Consistency of restricted forms of Martin's Axiom with the negation of the Continuum Hypothesis

Question

Consider $\mathsf{MA}(S)$, the forcing axiom for all ccc posets which preserve a Souslin tree $S$. is $\lnot \mathsf{CH}$ consistent with $\mathsf{ZFC}+\mathsf{MA}(S)$? Does there exist a model for $\mathsf{ZFC}+\mathsf{MA}_{\omega_1}(S)[S]+\lnot \mathsf{CH}$?

Answer 1

Yes, given a fixed Souslin tree $S$ here are models of $\mathsf{ZFC} + \mathsf{MA}(S) + \neg \mathsf{CH}$. Basically just take a regular cardinal $\theta > \aleph_1$ such that $2^\lambda \leq \theta$ for all $\lambda < \theta$ and proceed to force $\mathsf{MA} + \mathfrak{c} = \theta$ mostly as usual: a finite support iteration $\langle \mathbb{P}_\alpha , \dot{\mathbb{Q}}_\alpha \rangle_{\alpha \leq \theta}$ of length $\theta$ of all c.c.c. posets of cardinality ${<}\theta$; but skip those posets that don't preserve $S$.

• As Cohen forcing doesn't destroy any Souslin tree (and as the iteration doesn't collapse cardinals), in the extension we will have $\mathfrak{c} = \theta > \aleph_1$.

• By induction on $\beta \leq \theta$ we can show that $\mathbb{P}_\beta$ does not destroy $S$. For this note that since all our posets belong to the ground model, $\mathbb{P}_\beta \Vdash \check{S}\text{ is Souslin}$ iff $\mathbb{P}_\beta \times S$ is c.c.c. iff $S \Vdash \check{\mathbb{P}}_\beta\text{ is c.c.c.}$

• By repeating the standard argument, in the extension whenever $\mathbb{Q}$ is a c.c.c. poset of cardinality ${<}\mathfrak{c}$ which doesn't destroy $S$, and $\{ D_i : i < \lambda \}$ ($\lambda < \mathfrak{c}$) is a family of dense subsets of $\mathbb{Q}$, there is a $\{ D_i : i < \lambda \}$-generic filter for $\mathbb{Q}$.

• Basically repeating the standard argument showing that $\mathsf{MA}$ for c.c.c. posets of cardinality ${<}\mathfrak{c}$ implies the full $\mathsf{MA}$, we can show that $\mathsf{MA}(S)$ holds in the extension.

(Additionally, if you happen to have a supercompact cardinal laying around, you can get a model of $\mathsf{PFA}(S)$ using the same standard technique to get the consistency of $\mathsf{PFA}$ modulo that supercompact. It is a result of Tadatoshi Miyamoto that the countable support iteration of proper posets which do not destroy a fixed Souslin tree $S$ also doesn't destroy $S$.)