快速古德斯坦散步

IF 0.9 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-12-27 DOI:10.1112/blms.13210

David Fernández-Duque, Andreas Weiermann

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摘要

我们介绍一个家庭（a k） k ＆lt；ω $(\mathbb {A}_k)_{k<\omega }$基于ε 0 $\varepsilon _0$的快速增长函数，并使用这些来定义Goodstein过程的一个变体。我们证明了这种变体是终止的，并且这一事实在Kripke-Platek集合理论（或其他巴赫曼-霍华德强度理论）中是不可证明的。此外，我们还证明了这个Goodstein过程是最大长度的，因此基于相同快速增长函数的任何其他Goodstein过程也将终止。

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Fast Goodstein walks

We introduce a family ${(A_{k})}_{k < ω}$ of fast-growing functions based on $ε_{0}$ and use these to define a variant of the Goodstein process. We show that this variant terminates and that this fact is not provable in Kripke–Platek set theory (or other theories of Bachmann–Howard strength). We, moreover, show that this Goodstein process is of maximal length, so that any alternative Goodstein process based on the same fast-growing functions will also terminate.