Submission date: 2 February 2023

Abstract:

A complete theory T has the Schröder-Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and probability algebras with or without atoms. We also study a weaker notion, the SB-property up to perturbations. This property holds if any two bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. Finally we study how the SB-property behaves with respect to randomizations.

Mathematics Subject Classification: 03C45, 03C66

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Full text arXiv 2302.01220: pdf, ps.