Beth number

In mathematics, particularly in set theory, the beth numbers form a certain sequence of infinite cardinal numbers, conventionally written, where 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 that are not indexed by. On the other hand, beth numbers are cofinal in plain Zermelo-Fraenkel set theory.

Definition

Beth numbers are indexed by ordinal numbers and defined in terms of the cumulative hierarchy by, where is the cardinality of and is the first infinite ordinal number. In particular,, and it follows by Cantor's theorem and transfinite induction that the sequence of beth numbers is strictly increasing.
. In general, for ordinal, and for every limit ordinal. The axiom of choice implies that the inequality holds in general.
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.
Like aleph numbers, beth numbers are idempotent:. This follows by transfinite induction from two points:

for every limit ordinal ; and
for every idempotent cardinal number.

The axiom of choice implies that every set of cardinal numbers has a supremum and that for any set, the union set of all its members can be no larger than the supremum of its member cardinalities times its own cardinality. It follows that for every limit ordinal.
Note that this behavior is different from that of successor ordinals. Even with the axiom of choice, cardinalities less than but greater than can exist.

Relation to the aleph numbers

Even with the axiom of choice, little more is knowable about the relationship to aleph numbers than is stated above. For example, cannot be, but there is a model in which.
Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable. Since, the axiom implies that
for all ordinals.
Given the axiom of choice, the continuum hypothesis is equivalent to
Without the axiom of choice, there are statements that address Cantor's concern about subsets of the real line without obviously implying that the real line admits a well ordering. For example, one version of the hypothesis is that for every cardinal number.
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, i.e.,
for all ordinals. This assertion obviously implies the axiom of choice since beth numbers are cofinal. 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

Since this is equal to, or aleph null, sets with cardinality 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 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
the complex numbers
the uncomputable real numbers
Euclidean space
the power set of the natural numbers
the set of sequences of integers
the set of sequences of real numbers,
the set of all real analytic functions from to
the set of all continuous functions from to
the set of all functions from to with at most countable discontinuities
the set of finite subsets of real numbers
the set of all analytic functions from to
the set of all functions from the natural numbers to the natural numbers.
Beth two

is also referred to as .
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 to
the set of all functions from to
the set of all functions from to 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 of,, and
the set of deterministic fractals in
the set of random fractals in.
Beth omega

is the smallest uncountable strong limit cardinal.

Generalization

The more general symbol, for ordinals and cardinals, is occasionally used. Given the axiom of choice, it is defined by:
Without the axiom of choice, the definition is more complicated. The main difficulty is that the cardinalities of infinite disjoint unions cannot be calculated just from the cardinalities of the components. Still using recursion, define as follows:
Given an injection, we can construct recursively injections. If is a bijection then so is, so we can define the mapping for cardinal numbers by.
It follows that
where is an ordinary beth number, and
In particular,. For any cardinal and any ordinal number,. On the other hand, for some, and it follows that for large . Thus for every cardinal number there is an ordinal number such that
This also holds in Zermelo–Fraenkel set theory with ur-elements, provided that 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.

Borel determinacy

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