Mathematical Logic Quarterly 43 (1):103-120 (1997)

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Abstract
We first note that Gentzen's proof-reduction for his consistency proof of PA can be directly interpreted as moves of Kirby-Paris' Hydra Game, which implies a direct independence proof of the game . Buchholz's Hydra Game for labeled hydras is known to be much stronger than PA. However, we show that the one-dimensional version of Buchholz's Game can be exactly identified to Kirby-Paris' Game , by a simple and natural interpretation . Jervell proposed another type of a combinatorial game, by abstracting Gentzen's proof-reductions and showed that his game is independent of PA. We show that this Jervell's game is actually much stronger than PA, by showing that the critical ordinal of Jervell's game is φω = ϵ0) in the Veblen hierarchy of ordinals.
Keywords Peano arithmetic  Proof‐theoretic independence result  Cut elimination  Combinatorial independent game  Hydra game
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DOI 10.1002/malq.19970430113
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Proof Theory.K. Schütte - 1977 - Springer Verlag.
Proof Theory.Gaisi Takeuti - 1987 - Elsevier.
Proof Theory.Gaisi Takeuti - 1990 - Studia Logica 49 (1):160-161.

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