The standard proof of the completeness theorem in first-order Gödel logic is based on a first-order countable language. I want to know that is there any proof of the completeness theorem in first-order Gödel logic for uncountable language, or is there an example that refute the completeness of first-order Gödel logic in the case of uncountable languages?

Background: The proof of completeness theorem in first-order Gödel logic is based on a Henkin construction. Let $T$ be a complete consistent theory, i.e., all sentences can not be formally deduced from $T$ and for any pair $(\varphi,\psi)$ of sentences either $T\vdash\varphi\to\psi$ or $T\vdash\psi\to\varphi$. define the equivalence relation $\sim$ on the set of sentences as $$\varphi\sim\psi~~~~ \mbox{if and only if}~~~~ T\vdash\varphi\leftrightarrow\psi.$$ The set of equivalence classes of $\sim$ form a countable linearly ordered set $A$ which can be embedded continuously is $[0,1]$. Countability of the language that leads to the countability of $A$ is an essential part of the proof. Using this embedding one could construct a model for $T$ which show that $T$ is satisfiable. (See Hajek: Metamathematics of fuzzy logic).