In mathematics, particularly in set theory, the beth numbers form a certain (unset) sequence of infinite cardinal numbers (also known as transfinite numbers is the Hebrew letter beth. The beth numbers are related to the aleph numbers), but unless the generalized continuum hypothesis. On the other hand, beth numbers are cofinal (every cardinal number is less than a beth number) in plain Zermelo-Fraenkel set theory.

Definition

Beth numbers are indexed by ordinal numbers and defined in terms of the cumulative hierarchy, and it follows by Cantor's theorem is the cardinality of the power set.)

for every limit ordinal. The axiom of choice implies that the inequality holds in general.

, the cardinality of the continuum (the cardinality of the set of the real numbers is the cardinality of the power set of the continuum.

. This follows by transfinite induction from two points:

• ; and

  .

.

can exist. (In that case, the existence is undecidable in ZFC and controlled by the Generalized Continuum Hypothesis.)

Relation to the aleph numbers

.

Assuming the axiom of choice, infinite cardinalities are linearly ordered.

Given the axiom of choice, the continuum hypothesis is equivalent to

.

The generalized continuum hypothesis extends the assertion above to other indices. One formulation says the sequence of beth numbers is the same as the sequence of aleph numbers admits a well ordering.) Different formulations of the continuum hypothesis suggest different generalizations, but the known 'reasonable' generalizations turn out to be equivalent and to imply the axiom of choice, although the proofs are more difficult.

Specific cardinals

Beth null

, or aleph null 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 multisetss of integers

• the set of finite sequences of integers.

Beth one

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)

• the set of all real analytic functions

• the set of all continuous functions

• with at most countable discontinuities

• the set of finite subsets of real numbers

• the set of all analytic functions (the holomorphic functions)

• ).

Beth two

(pronounced beth two (pronounced ).

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)

• with uncountably many discontinuities

• 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

Beth omega

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

Generalization

, is occasionally used. Given the axiom of choice, it is defined by:

1. if λ is a limit ordinal.

as follows:

1. if λ is a limit ordinal.

.

It follows (easily with the axiom of choice or after more meticulous argument otherwise) that

is an ordinary beth number, and

such that

1. .

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.)

Borel determinacy

Borel determinacy is implied by the existence of all beths of countable index.

See also

• Uncountable set

References

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.

• See pages 6 and 204–205 for beth numbers.

• See page 109 for beth numbers.