# Xianghui Shi: A Posner-Robinson Theorem from Axiom $I_0$

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.

### One response to “Xianghui Shi: A Posner-Robinson Theorem from Axiom $I_0$”

1. saf

slides from the talk are now available:
Posner-Robinson Theorem from Axiom I0

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