In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). The second Hebrew letter (beth) is also used. To define the beth numbers, start by letting

be the cardinality of countably infinite sets; for concreteness, take the set 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. Note that the 1st beth number is equal to c, the cardinality of the continuum, and the 2nd beth number is the power set of c.

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.