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. The term transfinite was coined in 1895 by Georg Cantor, who wished to avoid some of the implications of the word infinite. In particular he believed that "truly infinite" is a perfect and thus divine quality and so refused to attribute this term to mathematical constructs comprehensible by humans. 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, described in his 1928 book Leçons sur les nombres transfinis, much expanded into Cardinal and Ordinal Numbers in 1958, with a second slightly revised edition in 1965..
Definition
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 (e.g., "the man from the left" or "the day of January"). For finite numbers these concepts are in one-to-one correspondence: five<==>fifth, but when extended to transfinite numbers, the concepts are no longer in one-to-one correspondence. A transfinite cardinal number is used to describe the size of an infinitely large set,
Examples
In Cantor's theory of ordinal numbers, every integer number must have a successor. The next integer after all the regular ones, that is the first infinite integer, is named <math>\omega</math>. In this context, <math>\omega+1</math> is larger than <math>\omega</math>, and <math>\omega\cdot2</math>, <math>\omega^{2}</math> and <math>\omega^{\omega}</math> are larger still. Arithmetic expressions containing <math>\omega</math> specify an ordinal number, and can be thought of as the set of all integers up to that number. A given number generally has multiple expressions that represent it, however, there is a unique Cantor normal form that represents it,