Set theory
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Set theory is the branch of
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
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 a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
– 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 Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
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), 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 Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
, and has various applications in
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
(such as in the theory of
relational algebra In database theory, relational algebra is a theory that uses algebraic structures for modeling data and defining queries on it with well founded semantics (computer science), semantics. The theory was introduced by Edgar F. Codd. The main applica ...
),
philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
, formal semantics, and evolutionary dynamics. Its foundational appeal, together with its paradoxes, and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
ians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
line to the study of the consistency of large cardinals.


History


Early history

The basic notion of grouping objects has existed since at least the emergence of numbers, and the notion of treating sets as their own objects has existed since at least the Tree of Porphyry, 3rd-century AD. The simplicity and ubiquity of sets makes it hard to determine the origin of sets as now used in mathematics, however, Bernard Bolzano's '' Paradoxes of the Infinite'' (''Paradoxien des Unendlichen'', 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded on Galileo's paradox, and introduced one-to-one correspondence of infinite sets, for example between the intervals ,5/math> and ,12/math> by the relation 5y = 12x. However, he resisted saying these sets were equinumerous, and his work is generally considered to have been uninfluential in mathematics of his time. Before mathematical set theory, basic concepts of
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
were considered to be solidly in the domain of philosophy (see: ''
Infinity (philosophy) In philosophy and theology, infinity is explored in articles under headings such as the Absolute (philosophy), Absolute, God, and Zeno's paradoxes. In Greek philosophy, for example in Anaximander, 'the Boundless' is the origin of all that is. He ...
'' and '). Since the 5th century BC, beginning with Greek philosopher
Zeno of Elea Zeno of Elea (; ; ) was a pre-Socratic Greek philosopher from Elea, in Southern Italy (Magna Graecia). He was a student of Parmenides and one of the Eleatics. Zeno defended his instructor's belief in monism, the idea that only one single en ...
in the West (and early
Indian mathematicians Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians in the modern era. One of such works is Hindu numeral system which is predominantly used today and is likely ...
in the East), mathematicians had struggled with the concept of infinity. With the development of calculus in the late 17th century, philosophers began to generally distinguish between actual and potential infinity, wherein mathematics was only considered in the latter.
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
famously stated: "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics." Development of mathematical set theory was motivated by several mathematicians.
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
's lecture ''On the Hypotheses which lie at the Foundations of Geometry'' (1854) proposed new ideas about
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, and about basing mathematics (especially geometry) in terms of sets or
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s in the sense of a
class Class, Classes, 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 d ...
(which he called ''Mannigfaltigkeit'') now called point-set topology. The lecture was published by Richard Dedekind in 1868, along with Riemann's paper on trigonometric series (which presented the Riemann integral), The latter was a starting point a movement in
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
for the study of “seriously” discontinuous functions. A young
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
entered into this area, which led him to the study of point-sets. Around 1871, influenced by Riemann, Dedekind began working with sets in his publications, which dealt very clearly and precisely with equivalence relations, partitions of sets, and homomorphisms. Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. However, he did not publish a formal explanation of his set theory until 1888.


Naive set theory

Set theory, as understood by modern mathematicians, is generally considered to be founded by a single paper in 1874 by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
titled '' On a Property of the Collection of All Real Algebraic Numbers''. In his paper, he developed the notion of
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
, comparing the sizes of two sets by setting them in one-to-one correspondence. His "revolutionary discovery" was that the set of all
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s is uncountable, that is, one cannot put all real numbers in a list. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. Cantor introduced fundamental constructions in set theory, such as the power set of a set ''A'', which is the set of all possible
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of ''A''. He later proved that the size of the power set of ''A'' is strictly larger than the size of ''A'', even when ''A'' is an infinite set; this result soon became known as Cantor's theorem. Cantor developed a theory of transfinite numbers, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter \aleph ( ,
aleph Aleph (or alef or alif, transliterated ʾ) is the first Letter (alphabet), letter of the Semitic abjads, including Phoenician alphabet, Phoenician ''ʾālep'' 𐤀, Hebrew alphabet, Hebrew ''ʾālef'' , Aramaic alphabet, Aramaic ''ʾālap'' � ...
) with a natural number subscript; for the ordinals he employed the Greek letter \omega (, omega). Set theory was beginning to become an essential ingredient of the new “modern” approach to mathematics. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, ...
and
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
and later from
Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
and L. E. J. Brouwer, while
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...
raised philosophical objections (see: '' Controversy over Cantor's theory''). Dedekind's algebraic style only began to find followers in the 1890s Despite the controversy, Cantor's set theory gained remarkable ground around the turn of the 20th century with the work of several notable mathematicians and philosophers. Richard Dedekind, around the same time, began working with sets in his publications, and famously constructing the real numbers using Dedekind cuts. He also worked with
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much Mathematical notati ...
in developing the Peano axioms, which formalized natural-number arithmetic, using set-theoretic ideas, which also introduced the epsilon symbol for set membership. Possibly most prominently,
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
began to develop his '' Foundations of Arithmetic''. In his work, Frege tries to ground all mathematics in terms of logical axioms using Cantor's cardinality. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept ''horse in the barn''. Frege attempted to explain our grasp of numbers through cardinality ('the number of...', or Nx: Fx ), relying on Hume's principle. However, Frege's work was short-lived, as it was found by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
that his axioms lead to a contradiction. Specifically, Frege's Basic Law V (now known as the axiom schema of unrestricted comprehension). According to Basic Law V, for any sufficiently well-defined
property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, re ...
, there is the set of all and only the objects that have that property. The contradiction, called Russell's paradox, is shown as follows: Let ''R'' be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If ''R'' is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: : \text R = \ \text R \in R \iff R \not \in R This came around a time of several paradoxes or counter-intuitive results. For example, that the parallel postulate cannot be proved, the existence of
mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
s that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with Peano arithmetic. The result was a
foundational crisis of mathematics Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particul ...
.


Basic concepts and notation

Set theory begins with a fundamental
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
between an object and a set . If is a '' member'' (or ''element'') of , the notation is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces . Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. 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, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
'' 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, variously denoted A\subset B, A\subsetneq B, or A\subsetneqq B (note however that the notation A\subset B is sometimes used synonymously with A\subseteq B; that is, allowing the possibility that and are equal). We call 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 . More complicated relations can exist; for example, the set is both a member and a proper subset of the set . Just as
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
features
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
s on
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
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 In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
'' 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 and is Some basic sets of central importance are the set of
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s, the set of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s and the
empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
– 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. The empty set can be denoted with empty braces " \ " or the symbol " \varnothing " or " \emptyset ". The power set of a set , denoted \mathcal(A), is the set whose members are all of the possible subsets of . For example, the power set of is . Notably, \mathcal(A) contains both and the empty set.


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, 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 In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
\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 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. Axiomatic set theory was originally devised to rid set theory of such paradoxes. The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose
ontology Ontology is the philosophical study of existence, being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of realit ...
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 set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...
s; ** 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 class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
es''. These include Von Neumann–Bernays–Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse–Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC. The above systems can be modified to allow '' 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 Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century" ...
, 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. Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
. 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 models of ZFC are a related subject. An enrichment of ZFC called internal set theory was proposed by Edward Nelson 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 space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s, and
relational algebra In database theory, relational algebra is a theory that uses algebraic structures for modeling data and defining queries on it with well founded semantics (computer science), semantics. The theory was introduced by Edgar F. Codd. The main applica ...
s 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 and
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms. Set theory as a foundation for
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
,
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
,
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, and
discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...
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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdős–Rado theorem.


Descriptive set theory

''Descriptive set theory'' is the study of subsets of the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
and, more generally, subsets of Polish spaces. It begins with the study of
pointclass In the mathematical field of descriptive set theory, a pointclass is a collection of Set (mathematics), sets of point (mathematics), points, where a ''point'' is ordinarily understood to be an element of some perfect set, perfect Polish space. In ...
es in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties of
Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
s can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals. The field of effective descriptive set theory is between set theory and
recursion theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ex ...
. 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 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, Classes, 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 d ...
that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe ''L'' developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model ''V'' of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model ''L'' constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent. The study of inner models is common in the study of determinacy and large 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 degrees have an elegant structure.


Forcing

Paul Cohen invented the method of '' forcing'' while searching for a
model A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...
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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s without changing any of the
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models.


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.


Controversy

From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop's influential book ''Foundations of Constructive Analysis''. A different objection put forth by
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
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 Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...
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'': Wittgenstein attempted to refute
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
after having only read the abstract. As reviewers Kreisel, Bernays, 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 have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory. Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone 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 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.


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 contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out Scholarly method, scholarly research into the transfer of mathematical know ...
. 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, India, the United Kingdom, Australia, New Zealand, Trinidad and Tobago, Jamaica, South Africa, and Singapore), elementary school, or grade school (in North America and the Philippines) is a school for primary ...
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 students (even though
John Venn John Venn, Fellow of the Royal Society, FRS, Fellow of the Society of Antiquaries of London, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in l ...
originally devised them as part of a procedure to assess the validity of
inference Inferences are steps in logical reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinct ...
s in term logic). Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition) of sets (e.g. "months starting with the letter ''A''"), which may be useful when learning
computer programming Computer programming or coding is the composition of sequences of instructions, called computer program, programs, that computers can follow to perform tasks. It involves designing and implementing algorithms, step-by-step specifications of proc ...
, since
Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
is used in various
programming language A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...
s. Likewise, sets and other collection-like objects, such as multisets and
list A list is a Set (mathematics), set of discrete items of information collected and set forth in some format for utility, entertainment, or other purposes. A list may be memorialized in any number of ways, including existing only in the mind of t ...
s, are common datatypes in computer science and programming. In addition to that, certain sets are commonly used in mathematical teaching, such as the sets \mathbb of
natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
, \mathbb of
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, \mathbb of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, etc.). These are commonly used when defining a
mathematical function In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. ...
as a relation from one set (the domain) to another set (the range).


See also

* Glossary of set theory *
Class (set theory) In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like ...
* List of set theory topics * Relational model – borrows from set theory * Venn diagram * Elementary Theory of the Category of Sets * Structural set theory


Notes


Citations


References

* * * * * * * * * *


External links

* Daniel Cunningham
Set Theory
article in the ''
Internet Encyclopedia of Philosophy The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia with around 900 articles about philosophy, philosophers, and related topics. The IEP publishes only peer review, peer-reviewed and blind-refereed original p ...
''. * 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. * * {{Authority control S Formal methods Georg Cantor