A non-arithmetical set? A set is called arithmetical if it can be defined by a first-order formula in Peano arithmetic. I first encountered these sets when exploring the arithmetical hierarchy in the context of computability theory. However, I have not encountered any examples of sets that are not arithmetical. Is there a canonical example of an non-arithmetical set? Thanks!

There are countably many first order formulas defining arithmetical sets. Let $\varphi_n$, $n\in\mathbb N$, be a list of those. Consider the set that contains a natural number $n$ iff $n$ is not contained in the set defined by the $n$-th formula.