The Virtual Large Cardinal Hierarchy

TL;DR

This paper advances the understanding of the virtual large cardinal hierarchy by analyzing virtual versions of superstrong, Woodin, and Berkeley cardinals, providing new characterizations and exploring their interrelations.

Contribution

It introduces new characterizations of virtually Woodin cardinals, explores the virtual Vopenka principle, and clarifies the hierarchy's structure and implications.

Findings

On is virtually Woodin iff for every class A, there is a proper class of virtually A-extendible cardinals.

The virtual Vopenka principle for finite languages is not equivalent but equiconsistent with the virtual Vopenka principle.

If no virtually Berkeley cardinals exist, then On is virtually Woodin iff On is virtually pre-Woodin.

Abstract

We continue the study of the virtual large cardinal hierarchy by analysing virtual versions of superstrong, Woodin, and Berkeley cardinals. Gitman and Schindler showed that virtualizations of strong and supercompact cardinals yield the same large cardinal notion. We provide various equivalent characterizations of virtually Woodin cardinals, including showing that On is virtually Woodin if and only if for every class A, there is a proper class of virtually A-extendible cardinals. We introduce the virtual Vopenka principle for finite languages and show that it is not equivalent to the virtual Vopenka principle (although the two principles are equiconsistent), but is equivalent to the assertion that On is virtually pre-Woodin, a weakening of virtually Woodin, which is equivalent to having for every class A, a weakly virtually A-extendible cardinal. We show that if there are no virtually Berkeley cardinals, then On is virtually Woodin if and only if On is virtually pre-Woodin (if and only if the virtual Vopenka principle for finite languages holds). In particular, if the virtual Vopenka principle holds and On is not Mahlo, then On is not virtually Woodin, and hence there is a virtually Berkeley cardinal.