# 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?

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