Assertion: projective set | /r/HasProperty | regular
Facets:
Source sentences:
text: There is, however, an axiom, called the axiom of Projective Determinacy, or PD, that is consistent with ZFC, modulo the consistency of some large cardinals (in fact, it follows from the existence of some large cardinals), and implies that all projective sets are regular.
text: It turns out that PD implies that all projective sets of reals are regular, and Woodin has shown that, in a certain sense, PD settles essentially all questions about the projective sets.
text: Further, under a weaker large-cardinal hypothesis, namely the existence of infinitely many Woodin cardinals, Martin and Steel (1989) proved that every projective set of real numbers is determined, i.e., the axiom of PD holds, hence all projective sets are regular.
text: Moreover, Woodin showed that the existence of infinitely many Woodin cardinals, plus a measurable cardinal above all of them, implies that every set of reals in \(L(\mathbb{R})\) is determined, i.e., the axiom \(AD^{L(\mathbb{R})}\) holds, hence all sets of real numbers that belong to \(L(\mathbb{R})\), and therefore all projective sets, are regular.