Generically extendible cardinals

Abstract

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.

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