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](http://www.math.unt.edu/~jkrueger/paper13.ps) (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However, he does not give a reference; the bibliography mentions a set of (unpublished) lectures by Magidor.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.



Getting PFA + GCH above $\omega$
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. 


