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# Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. [1]

53 relations: Aleph number, Algebraic number, Axiom of countable choice, Bijection, Cantor's diagonal argument, Cantor's theorem, Cardinal number, Cardinality, Cartesian product, Constructible universe, Coprime integers, Counting, Discrete mathematics, Disjoint sets, Domain of a function, Finite intersection property, Finite set, Fraction (mathematics), Function (mathematics), Georg Cantor, Georg Cantor's first set theory article, Hilbert's paradox of the Grand Hotel, Infinite set, Injective function, Inner model, Integer, ISO 31-11, Löwenheim–Skolem theorem, Map (mathematics), Mathematical induction, Mathematical logic, Mathematics, Naive set theory, Natural number, Ordered pair, Ordinal number, Power set, Prime number, Rational number, Real number, Recursion, Recursively enumerable set, Sequence, Set (mathematics), Skolem's paradox, Subset, Surjective function, Total order, Transcendental number, Uncountable set, ... Expand index (3 more) »

## Aleph number

In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered.

## Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

## Axiom of countable choice

The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Cantor's diagonal argument

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.

## Cantor's theorem

In elementary set theory, Cantor's theorem is a fundamental result that states that, for any set A, the set of all subsets of A (the power set of A, denoted by \mathcal(A)) has a strictly greater cardinality than A itself.

## Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Cartesian product

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.

## Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted L, is a particular class of sets that can be described entirely in terms of simpler sets.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Counting

Counting is the action of finding the number of elements of a finite set of objects.

## Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

## Disjoint sets

In mathematics, two sets are said to be disjoint sets if they have no element in common.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Finite intersection property

In general topology, a branch of mathematics, a collection A of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is nonempty.

## Finite set

In mathematics, a finite set is a set that has a finite number of elements.

## Fraction (mathematics)

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.

## Georg Cantor's first set theory article

Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties.

## Hilbert's paradox of the Grand Hotel

Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets.

## Infinite set

In set theory, an infinite set is a set that is not a finite set.

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Inner model

In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## ISO 31-11

ISO 31-11:1992 was the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology.

## Löwenheim–Skolem theorem

In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism.

## Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Naive set theory

Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects.

## Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

## Power set

In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Recursion

Recursion occurs when a thing is defined in terms of itself or of its type.

## Recursively enumerable set

In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

## Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

## Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

## Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.

## Union (set theory)

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

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Well-order

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

## References

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