Bounds on continuous Scott rank

Chan, William; Chen, Ruiyuan

doi:10.1090/proc/15056

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 $\hat {\mathscr {L}}\oplus \hat \Omega \oplus \mathcal {D}$.

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