Let be Peano Arithmetic. Gödel’s Second Incompleteness Theorem says that no consistent theory extending can prove its own consistency. (I’ll write for the statement asserting ‘s consistency; more on this later.)

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