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It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.

Is there a direct proof of "all projective sets of reals are regular" from PFA (i.e. without relying on determinacy)?

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    $\begingroup$ Using large cardinals: You use failure of square principles to get (a sufficiently iterable) $M_\omega^\sharp$, that allows you to realize $L(\mathbb R)$ essentially as a Solovay model. (But I imagine this is not what you are asking.) $\endgroup$
    – Andrés E. Caicedo
    Commented Feb 5, 2015 at 1:30
  • $\begingroup$ Interesting question; I feel like the answer is more likely to be yes if the conclusion is replaced by "after collapsing $\omega_1$, all projective sets of reals are regular" (but I still don't know what a proof would look like.) One could also consider the weakening where "all projective sets of reals are regular" is replaced by "there is no projective well-ordering of the reals." $\endgroup$
    – Trevor Wilson
    Commented Feb 6, 2015 at 1:42
  • $\begingroup$ So you're asking how to prove from $\sf PFA$ directly that every projective set is universally Baire? $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 6, 2015 at 10:33

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