TITLE: How fast can the base-bumping function in Goodstein's theorem grow?
QUESTION [14 upvotes]: In the usual presentation of Goodstein's theorem, the base is bumped up by the "add 1" function. Does the theorem still hold when we replace this function by a fast-growing one (e.g. Ackermann or busy beaver)? How far can we push this? For example, let's define $g_0(n)$ to be the number of Goodstein iterations needed to reach 0 when we start with base 2 and seed $n$ (so that $g_0(0)$ = 0). Then we can build a hierarchy of functions by defining $g_{k+1}(n)$ as the number of Goodstein iterations needed to reach 0 with seed $n$ and base-bumping function $g_k$ ($k$ = 0, 1, ...), continuing through the ordinals by diagonalization at each limit ordinal. Surely it's got to break down when we go past $\varepsilon_0$, if not long before that!

REPLY [9 votes]: Dear all,
let me give the following remark:
"Goodstein actually employed arbitrary increasing base-bumping functions. He showed that the convergence of all such is equivalent to transfinite induction below ϵ0."
This statement has to be taken with care when it comes to weakly increasing base bumping
functions. When we reach functions in the neighboorhood of log* then the Goodsteinprocess
becomes provable in PRA. 
But when we take a fixed iterate of log then of course termination of Goodstein
sequences is equivalent to the 1 consistency
of PA.
If the base bumping function growth faster than H_epsilon_0 then Goodstein
can of course yield more than the 1 consistency of PA.
Best,
Andreas