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Is all of mathemtics debatable?
By Andrew Smith (P2517) on Wednesday, July 19, 2000 - 10:44 pm:

Does the work done by Godel and others make all of mathematics debatable? Is it true to say that 1+1=2? What I'm trying to ask is; is there a point below which everything is OK and above which things cannot be proved or does the nonprovability of some things make nothing provable? Sorry if this is a bit of a mess but I can't really word it very well, any comments would be appreciated. Thanks.

By David Loeffler (P865) on Wednesday, July 19, 2000 - 11:33 pm:

Godel's theorems essentially state that there must exist some theorems that are unprovable in that they cannot be derived from axioms.
This does not mean that theorems already proven from these axioms are invalid; so within the axioms of set theory, it is still true that 1+1=2, since this does indeed follow from the axioms.

What one might suggest is that the theorems mean that the axiomatic model of proof is invalid, but to suggest that would be to put the entire mathematical community out of a job! Systems of axioms may not be all-poweful, but as far as they reach they are as sound as they ever were.

What does everyone else think?

By Dave Sheridan (Dms22) on Thursday, July 20, 2000 - 12:14 pm:

I'd prefer to look at it with the view that we misunderstand what proof should be. According to our current background in logic and set theory, there are serious deficiencies (for example, first order logic can't even define the real numbers and that's what they teach undergraduates) but it's better than nothing; anything we do prove in logic is fine but there may be things we can't prove, which could be done under a different system.

Andrew, do you know what Godel's theorems actually are? From the sound of the question, the answer is probably no. What you're referring to is that no matter which system of axioms we adopt, there will be some theorems which can neither be proved or disproved. On the other hand, such things can be resolved simply by adding sufficiently many axioms. A good example of this is number theory problems which require transfinite induction so can't be proved in Peano arithmetic. However, the theorems certainly are true, and are logically correct once the concept of infinity is understood.

-Dave