Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$).

Define several sets of total functions, in each case partially ordered by eventual domination:

- computable functions $\mathbb{N} \to \mathbb{N}$
- arithmetic functions $\mathbb{N} \to \mathbb{N}$ (i.e. the set of all pairs $(n, f(n))$ is an arithmetical set)
- all functions $\mathbb{N} \to \mathbb{N}$
- all functions $\mathbb{N} \to \mathbb{R}$
- elementary functions $\mathbb{R} \to \mathbb{R}$
- real analytic functions $\mathbb{R} \to \mathbb{R}$
- all functions $\mathbb{R} \to \mathbb{R}$
- all functions $\omega_1 \to \omega_1$
- Any other interesting cases?

What is the height of each of these posets? Is anything of these a known open problem? Has anything been proven to be independent of $ZFC$?

anything, right? Surely $c \neq \aleph_\omega$ is true in ZFC... $\endgroup$ – Maxime Bourrigan Nov 29 '11 at 16:12