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I'm not a mathematician so apologies if this question makes no sense.

My understanding of modern mathematics is that it is built on a set of axioms known as Zermelo–Fraenkel set theory + the axiom of choice. In other words everything else (such as the fundamental theorem of calculus) depends on ZFC being valid. However Godel's first incompleteness theorem states that there are statements that ZFC can neither prove nor disprove. That seems to imply that any mathematics project might be futile because the desired result is undecidable.

For example take Goldbach's conjecture. It's a well-known problem, easily posed, and seems like a problem that number theorists would be interested in. However if I'm understanding things right, for all we know, Goldbach's conjecture might be undecidable. In that case anyone attempting to prove Goldbach's conjecture is off on a wild goose's chase, as is anyone attempting to find a counterexample. Why bother then?

My best guess is that one of the following is true:

  • They knew beforehand (how?) that Goldbach's conjecture is not undecidable.
  • They think that proving that Goldbach's conjecture is undecidable is still a worthy result. This is weird to me since presumably the methods necessary to prove it's undecidable will be very different from the methods used to prove its truth/falsehood, so they won't even reach the proof in the first place. Instead it'll be "I'm working on it" for years and then someone proves that I've wasted years of my life trying to solve an undecidable problem.
  • Mathematicians first prove that a problem is decidable before attempting to solve it. This is also weird to me since from Wikipedia's page on Goldbach's conjecture, there's literally no mention on whether or not it is decidable. That implies that either the conjecture is trivially decidable, or this guess is wrong. The former seems impossible since there's nothing special about Goldbach's conjecture (it could equally be any other conjecture).
  • Something else?

Any explanation is appreciated!

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  • $\begingroup$ In fact, apart from rare exceptions, it is always possible that an unsolved problem is undecidable. In particular, noone can rule out that the Goldbach conjecture is undecidable. Nevertheless, it is interesting to study the conjecture, even if chances are small that it is ever solved. Goldbach's conjecture is also an example of a conjecture almost every mathematician is convinced that it must be true. $\endgroup$ – Peter Dec 31 '18 at 15:18
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Related thread: Could it be that Goldbach conjecture is undecidable?

I think that they don't know for sure that the Goldbach conjecture is decidable. They may have some evidence that makes them think it's decidable, and they just try to solve it anyway and hope for the best.

Ideally as a mathematician is working on such a problem, the research will be flowing, so that interesting ideas are being discovered along the way. These ideas have value and beauty of their own even if the ultimate goal of resolving the Goldbach conjecture is never reached. It would probably be a bad strategy to spend like 20 years trying to solve a famous math problem if interesting ideas and partial progress weren't flowing.

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