In mathematics, particularly in set theory, the **beth numbers** are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written , where is the Hebrew letter beth. The beth numbers are related to the aleph numbers (), but unless the generalized continuum hypothesis is true, there are numbers indexed by that are not indexed by .

Beth numbers are defined by transfinite recursion:

where is an ordinal and is a limit ordinal.^{[1]}

The cardinal is the cardinality of any countably infinite set such as the set of natural numbers, so that .

Let be an ordinal, and be a set with cardinality . Then,

- denotes the power set of (i.e., the set of all subsets of ),

- the set denotes the set of all functions from to ,

- the cardinal is the result of cardinal exponentiation, and

- is the cardinality of the power set of .

Given this definition,

are respectively the cardinalities of

so that the second beth number is equal to , the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number 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 , the corresponding beth number is defined to be the supremum of the beth numbers for all ordinals strictly smaller than :

One can show that this definition is equivalent to

For instance:

- is the cardinality of .
- is the cardinality of .
- is the cardinality of .

This equivalence can be shown by seeing that:

- for any set , the union set of all its members can be no larger than the supremum of its member cardinalities times its own cardinality,
- for any two non-zero cardinalities , if at least one of them is an infinite cardinality, then the product will be the larger of the two,
- the set will be smaller than most or all of its subsets for any limit ordinal
- therefore, for any limit ordinal

Note that this behavior is different from that of successor ordinals. Cardinalities less than but greater than any can exist when is a successor ordinal (in that case, the existence is undecidable in ZFC and controlled by the Generalized Continuum Hypothesis); but cannot exist when is a limit ordinal, even under the second definition presented.

One can also show that the von Neumann universes have cardinality .

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 and , it follows that

Repeating this argument (see transfinite induction) yields for all ordinals .

The continuum hypothesis is equivalent to

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

Since this is defined to be , or aleph null, sets with cardinality include:

- the natural numbers
- the rational numbers
- the algebraic numbers
- the computable numbers and computable sets
- the set of finite sets of integers or of rationals or of algebraic numbers
- the set of finite multisets of integers
- the set of finite sequences of integers.

Main article: cardinality of the continuum |

Sets with cardinality include:

- the transcendental numbers
- the irrational numbers
- the real numbers
- the complex numbers
- the uncomputable real numbers
- Euclidean space
- the power set of the natural numbers (the set of all subsets of the natural numbers)
- the set of sequences of integers (i.e., , which includes all functions from to )
- the set of sequences of real numbers,
- the set of all real analytic functions from to
- the set of all continuous functions from to
- the set of all functions from to with at most countable discontinuities
^{[2]} - the set of finite subsets of real numbers
- the set of all analytic functions from to (the holomorphic functions)
- the set of all functions from the natural numbers to the natural numbers ().

(pronounced *beth two*) is also referred to as (pronounced *two to the power of ).*

Sets with cardinality 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 to ()
- the set of all functions from to
- the set of all functions from to with uncountably many discontinuities
^{[2]} - the power set of the set of all functions from the set of natural numbers to itself, or the number of sets of sequences of natural numbers
- the Stone–Čech compactifications of , , and
- the set of deterministic fractals in
^{[3]} - the set of random fractals in .
^{[4]}

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

The more general symbol , for ordinals and cardinals , is occasionally used. It is defined by:

- if
*λ*is a limit ordinal.

So

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

And in ZF, for any cardinal and ordinals and :

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

holds for all sufficiently large ordinals . That is, there is an ordinal such that the equality holds for every ordinal .

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 is implied by the existence of all beths of countable index.^{[5]}