Can a mathematical proof always be objectively determined as correct or incorrect?

Question

Fields medalist Michael Atiyah claimed a simple proof of the Riemann hypothesis, but many mathematicians rejected his proof. Am I right in saying that Atiyah's proof is either objectively correct (then why did many mathematicians reject it?) or objectively incorrect (then as one of the leading mathematicians, Atiyah failed to find his own mistake(s)). Or is it possible that the truthfulness of Atiyah's proof cannot be determined?

So my question is,

In a certain axiomatic system, can a mathematical proof always be objectively determined as correct or incorrect? Or is it possible that there exists a proof to a result such that the truthfulness of the proof cannot be determined?

Here I am not considering the cases such as the proof contains typos or gramatical mistakes which can easily be fixed.

Answer 1

For proofs on paper one should always be able to check if its objectively correct.

The situation is different if its a computer proof such as the four-color problem. Here a program checks all the possible cases. Its hard for the individual to check if its objectively correct. At least, its questionable.

Answer 2

If you completely formalize the statement and the proof, it can be, in principle , checked automatically, whether the given proof shows the given statement. This is also true for computer proofs because they must have a finite running time. This does however not at all mean that such a verification can be done in a reasonable time.

A SIMPLE proof of the Riemann hypoethesis is so unlikely, that I heavily doubt every such proof whoever claims to have found it.