Countable models of weakly quasi-o-minimal theories I

TL;DR

This paper explores the properties of trivial and order-trivial types in weakly o-minimal theories, establishing their behavior and implications for the number of countable models in such theories.

Contribution

It introduces the concepts of triviality and order-triviality for types, proves their equivalence in weakly o-minimal types, and shows that theories with a definable shift have continuum many countable models.

Findings

Triviality is preserved under nonforking extensions.

In o-minimal theories, all complete 1-types are trivial.

Weakly quasi-o-minimal theories with a definable shift have continuum many countable models.

Abstract

We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global extensions of a weakly o-minimal type, in which case we say that the type is trivial. In the o-minimal case, we prove that every definable complete 1-type over a model is trivial. We prove that the triviality has several favorable properties; in particular, it is preserved in nonforking extensions of a weakly o-minimal type and under weak nonorthogonality of weakly o-minimal types. We introduce the notion of a shift in a linearly ordered structure that generalizes the successor function. Then we apply the techniques developed to prove that every weakly quasi-o-minimal theory that admits a definable shift has $2^{\aleph_0}$ countable models.