We show that the modal propositional logic $G$, originally introduced to describe the modality "it is provable that", is also sound for various interpretations using filters on ordinal numbers, for example the end-segment filters, the club filters, or the ineffable filters. We also prove that $G$ is complete for the interpretation using end-segment filters. In the case of club filters, we show that $G$ is complete if Jensen's principle $\square_\kappa$ holds for all $\kappa < \aleph_\omega$; on the other hand, it is consistent relative to a Mahlo cardinal that $G$ be incomplete for the club filter interpretation.