# User Contributed Dictionary

### Etymology

Coined by Georg Cantor### Adjective

transfinite- Beyond finite.
- Relating to transfinite numbers.

#### 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.- ω (omega) is defined as the lowest transfinite ordinal number.

- Aleph-null, , is defined as the first transfinite cardinal number and is the cardinality of the infinite set of the integers. If the axiom of choice holds, the next higher cardinal number is aleph-one, . If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-zero. But in any case, there are no cardinals between aleph-zero and aleph-one.

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.

## See also

## References

- Levy, Azriel, 2002 (1979) Basic Set Theory. Dover Publications. ISBN 0-486-42079-5
- O'Connor, J. J. and E. F. Robertson (1998) "Georg Ferdinand Ludwig Philipp Cantor," MacTutor History of Mathematics archive.
- Rubin, Jean E., 1967. "Set Theory for the Mathematician". San Francisco: Holden-Day. Grounded in Morse-Kelley set theory.
- Rudy Rucker, 2005 (1982) Infinity and the Mind. Princeton Univ. Press. Primarily an exploration of the philosophical implications of Cantor's paradise.
- Patrick Suppes, 1972 (1960) "Axiomatic Set Theory". Dover. ISBN 0-486-61630-4. Grounded in ZFC.

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