Bounds on continuous Scott rank

@article{Chan2020BoundsOC,
  title={Bounds on continuous Scott rank},
  author={William Chan and Ruiyuan Chen},
  journal={Proceedings of the American Mathematical Society},
  year={2020}
}
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 $\mathscr{L}$ 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… 
3 Citations
Polish G-spaces, the generalized model theory and complexity
Given Polish space ${\bf Y}$ and continuous language $L$ we study the corresponding logic $\mathsf{Iso}({\bf Y})$-space ${\bf Y}_L$. We build a framework of generalized model theory towards analysis
AN INTRODUCTION TO THE SCOTT COMPLEXITY OF COUNTABLE STRUCTURES AND A SURVEY OF RECENT RESULTS
Abstract Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a
Bounds on Scott ranks of some polish metric spaces
TLDR
The Scott rank of [Formula: see text] is countable and in fact less than the Church–Kleene ordinal in the natural first-order language of metric spaces.

References

SHOWING 1-10 OF 12 REFERENCES
The Scott rank of Polish metric spaces
We study the usual notion of Scott rank but in the setting of Polish metric spaces. The signature consists of distance relations: for each rational $q > 0$, there is a relation $R_{<q}(x,y)$ stating
An example concerning Scott heights
  • M. Makkai
  • Mathematics
    Journal of Symbolic Logic
  • 1981
Unless otherwise stated, every structure in this paper is countable in a countable, actually recursive language and every formula is one of . The definition of the so-called canonical Scott-sentence
Bounds on Scott ranks of some polish metric spaces
TLDR
The Scott rank of [Formula: see text] is countable and in fact less than the Church–Kleene ordinal in the natural first-order language of metric spaces.
Model theory with applications to algebra and analysis
Preface List of contributors 1. Conjugacy in groups of finite Morley rank Olivier Frecon and Eric Jaligot 2. Permutation groups of finite Morley rank Alexandre Borovik and Gregory Cherlin 3. A survey
Scott rank of Polish metric spaces
Model Theory with Applications to Algebra and Analysis: Model theory for metric structures
A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally there are functions (of several variables) between
Scott sentences and admissible sets
Admissible Sets and Structures: An Approach to Definability Theory
...
...