Categories: Set theory | Cardinal numbers

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.

Categories: Set theory | Cardinal numbers

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