Monthly Archives: March 2009

What Happens When You Iterate Gödel’s Theorem?

Let \mathrm{PA} be Peano Arithmetic.  Gödel’s Second Incompleteness Theorem says that no consistent theory T extending \mathrm{PA} can prove its own consistency. (I’ll write \mathrm{Con}(T) for the statement asserting T‘s consistency; more on this later.)

In particular, \mathrm{PA} + \mathrm{Con}(\mathrm{PA}) is stronger than \mathrm{PA}.  But certainly, given that we believe that everything \mathrm{PA} proves is true, we believe that \mathrm{PA} does not prove a contradiction, and hence is consistent.  Thus, we believe that everything that (\mathrm{PA} + \mathrm{Con}(\mathrm{PA})) proves is true.  But by a similar argument, we believe that everything that (\mathrm{PA} + \mathrm{Con}(\mathrm{PA} + \mathrm{Con}(\mathrm{PA}))) proves is true.  Where does this stop?  Once we believe that everything \mathrm{PA} proves is true, what, exactly, are we committed to believing?

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