Beth number


In mathematics, the beth numbers are a certain sequence of infinite cardinal numbers, conventionally written, where is the second Hebrew letter. 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 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.
Given this definition,
are respectively the cardinalities of
so that the second beth number is equal to, the cardinality of the continuum, 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 as 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 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 then sets with cardinality include:
Sets with cardinality include:
is also referred to as 2c.
Sets with cardinality include:
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:
So
In 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 Zermelo–Fraenkel set theory 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.
This also holds in Zermelo–Fraenkel set theory with ur-elements with or without the axiom of choice provided the ur-elements form a set which is equinumerous with a pure set. If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.