2022/09/03 by Asperó, David, Golshani, Mohammad
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2209.01395
We show that the Proper Forcing Axiom for forcing notions of size ℵ1 is consistent with the continuum being arbitrarily large. In fact, assuming GCH holds and κ≥ω2 is a regular cardinal, we prove that there is a proper and ℵ2-c.c. forcing giving rise to a model of this forcing axiom together with 2ℵ0=κ and which, in addition, satisfies all statements of the form H(ℵ2)\models ∃ yφ(a, y), where a∈ H(ℵ2) and φ(x, y) is a Σ0 formula with the property that for every ground model M of CH with a∈ M there is, in M, a suitably nice poset -- specifically, a poset ℚ\subseteqH(κ)M which is ω1-linked and symmetrically proper -- adding some b such that φ(a, b). In particular, ℙ forces Moore's Measuring principle, Baumgartner's Axiom for ℵ1-dense sets of reals, Todorčević's Open Colouring Axiom for sets of size ℵ1, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todorčević's P-ideal Dichotomy for ℵ1-generated ideals on ω1, 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 ℵ1.