A set of axioms is ("syntactically", or "negation"-) complete if, for any statement in the axioms' language, that statement or its negation is provable from the axioms (Smith 2007, p.  24). This is the notion relevant for Gödel's first Incompleteness theorem. It is not to be confused with "semantic" completeness, which means that the set of axioms proves all the semantic tautologies of the given language. In his completeness theorem, Gödel proved that first order logic is "semantically" complete. But it is not syntactically complete, since there are sentences expressible in the language of first order logic that can be neither proved nor disproved from the axioms of logic alone.