By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show (...) also that Vopěnka's proof can be reformulated in arithmetic by using the arithmetized completeness theorem. (shrink)

We argue that Anselm’s ontological argument (or at least one reconstruction of it) is based on an empirical version of Berry’s paradox. It is invalid, but it takes some understanding of trivalence to see why this is so. Under our analysis, Anselm’s use of the notion of existence is not the heart of the matter; rather, trivalence is.

This brief note corrects an error in one of the reduction steps in my paper 'Normalisation for Bilateral Classical Logic with some Philosophical Remarks' published in the Journal of Applied Logics 8/2 (2021): 531-556.

Après une étude du paradoxe de Richard, on considère plusieurs de ses solutions. On reformule ensuite le paradoxe grâce à la théorie de la complexité de Kolmogorov et on en donne une solution en partant de la démonstration par Chaitin du sens seulement relatif de la complexité de Kolmogorov.In this article, I study Richard’s paradox, and I consider several of its solutions. I then restate the paradox using Kolmogorov’s theory of complexity. Taking as a starting point Chaitin’s demonstration that Kolmogorov’s (...) understanding of «complexity» is only relative, I put forth a new solution to the paradox. (shrink)