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 then 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 is provable in PA, then " 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. Provability logic abstracts away from the details of encodings used in Gödel's incompleteness theorems by expressing the provability of in the given system in the language of modal logic, by means of the modality . Then we can formalize Löb's theorem by the axiom known as axiom GL, for Gödel–Löb. This is sometimes formalized by means of the inference rule: If then The provability logic GL that results from taking the modal logic K4 (or K, since the axiom schema 4, , then becomes redundant) and adding the above axiom GL is the most intensely investigated system in provability logic. Löb's theorem can be proved within modal logic using only some basic rules about the provability operator (the K4 system) plus the existence of modal fixed points. We will assume the following grammar for formulas: If is a propositional variable, then is a formula. If is a propositional constant, then is a formula. If is a formula, then is a formula. If and are formulas, then so are , , , , and A modal sentence is a modal formula that contains no propositional variables. We use to mean is a theorem. If is a modal formula with only one propositional variable , then a modal fixed point of is a sentence such that We will assume the existence of such fixed points for every modal formula with one free variable.