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

562

PROLEGOMENA TO CARDINAL ARITHMETIC

[PART II

(Some important equivalent forms cannot be given at this stage, as they
depend upon definitions not yet given, such as the definitions of cardinal
multiplication and of well-ordered series. Cf. *114'26 and *258'37.) Finally
we shall give propositions showing that various special classes of classes are
multipliable. Most of these propositions will not be used in the sequel, but
they illustrate the nature of the difficulties involved in proving that a class
of classes is multipliable, and some of them show that mere size does not
prevent a class from being multipliable. For example, *88'48 shows that,
given any class of classes Kc, if each member a is replaced by t"a u t'a, the
result is a multipliable class of classes; but the only effect of this change
is to increase the number of members of each member of our class of
classes by one.
The chief propositions in this number which are afterwards referred to
are the following:
*88'22.: K e Cls2 Mult. X C K. D. X e Cls2 Mult
*88'32. F:. Mult ax.: K e Cls ex2 excl. K ). a! e,&'
*88'33. F: Mult ax. (a). a! ea'Cl ex'a
*88-361. F:.Mult ax. =: C G'R. -R,K a! RC'K
*88'37. F:. Mult ax. -: AR e K. ). [! ec'K
The above is usually the most convenient form of the multiplicative
axiom.
*88372. F:. Multax. =: A e Kc. =. ec = A
This proposition is used in *114, to prove that the multiplicative axiom
is equivalent to the proposition that a cardinal product vanishes when, and
only when, one of its factors vanishes.
*88'01. Rel Mult= P {g! PaI'P} Df
*88-02. Cls2 Mult = {Ia! C'cK} Df
*88'03. Mult ax. =:. e Cls ex2 excl.,: (3tP): a K". a a e 1 Df
*881. F: P e Rel Mult. _. a! P,'('P [*203. (*88-01)]
*8811. F: P e Rel Mult. X C (PP...a! P,'X
Dem.. *80-6. H: R e PA'CP. x C a'P. D. R [X P&'\.
[*10-24] D. t! Pa:'
[*1011-23'35] D) F:! Pa( P. X C ('PP..! PA'X (1)
F. (1). *88-1. D F. Prop