Beth 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 second 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 . Beth numbers are defined by transfinite recursion: where is an ordinal and is a limit ordinal. 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 . 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 .