illustrating the intersection
of two sets
Set theory is a branch of mathematical logic
that studies sets
, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical object
The modern study of set theory was initiated by Georg Cantor
and Richard Dedekind
in the 1870s. After the discovery of paradoxes
in naive set theory
, such as Russell's paradox
, numerous axiom systems
were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms
, with or without the axiom of choice
, are the best-known.
Set theory is commonly employed as a foundational system for mathematics
, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics
in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number
line to the study of the consistency
of large cardinal
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
Since the 5th century BC, beginning with Greek
mathematician Zeno of Elea
in the West and early 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. Modern understanding of infinity began in 1870–1874, and was motivated by Cantor's work in real analysis
. An 1872 meeting between Cantor and Richard Dedekind
influenced Cantor's thinking, and culminated in Cantor's 1874 paper.
Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass
and Dedekind supported Cantor, Leopold Kronecker
, now seen as a founder of mathematical constructivism
, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence
among sets, his proof that there are more real number
s than integers, and the "infinity of infinities" ("Cantor's paradise
") resulting from the power set
operation. This utility of set theory led to the article "Mengenlehre", contributed in 1898 by Arthur Schoenflies
to Klein's encyclopedia
The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies
. Bertrand Russell
and Ernst Zermelo
independently found the simplest and best known paradox, now called Russell's paradox
: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899, Cantor had himself posed the question "What is the cardinal number
of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his ''The Principles of Mathematics
In 1906, English readers gained the book ''Theory of Sets of Points'' by husband and wife William Henry Young
and Grace Chisholm Young
, published by Cambridge University Press
The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and the work of Abraham Fraenkel
and Thoralf Skolem
in 1922 resulted in the set of axioms ZFC
, which became the most commonly used set of axioms for set theory. The work of analysts
, such as that of Henri Lebesgue
, demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas—such as algebraic geometry
and algebraic topology
is thought to be a preferred foundation.
Basic concepts and notation
Set theory begins with a fundamental binary relation
between an object and a set . If is a ''member
'' (or ''element'') of , the notation 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.
A derived binary relation between two sets is the subset relation, also called ''set inclusion''. If all the members of set are also members of set , then is a ''subset
'' of , denoted .
For example, is a subset of , and so is but 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. is called a ''proper subset'' of if and only if is a subset of , but is not equal to . Also, 1, 2, and 3 are members (elements) of the set , but are not subsets of it; and in turn, the subsets, such as , are not members of the set .
Just as arithmetic
features binary operation
s on number
s, set theory features binary operations on sets. The following is a partial list of them:
'' of the sets and , denoted ,
is the set of all objects that are a member of , or , or both. For example, the union of and is the set .
'' of the sets and , denoted ,
is the set of all objects that are members of both and . For example, the intersection of and is the set .
'' of and , denoted , is the set of all members of that are not members of . The set difference is , while conversely, the set difference is . When is a subset of , the set difference is also called the ''complement
'' of in . In this case, if the choice of is clear from the context, the notation is sometimes used instead of , particularly if is a universal set
as in the study of Venn diagram
'' of sets and , denoted or ,
is the set of all objects that are a member of exactly one of and (elements which are in one of the sets, but not in both). For instance, for the sets and , the symmetric difference set is . It is the set difference of the union and the intersection, or .
'' of and , denoted ,
is the set whose members are all possible ordered pair
s , where is a member of and is a member of . For example, the Cartesian product of
'' of a set , denoted
is the set whose members are all of the possible subsets of . For example, the power set of is .
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'', though this name is ambiguous and can lead to several interpretations.
thumb|right|300px|An initial segment of the von Neumann hierarchy.
A set is 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
, known as its ''rank''. The rank of a pure set
is defined to be the least upper bound
of all successors
of ranks of members of
. 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
Formalized 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 was originally devised to rid set theory of such paradoxes.
The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy
. Such systems come in two flavors, those whose ontology
*''Sets alone''. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory
with the Axiom of Choice
(ZFC). Fragments of ZFC include:
** Zermelo set theory
, which replaces the axiom schema of replacement
with that of separation
** General set theory
, a small fragment of Zermelo set theory
sufficient for the Peano axioms
and finite set
** Kripke–Platek set theory
, which omits the axioms of infinity, powerset
, and choice
, and weakens the axiom schemata of separation
*''Sets and proper class
es''. These include Von Neumann–Bernays–Gödel set theory
, which has the same 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) 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.
Systems of constructive set theory
, such as CST, CZF, and IZF, embed their set axioms in intuitionistic
instead of classical logic
. Yet other systems accept classical logic but feature a nonstandard membership relation. These include 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
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graph
, and vector space
s can all be defined as sets satisfying various (axiomatic) properties. Equivalence
and order relation
s are ubiquitous in mathematics, and the theory of mathematical relations
can be described in set theory.
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
or second-order logic
. For example, properties of the natural
and real number
s can be derived within set theory, as each number system can be identified with a set of equivalence class
es under a suitable equivalence relation
whose field is some infinite set
Set theory as a foundation for mathematical analysis
, 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
Areas of study
Set theory is a major area of research in mathematics, with many interrelated subfields.
Combinatorial set theory
''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
''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 invariants
in many fields of mathematics.
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
An ''inner model'' of Zermelo–Fraenkel set theory (ZF) is a transitive 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 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).
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'' 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.
invented the method of ''forcing
'' while searching for a model
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 relative consistency
by finitistic methods, the other method being Boolean-valued model
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'' 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 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.
Objections to set theory
From set theory's inception, some mathematicians have objected to it
as a foundation for mathematics
. The most common objection to set theory, one Kronecker
voiced in set theory's earliest years, starts from the constructivist
view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in 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''.
A different objection put forth by Henri Poincaré
is that defining sets using the axiom schemas of specification
, as well as the axiom of power set
, introduces impredicativity
, a type of 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 sing predicative methods
condemned set theory philosophically for its connotations of Mathematical platonism
. 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 constructivism
. Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics. 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 Kreisel
, and 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.
have proposed topos theory
as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism
, finite set theory, and computable
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 pointless topology
and Stone space
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.''Homotopy Type Theory: Univalent Foundations of Mathematics''
The Univalent Foundations Program. Institute for Advanced Study.
Set theory in mathematical education
As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing basic theory, or 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 grade 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.
Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition
) of sets, (e.g. "months starting with the letter ''A''"). This may be useful when learning computer programming, as sets and boolean logic are basic building blocks of many programming languages.
Sets are commonly referred to when teaching about different types of numbers (, , , ...), and when defining mathematical functions as a relationship between two sets.
* Glossary of set theory
* Class (set theory)
* List of set theory topics
* Relational model – borrows from set theory
* Daniel Cunningham
article in the ''Internet Encyclopedia of Philosophy''.
* Jose Ferreiros
The Early Development of Set Theory
article in the ''tanford Encyclopedia of Philosophy'.
* 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).
* Schoenflies, Arthur (1898)
in Klein's encyclopedia.