${\text{NS}}_{{\omega }_1}$ saturated, ${\Delta }_1(\{{\omega }_1\})$-definable and a ${\Delta }_4^1$-definable well-order of the reals

Abstract

Assuming $M_1$, the canonical inner model with one Woodin cardinal exists, we construct a model in which the nonstationary ideal on ${\omega }_1$ is ${\aleph }_2$-saturated, ${\Delta }_1$-definable with ${\omega }_1$ as the only parameter and there is a ${\Sigma }_4^1$-definable well-order of the reals. This implies that contrary to the assumption that $N{S}_{{\omega }_1}$ is ${\aleph }_1$-dense, the assumption of $N{S}_{{\omega }_1}$ being saturated and ${\Delta }_1$-definable does not imply any nice structural properties for the projective subsets of the reals.