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Metric Scott analysis. (English) Zbl 1390.03039

The article under review brings the Scott analysis from classical logic into the setting of continuous logic. It is proven that Scott sentences for separable structures exist and a classical result of S. Gao [J. Symb. Log. 63, No. 3, 891–896 (1998; Zbl 0922.03045)] characterizing those Scott sentences with only countable models is extended to the continuous setting (with countable replaced by separable). A continuous version of the Lopez-Escobar theorem [E. G. K. Lopez-Escobar, Fundam. Math. 57, 253–272 (1965; Zbl 0137.00701)] is also obtained.
The approach here involves a notion called weak modulus. When one uses the so-called universal weak modulus, one obtains a Scott analysis that describes isomorphism. In this way, the authors prove a continuous analog of the fact that isomorphism is Borel precisely when the Scott ranks are bounded below \(\omega_1\). If, instead, one uses the 1-Lipshitz modulus, then the Scott analysis specializes to familiar distances in analysis, such as Gromov-Hausdorff distance for metric spaces and Kadets distances for Banach spaces. As a result, the authors prove that the set of Polish metric spaces at Gromov-Hausdorff distance \(0\) from a particular Polish metric space is Borel.

MSC:

03C75 Other infinitary logic
03E15 Descriptive set theory
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