# NS saturated and $\Delta_1$-definable

## TL;DR

This paper constructs models of ZFC where the nonstationary ideal on ω₁ is both saturated and Δ₁-definable, using new coding techniques, under assumptions involving Woodin cardinals.

## Contribution

It introduces a novel coding method to achieve saturation and Δ₁-definability of NS in models with Woodin cardinals, answering longstanding questions.

## Key findings

- Existence of models with NS saturated and Δ₁-definable under certain large cardinal assumptions.
- Development of a new coding technique for set-theoretic model construction.
- Models can include parameters like ladder systems and Suslin trees to achieve definability.

## 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 $\ZFC$ in which $\NS$ is $\aleph_2$-saturated and $\Delta_1$-definable with $\omega_1$ as a parameter which answers a question of Sy-David Friedman and Liuzhen Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\NS$ saturated and $\Delta_1$-definable with a ladder system $\vec{C}$ and a full Suslin tree $T$ as parameters. Both results rely on a new coding technique whose presentation is the main goal of this article.