1288:. However, this step alone takes one to theories of sets which are considered too weak. So some of the power of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset, replacement, and infinity) which may be regarded as special cases of comprehension. So far, these axioms do not seem to lead to any contradiction. Subsequently, the axiom of choice and the axiom of regularity were added to exclude models with some undesirable properties. These two axioms are known to be relatively consistent.
2512:
25:
1295:. The axiom of regularity together with the axiom of pairing also prohibit such a universal set. However, Russell's paradox yields a proof that there is no "set of all sets" using the axiom schema of separation alone, without any additional axioms. In particular, ZF without the axiom of regularity already prohibits such a universal set.
1298:
If a theory is extended by adding an axiom or axioms, then any (possibly undesirable) consequences of the original theory remain consequences of the extended theory. In particular, if ZF without regularity is extended by adding regularity to get ZF, then any contradiction (such as
Russell's paradox)
665:
will contain elements, called non-standard natural numbers, that satisfy the definition of natural numbers in that model but are not really natural numbers. They are "fake" natural numbers which are "larger" than any actual natural number. This model will contain infinite descending sequences of
1426:
wrote that "Zermelo's system has the notational advantage of not containing any explicitly typed variables, although in fact it can be seen as having an implicit type structure built into it, at least if the axiom of regularity is included. The details of this implicit typing are spelled out in
1367:, is equal to the class of all sets. This statement is even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted). From any model which does not satisfy the axiom of regularity, a model which satisfies it can be constructed by taking only sets in
213:
263:), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.
1050:
357:
1622:
1528:â ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets; in later papers Mirimanoff also explored what are now called
1182:
1402:
1361:
1454:. That was at the basis of both Russell's and Zermelo's intuitions. Indeed the best way to regard Zermelo's theory is as a simplification and extension of Russell's. (We mean Russell's
465:
1671:
816:
131:
1478:
In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchy turns out to be equivalent to ZF, including regularity.
818:. This is an unending descending sequence of elements. But this sequence is not definable in the model and thus not a set. So no contradiction to regularity can be proved.
756:
1546:) and in the same publication von Neumann gives an axiom (p. 412 in translation) which excludes some, but not all, non-well-founded sets. In a subsequent publication,
1735:
706:
976:
46:
2969:
1284:. In early formalizations of sets, mathematicians and logicians have avoided that contradiction by replacing the axiom schema of comprehension with the much weaker
1462:. Thus mixing of types is easier and annoying repetitions are avoided. Once the later types are allowed to accumulate the earlier ones, we can then easily imagine
1691:
297:
1564:
1643:
are objects that are not sets, but which can be elements of sets. In ZF set theory, there are no urelements, but in some other set theories such as
2287:
2260:
2173:
2122:
2081:
2060:
2039:
2018:
1314:}, i.e. have themselves as their only elements) is consistent with the theory obtained by removing the axiom of regularity from ZFC. Various
623:. This contradicts the fact that they are disjoint sets. Since our supposition led to a contradiction, there must not be any such function,
2658:
2478:
1234:, meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent. For his proof in modern notation see
2986:
1242:
274:. Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of
74:
2964:
2411:
2128:
100:
3119:
2738:
2617:
1466:
the types into the transfinite—just how far we want to go must necessarily be left open. Now
Russell made his types
2981:
2974:
630:
The nonexistence of a set containing itself can be seen as a special case where the sequence is infinite and constant.
2612:
2575:
1754:
1529:
1315:
1143:
540:
377:
1249:
in 1941, although he did not publish a proof until 1954. The proof involves (and led to the study of) Rieger-Bernays
1450:
The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of the
1370:
1329:
1285:
1273:
370:
2663:
2555:
2543:
2538:
608:
256:
3124:
2471:
2211:
Sangiorgi, Davide (2011), "Origins of bisimulation and coinduction", in
Sangiorgi, Davide; Rutten, Jan (eds.),
2183:
2093:(1917), "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles",
894:
291:
1318:
allow "safe" circular sets, such as Quine atoms, without becoming inconsistent by means of
Russell's paradox.
432:
3083:
3001:
2876:
2828:
2642:
2565:
2323:(1928), "Ăber die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre",
1277:
638:
3035:
2916:
2728:
2548:
2151:
488:
208:{\displaystyle \forall x\,(x\neq \varnothing \rightarrow \exists y(y\in x\ \land y\cap x=\varnothing )).}
2951:
2865:
2785:
2765:
2743:
287:
1650:
1291:
In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is no
765:
3025:
3015:
2849:
2780:
2733:
2673:
2560:
1487:
1364:
1281:
2156:
711:
361:
Given the other axioms of
ZermeloâFraenkel set theory, the axiom of regularity is equivalent to the
3020:
2931:
2844:
2839:
2834:
2648:
2528:
2464:
2244:, van Heijenoort, 1967, in English translation by Stefan Bauer-Mengelberg, pp. 291–301.
1045:{\displaystyle \textstyle \operatorname {rank} (w)=\cup \{\operatorname {rank} (z)+1\mid z\in w\}}
56:
2943:
2938:
2723:
2678:
2585:
2371:
2340:
1963:
1932:
1924:
1886:
1878:
1696:
1550:, p. 231) gave an equivalent but more complex version of the Axiom of Class Foundation, cf.
1184:, which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (
965:
is ranked. Applying the axioms of replacement and union to combine the ranks of the elements of
673:
2387:"Ăber Grenzzahlen und Mengenbereiche. Neue Untersuchungen ĂŒber die Grundlagen der Mengenlehre."
1901:
2800:
2637:
2629:
2600:
2570:
2501:
2283:
2256:
2169:
2118:
2090:
2077:
2056:
2035:
2014:
1764:
1491:
1423:
1250:
650:
362:
122:
107:
33:
3088:
3078:
3063:
3058:
2926:
2580:
2401:
2363:
2351:
2332:
2320:
2296:
2198:
2161:
1955:
1916:
1870:
1759:
1407:
1269:
1253:(or method), which were used for other proofs of independence for non-well-founded systems (
1116:
The axiom of dependent choice and no infinite descending sequence of sets implies regularity
402:
219:
2957:
2895:
2713:
2533:
1245:
from the other axioms of ZF(C), assuming they are consistent. The result was announced by
260:
223:
1647:, there are. In these theories, the axiom of regularity must be modified. The statement "
1214:. As this is an infinite descending chain, we arrive at a contradiction and so, no such
352:{\displaystyle \{(n,\alpha )\mid n\in \omega \land \alpha {\text{ is an ordinal }}\}\,.}
3093:
2890:
2871:
2775:
2760:
2717:
2653:
2595:
2248:
2232:
2150:, The Western Ontario Series in Philosophy of Science, vol. 75, pp. 171â187,
2069:
1676:
496:
279:
270:; it was adopted in a formulation closer to the one found in contemporary textbooks by
39:
2386:
2148:
Logic, Mathematics, Philosophy, Vintage
Enthusiasms. Essays in Honour of John L. Bell.
2107:
3113:
3098:
3068:
2900:
2814:
2809:
2382:
2375:
2344:
2048:
1943:
1936:
1890:
1742:
1431:
1414:, p. 206) wrote that "The idea of rank is a descendant of Russell's concept of
1292:
662:
476:
283:
114:
2450:
2225:
Axiomatic set theory. Proceedings of
Symposia in Pure Mathematics Volume 13, Part II
3048:
3043:
2861:
2790:
2748:
2607:
2511:
1897:
1858:
1299:
which followed from the original theory would still follow in the extended theory.
1246:
847:
654:
366:
882:
This was actually the original form of the axiom in von
Neumann's axiomatization.
365:. The axiom of induction tends to be used in place of the axiom of regularity in
3073:
2708:
2354:(1929), "Uber eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre",
2165:
2027:
1617:{\displaystyle A\neq \emptyset \rightarrow \exists x\in A\,(x\cap A=\emptyset )}
1419:
850:
for specifics. This definition eliminates one pair of braces from the canonical
88:
3053:
2921:
2824:
2487:
2446:
2440:
2367:
2269:
Urquhart, Alasdair (2003), "The Theory of Types", in
Griffin, Nicholas (ed.),
2220:
1640:
1439:
1303:
1088:} (which exists by the axiom of pairing). We see there must be an element of {
851:
380:
have indeed postulated the existence of sets that are elements of themselves.
2203:
2856:
2819:
2770:
2668:
2423:
From Kant to
Hilbert: A Source Book in the Foundations of Mathematics Vol. 2
1644:
104:
2436:
1542:
pointed out that non-well-founded sets are superfluous (on p. 404 in
1124:
be a counter-example to the axiom of regularity; that is, every element of
2406:
2140:
227:
1226:
Regularity was shown to be relatively consistent with the rest of ZF by
2336:
2115:
One Hundred Years of Russell Ìs Paradox: Mathematics, Logic, Philosophy
1967:
1928:
1882:
917:
is ranked and we are done. Otherwise, apply the axiom of regularity to
637:
that can be represented as sets as opposed to undefinable classes. The
1458:
theory of types, of course.) The simplification was to make the types
2881:
2703:
1959:
1920:
1874:
2313:
From Frege to Gödel: A Source Book in Mathematical Logic, 1879â1931
661:, then it will also satisfy the axiom of regularity. The resulting
2753:
2520:
1693:
is not empty and is not an urelement. One suitable replacement is
282:
easier to prove; and it not only allows induction to be done on
2460:
2002:
Combinatorial Set Theory: With a Gentle Introduction to Forcing
2032:
Set Theory: The Third Millennium Edition, Revised and Expanded
646:
18:
2456:
2146:, in DeVidi, David; Hallett, Michael; Clark, Peter (eds.),
1068:
For every two sets, only one can be an element of the other
826:
The axiom of regularity enables defining the ordered pair (
645:, satisfy the axiom of regularity (and all other axioms of
1510:"regular" (French: "ordinaire") if every descending chain
1104:. By the definition of disjoint then, we must have either
1080:
be sets. Then apply the axiom of regularity to the set {
1628:
The contemporary and final form of the axiom is due to
51:
980:
1861:(1941), "A system of axiomatic set theory. Part II",
1699:
1679:
1653:
1567:
1373:
1332:
1146:
979:
768:
714:
676:
435:
300:
134:
1096:} which is also disjoint from it. It must be either
822:
Simpler set-theoretic definition of the ordered pair
633:
Notice that this argument only applies to functions
3034:
2997:
2909:
2799:
2687:
2628:
2519:
2494:
1506:, §4.4, esp. p. 186, 188). Mirimanoff called a set
2053:Set Theory: An Introduction to Independence Proofs
1729:
1685:
1665:
1616:
1396:
1355:
1176:
1044:
810:
750:
700:
459:
351:
207:
2108:"Predicativity, Circularity, and Anti-Foundation"
376:In addition to omitting the axiom of regularity,
2299:(1925), "Eine Axiomatisierung der Mengenlehre",
2184:"A contribution to Gödel's axiomatic set theory"
1532:("extraordinaire" in Mirimanoff's terminology).
1177:{\displaystyle aRb:\Leftrightarrow b\in S\cap a}
397:be a set, and apply the axiom of regularity to {
278:. However, regularity makes some properties of
2356:Journal fĂŒr die Reine und Angewandte Mathematik
2301:Journal fĂŒr die Reine und Angewandte Mathematik
2213:Advanced Topics in Bisimulation and Coinduction
1448:
1322:Regularity, the cumulative hierarchy, and types
953:by the definition of transitive closure. Since
1543:
483:No infinite descending sequence of sets exists
2472:
1673:" needs to be replaced with a statement that
1397:{\displaystyle \bigcup _{\alpha }V_{\alpha }}
1356:{\displaystyle \bigcup _{\alpha }V_{\alpha }}
1241:The axiom of regularity was also shown to be
8:
1902:"A system of axiomatic set theory. Part VII"
1038:
1002:
448:
442:
373:), where the two axioms are not equivalent.
342:
301:
2271:The Cambridge Companion to Bertrand Russell
2011:Cantorian set theory and limitation of size
1946:(1971), "The iterative conception of set",
1547:
1539:
1231:
758:, and so on. For any actual natural number
405:. We see that there must be an element of {
267:
2479:
2465:
2457:
2141:"Paradox, ZF, and the Axiom of Foundation"
1977:, Harvard University Press, pp. 13â29
1495:
487:Suppose, to the contrary, that there is a
218:The axiom of regularity together with the
2405:
2202:
2155:
2076:, Mineola, New York: Dover Publications,
1830:
1818:
1698:
1678:
1652:
1592:
1566:
1388:
1378:
1372:
1347:
1337:
1331:
1145:
978:
767:
713:
675:
539:, which can be seen to be a set from the
434:
345:
337:
299:
141:
133:
75:Learn how and when to remove this message
1794:
1636:Regularity in the presence of urelements
1411:
1306:(sets that satisfy the formula equation
460:{\displaystyle A\cap \{A\}=\varnothing }
1775:
1629:
1503:
1430:, and again in a well-known article of
1428:
1258:
1254:
1222:Regularity and the rest of ZF(C) axioms
1052:. This contradicts the conclusion that
670:is a non-standard natural number, then
454:
271:
193:
151:
1842:
1782:
1551:
1535:
1470:in his notation and Zermelo left them
1434:
1326:In ZF it can be proven that the class
1235:
1227:
543:. Applying the axiom of regularity to
369:theories (ones that do not accept the
1443:
384:Elementary implications of regularity
275:
55:. Parenthetical referencing has been
7:
1806:
1555:
1499:
1486:The concept of well-foundedness and
1056:is unranked. So the assumption that
286:but also on proper classes that are
2425:, Clarendon Press, pp. 1219â33
2223:(1974), "Axiomatizing set theory",
1703:
1660:
1608:
1580:
1574:
1128:has a non-empty intersection with
288:well-founded relational structures
157:
135:
14:
2191:Czechoslovak Mathematical Journal
1490:of a set were both introduced by
1446:) went further and claimed that:
653:). So if one forms a non-trivial
266:The axiom is the contribution of
259:(which is a weakened form of the
59:; convert to shortened footnotes.
2510:
1666:{\displaystyle x\neq \emptyset }
1265:Regularity and Russell's paradox
1060:was non-empty must be false and
909:consisting of unranked sets. If
811:{\displaystyle (n-k-1)\in (n-k)}
595:+1) which is also an element of
535:a natural number}, the range of
226:, and that there is no infinite
23:
2417:from the original on 2022-10-09
2134:from the original on 2022-10-09
666:elements. For example, suppose
1724:
1712:
1709:
1700:
1611:
1593:
1577:
1132:. We define a binary relation
1017:
1011:
993:
987:
805:
793:
787:
769:
751:{\displaystyle (n-2)\in (n-1)}
745:
733:
727:
715:
689:
677:
413:}. Since the only element of {
389:No set is an element of itself
316:
304:
224:no set is an element of itself
199:
196:
163:
154:
142:
1:
2311:van Heijenoort, Jean (1967),
2000:Halbeisen, Lorenz J. (2012),
1982:Enderton, Herbert B. (1977),
1909:The Journal of Symbolic Logic
1863:The Journal of Symbolic Logic
969:, we get an ordinal rank for
878:Every set has an ordinal rank
579:. However, we are given that
2273:, Cambridge University Press
2255:, Dover Publications, Inc.,
2215:, Cambridge University Press
1995:, Cambridge University Press
1544:van Heijenoort's translation
1316:non-wellfounded set theories
113:contains an element that is
2280:Set Theory: An Introduction
2166:10.1007/978-94-007-0214-1_9
2113:, in Link, Godehard (ed.),
2095:L'Enseignement Mathématique
2013:, Oxford University Press,
1755:Non-well-founded set theory
1730:{\displaystyle (\exists y)}
541:axiom schema of replacement
101:ZermeloâFraenkel set theory
3141:
2970:von NeumannâBernaysâGödel
2282:(2nd ed.), Springer,
2278:Vaught, Robert L. (2001),
2009:Hallett, Michael (1996) ,
1286:axiom schema of separation
1274:unrestricted comprehension
701:{\displaystyle (n-1)\in n}
575:) for some natural number
409:} which is disjoint from {
371:law of the excluded middle
2771:One-to-one correspondence
2508:
2421:Ewald, W.B., ed. (1996),
2368:10.1515/crll.1929.160.227
1993:Logic, induction and sets
1280:) is inconsistent due to
401:}, which is a set by the
378:non-standard set theories
339: is an ordinal
257:axiom of dependent choice
2204:10.21136/CMJ.1957.100254
639:hereditarily finite sets
292:lexicographical ordering
2451:the axiom of foundation
2394:Fundamenta Mathematicae
1973:Boolos, George (1998),
1278:axiom of extensionality
1264:
1238:, §10.1) for instance.
929:which is disjoint from
559:. By the definition of
555:which is disjoint from
103:that states that every
2729:Constructible universe
2556:Constructibility (V=L)
2237:Axiomatized set theory
1984:Elements of Set Theory
1975:Logic, Logic and Logic
1731:
1687:
1667:
1618:
1476:
1398:
1357:
1178:
1120:Let the non-empty set
1046:
812:
752:
702:
475:(by the definition of
461:
353:
209:
32:This article includes
2952:Principia Mathematica
2786:Transfinite induction
2645:(i.e. set difference)
2407:10.4064/fm-16-1-29-47
2325:Mathematische Annalen
2139:Rieger, Adam (2011),
2117:, Walter de Gruyter,
1948:Journal of Philosophy
1833:, pp. 17â19, 26.
1732:
1688:
1668:
1619:
1530:non-well-founded sets
1418:". Comparing ZF with
1399:
1358:
1272:(the axiom schema of
1261:, pp. 210â212).
1179:
1108:is not an element of
1047:
813:
753:
703:
462:
354:
210:
3120:Axioms of set theory
3026:Burali-Forti paradox
2781:Set-builder notation
2734:Continuum hypothesis
2674:Symmetric difference
2253:Axiomatic Set Theory
2182:Riegger, L. (1957),
2106:Rathjen, M. (2004),
1991:Forster, T. (2003),
1785:, pp. 175, 178.
1737:, which states that
1697:
1677:
1651:
1565:
1408:Herbert Enderton
1371:
1365:von Neumann universe
1330:
1144:
977:
766:
712:
674:
471:the only element of
433:
298:
132:
2987:TarskiâGrothendieck
2437:Axiom of foundation
2242:From Frege to Gödel
2221:Scott, Dana Stewart
1898:Bernays, Paul Isaac
1859:Bernays, Paul Isaac
961:, every element of
125:, the axiom reads:
97:axiom of foundation
95:(also known as the
93:axiom of regularity
16:Axiom of set theory
2576:Limitation of size
2337:10.1007/BF01459102
2315:, pp. 393â413
2227:, pp. 207â214
2091:Mirimanoff, Dmitry
1727:
1683:
1663:
1614:
1554:, p. 53) and
1540:von Neumann (1925)
1502:, p. 68) and
1394:
1383:
1353:
1342:
1257:, p. 193 and
1251:permutation models
1232:von Neumann (1929)
1174:
1042:
1041:
921:to get an element
895:transitive closure
808:
748:
698:
507:+1) an element of
457:
425:is disjoint from {
421:, it must be that
363:axiom of induction
349:
268:von Neumann (1925)
205:
40:properly formatted
3107:
3106:
3016:Russell's paradox
2965:ZermeloâFraenkel
2866:Dedekind-infinite
2739:Diagonal argument
2638:Cartesian product
2502:Set (mathematics)
2419:; translation in
2352:von Neumann, John
2321:von Neumann, John
2309:; translation in
2297:von Neumann, John
2289:978-0-8176-4256-3
2262:978-0-486-61630-8
2175:978-94-007-0213-4
2124:978-3-11-019968-0
2083:978-0-486-42079-0
2062:978-0-444-86839-8
2041:978-3-540-44085-7
2020:978-0-19-853283-5
1821:, pp. 62â63.
1765:Epsilon-induction
1686:{\displaystyle x}
1548:von Neumann (1929
1492:Dmitry Mirimanoff
1424:Alasdair Urquhart
1374:
1333:
1302:The existence of
1282:Russell's paradox
957:is disjoint from
905:be the subset of
651:axiom of infinity
551:be an element of
467:, we cannot have
340:
284:well-ordered sets
244:is an element of
177:
123:first-order logic
99:) is an axiom of
85:
84:
77:
3132:
3089:Bertrand Russell
3079:John von Neumann
3064:Abraham Fraenkel
3059:Richard Dedekind
3021:Suslin's problem
2932:Cantor's theorem
2649:De Morgan's laws
2514:
2481:
2474:
2467:
2458:
2426:
2418:
2416:
2409:
2391:
2378:
2362:(160): 227â241,
2347:
2316:
2308:
2292:
2274:
2265:
2239:
2228:
2216:
2207:
2206:
2188:
2178:
2159:
2145:
2135:
2133:
2112:
2102:
2086:
2074:Basic set theory
2065:
2044:
2023:
2005:
1996:
1987:
1986:, Academic Press
1978:
1970:
1939:
1906:
1893:
1846:
1840:
1834:
1828:
1822:
1816:
1810:
1804:
1798:
1792:
1786:
1780:
1736:
1734:
1733:
1728:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1623:
1621:
1620:
1615:
1558:, p. 72):
1403:
1401:
1400:
1395:
1393:
1392:
1382:
1362:
1360:
1359:
1354:
1352:
1351:
1341:
1270:Naive set theory
1183:
1181:
1180:
1175:
1064:must have rank.
1051:
1049:
1048:
1043:
889:is any set. Let
817:
815:
814:
809:
757:
755:
754:
749:
707:
705:
704:
699:
466:
464:
463:
458:
403:axiom of pairing
358:
356:
355:
350:
341:
338:
220:axiom of pairing
214:
212:
211:
206:
175:
80:
73:
69:
66:
60:
54:
49:this article by
34:inline citations
27:
26:
19:
3140:
3139:
3135:
3134:
3133:
3131:
3130:
3129:
3125:Wellfoundedness
3110:
3109:
3108:
3103:
3030:
3009:
2993:
2958:New Foundations
2905:
2795:
2714:Cardinal number
2697:
2683:
2624:
2515:
2506:
2490:
2485:
2433:
2420:
2414:
2389:
2381:
2350:
2319:
2310:
2295:
2290:
2277:
2268:
2263:
2249:Suppes, Patrick
2247:
2233:Skolem, Thoralf
2231:
2219:
2210:
2186:
2181:
2176:
2157:10.1.1.100.9052
2143:
2138:
2131:
2125:
2110:
2105:
2089:
2084:
2068:
2063:
2047:
2042:
2026:
2021:
2008:
1999:
1990:
1981:
1972:
1960:10.2307/2025204
1942:
1921:10.2307/2268864
1904:
1896:
1875:10.2307/2267281
1857:
1854:
1849:
1841:
1837:
1829:
1825:
1817:
1813:
1805:
1801:
1793:
1789:
1781:
1777:
1773:
1751:
1695:
1694:
1675:
1674:
1649:
1648:
1638:
1563:
1562:
1527:
1520:
1484:
1452:theory of types
1384:
1369:
1368:
1343:
1328:
1327:
1324:
1293:set of all sets
1267:
1224:
1204:
1200:
1189:
1142:
1141:
1118:
1112:or vice versa.
1070:
975:
974:
949:is a subset of
913:is empty, then
880:
824:
764:
763:
710:
709:
672:
671:
660:
644:
497:natural numbers
485:
431:
430:
391:
386:
296:
295:
261:axiom of choice
249:
242:
235:
130:
129:
81:
70:
64:
61:
52:correcting them
50:
44:
28:
24:
17:
12:
11:
5:
3138:
3136:
3128:
3127:
3122:
3112:
3111:
3105:
3104:
3102:
3101:
3096:
3094:Thoralf Skolem
3091:
3086:
3081:
3076:
3071:
3066:
3061:
3056:
3051:
3046:
3040:
3038:
3032:
3031:
3029:
3028:
3023:
3018:
3012:
3010:
3008:
3007:
3004:
2998:
2995:
2994:
2992:
2991:
2990:
2989:
2984:
2979:
2978:
2977:
2962:
2961:
2960:
2948:
2947:
2946:
2935:
2934:
2929:
2924:
2919:
2913:
2911:
2907:
2906:
2904:
2903:
2898:
2893:
2888:
2879:
2874:
2869:
2859:
2854:
2853:
2852:
2847:
2842:
2832:
2822:
2817:
2812:
2806:
2804:
2797:
2796:
2794:
2793:
2788:
2783:
2778:
2776:Ordinal number
2773:
2768:
2763:
2758:
2757:
2756:
2751:
2741:
2736:
2731:
2726:
2721:
2711:
2706:
2700:
2698:
2696:
2695:
2692:
2688:
2685:
2684:
2682:
2681:
2676:
2671:
2666:
2661:
2656:
2654:Disjoint union
2651:
2646:
2640:
2634:
2632:
2626:
2625:
2623:
2622:
2621:
2620:
2615:
2604:
2603:
2601:Martin's axiom
2598:
2593:
2588:
2583:
2578:
2573:
2568:
2566:Extensionality
2563:
2558:
2553:
2552:
2551:
2546:
2541:
2531:
2525:
2523:
2517:
2516:
2509:
2507:
2505:
2504:
2498:
2496:
2492:
2491:
2486:
2484:
2483:
2476:
2469:
2461:
2455:
2454:
2444:
2432:
2431:External links
2429:
2428:
2427:
2383:Zermelo, Ernst
2379:
2348:
2317:
2293:
2288:
2275:
2266:
2261:
2245:
2229:
2217:
2208:
2197:(3): 323â357,
2179:
2174:
2136:
2123:
2103:
2087:
2082:
2066:
2061:
2049:Kunen, Kenneth
2045:
2040:
2024:
2019:
2006:
1997:
1988:
1979:
1954:(8): 215â231,
1944:Boolos, George
1940:
1894:
1853:
1850:
1848:
1847:
1845:, p. 179.
1835:
1831:Sangiorgi 2011
1823:
1819:Halbeisen 2012
1811:
1799:
1797:, p. 305.
1787:
1774:
1772:
1769:
1768:
1767:
1762:
1757:
1750:
1747:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1682:
1662:
1659:
1656:
1637:
1634:
1630:Zermelo (1930)
1626:
1625:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1525:
1518:
1483:
1480:
1440:Dana Scott
1391:
1387:
1381:
1377:
1350:
1346:
1340:
1336:
1323:
1320:
1310: = {
1266:
1263:
1223:
1220:
1202:
1198:
1187:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1117:
1114:
1069:
1066:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
879:
876:
823:
820:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
697:
694:
691:
688:
685:
682:
679:
658:
642:
607:+1) is in the
484:
481:
456:
453:
450:
447:
444:
441:
438:
390:
387:
385:
382:
367:intuitionistic
348:
344:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
272:Zermelo (1930)
247:
240:
233:
216:
215:
204:
201:
198:
195:
192:
189:
186:
183:
180:
174:
171:
168:
165:
162:
159:
156:
153:
150:
147:
144:
140:
137:
83:
82:
65:September 2020
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3137:
3126:
3123:
3121:
3118:
3117:
3115:
3100:
3099:Ernst Zermelo
3097:
3095:
3092:
3090:
3087:
3085:
3084:Willard Quine
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3041:
3039:
3037:
3036:Set theorists
3033:
3027:
3024:
3022:
3019:
3017:
3014:
3013:
3011:
3005:
3003:
3000:
2999:
2996:
2988:
2985:
2983:
2982:KripkeâPlatek
2980:
2976:
2973:
2972:
2971:
2968:
2967:
2966:
2963:
2959:
2956:
2955:
2954:
2953:
2949:
2945:
2942:
2941:
2940:
2937:
2936:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2914:
2912:
2908:
2902:
2899:
2897:
2894:
2892:
2889:
2887:
2885:
2880:
2878:
2875:
2873:
2870:
2867:
2863:
2860:
2858:
2855:
2851:
2848:
2846:
2843:
2841:
2838:
2837:
2836:
2833:
2830:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2807:
2805:
2802:
2798:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2755:
2752:
2750:
2747:
2746:
2745:
2742:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2719:
2715:
2712:
2710:
2707:
2705:
2702:
2701:
2699:
2693:
2690:
2689:
2686:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2644:
2641:
2639:
2636:
2635:
2633:
2631:
2627:
2619:
2618:specification
2616:
2614:
2611:
2610:
2609:
2606:
2605:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2550:
2547:
2545:
2542:
2540:
2537:
2536:
2535:
2532:
2530:
2527:
2526:
2524:
2522:
2518:
2513:
2503:
2500:
2499:
2497:
2493:
2489:
2482:
2477:
2475:
2470:
2468:
2463:
2462:
2459:
2452:
2448:
2447:Inhabited set
2445:
2442:
2438:
2435:
2434:
2430:
2424:
2413:
2408:
2403:
2399:
2395:
2388:
2384:
2380:
2377:
2373:
2369:
2365:
2361:
2357:
2353:
2349:
2346:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2314:
2306:
2302:
2298:
2294:
2291:
2285:
2281:
2276:
2272:
2267:
2264:
2258:
2254:
2250:
2246:
2243:
2240:Reprinted in
2238:
2234:
2230:
2226:
2222:
2218:
2214:
2209:
2205:
2200:
2196:
2192:
2185:
2180:
2177:
2171:
2167:
2163:
2158:
2153:
2149:
2142:
2137:
2130:
2126:
2120:
2116:
2109:
2104:
2100:
2096:
2092:
2088:
2085:
2079:
2075:
2071:
2067:
2064:
2058:
2054:
2050:
2046:
2043:
2037:
2033:
2029:
2025:
2022:
2016:
2012:
2007:
2003:
1998:
1994:
1989:
1985:
1980:
1976:
1971:reprinted in
1969:
1965:
1961:
1957:
1953:
1949:
1945:
1941:
1938:
1934:
1930:
1926:
1922:
1918:
1914:
1910:
1903:
1899:
1895:
1892:
1888:
1884:
1880:
1876:
1872:
1868:
1864:
1860:
1856:
1855:
1851:
1844:
1839:
1836:
1832:
1827:
1824:
1820:
1815:
1812:
1809:, p. 73.
1808:
1803:
1800:
1796:
1795:Urquhart 2003
1791:
1788:
1784:
1779:
1776:
1770:
1766:
1763:
1761:
1760:Scott's trick
1758:
1756:
1753:
1752:
1748:
1746:
1744:
1740:
1721:
1718:
1715:
1706:
1680:
1657:
1654:
1646:
1642:
1635:
1633:
1631:
1605:
1602:
1599:
1596:
1589:
1586:
1583:
1571:
1568:
1561:
1560:
1559:
1557:
1553:
1549:
1545:
1541:
1537:
1536:Skolem (1923)
1533:
1531:
1524:
1517:
1513:
1509:
1505:
1504:Hallett (1996
1501:
1497:
1493:
1489:
1481:
1479:
1475:
1473:
1469:
1465:
1461:
1457:
1453:
1447:
1445:
1441:
1437:
1435:
1433:
1432:George Boolos
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1389:
1385:
1379:
1375:
1366:
1363:, called the
1348:
1344:
1338:
1334:
1321:
1319:
1317:
1313:
1309:
1305:
1300:
1296:
1294:
1289:
1287:
1283:
1279:
1275:
1271:
1262:
1260:
1256:
1252:
1248:
1244:
1239:
1237:
1233:
1229:
1228:Skolem (1923)
1221:
1219:
1217:
1213:
1209:
1205:
1194:
1190:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1139:
1135:
1131:
1127:
1123:
1115:
1113:
1111:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1067:
1065:
1063:
1059:
1055:
1035:
1032:
1029:
1026:
1023:
1020:
1014:
1008:
1005:
999:
996:
990:
984:
981:
972:
968:
964:
960:
956:
952:
948:
945:is unranked.
944:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
888:
883:
877:
875:
873:
869:
865:
861:
857:
853:
849:
845:
841:
837:
833:
829:
821:
819:
802:
799:
796:
790:
784:
781:
778:
775:
772:
761:
742:
739:
736:
730:
724:
721:
718:
695:
692:
686:
683:
680:
669:
664:
656:
652:
648:
640:
636:
631:
628:
626:
622:
618:
614:
610:
606:
602:
598:
594:
590:
586:
582:
578:
574:
570:
566:
562:
558:
554:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
490:
482:
480:
478:
474:
470:
451:
445:
439:
436:
429:}. So, since
428:
424:
420:
416:
412:
408:
404:
400:
396:
388:
383:
381:
379:
374:
372:
368:
364:
359:
346:
334:
331:
328:
325:
322:
319:
313:
310:
307:
293:
289:
285:
281:
277:
273:
269:
264:
262:
258:
254:
250:
243:
236:
229:
225:
222:implies that
221:
202:
190:
187:
184:
181:
178:
172:
169:
166:
160:
148:
145:
138:
128:
127:
126:
124:
120:
116:
112:
109:
106:
102:
98:
94:
90:
79:
76:
68:
58:
53:
48:
43:
41:
38:they are not
35:
30:
21:
20:
3049:Georg Cantor
3044:Paul Bernays
2975:MorseâKelley
2950:
2883:
2882:Subset
2829:hereditarily
2791:Venn diagram
2749:ordered pair
2664:Intersection
2608:Axiom schema
2590:
2422:
2397:
2393:
2359:
2355:
2328:
2324:
2312:
2304:
2300:
2279:
2270:
2252:
2241:
2236:
2224:
2212:
2194:
2190:
2147:
2114:
2098:
2094:
2073:
2070:LĂ©vy, Azriel
2055:, Elsevier,
2052:
2034:, Springer,
2031:
2028:Jech, Thomas
2010:
2001:
1992:
1983:
1974:
1951:
1947:
1915:(2): 81â96,
1912:
1908:
1866:
1862:
1838:
1826:
1814:
1802:
1790:
1778:
1738:
1639:
1627:
1552:Suppes (1972
1534:
1522:
1515:
1511:
1507:
1485:
1477:
1471:
1467:
1463:
1459:
1455:
1451:
1449:
1438:
1415:
1406:
1325:
1311:
1307:
1301:
1297:
1290:
1268:
1259:Forster 2003
1255:Rathjen 2004
1247:Paul Bernays
1240:
1236:Vaught (2001
1225:
1215:
1211:
1207:
1196:
1192:
1185:
1137:
1133:
1129:
1125:
1121:
1119:
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1071:
1061:
1057:
1053:
970:
966:
962:
958:
954:
950:
946:
942:
938:
934:
930:
926:
922:
918:
914:
910:
906:
902:
898:
890:
886:
884:
881:
871:
867:
863:
859:
855:
854:definition (
848:ordered pair
843:
839:
835:
831:
827:
825:
759:
667:
634:
632:
629:
624:
620:
616:
612:
609:intersection
604:
600:
596:
592:
588:
584:
580:
576:
572:
568:
564:
560:
556:
552:
548:
544:
536:
532:
528:
524:
520:
516:
512:
508:
504:
500:
492:
486:
472:
468:
426:
422:
418:
414:
410:
406:
398:
394:
392:
375:
360:
290:such as the
276:Kunen (1980)
265:
252:
245:
238:
237:) such that
231:
217:
118:
110:
96:
92:
86:
71:
62:
37:
3074:Thomas Jech
2917:Alternative
2896:Uncountable
2850:Ultrafilter
2709:Cardinality
2613:replacement
2561:Determinacy
2331:: 373â391,
1869:(1): 1â17,
1843:Rieger 2011
1783:Rieger 2011
1420:type theory
1304:Quine atoms
1243:independent
1195:satisfying
649:except the
587:) contains
515:) for each
255:. With the
89:mathematics
3114:Categories
3069:Kurt Gödel
3054:Paul Cohen
2891:Transitive
2659:Identities
2643:Complement
2630:Operations
2591:Regularity
2529:Adjunction
2488:Set theory
2441:PlanetMath
2004:, Springer
1771:References
1641:Urelements
1556:LĂ©vy (2002
1500:LĂ©vy (2002
1460:cumulative
1157::⇔
852:Kuratowski
655:ultrapower
57:deprecated
3002:Paradoxes
2922:Axiomatic
2901:Universal
2877:Singleton
2872:Recursive
2815:Countable
2810:Amorphous
2669:Power set
2586:Power set
2544:dependent
2539:countable
2400:: 29â47,
2376:199545822
2345:120784562
2307:: 219â240
2251:(1972) ,
2152:CiteSeerX
2072:(2002) ,
1937:250351655
1891:250344277
1807:LĂ©vy 2002
1743:inhabited
1719:∈
1704:∃
1661:∅
1658:≠
1609:∅
1600:∩
1587:∈
1581:∃
1578:→
1575:∅
1572:≠
1464:extending
1390:α
1380:α
1376:⋃
1349:α
1339:α
1335:⋃
1169:∩
1163:∈
1033:∈
1027:∣
1009:
1000:∪
985:
973:, to wit
800:−
791:∈
782:−
776:−
740:−
731:∈
722:−
693:∈
684:−
519:. Define
495:, on the
455:∅
440:∩
335:α
332:∧
329:ω
326:∈
320:∣
314:α
194:∅
185:∩
179:∧
170:∈
158:∃
155:→
152:∅
149:≠
136:∀
105:non-empty
3006:Problems
2910:Theories
2886:Superset
2862:Infinite
2691:Concepts
2571:Infinity
2495:Overview
2412:archived
2385:(1930),
2235:(1923),
2129:archived
2051:(1980),
2030:(2003),
1900:(1954),
1749:See also
1472:implicit
1468:explicit
1276:and the
1218:exists.
1206:for all
933:. Since
885:Suppose
846:}}; see
567:must be
489:function
477:disjoint
280:ordinals
251:for all
228:sequence
115:disjoint
2944:General
2939:Zermelo
2845:subbase
2827: (
2766:Forcing
2744:Element
2716: (
2694:Methods
2581:Pairing
2453:on nLab
2101:: 37â52
1968:2025204
1929:2268864
1883:2267281
1852:Sources
1482:History
1442: (
1410: (
901:}. Let
893:be the
47:improve
45:Please
2835:Filter
2825:Finite
2761:Family
2704:Almost
2549:global
2534:Choice
2521:Axioms
2374:
2343:
2286:
2259:
2172:
2154:
2121:
2080:
2059:
2038:
2017:
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1935:
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1456:simple
937:is in
862:) = {{
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547:, let
176:
91:, the
36:, but
2927:Naive
2857:Fuzzy
2820:Empty
2803:types
2754:tuple
2724:Class
2718:large
2679:Union
2596:Union
2415:(PDF)
2390:(PDF)
2372:S2CID
2341:S2CID
2187:(PDF)
2144:(PDF)
2132:(PDF)
2111:(PDF)
1964:JSTOR
1933:S2CID
1925:JSTOR
1905:(PDF)
1887:S2CID
1879:JSTOR
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499:with
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117:from
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2449:and
2360:1929
2284:ISBN
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2170:ISBN
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2078:ISBN
2057:ISBN
2036:ISBN
2015:ISBN
1538:and
1496:1917
1488:rank
1444:1974
1416:type
1412:1977
1230:and
1076:and
1072:Let
1006:rank
982:rank
897:of {
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708:and
657:of V
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2439:at
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1956:doi
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1436:."
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1100:or
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866:},{
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641:, V
611:of
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