On the logical structure of choice and bar induction principles

Nuria Brede (1) , Hugo Herbelin (2)

1 University of Potsdam = Universität Potsdam
2 PI.R2 - Design, study and implementation of languages for proofs and programs

We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "effective" or "intensional" view of respectively ill-and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain A, a codomain B and a "filter" T on finite approximations of functions from A to B, a generalised form GDC(A,B,T) of the axiom of dependent choice and dually a generalised bar induction principle GBI(A,B,T) such that: - GDC(A,B,T) intuitionistically captures • the strength of the general axiom of choice expressed as ∀a ∃b R(a, b) ⇒ ∃α ∀a R(a, α(a))) when T is a filter that derives point-wise from a relation R on A × B without introducing further constraints, • the strength of the Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set Bool, • the strength of the axiom of dependent choice if A = N, and up to weak classical reasoning • the (choice) strength of Weak König’s Lemma if A = N and B = Bool. - GBI(A,B,T) intuitionistically captures • the strength of Gödel’s completeness theorem in the form validity implies provability when B = Bool, • the strength of the Bar Induction when A = N, • the (choice) strength of the Weak Fan Theorem when A = N and B = Bool. Contrastingly, even though GDC(A,B,T) and GBI(A,B,T) smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when A is Bool^N and B is N.

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Nuria Brede, Hugo Herbelin. On the logical structure of choice and bar induction principles. 2021. ⟨hal-03144849v1⟩