basic constructions:
strong axioms
further
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
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:
for any proposition $P$ (“Löb’s axiom”).
This reduces to an incompleteness theorem when taking $P =$ false and using that
negation is $\not P = (P \to false)$;
consistency means that $\Box P \to \not \Box \not P$
Where the last line reads in words “It is not provable that false is not provable.”
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 $P$,
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 $P \to \blacktriangleright P$.
Jason Gross, Jack Gallagher, Benya Fallenstein, Löb’s theorem – A functional pearl of dependently typed quining, 2016 (pdf)
Neelakantan Krishnaswami, Löb’s theorem is (almost) the Y-combinator, 2016
Jason Gross, MO comment on incompletenss theorems in type theory, 2017
See also
Unified account of Löb's theorem/Gödel's second incompleteness theorem, Kripke semantics for provability logic, and guarded recursion, via essentially algebraic theories which “self-internalize”:
Last revised on December 20, 2023 at 04:06:24. See the history of this page for a list of all contributions to it.