$\mathsf{MA} (\mathcal{I}$) and a Failure of Separation on the third Level

TL;DR

The paper addresses the question of whether weaker forcing axioms determine third-level projective separation. By constructing a model of $MA(\mathcal{I})$ with $\aleph_1=\aleph_1^L$ and using independent sequences of Suslin trees together with an elaborate almost disjoint coding scheme, the authors force a failure of both $\Pi^1_3$- and $\Sigma^1_3$-separation. The approach hinges on a two-stage coding device that embeds information about branches in $\vec{S}^i$ into reals, while a carefully designed iteration with a notion of suitability preserves Suslin trees and yields the desired separation failings, showing that $MA(\mathcal{I})$ does not decide third-level separation. The work highlights a tension between forcing axioms and projective regularity, and lays groundwork for further exploration of separation at higher levels under restricted axioms.

Abstract

We present a method which forces the failure of $Π^1_3$ and $Σ^1_3$-separation, while $\mathsf{MA} (\mathcal{I}$) holds, for $\mathcal{I}$ the family of indestructible ccc forcings. This shows that, in contrast to the assumption $\mathsf{BPFA}$ and $\aleph_1=\aleph_1^L$ which implies $Π^1_3$-separation, that weaker forcing axioms do not decide separation on the third projective level.

Theorems & Definitions (18)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.5