Just for reference the important caveat is no logic that can encode basic Peano arithmetic can be consistent and complete. There are plenty of axiomatic systems that are complete and consistent, even complicated mathematical ones (the first order theory of complete ordered fields, a.k.a. the real numbers is complete and consistent). Also a stronger logic can prove the consistency of a subset contained within it. Thus the first order theory of Peano arithmetic can be proved to be consistent via second order logic. Finally you may also want to look up Tarski's indefinitability of truth -- a theorem which gives the results you have here as a corollary, but is simpler: in sufficiently power logical systems you can't even define a truth predicate.