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Cantor's theorem, the Glossary

In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A, the set of all subsets of A, known as the power set of A, has a strictly greater cardinality than A itself.^[1]

61 relations: Alonzo Church, Automated theorem proving, Axiom schema of specification, Bertrand Russell, Bijection, Cantor's diagonal argument, Cantor's first set theory article, Cantor's paradox, Cardinal number, Cardinality, Cardinality of the continuum, Category (mathematics), Contradiction, Controversy over Cantor's theory, Countable set, Element (mathematics), Empty set, Enumeration, Equinumerosity, Ernst Zermelo, Finite set, Georg Cantor, Gottlob Frege, Identity function, Image (mathematics), Indicator function, Infinite set, Injective function, Integer, Isabelle (proof assistant), Lawrence Paulson, Lawvere's fixed-point theorem, Mathematician, Naive Set Theory (book), Natural number, Negation, Otter (theorem prover), Paradoxes of set theory, Paul Halmos, Philosophy of mathematics, Power set, Product (category theory), Proof by contradiction, Propositional function, Real number, Reductio ad absurdum, Russell's paradox, Schröder–Bernstein theorem, Set (mathematics), Set theory, ... Expand index (11 more) »
1891 in science
1891 introductions
Cardinal numbers
Georg Cantor

Alonzo Church

Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science.

See Cantor's theorem and Alonzo Church

Automated theorem proving

Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs.

See Cantor's theorem and Automated theorem proving

Axiom schema of specification

In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderung Axiom), subset axiom or axiom schema of restricted comprehension is an axiom schema.

See Cantor's theorem and Axiom schema of specification

Bertrand Russell

Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, logician, philosopher, and public intellectual.

See Cantor's theorem and Bertrand Russell

Bijection

A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the first set (the domain) is mapped to exactly one element of the second set (the codomain).

See Cantor's theorem and Bijection

Cantor's diagonal argument

Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbersinformally, that there are sets which in some sense contain more elements than there are positive integers. Cantor's theorem and Cantor's diagonal argument are cardinal numbers, Georg Cantor, set theory and theorems in the foundations of mathematics.

See Cantor's theorem and Cantor's diagonal argument

Cantor's first set theory article

Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. Cantor's theorem and Cantor's first set theory article are Georg Cantor and set theory.

See Cantor's theorem and Cantor's first set theory article

Cantor's paradox

In set theory, Cantor's paradox states that there is no set of all cardinalities. Cantor's theorem and Cantor's paradox are cardinal numbers and Georg Cantor.

See Cantor's theorem and Cantor's paradox

Cardinal number

In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. Cantor's theorem and cardinal number are cardinal numbers.

See Cantor's theorem and Cardinal number

Cardinality

In mathematics, the cardinality of a set is a measure of the number of elements of the set. Cantor's theorem and cardinality are cardinal numbers.

See Cantor's theorem and Cardinality

Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. Cantor's theorem and cardinality of the continuum are cardinal numbers and set theory.

See Cantor's theorem and Cardinality of the continuum

Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows".

See Cantor's theorem and Category (mathematics)

Contradiction

In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact.

See Cantor's theorem and Contradiction

Controversy over Cantor's theory

In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Cantor's theorem and Controversy over Cantor's theory are Georg Cantor and set theory.

See Cantor's theorem and Controversy over Cantor's theory

Countable set

In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Cantor's theorem and countable set are cardinal numbers.

See Cantor's theorem and Countable set

Element (mathematics)

In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set.

See Cantor's theorem and Element (mathematics)

Empty set

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

See Cantor's theorem and Empty set

Enumeration

An enumeration is a complete, ordered listing of all the items in a collection.

See Cantor's theorem and Enumeration

Equinumerosity

In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from A to B such that for every element y of B, there is exactly one element x of A with f(x). Cantor's theorem and Equinumerosity are cardinal numbers.

See Cantor's theorem and Equinumerosity

Ernst Zermelo

Ernst Friedrich Ferdinand Zermelo (27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics.

See Cantor's theorem and Ernst Zermelo

Finite set

In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Cantor's theorem and finite set are cardinal numbers.

See Cantor's theorem and Finite set

Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (– 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics.

See Cantor's theorem and Georg Cantor

Gottlob Frege

Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician.

See Cantor's theorem and Gottlob Frege

Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged.

See Cantor's theorem and Identity function

Image (mathematics)

In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each element of a given subset A of its domain X produces a set, called the "image of A under (or through) f".

See Cantor's theorem and Image (mathematics)

Indicator function

In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero.

See Cantor's theorem and Indicator function

Infinite set

In set theory, an infinite set is a set that is not a finite set. Cantor's theorem and infinite set are cardinal numbers.

See Cantor's theorem and Infinite set

Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.

See Cantor's theorem and Injective function

Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

See Cantor's theorem and Integer

Isabelle (proof assistant)

The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala.

See Cantor's theorem and Isabelle (proof assistant)

Lawrence Paulson

Lawrence Charles Paulson (born 1955) is an American computer scientist.

See Cantor's theorem and Lawrence Paulson

Lawvere's fixed-point theorem

In mathematics, Lawvere's fixed-point theorem is an important result in category theory.

See Cantor's theorem and Lawvere's fixed-point theorem

Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.

See Cantor's theorem and Mathematician

Naive Set Theory (book)

Naive Set Theory is a mathematics textbook by Paul Halmos providing an undergraduate introduction to set theory. Cantor's theorem and Naive Set Theory (book) are set theory.

See Cantor's theorem and Naive Set Theory (book)

Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0. Cantor's theorem and natural number are cardinal numbers.

See Cantor's theorem and Natural number

Negation

In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition P to another proposition "not P", standing for "P is not true", written \neg P, \mathord P or \overline.

See Cantor's theorem and Negation

Otter (theorem prover)

Otter is an automated theorem prover developed by William McCune at Argonne National Laboratory in Illinois.

See Cantor's theorem and Otter (theorem prover)

Paradoxes of set theory

This article contains a discussion of paradoxes of set theory. Cantor's theorem and paradoxes of set theory are set theory.

See Cantor's theorem and Paradoxes of set theory

Paul Halmos

Paul Richard Halmos (Halmos Pál; 3 March 3 1916 – 2 October 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces).

See Cantor's theorem and Paul Halmos

Philosophy of mathematics

Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.

See Cantor's theorem and Philosophy of mathematics

Power set

In mathematics, the power set (or powerset) of a set is the set of all subsets of, including the empty set and itself.

See Cantor's theorem and Power set

Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.

See Cantor's theorem and Product (category theory)

Proof by contradiction

In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction.

See Cantor's theorem and Proof by contradiction

Propositional function

In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that is not defined or specified (thus being a free variable), which leaves the statement undetermined.

See Cantor's theorem and Propositional function

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

See Cantor's theorem and Real number

Reductio ad absurdum

In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.

See Cantor's theorem and Reductio ad absurdum

Russell's paradox

In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901.

See Cantor's theorem and Russell's paradox

Schröder–Bernstein theorem

In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and, then there exists a bijective function. Cantor's theorem and Schröder–Bernstein theorem are cardinal numbers and theorems in the foundations of mathematics.

See Cantor's theorem and Schröder–Bernstein theorem

Set (mathematics)

In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. Cantor's theorem and set (mathematics) are set theory.

See Cantor's theorem and Set (mathematics)

Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Cantor's theorem and set theory are Georg Cantor.

See Cantor's theorem and Set theory

Singleton (mathematics)

In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element.

See Cantor's theorem and Singleton (mathematics)

Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

See Cantor's theorem and Springer Science+Business Media

Subset

In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).

See Cantor's theorem and Subset

Surjective function

In mathematics, a surjective function (also known as surjection, or onto function) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that.

See Cantor's theorem and Surjective function

The Principles of Mathematics

The Principles of Mathematics (PoM) is a 1903 book by Bertrand Russell, in which the author presented his famous paradox and argued his thesis that mathematics and logic are identical.

See Cantor's theorem and The Principles of Mathematics

Total order

In mathematics, a total order or linear order is a partial order in which any two elements are comparable. Cantor's theorem and total order are set theory.

See Cantor's theorem and Total order

Universal instantiation

In predicate logic, universal instantiation (UI; also called universal specification or universal elimination, and sometimes confused with dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class.

See Cantor's theorem and Universal instantiation

Universal set

In set theory, a universal set is a set which contains all objects, including itself.

See Cantor's theorem and Universal set

Without loss of generality

Without loss of generality (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics.

See Cantor's theorem and Without loss of generality

Zermelo set theory

Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG).

See Cantor's theorem and Zermelo set theory

Zermelo–Fraenkel 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.

See Cantor's theorem and Zermelo–Fraenkel set theory

References

[1] https://en.wikipedia.org/wiki/Cantor's_theorem

Also known as Cantor theorem, Cantor's theory, Cantors theorem.

, Singleton (mathematics), Springer Science+Business Media, Subset, Surjective function, The Principles of Mathematics, Total order, Universal instantiation, Universal set, Without loss of generality, Zermelo set theory, Zermelo–Fraenkel set theory.

Cantor's theorem, the Glossary

Table of Contents

See also

1891 in science

1891 introductions

Cardinal numbers

Georg Cantor

References