We present a method which forces the failure of Π 3 1 and Σ 3 1 -separation, while MA ( I ) holds, for I the family of indestructible ccc forcings. This shows that, in contrast to the assumption BPFA and ℵ 1 = ℵ 1 L which implies Π 3 1 -separation, that weaker forcing axioms do not decide separation on the third projective level.

Abstract We produce, relative to a backslash textsf upper Z upper F upper C $\textsf {ZFC}$ \textsf Z F C model with a supercompact cardinal, a backslash textsf upper Z upper F upper C $\textsf {ZFC}$ \textsf Z F C model of the Proper Forcing Axiom in which the nonstationary ideal on omega 1 $\omega _1$ ω 1 is upper Pi 1 $\Pi _1$ Π 1 -definable in a parameter from upper H Subscript normal first transfinite cardinal 2 $H_{\aleph _2}$ H ℵ 2 .

We show that there are models of MAω1 where the Σ31-uniformization property holds. Further we show that “BPFA+ ℵ1 is not inaccessible to reals” outright implies that the Σ31-uniformization property is true.

Abstract We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal $M_1$ , there is a model of $\mathsf {ZFC}$ in which $\mbox {NS}_{\omega _{1}}$ is $\aleph _2$ -saturated and ${\Delta }_{1}$ -definable with $\omega _1$ as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\mbox {NS}_{\omega _{1}}$ saturated and ${\Delta }_{1}$ -definable with a ladder system $\vec {C}$ and a full Suslin treeTas parameters. Both results rely on a new coding technique whose presentation is the main goal of this article .

Abstract We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $ , there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $ -saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $ -wellorder.