The law of excluded middle in mathematics

Question

I want to make sure that I'm understanding this correctly.

Let ZFC denote the Zermelo-Frankel theory of sets with the axiom of choice. Let H denote the continuum hypothesis.

Let A be a formula of ZFC. Then (A or not A) is a theorem of ZFC. In particular, (H or not H) is a theorem of ZFC. However, neither H nor (not H) is a theorem of ZFC. That is, H is undecidable.

Is any of this incorrect?

Answer 1

Yes, correct.

(Assuming ZFC) one can construct various models of ZFC.

For any closed formula $A$ and any model $M$ we either have $M\models A$ or $M\models \lnot A$.
A formula is a theorem for ZFC iff every model of ZFC validates it.

Thus, $A\lor\lnot A$ is indeed always a theorem, but for the continuum hypothesis, there are models that satisfy it and there are also which don't.