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

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.
Comment: Converted to a pedagogical focus