a paragraph that stuck me on page 126 of the interesting book with the horrendous cover with the great title on it:

“It is often said that it is only by ‘going outside the system’ that one can prove the Gödel sentence of a theory such as PA, an image that reinforces the idea that ‘a system cannot understand itself fully’. This is correct in the sense that one cannot prove, or even truly postulate, the Gödel sentence of the system, or equivalently the consistency of the system, in the system, that is, by a proof formalizable in the theory. But the image of ‘going outside the system’ is a bit too seductive, in that it suggests that there is some generally applicable way of viewing a system ‘from outside’ so as to be able to prove things about it that are not provable in the system. We don’t know of any such general method. Consider the theory T obtained by ading Goldbach’s conjecture as a new axiom to PA. T is consistent if and only if Goldbach’s conjecture is true, but we have no inkling of any method of ‘going outside the system’ by which T might be proved or ‘seen’ to be consistent.”

The book is full of passages like these, in which general and sweeping claims based on Gödel’s theorems are carefully dismantled and in which the subtleties connected to them are clearly explained. Great reading.

The full details of the book:

Torkel Franzén, “Gödel’s Theorem – An Incomplete Guide to its Use and Abuse”,
A.K.Peters, Wellesley, Massachusetts, 2005.