Löb's theorem

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 ϕ in the given system in the language of modal logic, by means of the modality ◻ ϕ .

Then we can formalize Löb's theorem by the axiom

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

from

The provability logic GL that results from taking the modal logic K4 (or K, since the axiom schema 4, ◻ A → ◻ ◻ A , then becomes redundant) and adding the above axiom GL is the most intensely investigated system in provability logic.

Modal proof of Löb's theorem

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.

Modal formulas

We will assume the following grammar for formulas:

1. If X is a propositional variable, then X is a formula.
2. If K is a propositional constant, then K is a formula.
3. If A is a formula, then ◻ A is a formula.
4. If A and B are formulas, then so are ¬ A , A → B , A ∧ B , A ∨ B , and A ↔ B

A modal sentence is a modal formula that contains no propositional variables. We use ⊢ A to mean A is a theorem.

Modal fixed points

If F ( X ) is a modal formula with only one propositional variable X , then a modal fixed point of F ( X ) is a sentence Ψ such that

We will assume the existence of such fixed points for every modal formula with one free variable. This is of course not an obvious thing to assume, but if we interpret ◻ as provability in Peano Arithmetic, then the existence of modal fixed points is in fact true.

Modal rules of inference

In addition to the existence of modal fixed points, we assume the following rules of inference for the provability operator ◻ :

1. (necessitation) From ⊢ A conclude ⊢ ◻ A : Informally, this says that if A is a theorem, then it is provable.
2. (internal necessitation) ⊢ ◻ A → ◻ ◻ A : If A is provable, then it is provable that it is provable.
3. (box distributivity) ⊢ ◻ ( A → B ) → ( ◻ A → ◻ B ) : This rule allows you to do modus ponens inside the provability operator. If it is provable that A implies B, and A is provable, then B is provable.

Proof of Löb's theorem

1. Assume that there is a modal sentence P such that ⊢ ◻ P → P .
   Roughly speaking, it is a theorem that if P is provable, then it is, in fact true.
   This is a claim of soundness.
2. From the existence of modal fixed points for every formula (in particular, the formula X → P ) it follows there exists a sentence Ψ such that ⊢ Ψ ↔ ( ◻ Ψ → P ) .
3. From 2, it follows that ⊢ Ψ → ( ◻ Ψ → P ) .
4. From the necessitation rule, it follows that ⊢ ◻ ( Ψ → ( ◻ Ψ → P ) ) .
5. From 4 and the box distributivity rule, it follows that ⊢ ◻ Ψ → ◻ ( ◻ Ψ → P ) .
6. Applying the box distributivity rule with A = ◻ Ψ and B = P gives us ⊢ ◻ ( ◻ Ψ → P ) → ( ◻ ◻ Ψ → ◻ P ) .
7. From 5 and 6, it follows that ⊢ ◻ Ψ → ( ◻ ◻ Ψ → ◻ P ) .
8. From the internal necessitation rule, it follows that ⊢ ◻ Ψ → ◻ ◻ Ψ .
9. From 7 and 8, it follows that ⊢ ◻ Ψ → ◻ P .
10. From 1 and 9, it follows that ⊢ ◻ Ψ → P .
11. From 2, it follows that ⊢ ( ◻ Ψ → P ) → Ψ .
12. From 10 and 11, it follows that ⊢ Ψ
13. From 12 and the necessitation rule, it follows that ⊢ ◻ Ψ .
14. From 13 and 10, it follows that ⊢ P .

Examples

An immediate corollary 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. Given we know PA is consistent (but PA does not know PA is consistent), here are some simple examples:

In Doxastic logic, Löb's theorem shows that any system classified as a reflexive "type 4" reasoner must also be "modest": such a reasoner can never believe "my belief in P would imply that P is true", without first believing that P is true.

Converse: Löb's theorem implies the existence of modal fixed points

Not only does the existence of modal fixed points imply Löb's theorem, but the converse is valid, too. When Löb's theorem is given as an axiom (schema), the existence of a fixed point (up to provable equivalence) p ↔ A ( p ) for any formula A(p) modalized in p can be derived. Thus in normal modal logic, Löb's axiom is equivalent to the conjunction of the axiom schema 4, ( ◻ A → ◻ ◻ A , ) and the existence of modal fixed points.