In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written ${\displaystyle \beth _{0},\ \beth _{1},\ \beth _{2},\ \beth _{3},\ \dots }$, where ${\displaystyle \beth }$ is the Hebrew letter beth. The beth numbers are related to the aleph numbers (${\displaystyle \aleph _{0},\ \aleph _{1},\ \dots }$), but unless the generalized continuum hypothesis is true, there are numbers indexed by ${\displaystyle \aleph }$ that are not indexed by ${\displaystyle \beth }$.

Definition

Beth numbers are defined by transfinite recursion:

• ${\displaystyle \beth _{0}=\aleph _{0},}$
• ${\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha )),}$
• ${\displaystyle \beth _{\lambda }=\sup\{\beth _{\alpha }:\alpha <\lambda \},}$

where ${\displaystyle \alpha }$ is an ordinal and ${\displaystyle \lambda }$ is a limit ordinal.^[1]

The cardinal ${\displaystyle \beth _{0}=\aleph _{0))$ is the cardinality of any countably infinite set such as the set ${\displaystyle \mathbb {N} }$ of natural numbers, so that ${\displaystyle \beth _{0}=|\mathbb {N} |}$.

Let ${\displaystyle \alpha }$ be an ordinal, and ${\displaystyle A_{\alpha ))$ be a set with cardinality ${\displaystyle \beth _{\alpha }=|A_{\alpha }|}$. Then,

• ${\displaystyle {\mathcal {P))(A_{\alpha })}$ denotes the power set of ${\displaystyle A_{\alpha ))$ (i.e., the set of all subsets of ${\displaystyle A_{\alpha ))$),
• the set ${\displaystyle 2^{A_{\alpha ))\subset {\mathcal {P))(A_{\alpha }\times 2)}$ denotes the set of all functions from ${\displaystyle A_{\alpha ))$ to {0,1},
• the cardinal ${\displaystyle 2^{\beth _{\alpha ))}$ is the result of cardinal exponentiation, and
• ${\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha ))=|2^{A_{\alpha ))|=|{\mathcal {P))(A_{\alpha })|}$ is the cardinality of the power set of ${\displaystyle A_{\alpha ))$.

Given this definition,

${\displaystyle \beth _{0},\ \beth _{1},\ \beth _{2},\ \beth _{3},\ \dots }$

are respectively the cardinalities of

${\displaystyle \mathbb {N} ,\ {\mathcal {P))(\mathbb {N} ),\ {\mathcal {P))({\mathcal {P))(\mathbb {N} )),\ {\mathcal {P))({\mathcal {P))({\mathcal {P))(\mathbb {N} ))),\ \dots .}$

so that the second beth number ${\displaystyle \beth _{1))$ is equal to ${\displaystyle {\mathfrak {c))}$, the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number ${\displaystyle \beth _{2))$ 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 to be the supremum of the beth numbers for all ordinals strictly smaller than λ:

${\displaystyle \beth _{\lambda }=\sup\{\beth _{\alpha }:\alpha <\lambda \}.}$

One can also show that the von Neumann universes ${\displaystyle V_{\omega +\alpha ))$ have cardinality ${\displaystyle \beth _{\alpha ))$.

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 ${\displaystyle \aleph _{0))$ and ${\displaystyle \aleph _{1))$, it follows that

${\displaystyle \beth _{1}\geq \aleph _{1}.}$

Repeating this argument (see transfinite induction) yields ${\displaystyle \beth _{\alpha }\geq \aleph _{\alpha ))$ for all ordinals ${\displaystyle \alpha }$.

The continuum hypothesis is equivalent to

${\displaystyle \beth _{1}=\aleph _{1}.}$

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., ${\displaystyle \beth _{\alpha }=\aleph _{\alpha ))$ for all ordinals ${\displaystyle \alpha }$.

Specific cardinals

Beth null

Since this is defined to be ${\displaystyle \aleph _{0))$, or aleph null, sets with cardinality ${\displaystyle \beth _{0))$ 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 ${\displaystyle \beth _{1))$ 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 all functions from R to R with at most countable discontinuities ^[2]
• the set of finite subsets of real numbers
• the set of all analytic functions from C to C (the holomorphic functions)
• the set of all functions from the natural numbers to the natural numbers (${\displaystyle \mathbb {N^{N)) }$)

Beth two

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

Sets with cardinality ${\displaystyle \beth _{2))$ 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 (R^R)
• The set of all functions from R^m to Rⁿ
• The set of all functions from R to R with uncountable discontinuities ^[2]
• 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
• The set of deterministic fractals in R^{n [3]}
• The set of random fractals in R^{n [4]}

Beth omega

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

Generalization

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

${\displaystyle \beth _{0}(\kappa )=\kappa ,}$

${\displaystyle \beth _{\alpha +1}(\kappa )=2^{\beth _{\alpha }(\kappa )},}$

${\displaystyle \beth _{\lambda }(\kappa )=\sup\{\beth _{\alpha }(\kappa ):\alpha <\lambda \))$ if λ is a limit ordinal.

So

${\displaystyle \beth _{\alpha }=\beth _{\alpha }(\aleph _{0}).}$

In Zermelo–Fraenkel set theory (ZF), for any cardinals κ and μ, there is an ordinal α such that:

${\displaystyle \kappa \leq \beth _{\alpha }(\mu ).}$

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

${\displaystyle \beth _{\beta }(\beth _{\alpha }(\kappa ))=\beth _{\alpha +\beta }(\kappa ).}$

Consequently, in ZF absent ur-elements with or without the axiom of choice, for any cardinals κ and μ, the equality

${\displaystyle \beth _{\beta }(\kappa )=\beth _{\beta }(\mu )}$

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 that 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.

Borel determinacy

Borel determinacy is implied by the existence of all beths of countable index.^[5]