# transfinite number

## English

English Wikipedia has an article on:
Wikipedia

### Noun

transfinite number (plural transfinite numbers)

1. (set theory) Any cardinal or ordinal number which is larger than any finite, i.e. natural number; often represented by the Hebrew letter aleph () with a subscript 0, 1, etc.
• 1961, Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, Courier Dover Publications →ISBN, page 228
It will be recalled that Cantor called the first transfinite number0. He called the second transfinite number—the one describing the set of all real numbers— C. It has not been proved whether C is the next transfinite number after ℵ0 or whether another number exists between them.
• 1968, B. T. Levšenko, "Spaces of transfinite dimensionality", Fourteen Papers on Algebra, Topology, Algebraic and Differential Geometry, American Mathematical Soc. →ISBN, page 141
Let ${\displaystyle R}$  be a bicompact of dimensionality ${\displaystyle \operatorname {ind} (R)\leq \alpha }$ . If ${\displaystyle \alpha }$  is an isolated transfinite number, than [sic] at any point ${\displaystyle x\in R}$  there exist arbitrarily small neighborhoods ${\displaystyle Vx}$  with boundaries of dimensionality ${\displaystyle \operatorname {ind} {\overline {Vx}}\leq \alpha -1}$ .
• 1990, Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press →ISBN, page 180
After all, it was the ordinals that made precise definition of the transfinite cardinals possible. And until Cantor had introduced the order types of transfinite number classes, he could not define precisely any transfinite cardinal beyond the first power.
• 2009, John Tabak, Numbers: Computers, Philosophers, and the Search for Meaning, Infobase Publishing →ISBN, page 153
For example, does there exist a transfinite number that is strictly bigger than ℵ0 and strictly smaller than ℵ1? In this case an instance of this in between number is too big to be put into one-to-one correspondence with the set of natural numbers, and too small to be put into one-to-one correspondence with the set of real numbers.
• 2012, Benjamin Wardhaugh, A Wealth of Numbers: An Anthology of 500 Years of Popular Mathematics Writing, Princeton University Press →ISBN, page 136
Having demonstrated the existence of a one-to-one correspondence, we can conclude that the class of the squares of all the natural numbers has the same transfinite number as the class of all the natural numbers! This result is not what might have been anticipated, seeing that the second class is a proper subset of the first.

#### Usage notes

Some related concepts:

• The aleph numbers, ${\displaystyle \aleph _{0},\aleph _{1},\dots }$ , represent an enumeration of the transfinite numbers.
• The smallest transfinite number, ${\displaystyle \aleph _{0}}$  (aleph-null) — also denoted ${\displaystyle \omega }$  — is the cardinality of the natural numbers. Each succeeding ${\displaystyle \aleph _{n}}$  is defined to be the smallest transfinite number greater than ${\displaystyle \aleph _{n-1}}$ .
• The beth numbers, ${\displaystyle \beth _{0},\beth _{1}\dots }$ , 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, ${\displaystyle \beth _{0}=\aleph _{0}}$  and, for ${\displaystyle n>0}$ , ${\displaystyle \beth _{n}}$  is the power set of ${\displaystyle \beth _{n-1}}$ . A consequence of this definition is that ${\displaystyle \beth _{1}}$  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 ${\displaystyle \aleph _{0}}$  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 ${\displaystyle \aleph _{1}}$ .
• The continuum hypothesis implies that ${\displaystyle \beth _{1}=\aleph _{1}}$ .
• The generalized continuum hypothesis states that ${\displaystyle \beth _{n}=\aleph _{n}}$ .