The present paper attempts to state and prove a completeness theorem for the system S5 of , supplemented by first-order quantifiers and the sign of equality. We assume that we possess a denumerably infinite list of individual variables a, b, c, …, x, y, z, …, xm, ym, zm , … as well as a denumerably infinite list of n-adic predicate variables Pn , Qn , Rn , …, Pm n , Qm n , Rm n ,…; if n=0, an n-adic predicate variable is often called a “propositional variable.” A formula Pn (x1 , …,xn ) is an n-adic prime formula; often the superscript will be omitted if such an omission does not sacrifice clarity.
(Received August 25 1958)
1 My thanks to the referee and to Professor H. B. Curry for their helpful comments on this paper and their careful reading of it. I must express an added debt of gratitude to Curry; without his constant encouragement of my research, publication of these results might have been delayed for years.