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

Actually, Arithmetic is considered as syntactically incomplete. How-
ever, there are different types of arithmetical theories. One of the most
important is the second-order Categorical Arithmetic (AR), which inter-
prets the principle of induction with the so-called full semantics. Now,
who ever concluded that AR is sintactically (or semantically, since cate-
goricity implies equivalence of the two types of completeness) incomplete
Since this theory is not effectively axiomatizable, the incompleteness The-
orems cannot be applied to it. Nor is it legitimate to assert that the un-
decidability of the statements is generally kept in passing from a certain
theory (such as PA) to another that includes it (such as 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 aims to present
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.