There are many consequences of the Proper Forcing Axiom (PFA) which essentially say that $\omega_2$ has a lot of compactness.
For instance, Matteo Viale and Christoph Weiss have a few papers in which the combinatorial properties $\mathrm{SP}(\kappa)$ and $\mathrm{ISP}(\kappa)$ (which can be seen as generalizations of the tree property) are isolated, and the following theorems (which are modification of older results of Jech and Magidor, respectively) are stated:
- $\kappa$ is strongly compact iff $\kappa$ is inaccessible and $\mathrm{SP}(\kappa)$
- $\kappa$ is supercompact iff $\kappa$ is inaccessible and $\mathrm{ISP}(\kappa)$
They also show that $\mathrm{PFA} \rightarrow \mathrm{ISP}(\omega_2)$ (and $\mathrm{SP}(\omega_2)$), so one can read this as saying PFA implies $\omega_2$ is as "compact" as a supercompact, minus the inaccessibility.
On the other hand, there are consequences of PFA (including the statement of PFA itself) which seem to say that $\omega_2$ is very much incompact. For instance, consider Rado's Conjecture (RC). RC is the statement: If $T$ is a tree such that every subtree of size $< \omega_2$ can be decomposed into countably many antichains, then so can $T$. It's not hard to see how RC is, in some sense, saying that $\omega_2$ is compact: If we replace $\omega_2$ with a compact cardinal $\kappa$ in the statement of RC, then the statement is true by the "compactness of the language $\mathcal{L}_{\kappa,\kappa}$" characterization of $\kappa$. But PFA contradicts RC. Nonetheless, both PFA and RC are usually obtained by proper forcings which collapse a supercompact to $\omega_2$.
PFA itself seems to say $\omega_2$ is incompact: Let $\mathbb{P}$ be a proper forcing, and consider the language which has a constant symbol for every element of $\mathbb{P}$, a binary relation symbol (for the relation on $\mathbb{P}$), a unary predicate for each dense subset of $\mathbb{P}$, and a unary predicate which will stand for a generic filter. Consider the theory consisting of the positive diagram of $(\mathbb{P},\leq,p\ (p\in \mathbb{P}), D\ (D \subseteq \mathbb{P}\mbox{ dense}))$ together with formulas saying that $G$ is a filter and $G$ meets every $D$ (a new formula for each $D$). PFA says that any subtheory of size $< \omega_2$ has a model, but there's no filter meeting every dense set in the ground model, so loosely speaking, this theory doesn't have a model (although I suppose there could be a model which adds unnamed elements to each dense subset and has a generic meeting each dense subset at an unnamed condition).
My question is:
Is there some way to reconcile the fact that PFA seems to simultaneously say that $\omega_2$ has properties strongly indicative of some sort of compactness, and also has properties strongly indicative of some sort of incompactness?
