Maximal ordinal embeddable into poset of functions under eventual Domination

Question

Consider the set of functions from $\mathbb{N}\to\mathbb{N}$. We can impose a partial order on this set by saying that $f>g$ if $f(n)>g(n)$ for all sufficiently large $n$. By a diagonalization argument, it can be shown that one can embed $\omega_1$, the first uncountable ordinal in this Poset. Since the set of functions from $\mathbb{N}\to \mathbb{N}$ has the cardinality of $\mathbb{R}$ it is clear that one can not embed $\omega_2$ if the continuum hypothesis is true. In general what is the smallest ordinal one can not embed. In particular, is there an embedding of $\omega_1+1$ in general?

Answer 1

The smallest such ordinal is $\omega_2$. I do not want to type much so I'll give you an excellent reference: Problem 24, Chapter 11, P. Komjath and V. Totik - Problems and theorems in classical set theory.