[Filligree]

The latter would be an axiom. A disproof would be a proof that there is no proof, so if you’d proven that no proof exists one way or the other then you’ve proven it can’t be disproven _or_ proven.

Which means you’ve hit a branch in mathematics. You can assume it to be either true or false, and you’ll get new results based on that; both branches are equally valid.

[LegionMammal978]

Constructively speaking, a disproof is a "proof that a statement leads to a contradiction". A "proof that there is no proof (assuming consistency)" can exist just fine, and that's exactly what the incompleteness theorem is, alongside a "proof that there is no disproof, i.e., that a contradiction cannot be derived from the statement (assuming consistency)".