The first omitting cardinal for Magidority

TL;DR

This paper proves that the first omitting cardinal for Magidority can be a successor of a supercompact or singular cardinal, advancing understanding of large cardinal hierarchies and their relationships.

Contribution

It establishes the consistency of the first omitting cardinal for Magidority being a successor of a supercompact or singular cardinal, a novel result in large cardinal theory.

Findings

$oldsymbol{ ext{α}_M(oldsymbol{ ext{λ}})}$ can be a successor of a supercompact cardinal

$oldsymbol{ ext{α}_M(oldsymbol{ ext{λ}})}$ can be a successor of a singular cardinal with uncountable cofinality

Advances the understanding of the structure of Magidor cardinals and their associated omitting cardinals.

Abstract

We prove that $\alpha_M(\lambda)$ can be successor of a supercompact cardinal, when $\lambda$ is a Magidor cardinal. From this result we obtain the consistency of $\alpha_M(\lambda)$ being a successor of a singular cardinal with uncountable cofinality.