Kurt G\"odel and the Logic of Concepts

TL;DR

This paper explores G"odel's philosophical views on the objectivity of mathematics, emphasizing the importance of the logic of concepts and intensional considerations to address incompleteness.

Contribution

It offers a reinterpretation of G"odel's philosophy, highlighting the role of the logic of concepts in overcoming mathematical incompleteness.

Findings

G"odel's incompleteness relates to extensionality in set theory.

Incompleteness can be addressed through intensional considerations.

The logic of concepts is key to understanding G"odel's philosophical approach.

Abstract

The literature dealing with G\"{o}del's legacy is largely preoccupied with challenging his philosophical views, regarding them as outdated. We believe that such an approach prevents us from seeing G\"{o}del'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 G\"{o}del'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 G\"{o}del 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.