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arXiv · 2305.01066

Provable better quasi orders

Anton Freund·Alberto Marcone·Fedor Pakhomov·Giovanni Soldà
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Abstract

It has recently been shown that fairly strong axiom systems such as $\mathsf{ACA}_0$ cannot prove that the antichain with three elements is a better quasi order ($\mathsf{bqo}$). In the present paper, we give a complete characterization of the finite partial orders that are provably $\mathsf{bqo}$ in such axiom systems. The result will also be extended to infinite orders. As an application, we derive that a version of the minimal bad array lemma is weak over $\mathsf{ACA_0}$. In sharp contrast, a recent result shows that the same version is equivalent to $\Pi^1_2$-comprehension over the stronger base theory $\mathsf{ATR}_0$.

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Anton Freund, Alberto Marcone, Fedor Pakhomov, Giovanni Soldà. 2023-05-01. Provable better quasi orders. https://arxiv.org/abs/2305.01066

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