 Russell’s definitions of ordinal numbers and real numbers resemble the definitions of natural numbers. Ordinal numbers are a special case of  relation numbers. Just as a cardinal number can be defined as a class of similar classes where the similarity is simply equinumerosity, the existence of a one to one mapping between the two classes, a relation number is a class of similar classes which are ordered by some relation. Ordinal numbers are the relation numbers of well-ordered classes. “Relation-Arithmetic” is the subject of Part IV of Volume II of Principia Mathematica, chapters *150 to *186. All of the properties of the arithmetic of ordinal numbers are derived from the more general arithmetic of relation numbers. Thus, for example, the addition of ordinal numbers is not commutative. The first infinite ordinal \(\omega\) is the relation number of the well-ordered classes similar to \(1, 2, 3, \ldots\) etc. The sum \(1 + \omega\) will be the relation number of ordered classes which result from adding one element at the beginning of the ordering, say \(0, 1, 2, 3, \ldots\) etc., which has the same ordinal number \(\omega\). Thus \(1 + \omega = \omega\). On the other hand, adding an element at the “end” of such a well ordered class will give an ordering that is not similar: \(1, 2, 3, \ldots \text{etc.}, 0\). Consequently, \(1 + \omega \ne \omega + 1\). On the other hand addition of ordinals, and indeed relation numbers in general, is associative, that is, \((\alpha + \beta) + \gamma = \alpha + (\beta + \gamma)\), which is proved with certain restrictions in *174. Ordinal numbers are thus defined exactly as natural numbers, as classes of similar classes, in such a way that all the desired theorems can be proved. Describing ordinal numbers as “fictions”, “incomplete symbols” and “constructions” applies in the same way as in the case of natural numbers.