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# Set theory  ( Formal Methods )

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Set theory is the branch of mathematical logic 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 , 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 and 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 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 , and has various applications in (such as in the theory of relational algebra), and formal semantics. Its foundational appeal, together with its , its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the line to the study of the of .

History
Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by : "On a Property of the Collection of All Real Algebraic Numbers".

Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of . Especially notable is the work of in the first half of the 19th century. Modern understanding of infinity began in 1870–1874, and was motivated by Cantor's work in .. An 1872 meeting between Cantor and influenced Cantor's thinking, and culminated in Cantor's 1874 paper.

Cantor's work initially polarized the mathematicians of his day. While and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the operation. This utility of set theory led to the article "Mengenlehre", contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.

The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called or . and independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899, Cantor had himself posed the question "What is the of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. Rather than the term set, Russell used the term class, which has subsequently been used more technically.

In 1906, the term set appeared in the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.

The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and the work of and 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 , demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology— is thought to be a preferred foundation.

Basic concepts and notation
Set theory begins with a fundamental between an object and a set . If is a (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 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 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 features on , 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 .
• 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 as in the study of .
• Symmetric difference of sets and , denoted or , is the set of all objects that are a member of exactly one of and (elements which are in one of the sets, but not in both). For instance, for the sets and , the symmetric difference set is . It is the set difference of the union and the intersection, or .
• Cartesian product of and , denoted , is the set whose members are all possible , where is a member of and is a member of . For example, the Cartesian product of
• of a set , denoted $\mathcal\left\{P\right\}\left(A\right)$, 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 , the set of and the —the unique set containing no elements. The empty set is also occasionally called the null set, though this name is ambiguous and can lead to several interpretations.

Ontology
A set is 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 $\alpha$, 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 $V_\left\{\alpha\right\}$ 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 . 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 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 and ;
• Kripke–Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
• Sets and . These include Von Neumann–Bernays–Gödel set theory, 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 , objects that can be members of sets but that are not themselves sets and do not have any members.

The systems of NFU (allowing ) and NF (lacking them) are not based on a cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. 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 . Yet other systems accept classical logic but feature a nonstandard membership relation. These include and fuzzy set theory, in which the value of an embodying the membership relation is not simply True or False. The Boolean-valued models of are a related subject.

An enrichment of called internal set theory was proposed by in 1977.

Applications
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, , rings, , and relational algebras can all be defined as sets satisfying various (axiomatic) properties. Equivalence and 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 and 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 .

Set theory as a foundation for mathematical analysis, , , and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, , includes human-written, computer-verified derivations of more than 12,000 theorems starting from 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 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 and, more generally, subsets of . It begins with the study of in the and extends to the study of more complex hierarchies such as the projective hierarchy and the . Many properties of 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 . 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 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 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 , 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 have an elegant structure.

Forcing
Paul Cohen invented the method of forcing while searching for a of 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 without changing any of the 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 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 that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

Objections to set theory
From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by 's influential book Foundations of Constructive Analysis.

A different objection put forth by Henri Poincaré is that defining sets using the axiom schemas of specification and replacement, as well as the axiom of power set, introduces , 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 has said that "all of scientifically applicable analysis can be developed using".

Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism. He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and . 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 , , , and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as begun to rehabilitate Wittgenstein's arguments.

have proposed as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and 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 .

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. 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 early in mathematics education.

In the US in the 1960s, the experiment aimed to teach basic set theory, among other abstract concepts, to 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. are widely employed to explain basic set-theoretic relationships to students (even though originally devised them as part of a procedure to assess the validity of in ).

Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition

(2011). 9781441174130, Bloomsbury Publishing. .
) of sets (e.g. "months starting with the letter A"), which may be useful when learning computer programming, since is used in various programming languages. Likewise, sets and other collection-like objects, such as and lists, are common in and programming.

In addition to that, sets are commonly referred to in mathematical teaching when talking about different types of numbers (the sets $\mathbb\left\{N\right\}$ of , $\mathbb\left\{Z\right\}$ of , $\mathbb\left\{R\right\}$ of , etc.), and when defining a mathematical function as a relation from one set (the domain) to another set (the range).

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