Abstract

The author here constructs a system of simple type theory in which the type hierarchy does not extend merely to any finite height, but to an infinite height; this added part allows him to prove the existence of infinite sets within the theory, instead of taking it as an axiom in the usual simple type theory. The system has been presented in such sufficient generality so as to make it able to accommodate current scientific theories; the author has turned in the direction of using type theory rather than set theory as the underlying logic of scientific theories. The system is formalized in Church's lambda-calculus notation; the semantics of the system are treated in great detail and appear to be the most complete study so far in print. The system is shown to contain at each type a "basic logic" somewhat similar to first-order logic; the author also shows that Peano's axioms for arithmetic appear as theorems when cast in suitable form. This book is one of a number of recent studies of type theory; together they represent a revival of interest in the subject, both as a foundation for mathematics and as an object of interest in itself. The book stems from Andrews' Princeton doctoral thesis.—P. J. M.