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

SECTION Dj

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analogous to the infinite cardinal product. This will be explained at a later
stage (*172).
Although it is chiefly eA' and FA'K that will be required in the sequel,
we shall treat P,'K generally, because this introduces little extra complication, and most of the theorems which hold for e,'c or F,'Kc have exact
analogues for Pa,'c.
P'tc, as above defined, is the class of one-many relations contained in
P and having K for their converse domain. We know of no proof that there
always are such relations when K C ('P. In fact, the proposition
C 'P. p,,. g! PA'e
is equivalent to the "multiplicative axiom," i.e. to the axiom that, given any
class of mutually exclusive classes, none of which is null, there is at least one
class formed of one member from each of these classes. (This equivalence
is proved in *88'38, below.) It is also equivalent to Zermelo's axiom*,
which is
(a).! EA'C1 ex l;
hence also it is equivalent to the proposition that every class can be wellordered. In the absence of evidence as to the truth or falsehood of these
various propositions, we shall not assume their truth, but shall explicitly
introduce them as hypotheses wherever they are relevant.
In the present section, we shall begin (*80) by considering such properties
of P'Kc as do not depend upon any hypothesis as to P. We shall then
(*81) proceed to consider such further properties of P,'K as result from the
hypothesis P r Kc - Cls - 1. This hypothesis is important, because it is
verified in many of the applications we wish to make, and because it leads
to important properties of PA'K which are not true in general when P is not
subject to any hypothesis. These special properties are mostly due to the
fact that when P KE is a many-one relation, P)A'K consists of one-one relations
(not merely of one-many relations, as it does in the general case). This
is proved in *81'1. We then (*82) proceed to consider the case of relative
products, i.e. (P Q),'X. It will appear that, with a suitable hypothesis,
(P Q)A'x = IQ"PA'Q" and D"(P Q),A'= D"P,'/Q"X. In the following
number (*83) we apply the results of *80 to the particular case where P is
replaced by e, which is the important case for cardinal arithmetic. In *84
we apply the propositions of *81 to the case where P is replaced by e, and
where, therefore, we have the hypothesis e Kg e Cls -* 1. This hypothesis is
equivalent to the hypothesis that no two members of Kc have any members in
common, i.e. that
a, 3e K. a+, 4. a, S. a r, /3= A.
* See his "Beweis, dass jede Menge wohlgeordnet werden kann," Math. Annalen, Vol. LIX.
pp. 514-516.