Singular cardinals and strong extenders

TL;DR

This paper explores conditions where a singular cardinal and a short extender witnessing strongness lead to the singularity of the cardinal in the ultrapower, deepening understanding of large cardinal embeddings.

Contribution

It introduces new conditions under which a singular cardinal remains singular in the ultrapower generated by a short extender witnessing strongness.

Findings

Identifies scenarios where $$ remains singular in the ultrapower

Provides criteria linking singular cardinals and strong extenders

Enhances understanding of the structure of ultrapowers in large cardinal theory

Abstract

We investigate the circumstances under which there exist a singular cardinal $\mu$ and a short $(\kappa, \mu)$-extender $E$ witnessing "$\kappa$ is $\mu$-strong", such that $\mu$ is singular in $\Ult(V, E)$.