The author of the article does not stipulate that the axiom system has to be recursively enumerable. From the article: We assume it to contain the basic Peano

The author of the article does not stipulate that the axiom system has to be recursively enumerable. From the article: We assume it to contain the basic Peano axioms of arithmetic. An axiom system is complete, if every true statement can be proven within the system. The theorem in the article is true for recursively enumerable axioms systems that contain first order arithmetic. EDIT: Note that the second order Peano Axioms are categorical but not recursively enumerable. This is also an axiomatic system that has finitely many axiom schema.