Generalized Decidability via Brouwer Trees
This work provides a new theoretical framework for constructive mathematicians to classify undecidable properties with finer granularity.
The authors introduce a framework for graded decidability using Brouwer ordinals in homotopy type theory, generalizing classical decidability and semidecidability. They prove closure properties and show that countable meets of semidecidable propositions are ω²-decidable, with all results formalized in Cubical Agda.
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 $α$-decidable, for a Brouwer ordinal $α$, and show that it generalizes decidability and semidecidability. Further generalizing known results, we show that $α$-decidable propositions are closed under binary conjunction, and discuss for which $α$ 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 $ω^2$-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.