Approximate categoricity in continuous logic

Archive for Mathematical Logic:1-31 (forthcoming)
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Abstract

We explore approximate categoricity in the context of distortion systems, introduced in our previous paper (Hanson in Math Logic Q 69(4):482–507, 2023), which are a mild generalization of perturbation systems, introduced by Yaacov (J Math Logic 08(02):225–249, 2008). We extend Ben Yaacov’s Ryll-Nardzewski style characterization of separably approximately categorical theories from the context of perturbation systems to that of distortion systems. We also make progress towards an analog of Morley’s theorem for inseparable approximate categoricity, showing that if there is some uncountable cardinal $$\kappa $$ such that every model of size $$\kappa $$ is ‘approximately saturated,’ in the appropriate sense, then the same is true for all uncountable cardinalities. Finally we present some examples of these phenomena and highlight an apparent interaction between ordinary separable categoricity and inseparable approximate categoricity.

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Metric spaces are universal for bi-interpretation with metric structures.James Hanson - 2023 - Annals of Pure and Applied Logic 174 (2):103204.
On perturbations of continuous structures.Itaï Ben Yaacov - 2008 - Journal of Mathematical Logic 8 (2):225-249.
Uncountable Dense Categoricity in Cats.Itay Ben-Yaacov - 2005 - Journal of Symbolic Logic 70 (3):829 - 860.
Approximate isomorphism of metric structures.James E. Hanson - forthcoming - Mathematical Logic Quarterly.

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