Slobodan Vujošević | Searcharxiv

arXiv2024

Kurt Gödel and the Logic of Concepts

The literature dealing with Gödel's legacy is largely preoccupied with challenging his philosophical views, regarding them as outdated. We believe that such an approach prevents us from seeing Gödel's views in the right light and understanding their rationale. In this article, his views are discussed in the philosophical realm in which he himself understood them. We explore the consequences of Gödel's incompleteness theorems for the question of the objectivity of mathematics and its epistemology. Taking set theory as the paradigm of formal mathematical theories, we examine the relationship between its incompleteness and extensionality. We argue, based on his philosophical views, that Gödel believed incompleteness can be overcome only by some intensional considerations about concepts from the basis of mathematical theories. These considerations should eventually lead to founding the so-called logic of concepts.

math.HO↗

arXiv2022

Logic of the Ontological Argument

In his ontological argument Gödel says nothing about its underlying logic. The argument is modal and at least of second-order and since S5 axiom is used so it is widely accepted that the logic of the argument is the S5 second-order modal logic. However, there is a step in the proof in which Gödel applies the necessitation rule on the assumptions of the argument. This is repeated by all of his followers. This application of the necessitation rule can seriously harm the consequence relation of the logic of the ontological argument. It seems that the only way to preserve the modal logic S5 for the ontological argument is to assume some of its axioms in the necessitated form.

math.LO↗