[2608.14488] Strongly relativizing reals Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Logic arXiv:2608.14488 (math) [Submitted on 14 Aug 2026] Title: Strongly relativizing reals Authors: Tyler Arant View a PDF of the paper titled Strongly relativizing reals, by Tyler Arant View PDF HTML (experimental) Abstract: For a real $f\in \mathbb{N}^\mathbb{N}$, it is in general not the case that every set which is both $\Sigma^1_1(f)$ and $\Pi^1_1(f)$ is the $f$-section of a $\Delta^1_1$ set. However, there are reals $f$ for which this, in fact, does happen; we say that such a real strongly relativizes $\Delta^1_1$. In this paper, we will prove that the reals which strongly relativize $\Delta^1_1$ are exactly the hyperlow reals, i.e., the $f\in \mathbb{N}^\mathbb{N}$ with $\omega_1^f=\omega_1^{\text{ck}}$. We will also study reals that strongly relativize the classes $\Delta^0_\alpha$ for $1\leq \alpha<\omega_1^{\text{ck}}$. We characterize the reals that strongly relativize $\Delta^0_1$ subsets of $\mathbb{N}$ as the reals which are computably dominated. For $1\leq \alpha<\omega_1^{\text{ck}}$, we show that there are continuum many reals which strongly relativize $\Delta^0_\alpha$ subsets of $\mathbb{N}$. We will also find non-trivial examples of reals which strongly relativize $\Delta^0_{\alpha}$ ($1\leq \alpha<\omega_1^{\text{ck}}$) subsets of Baire space. Comments: 19 pages Subjects: Logic (math.LO) MSC classes: 03E15 (primary) 03D65 (secondary) Cite as: arXiv:2608.14488 [math.LO] (or arXiv:2608.14488v1 [math.LO] for this version) https://doi.org/10.48550/arXiv.2608.14488 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Tyler Arant [ view email ] [v1] Fri, 14 Aug 2026 17:03:45 UTC (20 KB) Full-text links: Access Paper: View a PDF of the paper titled Strongly relativizing reals, by Tyler Arant View PDF HTML (experimental) TeX Source view license Current browse context: math.LO < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from