# transfinite number

## English

### Noun

transfinite number ‎(plural transfinite numbers)

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1. (set theory) 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 9780486289731), page 228
It will be recalled that Cantor called the first transfinite number ℵ0. 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 9780821817735), 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 9780691024479), 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 9780816068746), 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 9781400841981), 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.