Ramsey-like cardinals

TL;DR

This paper introduces new large cardinal axioms that extend the Ramsey elementary embeddings characterization, establishing a hierarchy between weakly compact and measurable cardinals, and enhancing understanding of elementary embedding properties.

Contribution

It proposes a hierarchy of large cardinal axioms generalizing Ramsey cardinals, bridging the gap between weakly compact and measurable cardinals, and exploring their consistency with V=L.

Findings

New axioms form a natural hierarchy of large cardinals.

Hierarchy lies between weakly compact and measurable cardinals.

Results are consistent with V=L.

Abstract

One of the numerous characterizations of a Ramsey cardinal kappa involves the existence of certain types of elementary embeddings for transitive sets of size \kappa satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L.