On the logical structure of choice and bar induction principles

Abstract

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 functions 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• the (choice) strength of Weak Kőnig's Lemma if A = N and B = B. 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.

Stateful Realizers for Nonstandard Analysis

“…Finally, we would like to mention that Brede and Herbelin's [BH21] establishes a hierarchy of choice principles, relating in particular tree-based choices principles and their dual bar induction-based principles. Many of the principles they study are not attached to a precise computational content so far, and so it would be interesting to see if there exist specific interpretations that could capture exactly each of these principles, and, in particular, "lower" instances of their generalized dependent choice or generalized bar induction principles.…”

Separating Markov's Principles

“…Pédrot [44] states that MP B is equivalent to "a Turing machine that does not loop necessarily terminates", which again is only equivalent to MP PR . An earlier version of [49] contained a similar formulation.…”