Set theory is the branch of
mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
that studies
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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 within
naive set theory (such as
Russell's paradox,
Cantor's paradox and the
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after C ...
) various
axiomatic systems 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
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
(such as in the theory of
relational algebra),
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
and
formal semantics. Its foundational appeal, together with its
paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for
logicians and
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 cardinals.
History
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 numbers 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
Felix Klein's ''Encyclopedia of Mathematical Sciences'' is a German mathematical encyclopedia published in six volumes from 1898 to 1933. Klein and Wilhelm Franz Meyer were organizers of the encyclopedia. Its full title in English is ''Encyclope ...
.
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
Antinomy (Greek ἀντί, ''antí'', "against, in opposition to", and νόμος, ''nómos'', "law") refers to a real or apparent mutual incompatibility of two laws. It is a term used in logic and epistemology, particularly in the philosophy of I ...
or
paradoxes.
Bertrand Russell and
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic ...
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
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
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''. Rather than the term ''set'', Russell used the term ''
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
'', which has subsequently been used more technically.
In 1906, the term ''set'' appeared in the book ''Theory of Sets of Points'' by husband and wife
William Henry Young
William Henry Young FRS (London, 20 October 1863 – Lausanne, 7 July 1942) was an English mathematician. Young was educated at City of London School and Peterhouse, Cambridge. He worked on measure theory, Fourier series, differential calcul ...
and
Grace Chisholm Young
Grace Chisholm Young (née Chisholm, 15 March 1868 – 29 March 1944) was an English mathematician. She was educated at Girton College, Cambridge, England and continued her studies at Göttingen University in Germany, where in 1895 she receive ...
, published by
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pr ...
.
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
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
, 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—
category theory 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
Member may refer to:
* Military jury, referred to as "Members" in military jargon
* Element (mathematics), an object that belongs to a mathematical set
* In object-oriented programming, a member of a class
** Field (computer science), entries in ...
'' (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 operations on
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s, set theory features binary operations on sets. The following is a partial list of them:
*''
Union'' 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 .
*''
Intersection'' 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 .
*''
Set difference'' 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 diagrams.
*''
Symmetric difference'' 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 .
*''
Cartesian product'' of and , denoted , is the set whose members are all possible
ordered pairs , where is a member of and is a member of . For example, the Cartesian product of
*''
Power set'' 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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, the set of
real numbers 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.
Ontology
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
In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W_\alpha indexed by ordinals \alpha such that
* W_\alpha \subseteq W_
* If \lambda is a limit ordinal, then W_\lambda = \bigcup_ W_
Some authors additionally r ...
, 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 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
.
Formalized set theory
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using
Venn 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 are
Russell's paradox and the
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after C ...
. 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
In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W_\alpha indexed by ordinals \alpha such that
* W_\alpha \subseteq W_
* If \lambda is a limit ordinal, then W_\lambda = \bigcup_ W_
Some authors additionally r ...
. Such systems come in two flavors, those whose
ontology
In metaphysics, ontology is the philosophy, philosophical study of being, as well as related concepts such as existence, Becoming (philosophy), becoming, and reality.
Ontology addresses questions like how entities are grouped into Category ...
consists of:
*''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 sets;
**
Kripke–Platek set theory, which omits the axioms of infinity,
powerset, and
choice, and weakens the axiom schemata of
separation and
replacement.
*''Sets and
proper classes''. 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 ''
urelements'', 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
urelements) 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 has argued that it does reflect an
iterative conception of set
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
.
Systems of
constructive set theory, such as CST, CZF, and IZF, embed their set axioms in
intuitionistic
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
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
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take v ...
s of
ZFC are a related subject.
An enrichment of
ZFC called
internal set theory was proposed by
Edward Nelson
Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematic ...
in 1977.
Applications
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as
graphs,
manifolds,
rings,
vector spaces, and
relational algebras can all be defined as sets satisfying various (axiomatic) properties.
Equivalence and
order relations 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
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
and
real numbers can be derived within set theory, as each number system can be identified with a set of
equivalence classes under a suitable
equivalence relation whose field is some
infinite set.
Set theory as a foundation for
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
,
topology,
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, 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
In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets. It is named after Paul Erdős and Richard Rado
Richard Rado FRS (28 ...
.
Descriptive set theory
''Descriptive set theory'' is the study of subsets of the
real line and, more generally, subsets of
Polish spaces. It begins with the study of
pointclasses 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 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 of
effective descriptive set theory
Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightface definitions; that is, definitions that do not require an arbitrary real parameter (Moschovakis 1980). Thus effective descriptiv ...
is between set theory and
recursion theory. It includes the study of
lightface pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by ...
es, and is closely related to
hyperarithmetical theory In recursion theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an import ...
. 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 relations and more complicated definable
equivalence relations. 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
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
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
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and si ...
and
large 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).
Large cardinals
A ''large cardinal'' is a cardinal number with an extra property. Many such properties are studied, including
inaccessible 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 in
Zermelo–Fraenkel set theory.
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 In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wa ...
s have an elegant structure.
Forcing
Paul Cohen invented the method of ''
forcing'' while searching for a
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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s without changing any of the
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
s of the original model. Forcing is also one of two methods for proving
relative consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
by finitistic methods, the other method being
Boolean-valued model
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take v ...
s.
Cardinal invariants
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 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
''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
Naivety (also spelled naïvety), naiveness, or naïveté is the state of being naive. It refers to an apparent or actual lack of experience and sophistication, often describing a neglect of pragmatism in favor of moral idealism. A ''naïve'' may ...
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 and
replacement, 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.
Ludwig Wittgenstein 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 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. Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in ''
Remarks on the Foundations of Mathematics
''Remarks on the Foundations of Mathematics'' (german: Bemerkungen über die Grundlagen der Mathematik) is a book of Ludwig Wittgenstein's notes on the philosophy of mathematics. It has been translated from German to English by G.E.M. Anscombe, ...
'': Wittgenstein attempted to refute
Gödel's incompleteness theorems after having only read the abstract. As reviewers
Kreisel
Kreisel (German for gyro) is the name of a turn found on some bobsleigh, luge, and skeleton tracks around the world. It is a single continuous curve which turns far enough around that the track must go under itself on the exit of the curve.
Ther ...
,
Bernays Bernays is a surname. Notable people with the surname include:
* Adolphus Bernays (1795–1864), professor of German in London; brother of Isaac Bernays and father of:
** Lewis Adolphus Bernays (1831–1908), public servant and agricultural write ...
,
Dummett, 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.
Category theorists
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
*Categories (Peirce)
* ...
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
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is clos ...
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 In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this appr ...
and
Stone spaces.
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
In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory ...
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
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
.
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 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
A primary school (in Ireland, the United Kingdom, Australia, Trinidad and Tobago, Jamaica, and South Africa), junior school (in Australia), elementary school or grade school (in North America and the Philippines) is a school for primary e ...
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 diagrams are widely employed to explain basic set-theoretic relationships to primary school
A primary school (in Ireland, the United Kingdom, Australia, Trinidad and Tobago, Jamaica, and South Africa), junior school (in Australia), elementary school or grade school (in North America and the Philippines) is a school for primary e ...
students (even though John Venn originally devised them as part of a procedure to assess the validity of inferences in term logic).
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''"), which may be useful when learning computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
, since boolean logic is used in various programming languages. Likewise, set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s and other collection-like objects, such as multisets and lists, are common datatypes in computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
and programming.
In addition to that, set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s are commonly referred to in mathematical teaching when talking about different types of numbers (the sets of natural numbers, of integers, of real numbers, etc.), and when defining a mathematical function as a relation from one set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
(the domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
) to another set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
(the range).
See also
*
* Class (set theory)
* List of set theory topics
* Relational model – borrows from set theory
* Venn diagram
Notes
References
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Further reading
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External links
* Daniel Cunningham
Set Theory
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)
Mengenlehre
in Klein's encyclopedia
Felix Klein's ''Encyclopedia of Mathematical Sciences'' is a German mathematical encyclopedia published in six volumes from 1898 to 1933. Klein and Wilhelm Franz Meyer were organizers of the encyclopedia. Its full title in English is ''Encyclope ...
.
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{{Authority control
S
Formal methods
Georg Cantor