The Simple Theory of Types

Abstract

This chapter traces the history of the simple theory of types and discusses several questions on type theory. The simple theory of types provides a straightforward, reasonably secure foundation for the greater part of classical mathematics. The theory is straightforward because it embodies two principles that (at least before the advent of modern abstract concepts) were part of the mathematicians' normal code of practice—namely, a variable always has a precisely delimited range, and a distinction must always be made between a function and its arguments. The chapter presents a proof of the completeness of the cumulative type system when the axiom of infinity is replaced by axioms of finitude, and a proof of the equivalence asserted. The distinction between free and bound variables is further discussed in the chapter. The chapter also describes the use of hypotheses and the deduction theorem, and the role played by variables that are restricted by hypothesis. The chapter also discusses the application of the theory of types in ordinary mathematical practice.