What are the implications of Gödel's Incompleteness Theorem?

For those not familiar with it, Gödel’s Incompleteness Theorem says that for any set of mathematical axioms that includes arithmetic, there are statements that can be formulated that are not provable in the system. What I know about the theorem, I got from the book Gödel, Escher, Bach

On one extreme, there are those who say that since the true (outside the axiom system) but unprovable (within the system) statement that Gödel came up with was so convoluted, that there is no serious implication. On the other end, there are those who say that the theorem shows that there are limitations to any computational system and therefore true artificial intelligence is impossible.

I go along with Hofstadter’s middle position. There seems to be an implication that it is sometimes necessary to think outside the box and to be able to think about the thought process. This does not preclude thinking robots, but it supports the idea that these robots might have to feel emotions and to respond to the world more like humans than Mr. Spock.