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 $\mathsf {ZFC}$ in which $\mbox {NS}_{\omega _{1}}$ is $\aleph _2$ -saturated and ${\Delta }_{1}$ -definable with $\omega _1$ as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\mbox {NS}_{\omega _{1}}$ 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 .

中文翻译：

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NS 饱和且可定义

我们表明，假设存在具有一个伍丁基数的规范内部模型$M_1$, 有一个模型$\mathsf {ZFC}$其中$\mbox {NS}_{\omega _{1}}$是$\阿莱夫_2$-饱和和${\Delta }_{1}$- 可定义$\欧米茄_1$作为回答 SD Friedman 和 L. Wu 问题的参数。我们还表明，从具有伍丁红衣主教的任意宇宙开始，有一个模型具有$\mbox {NS}_{\omega _{1}}$饱和和${\Delta }_{1}$- 可使用梯子系统定义$\vec {C}$和一棵完整的苏斯林树吨作为参数。这两个结果都依赖于一种新的编码技术，其介绍是本文的主要目标。