Nam Trang: A hierarchy of supercompactness measures in ZF+DC

Thursday, May 24, 2012, 4:00 PM
RH 440R

Speaker: Nam Trang (UC Berkeley)

Title: A hierarchy of supercompactness measures in ZF+DC

Abstract:

For each $\alpha < \omega_1$, let $$X_\alpha = \{f : \omega^\alpha \rightarrow\mathcal P_{\omega_1}(\mathbb{R})\mid f\text{ is increasing and continuous}\},$$

and $\mu_\alpha$ be a normal fine measure on $X_\alpha$. We identify $X_0$ with $\mathcal P_{\omega_1}(\mathbb{R})$. Martin and Woodin independently showed that these measures exist assuming $(ZF + DC_{\mathbb{R}})$ + AD + Every set is Suslin ($\mu_0$’s existence was originally shown by Solovay from $AD_{\mathbb{R}}$). We sketch the proof of the derived model construction giving the existence of these measures (+ $AD^+$) from large cardinals and the Prikry forcing construction which gives back the exact large cardinal strength from $AD^+$ and the measure. If time allows, we will survey some theorems on the structure theory of the model $L(\mathbb{R},\mu_\alpha)$ assuming the model satisfies $\Theta > \omega_2$ and $\mu_\alpha$ is a normal fine measure on $X_\alpha$. Here the main theorem is that our assumption implies $L(\mathbb{R},\mu_\alpha)$ satisfies $AD^+$.

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