Is the first-order incompleteness of a theory (like arithmetics, set theory or logic itself) avoidable in a second or higher-order axiomatizations? Can we avoid the first-order incompleteness of a theory (like arithmetics or set theory) in a second-order theory which contains the previous? How does it depend on the chosen semantics or models? If we can't, can we, at least, construct a second or higher-order theory which contains the previous and avoids the undecidability of its first-order propositions? Can we do it in general?, i.e., if we have a n-order theory with undecidable propositions, can we construct a n+k-order theory which contains the last and avoids (that is, in where these are decidable) its n or lower-order undecidable propositions? How does it depend on the chosen semantics or models?
What if we change the theory by the logic itself? If we have a n-order logic (with n>1 to ensure the incompleteness under some semantics) with undecidable propositions, can we construct a n+k-order logic which contains the previous and avoids (that is, in where these are decidable) its n or lower-order undecidable propositions? How does it depend on the chosen semantics or models?
Can we construct a second or higher-order complete arithmetics? How does it depend on the chosen semantics or models?

Answer is YES, but the higher order Theory have to be NOT FORMAL, namely not denumerable (condition necessary but not sufficient). What Mummert says is true only for FORMAL System (higher order System CAN be formal: in this case they will be denumerable (although denumerability does not imply always formality)).
For, incompleteness theorem demands formality for the System (and the goedelian number is a denumeration kind for the propositions).
An example is Second order Arthmetics, not formal System (not denumerable propositions), cathegoric and maybe sintactly complete.
Incompleteness theorem also demands effective axiomatizability for the System. This is why a first-order complete axiomatization of arithmetic can be built (OK HURKIL): we can add to PA's axioms all its true propositions in the standard model. The new System is sintactly complete for construction but not effectively axiomatizable.
Giuseppe Raguní
