TITLE: A non-arithmetical set?
QUESTION [5 upvotes]: 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!

REPLY [3 votes]: The usual examples are things like:
$0^{\omega}$ or anything bigger.
Any arithmetically generic set.
The set of ordinal notations or equivalently the indexes for computable well-orderings or even the indexes of well-founded computable trees.
I figured I'd add these because these are natural examples not merely a diagonalization. 
Though the godel numbers of true statements of arithmetic is quite natural.