TITLE: Is any true sentence in the second-order Peano Axioms provable
QUESTION [11 upvotes]: Forgive the elementary nature of the question. I understand that the second order Peano Axioms are categorical in the sense that all their models are isomorphic. This equivalence class of models is taken to be the definition of natural numbers. 
My question is that: Is the theory defined by those second order axioms complete? i.e. is every second-order statement in this theory provable or disprovable from the axioms?
For the first order Peano Arithmetic axioms any statement that is true in all models is provable or disprovable by the completeness of first-order logic, but of course this is in applicable to the second-order axioms.

REPLY [4 votes]: Between them, the previous comments and replies have already covered all of the relevant points, but here in any case is a direct, unsophisticated answer:
Any effective procedure (i.e. anything you could program up) for proving statements of second-order logic must be incomplete (unless it proves false statements). Otherwise, because second-order arithmetic is categorical, we would have an effective procedure for determining the arithmetic truth. But this is impossible by Gödel's first incompleteness theorem.