Principia mathematica, vol. 1 (of 3)

For this number, we introduce three new notations, of which the first two will be used constantly, especially in the theory of series, while the third will be seldom used except in the present section. The two which are constantly used are i.e. is a member of and of , and no member of precedes in .

The letter may be regarded as standing for "begins." Thus if we take any member of , and proceed backwards and forwards as far as possible by -steps, we obtain a series which may be called the "family" of : this series, if it has a first term, has one which is a member of ; thus the members of are the beginners of families. For example, if is the relation of a peer to his heir, "" will mean " is a peer who is not the heir of a peer"; thus is the first of his family. If is the relation of parent and child, "" will be satisfied only by Adam and Eve; and so for other relations.

If is serial, reduces to a single term if it is not null; thus if a class has a first term, this term is . We also put and then , if it exists, is the last term of in the -series. Thus if is the class of peers, and is the relation of father to son, consists of those peers who are the first of their line, while consists of those peers who are the last of their line. If is a class of numbers, and is the relation of less to greater, is the smallest member of (if it exists), and is the largest (if it exists).

and "" and "" will be used constantly in connection with series, where the two latter will be considered in detail, but the present number is more specially concerned with a less general idea, namely that of generations. Take, e.g., the relation of parent and child; let us call it . Then the first generation consists of those who are parents but not children, i.e. ; the second consists of those who are children but not grandchildren, i.e. , i.e. , i.e. ; the third consists of those who are grandchildren but not great-grandchildren, i.e. , i.e. , i.e. ; and so on. Also we have hence the generations of are . Thus we put where "" stands for "generation."

The generations of do not in general exhaust the field of . For will only belong to a generation of if can be reached by successive -steps starting from a member of . If some of the families constituting the field of have no beginning, the members of these families will not belong to any generation of . Such terms together constitute the class which is the same class.