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 ℝ, and let C be a (complexity) class of relations on X. A C-presentation of a structure A is a tuple of relations S = (D, E, R1,…, Rn) such that
о All D, E, R1,…, Rn are in C;
о D ⊆ X and E is an equivalence relation on D (D is called the domain);
о R1, …, Rn are relations compatible with E.
S is a C-representation of A if A ≅ S/E. When E is the identity on D, we say that S is an injective C-presentation of A.
There are various possible choices for C. In this paper we concentrate on the case that C is the class of Borel relations. Given a topological space X as above, the σ-algebra of Borel sets is the smallest σ-algebra containing the open sets.