So this is a follow-up to this question of mine. According to the answers there so far, the law of excluded middle (i.e. the notion that $\varphi\vee\neg\varphi$ is true for all wffs $\varphi$ for every interpretation/structure/model) holds in first-order logic.
My question now is, how is this compatible with the independence of the continuum hypothesis from ZFC set theory? The continuum hypothesis presumably can be written as a wff in the language of first-order logic, so it has to have a true or false assignment to it. However, it is also independent from the axioms of ZFC, so as far as standard set theory is concerned, it's not determined whether the statement is true or false.
How is this conundrum resolved or dealt with? What's the right way to think about this dilemma?