$\text{NS}_{\omega_1}$ saturated, $\Delta_1 ( \{ \omega_1 \} )$-definable and a $\Delta^1_4$-definable well-order of the reals

TL;DR

The paper constructs a model assuming $M_1$ where the nonstationary ideal on $oldsymbol{\omega_1}$ is saturated and definable, and there exists a well-order of the reals with high definability, challenging previous assumptions.

Contribution

It demonstrates the consistency of a model with a saturated, definable nonstationary ideal and a highly definable well-order of the reals under $M_1$.

Findings

The nonstationary ideal on $oldsymbol{\omega_1}$ can be saturated and definable.

A $oldsymbol{\Sigma^1_4}$-definable well-order of the reals exists in the model.

This challenges the idea that saturation and definability imply structural regularity of projective sets.

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 $\Sigma^1_{4}$-definable well-order of the reals. This implies that contrary to the assumption that $NS_{\omega_1}$ is $\aleph_1$-dense, the assumption of $NS_{\omega_1}$ being saturated and $\Delta_1$-definable does not imply any nice structural properties for the projective subsets of the reals.