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Number that is larger than all finite numbers
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 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 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 are no longer in one to one correspondence. A transfinite cardinal number is used to describe the size of an infinitely large set,

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 In this context, is larger than , and , and are larger still. Arithmetic expressions containing 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, essentially a finite sequence of digits that give coefficients of descending powers of
Not all infinite integers can be represented by a Cantor normal form however, and the first one that cannot is given by the limit and is termed is the smallest solution to , and the following solutions give larger ordinals still, and can be followed until one reaches the limit , which is the first solution to This means that in order to be able to specify all transfinite integers, one must think up an infinite sequence of names: because if one were to specify a single largest integer, one would then always be able to mention its larger successor. But as noted by Cantor, even this only allows one to reach the lowest class of transfinite numbers: those whose size of sets correspond to the cardinal number .