Gödel Without Tears, slowly, 7

Next, we have the shortest chapter so far, on Quantifier Complexity, which introduces the notions of Delta_o, Sigma_1 and 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 Pi_1 sentence, that sentence must be true. You don’t have to believe the theory T is true to accept its 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 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.

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