arXiv:2608.07578 · v4
Demodalising Gödel's God? Ultrafilters, Rigidity and Modal Collapse Revisited
摘要
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.
作者备注：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"