# $\mathbb{P}_{\kappa}$ forces non$(\mathcal{M})$=cov$(\mathcal{M})=\kappa$

Let $\kappa$ be an uncountable regular cardibnal. Consider the finite support iteration $(\langle \mathbb{P}_{\alpha}\rangle _{\alpha \leq \kappa},\langle \mathbb{\dot{Q}}_{\alpha}\rangle _{\alpha \leq \kappa})$ where $\mathbb{\dot{Q}}_{\alpha}$ is a $\mathbb{P}_{\alpha}$-name for $\mathbb{E}$ for all $\alpha< \kappa$. Then $\mathbb{P}_{\kappa}$ forces non$(\mathcal{M})$=cov$(\mathcal{M})=\kappa$

$\mathbb{E}=\{\langle s,H \rangle:s \in \omega^{< \omega}\wedge H\subseteq [\omega^{\omega}]^{<\omega}\}$, ordered by $(s',H')\leq (s,H)$ iff $s \subseteq s'$ and $H \subseteq H'$ and $\forall{i \in \text{dom}(s'\setminus s)}\forall{g \in H'}(s(i)\neq g(i))$

How can I show that $\mathbb{P}_{\kappa}$ forces non$(\mathcal{M})$=cov$(\mathcal{M})=\kappa$.

• In the definition of $\mathbb{E}$, do you mean $s\in \omega^{<\omega}$ rather than $s\in \omega^\omega$? Jun 20 '15 at 21:40
First check that the set of reals which are eventually different from a given real is meager. Hence you can cover all reals by $\kappa$ meager sets coded by the eventually different reals you added. Less than $\kappa$ meager sets cannot cover because new Cohen reals appear at stages of countably cofinality. The Cohen reals also form a non meager set of size $\kappa$. Finally any set of reals of size less than $\kappa$ is made meager by any eventually different real appearing later.