Preserving levels of projective determinacy by tree forcings

TL;DR

This paper demonstrates that several classical tree forcings preserve projective determinacy and related properties, ensuring the stability of these set-theoretic principles under such forcings.

Contribution

It establishes that key tree forcings preserve projective determinacy and generic absoluteness, extending the understanding of their impact on inner model theory and descriptive set theory.

Findings

Preservation of real sharp existence by tree forcings

Level-by-level preservation of projective determinacy

No new equivalence classes added to thin projective relations

Abstract

We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.