On the logical structure of choice and bar induction principles
Résumé
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 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 $\forall a\exists\beta R(a, b) \Rightarrow\exists\alpha\forall a R(\alpha,(a \alpha (a)))$ 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 König’s Lemma if $A = \mathbb{N}$ and $B = \mathbb{B}$ (up to weak classical reasoning)
- GBI$_{A,B,T}$ intuitionistically captures the strength of
• Gödel’s completeness theorem in the form validity implies provability for entailment relations if $B = \mathbb{B}$,
• bar induction when $A = \mathbb{N}$,
• the Weak Fan Theorem when $A = \mathbb{N}$ and $B = \mathbb{B}$.
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 $\mathbb{B}^\mathbb{N}$ and $B$ is $\mathbb{N}$.
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