Why does the existence of independent statements not prove consistency? I've read before that, by the Principle of Explosion, if a theory is inconsistent, then absolutely any statement can be proven within it.
Obviously, there are statements which are independent of ZFC (Continuum Hypothesis, etc). It seems to me that this proves that ZFC is incomplete. Why does it not then follow that ZFC is consistent? It seems to me that we could say "Assume ZFC is inconsistent. Then the the Continuum Hypothesis is provable in ZFC. But the Continuum Hypothesis is neither provable nor disprovable in ZFC. Therefore ZFC is consistent."
What am I missing here?

Independence proofs generally have as an explicit premise that ZFC is consistent. Thus, what is proved is

If ZFC is consistent, then the Continuum Hypothesis is independent of ZFC.

Therefore the reasoning you sketch is not available.
