Cofinality of functions $\mathbb N\to\mathbb N$ under eventual domination

Question

As the question suggests, I'm interested in the cofinality of functions $\mathbb N\to\mathbb N$ under eventual domination, i.e. $$f\prec g \quad\Longleftrightarrow\quad\exists N,\ \forall n>N,\ f(n)<g(n)$$ Obviously, the cofinality of this set lies in $[\omega,\mathfrak c]$. Moreover, the cofinality must be at least $\omega_1$, since for any denumerable set of functions $\{f_1,f_2,\ldots\}$, one can define $$f(n)=\max\{f_m(n):m\le n\}+1,$$ which will eventually dominate all functions in the set.

Without assuming the Continuum Hypothesis, can these bounds be made any stricter?

Answer 1

This is the dominating number $\mathfrak{d}$. Basically $\omega<\mathfrak{d}\le 2^{\omega}$ is all that can be said in $\mathsf{ZFC}$ alone. For example, it is consistent with $\mathsf{ZFC}$ that $\mathfrak{d}=\aleph_{17}$ and $2^\omega=\aleph_{42}$. Meanwhile, $\mathsf{MA}$ (which is compatible with $\neg\mathsf{CH}$) implies $\mathfrak{d}=2^{\aleph_0}$.

As a hopefully-useful bit of context, $\mathfrak{d}$ is an example of a cardinal characteristic of the continuum; I've written a bit about these here, and Blass' survey paper is excellent. The really interesting questions about CCCs are usually around their relations to one another. For example, a related cardinal is the bounding number $\mathfrak{b}$, the smallest size of a set $F$ of functions $\mathbb{N}\rightarrow\mathbb{N}$ such that no single $g:\mathbb{N}\rightarrow\mathbb{N}$ eventually dominates every $f\in F$; we trivially have $\mathfrak{b}\le\mathfrak{d}$, and it's consistent with $\mathsf{ZFC}$ that in fact $\mathfrak{b}<\mathfrak{d}$. Until relatively recently it was largely unknown how to prove the consistency of a separation of three or more cardinal characteristics; see this old MO post of mine.