Transfinite number

Encyclopedia : T : TR : TRA : Transfinite number

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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.

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 lowest transfinite ordinal number is ω.

• The first transfinite cardinal number is aleph-null, [\aleph_0], the cardinality of the infinite set of the integers. The next higher cardinal number is aleph-one, [\aleph_1].

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.

References

• O'Connor, J. J. and E. F. Robertson (1998) ["Georg Ferdinand Ludwig Philipp Cantor"], MacTutor History of Mathematics archive.

See also

• Beth number
• Large cardinal
• Mahlo cardinal
• Infinitesimal

 

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