On the complexity of the theory of a computably presented metric structure

Abstract

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form φ M ≤ r, and the open diagram, which encapsulates strict inequalities of the form φ M < r. We show that the closed and open Σ N diagrams are Π 0 N+1 and Σ N respectively, and that the closed and open Π N diagrams are Π 0 N and Σ 0 N+1 respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.

Computably Compact Metric Spaces

“…Finally, the last item (viii) of the theorem is new; it is inspired by [10, Definition 9.9] and the very recent paper [25]. It says that any formula of continuous logic formed in the language of pure metric uniformly defines a computable function M n Ñ [0, 1], where n is the number of free parameters in the formula.…”

“…The significance of this fact is that, in general, even if a computably metrized manifold admits a triangulation, it is not known whether it always admits an arithmetical triangulation. 25 Indeed, even in the seemingly trivial case of compact surfaces, producing an arithmetical triangulation based entirely on the given metric takes some 18 Turing jumps [58]. It is believed that complexity is likely close to being optimal.…”