A Proof Of Syntactic Incompleteness Of The Second-Order Categorical Arithmetic

Actually, Arithmetic is considered as syntactically incomplete. However there are different types of arithmetical theories. One of the most important is the SecondOrder Categorical Arithmetic(AR), which interprets the induction principle by the socalled full semantics.Now who ever concluded that AR is sintactically (or semantically, from the categoricity)incomplete? Since this theory is not effectively axiomatizable, the incompleteness Theorems cannot be applied to it. Nor is it legitimate to assert that the undecidability for a statements is generally kept in passing from a certain theory (e.g. PA) to another that includes it (e.g. AR). Of course, although the language of AR is semantically incomplete, this does not imply that the same AR is semantically/sintactically incomplete. Pending a response to the previous question, this paper presents a proof of the syntactic/semantical incompleteness of AR, by examples based on the different modes of representation (i.e. codes) of the natural numbers in computation