The proper forcing axiom for $\aleph_1$-sized posets, $\omega_1$-linked symmetrically proper forcing, and the size of the continuum

Abstract

We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$ is consistent with the continuum being arbitrarily large. In fact, assuming $GCH$ holds and $\kappa\geq\omega_2$ is a regular cardinal, we prove that there is a proper and $\aleph_2$-c.c.\ forcing giving rise to a model of this forcing axiom together with $2^{\aleph_0}=\kappa$ and which, in addition, satisfies all statements of the form $\mathcal{H}(\aleph_2)\models \exists y\varphi(a, y)$, where $a\in \mathcal{H}(\aleph_2)$ and $\varphi(x, y)$ is a $\Sigma_0$ formula with the property that for every ground model $M$ of $CH$ with $a\in M$ there is, in $M$, a suitably nice poset -- specifically, a poset $\mathbb{Q}\subseteq\mathcal{H}(\kappa)^M$ which is $\omega_1$-linked and symmetrically proper -- adding some $b$ such that $\varphi(a, b)$. In particular, $\mathbb{P}$ forces Moore's Measuring principle, Baumgartner's Axiom for $\aleph_1$-dense sets of reals, Todor\v{c}evi\'{c}'s Open Colouring Axiom for sets of size $\aleph_1$, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todor\v{c}evi\'{c}'s P-ideal Dichotomy for $\aleph_1$-generated ideals on $\omega_1$, among other statements. Hence, all these statements are simultaneously compatible with a large continuum. Finally, we show that a further small variation of our construction yields a model satisfying, in addition to all the earlier conclusions, Martin's Maximum for posets of size $\aleph_1$.