Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

608

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

— + v
We have min,'a = a n C'P - P"a.
If P is serial, minp'a reduces to a single term if it is not null; thus if a
class a has a first term, this term is minp'a. We also put
maxp min (P) Df,
and then maxp'a, if it exists, is the last term of a in the P-series. Thus if a
is the class of peers, and P is the relation of father to son, minp'a consists of
those peers who are the first of their line, while maxp'a consists of those
peers who are the last of their line. If a is a class of numbers, and P is
the relation of less to greater, minp'a is the smallest member of a (if it
exists), and maxp'a is the largest (if it exists).
B and "maxp" and "min " 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 P. Then
the first generation consists of those who are parents but not children,
i.e. B'P; the second consists of those who are children but not grandchildren,
i.e. (C'P- C[P2, i.e. (I'P -P(IP, i.e. minpa'I'P; the third consists of those
who are grandchildren but not great-grandchildren, i.e. (aP2 -a 'P3, i.e.
(I'P -P"' P2, i.e. minp'a'P2; and so on. Also we have
B'P = minp''([I C'P);
hence the generations of P are minp"(I"Potid'P. Thus we put
gen'P= minp"C["Potid'P Df,
where "gen " stands for "generation."
When P is a one-many relation, such as that of father and son, every
v --
generation is of the form T"B'P, where T is a power of P (including
I C'P). When P is not a one-many relation, this is not in general the
case.
The generations of P do not in general exhaust the field of P. For x will
only belong to a generation of P if x can be reached by successive P-steps
starting from a member of B'P. If some of the families constituting the
field of P have no beginning, the members of these families will not belong
to any generation of P. Such terms together constitute the class
p'("E'ot'P,
or p'I"Potid'P,
which is the same class.