A common Misconception about the Categorical Arithmetic

TL;DR

This paper clarifies misconceptions about applying incompleteness theorems to categorical arithmetic and offers an alternative interpretation of Gödel's statement to correct common misunderstandings.

Contribution

It corrects widespread misconceptions regarding the applicability of incompleteness theorems to categorical arithmetic and provides a new interpretation of Gödel's original comments.

Findings

Incompleteness theorems do not apply to categorical arithmetic.

The semantic incompleteness of second-order logic is misunderstood.

An alternative interpretation of Gödel's statement is proposed.

Abstract

Although the categorical arithmetic is not effectively axiomatizable, the belief that the incompleteness Theorems can be apply to it is fairly common. Furthermore, the so-called "essential" (or "inherent") semantic incompleteness of the second-order Logic that can be deduced by these same Theorems does not imply the standard semantic incompleteness that can be derived using the Loewenheim-Skolem or the compactness Theorem. This state of affairs has its origins in an incorrect and misinterpreted Goedel's comment at the Koenigsberg congress of 1930 and has consolidated due to different circumstances. This paper aims to clear up these questions and proposes an alternative interpretation for the Goedel's statement.