TITLE: Nontrivial Borel Sets
QUESTION [5 upvotes]: We are covering Measure Theory and the Borel Algebra $B(\mathbb{R})$ and I am trying to find a nontrivial example of a Borel Set. With the notions of $F_\sigma$ and $G_\delta$ sets, I am wondering if we can find a borel set which is not the countable union or countable intersection of $F_\sigma$ or $G_\delta$ sets. 
Mainly, I want to know how "bad" can a set in $B(\mathbb{R})$ be?

REPLY [2 votes]: As an addendum to Noah Schweber's answer, here's a concrete example of a set that is in both $\Sigma_3^0$ and $\Pi_3^0$ (and hence in $\Delta_3^0$) but neither in $\Sigma_2^0$ nor in $\Pi_2^0$.  The set of all real numbers in $[0,1]$ whose base-$10$ representations contain (a) infinitely many $1$'s and (b) finitely many digits that are not in $\{1,2\}$.
Requirement (a) gives a set in $\Pi_2^0$ that is not in $\Sigma_2^0$, while requirement (b) gives a set in $\Sigma_2^0$ that is not in $\Pi_2^0$.  Finite unions and intersections of sets in $\Sigma_n^0$ and $\Pi_n^0$ produce sets that are provably in $\Delta_{n+1}^0 = \Sigma_{n+1}^0 \cap \Pi_{n+1}^0$.