TheInfoList

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 formal sys ...
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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, is mostly concerned with those that are relevant to
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
as a whole. The modern study of set theory was initiated by the German mathematicians
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
and
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
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 Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...
''. After the discovery of
paradoxes A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-con ...
within
naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...
(such as
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

,
Cantor's paradox In set theory Set theory is the branch of that studies , 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 , is mostly concerned with those that ar ...
and
Burali-Forti paradox In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
) various axiomatic systems were proposed in the early twentieth century, of which
Zermelo–Fraenkel set theory In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...
(with or without the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

) 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 Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

, and has various applications in
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
(such as in the theory of
relational algebra In database theory, relational algebra is a theory that uses algebraic structures with a well-founded semantics for modeling data, and defining queries on it. The theory has been introduced by Edgar F. Codd. The main application of relational ...
),
philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such questio ...

and formal semantics. Its foundational appeal, together with its
paradoxes A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-con ...
, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for
logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact A fact is something that is true True most commonly refers to truth Truth is the property of being in accord with fac ...

s and
philosophers of mathematics A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek thinker Pythagoras ( ...
. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
line to the study of the
consistency In classical logic, classical deductive logic, a consistent theory (mathematical logic), 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 def ...

of
large cardinal In the mathematical field of set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections o ...
s.

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 Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
: "
On a Property of the Collection of All Real Algebraic Numbers Cantor's first set theory article contains Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor e ...
". Since the 5th century BC, beginning with Greek mathematician in the West and early
Indian mathematicians The chronology of Indian mathematicians spans from the Indus Valley Civilization oxen for pulling a cart and the presence of the chicken The chicken (''Gallus gallus domesticus''), a subspecies of the red junglefowl, is a type of d ...
in the East, mathematicians had struggled with the concept of
infinity Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

. Especially notable is the work of
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian A Bohemian () is a resident of Bohemia Bohemia ( ; cs, Čechy ; ; hsb, Čěska; szl, Czechy) is the westernmost a ...

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 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

. An 1872 meeting between Cantor and
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
influenced Cantor's thinking, and culminated in Cantor's 1874 paper. Cantor's work initially polarized the mathematicians of his day. While
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

and Dedekind supported Cantor,
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

, now seen as a founder of
mathematical constructivism In the philosophy of mathematics Philosophy (from , ) is the study of general and fundamental questions, such as those about reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient ...
, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as
one-to-one correspondence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
among sets, his proof that there are more
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s than integers, and the "infinity of infinities" ("
Cantor's paradise ''Cantor's paradise'' is an expression used by in describing set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, ...
") resulting from the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
operation. This utility of set theory led to the article "Mengenlehre", contributed in 1898 by
Arthur Schoenflies Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a Germany, German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. Schoen ...
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 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 or
paradoxes A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-con ...
.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
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 set theory, Zer ...
independently found the simplest and best known paradox, now called
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

: 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 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
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 ''The Principles of Mathematics'' (''PoM'') is a 1903 book by Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath, philosopher, Mathematical logic, logician, mathemat ...
''. 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 f ...
, 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 and Grace Chisholm Young, published by
Cambridge University Press Cambridge University Press (CUP) is the publishing business of the University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowled ...
. 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 Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israel Israel (; he, יִשְׂרָאֵל; ar, إِسْرَائِيل), officially known as the State of Israe ...
and
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set ...
in 1922 resulted in the set of axioms , which became the most commonly used set of axioms for set theory. The work of , such as that of
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (a ...
, 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 Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

and
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
is thought to be a preferred foundation.

Basic concepts and notation

Set theory begins with a fundamental
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
between an object and a set . If is a ''
member Member may refer to: * Military juryA United States military "jury" (or "Members", in military parlance) serves a function similar to an American civilian jury, but with several notable differences. Only a Courts-martial in the United States, Gene ...
'' (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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

'' 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
'' 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 Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
features
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s on
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

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 The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
'' 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 In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

'' 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 A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
'' 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 In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
as in the study of
Venn diagram A Venn diagram is a widely used diagram A diagram is a symbolic representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy ...
s. *''
Symmetric difference In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'' 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
'' of and , denoted , is the set whose members are all possible
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s , where is a member of and is a member of . For example, the Cartesian product of *''
Power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
'' of a set , denoted $\mathcal\left(A\right)$, 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s, the set of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

—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.

Some ontology

A set is
pure Pure may refer to: Computing * A pure function * A virtual function, pure virtual function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify too ...
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 In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
'' of pure sets, and many systems of
axiomatic set theory 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, i ...
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 hierarchyIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by
transfinite recursion Transfinite induction is an extension of mathematical induction Image:Dominoeffect.png, Mathematical induction can be informally illustrated by reference to the sequential effect of falling Domino effect, dominoes. Mathematical induction is a m ...
) an
ordinal number In set theory Set theory is the branch of that studies , 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 , is mostly concerned with those that ...
$\alpha$, known as its ''rank.'' The rank of a pure set $X$ 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 $\alpha$, the set $V_$ is defined to consist of all pure sets with rank less than $\alpha$. The entire von Neumann universe is denoted $V$.

Formalized set theory

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using
Venn diagram A Venn diagram is a widely used diagram A diagram is a symbolic representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy ...
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 In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

and the
Burali-Forti paradox In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ty ...
. 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 hierarchyIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. Such systems come in two flavors, those whose
ontology Ontology is the branch of philosophy that studies concepts such as existence, being, Becoming (philosophy), becoming, and reality. It includes the questions of how entities are grouped into Category of being, basic categories and which of these ...

consists of: *''Sets alone''. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory with the (ZFC). Fragments of ZFC include: **
Zermelo set theory Zermelo set theory (sometimes denoted by Z-), as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo-Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It ...
, which replaces the
axiom schema of replacement In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...

with that of separation; **
General set theoryGeneral set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Zermelo set theory, Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include th ...
, a small fragment of
Zermelo set theory Zermelo set theory (sometimes denoted by Z-), as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo-Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It ...
sufficient for the
Peano axioms In 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 p ...
and
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s; **
Kripke–Platek set theoryThe Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. KP is considerably weaker than Zermelo–Fraenkel set theory (ZFC), and can be thought of as roughly the impredicativity, predi ...
, which omits the axioms of infinity,
powerset Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion. In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ...
, and
choice A choice is the range of different things from which you can choose. The arrival at a choice may incorporate Motivation, motivators and Choice modelling, models. For example, a traveler might choose a route for a journey based on the preferenc ...

, and weakens the axiom schemata of separation and . *''Sets and
proper class Proper may refer to: Mathematics * Proper map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and qu ...
es''. These include
Von Neumann–Bernays–Gödel set theory In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theorie ...
, which has the same strength as 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 In set theory Set theory is the branch of that studies , 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 , is mostly concerned with those that ar ...
s'', objects that can be members of sets but that are not themselves sets and do not have any members. The ''
New FoundationsIn mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
'' systems of NFU (allowing
urelement In set theory Set theory is the branch of that studies , 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 , is mostly concerned with those that ar ...
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 In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

does not hold. Systems of
constructive set theoryConstructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "=" and "\in" of classical set theory is usually used, so this is not to be confused with a co ...
, 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 funda ...
classical logic Classical logic (or standard logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world. Char ...
. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and
fuzzy set theoryIn mathematics, fuzzy sets (a.k.a. uncertain sets) are somewhat like Set (mathematics), sets whose Element (mathematics), elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi Asker Zadeh, Lotfi A. Zadeh and in 1965 ...
, in which the value of an
atomic formula In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...
embodying the membership relation is not simply True or False. The
Boolean-valued model In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
s of are a related subject. An enrichment of called
internal set theoryInternal set theory (IST) is a mathematical theory of Set (mathematics), sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the ...
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 Princeton University is a private university, private Ivy League research university in ...
in 1977.

Applications

Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
s,
manifolds The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
,
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
,
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s, and
relational algebra In database theory, relational algebra is a theory that uses algebraic structures with a well-founded semantics for modeling data, and defining queries on it. The theory has been introduced by Edgar F. Codd. The main application of relational ...
s can all be defined as sets satisfying various (axiomatic) properties.
Equivalence Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit The album-equivalent unit is a measurement unit in music industry to define the consumption of music that equals the purchase of one album copy. This consumpti ...
and
order relation Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
s are ubiquitous in mathematics, and the theory of mathematical
relations Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
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 Image:Principia Mathematica 54-43.png, 500px, ✸54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st editionp. 379(p. 362 in 2nd edition; p. 360 in abridged v ...
'', 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 First or 1st is the ordinal form of the number one (#1). First or 1st may also refer to: *World record A world record is usually the best global and most important performance that is ever recorded and officially verified in a specific skill ...
or
second-order logic In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
. For example, properties of the
natural Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and ...
and
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s can be derived within set theory, as each number system can be identified with a set of
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es under a suitable
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
whose field is some
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
. Set theory as a foundation for
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
,
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

,
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, and
discrete mathematics Discrete mathematics is the study of mathematical structures In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
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 Metamath is a formal language and an associated computer program (a proof checker) for archiving, verifying, and studying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in ...
, includes human-written, computer-verified derivations of more than 12,000 theorems starting from set theory,
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
and
propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...
.

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 Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
to infinite sets. This includes the study of
cardinal arithmetic 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
and the study of extensions of
Ramsey's theorem In combinatorial mathematics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph Complete may refer to: Logic * Co ...
such as the Erdős–Rado theorem. Double extension set theory (DEST) is an
axiomatic set theory 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, i ...
proposed by Andrzej Kisielewicz consisting of two separate membership relations on the universe of sets.

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 is between set theory and recursion theory. It includes the study of lightface pointclasses, 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 relations and more complicated definable
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s. This has important applications to the study of invariant (mathematics), invariants in many fields of mathematics.

Fuzzy set theory

In set theory as Georg Cantor, Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In ''
fuzzy set theoryIn mathematics, fuzzy sets (a.k.a. uncertain sets) are somewhat like Set (mathematics), sets whose Element (mathematics), elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi Asker Zadeh, Lotfi A. Zadeh and in 1965 ...
'' 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 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 In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

, 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 In the mathematical field of set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections o ...
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).

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 In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...
.

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 degrees have an elegant structure.

Forcing

Paul Cohen (mathematician), Paul Cohen invented the method of ''forcing (mathematics), forcing'' while searching for a model theory, model of in which the continuum hypothesis fails, or a model of ZF in which the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s without changing any of the
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
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 In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
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 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.

Objections to set theory

From set theory's inception, some mathematicians controversy over Cantor's theory, 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''. 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]". 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 (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. 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. 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. 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 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 types. Principles such as the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

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 the basics of
naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...
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 A Venn diagram is a widely used diagram A diagram is a symbolic representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy ...
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 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, since boolean logic is used in various programming languages. Likewise, set (mathematics), sets and other collection-like objects, such as multisets and list (abstract data type), lists, are common datatypes in
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
and computer programming, programming. In addition to that, set (mathematics), sets are commonly referred to in mathematical teaching when talking about different types of numbers (, , , ...), 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).

* Glossary of set theory * Class (set theory) * List of set theory topics * Relational model – borrows from set theory

* * * * * * * *