Infinitary provability logic
TL;DR
A non-well-founded deep inference proof system for the modal language with at most countably infinite conjunctions and disjunctions is developed and it is shown that the infinitary provability logic arising from certain admissible sets lies between the set of theorems of the Hilbert-style calculus and the non-well-founded deep inference system.
Abstract
G\"odel-L\"ob provability logic $\GL$ is a propositional modal system that on one hand enjoys completeness with respect to conversely well-founded Kripke frames and on the other hand captures all modal principles about $\PA$-provability that are provable in $\PA$ itself. In the present paper we carry out an initial investigation into the question of what the infinitary counterpart of $\GL$ is. We develop a non-well-founded deep inference proof system $\dgla$ for the modal language with at most countably infinite conjunctions and disjunctions. We show that the calculus is sound and complete for well-founded transitive Kripke frames. Using Kripke-Platek set theory we develop an interpretation of the infinitary modal language in terms of infinitary provability over admissible sets. Then we show that a natural Hilber-style variant of infinitary $\GL$ is sound for this interpretation. We leave open, however, the question if $\dgla$ proves any additional theorems in comparison with the Hilbert-style calculus. Nevertheless, under certain conditions we do show that the infinitary provability logic arising from certain admissible sets lies between the set of theorems of the Hilbert-style calculus and the non-well-founded deep inference system.