Capturing sets of ordinals by normal ultrapowers

We investigate the extent to which ultrapowers by normal measures on $\kappa$ can be correct about powersets $\mathcal{P}(\lambda)$ for $\lambda>\kappa$. We consider two versions of this questions, the capturing property $\mathrm{CP}(\kappa,\lambda)$ and the local capturing property $\mathrm{LCP}(\kappa,\lambda)$. $\mathrm{CP}(\kappa,\lambda)$ holds if there is an ultrapower by a normal measure on $\kappa$ which correctly computes $\mathcal{P}(\lambda)$. $\mathrm{LCP}(\kappa,\lambda)$ is a weakening of $\mathrm{CP}(\kappa,\lambda)$ which holds if every subset of $\lambda$ is contained in some ultrapower by a normal measure on $\kappa$. After examining the basic properties of these two notions, we identify the exact consistency strength of $\mathrm{LCP}(\kappa,\kappa^+)$. Building on results of Cummings, who determined the exact consistency strength of $\mathrm{CP}(\kappa,\kappa^+)$, and using a forcing due to Apter and Shelah, we show that $\mathrm{CP}(\kappa,\lambda)$ can hold at the least measurable cardinal.Comment: 20 page

Transferring Compactness

We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a cardinal $\kappa$ that is $n$-$d$-stationary for all $n\in \omega$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $\kappa$ being $(n+1)$-$d$-stationary is equivalent to $\kappa$ being $\mathbf{\Pi}^1_n$-indescribable. We also show that it is consistent that there is a cardinal $\kappa\leq 2^\omega$ such that $P_\kappa(\lambda)$ is $n$-stationary for all $\lambda\geq \kappa$ and $n\in \omega$, answering a question of Sakai.Comment: Corrected some typo

Strong ultrapowers and long core models

In his paper [4] Steel asked whether there can exist a normal measure U on a cardinal ^ such that P^+ ` U lt(V; U)