Generically supercompact cardinals by forcing with chain conditions

Abstract

A ccc-generically supercompact cardinal $\kappa$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $\kappa$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically measurable cardinals (Theorem 4.1). This is in a strong contrast to $\mathcal{P}$-generically supercompact cardinals for the class $\mathcal{P}$ of all $\sigma$-closed posets, which can be ${\aleph }_n$ for any n>1.