Introduction and Proof of the Goodstein Sequence and Hydra Game

Abstract: This thesis will introduce and prove the theorem of the Hydra Game as well as the theorem surrounding the Goodstein sequence. The two problems which are clearly intertwined will be introduced and solved with the assumption that the reader has no previous knowledge of them and little to no knowledge of ordinal numbers. All results have previusly been reached and presented, most notably by Kirby and Paris (Kirby et al, 1982). The aim of the paper is not to break ground but educate. The end of the paper will mention how these theorems show limitations of Peano Arithmetic but will not attempt to prove the statement.