Abstract

Two philosophical arguments, e.g. that the meaning of an expression transcends its use and that the human arithmetical thinking is not entirely algorithmic base their theses on Gödel’s first incompleteness theorem. But in both these arguments and in some of their criticisms the word “true” is often used ambiguously: it swings between a licit metamathematical use and an illicit transfer of it in a formal system. The aim of this paper is to show the way these arguments are connected, via G-type sentences, and how we argue that the sentence G, albeit unprovable in PA, is true, by using non-conservative extensions of PA with reflections. And this without any illicit use of “true”.