A Model in Which the Separation Principle Holds for a Given Effective Projective Sigma-Class

Abstract

In this paper, we prove the following: If n≥3, there is a generic extension of L—the constructible universe—in which it is true that the Separation principle holds for both effective (lightface) classes Σn1 and Πn1 of sets of integers. The result was announced long ago by Leo Harrington with a sketch of the proof for n=3; its full proof has never been presented. Our methods are based on a countable product of almost-disjoint forcing notions independent in the sense of Jensen–Solovay.