We know $PFA$ implies $2^{\aleph_0}=\aleph_2$. 

**Q1.** What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum function in other cardinals greater than $\aleph_{0}$? In the other words, is there any other known non-trivial result in the form "$ZFC+PFA\vdash 2^{\aleph_{\alpha}}=\aleph_{\beta}$" for some ordinals $\alpha , \beta$, or all other situations are consistent with $PFA$ like Easton's theorem in presence of some large cardinal?

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In the direction of my [previous question](http://mathoverflow.net/questions/155428/very-large-cardinal-axioms-and-continuum-hypothesis) on Godel's program for deciding $CH$ and $GCH$ using adding large cardinal axioms to $ZFC$ and based on the impact of $PFA$ on $CH$ which is compatible with Godel's conjecture on refuting $CH$ and $GCH$ using large cardinal axioms. One can consider existence of another large cardinal tree in the **shadow** of the current standard tree of large cardinal assumptions including $PFA$ in a rank near supercompacts. (The equiconsistency of PFA and supercompacts is not proved yet but we assume it for straightforwardness of the below diagram.)

Maybe Godel's conjecture and program are true if we replace the standard tree of large cardinals with another tree with equiconsistent steps. Please note the following imaginary **large cardinal ladder** which includes **two parallel trees** of "large cardinal" assumptions: 

**Q2.** Can we complete the right part of the following ladder with other axioms like $A$, $B$, $C$, ..., $X$ which are equiconsistent with usual large cardinal axioms and also refute $CH$ $GCH$ and $V=L$ in direction of Godel's program? 

![enter image description here][1]  


  [1]: https://i.sstatic.net/eaZQs.png