Textbooks about non-RE languages for undergraduate students
Despite its name, Soare's book Recursively enumerable sets and degrees has lots of information about this. In particular, it covers in pretty good detail the arithmetical hierarchy.
(Incidentally, note that the non-r.e. elements of the arithmetical hierarchy aren't all &quot;r.e.-hard.&quot; For example, consider any minimal $\Delta^0_2$ degree, that is any Turing degree $\le_T{\bf 0'}$ which is nonzero but not strictly above any nonzero Turing degree. There are infinitely many minimal $\Delta^0_2$ degrees; by the Sacks Density Theorem, they are all non-r.e. and hence by minimality don't compute any nonrecursive r.e. sets. The way in which the r.e. degrees sit inside the $\Delta^0_2$ degrees is actually fairly complicated.)