This primary purpose of this thesis is to show that a certain, interesting set of axioms does indeed yield 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 approximated from of strong completeness, whereby ?? ? (if and) only if ?? ?? for all n ?. This approximated from of strong completeness asserts that if ?? ?, then proofs from ?, being finite, can provide arbitrary better approximations of the truth of ?. We offer a definition of the language of continuous first-order logic and then supply a precise formulatio of its semantics. Indeed, this thesis can also be seen as an effort to precisely organize and unify the various presentations of continuous first-order logic found in the literature whichare often intimated in a rough-and-ready from. Finally, we state and prove various results needed to reach the goal of thesis: to state and prove the completeness theorem for continuous first-order logic. To follow our intuitions to this end, the structure of our approach is largely borrowed from the 31.8pt; HEIGHT: 45pt" id=_x0000_i1025 type="#_x0000_t75" is a recursive real,and moreover, uniformly computable from ?. If T is incomplete,we say it is decidable if for every sentence ? the real number is uniformly recursive from ?, where is the maximal value of ? consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) aximatization then it is decidable.