# Capturing sets of ordinals by normal ultrapowers

## TL;DR

This paper explores how ultrapowers by normal measures can accurately reflect powersets for larger cardinals, introducing and analyzing the capturing properties and their consistency strengths in set theory.

## Contribution

It introduces the local capturing property, analyzes its properties, and determines its exact consistency strength, extending previous results on the capturing property.

## Key findings

- $ ext{LCP}(oldsymbol{ ext{kappa}}, oldsymbol{ ext{kappa}^+})$'s consistency strength is identified.
- $ ext{CP}(oldsymbol{ ext{kappa}}, oldsymbol{ extlambda})$ can hold at the least measurable cardinal.
- The paper extends the understanding of ultrapower correctness properties in set theory.

## Abstract

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.