# 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?