A formulation of the simple theory of types

TLDR

Full integration of λ‑conversion into type theory is impossible while preserving λx as abstraction and juxtaposition as application, and the formulation draws on foundational work by Whitehead, Russell, Hilbert, Ackermann, and Bernays. The purpose of the present paper is to give a formulation of the simple theory of types that incorporates certain features of the calculus of λ‑conversion. The formulation defines type symbols inductively: ı and o are base types, and if α and β are types then (αβ) is a type, yielding the smallest closed class. The partial incorporation provides advantages for type theory and is presented as of interest, independent of its ultimate status as a foundation for logic and mathematics.

Abstract

The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λ x and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics). For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize. The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then ( αβ ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol ( αβ ) from the symbols α and β .