Indestructibility of Vopenka's Principle

TL;DR

This paper proves that Vopenka's Principle and Vopenka cardinals are indestructible under certain forcing iterations, enabling their consistency with various independent set-theoretic statements without additional preparatory forcing.

Contribution

It establishes the indestructibility of Vopenka's Principle and Vopenka cardinals under reverse Easton forcing, simplifying their preservation in set-theoretic extensions.

Findings

Vopenka's Principle is indestructible under specified forcing.

Vopenka cardinals are preserved without preparatory forcing.

Consistency of large cardinals with GCH, definable well-order, and morasses.

Abstract

We show that Vopenka's Principle and Vopenka cardinals are indestructible under reverse Easton forcing iterations of increasingly directed-closed partial orders, without the need for any preparatory forcing. As a consequence, we are able to prove the relative consistency of these large cardinal axioms with a variety of statements known to be independent of ZFC, such as the generalised continuum hypothesis, the existence of a definable well-order of the universe, and the existence of morasses at many cardinals.