Well the issue I see is the very nature of Proof itself.
Axioms and proofs.
I can’t remember off the top of my head completely, but I believe that set theory, which I *think* is at the head of the families of mathematics generally, has the paradox of the complete and incomplete set and their relationship.
I don’t remember the verbage: I can find it though.
But here’s the resolution of the paradox: You ultimately have to move outside of the system. Because, here’s the thing:
It’s a system.
There are other systems.
You reach the limits of one then you have to move to another.
It’s unsatisfying of course but it’s how you get around paradoxes.
Of course the simplest way to get around a paradox?
“I don’t know.”
Accepting that as The Answer. Not “It can’t be known” – that’s a step too far. Rather it’s “I don’t know”. This eliminates the possibility that maybe somebody ELSE knows or that someday it might be known.
Of course, there’s currently no room for the Subjective “I” in the system itself.
It’s not a flaw of the system, just a limitation of it. In other systems, unique, personal subjectivity is welcome.