This question already has an answer here:
I am aware of Gödel's second incompleteness theorem, the proven existence of several unprovable statements (in ZFC), and the possibility that a formal system may include statements that are unprovably unprovable, that is, that there are statements in them which cannot be proven by them, but additionally, they cannot even be proven to be unprovable by them. My question is this:
Can the existence of unprovably unprovable statements be proven/disproven in ZFC or a (not necessarily conservative) extension like TG or MK?
(I am aware of a similar question which asks if there can be unprovably unprovable statements in formal systems, however this asks if the existence of such statements is provable in ZFC, which is a different matter)