On the depth of G\"{o}del's incompleteness theorem

TL;DR

This paper investigates the philosophical and methodological aspects of G"{o}del's incompleteness theorem, proposing criteria like influence, fruitfulness, and unity to explain its perceived depth in mathematical logic.

Contribution

It introduces a novel philosophical framework with three criteria to analyze the depth of G"{o}del's incompleteness theorem, focusing on its influence, fruitfulness, and unity.

Findings

G"{o}del's theorem is considered deep due to its influence, fruitfulness, and unity.

The proposed criteria offer a new philosophical perspective on mathematical depth.

Explanations support the validity of the criteria in assessing the theorem's depth.

Abstract

In this paper, we use G\"{o}del's incompleteness theorem as a case study for investigating mathematical depth. We take for granted the widespread judgment by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in. We focus on the methodological study of the depth of G\"{o}del's incompleteness theorem, and propose three criteria to account for its depth: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of G\"{o}del's incompleteness theorem.