All consistent axiomatic formulations of number theory include undecidable propositions …

Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved …

How can you figure out if you are sane? … Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is “peculiar” or not, given that you have only your own logic to judge itself? I don’t see any answer. I am reminded of Godel’s second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent.

— Godel, Escher, Bach: An Eternal Golden Braid

— Douglas Hofstadter

2013.10.02 Wednesday ACHK