We will use Goldbach's conjecture as an example$^1$. It is either true or false that every even number greater than 2 is the sum of two primes. Let's take a look at these two scenarios.
Goldbach's conjecture is true: If it is true it might be provable or it might not be. We currently don't know.
Goldbach's conjecture is false: If it is false, it definitely has a disproof as a counterexample must exist. A counterexample would be an even number that is not the sum of two primes.
If someone claims to have proven that Goldbach's conjecture has no proof, we can infer that Goldbach's conjecture must be true. This is because it can only have no proof if the conjecture is true. As a result, by proving that the conjecture has no proof, the conjecture has been shown to be true and hence proven.
There appears to be a clear contradiction here and hence I believe that it is impossible to prove that Goldbach's conjecture has no proof. Is there an issue with my reasoning or is it possible to prove that Goldbach's conjecture has no proof?
1 - Any conjecture that can be proven false with a counterexample would be applicable here.