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.KeywordsNormal FormFinite IndexAtomic FormulaRegular LanguageWinning StrategyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.