TITLE: Higher-rank Borel sets
QUESTION [5 upvotes]: What are interesting, illustrative examples of Borel sets, situated in Borel hierarchy higher than $\Sigma^{0}_{2}$ /$\Pi^{0}_{2}$?

REPLY [4 votes]: My personal favourite is the (F-sigma-delta) set of all real numbers x for which limit as n -> infinity sin(n! pi x) exists. There is a natural way to recursively iterate this construction to get Borel additive subgroups of reals at all finite levels of Borel hierarchy.