Meaning of transfinite number | Babel Free

Definitions

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.

Equivalents

Examples

“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.”

“Let⟳ R be a bicompact of dimensionality #92;operatorname#123;ind#125;(R)#92;le#92;alpha. If #92;alpha is an isolated transfinite number⟳, than ^([sic]) at any point⟳ x#92;inR there exist⟳ arbitrarily small neighborhoods Vx with boundaries of dimensionality #92;operatorname#123;ind#125;#92;overline#123;Vx#125;#92;le#92;alpha-1.”

“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.”

“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.”

“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.”

CEFR level