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Effective Mathematics of the Uncountable by Joel David Hamkins, Denis Hirschfeldt, Noam Greenberg

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EFFECTIVE MODEL THEORY: AN APPROACH VIA Σ-DEFINABILITY

ALEXEY STUKACHEV

Abstract We present an approach to abstract computability based on the notion of Σ-definability, and survey some results on effective model theory which are obtained within this framework.

§1. Introduction. We look at computability over abstract structures via the formalism based on the notion of Σ-definability in admissible sets and, in particular, in hereditarily finite (HF) superstructures. HF-computability, as well as some other different (yet equivalent) approaches to computability over abstract structures provide a natural framework for considering “computability” (“effectiveness”) relative to this structure. A survey of results on HF-computability in general can be ...

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