Hyperarithmetical Worm Battles

Abstract

Japaridze's provability logic GLP has one modality [n] for each natural number and has been used by Beklemishev for a proof theoretic analysis of Peano arithmetic (PA) and related theories. Among other benefits, this analysis yields the so-called Every Worm Dies (EWD) principle, a natural combinatorial statement independent of PA. Recently, Beklemishev and Pakhomov have studied notions of provability corresponding to transfinite modalities in GLP. We show that indeed the natural transfinite extension of GLP is sound for this interpretation, and yields independent combinatorial principles for the second order theory ACA of arithmetical comprehension with full induction. We also provide restricted versions of EWD related to the fragments IΣn of Peano arithmetic.