A lot of independence results (Continuum Hypothesis, existence of Suslin trees, etc.) can seem (to my mind anyway) a bit “esoteric”, i.e. not impacting in an obvious way to “normal” mathematics (areas of math other than “foundational” subjects like set theory).
But the question of whether all projective subsets of $\mathbb{R}^{n}$ are Lebesgue measurable stands out to me as an independence result that seems not “esoteric” at all, being a pretty basic question about which sets in $\mathbb{R}^{n}$ have a “volume” (according to Lebesgue measure).
I’m wondering to what degree this question is truly “undecided”. That is, is there anything like a consensus leaning in one direction as to what the “right” answer “should” be?
- This article by Hugh Woodin seems to argue pretty emphatically for Projective Determinacy as an additional “canonical” axiom, which would settle the Lebesgue measurability of projective sets question. But I wouldn’t assume that Woodin’s view represents a “consensus”.