Is the dominating number $\frak d$ regular?

Question

For a poset $(P, \leq)$ we say a subset $C\subseteq P$ is cofinal if for all $p\in P$ there is $c\in C$ such that $p\leq c$. We set $$\text{cf}(P,\leq) = \min\{|C|: C\subseteq P \text{ and } C \text{ is cofinal in } P\}.$$

Let $\omega^\omega$ be the set of all functions $f:\omega\to\omega$ ordered pointwise. Set ${\frak d} = \text{cf}(\omega^\omega)$.

Under $\sf CH$ it is known that ${\frak d} = 2^\omega$ which is regular. But are there other models of $\sf ZFC$ such that $\frak d$ is singular?

Answer 1

The dominating number $\mathfrak d$ is not necessarily regular. You can see in Theorem 2.5 of the reference below that it is consistent for $\mathfrak d$ to be any cardinal greater than or equal to $\mathfrak b$ (which, in turn, is regular) and less than or equal to $\mathfrak c=2^{\aleph_0}$ (which, against what you write in your question, is not necessarily regular).

MR2768685. Blass, Andreas(1-MI). Combinatorial cardinal characteristics of the continuum. Handbook of set theory. Vols. 1, 2, 3, 395–489, Springer, Dordrecht, 2010.

The argument essentially goes back to Hechler, and Andreas provides references and details.