The reverse mathematics of non-decreasing subsequences

TL;DR

This paper investigates the reverse mathematical strength of the statement that every function has an infinite non-decreasing subdomain, revealing its position between known logical principles and its computational properties.

Contribution

It characterizes the reverse mathematical strength of the non-decreasing subsequences principle, showing it is weaker than some classical theorems but not implied by Ramsey's theorem for pairs.

Findings

Weak computational strength for computably bounded functions

Does not imply the halting set

Independent of Ramsey's theorem for pairs

Abstract

Every function over the natural numbers has an infinite subdomain on which the function is non-decreasing. Motivated by a question of Dzhafarov and Schweber, we study the reverse mathematics of variants of this statement. It turns out that this statement restricted to computably bounded functions is computationally weak and does not imply the existence of the halting set. On the other hand, we prove that it is not a consequence of Ramsey's theorem for pairs. This statement can therefore be seen as an arguably natural principle between the arithmetic comprehension axiom and stable Ramsey's theorem for pairs.