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Abstract

The subject of this paper is a topic in that branch of universal algebra called the theory of models. The main trait that distinguishes model theory from other approaches to algebra is the fact that in model theory the language in which theorems and definitions are to be coined is explicitly, indeed formally, specified. This gives, of course, a peculiar slant to the type of problems that are of immediate interest to model theorists. The generalities in which we are interested concern the exact extent of definability of mathematical notions and the characterizability of types of mathematical structures in various formal languages, the existence of structures with particular properties in formally characterizable classes of structures, formal descriptions of types of properties preserved under various mathematical constructions, and the like. In a nutshell, the difference between an algebraist and a model theorist is the following: To an algebraist two mathematical structures A, ℬ are “essentially the same” if they are isomorphic,

$$ A\, \cong \,B, $$

while to a model theorist they are essentially the same in case they are elementarily equivalent,

$$ A\, \equiv \,B, $$

(if A and ℬ have the same first-order properties).

With partial support from NSF Grant GP 1612.

Received September 16, 1965.