Why do Gödel's incompleteness theorems make mathematicians so sad? There are complete, decidable and consistent fragments of mathematics like the arithmetic of real numbers, complex numbers, quaternions, etc., Euclidean geometry and the non-Euclidean geometries (as they are interpretable inside Euclidean geometry). Is this big portion of mathematics not enough or not worthy of studying?
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How would you know whether any of those things are complete? And only in a few cases can they be considered consistent, much less decidable. Most of them can encode number theory indirectly, and since it is incomplete, so are they. To the extent you restrict them so that it does not admit this, (e.g. not allowing you to pick out the set of integers as special) you lose most of the interesting problems. Yes, we can optimize perfectly over the reals if we can't be fussy about integer answers, but we care about integer answers...– user9166Apr 14, 2016 at 18:53
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@jobermark I don't understand your point. Some theories, e.g. Euclidean geometry, are mathematically proven to be complete.– E...Apr 14, 2016 at 19:27
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Euclidean Geometry is complete, but only in a way that leaves out any problem otherwise associated with number theory -- which means the classical axiomatization it is a toy version of the theory that is not actually useful or interesting.– user9166Apr 14, 2016 at 19:30
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@EliranH (Which makes it clear you did not read the whole comment.) Euclidean Geometry as used is EG + all of arithmetic, not just EG. And that is incomplete (except in the sense of the 'Completeness Theorem')– user9166Apr 14, 2016 at 19:39
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@EliranH Well, the reals without the integers is not useful, whether or not it is interesting. I gave an example of why. Arithmetic is pretty basic to most useful questions. I cannot answer questions about the complexity of optimizing problems with integer restrictions using the theory of the reals alone. OK, fine, so general linear programming is quadratic in time. But we don't use it for so much, as most real problems involve some kind of integer restriction. I am not going to repeat myself again (thanks)– user9166Apr 14, 2016 at 19:55
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1 Answer
One possible answer is this: Hilbert's program was to prove the consistency of math (including e.g. set theory) using a very modest set of assumptions. Gödel's second incompleteness theorem shows this to be impossible. In fact, it shows that the method should be reversed -- we must use stronger theories to prove consistency, rather than weaker ones.
Peter Smith has a clear written passage about this in his great Introduction to Gödel's Theorems:
The real impact of the Second Theorem isn't in the limitations it places on a theory's proving its own consistency. The key point is this. If a nice arithmetical theory T can't even prove itself to be consistent, it certainly can't prove that a richer theory T+ is consistent. Hence we can't use 'safe' reasoning to prove that other more 'risky' mathematical theories are in good shape. For example, we can't use unproblematic mathematical reasoning to convince ourselves of the consistency of set theory (with its postulation of a universe of wildly infinite sets).
And that is a very interesting result, for it seems to sabotage what is called Hilbert's Programme, which is precisely the project of defending the wilder reaches of infinitistic mathematics by giving consistency proofs which use only 'safe' methods.
Does this actually trouble mathematicians? Probably not in practice.
As for the first incompleteness theorem, again I don't think it troubles mathematicians. Even if confronted with an unprovable sentence, they can try and prove it (or its negation) in a stronger theory. See, for example, the case of Goodstein's Theorem.
