Algorítmico e não-algorítmico (2)

what Godel proved, beyond any doubt, is that when comes to axiomatizing simple arithmetic (not plane geometry), there are truths that "we can see" to be true but that can never be formally proved to be true. Actually, this claim must be carefully hedged: for any particular axiom system that is consistent (not subtly self-contradictory - a disqualifying flaw), there must be a sentence of arithmetic, not known as the Godel sentence of that system, that is not provable within the system but is true. (...) So that is what Godel, anchored by Turing to the world of computers, tells us: every computer that is a consistent truth-of-arithmetic-prover has an Achilles's heel, a truth it can never prove, even if it runs till doomsday. But so what? // Godel himself thought that the implication of his theorem was that human beings - at least the mathematicians among us - cannot, then be just machines, because they can do things no machine could do. More pointedly, at least some part of such a human being cannot be a mere machine, or even a hudge collection of gadgets. If hearts are pumping machines, and lungs are air-exchanging machines, and brains are computing machines, then mathematicians' minds cannot be their brains, Godel thought, since mathematicians' minds can do something that no mere computing machine can do.

Daniel Dennett, Darwin's dangerous idea, pp. 429-31.