On local and non-local properties

This chapter presents a method that is an analysis of first-order formulas in terms of local properties. A natural simple metric is used in model and the concept of a k-local formula is defined, where k is any natural number. The main theorem asserts that every first-order sentence, ϕ, is logically equivalent to a Boolean combination of sentences that assert, each, something of the following form: There exist “s” disjoint r-neighborhoods, each satisfying the r-local formula Ψ. If ϕ is a formula, one has to add to the combination r-local formulas in the free variables of ϕ. The theorem is proved by quantifier elimination.