Unprepared Indestructibility

TL;DR

This paper surveys a forcing indestructibility theorem for Vopenka's Principle, a large cardinal axiom, highlighting its unique feature of requiring no preparatory forcing to ensure indestructibility.

Contribution

It introduces a novel forcing indestructibility theorem for Vopenka's Principle without the need for preparatory forcing, advancing large cardinal indestructibility theory.

Findings

Vopenka's Principle is indestructible under certain forcings

No preparatory forcing is needed for this indestructibility

The paper provides a proof sketch of the main theorem

Abstract

This article is based on the talk of the same name which I gave at the "Aspects of Descriptive Set Theory" RIMS Symposium in Kyoto in October 2011; it is essentially just a survey of my article "Indestructibility of Vopenka's Principle". In particular, I present (with a sketch of the proof) a forcing indestructibility theorem for the large cardinal axiom Vopenka's Principle. It is notable in that there is no preparatory forcing required to make the axiom indestructible, unlike other indestructibility results.