Principia mathematica, vol. 1 (of 3)

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. As this Part advances, the approach to cardinal arithmetic becomes gradually more marked, until at last nothing is lacking except the definition of cardinal numbers, with which Part III opens.

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. 80 to 83, the class whose only member is is not identical with .) We define as the class of all unit classes, leaving it to Part III to show that , so defined, is a cardinal number. In like manner, we define a (cardinal or ordinal) couple, and then define as the class of all couples. The propositions on couples will not be much referred to in the remainder of the present Part, since their use belongs chiefly to arithmetic (Parts III and IV). On the other hand, the properties of unit classes are constantly required in Sections C, D, E of this Part.

Section B deals, first, with the class of sub-classes of a given class, i.e. of classes contained in a given class. The sub-classes of a given class are often important in arithmetic. Next we consider the class of sub-relations of a given relation, i.e. relations contained in a given relation. The propositions on this subject are analogous to those on sub-classes, but less important. Next we consider the question of "relative types," i.e. taking any object , and calling its type , we give a notation for expressing in terms of the type of classes of which is a member, or of relations in which may be either referent or relatum, and so on. The notations introduced in this connection are very useful in arithmetic, especially in connection with existence-theorems. But the propositions of Section B are very seldom required in the later sections of the present Part.

Section C, which deals with one-many, many-one and one-one relations, is very important, and is constantly relevant in the sequel. A relation is one-many when no term has more than one referent, many-one if no term has more than one relatum, and one-one if it is both one-many and many-one. In this section, we define the notion of similarity, upon which all cardinal arithmetic is based: two classes are said to be similar when there is a one-one relation whose domain is the one and whose converse domain is the other. We prove the elementary properties of similarity, including the Schröder-Bernstein theorem, namely: If is similar to part of , and is similar to part of , then is similar to .

Section D deals with the notion of selections, upon which both cardinal and ordinal multiplication are based. A selection from a set of classes is a class consisting of one member from each class of the set. Thus a selective relation may be defined as one which, for a given class of classes , makes a member of whenever is a member of . More exactly, a selective relation for a class of classes is one which is one-many, which has for its converse domain, and is such that, if , then . Such a relation may be called an -selector from . More generally, we may define a -selector from as a relation which is one-many, which has for its converse domain, and which is contained in . The theory of selectors is very important in arithmetic. But until we come to cardinal multiplication in Part III, Section B, the propositions of this fourth section will seldom be relevant.

Section E deals with mathematical induction, not in the special form in which it applies to finite integers (this is considered in Part III, Section C), but in a general form in which it applies to all relations. The propositions of this section are of very great importance, primarily in the theory of finite and infinite (Part III, Section C, and Part V, Section E), but also in many other subjects, and especially in the derivation of series from one-many, many-one or one-one relations—for example, in ordering the "rational" points of a projective space by means of successive constructions of harmonic points. The ideas involved in this section are somewhat complicated, and we must refer the reader to the section itself for an account of them.