# Beth number

In mathematics, the infinite cardinal numbers are represented by the Hebrew letter $\aleph$ (aleph) indexed with a subscript that runs over the ordinal numbers (see aleph number). The second Hebrew letter $\beth$ (beth) is used in a related way, but does not necessarily index all of the numbers indexed by $\aleph$ .

## Definition

To define the beth numbers, start by letting

$\beth _{0}=\aleph _{0}$ be the cardinality of any countably infinite set; for concreteness, take the set $\mathbb {N}$ 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

$\beth _{\alpha +1}=2^{\beth _{\alpha }},$ which is the cardinality of the power set of A if $\beth _{\alpha }$ is the cardinality of A.

Given this definition,

$\beth _{0},\ \beth _{1},\ \beth _{2},\ \beth _{3},\ \dots$ are respectively the cardinalities of

$\mathbb {N} ,\ P(\mathbb {N} ),\ P(P(\mathbb {N} )),\ P(P(P(\mathbb {N} ))),\ \dots .$ Because of Cantor's theorem each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals λ the corresponding beth number is defined as the supremum of the beth numbers for all ordinals strictly smaller than λ:

$\beth _{\lambda }=\sup\{\beth _{\alpha }:\alpha <\lambda \}.$ ## Relation to the aleph numbers

Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between $\aleph _{0}$ and $\aleph _{1}$ , it follows that

$\beth _{1}\geq \aleph _{1}.$ The continuum hypothesis is equivalent to

$\beth _{1}=\aleph _{1}.$ ## Specific cardinals

### Beth one

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### Beth two

$\beth _{2}$ (pronounced beth two) is also referred to as 2c (pronounced two to the power of c).

• The power set of the set of real numbers, so it is the number of subsets of the real line, or the number of sets of real numbers
• The power set of the power set of the set of natural numbers
• The set of all functions from R to R (RR)
• The set of all functions from Rm to Rn
• The power set of the set of all functions from the set of natural numbers to itself, so it is the number of sets of sequences of natural numbers
• The Stone–Čech compactifications of R, Q, and N

### Beth omega

$\beth _{\omega }$ (pronounced beth omega) is the smallest uncountable strong limit cardinal.

## Generalization

The more general symbol $\beth _{\alpha }(\kappa )$ , for ordinals α and cardinals κ, is occasionally used. It is defined by:

$\beth _{0}(\kappa )=\kappa ,$ $\beth _{\alpha +1}(\kappa )=2^{\beth _{\alpha }(\kappa )},$ $\beth _{\lambda }(\kappa )=\sup\{\beth _{\alpha }(\kappa ):\alpha <\lambda \}$ if λ is a limit ordinal.

So

$\beth _{\alpha }=\beth _{\alpha }(\aleph _{0}).$ In ZF, for any cardinals κ and μ, there is an ordinal α such that:

$\kappa \leq \beth _{\alpha }(\mu ).$ And in ZF, for any cardinal κ and ordinals α and β:

$\beth _{\beta }(\beth _{\alpha }(\kappa ))=\beth _{\alpha +\beta }(\kappa ).$ Consequently, in Zermelo–Fraenkel set theory absent ur-elements with or without the axiom of choice, for any cardinals κ and μ, the equality

$\beth _{\beta }(\kappa )=\beth _{\beta }(\mu )$ holds for all sufficiently large ordinals β (that is, there is an ordinal α such that the equality holds for every ordinal β ≥ α).

This also holds in Zermelo–Fraenkel set theory with ur-elements with or without the axiom of choice provided the ur-elements form a set which is equinumerous with a pure set (a set whose transitive closure contains no ur-elements). If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.