Are there unsolved problems known to be not independent of the axiomatic system it is proposed in?
For example, is Goldbach's conjecture known to be provable using the axioms of PA?
I believe I have heard something like this, but I can't find it anywhere on the internet.
If there is such problem, can you give a reference to the proof that it is not independent? (If it is not very long, you can also place it as an answer of course)
Note: It is also fine if it is a problem that has been solved, as long as the proof that the problem isn't independent doesn't use the fact that is has been proven. I'm just interested in how such proofs look.
Thank you in advance.
I already have the category where solving the problem is reduced to a finite, though too long, computation, thanks to user41404. Are there other problems known?
There is a counterexample to Goldbach somewhere between $10^{99999}$ and $10^{999999}$. There is an odd number of twin primes in the same interval. To prove or disprove just check all cases.