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 $Ω$ be a weak modulus of uniform continuity with code $\hatΩ$. Let $\mathcal{D}$ be a countable $\mathscr{L}$-pre-structure. Let $\bar{\mathcal{D}}$ denote the completion structure of $\mathcal{D}$. Then $\mathrm{SR}_Ω(\bar{D}) \leq ω_1^{\hat{\mathscr{L}}\oplus\hatΩ\oplus\mathcal{D}}$, the Church-Kleene ordinal relative to $\hat{\mathscr{L}}\oplus\hatΩ\oplus\mathcal{D}$.
The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of $\mathcal{L}_{ω_1 ω}$) is the set of Scott ranks of countable models of that theory. In $ZFC + PD$ we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular $\boldsymbolΣ^1_1$ classes of ordinals. Our investigation of Scott spectra leads to the resolution (in $ZFC$) of a number of open problems about Scott ranks…
We show that if $γ\in ω\cup \{\aleph_{0}\}$ and $\mathcal{A}$ is a counterexample to the analytic Vaught conjecture having exactly $γ$ many models of Scott rank $ω_{1}$, then there exists a club $C \subseteq ω_{1}$ such that $\mathcal{A}$ has exactly $γ$ many models of Scott rank $α$, for each $α\in C$.
Following the work of Friedman, Koerwien, Nies and Schlicht we positively answer their question whether the Scott rank of Polish metric spaces is countable.
If $\mathcal{N}$ is a proper Polish metric space and $\mathcal{M}$ is any countable dense submetric space of $\mathcal{N}$, then the Scott rank of $\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most $ω_1^{\mathcal{M}} + 1$, where $ω_1^{\mathcal{M}}$ is the Church-Kleene ordinal of $\mathcal{M}$ (construed as a subset of $ω$) which is the least ordinal with no presentation on $ω$ computable from $\mathcal{M}$. If $\mathcal{N}$ is a rigid Polish metric space and $\mathcal{M}$ is any countable dense submetric space, then the Scott rank of $\mathcal{…
We prove that every quasi-Hopfian finitely presented structure $A$ has a $d$-$Σ_2$ Scott sentence, and that if in addition $A$ is computable and $Aut(A)$ satisfies a natural computable condition, then $A$ has a computable $d$-$Σ_2$ Scott sentence. This unifies several known results on Scott sentences of finitely presented structures and it is used to prove that other not previously considered algebraic structures of interest have computable $d$-$Σ_2$ Scott sentences. In particular, we show that every right-angled Coxeter group of finite rank has a computable $d$-$Σ_2$ Scott sentence, as well a…
Let $\mathscr{L}$ be a recursive language. Let $S(\mathscr{L})$ be the set of $\mathscr{L}$-structures with domain $ω$. Let $Φ: {}^ω2 \rightarrow S(\mathscr{L})$ be a $Δ_1^1$ function with the property that for all $x,y \in {}^ω2$, $ω_1^x = ω_1^y$ if and only if $Φ(x) \approx_{\mathscr{L}} Φ(y)$. Then there is some $x \in {}^ω2$ so that $\mathrm{SR}(Φ(x)) = ω_1^x + 1$.
The Scott process of a relational structure $M$ is the sequence of sets of formulas given by the Scott analysis of $M$. We present axioms for the class of Scott processes of structures in a relational vocabulary $τ$, and use them to give a proof of an unpublished theorem of Leo Harrington from the 1970's, showing that a counterexample to Vaught's Conjecture has models of cofinally many Scott ranks below $ω_{2}$. Our approach also gives a theorem of Harnik and Makkai, showing that if there exists a counterexample to Vaught's Conjecture, then there is a counterexample whose uncountable models ha…
We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank $ω_1^{CK}$, the computable infinitary theory is $\aleph_0$-categorical. Millar and Sacks asked whether this was always the case. We answer this question by constructing an example whose computable infinitary theory has non-isomorphic countable models. The standard known computable structures of Scott rank $ω_1^{CK}+1$ have infinite indiscernible sequences. We give two constructions with no indiscernible ordered triple.
We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $ω$ and that non-standard models of true arithmetic must have Scott rank greater than $ω$. Other than that there are no restrictions. By giving a reduction via $Δ^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $ω$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $α>ω$ is realized as the Scott rank of a model of $T$.
Given a countable scattered linear order $L$ of Hausdorff rank $α< ω_1$ we show that it has a $d\text{-}Σ_{2α+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $α< ω_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $α$ the class of Hausdorff rank $α$ linear orders is $\pmb Σ_{2α+2}$ complete.
Hjorth has introduced a Scott analysis for general Polish group actions, and has asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank - namely, the orbit equivalence relation is Borel if and only if Hjorth ranks are bounded. We present the principles of Hjorth analysis and Hjorth rank, and answer Hjorth's question positively. From that we get a positive answer to a conjecture due to Hjorth - for every limit ordinal $α$ , the set of elements whose orbit is of complexity less than $α$ is a Borel set. We then show Nadel's theorem for Hjorth rank - th…
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 Lopez-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 s…
We study continuous actions of Polish groups on Polish spaces. We develop Scott analysis introduced by Hjorth for studying orbit equivalence relations. We define eventually open actions and prove that this property characterizes the actions endowed with a complete system of hereditarily countable invariant structures.
