Set theory is the branch of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions that studies sets A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In, which are collections of objects. Although any type of object 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 Georg Cantor Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the and Richard Dedekind Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers in the 1870s. After the discovery of paradoxes This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory in naive set theory Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most contemporary mathematics, numerous axiom systems In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins and there is no "rule" for which object to, are the best-known.

The language of set theory is used in the definitions of nearly all mathematical objects, such as functions The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain, 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 Venn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations between a finite collection of sets . Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer and Euler diagrams An Euler diagram is a diagrammatic means of representing sets and their relationships. The first use of "Eulerian circles" is commonly attributed to Leonhard Euler . They are closely related to 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 In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers are a standard part of the undergraduate mathematics curriculum.

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins and there is no "rule" for which object to. Beyond its foundational role, set theory is a branch of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions 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 In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an line to the study of the consistency In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term of large cardinals 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 are, as the name suggests, generally very "large" . The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and.

History

Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the: "On a Characteristic Property of All Real Algebraic Numbers".^[1][2]

Beginning with the work of Zeno Zeno of Elea (ca. 490 BC? – ca. 430 BC?) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound" around 450 BC, mathematicians had been struggling with the concept of infinity Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the Latin infinitas or "unboundedness". Especially notable is the work of Bernard Bolzano Bernhard Placidus Johann Nepomuk Bolzano – December 18, 1848), Bernard Bolzano in English, was a Bohemian mathematician, logician, philosopher, theologian, Catholic priest and antimilitarist of German mother tongue in the first half of the 19th century. The modern understanding of infinity began 1867-71, with Cantor's work on number theory. An 1872 meeting between Cantor and Richard Dedekind Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers influenced Cantor's thinking and culminated in Cantor's 1874 paper.

Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass Karl Theodor Wilhelm Weierstrass (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the "father of modern analysis" and Dedekind supported Cantor, Leopold Kronecker Leopold Kronecker was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole numbers", saying, "God made the integers; all else is the work of man" (Bell 1986, p. 477). This put Kronecker in bitter opposition to some of the mathematical extensions of Georg Cantor, Kronecker's, now seen as a founder of mathematical constructivism In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivists, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y and no unmapped element exists in either X or Y among sets, his proof that there are more real numbers In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an than integers, and the "infinity of infinities" ("Cantor's paradise") the power set In mathematics, given a set S, the power set of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set operation gives rise to.

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. The term is also used for an apparent contradiction that actually expresses a non-dual truth . Typically, the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or. Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, socialist, pacifist, and social critic. He spent most of his life in England; he was born in Wales where he also died, aged 97 and Ernst Zermelo Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his proof of the well-ordering theorem and his axiomatization of set theory independently found the simplest and best known paradox, now called Russell's paradox In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard Dedekind and Frege leads to a contradiction. The very same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other and involving "the set of all sets that are not members of themselves." This leads to a contradiction, since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question: "what is the cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets of the set of all sets?" and obtained a related paradox.

The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his proof of the well-ordering theorem and his axiomatization of set theory in 1908 and Abraham Fraenkel Abraham Halevi Fraenkel (Hebrew: אברהם הלוי (אדולף) פרנקל‎; February 17, 1891 Munich, Germany – October 15, 1965 Jerusalem, Israel), known as Abraham Fraenkel, was an Israeli mathematician born in Germany in 1922 resulted in the canonical axiomatic set theory ZFC In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes of naive set theory like Russell's paradox, which is thought to be free of paradoxes. The work of analysts Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and such as Henri Lebesgue Henri Léon Lebesgue was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis. His theory was published originally in his dissertation Intégrale, longueur, aire (" 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

Set theory begins with a fundamental binary relation In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations between an object o and a set A. If o is a member In mathematics, an element or member of a set is any one of the distinct objects that make up that set (or element) of A, we write o ∈ A. Since sets are objects, the membership relation can relate sets as well.

A derived binary relation In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not. From this definition, it is clear that a set is a subset of itself; in cases where one wishes to avoid this, the term proper subset In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment is defined to exclude this possibility.

Just as arithmetic Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers. In common usage, it refers to the simpler properties when features binary operations In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division on numbers A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number. In addition to their use in counting and measuring, numerals are often used for labels ,, set theory features binary operations on sets. The:

• Union In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets gives a set of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .
• Intersection 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, all elements of B that also belong to A), but no other elements of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .
• Complement of set A relative to set U, denoted A^c, is the set of all members of U that are not members of A. This terminology is most commonly employed when U is a universal set, as in the study of Venn diagrams. This operation is also called the set difference of U and A, denoted U \ A. The complement of {1,2,3} relative to {2,3,4} is {4} , while, conversely, the complement of {2,3,4} relative to {1,2,3} is {1} .
• Symmetric difference of sets A and B is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B).
• Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b) where a is a member of A and b is a member of B.
• Power set of a set A is the set whose members are all possible subsets of A. For example, the powerset of {1, 2} is { {}, {1}, {2}, {1,2} } .

Some 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. When doing set theory, it is common to restrict attention to the pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only.

A key idea in set theory is the von Neumann universe of pure sets. Sets in this universe are arranged in a 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. A set is assigned a rank by transfinite recursion: if the least upper bound on the ranks of the members of a set X is α then X is assigned rank α+1. Also, for each ordinal α, the set V_α contains all sets assigned a rank less than α.

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