nLab Löb's theorem




Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




Löb’s theorem in its original form is a generalization of the Gödel incompleteness theorem whose formulation lends itself to the tools of type theory and modal logic.

Löb’s theorem states that to prove that a proposition is provable, it is sufficient to prove the proposition under the assumption that it is provable. Since the Curry-Howard isomorphism identifies formal proofs with abstract syntax trees of programs; Löb’s theorem implies, for total languages which validate it, that self-interpreters are impossible. (Gross-Gallagher-Fallenstein 16)

In provability logic the abstract statement is considered in itself as an axiom on a modal operator \Box interpreted as the modality “is provable”. In this form the statement reads formally:

(PP)P \Box(\Box P \to P) \to \Box P

for any proposition PP (“Löb’s axiom”).

This reduces to an incompleteness theorem when taking P=P = false and using that

  1. negation is ¬P=(Pfalse)\not P = (P \to false);

  2. consistency means that P¬¬P\Box P \to \not \Box \not P

(falsefalse)false (¬false)false (¬false)¬¬false (¬false)false ¬¬false \begin{aligned} & \Box (\Box false \to false) \to \Box false \\ \Rightarrow \;\; & \Box ( \not \Box false ) \to \Box false \\ \Rightarrow \;\; & \Box ( \not \Box false ) \to \not \Box \not false \\ \Rightarrow \;\; & \Box ( \not \Box false ) \to false \\ \Rightarrow \;\; & \not \Box \not \Box false \end{aligned}

Where the last line reads in words “It is not provable that false is not provable.”

Guarded Recursion Variant

A variant of the Löb axiom is used in guarded recursion and synthetic guarded domain theory, which uses a modality \blacktriangleright, usually pronounced “later”. Then the Löb induction axiom is for any proposition PP,

(PP)P(\blacktriangleright P \to P) \to P

In a setting with the principle of unique choice, this can be used to prove the existence of all guarded fixed points.

Note that the assumptions about the later modality \blacktriangleright are usually quite different from the provability modality \Box. For instance, \Box is usually assumed to have some subset of the properties of a comonadic modality, but \blacktriangleright typically satisfies PPP \to \blacktriangleright P.


See also

Last revised on August 6, 2022 at 12:00:04. See the history of this page for a list of all contributions to it.