Bounds on Continuous Scott Rank

An analog of Nadel’s effective bound for the continuous Scott rank of metric structures, developed by Ben Yaacov, Doucha, Nies, and Tsankov, will be established: Let $\mathcal{L}$ be a language of continuous logic with code $\widehat{\mathcal{L}}$. Let $\mathrm{\Omega }$ be a weak modulus of uniform continuity with code $\widehat{\mathrm{\Omega }}$. Let $\mathcal{D}$ be a countable $\mathcal{L}$-pre-structure. Let $\bar{\mathcal{D}}$ denote the completion structure of $\mathcal{D}$. Then $$\mathrm{S}\mathrm{R}$_\Omega(\bar{D}) \leq \omega_1^{\hat{\mathscr{L}}\oplus\hat{\Omega}\oplus$\mathcal{D}$}$, the Church-Kleene ordinal relative to $\widehat{\mathcal{L}}\oplus \widehat{\mathrm{\Omega }}\oplus \mathcal{D}$.