On the depth of Gödel's incompleteness theorem

Yong Cheng

Published 2020-08-30Version 1

Is G\"{o}del's incompleteness theorem deep? For us, the answer of this question depends on our view of mathematical depth. This paper is motivated by the following question: if G\"{o}del's incompleteness theorem is considered as deep, what its depth consists in? For us, an account of the depth of a specific mathematical theorem consists of the general criteria of mathematical depth and the specific justification that the theorem we consider satisfies these general criteria. In this paper, we propose three general criteria of mathematical depth: influence, fruitfulness and unity. We show that if adopting our Influence-Fruitfulness-Unity criteria, G\"{o}del's incompleteness theorem can be viewed as deep. As an application of our criteria, we justify for the depth of G\"{o}del's incompleteness theorem by showing that it satisfies the Influence-Fruitfulness-Unity criteria. Finally, we give some explanations of our account of the depth of G\"{o}del's incompleteness theorem.