So these are mostly doubts I have, but:
What's the difference (if any) between the meta statement "ZFC is consistent" and the formal statement Con(ZFC) expressed in ZFC? I've heard that they're different, but how? Con(ZFC) cannot be proven in ZFC, is there any way either Con(ZFC) or "ZFC is consistent" can be proven, perhaps in another axiomatic system? And if not, what's the difference between the two?
Can a statement be true but unprovable? What does it mean to say a statement is true if it is unprovable?
Is it possible to prove within ZFC the equivalence of a statement X and Con(ZFC)?
What does it mean for Con(ZFC) to be false? Can there exist a proof in ZFC that Con(ZFC) is false?
If Con(ZFC) is false, does it also simultaneously become true? In general if a system of axioms is inconsistent, doesn't it mean that every statement expressible in that system can be proven both true and false?
Is the set of statements equivalent to Con(ZFC) a strict subset of the set of statements independent of ZFC? Does the notion of statements independent of ZFC exist if ZFC is inconsistent?
Any reading material that you would recommend to understand this topic better?