4 research outputs found

    Reverse Mathematics in Bishop’s Constructive Mathematics

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    We will overview the results in an informal approach to constructive reverse mathematics, that is reverse mathematics in Bishop’s constructive mathematics, especially focusing on compactness properties and continuous properties

    On the logical structure of choice and bar induction principles

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    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 AA, a codomain BB and a "filter" TT on finite approximations of functions from AA to BB, a generalised form GDCA,B,T_{A,B,T} of the axiom of dependent choice and dually a generalised bar induction principle GBIA,B,T_{A,B,T} such that: GDCA,B,T_{A,B,T} intuitionistically captures the strength of ∙\bullet the general axiom of choice expressed as ∀a∃bR(a,b)⇒∃α∀αR(α,α(a))\forall a\exists b R(a, b) \Rightarrow\exists\alpha\forall \alpha R(\alpha,\alpha(a)) when TT is a filter that derives point-wise from a relation RR on A×BA \times B without introducing further constraints, ∙\bullet the Boolean Prime Filter Theorem / Ultrafilter Theorem if BB is the two-element set B\mathbb{B} (for a constructive definition of prime filter), ∙\bullet the axiom of dependent choice if A=NA = \mathbb{N}, ∙\bullet Weak K{\"o}nig's Lemma if A=NA = \mathbb{N} and B=BB = \mathbb{B} (up to weak classical reasoning) GBIA,B,T_{A,B,T} intuitionistically captures the strength of ∙\bullet G{\"o}del's completeness theorem in the form validity implies provability for entailment relations if B=BB = \mathbb{B}, ∙\bullet bar induction when A=NA = \mathbb{N}, ∙\bullet the Weak Fan Theorem when A=NA = \mathbb{N} and B=BB = \mathbb{B}. Contrastingly, even though GDCA,B,T_{A,B,T} and GBIA,B,T_{A,B,T} smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when AA is BN\mathbb{B}^\mathbb{N} and BB is N\mathbb{N}.Comment: LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun 2021, Rome / Virtual, Ital
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