Ordinal number
Ordinal number
A class b is well-ordered by a dyadic relation R if it is ordered by R (see order) and, for every class a such that a ? b, there is a member x of a, such that xRy holds for every member y of a; and R is then called a well-ordering relation. The ordinal number of a class b well-ordered by a relation R, or of a well-ordering relation R, is defined to be the relation-number (q. v.) of R.
The ordinal numbers of finite classes (well-ordered by appropriate relations) are called finite ordinal numbers. These are 0, 1, 2, … (to be distinguished, of course, from the finite cardinal numbers 0, 1, 2, . . .).
The first non-finite (transfinite or infinite) ordinal number is the ordinal number of the class of finite ordinal numbers, well-ordered in their natural order, 0, 1, 2, . . .; it is usually denoted by the small Greek letter omega. — A.C.
G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, translated and with an introduction by P. E. B. Jourdain, Chicago and London, 1915. (new ed. 1941); Whitehead and Russell, Princtpia Mathematica. vol. 3.