Transfinite number

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"). 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, ==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 \omega. In this context, \omega+1 is larger than \omega, and \omega\cdot2, \omega^{2} and \omega^{\omega} are larger still. Arithmetic expressions containing \omega 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 \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^{\omega^{\omega^{...}}} and is termed \varepsilon_{0}. \varepsilon_{0} is the smallest solution to \omega^{\varepsilon}=\varepsilon, and the following solutions \varepsilon_{1}, ...,\varepsilon_{\omega}, ...,\varepsilon_{\varepsilon_{0}}, ... give larger ordinals still, and can be followed until one reaches the limit \varepsilon_{\varepsilon_{\varepsilon_{...}}}, which is the first solution to \varepsilon_{\alpha}=\alpha. 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 \aleph_{0}. ==See also==