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.
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Notes
Here, the term ‘open’ is derived from topological consideration rather than the absence of quantifiers.
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Acknowledgements
Isaac Goldbring was partially supported by the National Science Foundation, grant DMS-2054477. We thank the referees for their helpful suggestions and corrections.
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Camrud, C., Goldbring, I. & McNicholl, T.H. On the complexity of the theory of a computably presented metric structure. Arch. Math. Logic 62, 1111–1129 (2023). https://doi.org/10.1007/s00153-023-00884-4
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DOI: https://doi.org/10.1007/s00153-023-00884-4
Keywords
- Computable structure theory
- Continuous logic
- Computable analysis
- Metric structures
- Hyperarithmetical hierarchy