2518:, von Neumann called the "overall effect of their activity . . . devastating". With regards to the axiomatic method employed by second group composed of Zermelo, Fraenkel and Schoenflies, von Neumann worried that "We see only that the known modes of inference leading to the antinomies fail, but who knows where there are not others?" and he set to the task, "in the spirit of the second group", to "produce, by means of a finite number of purely formal operations . . . all the sets that we want to see formed" but not allow for the antinomies. (All quotes from von Neumann 1925 reprinted in van Heijenoort, Jean (1967, third printing 1976),
6763:
4967:
2440:
231:
3801:
1150:
7763:
4979:
47:
2129:. He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical
7773:
1888:, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).
7783:
5003:
4991:
392:
1168:
of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von
Neumann universe are
2501:
of the set of all sets that do not contain themselves (Russell), of the set of all transfinite ordinal numbers (Burali-Forti), and the set of all finitely definable real numbers (Richard)." He goes on to observe that two "tendencies" were attempting to "rehabilitate" set theory. Of the first effort,
2220:
can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for
3175:: "When we prove a theorem or decide a proposition, we operate in a purely formal, syntactical manner. In doing mathematics, we do not discover pre-existing truths that were 'already there without one knowing' (PG 481)âwe invent mathematics, bit-by-little-bit." Note, however, that Wittgenstein does
1654:
is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since
2022:
of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set
1935:
refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption
334:
Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of
ZermeloâFraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of
2137:. Meta-mathematical statements â which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory â are not mathematics. Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in
3207:
of techniques of proof' (RFM III, §46), it does not require a foundation (RFM VII, §16) and it cannot be given a self-evident foundation (PR §160; WVC 34 & 62; RFM IV, §3). Since set theory was invented to provide mathematics with a foundation, it is, minimally,
1983:
fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the
1876:
constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.
1940:(AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the
2115:, into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted ZermeloâFraenkel theory, is much greater than that of constructive mathematics, to the point that
1788:. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.
1327:. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are
2079:
and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by
1500:, are not based on a cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the
1161:
if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the
2044:
that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the
498:
is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. Since sets are objects, the membership relation can relate sets as well.
1831:
in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.
1860:
developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model
3254:
Ferro, Alfredo; Omodeo, Eugenio G.; Schwartz, Jacob T. (September 1980), "Decision
Procedures for Elementary Sublanguages of Set Theory. I. Multi-Level Syllogistic and Some Extensions",
297:
in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of
2883:
1619:, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using
1099:
1509:
406:
2248:
students, but was met with much criticism. The math syllabus in
European schools followed this trend, and currently includes the subject at different levels in all grades.
2402:
2376:
2350:
1273:
2049:, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.
3651:. 3 vols., 2010. Each chapter surveys some aspect of contemporary research in set theory. Does not cover established elementary set theory, on which see Devlin (1993).
1293:
1246:
1198:
1914:, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in
1313:
1222:
562:
is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term
1224:
is defined to be the least ordinal that is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set
1769:
can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.
3224:: "An expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular number
401:
Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by
5142:
3190:
2185:
set theory. Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for
3414:
2644:, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich Frommann Verlag, p. 152,
2575:
282:, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory â as a branch of
2018:
is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of
5817:
4265:
1446:
4427:
3256:
258:
5900:
5041:
1635:
can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms.
1655:
such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project,
1578:
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as
3231:
2229:
As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics of
6799:
3723:
3162:
2139:
2009:
363:, and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for
3200:
428:
in the first half of the 19th century. Modern understanding of infinity began in 1870â1874, and was motivated by Cantor's work in
6214:
3211:
2530:(pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann's 1925 paper.
7516:
7488:
2177:
as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as
2075:
view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in
7541:
6372:
3600:
3573:
3548:
3524:
3502:
3394:
3332:
3039:
1936:
that a broader class of games is determined often implies that a broader class of sets will have a topological property. The
5160:
2144:
7392:
6227:
5550:
4790:
3954:
3767:
194:
2963:
7546:
6818:
3621:
3130:
2058:
1613:
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of
1504:
does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness,
2497:
observed that "set theory in its first, "naive" version, due to Cantor, led to contradictions. These are the well-known
7051:
6232:
6222:
5959:
5812:
5165:
4935:
4420:
4282:
1817:
In set theory as Cantor defined and
Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In
1462:
5156:
2907:
7698:
7526:
7056:
6368:
5007:
3731:
3482:
3445:
3423:
3303:
3154:
3105:
3076:
2676:
2649:
2622:
2527:
1173:, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by
251:
5710:
3241:
2163:
all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as
7786:
6880:
6465:
6209:
5034:
4471:
1993:
1773:
7174:
5770:
5463:
4885:
4260:
3854:
1915:
1672:
1358:
324:
5204:
4140:
3172:
7465:
7427:
7084:
6792:
6726:
6428:
6191:
6186:
6011:
5432:
5116:
4983:
2739:
1505:
1323:
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using
7607:
7584:
7314:
7304:
6721:
6504:
6421:
6134:
6065:
5942:
5184:
4413:
4034:
3913:
3683:
3665:
2096:
1450:
17:
3221:
3184:
7688:
7276:
7184:
7089:
6865:
6850:
6646:
6472:
6158:
5792:
5391:
4995:
4277:
3370:
1579:
1415:
352:
244:
3655:
2330:
are commonly referred to in mathematical teaching when talking about different types of numbers (the sets
7812:
7776:
7511:
7009:
6524:
6519:
6129:
5868:
5797:
5126:
5027:
4910:
4466:
4270:
3908:
3871:
3678:
3660:
2178:
2100:
2072:
1458:
1431:
1387:
371:. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the
87:
31:
3513:
1527:. Yet other systems accept classical logic but feature a nonstandard membership relation. These include
7748:
7397:
6453:
6043:
5437:
5405:
5096:
4481:
2217:
2064:
1958:
1792:
1427:
1391:
1071:
3925:
7766:
7693:
7668:
7531:
7179:
6785:
6743:
6692:
6589:
6087:
6048:
5525:
5170:
4895:
4867:
4504:
3959:
3844:
3832:
3827:
2463:
2308:
694:
368:
209:
199:
189:
55:
5199:
3712:
2570:
1721:
1475:, objects that can be members of sets but that are not themselves sets and do not have any members.
7617:
7450:
7036:
6905:
6584:
6514:
6053:
5905:
5888:
5611:
5091:
4940:
3760:
3290:, International Series of Monographs on Computer Science, Oxford Science Publications, Oxford, UK:
1785:
1700:
1659:, includes human-written, computer-verified derivations of more than 12,000 theorems starting from
2786:
2385:
2359:
2333:
2221:
a detailed analysis of the formulations required in order to derive various mathematical results.
7817:
7678:
7612:
7503:
7319:
6979:
6416:
6393:
6354:
6240:
6181:
5827:
5747:
5591:
5535:
5148:
4825:
4815:
4785:
4719:
4454:
4379:
4297:
4172:
4124:
3938:
3861:
3735:
3698:
3540:
1897:
1800:
1516:
1497:
1251:
826:
304:
3704:
3673:
1853:
that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the
7743:
7574:
7455:
7222:
7212:
7207:
6706:
6433:
6411:
6378:
6271:
6117:
6102:
6075:
6026:
5910:
5845:
5670:
5636:
5631:
5505:
5336:
5313:
4923:
4820:
4800:
4795:
4724:
4449:
4331:
4212:
4024:
3837:
3121:
2453:
2280:
2126:
2046:
2032:
1854:
1733:
1607:
204:
179:
82:
3694:
3384:
3025:
2731:
2724:
7822:
7713:
7683:
7673:
7569:
7483:
7359:
7299:
7266:
7256:
7139:
7104:
7094:
7031:
6900:
6875:
6870:
6835:
6636:
6489:
6281:
5999:
5735:
5641:
5500:
5485:
5366:
5341:
4950:
4880:
4757:
4681:
4620:
4605:
4600:
4577:
4459:
4247:
4161:
4081:
4061:
4039:
2409:
2234:
2209:
2197:
2130:
1963:
1953:
1907:
1680:
1639:
1615:
1278:
1231:
1183:
1174:
356:
214:
128:
2611:
7473:
7445:
7417:
7412:
7241:
7217:
7169:
7152:
7147:
7129:
7119:
7114:
7076:
7026:
7021:
6938:
6884:
6609:
6571:
6448:
6252:
6092:
6016:
5994:
5822:
5780:
5679:
5646:
5510:
5298:
5209:
4930:
4810:
4805:
4729:
4630:
4321:
4311:
4145:
4076:
4029:
3969:
3849:
3295:
3284:
2942:
2417:
2320:
2296:
2288:
2205:
2201:
1997:
1976:
1937:
1881:
1865:
1796:
1781:
1758:
1651:
1599:
1548:
1520:
1343:
1332:
1328:
1170:
1164:
1144:
882:
316:
308:
174:
133:
102:
3049:
2950:
2811:
635:, set theory features binary operations on sets. The following is a partial list of them:
8:
7807:
7738:
7663:
7579:
7564:
7329:
7109:
7066:
7061:
6958:
6948:
6920:
6738:
6629:
6614:
6594:
6551:
6438:
6388:
6314:
6259:
6196:
5989:
5984:
5932:
5700:
5689:
5361:
5261:
5189:
5180:
5176:
5111:
5106:
4945:
4855:
4777:
4676:
4610:
4567:
4557:
4537:
4316:
4227:
4135:
4130:
3944:
3886:
3817:
3753:
3180:
2692:
2425:
2122:
2112:
1911:
1906:
is a cardinal number with an extra property. Many such properties are studied, including
1717:
1713:
1668:
1563:
312:
1691:
Set theory is a major area of research in mathematics with many interrelated subfields:
7703:
7602:
7478:
7435:
7344:
7286:
7271:
7261:
7046:
6845:
6767:
6536:
6499:
6484:
6477:
6460:
6264:
6246:
6112:
6038:
6021:
5974:
5787:
5696:
5530:
5515:
5475:
5427:
5412:
5400:
5356:
5331:
5101:
5050:
4971:
4890:
4830:
4762:
4752:
4691:
4666:
4542:
4499:
4494:
4239:
4234:
4019:
3974:
3881:
3690:
3592:
3565:
3434:
2834:
2592:
2458:
2445:
2186:
2104:
1840:
1683:, enhancing the understanding of well-established models of evolution and interaction.
1624:
1595:
1587:
1419:
1401:
1397:
1383:
1298:
1207:
640:
417:
344:
275:
234:
138:
38:
5720:
3629:
2991:"Unifying evolutionary dynamics: a set theory exploration of symmetry and interaction"
1131:âthe unique set containing no elements. The empty set is also occasionally called the
7723:
7653:
7632:
7594:
7402:
7369:
7349:
7041:
6953:
6827:
6762:
6702:
6509:
6319:
6309:
6201:
6082:
5917:
5893:
5674:
5658:
5563:
5540:
5417:
5386:
5351:
5246:
5081:
4966:
4686:
4671:
4615:
4562:
4096:
3933:
3896:
3866:
3790:
3596:
3569:
3544:
3520:
3498:
3478:
3441:
3419:
3390:
3346:
3328:
3318:
3299:
3150:
3101:
3072:
3035:
2745:
2735:
2715:
2672:
2645:
2618:
2596:
2523:
2439:
2421:
2413:
2327:
2300:
2276:
2068:
1664:
1620:
986:
930:(elements which are in one of the sets, but not in both). For instance, for the sets
441:
279:
230:
59:
3030:, Springer Monographs in Mathematics (Third Millennium ed.), Berlin, New York:
2869:
2852:
2838:
2092:
7556:
7440:
7407:
7202:
7124:
7013:
6999:
6994:
6943:
6930:
6855:
6808:
6716:
6711:
6604:
6561:
6383:
6344:
6339:
6324:
6150:
6107:
6004:
5802:
5752:
5326:
5288:
4900:
4875:
4747:
4595:
4532:
4384:
4374:
4359:
4354:
4222:
3876:
3642:
3557:
3470:
3265:
3091:
3045:
2998:
2946:
2864:
2826:
2584:
2515:
2503:
2494:
2468:
2316:
2261:
2230:
2116:
2076:
2041:
1819:
1812:
1777:
1647:
1532:
1342:
The most widely studied systems of axiomatic set theory imply that all sets form a
628:
340:
320:
299:
290:
7627:
7521:
7493:
7387:
7339:
7324:
7309:
7164:
7159:
7099:
6989:
6963:
6915:
6860:
6697:
6687:
6641:
6624:
6579:
6541:
6443:
6363:
6170:
6097:
6070:
6058:
5964:
5878:
5852:
5807:
5775:
5576:
5378:
5321:
5271:
5236:
5194:
4840:
4767:
4696:
4489:
4253:
4191:
4009:
3822:
3638:
3586:
3534:
3492:
3322:
3291:
3125:
3095:
3066:
3031:
2938:
2637:
2353:
2213:
2170:
2160:
2156:
2108:
1989:
1980:
1869:
1762:
1754:
1676:
1524:
1501:
1480:
1423:
1369:
451:
445:
425:
328:
156:
3494:
Labyrinth of
Thought: A History of Set Theory and Its Role in Modern Mathematics
2762:
2507:
2204:. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with
7733:
7637:
7536:
7382:
7354:
6682:
6661:
6619:
6599:
6494:
6349:
5947:
5937:
5927:
5922:
5856:
5730:
5606:
5495:
5490:
5468:
5069:
4918:
4845:
4552:
4389:
4186:
4167:
4071:
4056:
4013:
3949:
3891:
3708:
2990:
2253:
2245:
2182:
2164:
1985:
1885:
1824:
1628:
1603:
1536:
1178:
1120:
748:
502:
A derived binary relation between two sets is the subset relation, also called
474:
380:
184:
3474:
3002:
2830:
2244:
experiment aimed to teach basic set theory, among other abstract concepts, to
286:â is mostly concerned with those that are relevant to mathematics as a whole.
7801:
7622:
6910:
6656:
6334:
5841:
5626:
5616:
5586:
5571:
5241:
4706:
4638:
4590:
4394:
4196:
4110:
4105:
3588:
The
Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise
3409:
3062:
2664:
2588:
2292:
2148:
2119:
has said that "all of scientifically applicable analysis can be developed ".
2081:
1709:
1567:
871:
564:
429:
109:
4364:
7718:
7377:
6556:
6403:
6304:
6296:
6176:
6124:
6033:
5969:
5952:
5883:
5742:
5601:
5303:
5086:
4648:
4643:
4547:
4344:
4339:
4157:
4086:
4044:
3903:
3800:
3719:
3462:
3269:
2749:
2719:
2571:"Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"
2566:
2511:
2474:
2249:
2174:
2152:
1968:
1941:
1850:
1746:
1591:
1441:
1405:
1324:
1013:
875:
413:
402:
395:
294:
51:
4405:
2063:
From set theory's inception, some mathematicians have objected to it as a
289:
The modern study of set theory was initiated by the German mathematicians
7708:
7334:
7246:
6666:
6546:
5725:
5715:
5662:
5346:
5266:
5251:
5131:
5076:
4850:
4514:
4437:
4369:
4004:
3021:
2642:
Einleitung zur GröĂenlehre und erste
Begriffe der allgemeinen GröĂenlehre
2405:
2190:
1927:
1632:
1135:, though this name is ambiguous and can lead to several interpretations.
1124:
376:
372:
283:
147:
73:
1149:
1053:{1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
7728:
7658:
7251:
6984:
6840:
5596:
5422:
5228:
4835:
4714:
4349:
4120:
3582:
2269:
2019:
1750:
1409:
624:
348:
152:
3469:, Undergraduate Texts in Mathematics (2nd ed.), Springer Verlag,
3364:
1992:
of the original model. Forcing is also one of two methods for proving
1598:
can all be defined as sets satisfying various (axiomatic) properties.
7233:
7194:
6748:
6651:
5704:
5621:
5581:
5545:
5481:
5293:
5283:
5256:
5019:
4152:
4115:
4066:
3964:
2793:(Spring 2020 ed.), Metaphysics Research Lab, Stanford University
2265:
2257:
1766:
1742:
1583:
1528:
1489:
1471:
1128:
1058:
360:
46:
30:
This article is about the branch of mathematics. For other uses, see
2252:
are widely employed to explain basic set-theoretic relationships to
391:
7294:
6777:
6733:
6531:
5979:
5684:
5278:
4739:
4658:
4585:
3616:
3324:
Sheaves in
Geometry and Logic: A First Introduction to Topos Theory
2520:
From Frege to Gödel: A Source Book in
Mathematical Logic, 1879â1931
2498:
2312:
2304:
2241:
2134:
1656:
1643:
1347:
1228:
containing only the empty set is assigned rank 1. For each ordinal
1158:
421:
336:
124:
114:
3647:
6329:
5121:
4524:
3739:
3386:
Hegel's Rabble: An Investigation into Hegel's Philosophy of Right
2379:
1757:
and extends to the study of more complex hierarchies such as the
119:
3100:, New York: Oxford University Press, pp. 280â283, 293â294,
2730:(Rev. English ed.), New York: Dover Publications, pp.
918:, is the set of all objects that are a member of exactly one of
4177:
3999:
2208:
of sets arising from the inductive and recursive properties of
632:
612:, but are not subsets of it; and in turn, the subsets, such as
526:
3282:
Cantone, Domenico; Ferro, Alfredo; Omodeo, Eugenio G. (1989),
2279:(NOT, AND, OR), and semantic or rule description (technically
1606:
are ubiquitous in mathematics, and the theory of mathematical
1101:, is the set whose members are all of the possible subsets of
942:. It is the set difference of the union and the intersection,
420:
in the East, mathematicians had struggled with the concept of
5873:
5219:
5064:
4049:
3809:
3745:
2853:"Internal Set Theory: a New Approach to Nonstandard Analysis"
2260:
originally devised them as part of a procedure to assess the
2125:
condemned set theory philosophically for its connotations of
412:
Since the 5th century BC, beginning with Greek mathematician
407:
On a Property of the Collection of All Real Algebraic Numbers
364:
2669:
Georg Cantor: His Mathematics and Philosophy of the Infinite
1339:
was originally devised to rid set theory of such paradoxes.
3350:
1275:
is defined to consist of all pure sets with rank less than
3366:
Homotopy Type Theory: Univalent Foundations of Mathematics
1972:
1660:
1559:
1552:
1454:
3467:
The Joy of Sets: Fundamentals of Contemporary Set Theory
2071:
voiced in set theory's earliest years, starts from the
1519:, such as CST, CZF, and IZF, embed their set axioms in
323:
were proposed in the early twentieth century, of which
3536:
Set Theory and Its Philosophy: A Critical Introduction
2493:
In his 1925 paper ""An Axiomatization of Set Theory",
1357:. This includes the most common axiomatic set theory,
608:. Also, 1, 2, and 3 are members (elements) of the set
2388:
2362:
2336:
1301:
1281:
1254:
1234:
1210:
1186:
1119:
Some basic sets of central importance are the set of
1074:
720:, is the set of all objects that are members of both
2989:
Berkemeier, Francisco; Page, Karen M. (2023-09-29),
2435:
1880:
The study of inner models is common in the study of
1849:
of ZermeloâFraenkel set theory (ZF) is a transitive
3253:
2147:after having only read the abstract. As reviewers
3512:
3433:
3415:Set Theory: An Introduction to Independence Proofs
3283:
3281:
2723:
2610:
2545:This is the converse for ZFC; V is a model of ZFC.
2396:
2370:
2344:
1799:. This has important applications to the study of
1307:
1295:. The entire von Neumann universe is denoted
1287:
1267:
1240:
1216:
1192:
1093:
2283:) of sets (e.g. "months starting with the letter
2095:is that defining sets using the axiom schemas of
666:, is the set of all objects that are a member of
7799:
3317:
2167:begun to rehabilitate Wittgenstein's arguments.
1539:embodying the membership relation is not simply
1346:. Such systems come in two flavors, those whose
3556:
2763:"set theory | Basics, Examples, & Formulas"
2714:
2576:Journal fĂŒr die reine und angewandte Mathematik
2067:. The most common objection to set theory, one
1153:An initial segment of the von Neumann hierarchy
3257:Communications on Pure and Applied Mathematics
2988:
2935:Number Systems and the Foundations of Analysis
1996:by finitistic methods, the other method being
435:
6817:Note: This template roughly follows the 2012
6793:
5035:
4421:
3761:
2857:Bulletin of the American Mathematical Society
1712:to infinite sets. This includes the study of
359:. Its foundational appeal, together with its
252:
3738:from the original on 2021-10-31 – via
3144:
2671:, Harvard University Press, pp. 30â54,
2275:Set theory is used to introduce students to
1012:, is the set whose members are all possible
331:) is still the best-known and most studied.
4435:
2884:"6.3: Equivalence Relations and Partitions"
2303:and other collection-like objects, such as
1694:
1469:The above systems can be modified to allow
6800:
6786:
5227:
5042:
5028:
4428:
4414:
4217:
3768:
3754:
3382:
3122:"Wittgenstein's Philosophy of Mathematics"
2522:, Harvard University Press, Cambridge MA,
259:
245:
3490:
2932:
2868:
2559:
2390:
2364:
2338:
2224:
2140:Remarks on the Foundations of Mathematics
2026:
2010:Cardinal characteristics of the continuum
1727:
3376:
3090:
1318:
1148:
1051:. For example, the Cartesian product of
848:is clear from the context, the notation
390:
45:
3431:
2809:
2791:The Stanford Encyclopedia of Philosophy
2784:
2636:
2608:
794:, while conversely, the set difference
27:Branch of mathematics that studies sets
14:
7800:
7517:Knowledge representation and reasoning
5049:
3724:"Set Theory: An Offspring of Analysis"
3532:
3461:
3389:, Bloomsbury Publishing, p. 151,
3237:
3217:
3196:
3168:
3119:
3061:
2850:
2663:
2565:
2287:"), which may be useful when learning
2003:
1465:, both of which are stronger than ZFC.
1418:, which omits the axioms of infinity,
7542:Philosophy of artificial intelligence
6781:
5023:
4409:
3749:
3718:
3630:"The Early Development of Set Theory"
3581:
3408:
3369:. The Univalent Foundations Program.
1979:fails, or a model of ZF in which the
1834:
678:, or both. For example, the union of
450:Set theory begins with a fundamental
6861:Energy consumption (Green computing)
6807:
4990:
3562:Set Theory and the Continuum Problem
3510:
3068:Foundations of Constructive Analysis
3020:
2964:"A PARTITION CALCULUS IN SET THEORY"
2086:Foundations of Constructive Analysis
1447:Von NeumannâBernaysâGödel set theory
1426:, and weakens the axiom schemata of
424:. Especially notable is the work of
7547:Distributed artificial intelligence
6819:ACM Computing Classification System
5002:
3622:Internet Encyclopedia of Philosophy
3131:Stanford Encyclopedia of Philosophy
2143:: Wittgenstein attempted to refute
2091:A different objection put forth by
1806:
1791:A recent area of research concerns
1679:have recently seen applications in
1508:has argued that it does reflect an
1457:for theorems about sets alone, and
732:. For example, the intersection of
24:
7052:Integrated development environment
3455:
2982:
2196:An active area of research is the
1891:
1077:
938:, the symmetric difference set is
339:, and has various applications in
25:
7834:
7527:Automated planning and scheduling
7057:Software configuration management
3732:University of Wisconsin-Milwaukee
3609:
3147:Philosophical Remarks, §129, §174
2812:"The iterative conception of set"
1686:
1627:. For example, properties of the
1094:{\displaystyle {\mathcal {P}}(A)}
842:. In this case, if the choice of
307:within naive set theory (such as
7781:
7771:
7762:
7761:
6761:
5001:
4989:
4978:
4977:
4965:
3799:
3440:, Prindle, Weber & Schmidt,
3203:: "Given that mathematics is a '
2851:Nelson, Edward (November 1977),
2617:, Prindle, Weber & Schmidt,
2438:
2059:Controversy over Cantor's theory
1774:effective descriptive set theory
1745:and, more generally, subsets of
1610:can be described in set theory.
1107:. For example, the power set of
229:
7772:
7175:Computational complexity theory
4886:Computational complexity theory
3358:
3340:
3311:
3275:
3247:
3138:
3120:Rodych, Victor (Jan 31, 2018),
3113:
3084:
3055:
3014:
2956:
2926:
2908:"Order Relations and Functions"
2900:
2876:
2870:10.1090/S0002-9904-1977-14398-X
2844:
2803:
2539:
2471: â borrows from set theory
2145:Gödel's incompleteness theorems
1803:in many fields of mathematics.
1795:and more complicated definable
1741:is the study of subsets of the
1716:and the study of extensions of
1638:Set theory as a foundation for
1573:
1496:(lacking them), associate with
774:, is the set of all members of
6959:Network performance evaluation
3775:
2778:
2755:
2708:
2685:
2657:
2630:
2602:
2487:
2052:
1921:
1823:this condition was relaxed by
1749:. It begins with the study of
1708:concerns extensions of finite
1463:TarskiâGrothendieck set theory
1088:
1082:
13:
1:
7330:Multimedia information system
7315:Geographic information system
7305:Enterprise information system
6894:Computer systems organization
6722:History of mathematical logic
3728:Marden Lecture in Mathematics
3383:Frank Ruda (6 October 2011),
3179:identify such deduction with
3145:Wittgenstein, Ludwig (1975),
2789:, in Zalta, Edward N. (ed.),
2552:
2408:, etc.), and when defining a
854:is sometimes used instead of
616:, are not members of the set
7689:Computational social science
7277:Theoretical computer science
7090:Software development process
6866:Electronic design automation
6851:Very Large Scale Integration
6647:Primitive recursive function
3519:, McGraw-Hill Book Company,
3371:Institute for Advanced Study
3071:, New York: Academic Press,
2819:The Review of Symbolic Logic
2397:{\displaystyle \mathbb {R} }
2371:{\displaystyle \mathbb {Z} }
2345:{\displaystyle \mathbb {N} }
2240:In the US in the 1960s, the
1988:without changing any of the
1784:, and is closely related to
506:. If all the members of set
7:
7512:Natural language processing
7300:Information storage systems
3679:Encyclopedia of Mathematics
3661:Encyclopedia of Mathematics
3228:has a particular property."
3149:, Oxford: Basil Blackwell,
2933:Mendelson, Elliott (1973),
2431:
2047:normal Moore space question
1944:have an elegant structure.
1916:ZermeloâFraenkel set theory
1793:Borel equivalence relations
1780:. It includes the study of
1535:, in which the value of an
1510:iterative conception of set
1388:axiom schema of replacement
1268:{\displaystyle V_{\alpha }}
1138:
436:Basic concepts and notation
369:philosophers of mathematics
325:ZermeloâFraenkel set theory
32:Set theory (disambiguation)
10:
7839:
7428:Humanâcomputer interaction
7398:Intrusion detection system
7310:Social information systems
7295:Database management system
5711:SchröderâBernstein theorem
5438:Monadic predicate calculus
5097:Foundations of mathematics
4936:Films about mathematicians
4266:von NeumannâBernaysâGödel
3560:; Fitting, Melvin (2010),
3515:Introduction to Set Theory
2726:Introductory Real Analysis
2218:law of the excluded middle
2065:foundation for mathematics
2056:
2030:
2007:
1951:
1947:
1925:
1895:
1838:
1810:
1776:is between set theory and
1731:
1698:
1142:
439:
386:
343:(such as in the theory of
169:Relationship with sciences
36:
29:
7757:
7694:Computational engineering
7669:Computational mathematics
7646:
7593:
7555:
7502:
7464:
7426:
7368:
7285:
7231:
7193:
7138:
7075:
7008:
6972:
6929:
6893:
6826:
6815:
6757:
6744:Philosophy of mathematics
6693:Automated theorem proving
6675:
6570:
6402:
6295:
6147:
5864:
5840:
5818:Von NeumannâBernaysâGödel
5763:
5657:
5561:
5459:
5450:
5377:
5312:
5218:
5140:
5057:
4959:
4909:
4866:
4776:
4738:
4705:
4657:
4629:
4576:
4523:
4505:Philosophy of mathematics
4480:
4445:
4330:
4293:
4205:
4095:
4067:One-to-one correspondence
3983:
3924:
3808:
3797:
3783:
3475:10.1007/978-1-4612-0903-4
3321:; Moerdijk, leke (1992),
3003:10.1101/2023.09.27.559729
2831:10.1017/S1755020308080064
2640:(1975), Berg, Jan (ed.),
2464:List of set theory topics
2212:. Principles such as the
2181:, finite set theory, and
375:line to the study of the
303:. After the discovery of
7704:Computational healthcare
7699:Differentiable computing
7618:Graphics processing unit
7037:Domain-specific language
6906:Computational complexity
4941:Recreational mathematics
3707:, and library resources
3533:Potter, Michael (2004),
3511:Monk, J. Donald (1969),
3491:FerreirĂłs, Jose (2001),
3432:Johnson, Philip (1972),
2609:Johnson, Philip (1972),
2589:10.1515/crll.1874.77.258
2480:
1786:hyperarithmetical theory
1706:Combinatorial set theory
1701:Infinitary combinatorics
1695:Combinatorial set theory
1416:KripkeâPlatek set theory
1113:{ {}, {1}, {2}, {1, 2} }
780:that are not members of
512:are also members of set
37:Not to be confused with
7679:Computational chemistry
7613:Photograph manipulation
7504:Artificial intelligence
7320:Decision support system
6394:Self-verifying theories
6215:Tarski's axiomatization
5166:Tarski's undefinability
5161:incompleteness theorems
4826:Mathematical statistics
4816:Mathematical psychology
4786:Engineering mathematics
4720:Algebraic number theory
3541:Oxford University Press
3436:A History of Set Theory
2810:Forster, T. E. (2008),
2767:Encyclopedia Britannica
2613:A History of Set Theory
2412:as a relation from one
1961:invented the method of
1898:Large cardinal property
1555:are a related subject.
1517:constructive set theory
1498:Willard Van Orman Quine
1459:MorseâKelley set theory
1288:{\displaystyle \alpha }
1241:{\displaystyle \alpha }
1204:The rank of a pure set
1193:{\displaystyle \alpha }
7744:Educational technology
7575:Reinforcement learning
7325:Process control system
7223:Computational geometry
7213:Algorithmic efficiency
7208:Analysis of algorithms
6856:Systems on Chip (SoCs)
6768:Mathematics portal
6379:Proof of impossibility
6027:propositional variable
5337:Propositional calculus
4972:Mathematics portal
4821:Mathematical sociology
4801:Mathematical economics
4796:Mathematical chemistry
4725:Analytic number theory
4606:Differential equations
4025:Constructible universe
3845:Constructibility (V=L)
3656:"Axiomatic set theory"
3648:Handbook of Set Theory
3270:10.1002/cpa.3160330503
3134:(Spring 2018 ed.)
2888:Mathematics LibreTexts
2785:Bagaria, Joan (2020),
2693:"Introduction to Sets"
2454:Glossary of set theory
2398:
2372:
2346:
2281:intensional definition
2256:students (even though
2225:Mathematical education
2210:higher inductive types
2127:mathematical platonism
2038:Set-theoretic topology
2033:Set-theoretic topology
2027:Set-theoretic topology
1967:while searching for a
1908:inaccessible cardinals
1855:constructible universe
1782:lightface pointclasses
1739:Descriptive set theory
1734:Descriptive set theory
1728:Descriptive set theory
1400:, a small fragment of
1309:
1289:
1269:
1242:
1218:
1194:
1154:
1095:
788:{1, 2, 3} \ {2, 3, 4}
416:in the West and early
398:
62:
7714:Electronic publishing
7684:Computational biology
7674:Computational physics
7570:Unsupervised learning
7484:Distributed computing
7360:Information retrieval
7267:Mathematical analysis
7257:Mathematical software
7140:Theory of computation
7105:Software construction
7095:Requirements analysis
6973:Software organization
6901:Computer architecture
6871:Hardware acceleration
6836:Printed circuit board
6637:Kolmogorov complexity
6590:Computably enumerable
6490:Model complete theory
6282:Principia Mathematica
5342:Propositional formula
5171:BanachâTarski paradox
4951:Mathematics education
4881:Theory of computation
4601:Hypercomplex analysis
4248:Principia Mathematica
4082:Transfinite induction
3941:(i.e. set difference)
3286:Computable Set Theory
3097:In the Light of Logic
2410:mathematical function
2399:
2373:
2347:
2326:In addition to that,
2297:programming languages
2235:mathematics education
2198:univalent foundations
2040:studies questions of
1998:Boolean-valued models
1954:Forcing (mathematics)
1797:equivalence relations
1765:. Many properties of
1681:evolutionary dynamics
1640:mathematical analysis
1616:Principia Mathematica
1549:Boolean-valued models
1449:, which has the same
1386:, which replaces the
1319:Formalized set theory
1310:
1290:
1270:
1243:
1219:
1195:
1175:transfinite recursion
1152:
1096:
814:, the set difference
796:{2, 3, 4} \ {1, 2, 3}
786:. The set difference
418:Indian mathematicians
394:
357:evolutionary dynamics
327:(with or without the
49:
7474:Concurrent computing
7446:Ubiquitous computing
7418:Application security
7413:Information security
7242:Discrete mathematics
7218:Randomized algorithm
7170:Computability theory
7148:Model of computation
7120:Software maintenance
7115:Software engineering
7077:Software development
7027:Programming language
7022:Programming paradigm
6939:Network architecture
6585:ChurchâTuring thesis
6572:Computability theory
5781:continuum hypothesis
5299:Square of opposition
5157:Gödel's completeness
4931:Informal mathematics
4811:Mathematical physics
4806:Mathematical finance
4791:Mathematical biology
4730:Diophantine geometry
4322:Burali-Forti paradox
4077:Set-builder notation
4030:Continuum hypothesis
3970:Symmetric difference
3699:Klein's encyclopedia
3558:Smullyan, Raymond M.
3497:, Berlin: Springer,
3347:homotopy type theory
2386:
2360:
2334:
2289:computer programming
2206:universal properties
2202:homotopy type theory
2084:'s influential book
1994:relative consistency
1977:continuum hypothesis
1938:axiom of determinacy
1912:measurable cardinals
1866:continuum hypothesis
1864:of ZF satisfies the
1829:degree of membership
1759:projective hierarchy
1652:discrete mathematics
1376:(ZFC). Fragments of
1344:cumulative hierarchy
1337:Axiomatic set theory
1333:Burali-Forti paradox
1299:
1279:
1252:
1232:
1208:
1184:
1171:cumulative hierarchy
1165:von Neumann universe
1145:von Neumann universe
1072:
883:Symmetric difference
317:Burali-Forti paradox
134:Discrete mathematics
7749:Document management
7739:Operations research
7664:Enterprise software
7580:Multi-task learning
7565:Supervised learning
7287:Information systems
7110:Software deployment
7067:Software repository
6921:Real-time computing
6739:Mathematical object
6630:P versus NP problem
6595:Computable function
6389:Reverse mathematics
6315:Logical consequence
6192:primitive recursive
6187:elementary function
5960:Free/bound variable
5813:TarskiâGrothendieck
5332:Logical connectives
5262:Logical equivalence
5112:Logical consequence
4946:Mathematics and art
4856:Operations research
4611:Functional analysis
4283:TarskiâGrothendieck
3691:Schoenflies, Arthur
3615:Daniel Cunningham,
3327:, Springer-Verlag,
3181:philosophical logic
2295:is used in various
2123:Ludwig Wittgenstein
2004:Cardinal invariants
1827:so an object has a
1714:cardinal arithmetic
1669:propositional logic
1596:relational algebras
1564:internal set theory
1404:sufficient for the
874:as in the study of
824:is also called the
68:Part of a series on
7813:Mathematical logic
7532:Search methodology
7479:Parallel computing
7436:Interaction design
7345:Computing platform
7272:Numerical analysis
7262:Information theory
7047:Software framework
7010:Software notations
6949:Network components
6846:Integrated circuit
6537:Transfer principle
6500:Semantics of logic
6485:Categorical theory
6461:Non-standard model
5975:Logical connective
5102:Information theory
5051:Mathematical logic
4891:Numerical analysis
4500:Mathematical logic
4495:Information theory
3872:Limitation of size
3713:in other libraries
3593:Dover Publications
3566:Dover Publications
3319:Mac Lane, Saunders
2937:, Academic Press,
2697:www.mathsisfun.com
2459:Class (set theory)
2446:Mathematics portal
2394:
2368:
2342:
2200:and related to it
2187:pointless topology
2171:Category theorists
2105:axiom of power set
2016:cardinal invariant
1872:, the inner model
1841:Inner model theory
1835:Inner model theory
1722:ErdĆsâRado theorem
1625:second-order logic
1402:Zermelo set theory
1398:General set theory
1384:Zermelo set theory
1366:raenkel set theory
1305:
1285:
1265:
1238:
1214:
1190:
1155:
1091:
864:, particularly if
454:between an object
399:
345:relational algebra
276:mathematical logic
235:Mathematics Portal
63:
39:Set theory (music)
7795:
7794:
7724:Electronic voting
7654:Quantum Computing
7647:Applied computing
7633:Image compression
7403:Hardware security
7393:Security services
7350:Digital marketing
7130:Open-source model
7042:Modeling language
6954:Network scheduler
6775:
6774:
6707:Abstract category
6510:Theories of truth
6320:Rule of inference
6310:Natural deduction
6291:
6290:
5836:
5835:
5541:Cartesian product
5446:
5445:
5352:Many-valued logic
5327:Boolean functions
5210:Russell's paradox
5185:diagonal argument
5082:First-order logic
5017:
5016:
4616:Harmonic analysis
4403:
4402:
4312:Russell's paradox
4261:ZermeloâFraenkel
4162:Dedekind-infinite
4035:Diagonal argument
3934:Cartesian product
3791:Set (mathematics)
3722:(April 6, 1990),
3602:978-0-486-43520-6
3575:978-0-486-47484-7
3550:978-0-191-55643-2
3526:978-0-898-74006-6
3504:978-3-7643-5749-8
3418:, North-Holland,
3396:978-1-4411-7413-0
3334:978-0-387-97710-2
3092:Feferman, Solomon
3041:978-3-540-44085-7
2277:logical operators
2103:, as well as the
1665:first-order logic
1558:An enrichment of
1329:Russell's paradox
1308:{\displaystyle V}
1217:{\displaystyle X}
1169:organized into a
987:Cartesian product
629:binary operations
442:Set (mathematics)
321:axiomatic systems
309:Russell's paradox
274:is the branch of
269:
268:
224:
223:
54:illustrating the
16:(Redirected from
7830:
7785:
7784:
7775:
7774:
7765:
7764:
7585:Cross-validation
7557:Machine learning
7441:Social computing
7408:Network security
7203:Algorithm design
7125:Programming team
7085:Control variable
7062:Software library
7000:Software quality
6995:Operating system
6944:Network protocol
6809:Computer science
6802:
6795:
6788:
6779:
6778:
6766:
6765:
6717:History of logic
6712:Category of sets
6605:Decision problem
6384:Ordinal analysis
6325:Sequent calculus
6223:Boolean algebras
6163:
6162:
6137:
6108:logical/constant
5862:
5861:
5848:
5771:ZermeloâFraenkel
5522:Set operations:
5457:
5456:
5394:
5225:
5224:
5205:LöwenheimâSkolem
5092:Formal semantics
5044:
5037:
5030:
5021:
5020:
5005:
5004:
4993:
4992:
4981:
4980:
4970:
4969:
4901:Computer algebra
4876:Computer science
4596:Complex analysis
4430:
4423:
4416:
4407:
4406:
4385:Bertrand Russell
4375:John von Neumann
4360:Abraham Fraenkel
4355:Richard Dedekind
4317:Suslin's problem
4228:Cantor's theorem
3945:De Morgan's laws
3803:
3770:
3763:
3756:
3747:
3746:
3742:
3720:Rudin, Walter B.
3715:about set theory
3687:
3669:
3643:Akihiro Kanamori
3639:Foreman, Matthew
3628:Jose Ferreiros,
3605:
3578:
3553:
3529:
3518:
3507:
3487:
3450:
3439:
3428:
3400:
3399:
3380:
3374:
3362:
3356:
3344:
3338:
3337:
3315:
3309:
3308:
3289:
3279:
3273:
3272:
3251:
3245:
3235:
3229:
3227:
3215:
3209:
3206:
3194:
3188:
3166:
3160:
3159:
3142:
3136:
3135:
3126:Zalta, Edward N.
3117:
3111:
3110:
3088:
3082:
3081:
3059:
3053:
3052:
3018:
3012:
3011:
3010:
3009:
2986:
2980:
2979:
2978:
2977:
2968:
2960:
2954:
2953:
2930:
2924:
2923:
2922:
2921:
2915:Web.stanford.edu
2912:
2904:
2898:
2897:
2896:
2895:
2880:
2874:
2873:
2872:
2848:
2842:
2841:
2816:
2807:
2801:
2800:
2799:
2798:
2782:
2776:
2775:
2774:
2773:
2759:
2753:
2752:
2729:
2716:Kolmogorov, A.N.
2712:
2706:
2705:
2704:
2703:
2689:
2683:
2681:
2661:
2655:
2654:
2638:Bolzano, Bernard
2634:
2628:
2627:
2616:
2606:
2600:
2599:
2563:
2546:
2543:
2531:
2516:L. E. J. Brouwer
2504:Bertrand Russell
2495:John von Neumann
2491:
2469:Relational model
2448:
2443:
2442:
2403:
2401:
2400:
2395:
2393:
2377:
2375:
2374:
2369:
2367:
2351:
2349:
2348:
2343:
2341:
2317:computer science
2231:naive set theory
2117:Solomon Feferman
2042:general topology
1990:cardinal numbers
1820:fuzzy set theory
1813:Fuzzy set theory
1807:Fuzzy set theory
1778:recursion theory
1718:Ramsey's theorem
1648:abstract algebra
1566:was proposed by
1533:fuzzy set theory
1529:rough set theory
1445:. These include
1314:
1312:
1311:
1306:
1294:
1292:
1291:
1286:
1274:
1272:
1271:
1266:
1264:
1263:
1247:
1245:
1244:
1239:
1227:
1223:
1221:
1220:
1215:
1199:
1197:
1196:
1191:
1114:
1110:
1106:
1100:
1098:
1097:
1092:
1081:
1080:
1067:
1054:
1050:
1044:
1038:
1032:
1026:
1011:
1001:
995:
981:
961:
941:
937:
933:
929:
923:
917:
907:
897:
891:
869:
863:
853:
847:
841:
835:
823:
813:
807:
801:
797:
793:
789:
785:
779:
773:
763:
757:
743:
739:
735:
731:
725:
719:
709:
703:
689:
685:
681:
677:
671:
665:
655:
649:
619:
615:
611:
607:
602:is not equal to
601:
595:
589:
583:
573:
561:
557:
553:
549:
545:
535:
523:
517:
511:
497:
487:
471:
465:
459:
353:formal semantics
341:computer science
313:Cantor's paradox
300:naive set theory
291:Richard Dedekind
261:
254:
247:
233:
97:
96:
65:
64:
21:
7838:
7837:
7833:
7832:
7831:
7829:
7828:
7827:
7798:
7797:
7796:
7791:
7782:
7753:
7734:Word processing
7642:
7628:Virtual reality
7589:
7551:
7522:Computer vision
7498:
7494:Multiprocessing
7460:
7422:
7388:Security hacker
7364:
7340:Digital library
7281:
7232:Mathematics of
7227:
7189:
7165:Automata theory
7160:Formal language
7134:
7100:Software design
7071:
7004:
6990:Virtual machine
6968:
6964:Network service
6925:
6916:Embedded system
6889:
6822:
6811:
6806:
6776:
6771:
6760:
6753:
6698:Category theory
6688:Algebraic logic
6671:
6642:Lambda calculus
6580:Church encoding
6566:
6542:Truth predicate
6398:
6364:Complete theory
6287:
6156:
6152:
6148:
6143:
6135:
5855: and
5851:
5846:
5832:
5808:New Foundations
5776:axiom of choice
5759:
5721:Gödel numbering
5661: and
5653:
5557:
5442:
5392:
5373:
5322:Boolean algebra
5308:
5272:Equiconsistency
5237:Classical logic
5214:
5195:Halting problem
5183: and
5159: and
5147: and
5146:
5141:Theorems (
5136:
5053:
5048:
5018:
5013:
4964:
4955:
4905:
4862:
4841:Systems science
4772:
4768:Homotopy theory
4734:
4701:
4653:
4625:
4572:
4519:
4490:Category theory
4476:
4441:
4434:
4404:
4399:
4326:
4305:
4289:
4254:New Foundations
4201:
4091:
4010:Cardinal number
3993:
3979:
3920:
3804:
3795:
3779:
3774:
3709:in your library
3672:
3654:
3632:article in the
3619:article in the
3612:
3603:
3576:
3551:
3527:
3505:
3485:
3458:
3456:Further reading
3453:
3448:
3426:
3404:
3403:
3397:
3381:
3377:
3363:
3359:
3345:
3341:
3335:
3316:
3312:
3306:
3292:Clarendon Press
3280:
3276:
3252:
3248:
3236:
3232:
3225:
3216:
3212:
3204:
3195:
3191:
3167:
3163:
3157:
3143:
3139:
3118:
3114:
3108:
3089:
3085:
3079:
3060:
3056:
3042:
3034:, p. 642,
3032:Springer-Verlag
3019:
3015:
3007:
3005:
2987:
2983:
2975:
2973:
2966:
2962:
2961:
2957:
2931:
2927:
2919:
2917:
2910:
2906:
2905:
2901:
2893:
2891:
2882:
2881:
2877:
2849:
2845:
2814:
2808:
2804:
2796:
2794:
2783:
2779:
2771:
2769:
2761:
2760:
2756:
2742:
2713:
2709:
2701:
2699:
2691:
2690:
2686:
2679:
2662:
2658:
2652:
2635:
2631:
2625:
2607:
2603:
2583:(77): 258â262,
2564:
2560:
2555:
2550:
2549:
2544:
2540:
2535:
2534:
2502:exemplified by
2492:
2488:
2483:
2444:
2437:
2434:
2389:
2387:
2384:
2383:
2363:
2361:
2358:
2357:
2354:natural numbers
2337:
2335:
2332:
2331:
2227:
2214:axiom of choice
2109:impredicativity
2061:
2055:
2035:
2029:
2012:
2006:
1986:natural numbers
1981:axiom of choice
1956:
1950:
1930:
1924:
1900:
1894:
1892:Large cardinals
1886:large cardinals
1870:axiom of choice
1843:
1837:
1815:
1809:
1763:Wadge hierarchy
1755:Borel hierarchy
1736:
1730:
1703:
1697:
1689:
1677:Axiom of Choice
1604:order relations
1576:
1525:classical logic
1502:axiom of choice
1481:New Foundations
1321:
1300:
1297:
1296:
1280:
1277:
1276:
1259:
1255:
1253:
1250:
1249:
1233:
1230:
1229:
1225:
1209:
1206:
1205:
1200:, known as its
1185:
1182:
1181:
1147:
1141:
1121:natural numbers
1112:
1108:
1102:
1076:
1075:
1073:
1070:
1069:
1063:
1052:
1046:
1045:is a member of
1040:
1034:
1033:is a member of
1028:
1016:
1003:
997:
991:
963:
943:
939:
935:
931:
925:
919:
909:
899:
893:
887:
865:
855:
849:
843:
837:
831:
815:
809:
808:is a subset of
803:
799:
795:
791:
787:
781:
775:
765:
759:
753:
741:
737:
733:
727:
721:
711:
705:
699:
687:
683:
679:
673:
667:
657:
651:
645:
617:
613:
609:
603:
597:
591:
590:is a subset of
585:
584:if and only if
579:
569:
559:
555:
551:
550:is a subset of
547:
546:. For example,
537:
531:
519:
513:
507:
489:
488:, the notation
483:
467:
461:
455:
452:binary relation
448:
446:Algebra of sets
440:Main articles:
438:
426:Bernard Bolzano
389:
381:large cardinals
329:axiom of choice
265:
220:
219:
170:
162:
161:
157:Decision theory
105:
42:
35:
28:
23:
22:
15:
12:
11:
5:
7836:
7826:
7825:
7820:
7818:Formal methods
7815:
7810:
7793:
7792:
7790:
7789:
7779:
7769:
7758:
7755:
7754:
7752:
7751:
7746:
7741:
7736:
7731:
7726:
7721:
7716:
7711:
7706:
7701:
7696:
7691:
7686:
7681:
7676:
7671:
7666:
7661:
7656:
7650:
7648:
7644:
7643:
7641:
7640:
7638:Solid modeling
7635:
7630:
7625:
7620:
7615:
7610:
7605:
7599:
7597:
7591:
7590:
7588:
7587:
7582:
7577:
7572:
7567:
7561:
7559:
7553:
7552:
7550:
7549:
7544:
7539:
7537:Control method
7534:
7529:
7524:
7519:
7514:
7508:
7506:
7500:
7499:
7497:
7496:
7491:
7489:Multithreading
7486:
7481:
7476:
7470:
7468:
7462:
7461:
7459:
7458:
7453:
7448:
7443:
7438:
7432:
7430:
7424:
7423:
7421:
7420:
7415:
7410:
7405:
7400:
7395:
7390:
7385:
7383:Formal methods
7380:
7374:
7372:
7366:
7365:
7363:
7362:
7357:
7355:World Wide Web
7352:
7347:
7342:
7337:
7332:
7327:
7322:
7317:
7312:
7307:
7302:
7297:
7291:
7289:
7283:
7282:
7280:
7279:
7274:
7269:
7264:
7259:
7254:
7249:
7244:
7238:
7236:
7229:
7228:
7226:
7225:
7220:
7215:
7210:
7205:
7199:
7197:
7191:
7190:
7188:
7187:
7182:
7177:
7172:
7167:
7162:
7157:
7156:
7155:
7144:
7142:
7136:
7135:
7133:
7132:
7127:
7122:
7117:
7112:
7107:
7102:
7097:
7092:
7087:
7081:
7079:
7073:
7072:
7070:
7069:
7064:
7059:
7054:
7049:
7044:
7039:
7034:
7029:
7024:
7018:
7016:
7006:
7005:
7003:
7002:
6997:
6992:
6987:
6982:
6976:
6974:
6970:
6969:
6967:
6966:
6961:
6956:
6951:
6946:
6941:
6935:
6933:
6927:
6926:
6924:
6923:
6918:
6913:
6908:
6903:
6897:
6895:
6891:
6890:
6888:
6887:
6878:
6873:
6868:
6863:
6858:
6853:
6848:
6843:
6838:
6832:
6830:
6824:
6823:
6816:
6813:
6812:
6805:
6804:
6797:
6790:
6782:
6773:
6772:
6758:
6755:
6754:
6752:
6751:
6746:
6741:
6736:
6731:
6730:
6729:
6719:
6714:
6709:
6700:
6695:
6690:
6685:
6683:Abstract logic
6679:
6677:
6673:
6672:
6670:
6669:
6664:
6662:Turing machine
6659:
6654:
6649:
6644:
6639:
6634:
6633:
6632:
6627:
6622:
6617:
6612:
6602:
6600:Computable set
6597:
6592:
6587:
6582:
6576:
6574:
6568:
6567:
6565:
6564:
6559:
6554:
6549:
6544:
6539:
6534:
6529:
6528:
6527:
6522:
6517:
6507:
6502:
6497:
6495:Satisfiability
6492:
6487:
6482:
6481:
6480:
6470:
6469:
6468:
6458:
6457:
6456:
6451:
6446:
6441:
6436:
6426:
6425:
6424:
6419:
6412:Interpretation
6408:
6406:
6400:
6399:
6397:
6396:
6391:
6386:
6381:
6376:
6366:
6361:
6360:
6359:
6358:
6357:
6347:
6342:
6332:
6327:
6322:
6317:
6312:
6307:
6301:
6299:
6293:
6292:
6289:
6288:
6286:
6285:
6277:
6276:
6275:
6274:
6269:
6268:
6267:
6262:
6257:
6237:
6236:
6235:
6233:minimal axioms
6230:
6219:
6218:
6217:
6206:
6205:
6204:
6199:
6194:
6189:
6184:
6179:
6166:
6164:
6145:
6144:
6142:
6141:
6140:
6139:
6127:
6122:
6121:
6120:
6115:
6110:
6105:
6095:
6090:
6085:
6080:
6079:
6078:
6073:
6063:
6062:
6061:
6056:
6051:
6046:
6036:
6031:
6030:
6029:
6024:
6019:
6009:
6008:
6007:
6002:
5997:
5992:
5987:
5982:
5972:
5967:
5962:
5957:
5956:
5955:
5950:
5945:
5940:
5930:
5925:
5923:Formation rule
5920:
5915:
5914:
5913:
5908:
5898:
5897:
5896:
5886:
5881:
5876:
5871:
5865:
5859:
5842:Formal systems
5838:
5837:
5834:
5833:
5831:
5830:
5825:
5820:
5815:
5810:
5805:
5800:
5795:
5790:
5785:
5784:
5783:
5778:
5767:
5765:
5761:
5760:
5758:
5757:
5756:
5755:
5745:
5740:
5739:
5738:
5731:Large cardinal
5728:
5723:
5718:
5713:
5708:
5694:
5693:
5692:
5687:
5682:
5667:
5665:
5655:
5654:
5652:
5651:
5650:
5649:
5644:
5639:
5629:
5624:
5619:
5614:
5609:
5604:
5599:
5594:
5589:
5584:
5579:
5574:
5568:
5566:
5559:
5558:
5556:
5555:
5554:
5553:
5548:
5543:
5538:
5533:
5528:
5520:
5519:
5518:
5513:
5503:
5498:
5496:Extensionality
5493:
5491:Ordinal number
5488:
5478:
5473:
5472:
5471:
5460:
5454:
5448:
5447:
5444:
5443:
5441:
5440:
5435:
5430:
5425:
5420:
5415:
5410:
5409:
5408:
5398:
5397:
5396:
5383:
5381:
5375:
5374:
5372:
5371:
5370:
5369:
5364:
5359:
5349:
5344:
5339:
5334:
5329:
5324:
5318:
5316:
5310:
5309:
5307:
5306:
5301:
5296:
5291:
5286:
5281:
5276:
5275:
5274:
5264:
5259:
5254:
5249:
5244:
5239:
5233:
5231:
5222:
5216:
5215:
5213:
5212:
5207:
5202:
5197:
5192:
5187:
5175:Cantor's
5173:
5168:
5163:
5153:
5151:
5138:
5137:
5135:
5134:
5129:
5124:
5119:
5114:
5109:
5104:
5099:
5094:
5089:
5084:
5079:
5074:
5073:
5072:
5061:
5059:
5055:
5054:
5047:
5046:
5039:
5032:
5024:
5015:
5014:
5012:
5011:
4999:
4987:
4975:
4960:
4957:
4956:
4954:
4953:
4948:
4943:
4938:
4933:
4928:
4927:
4926:
4919:Mathematicians
4915:
4913:
4911:Related topics
4907:
4906:
4904:
4903:
4898:
4893:
4888:
4883:
4878:
4872:
4870:
4864:
4863:
4861:
4860:
4859:
4858:
4853:
4848:
4846:Control theory
4838:
4833:
4828:
4823:
4818:
4813:
4808:
4803:
4798:
4793:
4788:
4782:
4780:
4774:
4773:
4771:
4770:
4765:
4760:
4755:
4750:
4744:
4742:
4736:
4735:
4733:
4732:
4727:
4722:
4717:
4711:
4709:
4703:
4702:
4700:
4699:
4694:
4689:
4684:
4679:
4674:
4669:
4663:
4661:
4655:
4654:
4652:
4651:
4646:
4641:
4635:
4633:
4627:
4626:
4624:
4623:
4621:Measure theory
4618:
4613:
4608:
4603:
4598:
4593:
4588:
4582:
4580:
4574:
4573:
4571:
4570:
4565:
4560:
4555:
4550:
4545:
4540:
4535:
4529:
4527:
4521:
4520:
4518:
4517:
4512:
4507:
4502:
4497:
4492:
4486:
4484:
4478:
4477:
4475:
4474:
4469:
4464:
4463:
4462:
4457:
4446:
4443:
4442:
4433:
4432:
4425:
4418:
4410:
4401:
4400:
4398:
4397:
4392:
4390:Thoralf Skolem
4387:
4382:
4377:
4372:
4367:
4362:
4357:
4352:
4347:
4342:
4336:
4334:
4328:
4327:
4325:
4324:
4319:
4314:
4308:
4306:
4304:
4303:
4300:
4294:
4291:
4290:
4288:
4287:
4286:
4285:
4280:
4275:
4274:
4273:
4258:
4257:
4256:
4244:
4243:
4242:
4231:
4230:
4225:
4220:
4215:
4209:
4207:
4203:
4202:
4200:
4199:
4194:
4189:
4184:
4175:
4170:
4165:
4155:
4150:
4149:
4148:
4143:
4138:
4128:
4118:
4113:
4108:
4102:
4100:
4093:
4092:
4090:
4089:
4084:
4079:
4074:
4072:Ordinal number
4069:
4064:
4059:
4054:
4053:
4052:
4047:
4037:
4032:
4027:
4022:
4017:
4007:
4002:
3996:
3994:
3992:
3991:
3988:
3984:
3981:
3980:
3978:
3977:
3972:
3967:
3962:
3957:
3952:
3950:Disjoint union
3947:
3942:
3936:
3930:
3928:
3922:
3921:
3919:
3918:
3917:
3916:
3911:
3900:
3899:
3897:Martin's axiom
3894:
3889:
3884:
3879:
3874:
3869:
3864:
3862:Extensionality
3859:
3858:
3857:
3847:
3842:
3841:
3840:
3835:
3830:
3820:
3814:
3812:
3806:
3805:
3798:
3796:
3794:
3793:
3787:
3785:
3781:
3780:
3773:
3772:
3765:
3758:
3750:
3744:
3743:
3716:
3702:
3688:
3670:
3652:
3636:
3626:
3611:
3610:External links
3608:
3607:
3606:
3601:
3579:
3574:
3554:
3549:
3530:
3525:
3508:
3503:
3488:
3483:
3457:
3454:
3452:
3451:
3446:
3429:
3424:
3410:Kunen, Kenneth
3405:
3402:
3401:
3395:
3375:
3357:
3339:
3333:
3310:
3304:
3274:
3264:(5): 599â608,
3246:
3230:
3210:
3189:
3187:, paras. 7-12.
3183:; c.f. Rodych
3161:
3155:
3137:
3112:
3106:
3083:
3077:
3063:Bishop, Errett
3054:
3040:
3013:
2981:
2955:
2925:
2899:
2875:
2843:
2802:
2777:
2754:
2740:
2707:
2684:
2677:
2665:Dauben, Joseph
2656:
2650:
2629:
2623:
2601:
2557:
2556:
2554:
2551:
2548:
2547:
2537:
2536:
2533:
2532:
2485:
2484:
2482:
2479:
2478:
2477:
2472:
2466:
2461:
2456:
2450:
2449:
2433:
2430:
2392:
2366:
2340:
2254:primary school
2246:primary school
2226:
2223:
2179:constructivism
2173:have proposed
2165:Crispin Wright
2131:constructivism
2093:Henri Poincaré
2073:constructivist
2057:Main article:
2054:
2051:
2031:Main article:
2028:
2025:
2008:Main article:
2005:
2002:
1952:Main article:
1949:
1946:
1926:Main article:
1923:
1920:
1904:large cardinal
1896:Main article:
1893:
1890:
1839:Main article:
1836:
1833:
1825:Lotfi A. Zadeh
1811:Main article:
1808:
1805:
1732:Main article:
1729:
1726:
1699:Main article:
1696:
1693:
1688:
1687:Areas of study
1685:
1575:
1572:
1537:atomic formula
1521:intuitionistic
1506:Thomas Forster
1467:
1466:
1442:proper classes
1437:
1436:
1435:
1413:
1395:
1320:
1317:
1304:
1284:
1262:
1258:
1237:
1213:
1189:
1179:ordinal number
1143:Main article:
1140:
1137:
1117:
1116:
1090:
1087:
1084:
1079:
1055:
983:
879:
749:Set difference
745:
691:
437:
434:
388:
385:
267:
266:
264:
263:
256:
249:
241:
238:
237:
226:
225:
222:
221:
218:
217:
212:
207:
202:
197:
192:
187:
182:
177:
171:
168:
167:
164:
163:
160:
159:
150:
145:
136:
131:
122:
117:
112:
106:
101:
100:
93:
92:
91:
90:
85:
77:
76:
70:
69:
26:
9:
6:
4:
3:
2:
7835:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7806:
7805:
7803:
7788:
7780:
7778:
7770:
7768:
7760:
7759:
7756:
7750:
7747:
7745:
7742:
7740:
7737:
7735:
7732:
7730:
7727:
7725:
7722:
7720:
7717:
7715:
7712:
7710:
7707:
7705:
7702:
7700:
7697:
7695:
7692:
7690:
7687:
7685:
7682:
7680:
7677:
7675:
7672:
7670:
7667:
7665:
7662:
7660:
7657:
7655:
7652:
7651:
7649:
7645:
7639:
7636:
7634:
7631:
7629:
7626:
7624:
7623:Mixed reality
7621:
7619:
7616:
7614:
7611:
7609:
7606:
7604:
7601:
7600:
7598:
7596:
7592:
7586:
7583:
7581:
7578:
7576:
7573:
7571:
7568:
7566:
7563:
7562:
7560:
7558:
7554:
7548:
7545:
7543:
7540:
7538:
7535:
7533:
7530:
7528:
7525:
7523:
7520:
7518:
7515:
7513:
7510:
7509:
7507:
7505:
7501:
7495:
7492:
7490:
7487:
7485:
7482:
7480:
7477:
7475:
7472:
7471:
7469:
7467:
7463:
7457:
7456:Accessibility
7454:
7452:
7451:Visualization
7449:
7447:
7444:
7442:
7439:
7437:
7434:
7433:
7431:
7429:
7425:
7419:
7416:
7414:
7411:
7409:
7406:
7404:
7401:
7399:
7396:
7394:
7391:
7389:
7386:
7384:
7381:
7379:
7376:
7375:
7373:
7371:
7367:
7361:
7358:
7356:
7353:
7351:
7348:
7346:
7343:
7341:
7338:
7336:
7333:
7331:
7328:
7326:
7323:
7321:
7318:
7316:
7313:
7311:
7308:
7306:
7303:
7301:
7298:
7296:
7293:
7292:
7290:
7288:
7284:
7278:
7275:
7273:
7270:
7268:
7265:
7263:
7260:
7258:
7255:
7253:
7250:
7248:
7245:
7243:
7240:
7239:
7237:
7235:
7230:
7224:
7221:
7219:
7216:
7214:
7211:
7209:
7206:
7204:
7201:
7200:
7198:
7196:
7192:
7186:
7183:
7181:
7178:
7176:
7173:
7171:
7168:
7166:
7163:
7161:
7158:
7154:
7151:
7150:
7149:
7146:
7145:
7143:
7141:
7137:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7111:
7108:
7106:
7103:
7101:
7098:
7096:
7093:
7091:
7088:
7086:
7083:
7082:
7080:
7078:
7074:
7068:
7065:
7063:
7060:
7058:
7055:
7053:
7050:
7048:
7045:
7043:
7040:
7038:
7035:
7033:
7030:
7028:
7025:
7023:
7020:
7019:
7017:
7015:
7011:
7007:
7001:
6998:
6996:
6993:
6991:
6988:
6986:
6983:
6981:
6978:
6977:
6975:
6971:
6965:
6962:
6960:
6957:
6955:
6952:
6950:
6947:
6945:
6942:
6940:
6937:
6936:
6934:
6932:
6928:
6922:
6919:
6917:
6914:
6912:
6911:Dependability
6909:
6907:
6904:
6902:
6899:
6898:
6896:
6892:
6886:
6882:
6879:
6877:
6874:
6872:
6869:
6867:
6864:
6862:
6859:
6857:
6854:
6852:
6849:
6847:
6844:
6842:
6839:
6837:
6834:
6833:
6831:
6829:
6825:
6820:
6814:
6810:
6803:
6798:
6796:
6791:
6789:
6784:
6783:
6780:
6770:
6769:
6764:
6756:
6750:
6747:
6745:
6742:
6740:
6737:
6735:
6732:
6728:
6725:
6724:
6723:
6720:
6718:
6715:
6713:
6710:
6708:
6704:
6701:
6699:
6696:
6694:
6691:
6689:
6686:
6684:
6681:
6680:
6678:
6674:
6668:
6665:
6663:
6660:
6658:
6657:Recursive set
6655:
6653:
6650:
6648:
6645:
6643:
6640:
6638:
6635:
6631:
6628:
6626:
6623:
6621:
6618:
6616:
6613:
6611:
6608:
6607:
6606:
6603:
6601:
6598:
6596:
6593:
6591:
6588:
6586:
6583:
6581:
6578:
6577:
6575:
6573:
6569:
6563:
6560:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6526:
6523:
6521:
6518:
6516:
6513:
6512:
6511:
6508:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6488:
6486:
6483:
6479:
6476:
6475:
6474:
6471:
6467:
6466:of arithmetic
6464:
6463:
6462:
6459:
6455:
6452:
6450:
6447:
6445:
6442:
6440:
6437:
6435:
6432:
6431:
6430:
6427:
6423:
6420:
6418:
6415:
6414:
6413:
6410:
6409:
6407:
6405:
6401:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6374:
6373:from ZFC
6370:
6367:
6365:
6362:
6356:
6353:
6352:
6351:
6348:
6346:
6343:
6341:
6338:
6337:
6336:
6333:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6308:
6306:
6303:
6302:
6300:
6298:
6294:
6284:
6283:
6279:
6278:
6273:
6272:non-Euclidean
6270:
6266:
6263:
6261:
6258:
6256:
6255:
6251:
6250:
6248:
6245:
6244:
6242:
6238:
6234:
6231:
6229:
6226:
6225:
6224:
6220:
6216:
6213:
6212:
6211:
6207:
6203:
6200:
6198:
6195:
6193:
6190:
6188:
6185:
6183:
6180:
6178:
6175:
6174:
6172:
6168:
6167:
6165:
6160:
6154:
6149:Example
6146:
6138:
6133:
6132:
6131:
6128:
6126:
6123:
6119:
6116:
6114:
6111:
6109:
6106:
6104:
6101:
6100:
6099:
6096:
6094:
6091:
6089:
6086:
6084:
6081:
6077:
6074:
6072:
6069:
6068:
6067:
6064:
6060:
6057:
6055:
6052:
6050:
6047:
6045:
6042:
6041:
6040:
6037:
6035:
6032:
6028:
6025:
6023:
6020:
6018:
6015:
6014:
6013:
6010:
6006:
6003:
6001:
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5981:
5978:
5977:
5976:
5973:
5971:
5968:
5966:
5963:
5961:
5958:
5954:
5951:
5949:
5946:
5944:
5941:
5939:
5936:
5935:
5934:
5931:
5929:
5926:
5924:
5921:
5919:
5916:
5912:
5909:
5907:
5906:by definition
5904:
5903:
5902:
5899:
5895:
5892:
5891:
5890:
5887:
5885:
5882:
5880:
5877:
5875:
5872:
5870:
5867:
5866:
5863:
5860:
5858:
5854:
5849:
5843:
5839:
5829:
5826:
5824:
5821:
5819:
5816:
5814:
5811:
5809:
5806:
5804:
5801:
5799:
5796:
5794:
5793:KripkeâPlatek
5791:
5789:
5786:
5782:
5779:
5777:
5774:
5773:
5772:
5769:
5768:
5766:
5762:
5754:
5751:
5750:
5749:
5746:
5744:
5741:
5737:
5734:
5733:
5732:
5729:
5727:
5724:
5722:
5719:
5717:
5714:
5712:
5709:
5706:
5702:
5698:
5695:
5691:
5688:
5686:
5683:
5681:
5678:
5677:
5676:
5672:
5669:
5668:
5666:
5664:
5660:
5656:
5648:
5645:
5643:
5640:
5638:
5637:constructible
5635:
5634:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5605:
5603:
5600:
5598:
5595:
5593:
5590:
5588:
5585:
5583:
5580:
5578:
5575:
5573:
5570:
5569:
5567:
5565:
5560:
5552:
5549:
5547:
5544:
5542:
5539:
5537:
5534:
5532:
5529:
5527:
5524:
5523:
5521:
5517:
5514:
5512:
5509:
5508:
5507:
5504:
5502:
5499:
5497:
5494:
5492:
5489:
5487:
5483:
5479:
5477:
5474:
5470:
5467:
5466:
5465:
5462:
5461:
5458:
5455:
5453:
5449:
5439:
5436:
5434:
5431:
5429:
5426:
5424:
5421:
5419:
5416:
5414:
5411:
5407:
5404:
5403:
5402:
5399:
5395:
5390:
5389:
5388:
5385:
5384:
5382:
5380:
5376:
5368:
5365:
5363:
5360:
5358:
5355:
5354:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5319:
5317:
5315:
5314:Propositional
5311:
5305:
5302:
5300:
5297:
5295:
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5273:
5270:
5269:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5242:Logical truth
5240:
5238:
5235:
5234:
5232:
5230:
5226:
5223:
5221:
5217:
5211:
5208:
5206:
5203:
5201:
5198:
5196:
5193:
5191:
5188:
5186:
5182:
5178:
5174:
5172:
5169:
5167:
5164:
5162:
5158:
5155:
5154:
5152:
5150:
5144:
5139:
5133:
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5071:
5068:
5067:
5066:
5063:
5062:
5060:
5056:
5052:
5045:
5040:
5038:
5033:
5031:
5026:
5025:
5022:
5010:
5009:
5000:
4998:
4997:
4988:
4986:
4985:
4976:
4974:
4973:
4968:
4962:
4961:
4958:
4952:
4949:
4947:
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4925:
4922:
4921:
4920:
4917:
4916:
4914:
4912:
4908:
4902:
4899:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4873:
4871:
4869:
4868:Computational
4865:
4857:
4854:
4852:
4849:
4847:
4844:
4843:
4842:
4839:
4837:
4834:
4832:
4829:
4827:
4824:
4822:
4819:
4817:
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4789:
4787:
4784:
4783:
4781:
4779:
4775:
4769:
4766:
4764:
4761:
4759:
4756:
4754:
4751:
4749:
4746:
4745:
4743:
4741:
4737:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4712:
4710:
4708:
4707:Number theory
4704:
4698:
4695:
4693:
4690:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4668:
4665:
4664:
4662:
4660:
4656:
4650:
4647:
4645:
4642:
4640:
4639:Combinatorics
4637:
4636:
4634:
4632:
4628:
4622:
4619:
4617:
4614:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4592:
4591:Real analysis
4589:
4587:
4584:
4583:
4581:
4579:
4575:
4569:
4566:
4564:
4561:
4559:
4556:
4554:
4551:
4549:
4546:
4544:
4541:
4539:
4536:
4534:
4531:
4530:
4528:
4526:
4522:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4491:
4488:
4487:
4485:
4483:
4479:
4473:
4470:
4468:
4465:
4461:
4458:
4456:
4453:
4452:
4451:
4448:
4447:
4444:
4439:
4431:
4426:
4424:
4419:
4417:
4412:
4411:
4408:
4396:
4395:Ernst Zermelo
4393:
4391:
4388:
4386:
4383:
4381:
4380:Willard Quine
4378:
4376:
4373:
4371:
4368:
4366:
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4338:
4337:
4335:
4333:
4332:Set theorists
4329:
4323:
4320:
4318:
4315:
4313:
4310:
4309:
4307:
4301:
4299:
4296:
4295:
4292:
4284:
4281:
4279:
4278:KripkeâPlatek
4276:
4272:
4269:
4268:
4267:
4264:
4263:
4262:
4259:
4255:
4252:
4251:
4250:
4249:
4245:
4241:
4238:
4237:
4236:
4233:
4232:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4210:
4208:
4204:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4181:
4176:
4174:
4171:
4169:
4166:
4163:
4159:
4156:
4154:
4151:
4147:
4144:
4142:
4139:
4137:
4134:
4133:
4132:
4129:
4126:
4122:
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4103:
4101:
4098:
4094:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4051:
4048:
4046:
4043:
4042:
4041:
4038:
4036:
4033:
4031:
4028:
4026:
4023:
4021:
4018:
4015:
4011:
4008:
4006:
4003:
4001:
3998:
3997:
3995:
3989:
3986:
3985:
3982:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3958:
3956:
3953:
3951:
3948:
3946:
3943:
3940:
3937:
3935:
3932:
3931:
3929:
3927:
3923:
3915:
3914:specification
3912:
3910:
3907:
3906:
3905:
3902:
3901:
3898:
3895:
3893:
3890:
3888:
3885:
3883:
3880:
3878:
3875:
3873:
3870:
3868:
3865:
3863:
3860:
3856:
3853:
3852:
3851:
3848:
3846:
3843:
3839:
3836:
3834:
3831:
3829:
3826:
3825:
3824:
3821:
3819:
3816:
3815:
3813:
3811:
3807:
3802:
3792:
3789:
3788:
3786:
3782:
3778:
3771:
3766:
3764:
3759:
3757:
3752:
3751:
3748:
3741:
3737:
3733:
3729:
3725:
3721:
3717:
3714:
3710:
3706:
3703:
3700:
3696:
3692:
3689:
3685:
3681:
3680:
3675:
3671:
3667:
3663:
3662:
3657:
3653:
3650:
3649:
3644:
3640:
3637:
3634:
3631:
3627:
3624:
3623:
3618:
3614:
3613:
3604:
3598:
3594:
3590:
3589:
3584:
3580:
3577:
3571:
3567:
3563:
3559:
3555:
3552:
3546:
3542:
3538:
3537:
3531:
3528:
3522:
3517:
3516:
3509:
3506:
3500:
3496:
3495:
3489:
3486:
3484:0-387-94094-4
3480:
3476:
3472:
3468:
3464:
3463:Devlin, Keith
3460:
3459:
3449:
3447:0-87150-154-6
3443:
3438:
3437:
3430:
3427:
3425:0-444-85401-0
3421:
3417:
3416:
3411:
3407:
3406:
3398:
3392:
3388:
3387:
3379:
3372:
3368:
3367:
3361:
3355:
3353:
3348:
3343:
3336:
3330:
3326:
3325:
3320:
3314:
3307:
3305:0-198-53807-3
3301:
3297:
3293:
3288:
3287:
3278:
3271:
3267:
3263:
3259:
3258:
3250:
3243:
3239:
3234:
3223:
3219:
3214:
3208:unnecessary."
3202:
3198:
3193:
3186:
3182:
3178:
3174:
3170:
3165:
3158:
3156:0-631-19130-5
3152:
3148:
3141:
3133:
3132:
3127:
3123:
3116:
3109:
3107:0-195-08030-0
3103:
3099:
3098:
3093:
3087:
3080:
3078:4-87187-714-0
3074:
3070:
3069:
3064:
3058:
3051:
3047:
3043:
3037:
3033:
3029:
3028:
3023:
3017:
3004:
3000:
2996:
2992:
2985:
2972:
2965:
2959:
2952:
2948:
2944:
2940:
2936:
2929:
2916:
2909:
2903:
2889:
2885:
2879:
2871:
2866:
2862:
2858:
2854:
2847:
2840:
2836:
2832:
2828:
2824:
2820:
2813:
2806:
2792:
2788:
2781:
2768:
2764:
2758:
2751:
2747:
2743:
2737:
2733:
2728:
2727:
2721:
2717:
2711:
2698:
2694:
2688:
2680:
2678:0-674-34871-0
2674:
2670:
2666:
2660:
2653:
2651:3-7728-0466-7
2647:
2643:
2639:
2633:
2626:
2624:0-87150-154-6
2620:
2615:
2614:
2605:
2598:
2594:
2590:
2586:
2582:
2579:(in German),
2578:
2577:
2572:
2568:
2567:Cantor, Georg
2562:
2558:
2542:
2538:
2529:
2528:0-674-32449-8
2525:
2521:
2517:
2513:
2509:
2505:
2500:
2496:
2490:
2486:
2476:
2473:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2452:
2451:
2447:
2441:
2436:
2429:
2427:
2423:
2420:) to another
2419:
2415:
2411:
2407:
2381:
2355:
2329:
2324:
2322:
2318:
2314:
2311:, are common
2310:
2306:
2302:
2298:
2294:
2293:Boolean logic
2290:
2286:
2282:
2278:
2273:
2271:
2267:
2263:
2259:
2255:
2251:
2250:Venn diagrams
2247:
2243:
2238:
2236:
2232:
2222:
2219:
2215:
2211:
2207:
2203:
2199:
2194:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2141:
2136:
2132:
2128:
2124:
2120:
2118:
2114:
2110:
2107:, introduces
2106:
2102:
2098:
2097:specification
2094:
2089:
2087:
2083:
2082:Errett Bishop
2078:
2074:
2070:
2066:
2060:
2050:
2048:
2043:
2039:
2034:
2024:
2021:
2017:
2011:
2001:
1999:
1995:
1991:
1987:
1982:
1978:
1975:in which the
1974:
1970:
1966:
1965:
1960:
1955:
1945:
1943:
1942:Wadge degrees
1939:
1934:
1929:
1919:
1917:
1913:
1909:
1905:
1899:
1889:
1887:
1883:
1878:
1875:
1871:
1867:
1863:
1859:
1856:
1852:
1848:
1842:
1832:
1830:
1826:
1822:
1821:
1814:
1804:
1802:
1798:
1794:
1789:
1787:
1783:
1779:
1775:
1772:The field of
1770:
1768:
1764:
1760:
1756:
1752:
1748:
1747:Polish spaces
1744:
1740:
1735:
1725:
1723:
1719:
1715:
1711:
1710:combinatorics
1707:
1702:
1692:
1684:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1653:
1649:
1645:
1641:
1636:
1634:
1630:
1626:
1622:
1618:
1617:
1611:
1609:
1605:
1601:
1597:
1593:
1592:vector spaces
1589:
1585:
1581:
1571:
1569:
1568:Edward Nelson
1565:
1561:
1556:
1554:
1550:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1518:
1513:
1511:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1482:
1476:
1474:
1473:
1464:
1460:
1456:
1452:
1448:
1444:
1443:
1438:
1433:
1429:
1425:
1421:
1417:
1414:
1411:
1407:
1403:
1399:
1396:
1393:
1390:with that of
1389:
1385:
1382:
1381:
1379:
1375:
1373:
1367:
1365:
1361:
1356:
1353:
1352:
1351:
1350:consists of:
1349:
1345:
1340:
1338:
1334:
1330:
1326:
1325:Venn diagrams
1316:
1302:
1282:
1260:
1256:
1235:
1211:
1203:
1187:
1180:
1176:
1172:
1167:
1166:
1160:
1151:
1146:
1136:
1134:
1130:
1126:
1123:, the set of
1122:
1105:
1085:
1066:
1061:
1060:
1056:
1049:
1043:
1037:
1031:
1024:
1020:
1015:
1014:ordered pairs
1010:
1006:
1000:
994:
989:
988:
984:
979:
975:
971:
967:
959:
955:
951:
947:
928:
922:
916:
912:
906:
902:
896:
890:
885:
884:
880:
877:
876:Venn diagrams
873:
872:universal set
868:
862:
858:
852:
846:
840:
834:
829:
828:
822:
818:
812:
806:
784:
778:
772:
768:
762:
756:
751:
750:
746:
730:
724:
718:
714:
708:
702:
697:
696:
692:
676:
670:
664:
660:
654:
648:
643:
642:
638:
637:
636:
634:
630:
626:
621:
606:
600:
594:
588:
582:
577:
576:proper subset
572:
567:
566:
565:proper subset
544:
540:
534:
529:
528:
522:
516:
510:
505:
504:set inclusion
500:
496:
492:
486:
481:
477:
476:
470:
464:
458:
453:
447:
443:
433:
431:
430:real analysis
427:
423:
419:
415:
410:
408:
404:
397:
393:
384:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
342:
338:
332:
330:
326:
322:
318:
314:
310:
306:
302:
301:
296:
292:
287:
285:
281:
278:that studies
277:
273:
262:
257:
255:
250:
248:
243:
242:
240:
239:
236:
232:
228:
227:
216:
213:
211:
208:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
172:
166:
165:
158:
154:
151:
149:
146:
144:
140:
137:
135:
132:
130:
126:
123:
121:
118:
116:
113:
111:
110:Number theory
108:
107:
104:
99:
98:
95:
94:
89:
86:
84:
81:
80:
79:
78:
75:
72:
71:
67:
66:
61:
57:
53:
48:
44:
40:
33:
19:
18:Set-theoretic
7823:Georg Cantor
7719:Cyberwarfare
7378:Cryptography
6759:
6557:Ultraproduct
6404:Model theory
6369:Independence
6305:Formal proof
6297:Proof theory
6280:
6253:
6210:real numbers
6182:second-order
6093:Substitution
5970:Metalanguage
5911:conservative
5884:Axiom schema
5828:Constructive
5798:MorseâKelley
5764:Set theories
5743:Aleph number
5736:inaccessible
5642:Grothendieck
5526:intersection
5451:
5413:Higher-order
5401:Second-order
5347:Truth tables
5304:Venn diagram
5087:Formal proof
5006:
4994:
4982:
4963:
4896:Optimization
4758:Differential
4682:Differential
4649:Order theory
4644:Graph theory
4548:Group theory
4509:
4345:Georg Cantor
4340:Paul Bernays
4271:MorseâKelley
4246:
4179:
4178:Subset
4125:hereditarily
4087:Venn diagram
4045:ordered pair
3960:Intersection
3904:Axiom schema
3776:
3727:
3705:Online books
3677:
3674:"Set theory"
3659:
3646:
3633:
3620:
3587:
3561:
3535:
3514:
3493:
3466:
3435:
3413:
3385:
3378:
3365:
3360:
3351:
3342:
3323:
3313:
3285:
3277:
3261:
3255:
3249:
3233:
3213:
3192:
3176:
3164:
3146:
3140:
3129:
3115:
3096:
3086:
3067:
3057:
3026:
3022:Jech, Thomas
3016:
3006:, retrieved
2994:
2984:
2974:, retrieved
2970:
2958:
2934:
2928:
2918:, retrieved
2914:
2902:
2892:, retrieved
2890:, 2019-11-25
2887:
2878:
2860:
2856:
2846:
2822:
2818:
2805:
2795:, retrieved
2790:
2787:"Set Theory"
2780:
2770:, retrieved
2766:
2757:
2725:
2710:
2700:, retrieved
2696:
2687:
2668:
2659:
2641:
2632:
2612:
2604:
2580:
2574:
2561:
2541:
2519:
2512:Hermann Weyl
2508:Julius König
2489:
2475:Venn diagram
2406:real numbers
2325:
2299:. Likewise,
2284:
2274:
2239:
2228:
2195:
2191:Stone spaces
2175:topos theory
2169:
2138:
2121:
2111:, a type of
2090:
2085:
2062:
2037:
2036:
2015:
2013:
1962:
1957:
1932:
1931:
1903:
1901:
1879:
1873:
1861:
1857:
1846:
1844:
1828:
1818:
1816:
1790:
1771:
1751:pointclasses
1738:
1737:
1720:such as the
1705:
1704:
1690:
1663:set theory,
1637:
1633:real numbers
1614:
1612:
1577:
1574:Applications
1557:
1544:
1540:
1514:
1493:
1485:
1479:
1477:
1470:
1468:
1439:
1406:Peano axioms
1377:
1371:
1363:
1359:
1354:
1341:
1336:
1322:
1201:
1163:
1156:
1132:
1125:real numbers
1118:
1103:
1064:
1057:
1047:
1041:
1035:
1029:
1022:
1018:
1008:
1004:
998:
992:
985:
977:
973:
969:
965:
957:
953:
949:
945:
926:
920:
914:
910:
904:
900:
894:
888:
881:
866:
860:
856:
850:
844:
838:
832:
825:
820:
816:
810:
804:
782:
776:
770:
766:
760:
754:
747:
728:
722:
716:
712:
706:
700:
698:of the sets
695:Intersection
693:
688:{1, 2, 3, 4}
674:
668:
662:
658:
652:
646:
644:of the sets
639:
622:
604:
598:
592:
586:
580:
575:
574:is called a
570:
568:is defined.
563:
554:, and so is
542:
538:
532:
525:
520:
514:
508:
503:
501:
494:
490:
484:
479:
473:
468:
462:
456:
449:
414:Zeno of Elea
411:
403:Georg Cantor
400:
396:Georg Cantor
333:
298:
295:Georg Cantor
288:
271:
270:
142:
56:intersection
52:Venn diagram
43:
7729:Video games
7709:Digital art
7466:Concurrency
7335:Data mining
7247:Probability
6980:Interpreter
6667:Type theory
6615:undecidable
6547:Truth value
6434:equivalence
6113:non-logical
5726:Enumeration
5716:Isomorphism
5663:cardinality
5647:Von Neumann
5612:Ultrafilter
5577:Uncountable
5511:equivalence
5428:Quantifiers
5418:Fixed-point
5387:First-order
5267:Consistency
5252:Proposition
5229:Traditional
5200:Lindström's
5190:Compactness
5132:Type theory
5077:Cardinality
5008:WikiProject
4851:Game theory
4831:Probability
4568:Homological
4558:Multilinear
4538:Commutative
4515:Type theory
4482:Foundations
4438:mathematics
4370:Thomas Jech
4213:Alternative
4192:Uncountable
4146:Ultrafilter
4005:Cardinality
3909:replacement
3850:Determinacy
3695:Mengenlehre
3583:Tiles, Mary
3294:, pp.
3238:Rodych 2018
3218:Rodych 2018
3197:Rodych 2018
3169:Rodych 2018
2863:(6): 1165,
2720:Fomin, S.V.
2321:programming
2113:circularity
2101:replacement
2053:Controversy
2020:meagre sets
1933:Determinacy
1928:Determinacy
1922:Determinacy
1882:determinacy
1847:inner model
1600:Equivalence
1523:instead of
1515:Systems of
1484:systems of
1432:replacement
1410:finite sets
740:is the set
686:is the set
377:consistency
373:real number
319:), various
284:mathematics
200:Linguistics
190:Computation
185:Geosciences
148:Probability
74:Mathematics
7808:Set theory
7802:Categories
7787:Glossaries
7659:E-commerce
7252:Statistics
7195:Algorithms
7153:Stochastic
6985:Middleware
6841:Peripheral
6478:elementary
6171:arithmetic
6039:Quantifier
6017:functional
5889:Expression
5607:Transitive
5551:identities
5536:complement
5469:hereditary
5452:Set theory
4836:Statistics
4715:Arithmetic
4677:Arithmetic
4543:Elementary
4510:Set theory
4365:Kurt Gödel
4350:Paul Cohen
4187:Transitive
3955:Identities
3939:Complement
3926:Operations
3887:Regularity
3855:projective
3818:Adjunction
3777:Set theory
3617:Set Theory
3050:1007.03002
3027:Set Theory
3008:2023-12-07
2995:dx.doi.org
2976:2022-07-29
2951:0268.26001
2920:2022-07-29
2894:2022-07-27
2825:: 97â110,
2797:2020-08-20
2772:2020-08-20
2741:0486612260
2702:2020-08-20
2553:References
2499:antinomies
2270:term logic
2266:inferences
2183:computable
1959:Paul Cohen
1801:invariants
1767:Borel sets
1490:urelements
1488:(allowing
1472:urelements
1428:separation
1392:separation
1355:Sets alone
1248:, the set
1068:, denoted
1002:, denoted
898:, denoted
827:complement
764:, denoted
710:, denoted
656:, denoted
625:arithmetic
536:, denoted
460:and a set
349:philosophy
272:Set theory
210:Philosophy
153:Statistics
143:Set theory
7608:Rendering
7603:Animation
7234:computing
7185:Semantics
6876:Processor
6749:Supertask
6652:Recursion
6610:decidable
6444:saturated
6422:of models
6345:deductive
6340:axiomatic
6260:Hilbert's
6247:Euclidean
6228:canonical
6151:axiomatic
6083:Signature
6012:Predicate
5901:Extension
5823:Ackermann
5748:Operation
5627:Universal
5617:Recursive
5592:Singleton
5587:Inhabited
5572:Countable
5562:Types of
5546:power set
5516:partition
5433:Predicate
5379:Predicate
5294:Syllogism
5284:Soundness
5257:Inference
5247:Tautology
5149:paradoxes
4763:Geometric
4753:Algebraic
4692:Euclidean
4667:Algebraic
4563:Universal
4298:Paradoxes
4218:Axiomatic
4197:Universal
4173:Singleton
4168:Recursive
4111:Countable
4106:Amorphous
3965:Power set
3882:Power set
3833:dependent
3828:countable
3684:EMS Press
3666:EMS Press
2597:199545885
2313:datatypes
2305:multisets
2258:John Venn
2233:early in
2161:Goodstein
2069:Kronecker
1743:real line
1608:relations
1584:manifolds
1570:in 1977.
1440:Sets and
1380:include:
1370:axiom of
1368:with the
1283:α
1261:α
1236:α
1188:α
1157:A set is
1129:empty set
1062:of a set
1059:Power set
936:{2, 3, 4}
932:{1, 2, 3}
738:{2, 3, 4}
734:{1, 2, 3}
684:{2, 3, 4}
680:{1, 2, 3}
627:features
618:{1, 2, 3}
610:{1, 2, 3}
552:{1, 2, 3}
365:logicians
361:paradoxes
305:paradoxes
215:Education
205:Economics
180:Chemistry
7767:Category
7595:Graphics
7370:Security
7032:Compiler
6931:Networks
6828:Hardware
6734:Logicism
6727:timeline
6703:Concrete
6562:Validity
6532:T-schema
6525:Kripke's
6520:Tarski's
6515:semantic
6505:Strength
6454:submodel
6449:spectrum
6417:function
6265:Tarski's
6254:Elements
6241:geometry
6197:Robinson
6118:variable
6103:function
6076:spectrum
6066:Sentence
6022:variable
5965:Language
5918:Relation
5879:Automata
5869:Alphabet
5853:language
5707:-jection
5685:codomain
5671:Function
5632:Universe
5602:Infinite
5506:Relation
5289:Validity
5279:Argument
5177:theorem,
4984:Category
4740:Topology
4687:Discrete
4672:Analytic
4659:Geometry
4631:Discrete
4586:Calculus
4578:Analysis
4533:Abstract
4472:Glossary
4455:Timeline
4302:Problems
4206:Theories
4182:Superset
4158:Infinite
3987:Concepts
3867:Infinity
3784:Overview
3736:archived
3693:(1898).
3585:(2004),
3465:(1993),
3412:(1980),
3296:xii, 347
3094:(1998),
3065:(1967),
3024:(2003),
2839:15231169
2722:(1970),
2667:(1979),
2569:(1874),
2432:See also
2380:integers
2291:, since
2262:validity
2242:New Math
2216:and the
2135:finitism
2023:theory.
1761:and the
1675:and the
1657:Metamath
1644:topology
1451:strength
1420:powerset
1348:ontology
1331:and the
1139:Ontology
1133:null set
1127:and the
1027:, where
886:of sets
623:Just as
422:infinity
337:infinity
315:and the
129:Analysis
125:Calculus
115:Geometry
7777:Outline
6676:Related
6473:Diagram
6371: (
6350:Hilbert
6335:Systems
6330:Theorem
6208:of the
6153:systems
5933:Formula
5928:Grammar
5844: (
5788:General
5501:Forcing
5486:Element
5406:Monadic
5181:paradox
5122:Theorem
5058:General
4996:Commons
4778:Applied
4748:General
4525:Algebra
4450:History
4240:General
4235:Zermelo
4141:subbase
4123: (
4062:Forcing
4040:Element
4012: (
3990:Methods
3877:Pairing
3740:YouTube
3686:, 2001
3668:, 2001
3645:, eds.
3349:at the
3128:(ed.),
2971:Ams.org
2943:0357694
2750:1527264
2157:Dummett
2153:Bernays
2149:Kreisel
1964:forcing
1948:Forcing
1868:or the
1753:in the
1629:natural
1562:called
1362:ermeloâ
802:. When
633:numbers
518:, then
480:element
387:History
195:Biology
175:Physics
120:Algebra
83:History
58:of two
6439:finite
6202:Skolem
6155:
6130:Theory
6098:Symbol
6088:String
6071:atomic
5948:ground
5943:closed
5938:atomic
5894:ground
5857:syntax
5753:binary
5680:domain
5597:Finite
5362:finite
5220:Logics
5179:
5127:Theory
4697:Finite
4553:Linear
4460:Future
4436:Major
4131:Filter
4121:Finite
4057:Family
4000:Almost
3838:global
3823:Choice
3810:Axioms
3599:
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1580:graphs
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1424:choice
1422:, and
1109:{1, 2}
940:{1, 4}
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596:, but
560:{1, 4}
548:{1, 2}
527:subset
475:member
355:, and
7180:Logic
7014:tools
6429:Model
6177:Peano
6034:Proof
5874:Arity
5803:Naive
5690:image
5622:Fuzzy
5582:Empty
5531:union
5476:Class
5117:Model
5107:Lemma
5065:Axiom
4924:lists
4467:Lists
4440:areas
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4153:Fuzzy
4116:Empty
4099:types
4050:tuple
4020:Class
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3975:Union
3892:Union
3124:, in
2967:(PDF)
2911:(PDF)
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2593:S2CID
2481:Notes
2426:range
2424:(the
2416:(the
2309:lists
2077:naive
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7012:and
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6552:Type
6355:list
6159:list
6136:list
6125:Term
6059:rank
5953:open
5847:list
5659:Maps
5564:sets
5423:Free
5393:list
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5070:list
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3711:and
3597:ISBN
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3329:ISBN
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3173:§2.1
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2746:OCLC
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2328:sets
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2307:and
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2189:and
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2099:and
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1667:and
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558:but
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