3864:)
3861:x
3858:=
3855:y
3849:)
3846:x
3843:(
3837:(
3831:)
3828:y
3825:,
3822:x
3819:(
3796:)
3793:a
3790:(
3767:A
3747:a
3727:B
3707:)
3704:a
3701:(
3678:a
3658:A
3635:B
3595:B
3552:)
3549:]
3546:)
3543:C
3540:(
3531:A
3525:C
3522:[
3516:B
3510:C
3507:(
3503:C
3496:B
3489:A
3443:y
3423:B
3403:y
3383:A
3363:x
3323:A
3303:x
3280:]
3277:)
3272:n
3268:w
3264:,
3258:,
3253:1
3249:w
3245:,
3242:y
3239:,
3236:x
3233:(
3226:B
3220:y
3211:)
3206:n
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3192:,
3187:1
3183:w
3179:,
3176:y
3173:,
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3167:(
3161:y
3155:[
3151:A
3145:x
3138:B
3131:A
3122:n
3118:w
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3103:1
3099:w
3032:B
3009:]
3006:)
3001:n
2997:w
2993:,
2987:,
2982:1
2978:w
2974:,
2971:y
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2965:x
2962:(
2955:B
2949:y
2942:A
2936:x
2929:B
2922:A
2913:)
2910:)
2905:n
2901:w
2897:,
2891:,
2886:1
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2878:,
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2860:y
2852:x
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2843:[
2837:n
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2767:B
2747:A
2727:y
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2721:x
2718:,
2713:n
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2705:,
2699:,
2694:1
2690:w
2646:x
2626:y
2606:B
2586:B
2566:y
2546:A
2540:x
2480:B
2444:.
2432:B
2412:f
2392:A
2372:]
2369:A
2366:[
2363:f
2331:)
2328:]
2325:)
2322:A
2319:,
2316:y
2313:,
2310:x
2307:(
2300:A
2294:x
2285:B
2279:y
2276:[
2272:y
2265:B
2250:]
2247:)
2244:A
2241:,
2238:y
2235:,
2232:x
2229:(
2222:y
2219:!
2212:x
2206:[
2203:(
2199:A
2116:.
2077:}
2072:N
2064:n
2056:n
2052:A
2048:{
2028:A
2008:A
2000:1
1994:n
1990:A
1986:=
1981:n
1977:A
1965:.
1944:1
1930:)
1927:A
1921:A
1918:(
1915:P
1895:)
1892:A
1886:A
1883:(
1880:P
1857:A
1851:A
1841:A
1815:)
1810:N
1800:N
1795:(
1792:P
1778:1
1763:n
1759:n
1685:]
1682:A
1679:[
1676:F
1656:B
1636:y
1616:A
1596:F
1576:y
1556:x
1509:)
1506:]
1503:)
1500:A
1497:,
1494:y
1491:,
1488:x
1485:(
1478:A
1472:x
1463:B
1457:y
1454:[
1450:y
1443:B
1428:]
1425:)
1422:A
1419:,
1416:y
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1410:x
1407:(
1400:y
1397:!
1388:A
1382:x
1376:[
1373:(
1369:A
1337:i
1333:w
1305:!
1275:)
1272:]
1269:)
1266:A
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1258:n
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1244:,
1239:1
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1222:x
1219:(
1212:A
1206:x
1197:B
1191:y
1188:[
1184:y
1177:B
1162:]
1159:)
1156:A
1153:,
1148:n
1144:w
1140:,
1134:,
1129:1
1125:w
1121:,
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1112:x
1109:(
1102:y
1099:!
1090:A
1084:x
1078:[
1075:(
1071:A
1062:n
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1043:1
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959:,
956:A
953:,
948:n
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940:,
934:,
929:1
925:w
877:]
874:A
871:[
868:F
848:A
828:]
825:A
822:[
819:F
799:A
779:F
752:}
749:A
743:x
740::
737:)
734:x
731:(
726:P
722:F
718:{
694:]
691:A
688:[
683:P
679:F
656:P
652:F
631:A
611:B
591:y
588:=
585:)
582:x
579:(
574:P
570:F
549:A
543:x
523:B
517:y
497:y
477:B
457:)
454:y
451:,
448:x
445:(
442:P
420:y
417:=
414:)
411:x
408:(
403:P
399:F
376:P
372:F
351:)
348:y
345:,
342:x
339:(
336:P
316:y
296:x
268:P
252:.
240:B
220:F
200:A
180:]
177:A
174:[
171:F
83:)
77:(
72:)
68:(
54:.
20:)
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