Beth number

Definition

To define the beth numbers, start by letting

ℶ 0 = ℵ 0

be the cardinality of any countably infinite set; for concreteness, take the set 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

ℶ α + 1 = 2 ℶ α ,

which is the cardinality of the power set of A if ℶ α is the cardinality of A.

Given this definition,

ℶ 0 ,   ℶ 1 ,   ℶ 2 ,   ℶ 3 ,   …

are respectively the cardinalities of

N ,   P ( N ) ,   P ( P ( N ) ) ,   P ( P ( P ( N ) ) ) ,   … .

so that the second beth number ℶ 1 is equal to c , the cardinality of the continuum, and the third beth number ℶ 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:

ℶ λ = sup { ℶ α : α < λ } .

One can also show that the von Neumann universes V ω + α have cardinality ℶ α .

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 ℵ 0 and ℵ 1 , it follows that

ℶ 1 ≥ ℵ 1 .

Repeating this argument (see transfinite induction) yields ℶ α ≥ ℵ α for all ordinals α .

The continuum hypothesis is equivalent to

ℶ 1 = ℵ 1 .

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., ℶ α = ℵ α for all ordinals α .

Beth null

Since this is defined to be ℵ 0 or aleph null then sets with cardinality ℶ 0 include:

the natural numbers N

the rational numbers Q

the algebraic numbers

the computable numbers and computable sets

the set of finite sets of integers

the set of finite multisets of integers

the set of finite sequences of integers

Beth one

Sets with cardinality ℶ 1 include:

the transcendental numbers

the irrational numbers

the real numbers R

the complex numbers C

the uncomputable real numbers

Euclidean space Rⁿ

the power set of the natural numbers (the set of all subsets of the natural numbers)

the set of sequences of integers (i.e. all functions N → Z, often denoted Z^N)

the set of sequences of real numbers, R^N

the set of all real analytic functions from R to R

the set of all continuous functions from R to R

the set of finite subsets of real numbers

the set of all analytic functions from C to C

Beth two

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

Sets with cardinality ℶ 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 (R^R)

The set of all functions from R^m 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

ℶ ω (pronounced beth omega) is the smallest uncountable strong limit cardinal.

Generalization

The more general symbol ℶ α ( κ ) , for ordinals α and cardinals κ, is occasionally used. It is defined by:

ℶ 0 ( κ ) = κ , ℶ α + 1 ( κ ) = 2 ℶ α ( κ ) , ℶ λ ( κ ) = sup { ℶ α ( κ ) : α < λ } if λ is a limit ordinal.

So

ℶ α = ℶ α ( ℵ 0 ) .

In ZF, for any cardinals κ and μ, there is an ordinal α such that:

κ ≤ ℶ α ( μ ) .

And in ZF, for any cardinal κ and ordinals α and β:

ℶ β ( ℶ α ( κ ) ) = ℶ α + β ( κ ) .

Consequently, in Zermelo–Fraenkel set theory absent ur-elements with or without the axiom of choice, for any cardinals κ and μ, the equality

ℶ β ( κ ) = ℶ β ( μ )

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.