Let us begin by recapitulating the possible theories of classes which have presented themselves. A class may be identified with (α) the predicate, (β) the class concept, (γ) the concept of the class, (δ) Frege's range, (ε) the numerical conjunction of the terms of the class, (ζ) the whole composed of the terms of the class.(§ 489 ¶ 1)

Of these theories, the first three, which are intensional, have the defect that they do not render a class determinate when its terms are given. The other three do not have this defect, but they have others. (δ) suffers from a doubt as to there being such an entity, and also from the fact that, if ranges are terms, the contradiction is inevitable. (ε) is logically unobjectionable, but is not a single entity, except when the class has only one member. (ζ) cannot always exist as a term, for the same reason as applies against (δ); also it cannot be identified with the class on account of Frege's argument^{[129]}.(§ 489 ¶ 2)

Nevertheless, without a single object^{[130]} to represent an extension, Mathematics
crumbles. Two propositional functions which are equivalent for all values of the
variable may not be identical, but it is necessary that there should be some
object determined by both. Any object that may be proposed, however, presupposes
the notion of *class*. We may define *class* optatively as
follows: A class is an object uniquely determined by a propositional function,
and determined equally by any equivalent propositional function. Now we cannot
take as this object (as in other cases of symmetric transitive relations) the
class of propositional functions equivalent to a given propositional function,
unless we already have the notion of *class*. Again, equivalent
relations, considered intensionally, may be distinct: we want therefore to find
some one object determined equally by any one of a set of equivalent relations.
But the only objects that suggest themselves are the class of relations or the
class of couples forming their common range; and these both presuppose
*class*. And without the notion of class, elementary problems, such as
how many combinations can be formed of

become meaningless. Moreover, it appears immediately evident that
there is some sense in saying that two class-concepts have the `m` objects `n` at a
time?*same*
extension, and this requires that there should be some object which can be
called the extension of a class-concept. But it is exceedingly difficult to
discover any such object, and the
contradiction proves conclusively that, even if there be such an object
sometimes, there are propositional functions for which the extension is not one
term.(§ 489 ¶ 3)

The class as many, which we numbered (ε) in the above enumeration, is unobjectionable, but is many and not one. We may, if we choose, represent this by a single symbol: thus `x`∈`u` will mean

This must not be taken as a relation of two terms, `x` is one of the `u`'s.`x` and `u`, because `u` as the numerical conjunction is not a single term, and we wish to have a meaning for `x`∈`u` which would be the same if for `u` we substituted an equal class `v`, which prevents us from interpreting `u` intensionally. Thus we may regard

as expressing a relation of `x` is one of the `u`'s`x` to many terms, among which `x` is included. The main objection to this view, if only single terms can be subjects, is that, if `u` is a symbol standing essentially for many terms, we cannot make `u` a logical subject without risk of error. We can no longer speak, one might suppose, of a class of classes; for what should be the terms of such a class are not single terms, but are each many terms^{[131]}. We cannot assert a predicate of many, one would suppose, except in the sense of asserting it of each of the many; but what is required here is the assertion of a predicate concerning the many as many, not concerning each nor yet concerning the whole (if any) which all compose. Thus a class of classes will be many many's; its constituents will each be only many, and cannot therefore in any sense, one might suppose, be single constituents. Now I find myself forced to maintain, in spite of the apparent logical difficulty, that this is precisely what is required for the assertion of number. If we have a class of classes, each of whose members has two terms, it is necessary that the members should each be genuinely two-fold, and should not be each one. Or again, Brown and Jones are two,

requires that we should not combine Brown and Jones into a single whole, and yet it has the form of a subject-predicate proposition. But now a difficulty arises as to the number of members of a class of classes. In what sense can we speak of two couples? This seems to require that each couple should be a single entity; yet if it were, we should have two units, not two couples. We require a sense for diversity of collections, meaning thereby, apparently, if `u` and `v` are the collections in question, that `x`∈`u` and `x`∈`v` are not equivalent for all values of `x`.(§ 489 ¶ 4)

§ 489 n. 1. Archiv I. p. 444. ↩

§ 489 n. 2. For the use of the word *object* in the following discussion, see § 58, note. ↩

§ 489 n. 3. Wherever the context requires it, the reader is to add provided the class in question (or all the classes in question) do not consist of a single term.

↩

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.