Pigeons do not jump high

Benoit Monin; Ludovic Patey

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arXiv · 1803.09771

Pigeons do not jump high

Benoit Monin·Ludovic Patey

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Abstract

The infinite pigeonhole principle for 2-partitions asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we develop a new notion of forcing enabling a fine analysis of the computability-theoretic features of the pigeonhole principle. We deduce various consequences, such as the existence, for every set $A$, of an infinite subset of it or its complement of non-high degree. We also prove that every $\Delta^0_3$ set has an infinite low${}_3$ solution and give a simpler proof of Liu's theorem that every set has an infinite subset in it or its complement of non-PA degree.

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Benoit Monin, Ludovic Patey. 2018-03-26. Pigeons do not jump high. https://arxiv.org/abs/1803.09771

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