*88. CONDITIONS FOR THE EXISTENCE OF SELECTIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
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:
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 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 ." This we shall define as the "multiplicative axiom."
We will call a multipliable relation (denoted by "Rel Mult") if exists, or, what is equivalent, if Thus we put
We will call a multipliable class of classes if exists, i.e. we put
The multiplicative axiom will be denoted by "." Thus we put
In the present number, we shall first give various equivalent forms of the assumption that 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). (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 , if each member is replaced by , 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.
*88·32.
*88·33.
*88·361.
*88·37.
The above is usually the most convenient form of the multiplicative axiom.
*88·372.
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.
*88·02.
*88·03.
*88·1.
*88·11.
Dem.
*88·12
Dem.
*88·13
*88·14
Dem.
*88·15.
*88·2.
*88·21.
*88·22.
Dem.
*88·23
*88·24
Dem.
*88·25
Dem.
*88·26.
Dem.