Metric Scott analysis

Abstract

We develop an analogue of the classical Scott analysis for metric structures and infinitary continuous logic. Among our results are the existence of Scott sentences for metric structures and a version of the López-Escobar theorem. We also derive some descriptive set theoretic consequences: most notably, that isomorphism on a class of separable structures is a Borel equivalence relation iff their Scott rank is uniformly bounded below ω 1 . Finally, we apply our methods to study the Gromov-Hausdorff distance between metric spaces and the Kadets distance between Banach spaces, showing that the set of spaces with distance 0 to a fixed space is a Borel set.

Bounds on continuous Scott rank

“…The Scott analysis for continuous logic of [3] is formulated using the back-and-forth pseudo-distance functions r α : α ∈ ON from Definition 4.1. Like the classical Scott analysis, the continuous Scott analysis can be expressed through a certain monotone operator.…”

“…The authors of [3] developed a new Scott analysis for continuous logic for metric structures. If L is a language of continuous logic, then an L -structure of continuous logic is a Polish metric space endowed with a suitable interpretation for each symbol of L .…”

“…An alternative logic that appears more suitable for structures on Polish metric spaces is continuous logic for metric structures. The reader should consult [2] and [3] for more details on continuous logic for metric structures. A language L of continuous logic consists of function, relation, and constant symbols.…”

“…The authors of [3] proposed and developed a Scott analysis for continuous logic for metric structures. They defined an analogous back-and-forth pseudo-distance which depends on one additional object called a weak modulus of continuity Ω.…”

“…From their Scott analysis, they derived a Scott sentence ϕ N so that for all separable L -structure M, M |= (ϕ N = 0) if and only N and M are L -isomorphic in continuous logic, in the case that Ω is a universal weak modulus. See [3] Theorem 3.8 and 5.5 for these results and more details. These results give strong evidence that their theory can justly be called a "Scott analysis" for continuous logic.…”