A completeness theorem in modal logic

The present paper attempts to state and prove a completeness theorem for the system S5 of 1, 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, …, x m , y m , z m , … as well as a denumerably infinite list of n -adic predicate variables P n , Q n , R n , …, P m n , Q m n , R m n ,…; if n =0, an n -adic predicate variable is often called a “propositional variable.” A formula P n ( x 1 , …, x n ) is an n -adic prime formula; often the superscript will be omitted if such an omission does not sacrifice clarity.