The Principles of Mathematics (1903)

§ 102

A similar result, which, however, does not lead to a contradiction, may be proved concerning any relation. Let R be a relation, and consider the class w of terms which do not have the relation R to themselves. Then it is impossible that there should be any term a to which all of them and no other terms have the relation R. For, if there were such a term, the propositional function x does not have the relation R to x would be equivalent to x has the relation R to a. Substituting a through x throughout, which is legitimate since the equivalence is formal, we find a contradiction. When in place of R we put , the relation of a term to a class-concept which can be asserted of it, we get the above contradiction. The reason that a contradiction emerges here is that we have taken it as an axiom that any propositional function containing only one variable is equivalent to asserting membership of a class defined by the propositional function. Either this axiom, or the principle that every class can be taken as one term, is plainly false, and there is no fundamental objection to dropping either. But having dropped the former, the question arises: Which propositional functions define classes which are single terms as well as many, and which do not? And with this question our real difficulties begin.(§ 102 ¶ 1)

Any method by which we attempt to establish a one-one or many-one correlation of all terms and all propositional functions must omit at least one propositional function. Such a method would exist if all propositional functions could be expressed in the form …∈u, since this form correlates u with …∈u. But the impossibility of any such correlation is proved as follows. Let ϕx be a propositional function correlated with x; then, if the correlation covers all terms, the denial of ϕx(x) will be a propositional function, since it is a proposition for all values of x. But it cannot be included in the correlation; for if it were correlated with a, ϕa(x) would be equivalent, for all values of x, to the denial of ϕx(x); but this equivalence is impossible for the value a, since it makes ϕa(a) equivalent to its own denial. It follows that there are more propositional functions than terms--a result which seems plainly impossible, although the proof is as convincing as any in Mathematics. We shall shortly see how the impossibility is removed by the doctrine of logical types.(§ 102 ¶ 2)