Arithmetical and Hyperarithmetical Worm Battles

Abstract

Abstract Japaridze’s provability logic ${\operatorname {GLP}}$ has one modality $[n]$ for each natural number and has been used by Beklemishev for a proof theoretic analysis of Peano arithmetic (${\operatorname {PA}}$) and related theories. Among other benefits, this analysis yields the so-called Every Worm Dies (${\operatorname {EWD}}$) principle, a natural combinatorial statement independent of ${\operatorname {PA}}$. Recently, Beklemishev and Pakhomov have studied notions of provability corresponding to transfinite modalities in ${\operatorname {GLP}}$. We show that indeed the natural transfinite extension of ${\operatorname {GLP}}$ is sound for this interpretation and yields independent combinatorial principles for the second-order theory ${\operatorname {ACA}}$ of arithmetical comprehension with full induction. We also provide restricted versions of ${\operatorname {EWD}}$ related to the fragments ${\operatorname {I\varSigma }}_n$ of PA. In order to prove the latter, we show that standard Hardy functions majorize their variants based on tree ordinals.