  It turns out that the interpretability order is exceedingly complex. For example, through the uses of metamathematical coding techniques, one can show that 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. And for any theory T, one can show that there are theories T1 and T2 such that T1 &gt; T and T2 &gt; T and yet neither T1 ≤ T2 nor T2 ≤ T1 (that is, T1 and T2 are above T and are incomparable in the interpretability order). Thus, the order on the degrees of interpretability is neither well-founded nor linearly ordered. 