Some statements can nether be proven nor disproven in the limit. This means a number of things, and the shades of those meanings are actually pretty profound for Math, Science, Logic and Software.

Some points:

[disclaimer, I am REALLY rusty on this stuff, so take it with a grain of salt]

As Al pointed out, that you can't prove 'that P' does not mean P (P is some proposition) is false, but it absolutely does not mean that its true. There are (infinite) classes of mathematical intuitions that we can't prove true, but really sure do sound true.

Beyond that, another very interesting related problem is provability in the limit. Can you prove that you can't prove something? Turing pointed out that even a 'perfect' computer, an Ideal computer (the greatest brain, or a computer that uses every bit of mass in the universe to store and process information), can not compute if some other machine (brain, whatever) will ever halt (come to solution).

[Machine here is loose. Think 'mechanical rule-driven procedure', like some types of proofs can be.]

Think of a machine here as a theorem prover.

So suppose ideal-machine A is trying to prove a certain mathematical proposition. We are curious to see if the proposition A is working on can ever be proven, so we ask ideal-machine 2 to prove that machine A can find a proof.

Its provably true that machine B can never compute the answer we asked of it.

So not only are there cases where we do we not know, we don't know if we can know.

But Obviously Mathematics/Logic must be in pretty good shape if we can build skyscrapers and plot the orbits of the planets.

What the limits on mathematics, logic and computing really means to human knowledge and even human thought is really an open question. Its not entirely clear (though its probably a safe bet) that the Human Brain thinking is a 'rules-driven mechanical procedure'. Its not clear how much Godel and Turings results will alter science and knowledge moving forwards.

Davin :.