Two cardinal characteristics (cardinals between $\aleph_1$ and $\mathfrak{c}$ are:
- $\mathfrak{b}$, the least size of an unbounded family in $\omega^{\omega}$ ordered under eventual domination
- $\mathfrak{p}$, the least cardinal for which Martin's Axiom of $\sigma$-centered posets fails
It holds in ZFC that $\mathfrak{p}\leq\mathfrak{b}$. How do you force that the inequality to be strict? I am working through Kunen's Set Theory and the forcings I have come across where these values are computed either force $\mathfrak{b}=\aleph_1$ or satisfy some Martin's Axiom and have $\mathfrak{p}=\mathfrak{c}$.
Since $\mathfrak{p} \leq$ non(Null), it is enough to get a model where non(Null) $= \omega_1 < \mathfrak{b} = \omega_2$. Start with a model of CH and do a countable support iteration of length $\omega_2$ of Laver forcing. In this model the ground model reals are non null. For a proof of this, see Judah, Shelah: The Kunen-Miller chart, J Symb. Logic 55 (1990). Since the Laver real dominates all ground model reals, $\mathfrak{b} = \omega_2$ in this model.
Edit: That was an overkill. Here's how Kunen would have done it in his book. Start with a model where MA holds and continuum is $\omega_3$. So $\mathfrak{b} = \omega_3$ in this model. Now add $\omega_1$ random reals. Then non(Null) $ = \omega_1$ (as witnessed by the random reals) hence also $\mathfrak{p} = \omega_1$. Since random forcing is $\omega^{\omega}$-bounding (this is easy to check), $\mathfrak{b} \geq \omega_3$.