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arXiv · 1905.08425

The weakness of the pigeonhole principle under hyperarithmetical reductions

Benoit Monin·Ludovic Patey
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Abstract

The infinite pigeonhole principle for 2-partitions ($\mathsf{RT}^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\mathsf{RT}^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.

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Benoit Monin, Ludovic Patey. 2019-05-21. The weakness of the pigeonhole principle under hyperarithmetical reductions. https://arxiv.org/abs/1905.08425

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