Consequences of a Goedel's misjudgment

TL;DR

This paper corrects a misinterpretation of Goedel's remark, clarifying its implications for second-order arithmetic and semantic incompleteness, and proposes an alternative interpretation to prevent misconceptions in logic.

Contribution

It offers a corrected understanding of Goedel's comment and its impact on the application of incompleteness theorems to second-order languages.

Findings

Clarifies the semantic properties of second-order arithmetic.

Highlights the misapplication of Goedel's theorems to semantic incompleteness.

Proposes an alternative interpretation of Goedel's remark.

Abstract

The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce dangerous fruits, as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a language and some question about the semantic completeness: in particular is highlighted the fact that a non-formal theory may be semantically complete despite using a language semantically incomplete. Finally, an alternative interpretation of the Goedel's unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic incompleteness, categoricity, arithmetic, second-order languages, paradoxes