Pigeons do not jump high

TL;DR

This paper introduces a new forcing method to analyze the computability aspects of the pigeonhole principle, revealing properties of infinite subsets related to degrees of unsolvability.

Contribution

It develops a novel forcing technique to study the computability-theoretic features of the pigeonhole principle and derives new results about the degrees of infinite subsets.

Findings

Existence of infinite subsets of non-high degree for any set

Every Δ³ set has an infinite low₃ solution

Simplified proof that every set has an infinite subset of non-PA degree

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.