Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. We write $f <^* g$ if there is $N\in\omega$ such that $f(n) < g(n)$ for all $n>N$. A set $D\subseteq \omega^\omega$ is said to be *dominating* if for all $f\in \omega^\omega$ there is $g\in D$ such that $f <^* g$. Set $$\frak{d} = \textrm{min}\{|\mathrm{D}|: \mathrm{D}\subseteq \omega^\omega \textrm{ and } \mathrm{D} \textrm{ is dominating}\}.$$

Is it consistent that $\frak{d} < 2^{\aleph_0}$?