Set theory is the branch ofmathematical logic that studiessets, 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 ofmathematics – is mostly concerned with those that are relevant to mathematics as a whole.
The basic notion of grouping objects has existed since at least theemergence of numbers, and the notion of treating sets as their own objects has existed since at least theTree of Porphyry in 3rd-century AD. The simplicity and ubiquity of sets makes it hard to determine the origin of sets as now used in mathematics; however,Bernard Bolzano'sParadoxes of the Infinite (Paradoxien des Unendlichen, 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded onGalileo's paradox, and introducedone-to-one correspondence of infinite sets, for example between theintervals and by the relation. However, he resisted saying these sets wereequinumerous, and his work is generally considered to have been uninfluential in mathematics of his time.[1][2]
Before mathematical set theory, basic concepts ofinfinity were considered to be in the domain of philosophy (see:Infinity (philosophy) andInfinity § History). Since the 5th century BC, beginning with Greek philosopherZeno of Elea in the West (and earlyIndian mathematicians in the East), mathematicians had struggled with the concept of infinity. With thedevelopment of calculus in the late 17th century, philosophers began to generally distinguish between potential andactual infinity, wherein mathematics was only considered in the latter.[3]Carl Friedrich Gauss famously stated: "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics."[4]
Development of mathematical set theory was motivated by several mathematicians.Bernhard Riemann's lectureOn the Hypotheses which lie at the Foundations of Geometry (1854) proposed new ideas abouttopology. His lectures also introduced the concept of basing mathematics in terms of sets ormanifolds in the sense of aclass (which he calledMannigfaltigkeit) now calledpoint-set topology. The lecture was published byRichard Dedekind in 1868, along with Riemann's paper ontrigonometric series (which presented theRiemann integral), The latter was the starting point for a movement inreal analysis of the study of “seriously”discontinuous functions. A youngGeorg Cantor entered into this area, which led him to the study ofpoint-sets. Around 1871, influenced by Riemann, Dedekind began working with sets in his publications, which dealt very clearly and precisely withequivalence relations,partitions of sets, andhomomorphisms. Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. However, he did not publish a formal explanation of his set theory until 1888.
Cantor introduced fundamental constructions in set theory, such as thepower set of a setA, which is the set of all possiblesubsets ofA. He later proved that the size of the power set ofA is strictly larger than the size ofA, even whenA is an infinite set; this result soon became known asCantor's theorem. Cantor developed a theory oftransfinite numbers, calledcardinals andordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter (ℵ,aleph) with a natural number subscript; for the ordinals he employed the Greek letter (ω,omega).
Despite the controversy, Cantor's set theory gained remarkable ground around the turn of the 20th century with the work of several notable mathematicians and philosophers. Richard Dedekind, around the same time, began working with sets in his publications, and famously constructing the real numbers usingDedekind cuts. He also worked withGiuseppe Peano in developing thePeano axioms, which formalized natural-number arithmetic, using set-theoretic ideas, which also introduced theepsilon symbol forset membership. Possibly most prominently,Gottlob Frege began to develop hisFoundations of Arithmetic.
In his work, Frege tries to ground all mathematics in terms of logical axioms using Cantor's cardinality. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concepthorse in the barn. Frege attempted to explain our grasp of numbers through cardinality ('the number of...', or), relying onHume's principle.
LetR be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) IfR is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols:
Set theory begins with a fundamentalbinary relation between an objecto and a setA. Ifo is amember (orelement) ofA, the notationo ∈A is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }.[8] Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets.
A derived binary relation between two sets is the subset relation, also calledset inclusion. If all the members of setA are also members of setB, thenA is asubset ofB, denotedA ⊆B. For example,{1, 2} is a subset of{1, 2, 3}, and so is{2} but{1, 4} 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 termproper subset is defined, variously denoted,, or (note however that the notation is sometimes used synonymously with; that is, allowing the possibility thatA andB are equal). We callA aproper subset ofB if and only ifA is a subset ofB, butA is not equal toB. Also, 1, 2, and 3 are members (elements) of the set{1, 2, 3}, but are not subsets of it; and in turn, the subsets, such as{1}, are not members of the set{1, 2, 3}. More complicated relations can exist; for example, the set{1} is both a member and a proper subset of the set{1, {1}}.
Union of the setsA andB, denotedA ∪B, is the set of all objects that are a member ofA, orB, or both.[10] For example, the union of{1, 2, 3} and{2, 3, 4} is the set{1, 2, 3, 4}.
Intersection of the setsA andB, denotedA ∩B, is the set of all objects that are members of bothA andB.[11] For example, the intersection of{1, 2, 3} and{2, 3, 4} is the set{2, 3}.
Set difference ofU andA, denotedU ∖A, is the set of all members ofU that are not members ofA.[12] The set difference{1, 2, 3} ∖ {2, 3, 4} is{1}, while conversely, the set difference{2, 3, 4} ∖ {1, 2, 3} is{4}. WhenA is a subset ofU, the set differenceU ∖A is also called thecomplement ofA inU. In this case, if the choice ofU is clear from the context, the notationAc is sometimes used instead ofU ∖A, particularly ifU is auniversal set as in the study ofVenn diagrams.[13]
Symmetric difference of setsA andB, denotedA △B orA ⊖B, is the set of all objects that are a member of exactly one ofA andB (elements which are in one of the sets, but not in both). For instance, for the sets{1, 2, 3} and{2, 3, 4}, the symmetric difference set is{1, 4}. It is the set difference of the union and the intersection,(A ∪B) ∖ (A ∩B) or(A ∖B) ∪ (B ∖A).
Cartesian product ofA andB, denotedA ×B, is the set whose members are all possibleordered pairs(a,b), wherea is a member ofA andb is a member ofB. For example, the Cartesian product of {1, 2} and {red, white} is{(1, red), (1, white), (2, red), (2, white)}.[14]
Some basic sets of central importance are the set ofnatural numbers, the set ofreal numbers and theempty set – the unique set containing no elements. The empty set is also occasionally called thenull set,[15] though this name is ambiguous and can lead to several interpretations. The empty set can be denoted with empty braces "" or the symbol "" or "".
Thepower set of a setA, denoted, is the set whose members are all of the possible subsets ofA. For example, the power set of{1, 2} is{ {}, {1}, {2}, {1, 2} }. Notably, contains bothA and the empty set.
A set ispure 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 thevon 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 acumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (bytransfinite recursion) anordinal number, known as itsrank. The rank of a pure set 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 containing only the empty set is assigned rank 1. For each ordinal, the set is defined to consist of all pure sets with rank less than. The entire von Neumann universe is denoted .
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools usingVenn diagrams. 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 areRussell's paradox and theBurali-Forti paradox.Axiomatic set theory was originally devised to rid set theory of such paradoxes.[note 1]
The most widely studied systems of axiomatic set theory imply that all sets form acumulative hierarchy.[b] Such systems come in two flavors, those whoseontology consists of:
The above systems can be modified to allowurelements, objects that can be members of sets but that are not themselves sets and do not have any members.Zermelo set theory was originally defined over a domain consisting of both sets and urelements.
TheNew Foundations systems ofNFU (allowingurelements) andNF (lacking them), associate withWillard 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 theaxiom of choice does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness,Thomas Forster has argued that it does reflect aniterative conception of set.[16]
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse asgraphs,manifolds,rings,vector spaces, andrelational algebras can all be defined as sets satisfying various (axiomatic) properties.Equivalence andorder relations are ubiquitous in mathematics, and the theory of mathematicalrelations can be described in set theory.[18][19]
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume ofPrincipia 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, usingfirst orsecond-order logic. For example, properties of thenatural andreal numbers can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms.[20]
Set theory as a foundation formathematical analysis,topology,abstract algebra, anddiscrete 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 fromZFC set theory,first-order logic andpropositional logic.[21]
Descriptive set theory is the study of subsets of thereal line and, more generally, subsets ofPolish spaces. It begins with the study ofpointclasses in theBorel hierarchy and extends to the study of more complex hierarchies such as theprojective hierarchy and theWadge hierarchy. Many properties ofBorel sets 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 ofeffective descriptive set theory is between set theory andrecursion theory. It includes the study oflightface pointclasses, and is closely related tohyperarithmetical 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.
In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. Infuzzy set theory this condition was relaxed byLotfi A. Zadeh so an object has adegree 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.
Aninner model of Zermelo–Fraenkel set theory (ZF) is a transitiveclass that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is theconstructible universeL 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 modelV of ZF satisfies thecontinuum hypothesis or theaxiom of choice, the inner modelL 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 ofdeterminacy andlarge cardinals, 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).[22]
Alarge cardinal is a cardinal number with an extra property. Many such properties are studied, includinginaccessible cardinals,measurable cardinals, 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 inZermelo–Fraenkel set theory.
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. Theaxiom 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 theWadge degrees have an elegant structure.
Paul Cohen invented the method offorcing while searching for amodel ofZFC in which thecontinuum hypothesis fails, or a model of ZF in which theaxiom 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 thenatural numbers without changing any of thecardinal numbers of the original model. Forcing is also one of two methods for provingrelative consistency by finitistic methods, the other method beingBoolean-valued models.
Acardinal 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 ofmeagre sets 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 studies questions ofgeneral 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 thenormal 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.
From set theory's inception, some mathematicians have objected to it as afoundation for mathematics. The most common objection to set theory, oneKronecker voiced in set theory's earliest years, starts from theconstructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both innaive 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 byErrett Bishop's influential bookFoundations of Constructive Analysis.[23]
A different objection put forth byHenri Poincaré is that defining sets using the axiom schemas ofspecification andreplacement, as well as theaxiom of power set, introducesimpredicativity, a type ofcircularity, 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 thatSolomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]".[24]
Ludwig Wittgenstein condemned set theory philosophically for its connotations ofmathematical platonism.[25] 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".[26] Wittgenstein identified mathematics with algorithmic human deduction;[27] the need for a secure foundation for mathematics seemed, to him, nonsensical.[28] Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radicalconstructivism andfinitism. Meta-mathematical statements – which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory – are not mathematics.[29] Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder inRemarks on the Foundations of Mathematics: Wittgenstein attempted to refuteGödel's incompleteness theorems after having only read the abstract. As reviewersKreisel,Bernays,Dummett, andGoodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such asCrispin Wright begun to rehabilitate Wittgenstein's arguments.[30]
Category theorists have proposedtopos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such asconstructivism, finite set theory, andcomputable set theory.[31][32] Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework forpointless topology andStone spaces.[33]
An active area of research is theunivalent foundations and related to ithomotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, withuniversal properties of sets arising from the inductive and recursive properties ofhigher inductive types. Principles such as theaxiom of choice and thelaw 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.[34][35]
As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics ofnaive set theory early inmathematics education.
In the US in the 1960s, theNew Math experiment aimed to teach basic set theory, among other abstract concepts, toprimary school students but was met with much criticism.[36] The math syllabus in European schools followed this trend and currently includes the subject at different levels in all grades.Venn diagrams are widely employed to explain basic set-theoretic relationships to primary school students (even thoughJohn Venn originally devised them as part of a procedure to assess thevalidity ofinferences interm logic).
^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-knownantinomies 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 byBertrand Russell,Julius König,Hermann Weyl andL. 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,ISBN0-674-32449-8 (pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann's 1925 paper.
^The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong".
^This is the converse for ZFC; V is a model of ZFC.
^Ferreirós, José (2024),"The Early Development of Set Theory", in Zalta, Edward N.; Nodelman, Uri (eds.),The Stanford Encyclopedia of Philosophy (Winter 2024 ed.), Metaphysics Research Lab, Stanford University,archived from the original on 2023-03-20, retrieved2025-01-04
^Bolzano, Bernard (1975), Berg, Jan (ed.),Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich Frommann Verlag, p. 152,ISBN3-7728-0466-7
^Kaplansky, Irving (1972), De Prima, Charles (ed.),Set Theory and Metric Spaces, Boston: Allyn and Bacon, p. 4
^Kaplansky, Irving (1972), De Prima, Charles (ed.),Set Theory and Metric Spaces, Boston: Allyn and Bacon, pp. 5–6
^Kaplansky, Irving (1972), De Prima, Charles (ed.),Set Theory and Metric Spaces, Boston: Allyn and Bacon, pp. 5–6
^Kaplansky, Irving (1972), De Prima, Charles (ed.),Set Theory and Metric Spaces, Boston: Allyn and Bacon, p. 19
^Bagaria, Joan (2020),"Set Theory", in Zalta, Edward N. (ed.),The Stanford Encyclopedia of Philosophy (Spring 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved2020-08-20
^Rodych 2018,§2.1: "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 doesnot identify such deduction withphilosophical logic; cf. Rodych§1, paras. 7-12.
^Rodych 2018,§3.4: "Given that mathematics is a '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."
^Rodych 2018,§2.2: "An expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular numbern has a particular property."
^Taylor, Melissa August, Harriet Barovick, Michelle Derrow, Tam Gray, Daniel S. Levy, Lina Lofaro, David Spitz, Joel Stein and Chris (14 June 1999),"The 100 Worst Ideas Of The Century",TIME,archived from the original on 12 April 2025, retrieved12 April 2025{{cite magazine}}: CS1 maint: multiple names: authors list (link)
Foreman, Matthew,Akihiro Kanamori, eds.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).
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