> I'm not sure I understand what this changes to the overall situation. Remember that a formal system also can't prove its own consistency, so you can't just ignore bits that you don't like and expect that it doesn't matter, it's possible that one of these "true but unprovable" propositions would actually render your system inconsistent, in which case it ought to be addressed.

Is that true? It seems to me that if a statement is unprovable, then even if it would in theory render your system inconsistent, there's no way that can actually happen.

Since, if you found any example of a chain of logic which leads to an inconsistency, that would count as a proof of the original statement. Which is unprovable, and as such can't have proofs.