$$\newcommand{\N}{\mathbb N}$$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.