Forcing with ideals of closed sets

TL;DR

This paper studies a specific type of forcing derived from sigma-ideals generated by projective collections of closed sets, demonstrating its properness and the nature of the added real.

Contribution

It introduces a forcing notion based on I-positive Borel sets associated with sigma-ideals generated by projective closed sets, analyzing its properties.

Findings

Forcing with I-positive Borel sets is proper.

Adds a real of almost minimal degree.

Any real in the extension is either Cohen generic or the extension is trivial.

Abstract

Let I be a sigma-ideal sigma-generated by a projective collection of closed sets. The forcing with I-positive Borel sets is proper and adds a single real r of an almost minimal degree: if s is a real in V[r] then s is Cohen generic over V or V[s]=V[r].