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