The Arithmetical Hierarchy: A Realizability-Theoretic Perspective

TL;DR

This paper explores the arithmetical hierarchy through realizability theory, revealing new classification methods for natural problems by combining realizability with many-one reducibility, beyond traditional quantifier-based measures.

Contribution

It introduces a novel realizability-theoretic approach to classifying arithmetical problems, extending beyond classical degrees of unsolvability.

Findings

Realizability combined with many-one reducibility enables complex classification.

Traditional degrees of unsolvability relate mainly to quantifier complexity.

New nontrivial classifications of arithmetical problems are possible.

Abstract

In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical decision problems only plays a role in counting the number of quantifiers, jumps, or mind-changes. In contrast, we reveal that when the realizability interpretation is combined with many-one reducibility, it becomes possible to classify natural arithmetical problems in a very nontrivial way.