Countable models of weakly quasi-o-minimal theories I
Slavko Moconja, Predrag Tanović
TL;DR
The paper develops a framework for countable-model analysis in weakly quasi-o-minimal theories by introducing triviality and order-triviality for global invariant types and linking them to weakly o-minimal types. It introduces shifts in linearly ordered structures and proves that the existence of a definable shift forces the maximal number of countable models, $I(T,\aleph_0)=2^{\aleph_0}$. Key contributions include showing the equivalence of triviality notions for weakly o-minimal types, establishing stability of triviality under nonforking and weak nonorthogonality, and using semiintervals and shifts to derive model-count results. The results provide a broad mechanism to diagnose when a weakly quasi-o-minimal theory has many countable models, advancing Vaught's conjecture in this tame setting. The paper also offers detailed constructions and examples to illustrate the phenomena across o-minimal, weakly o-minimal, and almost weakly o-minimal theories.
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.