# transfinite number

## Contents

## English[edit]

### Noun[edit]

**transfinite number** (*plural* **transfinite numbers**)

- (set theory) 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.
**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 number ℵ
_{0}. 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 ℵ_{0}or whether another number exists between them.

- It will be recalled that Cantor called the first transfinite number ℵ
**1968**, B. T. Levšenko, "Spaces of transfinite dimensionality",*Fourteen Papers on Algebra, Topology, Algebraic and Differential Geometry*, American Mathematical Soc. ↑ISBN, page 141- Let be a bicompact of dimensionality . If is an isolated transfinite number, than
^{[sic]}at any point there exist arbitrarily small neighborhoods with boundaries of dimensionality .

- Let be a bicompact of dimensionality . If is an isolated transfinite number, than
**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 ℵ
_{0}and strictly smaller than ℵ_{1}? 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.

- For example, does there exist a transfinite number that is strictly bigger than ℵ
**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.