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 [1]][1] 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.
[1] Bill Dubuque, sci.math, Dec 11, 1995. Goedel's theorem: about anything in real world?
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