Near model completeness and 0-1 laws

We work throughout in a finite relational language L. Our aim is to analyze in as purely a model-theoretic context as possible some recent results of Shelah et al in which 0 − 1-laws for random structures of various types are proved by a specific kind of quantifier elimination: near model completeness. In Section 2 we describe the major results of these methods ([12], [11] etc.) and some of their context. In Section 3 we describe the framework in which these arguments can be carried out and prove one form of the general quantification elimination argument. We conclude the section by sketching a general outline of the proof of a 0−1 law. The hypotheses of this theorem have a ‘back and forth’ character. Establishing the ‘forth’ part depends heavily on probability computations and is not expounded here. The ‘back’ part is purely model theory. Section 4 carries out the ‘back’ portion of the proof in one context with some simplification from Shelah’s original version.