Ackermannian Goodstein Sequences of Intermediate Growth

Abstract

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 two new Goodstein processes, obtaining new independence results for ACA 0 and ACA + 0 , theories of second order arithmetic related to the existence of Turing jumps.

Fast Goodstein Walks

“…In order to compare F (n) to the length of our fast Goodstein walks, the following property will be useful. It is shown to hold in general for any system of fundamental sequences with the Bachmann property in [6]. Proposition 9.6.…”

“…Moreover, Ωw+[↑γ|{τ |k}] Ω +θ ≥ Ω, so {ϑ(Ωw+↑γ+↑b)|k} Ω = ω {ϑ(Ωw+↑γ+↑b)|k}Ω , meaning that we do not need to check the side-condition for epsilon-numbers separately. So, it remains to prove (6). Write ωρ = ω ρ1 c + ρ 0 in coefficient ω-normal form and consider the following cases.…”

“…A parametrized version of the Ackermann function gives rise to independence results for theories between PA and the second-order arithmetical theory ATR 0 of arithmetical transfinite recursion [1]. This process is based on a notion of normal form based on a 'sandwiching' procedure, but other natural notions of normal forms can be considered [6]. More generally, Goodstein walks are also naturally defined in terms of the Ackermann function.…”

Giant and illusionary giant Goodstein principles

“…These articles are based on a very complicated iterated sandwiching procedure and it is quite natural to ask what happens if the sandwiching is reduced to a one step approximation. This question has been investigated for the Ackermann function which starts at the bottom level with the exponential function in [7]. It turned out that the strength of the resulting Goodstein principles dropped considerably and we arrived in these cases at intermediate Goodstein principles.…”