On a theorem of Magidor

TL;DR

This paper explores a method to construct a generic extension of set theory where a supercompact cardinal becomes the first infinite ordinal and an inaccessible cardinal becomes its immediate successor.

Contribution

It introduces a Magidor-inspired technique to realize specific cardinal configurations in generic extensions, connecting large cardinals with smaller infinite ordinals.

Findings

Achieves a model where  is _ and  is _{+1}.

Demonstrates a method to control cardinal characteristics via forcing.

Provides insights into the relationship between large cardinals and their collapses or reductions.

Abstract

Assuming $\kappa$ is a supercompact cardinal and $\lambda$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $\kappa=\aleph_\omega$ and $\lambda=\aleph_{\omega+1}.$