Axiom $\mathcal{A}$ and supercompactness

TL;DR

This paper introduces a new axiom, $ ext{A}$, that enhances the understanding of supercompact cardinals, demonstrating their properties in models and their compatibility with other large cardinal axioms.

Contribution

The paper presents a new axiom $ ext{A}$ that improves models of supercompact cardinals and explores their relationships with other large cardinal notions.

Findings

Every supercompact cardinal can be made $C^{(1)}$-supercompact with inaccessible targets.

The axiom $ ext{A}$ is compatible with Woodin's $I_0$ cardinals.

Supercompactness is the strongest large-cardinal notion preserved by Radin forcing.

Abstract

We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in \cite{HMP} and provides a definitive answer to a question of Bagaria \cite[p.19]{Bag}. This configuration is a consequence of a new axiom we introduce -- called $\mathcal{A}$ -- which is showed to be compatible with Woodin's $I_0$ cardinals. We also answer a question of V. Gitman and G. Goldberg on the relationship between supercompactness and cardinal-preserving extendibility. As an incidental result, we prove a theorem suggesting that supercompactness is the strongest large-cardinal notion preserved by Radin forcing.