## Journal of Symbolic Logic

### Infinitary Combinatorics and Modal Logic

Andreas Blass

#### Abstract

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.

#### Article information

Source
J. Symbolic Logic, Volume 55, Issue 2 (1990), 761-778.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743330

Mathematical Reviews number (MathSciNet)
MR1056387

Zentralblatt MATH identifier
0699.03008

JSTOR