A Note on the Incompleteness of G\"odel's Incompleteness Theorems

TL;DR

This paper discusses how automated theorem provers may fail to identify some valid theorems due to limitations similar to G"odel's incompleteness, offering pedagogical insights rather than practical implications.

Contribution

It highlights a parallel between ATP theorem enumeration failures and G"odel's incompleteness, providing educational value on recursive enumeration.

Findings

ATP theorem enumeration can miss valid theorems due to inherent limitations.

The failure mode is analogous to G"odel's original proof.

No significant practical implications, mainly pedagogical insights.

Abstract

In this note we observe that automated theorem provers (ATPs) that recursively enumerate theorems in a particular way will fail to identify some valid theorems for a reason that is analogous to how G\"odel proved the existence of what are now referred to as G\"odel statements. This observation has no significant practical or theoretical implications, but it may be of pedagogical value for honing the intuition of students about recursive enumeration in the context of ATP.