Effective metric model theory

Abstract

This paper is a further investigation of a project carried out in Didehvar and Ghasemloo (2009) to study effective aspects of the metric logic. We prove an effective version of the omitting types theorem. We also present some concrete computable constructions showing that both the separable atomless probability algebra and the rational Urysohn space are computable metric structures.

An effective version of definability in metric model theory

“…In the following, the concepts of computable and decidable metric structures are explained. This approach to study the effectiveness of the metric structures is firstly introduced in [12].…”

“…It means p ∈ Σ ω is a name for a continuous function η p :⊆ Σ ω → Σ ω with a G δ -domain which on input q returns the value η p (q). For more details of this representation, see [6] and [12].…”

“…Next, the notion of an effectively presented language L and then a computable and a decidable L-structure will be established [12]. M, d, A, α).…”

“…In section 2.3, an implementation of TTE to study effectiveness of metric model theory is expressed. These definitions first appeared in [12].…”

Computable structure theory of continuous logic

“…In the 2010s, three projects were undertaken on effectivity with respect to continuous logic and metric structures. Didehvar and Pourmahdian completed joint work with Ghasemloo (13) and Tavana (28), and Moody wrote a Ph.D. thesis (26). The first two of these implicitly used computable presentations, while the last did so somewhat explicitly.…”