Kolmogorov-Veloso Problems and Dialectica Categories

TL;DR

This paper explores the categorical relationships among various problem frameworks in mathematics, revealing how Dialectica constructions connect Kolmogorov, Veloso, and Blass problems, and analyzing their foundational set-theoretic requirements.

Contribution

It establishes a categorical framework linking different problem notions and clarifies the set-theoretic assumptions needed for their analysis.

Findings

Kolmogorov problems serve as a bridge between Veloso and Blass problems.

Veloso problems require the Axiom of Choice, unlike Kolmogorov and Blass problems.

Weaker forms of choice, like dependent and countable choice, are also applicable.

Abstract

We investigate the categorical connection between Dialectica constructions, Kolmogorov problems, Veloso problems and Blass problems. We show that the work of Kolmogorov can be regarded as a bridge between Veloso's abstract notion of a problem and the conceptual problems Blass discussed in his questions-and-answers framework. This bridge can be seen by means of the categorical Dialectica constructions introduced in de Paiva's dissertation and reformulated by da Silva to account for set-theoretical foundational assumptions. The use of categorical concepts allows us to provide several examples, connecting extremely different areas of mathematics, and using simple methods. This paper also shows that while Blass and Kolmogorov notions of problem can be investigated using the Zermelo-Fraenkel (ZF) set-theoretical framework, Veloso problems require the Axiom of Choice (AC). Moreover, weaker notions of choice (dependent choice and countable choice) can also be accounted for in the problems' framework.