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

550

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

But the numbers of the members of s'K are
H^al, /la2, /Ua3i P91 X A2, P'92, py3s, -l 4 y2, X y3 -Thus the number of e^'s'c is
/Ial X JUa2 X /La3 X /-31 X /LU2 > X UB3 X /Uyi X /,Ly2 X /Ly3
Hence *85'44 enables us to conclude that
(4Ual X,a2 X< l/u3) X (UL,1 X /Act X UN3) X (vnyl X Uy2 X (by)
=,al X Lua2 X B'3 X uhI X /u,2 X iL2 X X Jly Xi X uy2 X /y3,
which is a case of the associative law. In fact *85'44 gives us this law in its
general form, when the number of brackets, and of factors in each bracket,
may be infinite or finite indifferently.
Another important pair of propositions is *85'53'54. These enable us to
reduce the problem of selections for any relation to the problem of selections
from a class of classes. The method is as follows: Given any term x, form
the class of ordered couples of which x is relatum while the referent is a
term having the relation P to x. Call this class of couples P x. Form
this class for every x which is a member of a; we thus obtain a class of
classes, namely P "a. Then the number of selections from this class of
classes is the same as the number of P,'a.
We have one other important pair of propositions in this number, namely
*85'61'63. These show that what is called "Zermelo's axiom" is equivalent
to what is called the "multiplicative axiom." Zermelo's axiom* is to the
effect that if a is any class, ea'Cl ex'a is never null, i.e. (a).! ea'Cl ex'a.
The "multiplicative axiom" is to the effect that if K e Cls ex2 excl, there is
at least one class formed by taking one representative from each member of
K, which is equivalent to
C e Cls ex" excl. D.! ea'K.
In *85'63, these two axioms are shown to be equivalent. From Zermelo's
theorem t it follows that both are equivalent to the assumption that every
class can be well-ordered. This will be proved later (*258).
The above-mentioned propositions, stated symbolically, are as follows:
*85'1.: Q ' X e Cls - 1. D"Q'X = D="e'Q" X
*8514. F: Q r X e Cls - 1. D. Q'X sm e4'Q"X
*85-27. F: K Cls2 excl. D. P's'K = S"DcceaPK
*85'28. F: K Cls2 excl. D. ea's'K = S"D'Dea'e, C
*85-43. F: K e Cls2 excl. D. Pa's/K sm ec'P" K

* See Math. Annalen, Vol. LIX.
+ loc. cit.