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 [beth](https://bliptext.com/articles/bet-letter) 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 \aleph that are not indexed by \beth.

Definition

Beth numbers are defined by transfinite recursion: where \alpha is an ordinal and \lambda is a limit ordinal. The cardinal is the cardinality of any countably infinite set such as the set \mathbb{N} of natural numbers, so that. Let \alpha be an ordinal, and A_\alpha be a set with cardinality. Then, Given this definition, are respectively the cardinalities of so that the second beth number \beth_1 is equal to, the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number \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 \lambda, the corresponding beth number is defined to be the supremum of the beth numbers for all ordinals strictly smaller than \lambda: One can show that this definition is equivalent to For instance: This equivalence can be shown by seeing that: Note that this behavior is different from that of successor ordinals. Cardinalities less than \beth_\beta but greater than any can exist when \beta is a successor ordinal (in that case, the existence is undecidable in ZFC and controlled by the Generalized Continuum Hypothesis); but cannot exist when \beta is a limit ordinal, even under the second definition presented. 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 \aleph_0 and \aleph_1, it follows that Repeating this argument (see transfinite induction) yields for all ordinals \alpha. 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 \alpha.

Specific cardinals

Beth null

Since this is defined to be \aleph_0, or aleph null, sets with cardinality \beth_0 include:

Beth one

Sets with cardinality \beth_1 include:

Beth two

\beth_2 (pronounced beth two) is also referred to as (pronounced two to the power of ). Sets with cardinality \beth_2 include:

Beth omega

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

Generalization

The more general symbol, for ordinals \alpha and cardinals \kappa, is occasionally used. It is defined by: So In Zermelo–Fraenkel set theory (ZF), for any cardinals \kappa and \mu, there is an ordinal \alpha such that: And in ZF, for any cardinal \kappa and ordinals \alpha and \beta: Consequently, in ZF absent ur-elements, with or without the axiom of choice, for any cardinals \kappa and \mu, the equality holds for all sufficiently large ordinals \beta. That is, there is an ordinal \alpha 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

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