But we must now consider the problems of classes which have one member or none. The case of the null-class might be met by a bare denial--this is only inconvenient, not self-contradictory. But in the case of classes having only one term, it is still necessary to distinguish them from their sole members. This results from Frege's argument, which we may repeat as follows. Let `u` be a class having more than one term; let `ιu` be the class of classes whose only member is `u`. Then `ιu` has one member, `u` has many; hence `u` and `ιu` are not identical. It may be doubted, at first sight, whether this argument is valid. The relation of `x` to `u` expressed by `x`∈`u` is a relation of a single term to many terms; the relation of `u` to `ιu` expressed by `u`∈`ιu` is a relation of many terms (as subject) to many terms (as predicate)^{[133]}. This is, so an objector might contend, a different relation from the previous one; and thus the argument breaks down. It is in different senses that `x` is a member of `u` and that `u` is a member of `ιu`; thus `u` and `ιu` may be identical in spite of the argument.(§ 491 ¶ 1)

This attempt, however, to escape from Frege's argument, is capable of refutation. For all the purposes of Arithmetic, to begin with, and for many of the purposes of logic, it is necessary to have a meaning for ∈ which is equally applicable to the relation of a term to a class, of a class to a class of classes, and so on. But the chief point is that, if every single term is a class, the proposition `x`∈`x`, which gives rise to the Contradiction, must be admissible. It is only by distinguishing `x` and `ιx`, and insisting that in `x`∈`u` the `u` must always be of a type higher by one than `x`, that the contradiction can be avoided. Thus, although we may identify the class with the numerical conjunction of its terms, wherever there are many terms, yet where there is only one term we shall have to accept Frege's range as an object distinct from its only term. And having done this, we may of course also admit a range in the cass of a null propositional function. We shall differ from Frege only in regarding a range as in no case a term, but an object of a different logical type, in the sense that a propositional function `ϕ`(`x`), in which `x` may be any term, is in general meaningless if for `x` we substitute a range; and if `x` may be any range of terms, `ϕ`(`x`) will in general be meaningless if for `x` we substitute either a term or a range of ranges of terms. Ranges, finally, are what are properly to be called *classes*, and it is of them that cardinal numbers are asserted.(§ 491 ¶ 2)

§ 491 n. 1. The word *predicate* is here used loosely, not in the precise sense defined in § 48. ↩

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