A Goodstein independence result for $ID_2$
Oriola Gjetaj
Authors: David Fernandez-Duque, Oriola Gjetaj and Andreas Weiermann
The Goodstein principle is a natural number-theoretic theorem. The original process works by writing natural numbers into nested exponential k-base normal form, then successively raising the base to k + 1 and subtracting 1 from the result. Such sequences always reach zero, but this fact is unprovable in Peano arithmetic.
Drawing from previous results in the literature, we consider canonical representations with respect to the Fast-Growing Extended Grzegorczyk hierarchy \(\{F_{a}\}_{a<\psi_0(\varepsilon_{\Omega+1})}\). Normal forms are written as base-k representations and the component \(a\) is written as base-\(\psi_0(\cdot)\) collapsing function up to Bachmann-Howard ordinal. We use an ordinal assignment to show that this sequence terminates and yields an independence result from the theory of \(ID_2\). This is part of joint work with A. Weiermann and D. Fern'andez-Duque on exploring normal form notations for the Goodstein principle.
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