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Abstract We continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $$\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $$\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $$ if and only if the dimension of $$\mathfrak{M}_\beta $$ is at most the dimension of $$\mathfrak{M}_\gamma $$ and that each countable model is isomorphic to some $$\mathfrak{M}_\beta $$ . We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω -stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].
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This content will become publicly available on July 3, 2026
Étale structures and the Joyal–Tierney representation theorem in countable model theory
An étale structure over a topological space X is a continuous family of structures (in some first-order language) indexed over X. We give an exposition of this fundamental concept from sheaf theory and its relevance to countable model theory and invariant descriptive set theory. We show that many classical aspects of spaces of countable models can be naturally framed and generalized in the context of étale structures, including the Lopez-Escobar theorem on invariant Borel sets, an omitting types theorem, and various characterizations of Scott rank. We also present and prove the countable version of the Joyal–Tierney representation theorem, which states that the isomorphism groupoid of an étale structure determines its theory up to bi-interpretability; and we explain how special cases of this theorem recover several recent results in the literature on groupoids of models and functors between them.
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- Award ID(s):
- 2224709
- PAR ID:
- 10638115
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- The Bulletin of Symbolic Logic
- ISSN:
- 1079-8986
- Page Range / eLocation ID:
- 1 to 43
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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