Beth number

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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[edit]

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.

so that the second beth number \beth_1 is equal to \mathfrak c, the cardinality of the continuum, and the third beth number \beth_2 is the cardinality of the power set of the continuum.

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 \}.

One can also show that the von Neumann universes V_{\omega+\alpha} \! have cardinality \beth_{\alpha} \!.

Relation to the aleph numbers[edit]

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 \ge \aleph_1.

Repeating this argument (see transfinite induction) yields \beth_\alpha \ge \aleph_\alpha for all ordinals \alpha.

The continuum hypothesis is equivalent to

\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, i.e., \beth_\alpha = \aleph_\alpha for all ordinals \alpha.

Specific cardinals[edit]

Beth null[edit]

Since this is defined to be \aleph_0 or aleph null then sets with cardinality \beth_0 include:

Beth one[edit]

Sets with cardinality \beth_1 include:

Beth two[edit]

\beth_2 (pronounced beth two) is also referred to as 2c (pronounced two to the power of c).

Sets with cardinality \beth_2 include:

  • 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[edit]

\beth_\omega (pronounced beth omega) is the smallest uncountable strong limit cardinal.

Generalization[edit]

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 \le \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.

References[edit]