# Pigeons do not jump high

2 AGL - Algèbre, géométrie, logique
ICJ - Institut Camille Jordan [Villeurbanne]
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.
Document type :
Preprints, Working Papers, ...
Domain :

Cited literature [20 references]

https://hal.archives-ouvertes.fr/hal-01888793
Contributor : Ludovic Patey <>
Submitted on : Friday, October 5, 2018 - 12:49:46 PM
Last modification on : Friday, March 8, 2019 - 9:34:02 AM
Document(s) archivé(s) le : Sunday, January 6, 2019 - 3:23:21 PM

### File

pigeons-jump.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-01888793, version 1
• ARXIV : 1803.09771

### Citation

Benoit Monin, Ludovic Patey. Pigeons do not jump high. 2018. ⟨hal-01888793⟩

Record views