User blog comment:LittlePeng9/Long hierarchies of functions (on my other blog)/@comment-11227630-20171031155258

There are $$\aleph_0^{\aleph_0}=\beth_1$$ different functions: $$\mathbb{N}\mapsto\mathbb{N}$$. If continuum hypothesis holds, there'll be only $$|\omega_1|$$ different such functions, then a chain of length $$>\omega_1$$ cannot be build.