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

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BOREL STRUCTURES: A BRIEF SURVEY

ANTONIO MONTALBÁN AND ANDRÉ NIES

Abstract We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. We discuss the main open questions in the area.

§1. Introduction. When looking at structures of size the continuum from an effective viewpoint, the following definition is a natural generalization of ideas from computable model theory.

DEFINITION 1.1. Let X be either 2ω, ωω or image, and let ...

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