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

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 \(\phi ^\mathcal {M}\le r\), and the open diagram, which encapsulates strict inequalities of the form \(\phi ^\mathcal {M}< r\). We show that the closed and open \(\Sigma _N\) diagrams are \(\Pi ^0_{N+1}\) and \(\Sigma ^0_N\) respectively, and that the closed and open \(\Pi _N\) diagrams are \(\Pi ^0_N\) and \(\Sigma ^0_{N + 1}\) 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.