Generically supercompact cardinals by forcing with chain conditions

TL;DR

This paper explores the properties of generically supercompact cardinals under ccc forcing, showing they can be smaller than or equal to the continuum and still possess strong largeness features, contrasting with other forcing classes.

Contribution

It introduces the concept of ccc-generically supercompact cardinals and demonstrates their size and largeness properties, contrasting with $\sigma$-closed poset generics.

Findings

ccc-generically supercompact cardinals can be ≤ continuum

Such cardinals are stationary limits of ccc-generically measurable cardinals

Contrast with $\sigma$-closed posets where cardinals can be smaller

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 $\cal P$-generically supercompact cardinals for the class $\cal P$ of all $\sigma$-closed posets, which can be $\aleph_n$ for any n>1.