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 $\kappa \ge {\omega }_2$ 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 $\mathcal{H}({\aleph }_2)\models \exists y\phi (a,y)$, where $a\in \mathcal{H}({\aleph }_2)$ and $\phi (x,y)$ 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 $\phi (a,b)$. In particular, $\mathbb{P}$ forces Moore's Measuring principle, Baumgartner's Axiom for ${\aleph }_1$-dense sets of reals, Todorčević's Open Colouring Axiom for sets of size ${\aleph }_1$, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todorčević'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 ${\aleph }_1$.