There is no difficulty in manufacturing similar contradictions ad lib. The solution of such contradictions by the theory of types is set forth fully in Principia Mathematica,[27] and also, more briefly, in articles by the present author in the American Journal [Pg 136] of Mathematics,[28] and in the Revue de Metaphysique et de Morale.[29] For the present an outline of the solution must suffice. [27]Vol. I., Introduction, chap. II., * 12 and * 20; vol. II., Prefatory Statement [28]"Mathematical Logic as based on the Theory of Types," vol. XXX., 1908, pp. 222-262. [29]"Les paradoxes de la logique," 1906, pp. 627-650. The fallacy consists in the formation of what we may call "impure" classes, i.e. classes which are not pure as to "type." As we shall see in a later chapter, classes are logical fictions, and a statement which appears to be about a class will only be significant if it is capable of translation into a form in which no mention is made of the class. This places a limitation upon the ways in which what are nominally, though not really, names for classes can occur significantly: a sentence or set of symbols in which such pseudo-names occur in wrong ways is not false, but strictly devoid of meaning. The supposition that a class is, or that it is not, a member of itself is meaningless in just this way. And more generally, to suppose that one class of individuals is a member, or is not a member, of another class of individuals will be to suppose nonsense; and to construct symbolically any class whose members are not all of the same grade in the logical hierarchy is to use symbols in a way which makes them no longer symbolise anything. Thus if there are individuals in the world, and classes of individuals, we cannot form a new class, consisting of both individuals and classes and having members. In this way the attempt to escape from the need for the axiom of infinity breaks down. I do not pretend to have explained the doctrine of types, or done more than indicate, in rough outline, why there is need of such a doctrine. I have aimed only at saying just so much as was required in order to show that we cannot prove the existence of infinite numbers and classes by such conjurer's methods as we have been examining. There remain, however, certain other possible methods which must be considered.