Arithmetical and Hyperarithmetical Worm Battles

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