In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written
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
Beth numbers are defined by transfinite recursion: where
Then, Given this definition, are respectively the cardinalities of so that the second beth number
, 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
, the corresponding beth number is defined to be the supremum of the beth numbers for all ordinals strictly smaller than
: 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.
is a successor ordinal (in that case, the existence is undecidable in ZFC and controlled by the Generalized Continuum Hypothesis); but cannot exist when
is a limit ordinal, even under the second definition presented.
One can also show that the von Neumann universes
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
, it follows that Repeating this argument (see transfinite induction) yields
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.,
, or aleph null, sets with cardinality
include: Sets with cardinality
(pronounced beth two) is also referred to as
(pronounced beth omega) is the smallest uncountable strong limit cardinal.
It is defined by: So In Zermelo–Fraenkel set theory (ZF), for any cardinals
: Consequently, in ZF absent ur-elements, with or without the axiom of choice, for any cardinals
, the equality holds for all sufficiently large ordinals
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 is implied by the existence of all beths of countable index.