# Set theory

axiomatic set theoryset-theoreticsetaxiomatic set theoriesset theoristsetsset theoreticset theoristsset theoriesset-theoretical
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.wikipedia
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### Mathematical logic

formal logicsymbolic logiclogic
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.

### Georg Cantor

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers".
He created set theory, which has become a fundamental theory in mathematics.

### Set (mathematics)

setsetsmathematical set
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
Developed 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.

### Naive set theory

naïve set theoryset theoryparadoxes of naive set theory
After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known.
Unlike axiomatic set theories, which are defined using formal logic, naïve set theory is defined informally, in natural language.

After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known.
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.

### Axiom of choice

ChoiceACaxiom of non-choice
After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known.
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

### Foundations of mathematics

foundation of mathematicsfoundationsfoundational
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.
It went through a series of crises with paradoxical results, until the discoveries stabilized during the 20th century as a large and coherent body of mathematical knowledge with several aspects or components (set theory, model theory, proof theory, etc.), whose detailed properties and possible variants are still an active research field.

### Large cardinal

large cardinalslarge cardinal axiomlarge cardinal property
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 the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.

### Georg Cantor's first set theory article

his first set theory articleHis proof1874 uncountability proof
Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers".
Georg Cantor published his first set theory article in 1874, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties.

After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Bertrand Russell and Ernst Zermelo 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 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory.

### Infinity

infiniteinfinitely
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 infinity.
Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea is also used in physics and the other sciences.

### Leopold Kronecker

KroneckerKronecker, Leopold
While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not.
He criticized Georg Cantor's work on set theory, and was quoted by as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man.").

### Bertrand Russell

RussellRussell, Bertrand16 Questions on the Assassination
Bertrand Russell and Ernst Zermelo 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.
His work has had a considerable influence on mathematics, logic, set theory, linguistics, artificial intelligence, cognitive science, computer science (see type theory and type system) and philosophy, especially the philosophy of language, epistemology and metaphysics.

### Power set

powerset2all subsets
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 real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation.
. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.

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 antinomies or paradoxes.
One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself, and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed.

### Zermelo–Fraenkel set theory

ZFZFCZermelo–Fraenkel
After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory.
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

### Thoralf Skolem

SkolemSkolem, ThoralfThoralf Albert Skolem
The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory.
Thoralf Albert Skolem (23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory.

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 real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation.
Cantor's paradise is an expression used by in describing set theory and infinite cardinal numbers developed by Georg Cantor.

### Abraham Fraenkel

FraenkelA. FraenkelAbraham Halevy Fraenkel
The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory.
He is known for his contributions to axiomatic set theory, especially his additions to Ernst Zermelo's axioms, which resulted in the Zermelo–Fraenkel axioms.

### Category theory

categorycategoricalcategories
Set theory is commonly used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation.
The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups.

### Union (set theory)

unionset unionunions
*Union of the sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

### Complement (set theory)

complementset differencecomplements
*Set difference of
In set theory, the complement of a set

### Universal set

set of all setsnot a valid setuniversal
is a universal set as in the study of Venn diagrams.
In set theory, a universal set is a set which contains all objects, including itself.

### Cartesian product

product×Cartesian square
*Cartesian product of
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

### Empty set

emptynonemptynon-empty
Some basic sets of central importance are the empty set (the unique set containing no elements; occasionally called the null set though this name is ambiguous), the set of natural numbers, and the set of real numbers.
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is [[0|zero]].