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

The open and clopen Ramsey theorems in the Weihrauch lattice

We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to ATR_0 from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight di\ufb00erent multivalued functions (\ufb01ve corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be strictly stronger than any previously studied multivalued functions arising from statements around ATR_0

A journey through computability, topology and analysis

This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility, as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of the open and clopen Ramsey theorems. In particular, since there is not a canonical way to phrase these theorems as multi-valued functions, we identify 8 different multi-valued functions (5 corresponding to the open Ramsey theorem and 3 corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch and arithmetic Weihrauch reducibility. We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. These notions turn out to be extremely relevant when exploring the Weihrauch degree of the problem DS of computing descending sequences in ill-founded linear orders. They allow us to show that DS, and the Weihrauch equivalent problem BS of finding bad sequences through non-well quasi-orders, while being very "hard" to solve, are rather weak in terms of uniform computational strength. We then generalize DS and BS by considering Gamma-presented orders, where Gamma is a Borel pointclass or Delta11, Sigma11, Pi11. We study the obtained DS-hierarchy and BS-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. Finally, we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and of Salem sets. We first work in the hyperspace K([0,1]) of compact subsets of [0,1] and show that the closed Salem sets form a Pi03-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in K([0,1]^d). We also generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still Pi03-complete when we endow K(R^d) with the Fell topology. A similar result holds also for the Vietoris topology. We conclude by showing how these results can be used to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions