Projective Wellorders and the Nonstationary Ideal

Abstract

We show that, under the assumption of the existence of $M_1^{\#}$, there exists a model on which the restricted nonstationary ideal $NS\upharpoonright A$ is ${\aleph }_2$-saturated, for $A$ a stationary co-stationary subset of ${\omega }_1$, while the full nonstationary ideal $NS$ can be made ${\Delta }_1$ definable with $K_{{\omega }_1}$ as a parameter. Further we show, again under the assumption of the existence of $M_1^{\#}$ that there is a model of set theory such that $NS$ is ${\aleph }_2$-saturated and such that there is lightface ${\Sigma }_4^1$-definable well-order on the reals. This result is optimal in the presence of a measurable cardinal.

This is the authors PhD thesis