transfinite number

== English == === Noun === transfinite number (plural transfinite numbers) (set theory) Any cardinal or ordinal number which is larger than any finite, i.e. natural number; often represented by the Hebrew letter aleph (ℵ) with a subscript 0, 1, etc. ==== Usage notes ==== Some related concepts: The aleph numbers, ℵ 0 , ℵ 1 , … {\displaystyle \aleph _{0},\aleph _{1},\dots } , represent an enumeration of the transfinite numbers. The smallest transfinite number, ℵ 0 {\displaystyle \aleph _{0}} (aleph-null) — also denoted ω {\displaystyle \omega } — is the cardinality of the natural numbers. Each succeeding ℵ n {\displaystyle \aleph _{n}} is defined to be the smallest transfinite number greater than ℵ n − 1 {\displaystyle \aleph _{n-1}} . The beth numbers, ℶ 0 , ℶ 1 … {\displaystyle \beth _{0},\beth _{1}\dots } , are an enumerated subset of the transfinite numbers, defined in a different, in some ways more mathematically tractable way. It is hypothesised that the beth numbers are in fact precisely the aleph numbers. By definition, ℶ 0 = ℵ 0 {\displaystyle \beth _{0}=\aleph _{0}} and, for n > 0 {\displaystyle n>0} , ℶ n {\displaystyle \beth _{n}} is the power set of ℶ n − 1 {\displaystyle \beth _{n-1}} . A consequence of this definition is that ℶ 1 {\displaystyle \beth _{1}} is the cardinality of the real numbers. It is not immediately clear that an ordered enumeration of the transfinite numbers, such as the aleph numbers represent, is even possible. In particular, it depends upon the axiom of choice (historically controversial for infinite collections of sets), without which transfinite numbers greater than ℵ 0 {\displaystyle \aleph _{0}} might exist that are not mutually comparable. The continuum hypothesis states that there is no transfinite number between the cardinality of the natural numbers and that of the real numbers — i.e., that the cardinality of the real numbers is ℵ 1 {\displaystyle \aleph _{1}} . The continuum hypothesis implies that ℶ 1 = ℵ 1 {\displaystyle \beth _{1}=\aleph _{1}} . The generalized continuum hypothesis states that ℶ n = ℵ n {\displaystyle \beth _{n}=\aleph _{n}} . ==== Hyponyms ==== aleph number, beth number aleph-null, aleph-one ω ==== Translations ==== === See also === hyperreal number infinitesimal === Further reading === Cardinal number on Wikipedia.Wikipedia Ordinal number on Wikipedia.Wikipedia Aleph number on Wikipedia.Wikipedia Beth number on Wikipedia.Wikipedia Hyperreal number on Wikipedia.Wikipedia Infinitesimal on Wikipedia.Wikipedia Levi-Civita field on Wikipedia.Wikipedia Surreal number on Wikipedia.Wikipedia Continuum hypothesis on Wikipedia.Wikipedia Absolute Infinite on Wikipedia.Wikipedia