Is it consistent (from suitable large cardinals) that there is a ccc poset which forces PFA?


This seems quite implausible to me. If we could force PFA via ccc forcing, the ground model would have to be quite close to a model of PFA (having the correct continuum, no squares, SCH holding etc.). However, the ground model cannot be a model of full PFA (or even BPFA), since it follows from a result of Caicedo and Veličković that *any* ccc forcing over such models destroys BPFA.


The reference for the Caicedo-Veličković result is



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> Andrés Eduardo Caicedo and Boban Veličković, *The bounded proper forcing axiom and well orderings of the reals*, Math. Res. Lett. **13** (2006), no. 3, 393--408. ([link](http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2006/0013/0003/MRL-2006-0013-0003-a005.pdf))
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They show that if $V$ and an inner model $M$ agree on $\omega\_2$ and both satisfy BPFA then they have the same subsets of $\omega\_1$. The conclusion above follows since any ccc forcing adds a subset of $\omega\_1$.



Forcing PFA with ccc forcing
I think a negative answer can be derived from the following observations. 


One. A nontrivial c.c.c. forcing adds a subset of $\omega\_{1}$ (consider the least cardinal $\kappa$ for which it adds a subset of $\kappa$, and the tree of possible initial segments for this subset; the splitnodes in the tree give rise to a name for a new subset of $\omega\_{1}$). 


Two. A c.c.c. partial order $P$ has the property that for any ordinal $\delta$, if in a $P$-extension one has a continuous $\subseteq$-increasing 
chain $\bar{N} = \langle N\_{\alpha} : \alpha < \omega\_{1} \rangle$ of countable subsets of $\delta$ with union $\delta$, then for some club $C \subseteq \omega\_{1}$ in the ground model, $\langle N\_{\alpha} : \alpha \in C \rangle$ exists already in the ground model. Now if we look at Justin Moore's MRP coding for subsets of $\omega\_{1}$ as in Section 4 of this paper (<http://arxiv.org/pdf/math/0501526v1.pdf>) we see that every subset of $\omega\_{1}$ coded in a $P$-extension by such a sequence $\bar{N}$ would be coded already by restriction of $\bar{N}$ in the ground model.


