Beth Number - Definition

Definition

To define the beth numbers, start by letting

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

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

Given this definition,

are respectively the cardinalities of

so that the second beth number is equal to, the cardinality of the continuum, and the third beth number is the cardinality of the power set of the continuum.

Because of Cantor's theorem each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals λ the corresponding beth number is defined as the supremum of the beth numbers for all ordinals strictly smaller than λ:

One can also show that the von Neumann universes have cardinality .