Unfortunately I have never taken a class in mathematical logic. Shame on me. But I am reading here (https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC) that the consistency of the ZFC axioms is independent of the ZFC axioms itself.
But this tells me that ZFC must be consistent: indeed, suppose it was not, then there would exists a statement A such that both A and the negation of A can be justified in ZFC. But then ZFC would be inconsistent. Hence consistency of the ZFC axioms is not independent of ZFC, a contradiction.
Why is my justification wrong?