Typical Ambiguity

Abstract

This paper is on simple theory of types and some of its extensions. Simple theory of types is certainly one of the most natural systems of set theory; the reason for this is that it is defined rather by a family of structures than by a system of axioms. In order to describe such a structure let T0 be a nonempty — finite or infinite — set; elements of T0 are elements of type 0. T1 is the set of subsets of T0; T2 is the set of subsets of T1, and in general Tn+l is the set of subsets of Tn (n natural number). If the set T0 is finite, we can write down this tower to any level we want; let us consider the case where T0 has exactly one element a: T0 is (a), T1 is (⋀, (a)) (⋀ is the empty set) T2 is (⋀, (⋀), ((a)), (⋀, (a))), etc. If T0 has n0 elements, the number of elements of T1 is n1 = 2no and of nk is the number of elements of Tk, nk+1=2nk. It T0 is infinite, these relations still hold if the number nk of elements of Tk is interpreted as the cardinal of Tk and if powers of 2 are defined in the usual way. In the infinite case such definitions are only possible on the basis of a set theory; as we do not want to presuppose such a theory, we formalize and axiomatize type theory. As to formalization, there are essentially two ways of doing it. The first possibility is to introduce a predicate Pk for each type k: Pk(a) says that a is an element of type k. The second possibility (which will be chosen) is to introduce a separate sequence of variables for each type.