Abstract

We shall give a direct proof of the independence result of a Buchholz style-Hydra Game on labeled finite trees. We shall show that Takeuti-Arai's cut-elimination procedure of $(\Pi^{1}_{1}-CA) + BI$ and of the iterated inductive definition systems can be directly expressed by the reduction rules of Buchholz's Hydra Game. As a direct corollary the independence result of the Hydra Game follows.