Prime and atomic models

Abstract

This paper gives some simple existence results on prime and atomic models over sets. It also contains an example in which there is no prime model over a certain set even though there is an atomic model over the set. The existence results are “local” in that they deal with just one set rather than all sets contained in models of some theory. For contrast, see the “global” results in [6] or [7, p. 200].Throughout the paper, L is a countable language, and T is a complete L-theory with infinite models. There is a “large” model of T that contains the set X and any other sets and models to be used in a particular construction of a prime or atomic model over X.A model is said to be prime over X if and every elementary monomorphism on X can be extended to an elementary embedding on all of . This notion is used in a variety of ways in model theory. It aids in distinguishing between models that are not isomorphic, as in Vaught [10]. It also aids in showing that certain models are isomorphic, as in Baldwin and Lachlan [1].