Reverse Mathematics in Bishop’s Constructive Mathematics

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

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 $\bullet$ the general axiom of choice expressed as $\forall a\exists b R(a, b) \Rightarrow\exists\alpha\forall \alpha R(\alpha,\alpha(a))$ when $T$ is a filter that derives point-wise from a relation $R$ on $A \times B$ without introducing further constraints, $\bullet$ the Boolean Prime Filter Theorem / Ultrafilter Theorem if $B$ is the two-element set $\mathbb{B}$ (for a constructive definition of prime filter), $\bullet$ the axiom of dependent choice if $A = \mathbb{N}$, $\bullet$ Weak K{\"o}nig's Lemma if $A = \mathbb{N}$ and $B = \mathbb{B}$ (up to weak classical reasoning) GBI$_{A,B,T}$ intuitionistically captures the strength of $\bullet$ G{\"o}del's completeness theorem in the form validity implies provability for entailment relations if $B = \mathbb{B}$, $\bullet$ bar induction when $A = \mathbb{N}$, $\bullet$ 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 $\mathbb{N}$.Comment: LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun 2021, Rome / Virtual, Ital