Invited Talks and Tutorials A Constructive Version of the Lusin Separation Theorem

The Lusin Separation Theorem is one of the fundamental early results of classical descriptive set theory. It states that if A1, A2 are disjoint analytic subsets of Baire space then they are Borel separable. Yiannis Moschovakis gives two proofs in his book, “Descriptive Set Theory”. One proof is obviously highly non-constructive. The other would appear to be constructive and uses Bar Induction and Bar Recursion. But the exact constructive status of the result needs some care. By constructively strengthening both the assumption and, in compensation, the conclusion of the implication we are able to give a proof in the formal system CZF of constructive set theory. We avoid using Bar Induction by using a strong inductive point-free notion of disjointness. We also strengthen the conclusion and avoid using any choice principles by using a point-free notion of Borel separation. So our main result would be acceptable in Bishop style constructive mathematics, even without countable choice. Moreover our main result, combined with Bar Induction, gives a pointset version of the Lusin theorem that would be acceptable in Brouwer’s Intuitionism. As a corollary we also obtain a version of Wim Veldman’s result concerning an Intuitionistic formulation of the Lusin theorem for positively disjoint strictly analytic sets. Here the notion of positive disjointness is the natural constructive formulation of disjointness for subsets of a set with an apartness, obtained by removing the double negation from the statement that A1, A2 are disjoint if their intersection is empty; i.e. whenever x1 is in A1 and x2 is in A2 then x1, x2 are not not apart. On Baire space the infinite sequences x1, x2 are apart if they differ at some stage. On the Order of Strategy Elimination Procedures in Strategic Games