Countable models of weakly quasi-o-minimal theories I

2024/12/29 by Slavko Moconja, Moconja, Slavko, Predrag Tanović +1
Mathematics · Computer Science · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2412.20589

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^ℵ₀ countable models.

Related

• Simple product and locally o-minimal theories
• On minimal flows. definably amenable groups, and o-minimality
• The Borel Complexity of Isomorphism for O-Minimal Theories
• Constructing models of small ordered theories with maximal countable spectrum
• The Schröder-Bernstein property for weakly minimal theories