### Title

Universal models for provability logics GLP_{\Lambda} with transfinitely many modalities

### Abstract

Lev Beklemishev based a novel paradigm for ordinal analysis of formal theories on calculations performed in the closed fragment of poly-modal provability logics. He used GLP to carry out $\Pi_n^0$ ordinal analyses of Peano Arithmetic (PA) and subsystems. GLP used $\omega$ many modalities. For theories beyond PA it is natural to work with transfinitely many modalities. The arising logics are still decidable and the closed fragments actually allow a fairly easy decision procedure. Full GLP does not admit Kripke semantics. For the closed fragment however a universal model was given by Ignatiev. In this talk we present this model and discuss how to extend it to the setting with transfinitely many modalities.