In
mathematics, the
Hebrew letter (
aleph) with various subscripts represents various
infinite cardinal numbers, but it is less commonly known that the second Hebrew letter (
beth) also occurs and has significance. To define the
beth numbers, start by letting
be the cardinality of
countably infinite sets; for concreteness, take the set
N of
natural numbers to be the typical case. Denote by
P(
A) the
power set of
A, i.e., the set of all subsets of
A. Then define
= the cardinality of the power set of
A if is the cardinality of
A.
Then
are respectively the cardinalities of
Each set in this sequence has cardinality strictly greater than the one preceding it, because of
Cantor's theorem.
For infinite limit ordinals κ, we define
If we assume the
axiom of choice, then infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable, and so, since no infinite cardinalities are between and , the celebrated
continuum hypothesis can be stated in this notation by saying
The
generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers.
