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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?

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$\omega_2$ being a successor, of course it is going to be non-compact. I doubt this has much to do with PFA or anything else, it is just that it is a successor. This is why rather than looking at direct instances of compactness, you search for traces that have survived (and think of them as evidence that $\omega_2$ is genuinely compact in some inner model). –  Andrés Caicedo Sep 7 '11 at 1:22
I can't seem to recall the details, but I do recall a definition of being "$\lambda$-thinly ineffable" (or "thinly $\lambda$-ineffable"). I do recall being in a seminar recently about PFA implying $\omega_2$ is $\lambda$-thinly ineffable. This implies some sense of compactness and as Andres says being a successor implies otherwise. –  Asaf Karagila Sep 7 '11 at 8:08
@Asaf, yes in the paper of Viale and Weiss, $\mathrm{ITP}(\kappa,\lambda)$ is used to denote that $\kappa$ is $\lambda$-thinly ineffable. $\mathrm{ITP}(\kappa)$ is then used to denote that $\forall\lambda\geq\kappa,\mathrm{ITP}(\kappa,\lambda)$. $\mathrm{ISP}(\kappa,\lambda)$ denotes that $\kappa$ is $\lambda$-"slenderly" ineffable, and $\mathrm{ISP}(\kappa)$ denotes that $\forall\lambda\geq\kappa,\mathrm{ISP}(\kappa,\lambda)$. $\mathrm{ISP}(\kappa,\lambda)\rightarrow\mathrm{ITP}(\kappa,\lambda)$, so the result that $\mathrm{PFA}\rightarrow\mathrm{ISP}(\omega_2)$ subsumes the one you mention. –  Amit Kumar Gupta Sep 7 '11 at 9:14

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