In mathematics, the **beth numbers** are a certain sequence of infinite cardinal numbers, conventionally written , where is the second Hebrew letter (beth).^{[1]} The beth numbers are related to the aleph numbers (), but there may be numbers indexed by that are not indexed by .

## Definition

To define the beth numbers, start by letting

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

which is the cardinality of the power set of *A* (if is the cardinality of *A*).^{[2]}

Given this definition,

are respectively the cardinalities of

^{[1]}

so that the second beth number is equal to , the cardinality of the continuum (the cardinality of the set of the real numbers),^{[2]} 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, λ, the corresponding beth number, is defined to be the supremum of the beth numbers for all ordinals strictly smaller than λ:

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

## 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 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 .

## Specific cardinals

### Beth null

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

- the transcendental numbers
- the irrational numbers
- the real numbers
**R** - the complex numbers
**C** - the uncomputable real numbers
- Euclidean space
**R**^{n} - 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**(the holomorphic functions)

### Beth two

(pronounced *beth two*) is also referred to as **2 ^{c}** (pronounced

*two to the power of c*).

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
**R**to**R**(**R**^{R}) - The set of all functions from
**R**^{m}to**R**^{n} - 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

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

## Generalization

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

- if
*λ*is a limit ordinal.

So

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

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

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

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.

## See also

## References

- ^
^{a}^{b}"Comprehensive List of Set Theory Symbols".*Math Vault*. 2020-04-11. Retrieved 2020-09-05. - ^
^{a}^{b}"beth numbers".*planetmath.org*. Retrieved 2020-09-05.

## Bibliography

- T. E. Forster,
*Set Theory with a Universal Set: Exploring an Untyped Universe*, Oxford University Press, 1995 —*Beth number*is defined on page 5. - Bell, John Lane; Slomson, Alan B. (2006) [1969].
*Models and Ultraproducts: An Introduction*(reprint of 1974 ed.). Dover Publications. ISBN 0-486-44979-3. See pages 6 and 204–205 for beth numbers. - Roitman, Judith (2011).
*Introduction to Modern Set Theory*. Virginia Commonwealth University. ISBN 978-0-9824062-4-3. See page 109 for beth numbers.