View source for Set theory

{{Short description|Branch of mathematics that studies sets}}
{{About|the branch of mathematics}}
{{Distinguish|Set theory (music)}}
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[[Image:Venn A intersect B.svg|thumb|right|A [[Venn diagram]] illustrating the [[intersection (set theory)|intersection]] of two [[set (mathematics)|sets]]]]
{{Math topics TOC}}

'''Set theory''' is the branch of [[mathematical logic]] that studies [[Set (mathematics)|sets]], which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory — as a branch of [[mathematics]] — is mostly concerned with those that are relevant to mathematics as a whole.

The modern study of set theory was initiated by the German mathematicians [[Richard Dedekind]] and [[Georg Cantor]] 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 ''[[naive set theory]]''. After the discovery of [[Paradoxes of set theory|paradoxes]] within naive set theory (such as [[Russell's paradox]], [[Cantor's paradox]] and the [[Burali-Forti paradox]]), various [[axiomatic system]]s were proposed in the early twentieth century, of which [[Zermelo–Fraenkel set theory]] (with or without the [[axiom of choice]]) is still the best-known and most studied.

Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of [[infinity]], and has various applications in [[computer science]] (such as in the theory of [[relational algebra]]), [[philosophy]], [[Semantics (computer science)|formal semantics]], and [[evolutionary dynamics]]. Its foundational appeal, together with its [[paradoxes]], and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for [[logic]]ians and [[Philosophy of mathematics|philosophers of mathematics]]. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the [[real number]] line to the study of the [[consistency]] of [[large cardinal]]s.

==History==
[[File:Georg Cantor 1894.jpg|thumb|160px|[[Georg Cantor]]]]

Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by [[Georg Cantor]]: "[[On a Property of the Collection of All Real Algebraic Numbers]]".<ref name="cantor1874">{{citation|last=Cantor|first=Georg|author-link=Georg Cantor|year=1874|title=Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen|url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002155583|journal=[[Journal für die reine und angewandte Mathematik]]|language=de|volume=1874|issue=77|pages=258–262|doi=10.1515/crll.1874.77.258|s2cid=199545885}}</ref><ref>{{citation |first=Philip |last=Johnson |year=1972 |title=A History of Set Theory |publisher=Prindle, Weber & Schmidt |isbn=0-87150-154-6 |url-access=registration |url=https://archive.org/details/mathematicalcirc0000eves_x3z6 }}</ref>

Since the 5th century BC, beginning with Greek mathematician [[Zeno of Elea]] in the West and early [[Indian mathematics|Indian mathematicians]] in the East, mathematicians had struggled with the concept of [[infinity]]. Especially notable is the work of [[Bernard Bolzano]] in the first half of the 19th century.<ref>{{Citation|last=Bolzano|first=Bernard|author-link=Bernard Bolzano|editor-last=Berg|editor-first=Jan|title=Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre|page=152|series=Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al.|volume=II, A, 7|publisher=Friedrich Frommann Verlag|location=Stuttgart, Bad Cannstatt|isbn=3-7728-0466-7|year=1975}}</ref> Modern understanding of infinity began in 1870–1874, and was motivated by Cantor's work in [[real analysis]].<ref>{{Citation|last=Dauben|first=Joseph|author-link=Joseph Dauben|title=Georg Cantor: His Mathematics and Philosophy of the Infinite|publisher=Harvard University Press|year=1979|isbn=0-674-34871-0|pages=30–54}}.</ref>

==Basic concepts and notation==
{{Main|Set (mathematics)|Algebra of sets}}

Set theory begins with a fundamental [[binary relation]] between an object {{math|''o''}} and a set {{math|''A''}}. If {{math|''o''}} is a ''[[set membership|member]]'' (or ''element'') of {{math|''A''}}, the notation {{math|''o'' ∈ ''A''}} is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }.<ref>{{Cite web|title=Introduction to Sets|url=https://www.mathsisfun.com/sets/sets-introduction.html|access-date=2020-08-20|website=www.mathsisfun.com}}</ref> Since sets are objects, the membership relation can relate sets as well.

A derived binary relation between two sets is the subset relation, also called ''set inclusion''. If all the members of set {{math|''A''}} are also members of set {{math|''B''}}, then {{math|''A''}} is a ''[[subset]]'' of {{math|''B''}}, denoted {{math|''A'' ⊆ ''B''}}. For example, {{math|{1, 2<nowiki>}</nowiki>}} is a subset of {{math|{1, 2, 3<nowiki>}</nowiki>}}, and so is {{math|{2<nowiki>}</nowiki>}} but {{math|{1, 4<nowiki>}</nowiki>}} 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 ''[[proper subset]]'' is defined. {{math|''A''}} is called a ''proper subset'' of {{math|''B''}} if and only if {{math|''A''}} is a subset of {{math|''B''}}, but {{math|''A''}} is not equal to {{math|''B''}}. Also, 1, 2, and 3 are members (elements) of the set {{math|{1, 2, 3<nowiki>}</nowiki>}}, but are not subsets of it; and in turn, the subsets, such as {{math|{1<nowiki>}</nowiki>}}, are not members of the set {{math|{1, 2, 3<nowiki>}</nowiki>}}.

Just as [[arithmetic]] features [[binary operation]]s on [[number]]s, set theory features binary operations on sets.<ref>{{citation|url=https://archive.org/details/introductoryreal00kolm_0/page/2|title=Introductory Real Analysis|last1=Kolmogorov|first1=A.N.|last2=Fomin|first2=S.V.|publisher=Dover Publications|year=1970|isbn=0486612260|edition=Rev. English|location=New York|pages=[https://archive.org/details/introductoryreal00kolm_0/page/2 2–3]|oclc=1527264|author-link=Andrey Kolmogorov|author-link2=Sergei Fomin|url-access=registration}}</ref> The following is a partial list of them:
*''[[Union (set theory)|Union]]'' of the sets {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∪ ''B''}}, is the set of all objects that are a member of {{math|''A''}}, or {{math|''B''}}, or both.<ref>{{Cite web|title=set theory {{!}} Basics, Examples, & Formulas|url=https://www.britannica.com/science/set-theory|access-date=2020-08-20|website=Encyclopedia Britannica|language=en}}</ref> For example, the union of {{math|{1, 2, 3<nowiki>}</nowiki>}} and {{math|{2, 3, 4<nowiki>}</nowiki>}} is the set {{math|{1, 2, 3, 4<nowiki>}</nowiki>}}.
*''[[Intersection (set theory)|Intersection]]'' of the sets {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' ∩ ''B''}}, is the set of all objects that are members of both {{math|''A''}} and {{math|''B''}}. For example, the intersection of {{math|{1, 2, 3<nowiki>}</nowiki>}} and {{math|{2, 3, 4<nowiki>}</nowiki>}} is the set {{math|{2, 3<nowiki>}</nowiki>}}.
*''[[Set difference]]'' of {{math|''U''}} and {{math|''A''}}, denoted {{math|''U'' \ ''A''}}, is the set of all members of {{math|''U''}} that are not members of {{math|''A''}}. The set difference {{math|{1, 2, 3} \ {2, 3, 4} }} is {{math|{1<nowiki>}</nowiki>}}, while conversely, the set difference {{math|{2, 3, 4} \ {1, 2, 3<nowiki>}</nowiki>}} is {{math|{4<nowiki>}</nowiki>}}. When {{math|''A''}} is a subset of {{math|''U''}}, the set difference {{math|''U'' \ ''A''}} is also called the ''[[complement (set theory)|complement]]'' of {{math|''A''}} in {{math|''U''}}. In this case, if the choice of {{math|''U''}} is clear from the context, the notation {{math|''A''<sup>''c''</sup>}} is sometimes used instead of {{math|''U'' \ ''A''}}, particularly if {{math|''U''}} is a [[universal set]] as in the study of [[Venn diagram]]s.
*''[[Symmetric difference]]'' of sets {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' △ ''B''}} or {{math|''A'' ⊖ ''B''}}, is the set of all objects that are a member of exactly one of {{math|''A''}} and {{math|''B''}} (elements which are in one of the sets, but not in both). For instance, for the sets {{math|{1, 2, 3<nowiki>}</nowiki>}} and {{math|{2, 3, 4<nowiki>}</nowiki>}}, the symmetric difference set is {{math|{1, 4<nowiki>}</nowiki>}}. It is the set difference of the union and the intersection, {{math|(''A'' ∪ ''B'') \ (''A'' ∩ ''B'')}} or {{math|(''A'' \ ''B'') ∪ (''B'' \ ''A'')}}.
*''[[Cartesian product]]'' of {{math|''A''}} and {{math|''B''}}, denoted {{math|''A'' × ''B''}}, is the set whose members are all possible [[ordered pair]]s {{math|(''a'', ''b'')}}, where {{math|''a''}} is a member of {{math|''A''}} and {{math|''b''}} is a member of {{math|''B''}}. For example, the Cartesian product of {{nowrap|1={1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.}}
*''[[Power set]]'' of a set {{math|''A''}}, denoted <math>\mathcal{P}(A)</math>, is the set whose members are all of the possible subsets of {{math|''A''}}. For example, the power set of {{math|{1, 2<nowiki>}</nowiki>}} is {{math|{ {}, {1}, {2}, {1, 2} <nowiki>}</nowiki>}}.

Some basic sets of central importance are the set of [[natural number]]s, the set of [[real number]]s and the [[empty set]]—the unique set containing no elements. The empty set is also occasionally called the ''null set'',<ref>{{Citation|last=Bagaria|first=Joan|title=Set Theory|date=2020|url=https://plato.stanford.edu/archives/spr2020/entries/set-theory/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Spring 2020|publisher=Metaphysics Research Lab, Stanford University|access-date=2020-08-20}}</ref> though this name is ambiguous and can lead to several interpretations.

==Ontology==
{{Main|von Neumann universe}}
[[Image:Von Neumann Hierarchy.svg|thumb|right|300px|An initial segment of the von Neumann hierarchy]]

A set is [[pure set|pure]] 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 ''[[von Neumann universe]]'' 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 organized into a [[cumulative hierarchy]], based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by [[transfinite recursion]]) an [[ordinal number]] <math>\alpha</math>, known as its ''rank.'' The rank of a pure set <math>X</math> 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 {{math| <nowiki>{{}}</nowiki> }} containing only the empty set is assigned rank 1. For each ordinal <math>\alpha</math>, the set <math>V_{\alpha}</math> is defined to consist of all pure sets with rank less than <math>\alpha</math>. The entire von Neumann universe is denoted&nbsp;<math>V</math>.

== Formalized set theory<!--'Axiomatic set theory' redirects here--> ==
{{anchor|Axiomatic set theory}}
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using [[Venn diagram]]s. 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 [[Russell's paradox]] and the [[Burali-Forti paradox]]. '''Axiomatic set theory'''<!--boldface per WP:R#PLA--> was originally devised to rid set theory of such paradoxes.{{NoteTag|In his 1925 paper ""An Axiomatization of Set Theory", [[John von Neumann]] observed that "set theory in its first, "naive" version, due to Cantor, led to contradictions. These are the well-known [[antinomy|antinomies]] 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, exemplified by [[Bertrand Russell]], [[Julius König]], [[Hermann Weyl]] and [[L. E. J. Brouwer]], 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), ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931'', Harvard University Press, Cambridge MA, {{ISBN|0-674-32449-8}} (pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann's 1925 paper.}}

The most widely studied systems of axiomatic set theory imply that all sets form a [[cumulative hierarchy]].{{efn|This is the converse for ZFC; V is a model of ZFC.}} Such systems come in two flavors, those whose [[ontology]] consists of:
*''Sets alone''. This includes the most common axiomatic set theory, [[Zermelo–Fraenkel set theory|'''Z'''ermelo–'''F'''raenkel set theory]] with the [[Axiom of choice|axiom of '''c'''hoice]] (ZFC). Fragments of '''ZFC''' include:
** [[Zermelo set theory]], which replaces the [[axiom schema of replacement]] with that of [[axiom schema of separation|separation]];
** [[General set theory]], a small fragment of [[Zermelo set theory]] sufficient for the [[Peano axioms]] and [[finite set]]s;
** [[Kripke–Platek set theory]], which omits the axioms of infinity, [[axiom of power set|powerset]], and [[axiom of choice|choice]], and weakens the axiom schemata of [[axiom schema of separation|separation]] and [[axiom schema of replacement|replacement]].
*''Sets and [[proper class]]es''. These include [[Von Neumann–Bernays–Gödel set theory]], which has the same [[Strength (mathematical logic)|strength]] as [[ZFC]] for theorems about sets alone, and [[Morse–Kelley set theory]] and [[Tarski–Grothendieck set theory]], both of which are stronger than ZFC.
The above systems can be modified to allow ''[[urelement]]s'', objects that can be members of sets but that are not themselves sets and do not have any members.

The ''[[New Foundations]]'' systems of '''NFU''' (allowing [[urelement]]s) and '''NF''' (lacking them), associate with [[Willard Van Orman Quine]], 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 [[axiom of choice]] does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, [[Thomas Forster (mathematician)|Thomas Forster]] has argued that it does reflect an [[iterative conception of set]].<ref>{{cite journal | last=Forster |first=T. E. |date=2008 | title = The iterative conception of set | journal = The Review of Symbolic Logic | volume = 1 |  pages = 97–110 |doi=10.1017/S1755020308080064 |s2cid=15231169 |url=https://www.dpmms.cam.ac.uk/~tf/iterativeconception.pdf}}</ref>

Systems of [[constructive set theory]], such as CST, CZF, and IZF, embed their set axioms in [[intuitionistic logic|intuitionistic]] instead of [[classical logic]]. Yet other systems accept classical logic but feature a nonstandard membership relation. These include [[Rough set|rough set theory]] and [[fuzzy set theory]], in which the value of an [[atomic formula]] embodying the membership relation is not simply '''True''' or '''False'''. The [[Boolean-valued model]]s of [[ZFC]] are a related subject.

An enrichment of [[ZFC]] called [[internal set theory]] was proposed by [[Edward Nelson]] in 1977.<ref>{{cite journal |last1=Nelson |first1=Edward |title=Internal Set Theory: a New Approach to Nonstandard Analysis |journal=Bulletin of the American Mathematical Society |date=November 1977 |volume=83 |issue=6 |page=1165 |doi=10.1090/S0002-9904-1977-14398-X |doi-access=free }}</ref>

==Applications==
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as [[graph (discrete mathematics)|graph]]s, [[manifolds]], [[ring (mathematics)|rings]], [[vector space]]s, and [[relational algebra]]s can all be defined as sets satisfying various (axiomatic) properties. [[equivalence relation|Equivalence]] and [[order relation]]s are ubiquitous in mathematics, and the theory of mathematical [[relation (mathematics)|relations]] can be described in set theory.<ref>{{Cite web |date=2019-11-25 |title=6.3: Equivalence Relations and Partitions |url=https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/6%3A_Relations/6.3%3A_Equivalence_Relations_and_Partitions |access-date=2022-07-27 |website=Mathematics LibreTexts |language=en}}</ref><ref>{{cite web|url=https://web.stanford.edu/class/archive/cs/cs103/cs103.1132/lectures/06/Slides06.pdf|title=Order Relations and Functions|website=Web.stanford.edu|access-date=2022-07-29}}</ref>

Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of ''[[Principia Mathematica]]'', 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 [[first-order logic|first]] or [[second-order logic]]. For example, properties of the [[natural number|natural]] and [[real number]]s can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms.<ref>{{citation
 | last = Mendelson | first = Elliott
 | mr = 357694
 | publisher = Academic Press
 | title = Number Systems and the Foundations of Analysis
 | zbl = 0268.26001
 | year = 1973}}</ref>

Set theory as a foundation for [[mathematical analysis]], [[topology]], [[abstract algebra]], and [[discrete mathematics]] 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 such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, [[Metamath]], includes human-written, computer-verified derivations of more than 12,000 theorems starting from [[ZFC]] set theory, [[first-order logic]] and [[propositional logic]].<ref>{{cite web|url=https://www.ams.org/bull/1956-62-05/S0002-9904-1956-10036-0/S0002-9904-1956-10036-0.pdf|title=A PARTITION CALCULUS IN SET THEORY |website=Ams.org|access-date=2022-07-29}}</ref> [[Zermelo–Fraenkel set theory|ZFC]] and the [[Axiom of choice|Axiom of Choice]] have recently seen applications in [[evolutionary dynamics]],<ref name=":0">{{Cite journal |last1=Berkemeier |first1=Francisco |last2=Page |first2=Karen M. |date=2023-09-29 |title=Unifying evolutionary dynamics: a set theory exploration of symmetry and interaction |url=http://dx.doi.org/10.1101/2023.09.27.559729 |access-date=2023-12-07 |website=dx.doi.org|doi=10.1101/2023.09.27.559729 }}</ref> enhancing the understanding of well-established models of evolution and interaction.

== Areas of study ==
Set theory is a major area of research in mathematics with many interrelated subfields:

=== Combinatorial set theory ===
{{Main|Infinitary combinatorics}}

''Combinatorial set theory'' concerns extensions of finite [[combinatorics]] to infinite sets. This includes the study of [[cardinal arithmetic]] and the study of extensions of [[Ramsey's theorem]] such as the [[Erdős–Rado theorem]].

=== Descriptive set theory ===
{{Main|Descriptive set theory}}

''Descriptive set theory'' is the study of subsets of the [[real line]] and, more generally, subsets of [[Polish space]]s. It begins with the study of [[pointclass]]es in the [[Borel hierarchy]] and extends to the study of more complex hierarchies such as the [[projective hierarchy]] and the [[Wadge hierarchy]]. Many properties of [[Borel set]]s can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.

The field of [[effective descriptive set theory]] is between set theory and [[recursion theory]]. It includes the study of [[lightface pointclass]]es, and is closely related to [[hyperarithmetical theory]]. 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.

A recent area of research concerns [[Borel equivalence relation]]s and more complicated definable [[equivalence relation]]s. This has important applications to the study of [[invariant (mathematics)|invariants]] in many fields of mathematics.

=== Fuzzy set theory ===
{{Main|Fuzzy set theory}}

In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In ''[[fuzzy set theory]]'' this condition was relaxed by [[Lotfi A. Zadeh]] so an object has a ''degree of membership'' 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.

=== Inner model theory ===
{{Main|Inner model theory}}

An ''inner model'' of Zermelo–Fraenkel set theory (ZF) is a transitive [[proper class|class]] that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the [[constructible universe]] ''L'' 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 ''V'' of ZF satisfies the [[continuum hypothesis]] or the [[axiom of choice]], the inner model ''L'' 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.

The study of inner models is common in the study of [[axiom of determinacy|determinacy]] and [[large cardinal]]s, 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).<ref>{{citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | edition= Third Millennium | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003 | zbl=1007.03002 | page=642 | url=https://books.google.com/books?id=CZb-CAAAQBAJ&pg=PA642 }}</ref>

=== Large cardinals ===
{{Main|Large cardinal property}}

A ''large cardinal'' is a cardinal number with an extra property. Many such properties are studied, including [[inaccessible cardinal]]s, [[measurable cardinal]]s, 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 [[Zermelo–Fraenkel set theory]].

=== Determinacy ===
{{Main|Determinacy}}

''Determinacy'' 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 that a broader class of games is determined often implies that a broader class of sets will have a topological property. The [[axiom of determinacy]] (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 [[Wadge degree]]s have an elegant structure.

=== Forcing ===
{{Main|Forcing (mathematics)}}

[[Paul Cohen (mathematician)|Paul Cohen]] invented the method of ''[[forcing (mathematics)|forcing]]'' while searching for a [[model theory|model]] of [[ZFC]] in which the [[continuum hypothesis]] fails, or a model of ZF in which the [[axiom of choice]] 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 [[natural number]]s without changing any of the [[cardinal number]]s of the original model. Forcing is also one of two methods for proving [[consistency (mathematical logic)|relative consistency]] by finitistic methods, the other method being [[Boolean-valued model]]s.

=== Cardinal invariants ===
{{Main|Cardinal characteristics of the continuum}}

A ''cardinal invariant'' 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 [[meagre set]]s 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 theory.

=== Set-theoretic topology ===
{{Main|Set-theoretic topology}}

''Set-theoretic topology'' studies questions of [[general topology]] 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 [[Moore space (topology)|normal Moore space question]], 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.

== Controversy ==
{{main|Controversy over Cantor's theory}}

From set theory's inception, some mathematicians have objected to it as a [[foundations of mathematics|foundation for mathematics]]. The most common objection to set theory, one [[Leopold Kronecker|Kronecker]] voiced in set theory's earliest years, starts from the [[mathematical constructivism|constructivist]] view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in [[naive set theory|naive]] 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 [[Errett Bishop]]'s influential book ''Foundations of Constructive Analysis''.<ref>{{citation|title=Foundations of Constructive Analysis|last=Bishop|first=Errett|publisher=Academic Press|year=1967|isbn=4-87187-714-0|location=New York|author-link=Errett Bishop|url=https://books.google.com/books?id=o2mmAAAAIAAJ}}</ref>

A different objection put forth by [[Henri Poincaré]] is that defining sets using the axiom schemas of [[Axiom schema of specification|specification]] and [[Axiom schema of replacement|replacement]], as well as the [[axiom of power set]], introduces [[impredicativity]], a type of [[Circular definition|circularity]], into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that [[Solomon Feferman]] has said that "all of scientifically applicable analysis can be developed [using predicative methods]".<ref>{{citation|title=In the Light of Logic|last=Feferman|first=Solomon|publisher=Oxford University Press|year=1998|isbn=0-195-08030-0|location=New York|pages=280–283, 293–294|author-link=Solomon Feferman|url=https://books.google.com/books?id=1rjnCwAAQBAJ}}</ref>

[[Ludwig Wittgenstein]] condemned set theory philosophically for its connotations of [[mathematical platonism]].<ref>{{Cite SEP|url-id=wittgenstein-mathematics|last=Rodych|first=Victor|date=Jan 31, 2018|title=Wittgenstein's Philosophy of Mathematics|edition=Spring 2018}}</ref>  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".<ref>{{citation |last=Wittgenstein |first=Ludwig |year=1975 |title=Philosophical Remarks, §129, §174 |publisher=Oxford: Basil Blackwell |isbn=0-631-19130-5 }}</ref>  Wittgenstein identified mathematics with algorithmic human deduction;{{Sfn|Rodych|2018|loc=[https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittInteConsForm §2.1]|ps=: "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)—we invent mathematics, bit-by-little-bit."  Note, however, that Wittgenstein does ''not'' identify such deduction with [[philosophical logic]]; c.f. Rodych [https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittMathTrac §1], paras. 7-12.}} the need for a secure foundation for mathematics seemed, to him, nonsensical.{{Sfn|Rodych|2018|loc=[https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittLateCritSetTheoNonEnumVsNonDenu §3.4]|ps=: "Given that mathematics is a '{{small caps|motley}} 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, unnecessary."}}  Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical [[Constructivism (math)|constructivism]] and [[finitism]].  Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics.{{Sfn|Rodych|2018|loc=[https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittInteFini §2.2]|ps=: "An expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular number {{mvar|n}} has a particular property."}}  Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in ''[[Remarks on the Foundations of Mathematics]]'': Wittgenstein attempted to refute [[Gödel's incompleteness theorems]] after having only read the abstract.  As reviewers [[Georg Kreisel|Kreisel]], [[Paul Bernays|Bernays]], [[Michael Dummett|Dummett]], and [[R. L. Goodstein|Goodstein]] all pointed out, many of his critiques did not apply to the paper in full.  Only recently have philosophers such as [[Crispin Wright]] begun to rehabilitate Wittgenstein's arguments.{{Sfn|Rodych|2018|loc=[https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittGodeUndeMathProp §3.6]}}

[[category theory|Category theorists]] have proposed [[topos theory]] as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as [[mathematical constructivism|constructivism]], finite set theory, and [[Turing Machine|computable]] set theory.<ref>{{citation|last1=Ferro|first1=Alfredo|last2=Omodeo|first2=Eugenio G.|last3=Schwartz|first3=Jacob T.|date=September 1980|title=Decision Procedures for Elementary Sublanguages of Set Theory. I. Multi-Level Syllogistic and Some Extensions|journal=[[Communications on Pure and Applied Mathematics]]|volume=33|issue=5|pages=599–608|doi=10.1002/cpa.3160330503}}</ref><ref>{{citation|url=https://archive.org/details/computablesetthe00cant/page/|title=Computable Set Theory|last1=Cantone|first1=Domenico|last2=Ferro|first2=Alfredo|last3=Omodeo|first3=Eugenio G.|publisher=[[Clarendon Press]]|year=1989|isbn=0-198-53807-3|series=International Series of Monographs on Computer Science, Oxford Science Publications|location=Oxford, UK|pages=[https://archive.org/details/computablesetthe00cant/page/ xii, 347]|url-access=registration}}</ref> Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for [[pointless topology]] and [[Stone space]]s.<ref>{{citation|title=Sheaves in Geometry and Logic: A First Introduction to Topos Theory|last1=Mac Lane|first1=Saunders|last2=Moerdijk|first2=leke|publisher=Springer-Verlag|year=1992|isbn=978-0-387-97710-2|author-link=Saunders Mac Lane|url=https://books.google.com/books?id=SGwwDerbEowC}}</ref>

An active area of research is the [[univalent foundations]] and related to it [[homotopy type theory]]. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with [[universal properties]] of sets arising from the inductive and recursive properties of [[higher inductive type]]s. Principles such as the [[axiom of choice]] and the [[law of the excluded middle]] 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 a detailed analysis of the formulations required in order to derive various mathematical results.<ref>{{nlab|id=homotopy+type+theory|title=homotopy type theory}}</ref><ref>[http://homotopytypetheory.org/book/ ''Homotopy Type Theory: Univalent Foundations of Mathematics'']. The Univalent Foundations Program. [[Institute for Advanced Study]].</ref>

== Mathematical education ==
As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics of [[naive set theory]] early in [[mathematics education]].

In the US in the 1960s, the [[New Math]] experiment aimed to teach basic set theory, among other abstract concepts, to [[primary school]] 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. [[Venn diagram]]s are widely employed to explain basic set-theoretic relationships to [[primary school]] students (even though [[John Venn]] originally devised them as part of a procedure to assess the [[validity (logic)|validity]] of [[inference]]s in [[term logic]]).

Set theory is used to introduce students to [[logical operators]] (NOT, AND, OR), and semantic or rule description (technically [[intensional definition]]<ref name="Ruda2011">{{cite book|author=Frank Ruda|title=Hegel's Rabble: An Investigation into Hegel's Philosophy of Right|url=https://books.google.com/books?id=VV0SBwAAQBAJ&pg=PA151|date=6 October 2011|publisher=Bloomsbury Publishing|isbn=978-1-4411-7413-0|page=151}}</ref>) of sets (e.g. "months starting with the letter ''A''"), which may be useful when learning [[computer programming]], since [[Boolean logic]] is used in various [[programming language]]s. Likewise, [[set (mathematics)|set]]s and other collection-like objects, such as [[multiset]]s and [[list (abstract data type)|list]]s, are common [[datatype]]s in [[computer science]] and [[computer programming|programming]].

In addition to that, [[set (mathematics)|set]]s are commonly referred to in mathematical teaching when talking about different types of numbers (the sets <math>\mathbb{N}</math> of [[natural numbers]], <math>\mathbb{Z}</math> of [[integer]]s, <math>\mathbb{R}</math> of [[real number]]s, etc.), and when defining a [[mathematical function]] as a relation from one [[set (mathematics)|set]] (the [[domain of a function|domain]]) to another [[set (mathematics)|set]] (the [[range of a function|range]]).

==See also==
{{Portal|Mathematics}}
* [[Glossary of set theory]]
* [[Class (set theory)]]
* [[List of set theory topics]]
* [[Relational model]]&nbsp;– borrows from set theory
* [[Venn diagram]]

== Notes ==
{{NoteFoot}}
{{notelist}}

== References ==
{{Reflist}}
{{refbegin}}
* {{Citation |last=Kunen |first=Kenneth |author-link=Kenneth Kunen |year=1980 |title=[[Set Theory: An Introduction to Independence Proofs]] |publisher=North-Holland |isbn=0-444-85401-0}}
* {{Citation |last=Johnson |first=Philip |year=1972 |title=A History of Set Theory |url=https://archive.org/details/historyofsettheo0000unse |url-access=registration |publisher=Prindle, Weber & Schmidt |isbn=0-87150-154-6}}
{{refend}}

==Further reading==
* {{Citation |last=Devlin |first=Keith |author-link=Keith Devlin |year=1993 |title=The Joy of Sets: Fundamentals of Contemporary Set Theory |series=Undergraduate Texts in Mathematics |doi=10.1007/978-1-4612-0903-4 |edition=2nd |publisher=Springer Verlag |isbn=0-387-94094-4 }}
* {{Citation |last=Ferreirós |first=Jose |year=2001 |title=Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics |url=https://books.google.com/books?id=DITy0nsYQQoC|location=Berlin |publisher=Springer |isbn=978-3-7643-5749-8}}
* {{Citation |last=Monk |first=J. Donald |year=1969 |title=Introduction to Set Theory |url=https://archive.org/details/introductiontose0000monk/page/n5/mode/2up |url-access=registration |publisher=McGraw-Hill Book Company |isbn=978-0-898-74006-6}}
* {{Citation |last=Potter |first=Michael |year=2004 |title=Set Theory and Its Philosophy: A Critical Introduction |url=https://books.google.com/books?id=FxRoPuPbGgUC|publisher=[[Oxford University Press]] |isbn=978-0-191-55643-2}}
* {{Citation |last1=Smullyan |first1=Raymond M. |author-link=Raymond Smullyan |last2=Fitting |first2=Melvin |year=2010 |title=Set Theory and the Continuum Problem |publisher=[[Dover Publications]] |isbn=978-0-486-47484-7}}
* {{Citation |last=Tiles |first=Mary |author-link=Mary Tiles |year=2004|title=The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise |url=https://books.google.com/books?id=02ASV8VB4gYC |publisher=[[Dover Publications]] |isbn=978-0-486-43520-6}}

==External links==
{{Sister project links|collapsible=yes|commonscat=yes|n=no|s=no}}
{{Wikibooks|Discrete mathematics/Set theory}}
* Daniel Cunningham, [http://www.iep.utm.edu/set-theo/ Set Theory] article in the ''[[Internet Encyclopedia of Philosophy]]''.
* Jose Ferreiros, [https://plato.stanford.edu/entries/settheory-early/ "The Early Development of Set Theory"] article in the ''[Stanford Encyclopedia of Philosophy]''.
* [[Matthew Foreman|Foreman, Matthew]], [[Akihiro Kanamori]], eds. ''[http://handbook.assafrinot.com/ Handbook of Set Theory]''. 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).
* {{Springer |title=Axiomatic set theory |id=p/a014310}}
* {{Springer |title=Set theory |id=p/s084750}}
* [[Arthur Schoenflies|Schoenflies, Arthur]] (1898). [https://archive.org/stream/encyklomath101encyrich#page/n229 Mengenlehre] in [[Klein's encyclopedia]].
* {{Library resources about |onlinebooks=yes |lcheading=Set theory |label=set theory}}
* {{cite web |first=Walter B. |last=Rudin |author-link=Walter Rudin |date=April 6, 1990 |title=Set Theory: An Offspring of Analysis |work=Marden Lecture in Mathematics |location=[[University of Wisconsin-Milwaukee]] |url=https://www.youtube.com/watch?v=hBcWRZMP6xs&list=PLvAAmIFroksMKHv5O4lwpJJzfmUL0cQ7A&index=3 | archive-url=https://ghostarchive.org/varchive/youtube/20211031/hBcWRZMP6xs| archive-date=2021-10-31 | url-status=live|via=[[YouTube]] }}{{cbignore}}

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