Aleph number

An aleph number is a cardinal number in the sequence ℵ₀, ℵ₁, ℵ₂, … that represents the size of an infinite well-orderable set.

Cardinal number: A number that measures the size of a set, up to bijection. Aleph numbers are the infinite cardinals, extending this notion of set size.
Aleph-null: The cardinality of the natural numbers, the smallest infinite cardinal. It is the first member of the aleph sequence.
Beth number: A cardinal in a sequence defined by repeated power-set operations, beginning with ℶ₀ = ℵ₀. Beth numbers track powersets, while alephs enumerate well-orderable cardinalities.
Generalized continuum hypothesis: The assertion that 2^κ = κ⁺ for every infinite cardinal κ. It proposes a precise relationship between every infinite aleph and its powerset.
Ordinal number: A number describing the order type of a well-ordered set. Ordinal indices specify which aleph appears in the sequence.
Successor cardinal: The least cardinal strictly greater than a given cardinal. For each cardinal κ, its successor is denoted κ⁺ and is the next larger cardinal.
Hartogs number: The least ordinal that does not inject into a given set. It constructs a larger well-ordered cardinal even when the set itself is not well-orderable.
Zermelo–Fraenkel set theory: An axiomatic foundation for mathematics built around sets and membership. Aleph arithmetic is studied within this foundational framework, usually with choice added.
Well-ordering: A total order in which every nonempty subset has a least element. Alephs enumerate precisely the infinite cardinalities represented by well-orderable sets.
Limit cardinal: An infinite cardinal that is not the successor of a smaller cardinal. Some alephs occur at limit ordinal indices rather than as immediate successors.

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Cardinality of the continuumRelated: They provide the standard scale against which the continuum's cardinality is compared.