Beth number

Beth numbers are a sequence of infinite cardinal numbers in set theory, written beth-0, beth-1, beth-2 and so on using the Hebrew letter beth, indexed by ordinal numbers and defined through the cumulative hierarchy. Each successive beth number is the power set of the one before it, so beth of alpha plus one equals two raised to the power of beth of alpha, and the sequence strictly increases by Cantor's theorem. Beth-0 equals aleph-0, the cardinality of the countably infinite sets, but beth numbers are not simply identical to the aleph numbers unless the generalized continuum hypothesis holds; assuming the axiom of choice, beth of alpha is always greater than or equal to aleph of alpha. The continuum hypothesis itself, assuming the axiom of choice, is equivalent to the statement that beth-1 equals aleph-1, and the generalized continuum hypothesis is equivalent to the two sequences matching at every ordinal.