The least strongly compact can be the least strong and indestructible
The following three theorems are proved. (A) Suppose \(V\vDash \text{ZFC}+ \text{GCH}+\kappa\) is supercompact. Then, in some forcing extension, (a) \(\kappa\) is both the least strongly compact and least strong cardinal, and (b) each of \(\kappa\)'s strong compactness and strongness is indestructible under \(<\!\kappa\)-directed closed partial orderings which are also \(\kappa\)-strategically closed. (B) Suppose \(V\vDash \text{ZFC}+ \text{GCH}+\kappa_1\) is supercompact \(+\,\kappa_2 < \lambda\) and \(\kappa_2\) and \(\lambda\) are the smallest cardinals above \(\kappa_1\) such that \(\kappa_2\) is \(\lambda\)-supercompact and \(\lambda\) is inaccessible. There is then a partial ordering \(\mathbb{P}\in V\) such that, in \((V_\lambda)^{V^\mathbb{P}},\) (a) \(\kappa_1\) and \(\kappa_2\) are both the first two strongly compact and first two strong cardinals, (b) each of \(\kappa_1\)'s strong compactness and strongness is indestructible under \(<\!\kappa_1\)-directed closed partial orderings which are also \(\kappa_1\)-strategically closed, and (c) \(\kappa_2\)'s strongness, but not its strong compactness, is indestructible under \(\kappa_2\)-strategically closed forcing which is also \(<\!\kappa_1\)-directed closed. (C) Suppose \(V\vDash\text{ZFC}+ \text{GCH}+\kappa\) is supercompact with no Mahlo cardinal above it. Then, in some forcing extension, (a) \(\kappa\) is both the least strongly compact and least strong cardinal, (b) there is no Mahlo cardinal above \(\kappa\), (c) \(\kappa\)'s strong compactness is indestructible under \(<\!\kappa\)-directed closed forcing, and (d) \(\kappa\)'s strongness is indestructible under \(\kappa^+\)-weakly closed forcing notions satisfying the Prikry condition.
scientific article; zbMATH DE number 5121367