Actually, Arithmetic is
considered as syntactically incomplete. However, there are different types of
arithmetical theories. One of the most important is the second-order
Categorical Arithmetic (AR), which interprets the principle of induction
with the so-called full semantics. Now, whoever concluded that AR is sintactically (or semantically, since categoricity implies equivalence of
the two types of completeness) 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 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 fact 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.
Cite this paper
Raguní, G. (2017). A Proof of Syntactic Incompleteness of the Second-Order Categorical Arithmetic. Open Access Library Journal, 4, e3969. doi: http://dx.doi.org/10.4236/oalib.1103969.
Vaananen, J. and T. Wang (2015) Internal Categoricity in Arithmetic and Set Theory. Notre Dame Journal of Formal Logic, 56, 122. https://doi.org/10.1215/00294527-2835038
Raguní, G. (2011) Confines lógicos de la Matemática, revista cultural La Torre del Virrey Nexofía. [Logical Boundaries of Mathematics.] 266-280. on the Web: http://www.latorredelvirrey.es/nexofia2011-confines-logicos -de-la-matematica/
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