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

*88. CONDITIONS FOR THE EXISTENCE OF SELECTIONS.

Summary of *88.
The existence of selections cannot, so far as is known at present, be proved
in general. That is, we cannot prove any of the following:
(P, K): K C P..! PAD'c
(P, K): P E CIs -> 1. K C (P'P.. a! P'
(P). g!P '(C'P
(c): A ~ e 1. ). M! CA,'
(K): K e Cls ex2 excl. D.! ei'K
(a).! e'C1 lex 'a
(K):. K c Cls ex2 excl. 3: (,u): a E K. Da. n a El
These various propositions can be shown to be all equivalent inter se;
and in virtue of Zermelo's theorem (cf. *258), they are equivalent to the
proposition "every class can be well-ordered." In the present number we
have to prove the above equivalences, as well as certain propositions giving
the existence of selections in various particular cases.
The most apparently obvious of the above propositions is the last,
namely: "If K is a class of mutually exclusive classes, no one of which is
null, there is at least one class,/ which takes one and only one member from
each member of E." This we shall define as the " multiplicative axiom."
We will call P a multipliable relation (denoted by "Rel Mult ") if
Pa'(I'P exists, or, what is equivalent, if K C P'P.,. g! Pc'K. Thus
we put
Rel Mult = P {! Pa'P} Df.
We will call K a multipliable class of classes if e'Ki exists, i.e. we put
Cls2 Mult = E {a! e'K} Df.
The multiplicative axiom will be denoted by " Mult ax." Thus we put
Mult ax. =:. K e Cls ex2 excl. ): (t/L): a e K. 3D./L e a e I Df.
In the present number, we shall first give various equivalent forms of the
assumption that P is a multipliable relation (*88-1 —15); we shall then
do the same for multipliable classes of classes (*88'2 —26); next we shall
give various equivalent forms of the multiplicative axiom (*88-3 —39).
R. & W. 36