Full Idea
Second Incompleteness Theorem: roughly, nice theories that include enough basic arithmetic can't prove their own consistency.
Gist of Idea
Second Incompleteness: nice theories can't prove their own consistency
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.5
Book Reference
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.6
A Reaction
On the face of it, this sounds less surprising than the First Theorem. Philosophers have often noticed that it seems unlikely that you could use reason to prove reason, as when Descartes just relies on 'clear and distinct ideas'.