Abstract
We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the density character of the set instead of the cardinality, is \({\aleph_0}\). In these settings we prove an analogue of Morley’s categoricity transfer theorem. We also give concrete examples of homogeneous MAECs.
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Å. Hirvonen, supported by the Finnish Academy of Science and Letters (Vilho, Yrjö ja Kalle Väisälän rahasto) and the graduate school MALJA. T. Hyttinen, partially supported by the Academy of Finland, grant 1106753.
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Hirvonen, Å., Hyttinen, T. Categoricity in homogeneous complete metric spaces. Arch. Math. Logic 48, 269–322 (2009). https://doi.org/10.1007/s00153-009-0120-z
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DOI: https://doi.org/10.1007/s00153-009-0120-z