Generalized Decidability via Brouwer Trees

TL;DR

This paper introduces a framework in homotopy type theory for classifying the decidability of properties using Brouwer ordinals, extending traditional notions and exploring their closure properties.

Contribution

It generalizes decidability concepts with Brouwer ordinals, formalizes the framework in Cubical Agda, and analyzes closure properties under logical operations.

Findings

$orall i. P(i)$ is $ ext{omega}^2$-decidable if each $P(i)$ is semidecidable.

$ ext{alpha}$-decidable propositions are closed under binary conjunction.

Results are formalized in Cubical Agda.

Abstract

In the setting of constructive mathematics, we suggest and study a framework for decidability of properties, which allows for finer distinctions than just "decidable, semidecidable, or undecidable". We work in homotopy type theory and use Brouwer ordinals to specify the level of decidability of a property. In this framework, we express the property that a proposition is $\alpha$-decidable, for a Brouwer ordinal $\alpha$, and show that it generalizes decidability and semidecidability. Further generalizing known results, we show that $\alpha$-decidable propositions are closed under binary conjunction, and discuss for which $\alpha$ they are closed under binary disjunction. We prove that if each $P(i)$ is semidecidable, then the countable meet $\forall i\in \mathbb N. P(i)$ is $\omega^2$-decidable, and similar results for countable joins and iterated quantifiers. We also discuss the relationship with countable choice. All our results are formalized in Cubical Agda.