Propositional logic --- syntactical completeness
You are not doing anything "wrong", you just missed the point of the incompleteness theorem, a bit.

What you claim (correctly) is that a proof system with no axioms is incomplete. You also say that this is not very impressive...

What Godel showed is something much stronger: consider the theory of Peano arithmetic, then there does not exist a complete axiomatization for it. 

How does this relate to your post? Well, in your suggestion, even when you consider an empty set of axioms, there are still some true sentences (the tautologies), and there are axiom systems that capture exactly those sentences.