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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 $\frak{d} < 2^{\aleph_0}$?

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Theorem 5.1 in Eric van Douwen's paper "The integers and topology" (Handbook of Set-theoretic Topology) is an old reference for a positive answer to your question and will provide a fuller explanation of the proof mentioned by Guest1245. There are newer references too: see the relevant chapter in L. Halbeisen, Combinatorial Set Theory. With a gentle introduction to forcing.

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Use the fact that adding any number of random reals does not increase the dominating number since every new real is dominated by a ground model real.

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