Some theorems on definability and decidability

In this paper a theorem about numerical relations will be established and shown to have certain consequences concerning decidability in quantification theory, as well as concerning the foundation of number theory. The theorem is that relations of natural numbers are reducible in elementary fashion to symmetric ones; i.e.: Theorem I. For every dyadic relation R of natural numbers there is a symmetric dyadic relation H of natural numbers such that R is definable in terms of H plus just truth-functions and quantification over natural numbers . To state the matter more fully, there is a (well-formed) formula ϕ of pure quantification theory, or first-order functional calculus, which meets these conditions: (a) ϕ has ‘ x ’ and ‘ y ’ as sole free individual variables; (b) ϕ contains just one predicate letter, and it is dyadic; (c) for every dyadic relation R of natural numbers there is a symmetric dyadic relation H of natural numbers such that, when the predicate letter in ϕ is interpreted as expressing H , ϕ comes to agree in truth-value with ‘ x bears R to y ’ for all values of ‘ x ’ and ‘ y ’.