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- W3172934626 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." @default.
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- W3172934626 date "2023-06-09" @default.
- W3172934626 modified "2023-09-26" @default.
- W3172934626 title "On the complexity of the theory of a computably presented metric structure" @default.
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- W3172934626 doi "https://doi.org/10.1007/s00153-023-00884-4" @default.
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