GOODSTEIN'S GENERALIZED THEOREM: FROM ROOTED TREE REPRESENTATIONS TO THE HYDRA GAME

A hereditary base-b representation, used in the celebrated Good stein's theorem, can easily be converted into a labeled rooted tree. In this way it is possible to give a more elementary geometric proof of the afore-mentioned theorem and to establish a more general version, geometrically proved. This view is very useful for better understanding the underlying logical problems and the need to use transfinite induction in the proof. Similar problems will then be considered, such as the so-called "hydra game".