add site
services
publishers
database
health videos
toolbar
stats
live show
health store
more stuff
JOIN/LOGIN
| advertise add site services publishers database health videos
| | about toolbar stats live show health store more stuff JOIN/LOGIN |
- tear-resistant competitor's numbers for track and field... shapeupshop.com | Cardinal Health System, Inc. Cardinal Health Partners accesschs.org | Cardinal Style - Cardinal 1/1 van Encore Medical - Encore Medical -... encoremedical.com |
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. [edit] DefinitionAs 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 Zermelo–Fraenkel set theory (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:
[edit] See also
[edit] References
| ||||||||||||||||||||||||||||||||||||
| ↑ top of page ↑ | about thumbshots |
, is defined as the first transfinite 
. 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.
) · 
)
)
or 
) ·
)
)
)
)
)
)
)