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arXiv · 2604.00771

Goodstein at the Second Threshold: An Independence Result for $ID_2$

Oriola Gjetaj·Andreas Weiermann
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Abstract

The classical Goodstein process, defined via hereditary base-$k$ exponential normal form, is a well-known example of a principle unprovable in Peano Arithmetic. In this paper, we generalize this framework by constructing a new Goodstein process based on the Hardy hierarchy. We develop an ordinal notation system utilizing a two-step collapsing procedure, which yields a proof-theoretic ordinal of $ψ_0ψ_1(\varepsilon_{Ω_2+1})$. By defining $k$-normal forms for natural numbers within this system, we introduce a Goodstein-type process and demonstrate that the theory of non-iterated positive inductive definitions for two operators ($ID_2$) cannot prove its termination. This result establishes a new independence result at the second proof-theoretic threshold, further extending the reach of Goodstein-type principles beyond the Bachmann-Howard level.

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Oriola Gjetaj, Andreas Weiermann. 2026-04-01. Goodstein at the Second Threshold: An Independence Result for $ID_2$. https://arxiv.org/abs/2604.00771

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