idea

Löb’s theorem makes it rigorous

Take a system T that is strong enough to talk about itself, and read its theorems as its beliefs. Löb (1955) says that if T proves “if I prove P, then P is true”, then T proves P outright. So a system that trusts itself about a statement ends up asserting that statement, whether or not it is true. Gödel’s second theorem is the special case where P is a falsehood.

That gives a precise version of arrogance: blanket self-trust, “whatever I conclude is true”, is something only a broken system can have. That is a theorem, not a metaphor.

Connects to

• the asymmetry
• escape routes
• second theorem
• Löb, Feferman
• free certainty

Sources

1. Wikipedia: Löb’s theorem
2. Löb, M. H. (1955). Solution of a problem of Leon Henkin. Journal of Symbolic Logic 20(2), 115–118.