  The original Goodstein process proceeds by writing natural numbers in nested
exponential $k$-normal form, then successively raising the base to $k+1$ and
subtracting one from the end result. Such sequences always reach zero, but this
fact is unprovable in Peano arithmetic. In this paper we instead consider
notations for natural numbers based on the Ackermann function. We define three
new Goodstein processes, obtaining new independence results for $ {\sf ACA}_0$,
${\sf ACA}_0'$ and ${\sf ACA}_0^+$, theories of second order arithmetic related
to the existence of Turing jumps.
