# Forcing PFA with ccc forcing

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

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)

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 $\omega_1$.

• Do you have a reference for the Caicedo-Velickovic result? Jul 11, 2016 at 14:22
• @PaulMcKenney I've added a reference. Jul 11, 2016 at 14:36
• The only "nontrivial" that a ccc forcing can really do and not destroy PFA is to shoot branches through Suslin trees. (Or so I think, anyway.) Which is of course consistent with your observation from the paper of Andres and Boban. But it seems really weird if that might happen. E.g. if you add a Cohen real to a model of PFA and somehow revive PFA by shooting branches through the Suslin trees you've added and restoring all sort of cardinal invariants to their rightful size. Jul 11, 2016 at 16:43
• (And yes, that does seem a bit implausible!) Jul 11, 2016 at 16:43
• @AsafKaragila Forcing with a Suslin tree forces $\mathfrak{t}= \omega_1$. This is Lemma 2 in Farah, 'OCA and towers in P(N)/fin', but there it is mentioned that Booth would have known of this result.
– tci
Jul 12, 2016 at 17:41

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.