Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1907.00284

The large cardinal strength of Weak Vopěnka's Principle

Trevor M. Wilson
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that Weak Vopěnka's Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class C there is a C-strong cardinal. Weak Vopěnka's Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing C-strongness of some cardinal.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Trevor M. Wilson. 2020-01-23. The large cardinal strength of Weak Vopěnka's Principle. https://arxiv.org/abs/1907.00284

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

KEEP EXPLORING

Related papers

arXiv2026

On Borel sets in ideal topologies

We study the Borel and analytic subsets of the spaces ${}^κκ$ and ${}^κ2$ endowed with ideal topologies, where $κ$ is a regular uncountable cardinal, thereby addressing some open problems of the literature. We provide a systematic analysis of the Borel hierarchy for an arbitrary ideal topology. In particular, we formulate a sufficient condition ensuring that the hierarchy does not collapse, demonstrate that every Borel set in such a topology is analytic, and establish the existence of a set that is not Borel. Our main result shows that, when the underlying ideal contains an unbounded subset, the collection of analytic sets coincides with the full power set of the ambient space. Finally, we prove that the Approximation Lemma holds in the setting of ideal topologies.

math.LO↗
arXiv2026

Inquisitive first-order logic is neither compact nor recursively axiomatizable

Inquisitive first-order logic is an extension of classical first-order logic with formulas regimenting first-order questions, such as "whether all objects are P", "which objects are P", and "what is one object that is P". Since it was first developed in 2009, two major meta-theoretical questions about this logic have remained open, in spite of significant efforts. The first concerns compactness: if a conclusion follows from a set of premises, does it always follow from some finite subset? The second concerns the computational status of validity: is the set of validities recursively enumerable, or equivalently, does the logic admit a recursive axiomatization? We settle both questions in the negative, showing that inquisitive first-order logic is neither compact nor recursively axiomatizable. Furthermore, we prove that it violates another signature property of first-order logic, namely, Craig interpolation. We discuss the significance of our results, and show how to extend them to a closely related logic, viz., inquisitive team logic.

math.LO↗
arXiv2026

The Borel Distinguishing Number of Schreier Graphs

The Borel distinguishing number $D_B(\mathcal{G})$ of a Borel graph $\mathcal{G}$, recently introduced by Bilge and Kaya, is the minimum number of colors required to break the symmetry of $\mathcal{G}$ in a Borel way. In this paper, we investigate the Borel distinguishing number of Schreier graphs induced by the free part of the shift action $Γ\curvearrowright n^Γ$. We prove that $D_B(\mathcal{G})\le n+1$ for $Γ=\mathbb{Z}^d$ equipped with the standard generators. Moreover, we show that $D_B(\mathcal{G})\ge n+1$ if $ Γ$ is amenable and $ \{γ\in \mathrm{Aut}(\mathrm{Cay}(Γ,S)) \mid γ(e) = e \}$ is non-trivial. We also show that $D_B(\mathcal{G})$ is finite if $Γ$ is finitely generated, and give some applications of our results. These results answer some questions raised by Bilge and Kaya.

math.LO↗