, transfinite numbers are numbers that are "infinite
" in the sense that they are larger than all finite
numbers, yet not necessarily absolutely infinite
. These include the transfinite cardinals, which are cardinal number
s used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal number
s used to provide an ordering of infinite sets.
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 writers share these qualms; it is now accepted usage to refer to transfinite cardinals
as "infinite". Nevertheless, the term "transfinite" also remains in use.
Any finite number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set
(e.g., "the third man from the left" or "the twenty-seventh day of January"). When extended to transfinite numbers, these two concepts become distinct. A transfinite cardinal number is used to describe the size of an infinitely large set,
while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered.
The most notable ordinal and cardinal numbers are, respectively:
): the lowest transfinite ordinal number. It is also the order type
of the natural number
s under their usual linear ordering.
): the first transfinite cardinal number. It is also the cardinality
of the infinite set
of the natural numbers. 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-nought. Either way, there are no cardinals between aleph-nought and aleph-one.
The continuum hypothesis
is the proposition that there are no intermediate cardinal numbers between
and the cardinality of the continuum
(the cardinality of the set of real number
or equivalently that
is the cardinality of the set of real numbers. In Zermelo–Fraenkel set theory
, neither the continuum hypothesis nor its negation can be proven without violating consistency.
Some authors, including P. Suppes and J. 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:
is a transfinite cardinal. That is, there is a Dedekind infinite set
such that the cardinality of ''
* There is a cardinal
*Epsilon numbers (mathematics)
*Infinity plus one
*Levy, Azriel, 2002 (1978) ''Basic Set Theory''. Dover Publications.
*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
, 2005 (1982) ''Infinity and the Mind''. Princeton Univ. Press. Primarily an exploration of the philosophical implications of Cantor's paradise. .
, 1972 (1960)Axiomatic Set Theory
. Dover. . Grounded in ZFC
Category:Basic concepts in infinite set theory