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.



Is any true sentence in the second-order Peano Axioms provable
Note that there are two different types of models of second-order logic: standard models, where second-order quantified variables range over all subsets of the domain; and Henkin models, where second-order quantified variables are allowed to range over a proper subset of the full power-set.


Henkin proved the completeness theorem for second-order logic for Henkin models, so if a sentence is true in all Henkin models, it is derivable. For completeness to hold, it is not enough to consider only standard models, so even though second-order PA is categorical for standard models, it is not complete (as we would expect, this being a corollary Gödel's theorem).


