Arithmetical and Hyperarithmetical Worm Battles

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 aritmetic $(PA)$ 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 $IΣ_n$ of Peano arithmetic. In order to prove the latter, we show that standard Hardy functions majorize their variants based on tree ordinals.

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