How do we know axioms are sufficient?

[Avichal]

In math, we define new concepts and have certain axioms regarding the concept that are believe to be true.
Now how do we know that the given axioms are enough to give a proof to any problem?

I hope this makes sense.

 

[micromass]

In math, we define new concepts and have certain axioms regarding the axioms that are believe to be true.
Now how do we know that the given axioms are enough to give a proof to any problem?

I hope this makes sense.

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.

 

[johnqwertyful]

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.

 

[johnqwertyful]

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.

Ahh, beat me to it.

 

[Avichal]

So whenever there is a problem, there is a possibility that it cannot be solved by the present axioms? Has there been a case like that?
What do we do in case we encounter some problem like that?

 

[micromass]

So whenever there is a problem, there is a possibility that it cannot be solved by the present axioms? Has there been a case like that?

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.

What do we do in case we encounter some problem like that?

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.

 

[Avichal]

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.

Why don't we try to change our axioms? Obviously if we are not able to show a proof, our axioms are not sufficient and should be changed.

 

[SteamKing]

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.

 

[micromass]

Why don't we try to change our axioms? Obviously if we are not able to show a proof, our axioms are not sufficient and should be changed.

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.