A Comment on Modal Collapse and Ultrafilters in G\"odel's Ontological Argument

Abstract

G\"odel'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 G\"odel's argument with a perhaps more manageable alternative that may have little to do with G\"odel'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, with machine-verified counterexamples, that the structural claim does not hold: what drives the collapse is primarily not the filter structure but the rigidity of positivity -- itself a thoroughly modal condition. Moreover, two claims of Odifreddi and Gomes are corrected: one on the derivability of G\"odel's Theorem IV, one on the extensionality of positivity. All results are machine-checked in Isabelle/HOL.