Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of objects can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the most well known.

Set theory, formalized using first-order logic, is the most common foundational system for mathematics. The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects. Elementary operations such as set union and intersection can be studied in this context. More advanced concepts such as cardinality are a standard part of the undergraduate mathematics curriculum.

Beyond its use as a foundational system, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

History
See Johnson (1972) for a book-length treatment. Mathematical topics typically emerge and evolve through interactions among many researchers. The point of origin of set theory is somewhat unusual in that it can be identified as an 1874 paper by Georg Cantor: "On a Characteristic Property of All Real Algebraic Numbers".

Beginning with the work of Zeno around 450 BC, mathematicians had been struggling with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century. The modern understanding of infinity began 1867-71, with Georg Cantor's work on number theory. An 1872 meeting between Cantor and Dedekind much influenced Cantor's thinking and culminated in Cantor (1874).

Cantor's work initially polarized the mathematicians of his day. While Weierstrass and Dedekind supported Cantor, Kronecker, now seen as a founder of mathematical constructivism, did not. But the utility of Cantorian concepts such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") the power set operation gives rise to, eventually led to the widespread acceptance of Cantorian set theory.

The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Russell and Zermelo independently found the simplest and best known paradox, now called Russell's paradox and involving "the set of all sets that are not members of themselves." Clearly this set cannot be a member of itself, and hence it must be a member of itself! In 1899 Cantor had himself posed the question: "what is the cardinal number of the set of all sets?" and obtained a related paradox. It was later realized that these paradoxes are not merely set theoretic, and that in logic the sentence "this sentence is false" gives rise to a similar problem, for if the sentence is true, it must be false. Kurt Gödel used this fact in the 1931 proof of his celebrated incompleteness theorem.

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 Fraenkel in 1922 resulted in the canonical axiomatic set theory ZFC, which is free of paradoxes. The work of analysts such as Lebesgue demonstrated the great mathematical utility of set theory. Axiomatic set theory has become woven into the very fabric of mathematics as we know it today.

Basic concepts
The basic relationship between objects and sets is the membership or "elementhood" relation; given an object O and a set A, either O is a member of A or it is not a member. The basic relationship between two sets is the subset relation, also called set inclusion. For example, {a, b} is a subset of {a, b, c}, but {a, d} is not.

Just as there are arithmetical operations that operate on numbers, there are operations in set theory that operate on sets. For instance, starting with the sets {1, 2, 3} and {2, 3, 4}, the union operation produces a new set {1, 2, 3, 4} containing all elements that are in either set, and the intersection operation produces the set {2, 3} consisting of all elements that are in both of the original sets. Additional operations on sets include:
 * Complementation: the set of all elements of a set U that are not in a set A is called the complement of A relative to U, denoted $$A^c$$. This terminology is used most often when U is an implicit "universal" set, as in the study of Venn diagrams. The set of elements of U not in A is also called the set difference, denoted $$U \setminus A$$.
 * The symmetric difference of two sets consists of all elements that are in exactly one of the two sets.
 * The Cartesian product of two sets A and B consists of all ordered pairs (a,b) where a is a member of A and b is a member of B.
 * The powerset of a set A consists of all subsets of A. For example, the powerset of {1, 2} is { {}, {1}, {2}, {1,2} }.

Interpretations
A key idea in set theory is the von Neumann universe of pure sets. 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. It is common in set theory to restrict attention to the pure sets, rather than studying arbitrary sets, and many axiomatic systems of set theory are only intended to axiomatize the pure sets.

The pure sets are arranged in the cumulative hierarchy based on how deeply their members, members of members, etc. are nested. Each set is assigned an ordinal number α in this hierarchy, known as its rank. Conversely, for each ordinal α the set Vα is defined to contain all sets that are assigned rank no greater than α. The assignment of ranks is done by transfinite recursion: if the least upper bound on the ranks of the elements of a set X is α then X is assigned rank α + 1.

Axiomatic set theory
The basic concepts of set theory can be studied informally and intuitively rather than axiomatically. Hence very elementary set theory can be taught in primary schools using, say, Venn diagrams. This intuitive approach gives rise to antinomies, the simplest and best known of which being Russell's paradox. Axiomatic set theory was originally devised to banish these antinomies.

The most widely studied systems of set theory are based on the concept of a cumulative hierarchy of sets. Such systems come in two flavors, those whose ontology consists of:
 * Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory (ZFC), which includes the axiom of choice. Fragments of ZFC include:
 * Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
 * General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
 * Kripke-Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
 * Proper classes as well as sets. This includes Von Neumann-Bernays-Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse-Kelley set theory, which is stronger than ZFC.

The systems NFU (with urelements) and NF (lacking them) are not based on a cumulative hierarchy. NF and NFU include a "set of all sets", relative to which every set has a complement. On the other hand, NF (but not NFU) disproves the axiom of choice.

Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic logic instead of first order logic. Yet other systems accept standard first order 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 and False. The Boolean-valued models of ZFC are a related subject.

Applications
Nearly all mathematical concepts are now defined formally in terms of sets and set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces are all defined as sets having various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of relations is entirely grounded 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 numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.

Areas of study
Set theory is a major area of research in mathematics, with many interrelated subfields.

Combinatorial set theory
Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdos-Rado theorem.

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 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 Zadeh so an object has a degree of membership in a set, as number between 0 and 1. E.g. the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.

Inner model theory
An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. The study of inner models of extensions of ZF is of interest in set theory because it can be used to prove consistency results. For example, it can be shown that regardless 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 continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has any model whatsoever) 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 forcing while searching for a model of ZFC in which the continuum hypothesis fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers 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.

Objections to set theory
From set theory's inception, some mathematicians objected to it as a foundation for mathematics, arguing, for example, that it is just a game which included elements of fantasy. 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. Ludwig Wittgenstein questioned the way ZF handled infinities. Wittgenstein's views about the foundations of mathematics were later criticised by Paul Bernays, and closely investigated by Crispin Wright, among others. In the mid 20th century, Errett Bishop dismissed set theory as "God's mathematics, which we should leave for God to do."

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.