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

504

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

When ic fulfils this hypothesis, it is a class of mutually exclusive classes.
For classes of mutually exclusive classes we adopt the notation "Cls2excl."
It is shown in *84*14 that a Cls2 excl is one for which we have e K e Cls -+ 1.
When Ke is a Cls2excl, Dr PA'K is a one-one relation, and D"e',cKsm e'c.
Also in this case D"eA'c consists of all classes formed of one member from
each member of K, i.e. all classes / such that
/h C s8K: a e K. Da-,. n ra 1.
In *85, we prove various important propositions, of which the chief is a form
of the associative law*, namely: K e Cls2 excl. ). eAs'c sm eA^'E"c.
Finally, in *88, we consider the question of the existence of selections. This
cannot in general be proved when K is an infinite class. The assumption
that E,'K is never null unless one member of K is null is equivalent to
various other assumptions, for example to the assumption that every class
can be well-ordered. One of these equivalent assumptions is called the
"multiplicative axiom." This axiom is equivalent to the assumption that
an arithmetical product cannot be zero unless one of its factors is zero, and
is regarded by some mathematicians as a self-evident truth. This can
be proved when the number of factors is finite, i.e. when K is a finite class,
but not when the number of factors is infinite. We have not assumed its
truth in the general case where it cannot be proved, but have included it
in the hypotheses of all propositions which depend upon it.

* Cf. notes to *42-1-11.