Speaker
Fedor Pakhomov
Ghent University
Talks at this conference:
Tuesday, 17:45, J336 
The Logic of Correct Models 
Authors: Juan Aguilera and Fedor Pakhomov In his wellknown paper [4] Robert Solovay proved that the provability logic of \(\mathsf{PA}\) is the GödelLöb logic \(\mathsf{GL}\). Also in the same paper Solovay considered some settheoretic interpretations of provability logic which lead to stronger provability logics. Latter Japaridze [3] proved arithmetical completeness theorem for the polymodal provability logic \(\mathsf{GLP}\), where for a modernized version [2] of his result \([n]\phi\) is interpreted as “the sentence \(\phi\) is provable in \(\mathsf{PA}\) extended by all true \(\Pi_n\)sentences.” In our work [1] we investigated the set theoretic interpretation of \([n]\), where \([n]\phi\) mean “the sentence \(\phi\) is true in all \(\Sigma_{n+1}\)correct transitive sets.” Assuming Gödel’s axiom \(V = L\) we prove that the set of polymodal formulas valid under this interpretation is precisely the set of theorems of the logic \(\mathsf{GLP}.3\). Here \(\mathsf{GLP}.3\) is the extension of \(\mathsf{GLP}\) by the axioms of linearity for all \([n]\). We also show that this result is not provable in \(\mathsf{ZFS}\), so the hypothesis \(V = L\) cannot be removed. As part of the proof, we derive the following purely modallogical results which are of independent interest: the logic \(\mathsf{GLP}.3\) coincides with the logic of closed substitutions of \(\mathsf{GLP}\), and is the maximal normal extension of \(\mathsf{GLP}\) that doesn’t prove \([n]\bot\), for any \(n\). Bibliography
