Jiachen Yuan (University of Leeds)

How far is almost strong compactness from strong compactness.

Almost strong compactness of $\kappa$ can be characterized as follows: for every $\delta < \kappa < \lambda$, there is an elementary embedding $j_{\delta,\lambda}: V \rightarrow M$ with critical point $\geq \delta$, so that $j_{\delta,\lambda}" \lambda \subseteq D \in M$ and $M \vDash |D|< j_{\delta,\lambda}(\kappa)$. Boney and Brooke-Taylor were wondering whether almost strong compactness is essentially the same as strong compactness. Recently, Goldberg showed that if $\kappa$ is of uncountable cofinality and SCH holds from below then these two closely related concepts are the same. In this joint work with Zhixing You, we show that these two can be different in general cases.