# Singular cardinals and strong extenders

• A. W. Apter, J. Cummings, and J. D. Hamkins, “Singular cardinals and strong extenders,” Central European Journal of Mathematics. (to appear)
@ARTICLE{ApterCummingsHamkins:SingularCardinalsAndStrongExtenders,
author = {Arthur W. Apter and James Cummings and Joel David Hamkins},
title = {Singular cardinals and strong extenders},
journal = {Central European Journal of Mathematics},
year = {},
volume = {},
number = {},
pages = {},
month = {},
url = {http://arxiv.org/abs/1206.3703},
eprint = {1206.3703},
note = {to appear},
abstract = {},
keywords = {},
source = {},
}

Brent Cody asked the question whether the situation can arise that one has an elementary embedding $j:V\to M$ witnessing the $\theta$-strongness of a cardinal $\kappa$, but where $\theta$ is regular in $M$ and singular in $V$.

In this article, we investigate the various circumstances in which this does and does not happen, 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 $Ult(V, E)$.