  It turns out that each of these possibilities is realized.  For the first it suffices to take the Π01-sentence Con(T).  For the second it is easy to see that there is no example that is Π01; the simplest possible complexity of such a sentence is Δ02 and it turns out that there are such examples; examples of this type of independence are called Orey sentences.  For the third kind of independence there are Π01 instances.  (This is a corollary of Lemma 14 on pages 128–129 of Lindström (2003).)