Vaught's conjecture on analytic sets
英文摘要:
This paper obtains an algebraic characterization of the Polish groups that satisfy Vaught's conjecture on $\bfΣ^1_1$ sets.
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Greg Hjorth
This paper obtains an algebraic characterization of the Polish groups that satisfy Vaught's conjecture on $\bfΣ^1_1$ sets.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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We introduce the $ω$-Vaught's conjecture, a strengthening of the infinitary Vaught's conjecture. We believe that if one were to prove the infinitary Vaught's conjecture in a structural way without using techniques from higher recursion theory, then the proof would probably be a proof of the $ω$-Vaught's conjecture. We show the existence of an equivalent condition to the $ω$-Vaught's conjecture and use this tool to show that all infinitary sentences whose models are linear orders satisfy the $ω$-Vaught's conjecture.
We prove in ZF a recursive-theoretic characterization of the Topological Vaught Conjecture by revisiting the fact that orbits in Polish $G$-spaces are Borel sets.
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 $α\in C$.
The Type $ω$-Vaught's Conjecture for $ω$-stable Theories
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We prove that, for every theory $T$ which is given by an ${\mathcal L}_{ω_1,ω}$ sentence, $T$ has less than $2^{\aleph_0}$ many countable models if and only if we have that, for every $X\in 2^ω$ on a cone of Turing degrees, every $X$-hyperarithmetic model of $T$ has an $X$-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught's conjecture.
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该研究构造了$L_{ω_1,ω}$的一个句子,其可数模型数为$\aleph_1$且无完美非同构模型,反驳了Vaught猜想,相关证明已在Lean中形式化验证。
Sharp Vaught's Conjecture for Some Classes of Partial Orders
本文将 Rubin 关于线性序完全理论的精确版 Vaught 猜想推广至包含线性序且在有限积与不相交并下封闭的最小偏序类,以及有根树扩展和无限线性序不相交并的树理论。
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A structure ${\mathbb Y}$ of a relational language $L$ is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $<$ on the set $Y\setminus F$ such that for each partial automorphism $φ$ (i.e., local automorphism, in Fra\"ıssé's terminology) of the linear order $\langle Y\setminus F, < \rangle$ the mapping ${\mathrm{id}} _F \cup φ$ is a partial automorphism of ${\mathbb Y}$. By a theorem of Fra\"ıssé, if $|L|<ω$, then ${\mathbb Y}$ is almost chainable iff the profile of ${\mathbb Y}$ is bounded; namely, iff there is a positive integer $m$ such that ${\mathbb Y}$ ha…
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Vaught's Conjecture and Theories of Partial Order Admitting a Finite Lexicographic Decomposition
本文研究了偏序理论中的Vaught猜想,证明了某些理论满足VC和VC$^\sharp$的条件。
This paper considers "definable cardinalities" arising from Polish group actions. The first part of the paper answers a question of Becker-Kechris by showing that under suitable determinacy assumptions in ZF+DC, every action by a Polish group on a metric space is either as complicated as the Vitali equivalence relation or no more complicated than the equality relation on bounded subsets of \aleph_1. The same argument shows that if hereditarily wellorderabel sets can be used as complete invariants, then elements in HC will do. The second half gives a new proof of Howard Becker's recent theorem …
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