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

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Actually, Arithmetic is considered as syntactically incomplete. How-ever, there are different types of arithmetical theories. One of the mostimportant 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 certaintheory (such as PA) to another that includes it (such as AR). Of course,although the language of AR is semantically incomplete, this does notimply that the same AR is semantically/sintactically incomplete.Pending a response to the previous question, this paper aims to presenta proof of the syntactic/semantical incompleteness of AR, by examplesbased on the different modes of representation (i.e. codes) of the naturalnumbers in computation.

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