
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 , of an infinite subset of 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 , of an infinite subset of it or its complement of non-high degree. We also prove that every set has an infinite low 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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