Goodstein actually employed arbitrary increasing base-bumping functions. He showed that the convergence of all such is equivalent to transfinite induction below $\epsilon_0$. This is illustrated somewhat more graphically by the Hercules vs. Hydra game. See the references from my old post  of 1995 which helped serve to popularize these topics on (use)net. Curiously that post received far more feedback than any of my other posts - from popular science writers to researchers, teachers and students.
 Bill Dubuque, sci.math, Dec 11, 1995. Goedel's theorem: about anything in real world?