Speaker: Daisuke Ikegami，中山大学

Inviter: 吴刘臻副研究员

Title: Preservation of AD via forcings

Language: English

Time & Venue: 2024.08.21 14:00-16:00 思源楼803

Abstract: 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.