Projective sets, intuitionistically

Wim Veldman

Published 2011-04-15, updated 2018-10-28Version 2

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 $\mathcal{N}$.