Generically Extendible Cardinals
Toshimichi Usuba · Notre Dame Journal of Formal Logic · 2025
In this paper, we study the notion of a generically extendible cardinal, which is a generic version of an extendible cardinal. We prove that the generic extendibility of ω1 or ω2 has small consistency strength, but that of a cardinal >ω2 does not. We also consider some results concerned with generically extendible cardinals, such as indestructibility, generic absoluteness of the reals, and Boolean-valued second-order logic.