Stationarily ordered types and the number of countable models

Annals of Pure and Applied Logic 171 (3):102765 (2020)
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Abstract

We introduce the notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the former 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 techniques developed are applied to prove that in the case of a binary, stationarily ordered theory with fewer than 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.

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References found in this work

On dp-minimal ordered structures.Pierre Simon - 2011 - Journal of Symbolic Logic 76 (2):448 - 460.
Witnessing Dp-Rank.Itay Kaplan & Pierre Simon - 2014 - Notre Dame Journal of Formal Logic 55 (3):419-429.
End extensions and numbers of countable models.Saharon Shelah - 1978 - Journal of Symbolic Logic 43 (3):550-562.

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