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Bounds on continuous Scott rank


Authors: William Chan and Ruiyuan Chen
Journal: Proc. Amer. Math. Soc. 148 (2020), 3591-3605
MSC (2010): Primary 03C57, 03C75, 03D60, 03E15
DOI: https://doi.org/10.1090/proc/15056
Published electronically: May 11, 2020
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Abstract: An analog of Nadel's effective bound for the continuous Scott rank of metric structures, developed in [Adv. Math. 318 (2017), pp. 46-87], will be established: Let $ \mathscr {L}$ be a language of continuous logic with code $ \hat {\mathscr {L}}$. Let $ \Omega $ be a weak modulus of uniform continuity with code $ \hat \Omega $. Let $ \mathcal {D}$ be a countable $ \mathscr {L}$-pre-structure. Let $ \bar {\mathcal {D}}$ denote the completion structure of $ \mathcal {D}$. Then $ \mathrm {SR}_\Omega (\bar {D}) \leq \omega _1^{\hat {\mathscr {L}}\oplus \hat \Omega \oplus \mathcal {D}}$, the Church-Kleene ordinal relative to $ \hat {\mathscr {L}}\oplus \hat \Omega \oplus \mathcal {D}$.


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Additional Information

William Chan
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
Email: william.chan@unt.edu

Ruiyuan Chen
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: ruiyuan@illinois.edu

DOI: https://doi.org/10.1090/proc/15056
Received by editor(s): August 19, 2019
Received by editor(s) in revised form: January 2, 2020
Published electronically: May 11, 2020
Additional Notes: The first author was supported by NSF grant DMS-1703708.
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2020 American Mathematical Society