Stationarily ordered types and the number of countable models

TL;DR

This paper introduces stationarily ordered types and theories, generalizing weak o-minimality, and demonstrates their properties, invariants, and applications to Vaught's conjecture in binary theories with few countable models.

Contribution

It defines stationarily ordered types and theories, analyzes forking and non-orthogonality, and proves Vaught's conjecture for binary, stationarily ordered theories.

Findings

Forking is an equivalence relation on stationarily ordered types.

Invariants of non-orthogonal types are closely related.

Countable models are classified by sequences of invariants in certain theories.

Abstract

We introduce notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the first is a relaxed version of weak o-minimality localized at the locus of a single type. We show that forking, as a binary relation on elements realizing stationarily ordered types, is an equivalence relation and that each stationarily ordered type in a model determines some order-type as an invariant of the model. We study weak and forking non-orthogonality of stationarily ordered types, show that they are equivalence relations and prove that invariants of non-orthogonal types are closely related. The developed techniques are applied to prove that in the case of a binary, stationarily ordered theory with fewer than $2^{\aleph_0}$ countable models, the isomorphism type of a countable model is determined by a certain sequence of invariants of the model. In particular, we confirm Vaught's conjecture for binary, stationarily ordered theories.