Friday, February 24 from 1:30 to 3pm

Fields Institute, Room 210

**Speaker:** Xianghui Shi (Beijing Normal University)

**Title:** A Posner-Robinson Theorem from Axiom $I_0$

**Abstract:** Under a slightly stronger version of Axiom $I_0$:

there is a *proper* elementary embedding j from $L(V_{\lambda+1})$ to $L(V_{\lambda+1})$ with critical point $<\lambda$, we prove an analog of Perfect Set Theorem in the context of $V_{\lambda+1}$. And as a collorary, we obtain a version of Posner-Robinson Theorem at $V_{\lambda+1}$: for every $A\in V_{\lambda+1}$, and for almost every $B\in V_{\lambda+1}$ (i.e. except a set of size $\lambda$) that can compute $A$, there is a $G\in V_{\lambda+1}$ such that the joint of $G$ and $B$ can compute the sharp of $G$.

Here “compute” and “joint” are analogs of the notions in the structure of Turing degrees.

This is a part of the study on the impact of large cardinal hypotheses on various generalized degree structures.

slides from the talk are now available:

Posner-Robinson Theorem from Axiom I0