Löb's theorem

In mathematical logic, Löb's theorem states that in a theory with Peano arithmetic, for any formula P, if it is provable that "if P is provable then P", then P is provable. I.e.


 * if $$PA \vdash Bew(\# P) \rightarrow P$$, then $$PA \vdash P$$

where Bew(#P) means that the formula with Gödel number P is provable.

Löb's theorem is named for Martin Hugo Löb.

Löb's theorem in provability logic
Provability logic abstracts away from the details of encodings used in Gödel's incompleteness theorems by expressing the provability of $$\phi$$ in the given system in the language of modal logic, by means of the modality $$\Box \phi$$.

Then we can formalize Löb's theorem by the axiom


 * $$\Box(\Box P\rightarrow P)\rightarrow \Box P$$,

known as axiom GL, for Gödel-Löb. This is sometimes formalised by means of an inference rule that infers


 * $$\Box P$$

from


 * $$\Box P\rightarrow P$$.

The provability logic GL that results from taking the modal logic K4 and adding the above axiom GL is the most intensely investigated system in provability logic.