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Pigeons do not jump high

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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Submitted on : Friday, October 5, 2018 - 12:49:46 PM
Last modification on : Tuesday, November 19, 2019 - 2:44:59 AM
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  • HAL Id : hal-01888793, version 1
  • ARXIV : 1803.09771

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Benoit Monin, Ludovic Patey. Pigeons do not jump high. Advances in Mathematics, Elsevier, In press. ⟨hal-01888793⟩

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