I am currently discovering descriptive set theory—with much pleasure! It is something of a surprise to me that, while the Borel hierarchy is indexed by $\omega_1$, the projective hierarchy is only indexed by $\mathbf{N}$, and that classical descriptive set theory stops there…

In particular, it would seem natural to “merge” the respective ideas behind Borel and projective hierarchies, to consider the smallest algebra of subsets of $\mathbf{N}^{\mathbf{N}}$ stable under complementation, countable unions and projections. Has this been done; and if yes, what is it called? I wonder in particular if it would be consistent with ZFC for every set in this algebra to be universally measurable? (and, maybe, that being universally measurable is preserved by projections…?). Clearly this cannot be a theorem of ZFC, since assuming $V=L$ proves existence of a $\mathbf{\Delta}^1_2$ set not being universally measurable; but I have read nowhere that my question would be contradictory with ZFC…

(Also, subsidiary question: the notation for Borel and projective hierarchies, with respective exponents $0$ and $1$, suggest that that there would be a “$k$-th order hierarchy” for $k \geq 2$, giving rise to $\mathbf{\Sigma}^k_n$, $\mathbf{\Pi}^k_n$ and $\mathbf{\Delta}^k_n$ sets… But I have nowhere read such a definition! So, does it exist something continuing the Borel and projective hierarchies in this way?…).

e.g."How do you call the police?" vs. "What do you call the police?"). $\endgroup$ – Robert Furber Nov 8 at 12:46