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Set theory is the branch of mathematics that studies sets, which are collections of objects. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important 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. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

Set theory, formalized using first-order logic, is the most common foundational system for mathematics. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science 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. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects. Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups 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. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set"

Beyond its use as a foundational system, set theory is a branch of mathematics in its own right, with an active research community. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. In Mathematics, the real numbers may be described informally in several different ways In the mathematical field of Set theory, a large cardinal property is a certain kind of property of Transfinite Cardinal numbers Cardinals with such properties

Contents

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". Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. [1][2]

Beginning with the work of Zeno around 450 BC, mathematicians had been struggling with the concept of infinity. Zeno of Elea (ˈziːnoʊ əv ˈɛliə Greek: Ζήνων ὁ Ἐλεάτης (ca Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Bernard (Bernhard Placidus Johann Nepomuk Bolzano ( &ndash December 18, 1848) was a Bohemian Mathematician, theologian, The modern understanding of infinity began 1867-71, with Georg Cantor's work on number theory. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. An 1872 meeting between Cantor and Dedekind much influenced Cantor's thinking and culminated in Cantor (1874). Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important

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. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued In the Philosophy of mathematics 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, the real numbers may be described informally in several different ways In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S)

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. A paradox is a true statement or group of statements that leads to a Contradiction or a situation which defies intuition; or inversely 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. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian Ernst Friedrich Ferdinand Zermelo ( July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the " 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. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. 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. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most

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. Ernst Friedrich Ferdinand Zermelo ( July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau Fränkel (or Fraenkel) is a surname and may refer to Adolf Abraham Halevi Fraenkel (1891 – 1965 German-Israeli mathematician known for Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common The work of analysts such as Lebesgue demonstrated the great mathematical utility of set theory. Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real Henri Léon Lebesgue leɔ̃ ləˈbɛg ( June 28, 1875, Beauvais &ndash July 26, 1941, Paris) was a French Axiomatic set theory has become woven into the very fabric of mathematics as we know it today.

Basic concepts

Main articles: Set and Algebra of sets

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 algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the 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. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently Additional operations on sets include:

Interpretations

A key idea in set theory is the von Neumann universe of pure sets. In Set theory and related branches of Mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class In the mathematical field of Set theory, a hereditary set (or pure set) is a set all of whose elements are hereditary 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. In Set theory and related branches of Mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class 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. Transfinite induction is an extension of Mathematical induction to well-ordered sets, for instance to sets of ordinals or cardinals.

Axiomatic set theory

Main article: Set

The basic concepts of set theory can be studied informally and intuitively rather than axiomatically. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Hence very elementary set theory can be taught in primary schools using, say, Venn diagrams. Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups This intuitive approach gives rise to antinomies, the simplest and best known of which being Russell's paradox. Antinomia redirects here For the Brachiopod Genus, see Antinomia (brachiopod. Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the 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. In Set theory and related branches of Mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class Such systems come in two flavors, those whose ontology consists of:

The systems NFU (with urelements) and NF (lacking them) are not based on a cumulative hierarchy. In Mathematical logic, New Foundations ( NF) is an Axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of In Set theory, a branch of Mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial' is an object (concrete In Mathematical logic, New Foundations ( NF) is an Axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of 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) allows systems of sets for which the axiom of choice does not hold. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic logic instead of first order logic. Constructive set theory is an approach to mathematical constructivism following the program of Axiomatic set theory. Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science Yet other systems accept standard first order logic but feature a nonstandard membership relation. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science 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. A rough set originated by prof Zdzisław I Pawlak is a formal approximation of a Crisp set (i Fuzzy sets are sets whose elements have degrees of membership In Mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper Propositional structure that is a formula The Boolean-valued models of ZFC are a related subject. In Mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure or model, in which the Truth Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common

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. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Equivalence and order relations are ubiquitous in mathematics, and the theory of relations is entirely grounded in set theory. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations

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 ( See Metamath ). The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science In Logic and Mathematics second-order logic is an extension of First-order logic, which itself is an extension of Propositional logic. Metamath is a computer-assisted proof checker It hasno specific logic embedded and can simply be regarded as a device to apply inference rules to formulas 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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, the real numbers may be described informally in several different ways In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Set theory, an infinite set is a set that is not a Finite 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. Analysis has its beginnings in the rigorous formulation of Calculus. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the 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. Metamath is a computer-assisted proof checker It hasno specific logic embedded and can simply be regarded as a device to apply inference rules to formulas Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science

Areas of study

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

Combinatorial set theory

Main article: Infinitary combinatorics

Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. In mathematics infinitary combinatorics, or combinatorial set theory, is an extension of ideas in Combinatorics to infinite sets Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdos-Rado theorem. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. This article goes into technical details quite quickly For a slightly gentler introduction see Ramsey theory. In mathematics infinitary combinatorics, or combinatorial set theory, is an extension of ideas in Combinatorics to infinite sets

Descriptive set theory

Main article: Descriptive set theory

Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. In Mathematical logic, descriptive set theory is the study of certain classes of " Well-behaved " subsets of the Real line and other In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Mathematics, a Polish space is a separable completely metrizable Topological space; that is a space Homeomorphic to a complete 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. In the mathematical field of Descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to In Mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra In the mathematical field of Descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol{\Sigma}^1_n In Descriptive set theory, Wadge degrees XXXX Wadge (date of birth &ndash date of death --> are levels of complexity for sets of reals and more 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. Effective descriptive set theory is the branch of Descriptive set theory dealing with sets of reals having Lightface definitions that is definitions Recursion theory, also called computability theory, is a branch of Mathematical logic that originated in the 1930s with the study of Computable functions It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In the mathematical field of Descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to In Recursion theory, hyperarithmetic theory is a generalization of Turing computability 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. In Mathematics, a Borel equivalence relation on a Polish space X is an Equivalence relation on X that is a Borel subset In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" This has important applications to the study of invariants in many fields of mathematics. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance.

Fuzzy set theory

Main article: 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. Fuzzy sets are sets whose elements have degrees of membership Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Ernst Friedrich Ferdinand Zermelo ( July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau Fränkel (or Fraenkel) is a surname and may refer to Adolf Abraham Halevi Fraenkel (1891 – 1965 German-Israeli mathematician known for 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. Fuzzy sets are sets whose elements have degrees of membership Lotfali Askar Zadeh (, born February 4, 1921) is an Iranian 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

Main article: 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. In Set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously The canonical example is the constructible universe L developed by Gödel. Gödel universe redirects here For Kurt Gödel 's cosmological solution to the Einstein field equations, see Gödel metric. 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. In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. 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). [3]

Large cardinals

Main article: Large cardinal property

A large cardinal is a cardinal number with an extra property. In the mathematical field of Set theory, a large cardinal property is a certain kind of property of Transfinite Cardinal numbers Cardinals with such properties Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. In Set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a Weak limit cardinal, and strongly inaccessible In Mathematics, a measurable cardinal is a certain kind of Large cardinal number 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

Main article: 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). The axiom of determinacy (abbreviated as AD) is an Axiom in the language of Set theory (that is one that makes an assertion about sets) AD can be used to prove that the Wadge degrees have an elegant structure. In Descriptive set theory, Wadge degrees XXXX Wadge (date of birth &ndash date of death --> are levels of complexity for sets of reals and more

Forcing

Main article: Forcing (mathematics)

Paul Cohen invented forcing while searching for a model of ZFC in which the continuum hypothesis fails. In the mathematical discipline of Set theory, forcing is a technique invented by Paul Cohen, for proving Consistency and independence results Paul Cohen may refer to Paul Cohen (mathematician (1934&ndash2007 American (middle initial J professor at Stanford University Paul Cohen In the mathematical discipline of Set theory, forcing is a technique invented by Paul Cohen, for proving Consistency and independence results In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite 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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models. In Mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure or model, in which the Truth

Cardinal invariants

Main article: Cardinal invariant

A cardinal invariant is a property of the real line measured by a cardinal number. In set theory Cichoń's diagram or Cichon's diagram is a table of 10 infinite Cardinal numbers related to the Set theory of the reals displaying the provable 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. In the mathematical fields of General topology and Descriptive set theory, a meagre set (also called a meager set or a set of first category) 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

Main article: 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. In Mathematics, set-theoretic topology is a subject that combines Set theory and General topology. In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures 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. In Mathematics, particularly Topology, a Moore space is a Topological space satisfying an axiom that may be thought of as a Separation axiom. 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. In Mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory 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. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued In the Philosophy of mathematics 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. Naive set theory is one of several theories of sets used in the discussion of the Foundations of mathematics. Ludwig Wittgenstein questioned the way ZF handled infinities. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common Wittgenstein's views about the foundations of mathematics were later criticised by Paul Bernays, and closely investigated by Crispin Wright, among others. Paul Bernays ( 17 October 1888 London – 18 September 1977 Zurich) was a Swiss mathematician who made significant Crispin Wright (born 1942 is a British Philosopher, who has written on neo- Fregean Philosophy of mathematics, Wittgenstein 's later In the mid 20th century, Errett Bishop dismissed set theory as "God's mathematics, which we should leave for God to do. Errett Albert Bishop (1928–1983 was an American Mathematician known for his work on analysis God is the principal or sole Deity in Religions and other belief systems that worship one deity. "

Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a topos (plural "topoi" or "toposes" is a type of category that behaves like the category of sheaves of sets Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory. In the Philosophy of mathematics Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm

See also

Notes

  1. ^ Philip Johnson, 1972, A History of Set Theory, Prindle, Weber & Schmidt ISBN 0871501546
  2. ^ A History of Set Theory.
  3. ^ Jech, Thomas (2003), Set Theory: Third Millennium Edition, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7 , p. Thomas J Jech (Tomáš Jech ˈtɔmaːʃ ˈjɛx born January 29, 1944 in Prague) is a set theorist who was at Penn State for more Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic 642.

Further reading

External links

Dictionary

set theory

-noun

  1. (mathematics) The mathematical theory of sets.
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