The proper forcing axiom for ${\aleph }_1$-sized posets, ${\omega }_1$-linked symmetrically proper forcing, and the size of the continuum

David Asperó, Mohammad Golshani

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 $\kappa \ge {\omega }_2$…