On the logical structure of choice and bar induction principles

Nuria Brede, Hugo Herbelin
2021 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
We develop an approach to choice principles and their contra-positive 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
more » ... ions from A to B, a generalised form GDCABT of the axiom of dependent choice and dually a generalised bar induction principle GBIABT such that: GDCABT 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 B, • the strength of the axiom of dependent choice if A = N, and up to weak classical reasoning GBIABT intuitionistically captures • the strength of Gödel's completeness theorem in the form validity implies provability for entailment relations when B = B, • the strength of the Bar Induction when A = N, • the (choice) strength of the Weak Fan Theorem when A = N and B = B. Contrastingly, even though GDCABT and GBIABT smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when A is B N and B is N.
doi:10.1109/lics52264.2021.9470523 fatcat:bfwohvtglfb6jlijnuns653w3u