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Intermediate Goodstein principles

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.