Infinite Cardinal number
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 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 .
Definition
Beth numbers are defined by
transfinite recursion:
where is an ordinal and is a
limit ordinal.
[1]
The cardinal is the cardinality of any
countably infinite
set such as the set of
natural numbers, so that .
Let be an ordinal, and be a set with cardinality . Then,
- denotes the
power set of (i.e., the set of all subsets of ),
- the set denotes the set of all functions from to {0,1},
- the cardinal is the result of
cardinal exponentiation, and
- is the cardinality of the power set of .
Given this definition,
are respectively the cardinalities of
so that the second beth number is equal to , the
cardinality of the continuum (the cardinality of the set of the
real numbers), 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 to be 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 (see
transfinite induction) 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, sets with cardinality include:
Beth one
Sets with cardinality include:
Beth two
(pronounced beth two) is also referred to as 2c (pronounced two to the power of c).
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 R to R (RR)
- The set of all functions from Rm to Rn
- The set of all functions from R to R with uncountable discontinuities
[2]
- The power set of the set of all functions from the set of natural numbers to itself, so it is the number of sets of sequences of natural numbers
- The
Stone–Čech compactifications of R, Q, and N
- The set of deterministic
fractals in Rn
[3]
- The set of random
fractals in Rn
[4]
Beth omega
(pronounced beth omega) 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:
- if λ is a limit ordinal.
So
In
Zermelo–Fraenkel set theory (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 ZF 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.
Borel determinacy
Borel determinacy is implied by the existence of all beths of countable index.
[5]
See also
References
Bibliography