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 $$\phi ^\mathcal {M}\le r$$, 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.