Gödel's incompleteness theorems

First theorem: in any consistent formal system that can express basic arithmetic, some statements remain unprovable within the system. No complete coverage of all mathematical truths arises under these constraints.

Second theorem: a system of that kind cannot prove its own consistency. Any proof of consistency requires assumptions outside the system. A reminder of the inherent boundaries of formal reasoning and the interplay between logic, mathematics, and truth.

Essential reading:

Gödel's Proof · 2001 · Ernest Nagel

Additional reading:

Gödel, Escher, Bach · 1999 · Douglas R. Hofstadter
When Einstein Walked with Gödel · 2018 · Jim Holt