Projective Wellorders and the Nonstationary Ideal

TL;DR

The paper demonstrates models of set theory where the nonstationary ideal exhibits high saturation and definability properties, assuming the existence of a certain large cardinal, revealing deep interactions between large cardinals, saturation, and definability.

Contribution

It constructs models with specific saturation and definability features of the nonstationary ideal under the assumption of $M_1^{\#}$, advancing understanding of their interplay.

Findings

Existence of models with $ ext{NS} \upharpoonright A$ $ ext{aleph}_2$-saturation for stationary co-stationary $A$.

Full nonstationary ideal $ ext{NS}$ can be made $ riangle_1$ definable with $K_{\omega_1}$ as a parameter.

Models where $ ext{NS}$ is $ ext{aleph}_2$-saturated and admits a lightface $\Sigma^1_4$-definable well-order on the reals.

Abstract

We show that, under the assumption of the existence of $M_1^{\#}$, there exists a model on which the restricted nonstationary ideal $\hbox{NS} \upharpoonright A$ is $\aleph_2$-saturated, for $A$ a stationary co-stationary subset of $\omega_1$, while the full nonstationary ideal $\hbox{NS}$ 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 $\hbox{NS}$ is $\aleph_2$-saturated and such that there is lightface $\Sigma^1_4$-definable well-order on the reals. This result is optimal in the presence of a measurable cardinal.