nLab Löb's theorem

Contents

Context

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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

semantics

Contents

Idea

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.

References

See also

Last revised on December 28, 2022 at 21:52:58. See the history of this page for a list of all contributions to it.