Editing Transfinite number (section)

==Examples==
In Cantor's theory of ordinal numbers, every integer number must have a successor.<ref name="ONG">[[John Horton Conway]], (1976) ''[[On Numbers and Games]]''. Academic Press, ISBN 0-12-186350-6. ''(See Chapter 3.)''</ref> 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 [[Ordinal arithmetic#Cantor normal form|Cantor normal form]] that represents it,<ref name="ONG" /> essentially a finite sequence of digits that give coefficients of descending powers of <math>\omega</math>.

Not all infinite integers can be represented by a Cantor normal form however, and the first one that cannot is given by the limit <math>\omega^{\omega^{\omega^{...}}}</math> and is termed <math>\varepsilon_{0}</math>.<ref name="ONG" /> <math>\varepsilon_{0}</math> is the smallest solution to <math>\omega^{\varepsilon}=\varepsilon</math>, and the following solutions <math>\varepsilon_{1}, ...,\varepsilon_{\omega}, ...,\varepsilon_{\varepsilon_{0}}, ...</math> give larger ordinals still, and can be followed until one reaches the limit <math>\varepsilon_{\varepsilon_{\varepsilon_{...}}}</math>, which is the first solution to <math>\varepsilon_{\alpha}=\alpha</math>. 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,{{citation needed|date=May 2021}} even this only allows one to reach the lowest class of transfinite numbers: those whose size of sets correspond to the cardinal number <math>\aleph_{0}</math>.