# Getting PFA + GCH above $\omega$

The Proper Forcing Axiom kills CH in a particularly specific way: it implies that $2^{\aleph_0}=\aleph_2$. However, its impact on the continuum function above $\aleph_0$ is much less clear. It is known, for example, that it implies the Singular Cardinal Hypothesis, SCH.

In this paper (see page 2), Aspero writes that Magidor showed the following:

Any model of ZFC+PFA has a forcing extension satisfying ZFC+PFA+"$2^\kappa=\kappa^+$ for all uncountable $\kappa$".

However, he does not give a reference; the bibliography mentions a set of (unpublished) lectures by Magidor. Googling around, I haven't found anything more specific.

I was wondering whether a proof of this exists in the literature, or - failing that - if someone could sketch an outline of it.

The proper forcing axiom is known to be indestructible by ${<}\aleph_2$-directed closed forcing, and since the forcing of the GCH for uncountable cardinals admits this degree of closure (iteratively add a Cohen set to the successor cardinals that are still there), it follows that one can simply force the GCH above $\aleph_1$ while preserving the PFA.
In Koenig and Yoshinobu's paper, they prove (Theorem 6.1) that $\sf PFA$ is preserved under $\omega_2$-closed forcings, and this should give the wanted result, as described by Joel.