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),
6756:
4960:
2440:
231:
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
3801:
1150:
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).
7749:
4972:
47:
7759:
7769:
4996:
4984:
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.
406:
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:
5135:
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.
3414:
2575:
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.
5810:
4258:
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).
1446:
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
4420:
3256:
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
258:
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.
5893:
5034:
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
6792:
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
2644:, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich Frommann Verlag, p. 152,
2139:
2009:
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
6207:
7502:
7474:
7527:
6365:
3600:
3573:
3548:
3524:
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3332:
3039:
5153:
2144:
7378:
6220:
5543:
4783:
3947:
3767:
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.
194:
5000:
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
2229:
As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics of
7532:
6811:
3621:
3130:
2058:
5805:
4275:
1462:
7044:
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5703:
5149:
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3303:
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3105:
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251:
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in the first half of the 19th century. Modern understanding of infinity began in 1870â1874, and was motivated by Cantor's work in
7772:
6873:
6458:
6202:
5027:
4464:
1993:
1773:
363:, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for
5763:
5456:
5197:
4253:
2530:(pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann's 1925 paper.
1915:
1672:
1358:
324:
7160:
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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
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that a broader class of games is determined often implies that a broader class of sets will have a topological property. The
1505:
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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,
1415:
7674:
7262:
7170:
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6858:
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observed that "set theory in its first, "naive" version, due to Cantor, led to contradictions. These are the well-known
1579:
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244:
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1817:
In set theory as Cantor defined and
Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In
1458:
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2100:
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87:
31:
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1173:, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by
3241:
2163:
all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as
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Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using
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are commonly referred to in mathematical teaching when talking about different types of numbers (the sets
2280:
2126:
2046:
2032:
1854:
1733:
1607:
204:
179:
82:
371:. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the
7808:
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7659:
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7252:
7242:
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4570:
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4240:
4154:
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3513:
2409:
2234:
2209:
2197:
2130:
1963:
1953:
1907:
1680:
1639:
1615:
1527:. Yet other systems accept classical logic but feature a nonstandard membership relation. These include
1174:
356:
214:
128:
6755:
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2439:
1071:
230:
7459:
7431:
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1997:
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1520:
1343:
1332:
1328:
1170:
1164:
1144:
882:
316:
308:
174:
133:
102:
3712:
2570:
1475:, objects that can be members of sets but that are not themselves sets and do not have any members.
7793:
7724:
7649:
7565:
7550:
7315:
7102:
7059:
7054:
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6544:
6431:
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5925:
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5354:
5254:
5182:
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5169:
5104:
5099:
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4848:
4770:
4669:
4603:
4560:
4550:
4530:
4309:
4220:
4133:
4128:
4123:
3937:
3879:
3817:
3753:
3290:, International Series of Monographs on Computer Science, Oxford Science Publications, Oxford, UK:
3180:
2425:
2122:
2112:
1911:
1717:
1713:
1668:
1659:, includes human-written, computer-verified derivations of more than 12,000 theorems starting from
1563:
312:
2786:
2385:
2359:
2333:
2221:
a detailed analysis of the formulations required in order to derive various mathematical results.
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7257:
7247:
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6529:
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6105:
6031:
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1383:
1251:
640:
417:
344:
275:
234:
138:
38:
6257:
5713:
3704:
3673:
1853:
that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the
3121:
7709:
7639:
7618:
7580:
7388:
7355:
7335:
7034:
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6820:
6695:
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5871:
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5569:
5371:
5314:
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4002:
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3125:
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3031:
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2938:
2811:
2637:
2353:
2213:
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2108:
1989:
1980:
1869:
1762:
1754:
1676:
1524:
1501:
1480:
1423:
1369:
635:, set theory features binary operations on sets. The following is a partial list of them:
451:
445:
425:
328:
156:
2692:
2507:
1906:
is a cardinal number with an extra property. Many such properties are studied, including
1691:
Set theory is a major area of research in mathematics with many interrelated subfields:
7719:
7623:
7522:
7368:
7340:
6675:
6654:
6612:
6592:
6487:
6342:
5940:
5930:
5920:
5915:
5849:
5723:
5599:
5488:
5483:
5461:
5062:
4911:
4838:
4545:
4382:
4179:
4160:
4064:
4049:
4006:
3942:
3884:
3434:
2253:
2245:
2182:
2164:
1985:
1885:
1824:
1683:, enhancing the understanding of well-established models of evolution and interaction.
1628:
1603:
1536:
1298:
1207:
1178:
1120:
748:
474:
380:
184:
3629:
2991:"Unifying evolutionary dynamics: a set theory exploration of symmetry and interaction"
1149:
1131:âthe unique set containing no elements. The empty set is also occasionally called the
17:
7787:
7608:
6903:
6649:
6327:
5834:
5619:
5609:
5579:
5564:
5234:
4699:
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4583:
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4357:
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4103:
4098:
3409:
3346:
3062:
2664:
2596:
2292:
2148:
2081:
1709:
1567:
930:(elements which are in one of the sets, but not in both). For instance, for the sets
871:
564:
429:
109:
3030:, Springer Monographs in Mathematics (Third Millennium ed.), Berlin, New York:
2869:
2852:
2838:
7704:
7363:
6549:
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6169:
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2474:
2249:
2174:
2152:
1968:
1941:
1850:
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1591:
1441:
1405:
1342:
The most widely studied systems of axiomatic set theory imply that all sets form a
1324:
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413:
402:
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51:
46:
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3534:
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6659:
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5708:
5655:
5339:
5259:
5244:
5124:
5069:
4843:
4507:
4430:
4362:
3997:
3494:
Labyrinth of
Thought: A History of Set Theory and Its Role in Modern Mathematics
3021:
2762:
2405:
2204:. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with
2190:
1927:
1632:
1124:
391:
376:
372:
283:
147:
73:
3708:
2990:
502:
A derived binary relation between two sets is the subset relation, also called
7714:
7644:
7237:
6977:
6833:
5589:
5415:
5221:
4828:
4707:
4342:
4113:
3582:
3474:
3002:
2830:
2435:
2269:
2244:
experiment aimed to teach basic set theory, among other abstract concepts, to
2019:
1750:
1409:
624:
348:
286:â is mostly concerned with those that are relevant to mathematics as a whole.
152:
3588:
The
Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise
2588:
2119:
has said that "all of scientifically applicable analysis can be developed ".
7219:
7180:
6741:
6644:
5697:
5614:
5574:
5538:
5474:
5286:
5276:
5249:
4145:
4108:
4059:
3957:
2265:
2257:
1766:
1742:
1583:
1528:
1489:
1471:
1128:
1058:
360:
3269:
2749:
2571:"Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"
4398:
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
7280:
6726:
6524:
5972:
5677:
5271:
4732:
4651:
4578:
2642:
Einleitung zur GröĂenlehre und erste
Begriffe der allgemeinen GröĂenlehre
2498:
2312:
2304:
2241:
2134:
1656:
1643:
1347:
1158:
1135:, though this name is ambiguous and can lead to several interpretations.
421:
336:
124:
114:
1053:{1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
6322:
5114:
4517:
3739:
2379:
119:
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.
5012:
4170:
3992:
2793:(Spring 2020 ed.), Metaphysics Research Lab, Stanford University
632:
526:
30:
This article is about the branch of mathematics. For other uses, see
2252:
are widely employed to explain basic set-theoretic relationships to
6770:
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
1228:
containing only the empty set is assigned rank 1. For each ordinal
5866:
5212:
5057:
4042:
3809:
3647:
1148:
364:
45:
3386:
Hegel's Rabble: An Investigation into Hegel's Philosophy of Right
1757:
and extends to the study of more complex hierarchies such as the
3350:
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
6774:
5016:
4402:
3749:
2208:
of sets arising from the inductive and recursive properties of
612:, but are not subsets of it; and in turn, the subsets, such as
3282:
Cantone, Domenico; Ferro, Alfredo; Omodeo, Eugenio G. (1989),
2279:(NOT, AND, OR), and semantic or rule description (technically
1972:
1660:
1606:
are ubiquitous in mathematics, and the theory of mathematical
1559:
1552:
1454:
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
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
2669:
Georg Cantor: His Mathematics and Philosophy of the Infinite
1339:
was originally devised to rid set theory of such paradoxes.
1077:
1275:
is defined to consist of all pure sets with rank less than
3366:
Homotopy Type Theory: Univalent Foundations of Mathematics
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),
1880:
The study of inner models is common in the study of
1849:
of ZermeloâFraenkel set theory (ZF) is a transitive
7632:
7579:
7541:
7488:
7450:
7412:
7354:
7271:
7217:
7179:
7131:
7068:
7001:
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6922:
6886:
6819:
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6563:
6395:
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5833:
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5443:
5370:
5305:
5220:
5211:
5133:
5050:
4902:
4859:
4769:
4731:
4698:
4650:
4622:
4569:
4516:
4473:
4323:
4286:
4198:
4088:
3976:
3917:
3808:
3783:
2147:after having only read the abstract. As reviewers
3512:
3433:
3415:Set Theory: An Introduction to Independence Proofs
3283:
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
2167:begun to rehabilitate Wittgenstein's arguments.
1539:embodying the membership relation is not simply
1346:. Such systems come in two flavors, those whose
2763:"set theory | Basics, Examples, & Formulas"
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
2935:Number Systems and the Foundations of Analysis
1996:by finitistic methods, the other method being
6810:Note: This template roughly follows the 2012
6786:
5028:
4414:
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:
8:
3738:from the original on 2021-10-31 – via
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.
2884:"6.3: Equivalence Relations and Partitions"
2303:and other collection-like objects, such as
1469:The above systems can be modified to allow
6793:
6779:
6771:
5854:
5449:
5217:
5035:
5021:
5013:
4421:
4407:
4399:
4210:
3768:
3754:
3746:
3122:"Wittgenstein's Philosophy of Mathematics"
2522:, Harvard University Press, Cambridge MA,
259:
245:
96:
64:
2868:
2390:
2389:
2387:
2364:
2363:
2361:
2338:
2337:
2335:
2140:Remarks on the Foundations of Mathematics
2010:Cardinal characteristics of the continuum
1300:
1280:
1259:
1253:
1233:
1209:
1185:
1076:
1075:
1073:
1051:. For example, the Cartesian product of
848:is clear from the context, the notation
390:
2791:The Stanford Encyclopedia of Philosophy
2558:
2538:
2486:
794:, while conversely, the set difference
228:
72:
27:Branch of mathematics that studies sets
7503:Knowledge representation and reasoning
3724:"Set Theory: An Offspring of Analysis"
3389:, Bloomsbury Publishing, p. 151,
3237:
3217:
3196:
3168:
2287:"), which may be useful when learning
1465:, both of which are stronger than ZFC.
1418:, which omits the axioms of infinity,
7528:Philosophy of artificial intelligence
3630:"The Early Development of Set Theory"
3369:. The Univalent Foundations Program.
1979:fails, or a model of ZF in which the
678:, or both. For example, the union of
450:Set theory begins with a fundamental
7:
6854:Energy consumption (Green computing)
4983:
3562:Set Theory and the Continuum Problem
3068:Foundations of Constructive Analysis
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
7533:Distributed artificial intelligence
6812:ACM Computing Classification System
4995:
3622:Internet Encyclopedia of Philosophy
3131:Stanford Encyclopedia of Philosophy
2143:: Wittgenstein attempted to refute
2091:A different objection put forth by
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
7045:Integrated development environment
2196:An active area of research is the
938:, the symmetric difference set is
339:, and has various applications in
25:
7513:Automated planning and scheduling
7050:Software configuration management
3732:University of Wisconsin-Milwaukee
3147:Philosophical Remarks, §129, §174
2812:"The iterative conception of set"
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
7767:
7757:
7748:
7747:
6754:
4994:
4982:
4971:
4970:
4958:
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:
7758:
7161:Computational complexity theory
4879:Computational complexity theory
3120:Rodych, Victor (Jan 31, 2018),
2908:"Order Relations and Functions"
2870:10.1090/S0002-9904-1977-14398-X
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
1496:(lacking them), associate with
774:, is the set of all members of
6952:Network performance evaluation
1823:this condition was relaxed by
1749:. It begins with the study of
1708:concerns extensions of finite
1463:TarskiâGrothendieck set theory
1088:
1082:
1:
7316:Multimedia information system
7301:Geographic information system
7291:Enterprise information system
6887:Computer systems organization
6715: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.),
2408:, etc.), and when defining a
854:is sometimes used instead of
616:, are not members of the set
7675:Computational social science
7263:Theoretical computer science
7083:Software development process
6859:Electronic design automation
6844:Very Large Scale Integration
6640: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
7498:Natural language processing
7286:Information storage systems
3679:Encyclopedia of Mathematics
3661:Encyclopedia of Mathematics
3228:has a particular property."
3149:, Oxford: Basil Blackwell,
2933:Mendelson, Elliott (1973),
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 }}
436:Basic concepts and notation
369:philosophers of mathematics
325:ZermeloâFraenkel set theory
32:Set theory (disambiguation)
7825:
7414:Humanâcomputer interaction
7384:Intrusion detection system
7296:Social information systems
7281:Database management system
5704:SchröderâBernstein theorem
5431:Monadic predicate calculus
5090:Foundations of mathematics
4929:Films about mathematicians
4259: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:
1925:
1895:
1838:
1810:
1776:is between set theory and
1731:
1698:
1142:
439:
343:(such as in the theory of
169:Relationship with sciences
36:
29:
7743:
7680:Computational engineering
7655:Computational mathematics
6808:
6750:
6737:Philosophy of mathematics
6686:Automated theorem proving
5857:
5811:Von NeumannâBernaysâGödel
5452:
4952:
4498:Philosophy of mathematics
4438:
4060:One-to-one correspondence
3797:
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
7690:Computational healthcare
7685:Differentiable computing
7604:Graphics processing unit
7030:Domain-specific language
6899:Computational complexity
4934: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
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
7665:Computational chemistry
7599:Photograph manipulation
7490:Artificial intelligence
7306:Decision support system
6387:Self-verifying theories
6208:Tarski's axiomatization
5159:Tarski's undefinability
5154:incompleteness theorems
4819:Mathematical statistics
4809:Mathematical psychology
4779:Engineering mathematics
4713: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 }
7730:Educational technology
7561:Reinforcement learning
7311:Process control system
7209:Computational geometry
7199:Algorithmic efficiency
7194:Analysis of algorithms
6849:Systems on Chip (SoCs)
6761:Mathematics portal
6372:Proof of impossibility
6020:propositional variable
5330:Propositional calculus
4965:Mathematics portal
4814:Mathematical sociology
4794:Mathematical economics
4789:Mathematical chemistry
4718:Analytic number theory
4599:Differential equations
4018: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:
18:Axiomatic set theories
7700:Electronic publishing
7670:Computational biology
7660:Computational physics
7556:Unsupervised learning
7470:Distributed computing
7346:Information retrieval
7253:Mathematical analysis
7243:Mathematical software
7133:Theory of computation
7098:Software construction
7088:Requirements analysis
6966:Software organization
6894:Computer architecture
6864:Hardware acceleration
6829:Printed circuit board
6630:Kolmogorov complexity
6583:Computably enumerable
6483:Model complete theory
6275:Principia Mathematica
5335:Propositional formula
5164:BanachâTarski paradox
4944:Mathematics education
4874:Theory of computation
4594:Hypercomplex analysis
4241:Principia Mathematica
4075:Transfinite induction
3934:(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:
7460:Concurrent computing
7432:Ubiquitous computing
7404:Application security
7399:Information security
7228:Discrete mathematics
7204:Randomized algorithm
7156:Computability theory
7141:Model of computation
7113:Software maintenance
7108:Software engineering
7070:Software development
7020:Programming language
7015:Programming paradigm
6932:Network architecture
6578:ChurchâTuring thesis
6565:Computability theory
5774:continuum hypothesis
5292:Square of opposition
5150:Gödel's completeness
4924:Informal mathematics
4804:Mathematical physics
4799:Mathematical finance
4784:Mathematical biology
4723:Diophantine geometry
4315:Burali-Forti paradox
4070:Set-builder notation
4023:Continuum hypothesis
3963: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
7735:Document management
7725:Operations research
7650:Enterprise software
7566:Multi-task learning
7551:Supervised learning
7273:Information systems
7103:Software deployment
7060:Software repository
6914:Real-time computing
6732:Mathematical object
6623:P versus NP problem
6588:Computable function
6382:Reverse mathematics
6308:Logical consequence
6185:primitive recursive
6180:elementary function
5953:Free/bound variable
5806:TarskiâGrothendieck
5325:Logical connectives
5255:Logical equivalence
5105:Logical consequence
4939:Mathematics and art
4849:Operations research
4604:Functional analysis
4276: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
7799:Mathematical logic
7518:Search methodology
7465:Parallel computing
7422:Interaction design
7331:Computing platform
7258:Numerical analysis
7248:Information theory
7040:Software framework
7003:Software notations
6942:Network components
6839:Integrated circuit
6530:Transfer principle
6493:Semantics of logic
6478:Categorical theory
6454:Non-standard model
5968:Logical connective
5095:Information theory
5044:Mathematical logic
4884:Numerical analysis
4493:Mathematical logic
4488:Information theory
3865: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)
7781:
7780:
7710:Electronic voting
7640:Quantum Computing
7633:Applied computing
7619:Image compression
7389:Hardware security
7379:Security services
7336:Digital marketing
7123:Open-source model
7035:Modeling language
6947:Network scheduler
6768:
6767:
6700:Abstract category
6503:Theories of truth
6313:Rule of inference
6303:Natural deduction
6284:
6283:
5829:
5828:
5534:Cartesian product
5439:
5438:
5345:Many-valued logic
5320:Boolean functions
5203:Russell's paradox
5178:diagonal argument
5075:First-order logic
5010:
5009:
4609:Harmonic analysis
4396:
4395:
4305:Russell's paradox
4254:ZermeloâFraenkel
4155:Dedekind-infinite
4028:Diagonal argument
3927: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
7816:
7771:
7770:
7761:
7760:
7751:
7750:
7571:Cross-validation
7543:Machine learning
7427:Social computing
7394:Network security
7189:Algorithm design
7118:Programming team
7078:Control variable
7055:Software library
6993:Software quality
6988:Operating system
6937:Network protocol
6802:Computer science
6795:
6788:
6781:
6772:
6759:
6758:
6710:History of logic
6705:Category of sets
6598:Decision problem
6377:Ordinal analysis
6318:Sequent calculus
6216:Boolean algebras
6156:
6155:
6130:
6101:logical/constant
5855:
5841:
5764:ZermeloâFraenkel
5515:Set operations:
5450:
5387:
5218:
5198:LöwenheimâSkolem
5085:Formal semantics
5037:
5030:
5023:
5014:
4998:
4997:
4986:
4985:
4974:
4973:
4963:
4962:
4894:Computer algebra
4869:Computer science
4589:Complex analysis
4423:
4416:
4409:
4400:
4378:Bertrand Russell
4368:John von Neumann
4353:Abraham Fraenkel
4348:Richard Dedekind
4310:Suslin's problem
4221:Cantor's theorem
3938:De Morgan's laws
3803:
3770:
3763:
3756:
3747:
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:
65:
21:
7824:
7823:
7819:
7818:
7817:
7815:
7814:
7813:
7784:
7783:
7782:
7777:
7768:
7739:
7720:Word processing
7628:
7614:Virtual reality
7575:
7537:
7508:Computer vision
7484:
7480:Multiprocessing
7446:
7408:
7374:Security hacker
7350:
7326:Digital library
7267:
7218:Mathematics of
7213:
7175:
7151:Automata theory
7146:Formal language
7127:
7093:Software design
7064:
6997:
6983:Virtual machine
6961:
6957:Network service
6918:
6909:Embedded system
6882:
6815:
6804:
6799:
6769:
6764:
6753:
6746:
6691:Category theory
6681:Algebraic logic
6664:
6635:Lambda calculus
6573:Church encoding
6559:
6535:Truth predicate
6391:
6357:Complete theory
6280:
6149:
6145:
6141:
6136:
6128:
5848: and
5844:
5839:
5825:
5801:New Foundations
5769:axiom of choice
5752:
5714:Gödel numbering
5654: and
5646:
5550:
5435:
5385:
5366:
5315:Boolean algebra
5301:
5265:Equiconsistency
5230:Classical logic
5207:
5188:Halting problem
5176: and
5152: and
5140: and
5139:
5134:Theorems (
5129:
5046:
5041:
5011:
5006:
4957:
4948:
4898:
4855:
4834:Systems science
4765:
4761:Homotopy theory
4727:
4694:
4646:
4618:
4565:
4512:
4483:Category theory
4469:
4434:
4427:
4397:
4392:
4319:
4298:
4282:
4247:New Foundations
4194:
4084:
4003:Cardinal number
3986:
3972:
3913:
3804:
3795:
3779:
3774:
3718:
3709:in your library
3672:
3654:
3632:article in the
3619:article in the
3612:
3603:
3581:
3576:
3556:
3551:
3532:
3527:
3510:
3505:
3490:
3485:
3461:
3458:
3456:Further reading
3453:
3448:
3431:
3426:
3408:
3404:
3403:
3397:
3382:
3381:
3377:
3363:
3359:
3345:
3341:
3335:
3317:
3316:
3312:
3306:
3292:Clarendon Press
3281:
3280:
3276:
3253:
3252:
3248:
3236:
3232:
3225:
3216:
3212:
3204:
3195:
3191:
3167:
3163:
3157:
3144:
3143:
3139:
3119:
3118:
3114:
3108:
3090:
3089:
3085:
3079:
3061:
3060:
3056:
3042:
3034:, p. 642,
3032:Springer-Verlag
3020:
3019:
3015:
3007:
3005:
2988:
2987:
2983:
2975:
2973:
2966:
2962:
2961:
2957:
2932:
2931:
2927:
2919:
2917:
2910:
2906:
2905:
2901:
2893:
2891:
2882:
2881:
2877:
2850:
2849:
2845:
2814:
2809:
2808:
2804:
2796:
2794:
2784:
2783:
2779:
2771:
2769:
2761:
2760:
2756:
2742:
2714:
2713:
2709:
2701:
2699:
2691:
2690:
2686:
2679:
2663:
2662:
2658:
2652:
2636:
2635:
2631:
2625:
2608:
2607:
2603:
2583:(77): 258â262,
2565:
2564:
2560:
2555:
2550:
2549:
2544:
2540:
2535:
2534:
2502:exemplified by
2492:
2488:
2483:
2444:
2437:
2434:
2384:
2383:
2358:
2357:
2354:natural numbers
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:
1297:
1296:
1277:
1276:
1255:
1250:
1249:
1230:
1229:
1225:
1206:
1205:
1200:, known as its
1182:
1181:
1147:
1141:
1121:natural numbers
1112:
1108:
1102:
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:
7822:
7820:
7812:
7811:
7806:
7804:Formal methods
7801:
7796:
7786:
7785:
7779:
7778:
7776:
7775:
7765:
7755:
7744:
7741:
7740:
7738:
7737:
7732:
7727:
7722:
7717:
7712:
7707:
7702:
7697:
7692:
7687:
7682:
7677:
7672:
7667:
7662:
7657:
7652:
7647:
7642:
7636:
7634:
7630:
7629:
7627:
7626:
7624:Solid modeling
7621:
7616:
7611:
7606:
7601:
7596:
7591:
7585:
7583:
7577:
7576:
7574:
7573:
7568:
7563:
7558:
7553:
7547:
7545:
7539:
7538:
7536:
7535:
7530:
7525:
7523:Control method
7520:
7515:
7510:
7505:
7500:
7494:
7492:
7486:
7485:
7483:
7482:
7477:
7475:Multithreading
7472:
7467:
7462:
7456:
7454:
7448:
7447:
7445:
7444:
7439:
7434:
7429:
7424:
7418:
7416:
7410:
7409:
7407:
7406:
7401:
7396:
7391:
7386:
7381:
7376:
7371:
7369:Formal methods
7366:
7360:
7358:
7352:
7351:
7349:
7348:
7343:
7341:World Wide Web
7338:
7333:
7328:
7323:
7318:
7313:
7308:
7303:
7298:
7293:
7288:
7283:
7277:
7275:
7269:
7268:
7266:
7265:
7260:
7255:
7250:
7245:
7240:
7235:
7230:
7224:
7222:
7215:
7214:
7212:
7211:
7206:
7201:
7196:
7191:
7185:
7183:
7177:
7176:
7174:
7173:
7168:
7163:
7158:
7153:
7148:
7143:
7137:
7135:
7129:
7128:
7126:
7125:
7120:
7115:
7110:
7105:
7100:
7095:
7090:
7085:
7080:
7074:
7072:
7066:
7065:
7063:
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7027:
7022:
7017:
7011:
7009:
6999:
6998:
6996:
6995:
6990:
6985:
6980:
6975:
6969:
6967:
6963:
6962:
6960:
6959:
6954:
6949:
6944:
6939:
6934:
6928:
6926:
6920:
6919:
6917:
6916:
6911:
6906:
6901:
6896:
6890:
6888:
6884:
6883:
6881:
6880:
6871:
6866:
6861:
6856:
6851:
6846:
6841:
6836:
6831:
6825:
6823:
6817:
6816:
6809:
6806:
6805:
6800:
6798:
6797:
6790:
6783:
6775:
6766:
6765:
6751:
6748:
6747:
6745:
6744:
6739:
6734:
6729:
6724:
6723:
6722:
6712:
6707:
6702:
6693:
6688:
6683:
6678:
6676:Abstract logic
6672:
6670:
6666:
6665:
6663:
6662:
6657:
6655:Turing machine
6652:
6647:
6642:
6637:
6632:
6627:
6626:
6625:
6620:
6615:
6610:
6605:
6595:
6593:Computable set
6590:
6585:
6580:
6575:
6569:
6567:
6561:
6560:
6558:
6557:
6552:
6547:
6542:
6537:
6532:
6527:
6522:
6521:
6520:
6515:
6510:
6500:
6495:
6490:
6488:Satisfiability
6485:
6480:
6475:
6474:
6473:
6463:
6462:
6461:
6451:
6450:
6449:
6444:
6439:
6434:
6429:
6419:
6418:
6417:
6412:
6405:Interpretation
6401:
6399:
6393:
6392:
6390:
6389:
6384:
6379:
6374:
6369:
6359:
6354:
6353:
6352:
6351:
6350:
6340:
6335:
6325:
6320:
6315:
6310:
6305:
6300:
6294:
6292:
6286:
6285:
6282:
6281:
6279:
6278:
6270:
6269:
6268:
6267:
6262:
6261:
6260:
6255:
6250:
6230:
6229:
6228:
6226:minimal axioms
6223:
6212:
6211:
6210:
6199:
6198:
6197:
6192:
6187:
6182:
6177:
6172:
6159:
6157:
6138:
6137:
6135:
6134:
6133:
6132:
6120:
6115:
6114:
6113:
6108:
6103:
6098:
6088:
6083:
6078:
6073:
6072:
6071:
6066:
6056:
6055:
6054:
6049:
6044:
6039:
6029:
6024:
6023:
6022:
6017:
6012:
6002:
6001:
6000:
5995:
5990:
5985:
5980:
5975:
5965:
5960:
5955:
5950:
5949:
5948:
5943:
5938:
5933:
5923:
5918:
5916:Formation rule
5913:
5908:
5907:
5906:
5901:
5891:
5890:
5889:
5879:
5874:
5869:
5864:
5858:
5852:
5835:Formal systems
5831:
5830:
5827:
5826:
5824:
5823:
5818:
5813:
5808:
5803:
5798:
5793:
5788:
5783:
5778:
5777:
5776:
5771:
5760:
5758:
5754:
5753:
5751:
5750:
5749:
5748:
5738:
5733:
5732:
5731:
5724:Large cardinal
5721:
5716:
5711:
5706:
5701:
5687:
5686:
5685:
5680:
5675:
5660:
5658:
5648:
5647:
5645:
5644:
5643:
5642:
5637:
5632:
5622:
5617:
5612:
5607:
5602:
5597:
5592:
5587:
5582:
5577:
5572:
5567:
5561:
5559:
5552:
5551:
5549:
5548:
5547:
5546:
5541:
5536:
5531:
5526:
5521:
5513:
5512:
5511:
5506:
5496:
5491:
5489:Extensionality
5486:
5484:Ordinal number
5481:
5471:
5466:
5465:
5464:
5453:
5447:
5441:
5440:
5437:
5436:
5434:
5433:
5428:
5423:
5418:
5413:
5408:
5403:
5402:
5401:
5391:
5390:
5389:
5376:
5374:
5368:
5367:
5365:
5364:
5363:
5362:
5357:
5352:
5342:
5337:
5332:
5327:
5322:
5317:
5311:
5309:
5303:
5302:
5300:
5299:
5294:
5289:
5284:
5279:
5274:
5269:
5268:
5267:
5257:
5252:
5247:
5242:
5237:
5232:
5226:
5224:
5215:
5209:
5208:
5206:
5205:
5200:
5195:
5190:
5185:
5180:
5168:Cantor's
5166:
5161:
5156:
5146:
5144:
5131:
5130:
5128:
5127:
5122:
5117:
5112:
5107:
5102:
5097:
5092:
5087:
5082:
5077:
5072:
5067:
5066:
5065:
5054:
5052:
5048:
5047:
5042:
5040:
5039:
5032:
5025:
5017:
5008:
5007:
5005:
5004:
4992:
4980:
4968:
4953:
4950:
4949:
4947:
4946:
4941:
4936:
4931:
4926:
4921:
4920:
4919:
4912:Mathematicians
4908:
4906:
4904:Related topics
4900:
4899:
4897:
4896:
4891:
4886:
4881:
4876:
4871:
4865:
4863:
4857:
4856:
4854:
4853:
4852:
4851:
4846:
4841:
4839:Control theory
4831:
4826:
4821:
4816:
4811:
4806:
4801:
4796:
4791:
4786:
4781:
4775:
4773:
4767:
4766:
4764:
4763:
4758:
4753:
4748:
4743:
4737:
4735:
4729:
4728:
4726:
4725:
4720:
4715:
4710:
4704:
4702:
4696:
4695:
4693:
4692:
4687:
4682:
4677:
4672:
4667:
4662:
4656:
4654:
4648:
4647:
4645:
4644:
4639:
4634:
4628:
4626:
4620:
4619:
4617:
4616:
4614:Measure theory
4611:
4606:
4601:
4596:
4591:
4586:
4581:
4575:
4573:
4567:
4566:
4564:
4563:
4558:
4553:
4548:
4543:
4538:
4533:
4528:
4522:
4520:
4514:
4513:
4511:
4510:
4505:
4500:
4495:
4490:
4485:
4479:
4477:
4471:
4470:
4468:
4467:
4462:
4457:
4456:
4455:
4450:
4439:
4436:
4435:
4428:
4426:
4425:
4418:
4411:
4403:
4394:
4393:
4391:
4390:
4385:
4383:Thoralf Skolem
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4329:
4327:
4321:
4320:
4318:
4317:
4312:
4307:
4301:
4299:
4297:
4296:
4293:
4287:
4284:
4283:
4281:
4280:
4279:
4278:
4273:
4268:
4267:
4266:
4251:
4250:
4249:
4237:
4236:
4235:
4224:
4223:
4218:
4213:
4208:
4202:
4200:
4196:
4195:
4193:
4192:
4187:
4182:
4177:
4168:
4163:
4158:
4148:
4143:
4142:
4141:
4136:
4131:
4121:
4111:
4106:
4101:
4095:
4093:
4086:
4085:
4083:
4082:
4077:
4072:
4067:
4065:Ordinal number
4062:
4057:
4052:
4047:
4046:
4045:
4040:
4030:
4025:
4020:
4015:
4010:
4000:
3995:
3989:
3987:
3985:
3984:
3981:
3977:
3974:
3973:
3971:
3970:
3965:
3960:
3955:
3950:
3945:
3943:Disjoint union
3940:
3935:
3929:
3923:
3921:
3915:
3914:
3912:
3911:
3910:
3909:
3904:
3893:
3892:
3890:Martin's axiom
3887:
3882:
3877:
3872:
3867:
3862:
3857:
3855:Extensionality
3852:
3847:
3842:
3841:
3840:
3835:
3830:
3820:
3814:
3812:
3806:
3805:
3798:
3796:
3794:
3793:
3787:
3785:
3781:
3780:
3775:
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7821:
7810:
7807:
7805:
7802:
7800:
7797:
7795:
7792:
7791:
7789:
7774:
7766:
7764:
7756:
7754:
7746:
7745:
7742:
7736:
7733:
7731:
7728:
7726:
7723:
7721:
7718:
7716:
7713:
7711:
7708:
7706:
7703:
7701:
7698:
7696:
7693:
7691:
7688:
7686:
7683:
7681:
7678:
7676:
7673:
7671:
7668:
7666:
7663:
7661:
7658:
7656:
7653:
7651:
7648:
7646:
7643:
7641:
7638:
7637:
7635:
7631:
7625:
7622:
7620:
7617:
7615:
7612:
7610:
7609:Mixed reality
7607:
7605:
7602:
7600:
7597:
7595:
7592:
7590:
7587:
7586:
7584:
7582:
7578:
7572:
7569:
7567:
7564:
7562:
7559:
7557:
7554:
7552:
7549:
7548:
7546:
7544:
7540:
7534:
7531:
7529:
7526:
7524:
7521:
7519:
7516:
7514:
7511:
7509:
7506:
7504:
7501:
7499:
7496:
7495:
7493:
7491:
7487:
7481:
7478:
7476:
7473:
7471:
7468:
7466:
7463:
7461:
7458:
7457:
7455:
7453:
7449:
7443:
7442:Accessibility
7440:
7438:
7437:Visualization
7435:
7433:
7430:
7428:
7425:
7423:
7420:
7419:
7417:
7415:
7411:
7405:
7402:
7400:
7397:
7395:
7392:
7390:
7387:
7385:
7382:
7380:
7377:
7375:
7372:
7370:
7367:
7365:
7362:
7361:
7359:
7357:
7353:
7347:
7344:
7342:
7339:
7337:
7334:
7332:
7329:
7327:
7324:
7322:
7319:
7317:
7314:
7312:
7309:
7307:
7304:
7302:
7299:
7297:
7294:
7292:
7289:
7287:
7284:
7282:
7279:
7278:
7276:
7274:
7270:
7264:
7261:
7259:
7256:
7254:
7251:
7249:
7246:
7244:
7241:
7239:
7236:
7234:
7231:
7229:
7226:
7225:
7223:
7221:
7216:
7210:
7207:
7205:
7202:
7200:
7197:
7195:
7192:
7190:
7187:
7186:
7184:
7182:
7178:
7172:
7169:
7167:
7164:
7162:
7159:
7157:
7154:
7152:
7149:
7147:
7144:
7142:
7139:
7138:
7136:
7134:
7130:
7124:
7121:
7119:
7116:
7114:
7111:
7109:
7106:
7104:
7101:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7081:
7079:
7076:
7075:
7073:
7071:
7067:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7016:
7013:
7012:
7010:
7008:
7004:
7000:
6994:
6991:
6989:
6986:
6984:
6981:
6979:
6976:
6974:
6971:
6970:
6968:
6964:
6958:
6955:
6953:
6950:
6948:
6945:
6943:
6940:
6938:
6935:
6933:
6930:
6929:
6927:
6925:
6921:
6915:
6912:
6910:
6907:
6905:
6904:Dependability
6902:
6900:
6897:
6895:
6892:
6891:
6889:
6885:
6879:
6875:
6872:
6870:
6867:
6865:
6862:
6860:
6857:
6855:
6852:
6850:
6847:
6845:
6842:
6840:
6837:
6835:
6832:
6830:
6827:
6826:
6824:
6822:
6818:
6813:
6807:
6803:
6796:
6791:
6789:
6784:
6782:
6777:
6776:
6773:
6763:
6762:
6757:
6749:
6743:
6740:
6738:
6735:
6733:
6730:
6728:
6725:
6721:
6718:
6717:
6716:
6713:
6711:
6708:
6706:
6703:
6701:
6697:
6694:
6692:
6689:
6687:
6684:
6682:
6679:
6677:
6674:
6673:
6671:
6667:
6661:
6658:
6656:
6653:
6651:
6650:Recursive set
6648:
6646:
6643:
6641:
6638:
6636:
6633:
6631:
6628:
6624:
6621:
6619:
6616:
6614:
6611:
6609:
6606:
6604:
6601:
6600:
6599:
6596:
6594:
6591:
6589:
6586:
6584:
6581:
6579:
6576:
6574:
6571:
6570:
6568:
6566:
6562:
6556:
6553:
6551:
6548:
6546:
6543:
6541:
6538:
6536:
6533:
6531:
6528:
6526:
6523:
6519:
6516:
6514:
6511:
6509:
6506:
6505:
6504:
6501:
6499:
6496:
6494:
6491:
6489:
6486:
6484:
6481:
6479:
6476:
6472:
6469:
6468:
6467:
6464:
6460:
6459:of arithmetic
6457:
6456:
6455:
6452:
6448:
6445:
6443:
6440:
6438:
6435:
6433:
6430:
6428:
6425:
6424:
6423:
6420:
6416:
6413:
6411:
6408:
6407:
6406:
6403:
6402:
6400:
6398:
6394:
6388:
6385:
6383:
6380:
6378:
6375:
6373:
6370:
6367:
6366:from ZFC
6363:
6360:
6358:
6355:
6349:
6346:
6345:
6344:
6341:
6339:
6336:
6334:
6331:
6330:
6329:
6326:
6324:
6321:
6319:
6316:
6314:
6311:
6309:
6306:
6304:
6301:
6299:
6296:
6295:
6293:
6291:
6287:
6277:
6276:
6272:
6271:
6266:
6265:non-Euclidean
6263:
6259:
6256:
6254:
6251:
6249:
6248:
6244:
6243:
6241:
6238:
6237:
6235:
6231:
6227:
6224:
6222:
6219:
6218:
6217:
6213:
6209:
6206:
6205:
6204:
6200:
6196:
6193:
6191:
6188:
6186:
6183:
6181:
6178:
6176:
6173:
6171:
6168:
6167:
6165:
6161:
6160:
6158:
6153:
6147:
6142:Example
6139:
6131:
6126:
6125:
6124:
6121:
6119:
6116:
6112:
6109:
6107:
6104:
6102:
6099:
6097:
6094:
6093:
6092:
6089:
6087:
6084:
6082:
6079:
6077:
6074:
6070:
6067:
6065:
6062:
6061:
6060:
6057:
6053:
6050:
6048:
6045:
6043:
6040:
6038:
6035:
6034:
6033:
6030:
6028:
6025:
6021:
6018:
6016:
6013:
6011:
6008:
6007:
6006:
6003:
5999:
5996:
5994:
5991:
5989:
5986:
5984:
5981:
5979:
5976:
5974:
5971:
5970:
5969:
5966:
5964:
5961:
5959:
5956:
5954:
5951:
5947:
5944:
5942:
5939:
5937:
5934:
5932:
5929:
5928:
5927:
5924:
5922:
5919:
5917:
5914:
5912:
5909:
5905:
5902:
5900:
5899:by definition
5897:
5896:
5895:
5892:
5888:
5885:
5884:
5883:
5880:
5878:
5875:
5873:
5870:
5868:
5865:
5863:
5860:
5859:
5856:
5853:
5851:
5847:
5842:
5836:
5832:
5822:
5819:
5817:
5814:
5812:
5809:
5807:
5804:
5802:
5799:
5797:
5794:
5792:
5789:
5787:
5786:KripkeâPlatek
5784:
5782:
5779:
5775:
5772:
5770:
5767:
5766:
5765:
5762:
5761:
5759:
5755:
5747:
5744:
5743:
5742:
5739:
5737:
5734:
5730:
5727:
5726:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5699:
5695:
5691:
5688:
5684:
5681:
5679:
5676:
5674:
5671:
5670:
5669:
5665:
5662:
5661:
5659:
5657:
5653:
5649:
5641:
5638:
5636:
5633:
5631:
5630:constructible
5628:
5627:
5626:
5623:
5621:
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5601:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5581:
5578:
5576:
5573:
5571:
5568:
5566:
5563:
5562:
5560:
5558:
5553:
5545:
5542:
5540:
5537:
5535:
5532:
5530:
5527:
5525:
5522:
5520:
5517:
5516:
5514:
5510:
5507:
5505:
5502:
5501:
5500:
5497:
5495:
5492:
5490:
5487:
5485:
5482:
5480:
5476:
5472:
5470:
5467:
5463:
5460:
5459:
5458:
5455:
5454:
5451:
5448:
5446:
5442:
5432:
5429:
5427:
5424:
5422:
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5400:
5397:
5396:
5395:
5392:
5388:
5383:
5382:
5381:
5378:
5377:
5375:
5373:
5369:
5361:
5358:
5356:
5353:
5351:
5348:
5347:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5312:
5310:
5308:
5307:Propositional
5304:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5266:
5263:
5262:
5261:
5258:
5256:
5253:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5235:Logical truth
5233:
5231:
5228:
5227:
5225:
5223:
5219:
5216:
5214:
5210:
5204:
5201:
5199:
5196:
5194:
5191:
5189:
5186:
5184:
5181:
5179:
5175:
5171:
5167:
5165:
5162:
5160:
5157:
5155:
5151:
5148:
5147:
5145:
5143:
5137:
5132:
5126:
5123:
5121:
5118:
5116:
5113:
5111:
5108:
5106:
5103:
5101:
5098:
5096:
5093:
5091:
5088:
5086:
5083:
5081:
5078:
5076:
5073:
5071:
5068:
5064:
5061:
5060:
5059:
5056:
5055:
5053:
5049:
5045:
5038:
5033:
5031:
5026:
5024:
5019:
5018:
5015:
5003:
5002:
4993:
4991:
4990:
4981:
4979:
4978:
4969:
4967:
4966:
4961:
4955:
4954:
4951:
4945:
4942:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4918:
4915:
4914:
4913:
4910:
4909:
4907:
4905:
4901:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4870:
4867:
4866:
4864:
4862:
4861:Computational
4858:
4850:
4847:
4845:
4842:
4840:
4837:
4836:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4776:
4774:
4772:
4768:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4738:
4736:
4734:
4730:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4706:
4705:
4703:
4701:
4700:Number theory
4697:
4691:
4688:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4657:
4655:
4653:
4649:
4643:
4640:
4638:
4635:
4633:
4632:Combinatorics
4630:
4629:
4627:
4625:
4621:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4584:Real analysis
4582:
4580:
4577:
4576:
4574:
4572:
4568:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4523:
4521:
4519:
4515:
4509:
4506:
4504:
4501:
4499:
4496:
4494:
4491:
4489:
4486:
4484:
4481:
4480:
4478:
4476:
4472:
4466:
4463:
4461:
4458:
4454:
4451:
4449:
4446:
4445:
4444:
4441:
4440:
4437:
4432:
4424:
4419:
4417:
4412:
4410:
4405:
4404:
4401:
4389:
4388:Ernst Zermelo
4386:
4384:
4381:
4379:
4376:
4374:
4373:Willard Quine
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4330:
4328:
4326:
4325:Set theorists
4322:
4316:
4313:
4311:
4308:
4306:
4303:
4302:
4300:
4294:
4292:
4289:
4288:
4285:
4277:
4274:
4272:
4271:KripkeâPlatek
4269:
4265:
4262:
4261:
4260:
4257:
4256:
4255:
4252:
4248:
4245:
4244:
4243:
4242:
4238:
4234:
4231:
4230:
4229:
4226:
4225:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4203:
4201:
4197:
4191:
4188:
4186:
4183:
4181:
4178:
4176:
4174:
4169:
4167:
4164:
4162:
4159:
4156:
4152:
4149:
4147:
4144:
4140:
4137:
4135:
4132:
4130:
4127:
4126:
4125:
4122:
4119:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4096:
4094:
4091:
4087:
4081:
4078:
4076:
4073:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4044:
4041:
4039:
4036:
4035:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4008:
4004:
4001:
3999:
3996:
3994:
3991:
3990:
3988:
3982:
3979:
3978:
3975:
3969:
3966:
3964:
3961:
3959:
3956:
3954:
3951:
3949:
3946:
3944:
3941:
3939:
3936:
3933:
3930:
3928:
3925:
3924:
3922:
3920:
3916:
3908:
3907:specification
3905:
3903:
3900:
3899:
3898:
3895:
3894:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
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:
3609:
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:
3455:
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:
3376:
3372:
3368:
3367:
3361:
3358:
3355:
3353:
3348:
3343:
3340:
3336:
3330:
3326:
3325:
3320:
3314:
3311:
3307:
3305:0-198-53807-3
3301:
3297:
3293:
3288:
3287:
3278:
3275:
3271:
3267:
3263:
3259:
3258:
3250:
3247:
3243:
3239:
3234:
3231:
3223:
3219:
3214:
3211:
3208:unnecessary."
3202:
3198:
3193:
3190:
3186:
3182:
3178:
3174:
3170:
3165:
3162:
3158:
3156:0-631-19130-5
3152:
3148:
3141:
3138:
3133:
3132:
3127:
3123:
3116:
3113:
3109:
3107:0-195-08030-0
3103:
3099:
3098:
3093:
3087:
3084:
3080:
3078:4-87187-714-0
3074:
3070:
3069:
3064:
3058:
3055:
3051:
3047:
3043:
3037:
3033:
3029:
3028:
3023:
3017:
3014:
3004:
3000:
2996:
2992:
2985:
2982:
2972:
2965:
2959:
2956:
2952:
2948:
2944:
2940:
2936:
2929:
2926:
2916:
2909:
2903:
2900:
2889:
2885:
2879:
2876:
2871:
2866:
2862:
2858:
2854:
2847:
2844:
2840:
2836:
2832:
2828:
2824:
2820:
2813:
2806:
2803:
2792:
2788:
2781:
2778:
2768:
2764:
2758:
2755:
2751:
2747:
2743:
2737:
2733:
2728:
2727:
2721:
2717:
2711:
2708:
2698:
2694:
2688:
2685:
2680:
2678:0-674-34871-0
2674:
2670:
2666:
2660:
2657:
2653:
2651:3-7728-0466-7
2647:
2643:
2639:
2633:
2630:
2626:
2624:0-87150-154-6
2620:
2615:
2614:
2605:
2602:
2598:
2594:
2590:
2586:
2582:
2579:(in German),
2578:
2577:
2572:
2568:
2567:Cantor, Georg
2562:
2559:
2552:
2542:
2539:
2529:
2528:0-674-32449-8
2525:
2521:
2517:
2513:
2509:
2505:
2500:
2496:
2490:
2487:
2480:
2476:
2473:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2452:
2451:
2447:
2441:
2436:
2431:
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:
2224:
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:
2052:
2050:
2048:
2043:
2039:
2034:
2026:
2024:
2021:
2017:
2011:
2003:
2001:
1999:
1995:
1991:
1987:
1982:
1978:
1975:in which the
1974:
1970:
1966:
1965:
1960:
1955:
1947:
1945:
1943:
1942:Wadge degrees
1939:
1934:
1929:
1921:
1919:
1917:
1913:
1909:
1905:
1899:
1891:
1889:
1887:
1883:
1878:
1875:
1871:
1867:
1863:
1859:
1856:
1852:
1848:
1842:
1834:
1832:
1830:
1826:
1822:
1821:
1814:
1806:
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:
1727:
1725:
1723:
1719:
1715:
1711:
1710:combinatorics
1707:
1702:
1694:
1692:
1686:
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:
1573:
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
1318:
1316:
1302:
1282:
1260:
1256:
1235:
1211:
1203:
1187:
1180:
1176:
1172:
1167:
1166:
1160:
1151:
1146:
1138:
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:
435:
433:
431:
430:real analysis
427:
423:
419:
415:
410:
408:
404:
397:
393:
386:
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:
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:
71:
67:
66:
61:
57:
53:
48:
44:
40:
33:
19:
7809:Georg Cantor
7705:Cyberwarfare
7364:Cryptography
6752:
6550:Ultraproduct
6397:Model theory
6362:Independence
6298:Formal proof
6290:Proof theory
6273:
6246:
6203:real numbers
6175:second-order
6086:Substitution
5963:Metalanguage
5904:conservative
5877:Axiom schema
5821:Constructive
5791:MorseâKelley
5757:Set theories
5736:Aleph number
5729:inaccessible
5635:Grothendieck
5519:intersection
5444:
5406:Higher-order
5394:Second-order
5340:Truth tables
5297:Venn diagram
5080:Formal proof
4999:
4987:
4975:
4956:
4889:Optimization
4751:Differential
4675:Differential
4642:Order theory
4637:Graph theory
4541:Group theory
4502:
4338:Georg Cantor
4333:Paul Bernays
4264:MorseâKelley
4239:
4172:
4171:Subset
4118:hereditarily
4080:Venn diagram
4038:ordered pair
3953:Intersection
3897: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:
7715:Video games
7695:Digital art
7452:Concurrency
7321:Data mining
7233:Probability
6973:Interpreter
6660:Type theory
6608:undecidable
6540:Truth value
6427:equivalence
6106:non-logical
5719:Enumeration
5709:Isomorphism
5656:cardinality
5640:Von Neumann
5605:Ultrafilter
5570:Uncountable
5504:equivalence
5421:Quantifiers
5411:Fixed-point
5380:First-order
5260:Consistency
5245:Proposition
5222:Traditional
5193:Lindström's
5183:Compactness
5125:Type theory
5070:Cardinality
5001:WikiProject
4844:Game theory
4824:Probability
4561:Homological
4551:Multilinear
4531:Commutative
4508:Type theory
4475:Foundations
4431:mathematics
4363:Thomas Jech
4206:Alternative
4185:Uncountable
4139:Ultrafilter
3998:Cardinality
3902: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
7794:Set theory
7788:Categories
7773:Glossaries
7645:E-commerce
7238:Statistics
7181:Algorithms
6978:Middleware
6834:Peripheral
6471:elementary
6164:arithmetic
6032:Quantifier
6010:functional
5882:Expression
5600:Transitive
5544:identities
5529:complement
5462:hereditary
5445:Set theory
4829:Statistics
4708:Arithmetic
4670:Arithmetic
4536:Elementary
4503:Set theory
4358:Kurt Gödel
4343:Paul Cohen
4180:Transitive
3948:Identities
3932:Complement
3919:Operations
3880:Regularity
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
7594:Rendering
7589:Animation
7220:computing
7171:Semantics
6869:Processor
6742:Supertask
6645:Recursion
6603:decidable
6437:saturated
6415:of models
6338:deductive
6333:axiomatic
6253:Hilbert's
6240:Euclidean
6221:canonical
6144:axiomatic
6076:Signature
6005:Predicate
5894:Extension
5816:Ackermann
5741:Operation
5620:Universal
5610:Recursive
5585:Singleton
5580:Inhabited
5565:Countable
5555:Types of
5539:power set
5509:partition
5426:Predicate
5372:Predicate
5287:Syllogism
5277:Soundness
5250:Inference
5240:Tautology
5142:paradoxes
4756:Geometric
4746:Algebraic
4685:Euclidean
4660:Algebraic
4556:Universal
4291:Paradoxes
4211:Axiomatic
4190:Universal
4166:Singleton
4161:Recursive
4104:Countable
4099:Amorphous
3958:Power set
3875: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
7753:Category
7581:Graphics
7356:Security
7025:Compiler
6924:Networks
6821:Hardware
6727:Logicism
6720:timeline
6696:Concrete
6555:Validity
6525:T-schema
6518:Kripke's
6513:Tarski's
6508:semantic
6498:Strength
6447:submodel
6442:spectrum
6410:function
6258:Tarski's
6247:Elements
6234:geometry
6190:Robinson
6111:variable
6096:function
6069:spectrum
6059:Sentence
6015:variable
5958:Language
5911:Relation
5872:Automata
5862:Alphabet
5846:language
5700:-jection
5678:codomain
5664:Function
5625:Universe
5595:Infinite
5499:Relation
5282:Validity
5272:Argument
5170:theorem,
4977:Category
4733:Topology
4680:Discrete
4665:Analytic
4652:Geometry
4624:Discrete
4579:Calculus
4571:Analysis
4526:Abstract
4465:Glossary
4448:Timeline
4295:Problems
4199:Theories
4175:Superset
4151:Infinite
3980:Concepts
3860: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
7763:Outline
6669:Related
6466:Diagram
6364: (
6343:Hilbert
6328:Systems
6323:Theorem
6201:of the
6146:systems
5926:Formula
5921:Grammar
5837: (
5781:General
5494:Forcing
5479:Element
5399:Monadic
5174:paradox
5115:Theorem
5051:General
4989:Commons
4771:Applied
4741:General
4518:Algebra
4443:History
4233:General
4228:Zermelo
4134:subbase
4116: (
4055:Forcing
4033:Element
4005: (
3983:Methods
3870: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
6432:finite
6195:Skolem
6148:
6123:Theory
6091:Symbol
6081:String
6064:atomic
5941:ground
5936:closed
5931:atomic
5887:ground
5850:syntax
5746:binary
5673:domain
5590:Finite
5355:finite
5213:Logics
5172:
5120:Theory
4690:Finite
4546:Linear
4453:Future
4429:Major
4124:Filter
4114:Finite
4050:Family
3993:Almost
3838:global
3823:Choice
3810:Axioms
3599:
3572:
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3205:motley
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2159:, and
1650:, and
1594:, and
1580:graphs
1547:. The
1492:) and
1424:choice
1422:, and
1109:{1, 2}
940:{1, 4}
742:{2, 3}
596:, but
560:{1, 4}
548:{1, 2}
527:subset
475:member
355:, and
7166:Logic
7007:tools
6422:Model
6170:Peano
6027:Proof
5867:Arity
5796:Naive
5683:image
5615:Fuzzy
5575:Empty
5524:union
5469:Class
5110:Model
5100:Lemma
5058:Axiom
4917:lists
4460:Lists
4433:areas
4216:Naive
4146:Fuzzy
4109:Empty
4092:types
4043:tuple
4013:Class
4007:large
3968:Union
3885:Union
3124:, in
2967:(PDF)
2911:(PDF)
2835:S2CID
2815:(PDF)
2593:S2CID
2481:Notes
2426:range
2424:(the
2416:(the
2309:lists
2077:naive
1969:model
1851:class
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870:is a
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103:Areas
88:Index
7005:and
6878:Form
6874:Size
6545:Type
6348:list
6152:list
6129:list
6118:Term
6052:rank
5946:open
5840:list
5652:Maps
5557:sets
5416:Free
5386:list
5136:list
5063:list
4129:base
3711:and
3597:ISBN
3570:ISBN
3545:ISBN
3521:ISBN
3499:ISBN
3479:ISBN
3442:ISBN
3420:ISBN
3391:ISBN
3329:ISBN
3300:ISBN
3242:§3.6
3222:§2.2
3201:§3.4
3173:§2.1
3151:ISBN
3102:ISBN
3073:ISBN
3036:ISBN
2746:OCLC
2736:ISBN
2673:ISBN
2646:ISBN
2619:ISBN
2581:1874
2524:ISBN
2514:and
2328:sets
2319:and
2307:and
2301:sets
2189:and
2133:and
2099:and
1884:and
1667:and
1631:and
1602:and
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1478:The
1461:and
1430:and
1408:and
1159:pure
1039:and
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924:and
892:and
758:and
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650:and
558:but
478:(or
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293:and
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6232:of
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5457:Set
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