Where Pigeonhole Principles meet König Lemmas

Abstract

We study the pigeonhole principle for ${\Sigma }_2$-definable injections with domain twice as large as the codomain, and the weak König lemma for ${\Delta }_2^0$-definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of $2$-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for ${\Sigma }_2$-definable injections.

33 pages