  Notice that in the case of the third example the two theories above T are incomparable in the interpretability order.  To construct a pair of such Π01-statements one uses a reciprocal form of the diagonal lemma to construct two Π01-statements that refer to one another.  Using such techniques can show that the interpretability order is quite complex.  For example, for any two theories T1 and T2 such that T1 &lt; T2 there is a third theory T such that T1 &lt; T &lt; T2.  Thus, the order on the degrees of interpretability is neither linearly ordered nor well-founded.  (See Feferman (1960).)