A common Misconception about the Categorical Arithmetic
  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. 
Giuseppe Ragun\'i
math.GM