Let $\mathbb{D}$ Hechler forcing.
Let be $\kappa$ an uncountable regular cardinal. Consider the finite support iteration $(\langle \mathbb{P} \rangle _{\alpha < \kappa}, \langle \dot{\mathbb{Q}} \rangle _{\alpha < \kappa})$ where $\dot{\mathbb{Q}}$ is a $\mathbb{P}_{\alpha}$-name for $\mathbb{D}$ for all $\alpha < \kappa$.
- I want to show $\mathbb{P}_{\kappa}$ forces $\text{non}(\mathcal{M})\leq .\kappa$. To see $\text{non}(\mathcal{M})\leq \kappa$. Let $\mathcal{F}\subseteq \omega^{\omega}$ be of size than $\kappa$ in the extension, then there is $\alpha < \kappa$ such that $\mathcal{F}$ is contained in $V^{\mathbb{D}_{\alpha}}$. As I conclude that $\mathcal{F} \notin \mathcal{M}$.?
$\textbf{Observation}$ Hechler forcing adds a Cohen reals.
Let $\mathbb{E}$ notion eventually different real forcing.
Let be $\kappa$ an uncountable regular cardinal. Consider the finite support iteration $(\langle \mathbb{P} \rangle _{\alpha < \kappa}, \langle \dot{\mathbb{Q}} \rangle _{\alpha < \kappa})$ where $\dot{\mathbb{Q}}$ is a $\mathbb{P}_{\alpha}$-name for $\mathbb{E}$ for all $\alpha < \kappa$.
- A suggestion of how to show $\mathbb{P}_{\kappa}$ forces $\text{cov}(\mathcal{M})\leq \kappa$.
Thanks.
For Question 1, the Cohen reals adjoined at each of the $\kappa$ steps (the reductions mod 2 of the Hechler reals) constitute a non-meager set in the extension. The reason is that any code for a meager $F_\sigma$ set appears at some stage of the iteration strictly before $\kappa$, and the Cohen reals added after that step are outside that meager $F_\sigma$ set.
For Question 2, associate to each of the $\mathbb E$-generic reals $e_\alpha$ that were added in your iteration the meager set $M_\alpha=\{x:x\text{ is eventually different from }e_\alpha\}$. these $\kappa$ meager sets cover all the reals of the final extension, because every such real is added at some stage $\beta<\kappa$ and therefore belongs to $M_\alpha$ for all $\alpha>\beta$.