The objects to be studied in this Part are not sharply distinguished from those studied in Part I. The difference is one of degree, the objects in this Part being of somewhat less general importance than those of Part I, and being studied more on account of their bearing on cardinal arithmetic than on their own account. Although cardinal arithmetic is the goal which determines our course in Part II, all the objects studied will be found to be also required in ordinal arithmetic and the theory of series.

Section A of this Part deals with unit classes and couples. A unit class is the class of terms identical with a given term, i.e. the class whose only member is the given term. (As explained in the Introduction, Chapter III, pp. 76 to 79, the class whose only member is x is not identical with x.) We define 1 as the class of all unit classes. In like manner, we define a (cardinal or ordinal) couple, and then define 2 as the class of all couples.