A note on cardinal preserving embeddings

TL;DR

This paper establishes the consistency of a set theory with a strongly compact cardinal by linking it to the existence of a cardinal-preserving embedding, and discusses implications under the SCH hypothesis.

Contribution

It demonstrates that the existence of a cardinal-preserving embedding implies the consistency of ZFC with a strongly compact cardinal and explores related preservation properties.

Findings

Consistency of ZFC + strongly compact cardinal proven

Cardinal-preserving embeddings preserve cofinality and continuum functions under SCH

Provides new insights into the structure of inner models and large cardinals

Abstract

We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving embedding preserves the cofinality and continuum functions.