Next, we have the shortest chapter so far, on Quantifier Complexity, which introduces the notions of $latex \Delta_o, \Sigma_1$ and $latex \Pi_1$ wffs.
But there is an intriguing little result in this chapter. If the consistent theory T which includes Robinson Arithmetic Q proves a given $latex \Pi_1$ sentence, that sentence must be true. You don’t have to believe the theory T is true to accept its $latex \Pi_1$ consequences are true. For example, suppose T is the wildly infinitary apparatus that Wiles uses to prove Fermat’s Last Theorem, which is equivalent to a $latex \Pi_1$ sentence. Then you don’t have to believe that infinitary apparatus is actually correct (whatever exactly that means); all you need to believe to accept Fermat’s Last Theorem (assuming Wiles’s corrected derivation from T is correctly done) is that T is consistent. I still find that rather remarkable.[Link now removed]