Beth number

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In mathematics, the infinite cardinal numbers are represented by the Hebrew letter <math>\aleph</math> (aleph) indexed with a subscript that runs over the ordinal numbers (see aleph number). The second Hebrew letter <math>\beth</math> (beth) and the third Hebrew letter <math>\gimel</math> (gimel) are also used. To define the beth numbers, start by letting

<math>\beth_0=\aleph_0</math>

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

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

which is 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>

One can also show that the von Neumann universes <math>V_{\omega+\alpha}</math> have cardinality <math>\beth_\alpha</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 (or <math>\mathfrak c</math>), the cardinality of the continuum, and the 2nd beth number <math>\beth_2</math> is the cardinality of the power set of the continuum.

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 by definition 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.

The more general symbol <math>\beth_\kappa(\alpha)</math>, for ordinals κ and cardinals α, is occasionally used. It is defined by

<math>\beth_0(\alpha)=\alpha</math>

<math>\beth_{\kappa+1}(\alpha)=2^{\beth_\kappa(\alpha)},</math>

<math>\beth_\kappa(\alpha)=\sup\{\,\beth_\lambda(\alpha):\lambda<\kappa\,\}</math> if κ is a limit ordinal.

So <math>\beth_\kappa=\beth_\kappa(\aleph_0).</math>