\(\newcommand{\N}{\mathbb N}\)Consider the set \(\N^\N\) of all functions from \(\N\) to itself. We impose a partial order of *eventual domination* on this set: we say that \(f\) eventually dominates \(g\) if for large enough \(n\) we have \(f(n)< g(n)\). In this poset, we consider chains, in particular the well-ordered ones. One question we can ask is, how long can such chains be? Here we establish a lower bound \(\omega_2\) for the supremum of possible lengths.

All background necessary for this post is some basic knowledge about ordinals, in particular cofinalities.