The Principles of Mathematics (1903)

§ 100

Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predicable of themselves. Before attempting to solve this puzzle, it will be well to make some deductions connected with it, and to state it in various different forms. I may mention that I was led to it in the endeavour to reconcile Cantor's proof that there can be no greatest cardinal number with the very plausible supposition that the class of all terms (which we have seen to be essential to all formal propositions) has necessarily the greatest possible number of members[68].(§ 100 ¶ 1)

Let w be a class-concept which can be asserted of itself, i.e. such that w is a w. Instances are class-concept, and the negations of ordinary class-concepts, e.g. not-man. Then (α) if w be contained in another class v, since w is a w, w is a v; consequently there is a term of v which is a class-concept that can be asserted of itself. Hence by contraposition, (β) if u be a class-concept none of whose members are class-concepts that can be asserted of themselves, no class-concept contained in u can be asserted of itself. Hence further, (γ) if u be any class-concept whatever, and u′ the class-concept of those members of u which are not predicable of themselves, this class-concept is contained in itself, and none of its members are predicable of themselves; hence by (β) u′ is not predicable of itself. Thus u′ is not a u′, and is therefore not a u; for the terms of u that are not terms of u′ are all predicable of themselves, which u′ is not. Thus (δ) if u be any class-concept whatever, there is a class-concept contained in u which is not a member of u, and is also one of those class-concepts that are not predicable of themselves. So far, our deductions seem scarcely open to question. But if we now take the last of them, and admit the class of those class-concepts that cannot be asserted of themselves, we find that this class must contain a class-concept not a member of itself and yet not belonging to the class in question.(§ 100 ¶ 2)

We may observe also that, in virtue of what we have proved in (β), the class of class-concepts which cannot be asserted of themselves, which we will call w, contains as members of itself all its sub-classes, although it is easy to prove that every class has more sub-classes than terms. Again, if y be any term of w, and w′ be the whole of w except y, then w′, being a sub-class of w, is not a w′ but is a w, and therefore is y. Hence each class-concept which is a term of w has all other terms of w as its extension. It follows that the concept bicycle is a teaspoon, and teaspoon is a bicycle. This is plainly absurd, and any number of similar absurdities can be proved.(§ 100 ¶ 3)