# Poignancy because of Gödel's theorems - why?

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?

• 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...
– user9166
Apr 14, 2016 at 18:53
• @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
• 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.
– user9166
Apr 14, 2016 at 19:30
• @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')
– user9166
Apr 14, 2016 at 19:39
• @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)
– user9166
Apr 14, 2016 at 19:55