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Continuum function

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Source updated January 13, 2026

In mathematics, the continuum function is the function $\kappa \mapsto 2^{\kappa }$ on cardinals, i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, the cardinal function yields the cardinality of the power set of a set of the given cardinality.

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Related topics

Continuum hypothesis

In mathematics, specifically set theory, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and the real numbers.

Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers R {\displaystyle \mathbb {R} } , sometimes called the continuum. It is an infinite cardinal number and is denoted by c {\displaystyle {\mathbf {\mathfrak {c}}}} or | R | .

Beth number

In mathematics, particularly in set theory, the beth numbers form a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written ℶ 0 , ℶ 1 , ℶ 2 , ℶ 3 , … {\displaystyle \beth _{0},\beth _{1},\beth _{2},\beth _{3},\dots } , where ℶ {\displaystyle \beth } is the Hebrew letter beth. The beth numbers are related to the aleph numbers ( ℵ 0 , ℵ 1 , … {\displaystyle \aleph _{0},\aleph _{1},\dots } ), but unless the generalized continuum hypothesis is true, there are numbers indexed by ℵ {\displaystyle \aleph } that are not indexed by ℶ {\displaystyle \beth } .