Löb's theorem

In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. If Prov(P) means that the formula P is provable, we may express this more formally as

If

${\displaystyle {\mathit {PA}}\vdash {\mathrm {Prov} (P)\rightarrow P}}$

then

${\displaystyle {\mathit {PA}}\vdash P}$.

An immediate corollary (the contrapositive) of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. For example, "If ${\displaystyle 1+1=3}$ is provable in PA, then ${\displaystyle 1+1=3}$" is not provable in PA.

Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox.