Kurt Godel: The World's Most Incredible Mind.

"Either mathematics is too big for the human mind or the human mind is more than a machine" ~ Godel

Kurt Godel (1931) proved two important things about any axiomatic system rich enough to include all of number theory.

1) You'll never be able to prove every true result..... you'll never be able to prove every result that is true in your system.

2) Godel also proved that one of the results that you can never prove is the result that says that the system is consistent. More precisely: You cannot prove the consistency of any mathematical system rich enough to include the known theory of numbers.

Hence, any consistent mathematical system that is rich enough to include number theory is inherently incomplete.

Second, one of the propositions whose truth or falsity cannot be proved within the system is precisely the proposition that states that the system is consistent. "

What Godel's proof means, then, is that we can't prove that arithmetic—let alone any more-complicated system—is consistent.

For 2000 years, mathematics has been the model—the subject—that convinces us that certainty is possible. Yet Now there's no certainty anywhere—not even in mathematics.

More...
http://teachingcompany.12.forumer.com/viewtopic.php?t=2327

Goedel's Ontological Proof.

For those interested in a discussion of Goedel's reasoning for God, then I suggest starting with this heavily annotated work, which I recently stumbled upon.

www.scribd.com/doc/95364925/Goedel-s-God-Proof-Annotated-Version

"It's not that God is subject to the Freedom Proof or the Doubt Proof.
According to Gödel, He's not. But we have to be, or else we are not free. So
our truth game with God turns into something like Feynman had described.
Feynman's Gods, every time physicists think they have the rules of the game
figured out, throw in a new wrinkle. They let people like Feynman make
progress, but if the Feynmans of the world learn too much, physics will stop
being the joy and challenge that it is. The Gods don't let that happen.

Gödel's God has to be very careful about how he lets our universe unfold.
If the world becomes totally controllable and comprehensible, we'll be God.
God does not object to that. In fact, according to Gödel, that is our destiny.
But it is also the end of us as free human beings. And human freedom is an
essential part of the beauty of God's universe."

~ page 251