When learning basic computability theory we are usually given as examples of arithmetic sets sets which are complete for their level of the arithmetic hierarchy (like the halting set, the set of indices of total functions etc.). Our teachers are of course quick to point out that not all sets are complete in this way. However, one could still be led to believe that arithmetically complex sets have quite a lot of computational power. Surely this cannot be the case.
Can you give me an example of an arithmetic set $A$ of some (precise) complexity $\Sigma^0_n$ (or $\Pi^0_n$ or whatever) such that $A$ does not compute all the sets of lower complexity? Can you give me one where $A$ does not compute the halting problem?
I should point out that a friend (who likes models of PA) has already given me an example by constructing a particular model of PA whose standard system contains a completion of PA but omits the halting problem. While this example is perfectly fine, it feels like the problem should have a much more elementary solution.