Model-Theoretic Games

Abstract

We will begin to answer the question: What languages can be defined with first-order sentences? The answer, of course, depends on what numerical predicates we are allowed to use. Throughout this section we will assume that we are working with first-order formulas with some fixed finite set of numerical predicates and a fixed interpretation I. In the subsequent sections of this chapter we will make specific choices for these numerical predicates and prove some limitations on the power of first-order sentences. For example, we will show that the numerical predicate x < y cannot be defined by a first-order formula in which the only numerical predicates are of the form x = y and y = x + 1, and that there are regular languages that cannot be defined by first-order sentences in which x < y is the only numerical predicate allowed.