[bibtex key=ApterCummingsHamkins2013:SingularCardinalsAndStrongExtenders]

Brent Cody asked the question whether the situation can arise that one has an elementary embedding 𝑗 :𝑉 →𝑀 witnessing the 𝜃-strongness of a cardinal 𝜅, but where 𝜃 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 𝜇 and a short (𝜅,𝜇)-extender 𝐸 witnessing “𝜅 is 𝜇-strong”, such that 𝜇 is singular in 𝑈⁢𝑙⁢𝑡⁡(𝑉,𝐸).