Chapter 4 - A Formal Theorem for Church's Theory of Types (with M.H.A. Newman, J. Symbolic Logic, vol. 7 (1942), pp. 28–33)

Abstract

This chapter discusses about the paper, “A Formal theorem in Church's Theory of Types,” which was written by Newman and Turing. Before 1950, most logicians, other than Gödel and Tarski, adhered to the notion of purity of method and would only use purely formal methods when dealing with formal systems. Later, models were used not only to give a better understanding of formal systems, but also via completeness theorems to establish formal provability. Turing relished the existence of finitely constructed models for type theory without an axiom of infinity. At some time in 1940, Newman became interested in exploiting the device of typical ambiguity to construct a system of logic. In Church's system, the axioms and rules are, except for the axioms of infinity (axioms 7 and 8), stated uniformly for all types—although the type o of propositions has a special status. Therefore, in the paper, Newman enquires whether this uniformity could be extended to axioms of infinity.