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Conference Papers Year : 2021

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 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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Dates and versions

hal-03144849 , version 1 (17-02-2021)
hal-03144849 , version 2 (15-03-2021)
hal-03144849 , version 3 (12-05-2021)
hal-03144849 , version 4 (18-06-2021)
hal-03144849 , version 5 (05-07-2022)

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Cite

Nuria Brede, Hugo Herbelin. On the logical structure of choice and bar induction principles. LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun 2021, Rome / Virtual, Italy. ⟨hal-03144849v5⟩
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