Beth number

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

<math>\beth_0=\aleph_0<math>

be the cardinality of countably infinite sets; for concreteness, take the set <math>\mathbb{N}<math> 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

<math>\beth_{\kappa+1}=2^{\beth_\kappa}<math>

= the cardinality of the power set of A if <math>\beth_\kappa<math> is the cardinality of A.

Then

<math>\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots<math>

are respectively the cardinalities of

<math>\mathbb{N},\ P(\mathbb{N}),\ P(P(\mathbb{N})),\ P(P(P(\mathbb{N}))),\ \dots.<math>

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 <math>\beth_1<math> is equal to c, the cardinality of the continuum, and the 2nd beth number <math>\beth_2<math> is the power set of c.

For infinite limit ordinals κ, we define

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

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 <math>\aleph_0<math> and <math>\aleph_1<math>, the celebrated continuum hypothesis can be stated in this notation by saying

<math>\beth_1=\aleph_1.<math>

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