Principia mathematica, vol. 1 (of 3)

The purpose of this number may be explained as follows. Consider the series of relations it is required to find a means of defining this series without the use of numbers. If we used numbers, and had the definition given later (*301) of where is any finite integer, the general term of the series would be . But we have not yet defined numbers, and we therefore desire some means, not involving numbers, of expressing what is intended when we say that, in a given term of the series, the same power of and of is to be involved. This we do as follows. Using the definition of in *43, we have

Thus the general term of our series is got by taking any power of (, and forming . The whole of the terms of the series are therefore constituted by the terms which have to the relation (; i.e. they are . For convenience of notation we put[67]

To illustrate the nature of (, suppose is the relation "first cousin," while is the relation of child to parent and is the relation of parent to child. Then is the relation "second cousin," is the relation "third cousin," and so on. Thus ( is the class of all relations of cousinship which do not involve a difference of generation; and "" will mean " is a cousin of in the same generation."

Most of the propositions in this number are inserted because they are required in the proof of *95·52, which states that, under suitable circumstances, . This proposition itself is proved mainly because it is required in the proof of *95·63, which states that, if , are one-one's each of which has its converse domain contained in its domain, and if the first generation of is similar to the first generation of , then the sum of the generations of is similar to the sum of the generations of . This leads immediately to a proposition (*95·71) which is half of the Schröder-Bernstein theorem (the other half being *94·53 or *94·54), namely: "If and are one-one's each of which has its converse domain contained in the domain of the other, then the sum of the generations of is similar to the sum of the generations of ."