NS SATURATED AND -DEFINABLE

Abstract We show that under the assumption of existence canonical inner model with one Woodin cardinal $M_1$ , there is a $\mathsf {ZFC}$ in which $\mbox {NS}_{\omega _{1}}$ $\aleph _2$ -saturated and ${\Delta }_{1}$ -definable $\omega _1$ as parameter answers question S. D. Friedman L. Wu. also starting from an arbitrary universe cardinal, saturated ladder system $\vec {C}$ full Suslin tree T parameters. Both results rely on new coding technique whose presentation main goal this article .