beth number

In mathematics, the Hebrew letter

\aleph

(aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). The second Hebrew letter

\beth

(beth) is also used. To define the beth numbers, start by letting

\beth_0=\aleph_0

be the cardinality of countably infinite sets; for concreteness, take the set

\mathbb{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

\beth_{\kappa+1}=2^{\beth_\kappa}

= the cardinality of the power set of A if

\beth_\kappa

is the cardinality of A. Then

\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots

are respectively the cardinalities of

\mathbb{N},\ P(\mathbb{N}),\ P(P(\mathbb{N})),\ P(P(P(\mathbb{N}))),\ \dots.

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

\beth_1

is equal to c, the cardinality of the continuum, and the 2nd beth number

\beth_2

is the power set of c. For infinite limit ordinals κ, we define

\beth_\kappa=\sup\{\,\beth_\lambda:\lambda<\kappa\,\}.

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

\aleph_0

and

\aleph_1

, the celebrated continuum hypothesis can be stated in this notation by saying

\beth_1=\aleph_1.

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

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