A proof of completeness for continuous first-order logic

The primary purpose of this article is to show that a certain natural set of axioms yields a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an \emph{approximated} form of strong completeness, whereby $\Sigma\vDash\varphi$ (if and) only if $\Sigma\vdash\varphi\dotminus 2^{-n}$ for all $n<\omega$. This approximated form of strong completeness asserts that if $\Sigma\vDash\varphi$, then proofs from $\Sigma$, being finite, can provide arbitrary better approximations of the truth of $\varphi$.