AskDefine | Define transfinite

User Contributed Dictionary

English

Etymology

Coined by Georg Cantor

Adjective

transfinite
  1. Beyond finite.
  2. Relating to transfinite numbers.

Related terms

Translations

beyond finite
  • French: transfini
  • German: transfinit
  • Portuguese: transfinito

Extensive Definition

Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary workers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.

Definition

As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers. Unlike the finite ordinals and cardinals, the transfinite ordinals and cardinals define different classes of numbers.
The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers. (If ZFC is consistent, then neither the continuum hypothesis nor its negation can be proven from ZFC.)
Some authors, for example Suppes, Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set, in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition, the following are all equivalent:
  • m is a transfinite cardinal. That is, there is a Dedekind infinite set A such that the cardinality of A is m.
  • m + 1 = m.
  • ≤ m.
  • there is a cardinal n such that + n = m.

References

transfinite in Spanish: Número transfinito
transfinite in French: Nombre transfini
transfinite in Italian: Numero transfinito
transfinite in Dutch: Transfiniet getal
transfinite in Portuguese: Número transfinito
transfinite in Chinese: 超限数
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