Projective sets, intuitionistically

TL;DR

This paper investigates definable subsets of Baire space within an intuitionistic framework, exploring their properties, classifications, and the impact of Brouwer's axioms on their structure, leading to a collapse of the projective hierarchy.

Contribution

It introduces an intuitionistic approach to projective sets, avoiding complement operations and proving separation and boundedness theorems under Brouwer's axioms.

Findings

Brouwer's Thesis proves separation for strictly analytic sets.

The projective hierarchy collapses to  levels.

Certain classes of sets are likely not .

Abstract

We study `definable' subsets of Baire space $\mathcal{N}$. The logic of our arguments is intuitionistic and we use L.E.J.~Brouwer's Thesis on bars in $\mathcal{N}$ and his continuity axioms. We avoid the operation of taking the complement of a subset of $\mathcal{N}$. A subset of $\mathcal{N}$ is $\mathbf{\Sigma}^1_1$ or: analytic if it is the projection of a closed subset of $\mathcal{N}$. Important $\mathbf{\Sigma}^1_1$ set are the set of the codes of all closed and located subsets of $\mathcal{N}$ that are positively uncountable and the set of the codes of all located and closed subsets of $\mathcal{N}$ containing at least one member coding a (positively) infinite subset of $\mathbb{N}$. A subset of $\mathcal{N}$ is strictly analytic if it is the projection of a closed and located subset of $\mathcal{N}$. Brouwer's Thesis on bars in $\mathcal{N}$ proves separation and boundedness theorems for strictly analytic subsets of $\mathcal{N}$. A subset of $\mathcal{N}$ is $\mathbf{\Pi}^1_1$ or: co-analytic if it is the co-projection of an open subset of $\mathcal{N} \times \mathcal{N}=\mathcal{N}$. There is no symmetry between analytic and co-analytic sets like in classical descriptive set theory. An important $\mathbf{\Pi}^1_1$ set is the set of the codes of all closed and located subsets of $\mathcal{N}$ all of whose members code an almost-finite subset of $\mathbb{N}$. The set of the codes of closed and located subsets of $\mathcal{N}$ that are almost-countable, or, equivalently, \{reducible in Cantor's sense, is treated at some length. This set is probably not $\mathbf{\Pi}^1_1$. The projective hierarchy collapses: every (positively) projective set is $\mathbf{\Sigma}^1_2$: the projection of a co-analytic subset of $\mathcal{N}$.