Beth number

In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers, but it is less commonly known that the second Hebrew letter (beth) also occurs and has significance. To define the beth numbers, start by letting

be the cardinality of countably infinite sets; for concreteness, take the set N of natural numbers to be the typical case. Denote by P(A) the power set of A, i.e., the set of all subsets of A. Then define

= the cardinality of the power set of A if is the cardinality of A.

Then

are respectively the cardinalities of

Each set in this sequence has cardinality strictly greater than the one preceding it, because of Cantor's theorem.

For infinite limit ordinals κ, we define

If we assume the axiom of choice, then infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable, and so, since no infinite cardinalities are between and , the celebrated continuum hypothesis can be stated in this notation by saying

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers.