It is known since early studies on constructibility and forcing that
counterexamples to some classical theorems of descriptive set theory
consistently exist at suitable projective levels.
This includes, e.g.,
1) a non-measurable Delta^1_2 set
2) a nonconstructible Delta^1_3 real
3) sets that witness failure of Separation for Pi^1_3,
and many more.
This naturally led to a question whether counterexamples consistently exist at n-th level of the hierarchy under the assumption that they do not exist at levels below n.
Some results in this direction, for arbitrary n, related to definable nonconstructible reals, prewellorderings, Separation, Reduction, are known since mid-1970s, mainly to Leo Harrington, but remain unpublished. The main goal of the talk will be to present proofs of these theorems up to major details, and explain related difficulties.
No general results like this are known so far for measurability and other regularity properties. Although the steps from n=2 to n=3 and from n=3 to n=4 have been resolved by methods that do not generalize to higher levels.
Some other open problems will be discussed.