- A. W. Apter, J. Cummings, and J. D. Hamkins, “Singular cardinals and strong extenders,” Central European J.~Math., vol. 11, iss. 9, pp. 1628-1634, 2013.
`@article {ApterCummingsHamkins2013:SingularCardinalsAndStrongExtenders, AUTHOR = {Apter, Arthur W. and Cummings, James and Hamkins, Joel David}, TITLE = {Singular cardinals and strong extenders}, JOURNAL = {Central European J.~Math.}, FJOURNAL = {Central European Journal of Mathematics}, VOLUME = {11}, YEAR = {2013}, NUMBER = {9}, PAGES = {1628--1634}, ISSN = {1895-1074}, MRCLASS = {03E55 (03E35 03E45)}, MRNUMBER = {3071929}, MRREVIEWER = {Samuel Gomes da Silva}, DOI = {10.2478/s11533-013-0265-1}, URL = {http://jdh.hamkins.org/singular-cardinals-strong-extenders/}, eprint = {1206.3703}, archivePrefix = {arXiv}, primaryClass = {math.LO}, }`

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 $Ult(V, E)$.