How do we know axioms are sufficient?

We don't know it because it's not true. Given any (sufficiently large) axiom system, there are always true theorems that cannot be proven. So we cannot prove any problem.

This is basically Godel's incompleteness theorem.

 

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.

 

Ahh, beat me to it.

 

Yes, there have been cases like that. One of the most famous problems is the continuum hypothesis: http://en.wikipedia.org/wiki/Continuum_hypothesis. It has been shown that it cannot be proven or disproven using the usual ZFC-axioms.

We try to show that no proof exists to prove the claim and that no proof exists that disproves the claim. That is all we can do. We can also add some stronger axioms and then show that those imply the problem, but then we change our axioms.

 

Which axioms would you change? Why?

In the CH controversy, it was shown that even if certain axioms were adopted, CH could still not be proven or disproven; the CH problem was independent of the axioms which would be used to prove or disprove it.

Other axioms have been proposed, and the controversy continues, unresolved.

In any system which relies on certain statements taken as axioms, there will be problems which cannot be proven beyond a reasonable doubt, as the lawyers and the logicians say.

One other famous axiom which generated much controversy was Euclid's Fifth Postulate, about parallel lines.

http://en.wikipedia.org/wiki/Parallel_postulate

This axiom has kept divers mathematicians busy for hundreds of years, trying to prove or disprove it. It has even led to the development of various non-Euclidean geometries, which dispense with Euclid's parallel lines and substitute other axioms in its place.

 

Godel has proven that we can never find an consistent (= useful) axiom system in which you can prove every true statement. So there are no sufficient axiom systems.