# Beth number

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

[itex]\beth_0=\aleph_0[itex]

be the cardinality of countably infinite sets; for concreteness, take the set [itex]\mathbb{N}[itex] 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

[itex]\beth_{\kappa+1}=2^{\beth_\kappa}[itex]

= the cardinality of the power set of A if [itex]\beth_\kappa[itex] is the cardinality of A.

Then

[itex]\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots[itex]

are respectively the cardinalities of

[itex]\mathbb{N},\ P(\mathbb{N}),\ P(P(\mathbb{N})),\ P(P(P(\mathbb{N}))),\ \dots.[itex]

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 [itex]\beth_1[itex] is equal to c, the cardinality of the continuum, and the 2nd beth number [itex]\beth_2[itex] is the power set of c.

For infinite limit ordinals κ, we define

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

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 [itex]\aleph_0[itex] and [itex]\aleph_1[itex], the celebrated continuum hypothesis can be stated in this notation by saying

[itex]\beth_1=\aleph_1.[itex]

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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