Cardinal numbers

ambiguous mathematical term used either for non-negative or for strictly positive integers, depending on usage

thumb|318x318px|A one-to-one correspondence between a set of apples and a set of oranges shows they have the same cardinality. In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through one-to-one correspondences between sets. That is, if their objects can be paired such that each object has a pair, and no object is paired more than once.

finite or infinite number that measures cardinality (size) of sets

set that has a finite number of elements

set with the same cardinality as some subset of the set of natural numbers

hypothesis that no set has a cardinality between that of the integers and that of the real numbers

in set theory, the theorem that a set has a strictly smaller cardinality than its powerset

proof technique in set theory

set with cardinal number larger than that of the set of all natural numbers

alef symbol (U+2135) or aleph, written left-to-right as the mathemical symbol ‹ℵ› for the first transfinite cardinal (countable); ordered sequence of transfinite numbers used to represent the cardinality (or size) of infinite countable sets

set with infinite cardinality

number larger than all finite numbers

paradox in naïve set theory that there is no set of all cardinal numbers

theorem that, if there exist injective functions in both directions between two sets, then there exists a bijection between them

cardinality of the set of real numbers

In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. Formally, \operatorname{cf}(A) = \inf \{|B| : B \subseteq A, (\forall x \in A) (\exists y \in B) (x \leq y)\}

ordered sequence of transfinite numbers used to represent the cardinality (or size) of continuous sets

cardinal number that equals its own cofinality

theorem

given a set, the least ordinal such that there is no injection from the ordinal into the set

class of cardinal numbers

proper subset B of A that is equinumerous to A

theorem in axiomatic set theory

function that returns cardinal numbers

the assertion that the generalized continuum hypothesis holds for singular strong limit cardinals