树的Scott谱隙有界

线性序的 Scott 谱隙是有界的

Scott Spectral Gaps are Bounded for Linear Orderings

本文证明了扩展线性序理论的无穷逻辑 $L_{ω_1 ω}$ 中，理论复杂度与最简模型复杂度间的 Scott 谱隙始终以 4 为上界，并给出了新的下界构造。

Games and Scott sentences for positive distances between metric structures

We develop various Ehrenfeucht-Fra\"ıssé games for distances between metric structures. We study two forms of distances: pseudometrics stemming from mapping spaces onto each other with some form of approximate isomorphism, and metrics stemming from measuring the distances between two spaces isometrically embedded into a third space. Using an infinitary version of Henson's positive bounded logic with approximations, we form Scott sentences capturing fixed distances to a given space. The Scott sentences of separable spaces are in $\mathcal{L}_{ω_1ω}$ for 0-distances and in $\mathcal{L}_{ω_2ω}$ f…

Metric Scott analysis

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…

Hjorth Analysis of General Polish Group Actions

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…

一个完整的有界理论与无界类型

A Complete Bounded Theory with Unbounded Types

研究揭示在一阶逻辑中，有界理论可能包含无界类型，挑战了传统认知。

Characterizing the existence of a Borel complete expansion

We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $Φ$ as a class of structures in a related language. From this, we show that $Φ$ has a Borel complete expansion if and only if $S_\infty$ divides $Aut(M)$ for some countable model $M\models Φ$. Using this, we prove that for theories $T_h$ asserting that $\{E_n\}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded then $T_h$ is not Borel complete, providing a converse to Theorem~2.1 of \cite{LU}.