A model in which the Separation principle holds for a given effective projective Sigma-class

Vladimir Kanovei; Vassily Lyubetsky

doi:10.3390/axioms11030122

paper

A model in which the Separation principle holds for a given effective projective Sigma-class

arXiv:2204.03915 · doi:10.3390/axioms11030122

Abstract

In this paper, we prove the following: If $n \geq 3$ , 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 $Σ_{n}^{1}$ and $Π_{n}^{1}$ for 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.

17 pages