# Löb’s theorem

**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](https://mapideas.org/i/h_asym)
- [escape routes](https://mapideas.org/i/h_escape)
- [second theorem](https://mapideas.org/i/th_second)
- [Löb, Feferman](https://mapideas.org/i/b_lob)
- [free certainty](https://mapideas.org/i/w_free)

Sources:
- [Wikipedia: Löb’s theorem](https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem)
- Löb, M. H. (1955). Solution of a problem of Leon Henkin. Journal of Symbolic Logic 20(2), 115–118.

Source: https://mapideas.org/i/h_lob · open on the map: https://mapideas.org/#h_lob