EXTENSIONALITY, ATTRIBUTES, AND CLASSES (1958)

In Principia Mathematica, an extensional system embodying the theory of types, Peano’s postulate qdistinct (natural) numbers have distinct (immediate) successors” is not formally derivable from purely logical axioms. The axiom of infinity (“the number of individuals is infinite”) is required for the proof that there is no finite cardinal nsuch that n equals n+1. Russell argued as follows:1 if only n individuals existed, then the number n + 1, being defined as the class of all classes that have n + 1 members (or that would have exactly n members if exactly one element were withdrawn from them), would be equal to the null class; for in that case no classes with n + 1 members would exist. But by parity of reasoning, the successor of n+1 would also be equal to the null class; therefore n and n+1, which on the hypothesis made are distinct numbers, would have the same successor. The usual reaction to this argument is that, without abandoning Russell’s conception of numbers as classes of similar classes, we fortunately do not need to postulate the axiom of infinity after all. For there are other ways of solving the logical paradoxes besides the theory of types, and once our constructive efforts are unimpeded by the latter, we can construct an infinite sequence of abstract entities without presupposing the existence of a single concrete individual: the null class, the unit class whose only member is the null class, the class whose members are the foregoing two classes, and so on. And once we have an infinite set of such abstract, though typically impure, entities, we can rest assured that no natural number will collapse into the null class. This is the approach of set theory, where such ghostly classes as the one just mentioned can be postulated to exist provided their definitions do not give rise to contradiction. However, I would like to re-examine Russell’s argument in order to see whether it isperhaps possible to get rid of the axiom of infinity without abandoning the (simple) theory of types.²