Godel's Incompleteness Theorem Quotes

It was, Elvi thought, like finding a sea turtle who thoroughly understood Godel’s incompleteness theorem, but didn’t have any sea-turtley application for it.

But you can't prove God exists. And isn't that what all science is ultimately about? Proving theories about the universe?" "Provability is not truth, Caro. Godel's incompleteness theorem tells us that, if we didn't already know it intuitively, which we do.

[C]onsistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system.

In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert's program with a single stroke. So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it's worth, I'm not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night.

Explanation is always incomplete: we can always raise another Why-questions. And the new why-questions may lead to a new theory which not only "explains" the old theory but corrects it. This is why the evolution of Physics is likely to be an endless process of correction and better approximation. And even if one day we should reach a stage where our theories were no longer open to correction, because they are simply true, they would still not be complete - and we should know it. For Godel's famous incompleteness theorem would come into play: in view of the Mathematical background of Physics, at best an infinite sequence of such true theories would be needed in order to answer the problems which any given (formalized) theory would be undecidable. Such considerations do not prove that the objective physical world is incomplete, or undetermined: they only show the essential incompleteness of our efforts. But they also show that it's barely possible (if possible at all) for science to reach a stage in which it can provide genuine support for the view that the physical world is deterministic. Why, the, should we not accept the verdict of common sense- at least until these arguments have been refuted?

Gödel’s scheme has nothing to do with mathematics in and of itself. It concerns false approaches (i.e. non-ontological approaches) to the definition of what math is. The incompleteness theorems proved that such approaches are doomed to failure. Gödel didn’t prove a single thing about what math is. What he proved is what’s it’s not. He proved that it definitely isn’t manmade.