The Ultrapower Axiom and the equivalence between strong compactness and supercompactness

TL;DR

This paper shows that under the Ultrapower Axiom, strong compactness and supercompactness are equivalent, providing evidence for their equiconsistency, except for some specific counterexamples.

Contribution

It proves the equivalence of strong compactness and supercompactness under the Ultrapower Axiom, clarifying their relationship beyond ZFC.

Findings

Strongly compact and supercompact cardinals are equivalent under the Ultrapower Axiom.

The equivalence excludes a class of counterexamples identified by Menas.

Supports the idea that these large cardinal notions are equiconsistent.

Abstract

The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are equivalent except for a class of counterexamples identified by Menas. This is evidence that strongly compact and supercompact cardinals are equiconsistent.