Generically extendible cardinals

TL;DR

This paper introduces the concept of generically extendible cardinals, analyzing their consistency strength and properties like indestructibility and generic absoluteness, expanding understanding of large cardinal notions in set theory.

Contribution

It defines and studies generically extendible cardinals, showing their varying consistency strengths and exploring their implications in set-theoretic properties.

Findings

Generic extendibility of ω₁ and ω₂ has small consistency strength.

Generic extendibility of cardinals above ω₂ has large consistency strength.

Results on indestructibility and generic absoluteness related to these 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 $\omega_1$ or $\omega_2$ has small consistency strength, but that of a cardinal $>\omega_2$ 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.