Projective hierarchy

Summary

The projective hierarchy sorts the projective subsets of a Polish space into levels $\Sigma^1_n$, $\Pi^1_n$ and $\Delta^1_n$ for $n \ge 1$. It starts with the analytic sets, and alternates two operations: taking complements and taking projections along the Baire space. The first level $\Delta^1_1$ consists exactly of the Borel sets, by Suslin's theorem. ZFC does not decide whether the higher levels behave well, for example whether all their sets are Lebesgue measurable. They do under projective determinacy, and they do not under the axiom of constructibility.