What I am trying to understand is why this is considered a solvable problem, when essentially you need to devise a method to find a better solution to the probl

What I am trying to understand is why this is considered a solvable problem, when essentially you need to devise a method to find a better solution to the problem in order to prove that there isn't a solution. Aren't all proofs flawed in this way? Like for example, 2 = fish. Prove that there isn't an undiscovered mathematical axiom that makes this not nonsense?