transfinite number

transfinite number (plural transfinite numbers)

1. (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 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, 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.

• aleph-null
• aleph-one

• hyperreal number