中国科学院数学与系统科学研究院
数学研究所
学术报告
数理逻辑研讨班
报告人：Daisuke Ikegami（中山大学）
题 目：Preservation of AD via forcings
时 间：2024.08.21（星期三）14：00-16：00
地 点：思源楼S803
摘 要：The research in this talk was motivated by the following question:
Could there be an elementary embedding $j \colon V \to V[G]$ such that $G$ is set-generic over $V$, $(V[G], \in , j)$ is a model of $\mathsf{ZF}+\mathsf{AD}$, and the critical point of $j$ is $\omega_1^V$?
The positive answer to the above question would give us a partial order which preserves the truth of $\mathsf{AD}$ while adding a new real. However, we do not know if there is such a partial order. To see whether there could be such a partial order, we have been working on the question: what kind of partial orders preserve the truth of $\mathsf{AD}$.
In this talk, we present several results on partial orders preserving the truth of $\mathsf{AD}$. Using our main result, we obtain a model of $\mathsf{ZF}+\mathsf{AD}^+$ where Mouse Capturing fails. Here Mouse Capturing states that for all reals $x$ and $y$, $x$ is ordinal definable from $y$ if and only if $x$ is in a mouse over $y$.
This is joint work with Nam Trang.
附件：