Bounds on continuous Scott rank
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- by William Chan and Ruiyuan Chen PDF
- Proc. Amer. Math. Soc. 148 (2020), 3591-3605 Request permission
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 $\hat {\mathscr {L}}\oplus \hat \Omega \oplus \mathcal {D}$.References
- Jon Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer-Verlag, Berlin-New York, 1975. An approach to definability theory. MR 0424560
- Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov, Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315–427. MR 2436146, DOI 10.1017/CBO9780511735219.011
- Itaï Ben Yaacov, Michal Doucha, André Nies, and Todor Tsankov, Metric Scott analysis, Adv. Math. 318 (2017), 46–87. MR 3689736, DOI 10.1016/j.aim.2017.07.021
- William Chan, Bounds on Scott Ranks of Some Polish Metric Spaces, arXiv e-prints (2019), arXiv:1906.04351.
- Michal Doucha, Scott rank of Polish metric spaces, Ann. Pure Appl. Logic 165 (2014), no. 12, 1919–1929. MR 3256743, DOI 10.1016/j.apal.2014.08.002
- Michal Doucha, Erratum to: “Scott rank of Polish metric spaces” [Ann. Pure Appl. Logic 165 (12) (2014) 1919–1929], Ann. Pure Appl. Logic 168 (2017), no. 7, 1490, DOI 10.1016/j.apal.2017.03.006.
- Sy Friedman, Katia Fokina, Martin Koerwien, and Andre Nies, The Scott rank of Polish metric spaces, arXiv e-prints (2019), arXiv:1906.00351.
- R. Björn Jensen, Admissible sets, https://www.mathematik.hu-berlin.de/~raesch/org/jensen.html .
- M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 (1981), no. 2, 301–318. MR 613284, DOI 10.2307/2273623
- Mark Nadel, Scott sentences and admissible sets, Ann. Math. Logic 7 (1974), 267–294. MR 384471, DOI 10.1016/0003-4843(74)90017-5
Additional Information
- William Chan
- Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203
- MR Author ID: 1204234
- Email: william.chan@unt.edu
- Ruiyuan Chen
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- MR Author ID: 1012788
- Email: ruiyuan@illinois.edu
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3591-3605
- MSC (2010): Primary 03C57, 03C75, 03D60, 03E15
- DOI: https://doi.org/10.1090/proc/15056
- MathSciNet review: 4108863