A Comment on Modal Collapse and Ultrafilters in Gödel's Ontological Argument

[2608.07578] Demodalising Gödel's God? Ultrafilters, Rigidity and Modal Collapse Revisited Skip to main content Search Submit Donate Log in Mathematics > Logic arXiv:2608.07578 (math) [Submitted on 4 Aug 2026 ( v1 ), last revised 26 Oct 2026 (this version, v4)] Title: Demodalising Gödel's God? Ultrafilters, Rigidity and Modal Collapse Revisited Authors: Christoph Benzmüller View PDF HTML (experimental) Abstract: Gödel's ontological argument is modal by design. Possibility and necessity are among its central concerns, and connected interpretation decisions -- extensions or intensions for the positive properties, constant or varying quantifier domains, the choice of modal logic, etc. -- subtly shape the argument and, in particular, whether the modal collapse is implied or not. It is tempting to abstract the modality away and to read the positive properties as merely forming an ultrafilter -- which, however, may replace Gödel's argument with a perhaps more manageable alternative that may have little to do with Gödel's original intention. In this spirit, Odifreddi and Gomes argue that modal collapse is an intrinsic structural feature of any theory that characterises positive properties as an ultrafilter and identifies God as its principal generator. This comment shows that the structural claim is false. There is an explicit S5 model, with constant domains, in which positivity is at every world a modal ultrafilter principally generated by a unique and necessarily existing Godlike individual, and in which the modal collapse nevertheless fails -- already for an atomic formula. What drives the collapse is not the ultrafilter structure but the rigidity of positivity -- itself a thoroughly modal condition: given only that positivity is a modal ultrafilter, modal collapse and rigidity are equivalent. Rigidity is thus exactly what the collapse adds to the ultrafilter structure, and the disagreement is whether it is to be counted part of that structure at all. Two further claims of Odifreddi and Gomes are clarified: one on the derivability of Gödel's Theorem IV, one on the extensionality of positivity. All results are machine-checked in Isabelle/HOL, which distinguishes kernel-checked proofs from model-finder output. Comments: 21 pages. Accepted for publication in Logica Universalis (revised version). Companion Isabelle/HOL session included as ancillary files (directory anc/GoedelUltrafilterNote); builds with Isabelle2025-2 in about 15 s. Previous title: "A Comment on Modal Collapse and Ultrafilters in Gödel's Ontological Argument" Subjects: Logic (math.LO) ; Logic in Computer Science (cs.LO) MSC classes: 03A05 (Primary), 03B45, 03B15, 68T15 (Secondary) ACM classes: F.4.1; I.2.3 Cite as: arXiv:2608.07578 [math.LO] (or arXiv:2608.07578v4 [math.LO] for this version) https://doi.org/10.48550/arXiv.2608.07578 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Christoph Benzmüller [ view email ] [v1] Tue, 4 Aug 2026 20:44:18 UTC (54 KB) [v2] Tue, 11 Aug 2026 12:22:20 UTC (59 KB) [v3] Wed, 26 Aug 2026 21:44:05 UTC (60 KB) [v4] Mon, 26 Oct 2026 21:07:01 UTC (93 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : GoedelUltrafilterNote/AltEntailmentVariant.thy GoedelUltrafilterNote/AndersonVariant.thy GoedelUltrafilterNote/BaseDefs.thy GoedelUltrafilterNote/ExtensionalityTests.thy GoedelUltrafilterNote/FittingVariant.thy GoedelUltrafilterNote/GoedelAx1Gen.thy GoedelUltrafilterNote/GoedelInconsistency.thy GoedelUltrafilterNote/HOML.thy GoedelUltrafilterNote/LICENSE GoedelUltrafilterNote/MFilter.thy GoedelUltrafilterNote/README.md GoedelUltrafilterNote/ROOT GoedelUltrafilterNote/ScottVariant.thy GoedelUltrafilterNote/SimpleVariantHF.thy GoedelUltrafilterNote/Th4Underivability.thy GoedelUltrafilterNote/UFilterOnly.thy GoedelUltrafilterNote/UFilterS5.thy GoedelUltrafilterNote/UFilterVariant.thy GoedelUltrafilterNote/audit/Audit.thy GoedelUltrafilterNote/audit/Audit2.thy GoedelUltrafilterNote/audit/README.md GoedelUltrafilterNote/audit/ROOT GoedelUltrafilterNote/audit/isabelle-report-filtered.txt GoedelUltrafilterNote/audit/probe/Probe.thy GoedelUltrafilterNote/audit/probe/p-report-filtered.txt GoedelUltrafilterNote/document/root.bib GoedelUltrafilterNote/document/root.tex (22 additional files not shown) Current browse context: math.LO < prev | next > new | recent | 2026-08 Change to browse by: cs cs.LO math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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. 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