Full Satisfaction Classes for Sequential Models

This talk reports on results obtained in collaboration with Ali Enayat. I present the basics of satisfaction classes and briefly indicate the various pitfalls and fallacies surrounding them. I briefly sketch some of the fundamental results in the area. Consider a sequential model M and a designated interpretation N of the natural numbers in that model. A satisfaction class S for M is *full* iff S satisfies the Tarski conditions for all formulas (including the non-standard ones) of the language of M according to (M,N). Our main result is as follows. Any sequential model M with natural numbers N (satisfying a few conditions) has an elementary extension M* with a full satisfaction class S. We can do a bit better by adding some natural extra conditions for S. I will discuss the main idea of the proof.