Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.04172

Friedman--Stanley cofinality without Borel completeness

Ruijun Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Friedman and Stanley asked whether an invariant Borel class of countable structures must be Borel complete whenever every countable level of their hierarchy reduces to it. We construct a counterexample in one fixed countable language. The structural counterexample and its principal supporting results have been formalized in Lean~4 and it is available on github.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ruijun Wang. 2026-11-03. Friedman--Stanley cofinality without Borel completeness. https://arxiv.org/abs/2611.04172

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Amenable C-sequences and ineffability in L

We give a relative consistency counterexample to the conjecture of Inamdar and Rinot that every regular uncountable cardinal carrying no amenable C-sequence is ineffable in the constructible universe L. The existence of a $Π^2_1$-indescribable cardinal is equiconsistent with the existence of a $Π^2_1$-indescribable cardinal carrying no amenable C-sequence and not ineffable in L. For every ground-model C-sequence on a weakly compact cardinal $κ$, generalized Hechler forcing adds a club D with $D\capβ\subseteq C_β$ at stationarily many inaccessible indices. We use this capture theorem in an iteration of length $κ^+$ after Johnstone's local lottery preparation.

math.LO↗
arXiv2026

Omitting Types for Infinitary Real-Valued Logic: A Topological Perspective

For countable vocabularies, we extend the omitting types theorem for countable fragments of infinitary logic to Keisler's general real-valued structures by proving that the space of Henkin types of each such fragment is Čech-complete. The structures interpret function symbols arbitrarily and predicate symbols by bounded real-valued functions, while each formula carries a bound of its own. This extends the setting of Eagle's theorem from metric structures with a prescribed master metric to general structures whose vocabulary prescribes the sorts and bounds of the symbols. Following Keisler and Miller's treatment of logic without equality, we build models from closed terms. Morley's Baire category method then gives every satisfiable theory on a countable fragment a model with countable sorts omitting any countable family of types whose closed-term instances have empty interior in the space of Henkin types satisfying the theory.

math.LO↗
arXiv2026

A Side Condition Principle With Applications

We introduce a combinatorial principle $\textsf{SC}(κ)$, for any regular cardinal $κ\ge ω_2$, which asserts the existence of a family $\mathcal{Y} \subseteq [H(κ)]^ω$ of elementary substructures of $H(κ)$ which is closed under finite intersections and satisfies that for stationarily many $N \in \mathcal{Y}$, if $M \in \mathcal{Y}$ and $M \cap ω_1 < N \cap ω_1$, then $M \cap N \cap κ\in N$. We explore the idea of defining forcings with side conditions consisting of finite subsets of such a family $\mathcal{Y}$. If $\textsf{SC}(ω_2)$ holds, then there exists a thin stationary subset of $[ω_2]^ω$, and consequently $\textsf{SC}(ω_2)$ is independent of ZFC. Assuming that $μ$ is a Mahlo cardinal, the Lévy collapse $\mathrm{Col}(ω,<μ)$ forces that $\textsf{SC}(κ)$ holds for all regular cardinals $κ\ge ω_2$. Local instances of $\textsf{SC}$ can be forced using matrices of models or derived as consequences of square principles. We apply the principle $\textsf{SC}$ to define two kinds of forcings: a streamlined version of a strongly proper collapse, and a forcing which is strongly proper on a stationary set and adds arbitrarily many Baumgartner clubs. Finally, we prove two consistency results using forcings derived from $\textsf{SC}$ related to Aronszajn trees, the guessing model principle, strong failures of club guessing, and the higher dominating number.

math.LO↗