On the logical structure of choice and bar induction principles

Abstract

We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" 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∃bR(a,b) ⇒ ∃α∀aR(a,α(a))) when T is a filter that derives point-wise from a relation R on A × B without introducing further constraints,•the Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set $\mathbb{B}$ (for a constructive definition of prime filter),•the axiom of dependent choice if $A = \mathbb{N}$,•Weak Kőnig’s Lemma if $A = \mathbb{N}$ and $B = \mathbb{B}$ (up to weak classical reasoning).GBIABT intuitionistically captures the strength of•Gödel’s completeness theorem in the form validity implies provability for entailment relations if $B = \mathbb{B}$ (for a constructive definition of validity),•bar induction if $A = \mathbb{N}$,•the Weak Fan Theorem if $A = \mathbb{N}$ and $B = \mathbb{B}$.Contrastingly, even though GDCABT and GBIABT smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when A is ${\mathbb{B}^{\mathbb{N}}}$ and B is $\mathbb{N}$.