In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets.[1] [2] The term transfinite was coined in 1895 by Georg Cantor,[3] [4] [5] [6] 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 and ordinals as infinite numbers. Nevertheless, the term transfinite also remains in use.
Notable work on transfinite numbers was done by Wacław Sierpiński: Leçons sur les nombres transfinis (1928 book) much expanded into Cardinal and Ordinal Numbers (1958, 2nd ed. 1965).
Any finite natural 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 marbles), whereas ordinal numbers specify the order of a member within an ordered set[7] (e.g., "the man from the left" or "the day of January"). When extended to transfinite numbers, these two concepts are no longer in one-to-one correspondence. 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:
\omega
\aleph0
\aleph1.
The continuum hypothesis is the proposition that there are no intermediate cardinal numbers between
\aleph0
\aleph1
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:
ak{m}
A
A
ak{m}.
ak{m}+1=ak{m}.
\aleph0\leqak{m}.
ak{n}
\aleph0+ak{n}=ak{m}.
Although transfinite ordinals and cardinals both generalize only the natural numbers, other systems of numbers, including the hyperreal numbers and surreal numbers, provide generalizations of the real numbers.
In Cantor's theory of ordinal numbers, every integer number must have a successor.[8] The next integer after all the regular ones, that is the first infinite integer, is named
\omega
\omega+1
\omega
\omega ⋅ 2
\omega2
\omega\omega
\omega
\omega
Not all infinite integers can be represented by a Cantor normal form however, and the first one that cannot is given by the limit
| |||||
\omega |
\varepsilon0
\varepsilon0
\omega\varepsilon=\varepsilon
\varepsilon1,...,\varepsilon\omega,
...,\varepsilon | |
\varepsilon0 |
,...
\varepsilon | |||||
|
\varepsilon\alpha=\alpha
\aleph0