Beth number

In mathematics, the beth numbers are a certain sequence of infinite cardinal numbers, conventionally written formula_1, where formula_2 is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers (formula_3), but there may be numbers indexed by formula_4 that are not indexed by formula_2.
Definition.
To define the beth numbers, start by letting
be the cardinality of any countably infinite set; for concreteness, take the set formula_7 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 formula_9 is the cardinality of "A").
Given this definition,
are respectively the cardinalities of
so that the second beth number formula_12 is equal to formula_13, the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number formula_14 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 formula_16 have cardinality formula_17.
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 formula_18 and formula_19, it follows that 
Repeating this argument (see transfinite induction) yields 
formula_21 
for all ordinals formula_22.
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., 
formula_24 
for all ordinals formula_22.
Specific cardinals.
Beth null.
Since this is defined to be formula_18, or aleph null, sets with cardinality formula_27 include:
Beth one.
Sets with cardinality formula_12 include:
Beth two.
formula_14 (pronounced "beth two") is also referred to as 2"c" (pronounced "two to the power of c").
Sets with cardinality formula_14 include:
Beth omega.
formula_31 (pronounced "beth omega") is the smallest uncountable strong limit cardinal.
Generalization.
The more general symbol formula_32, 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." 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.