Small Embedding Characterizations for Large Cardinals

TL;DR

This paper introduces new elementary embedding characterizations for various large cardinals, including subtle and ineffable ones, and explores their internal large cardinal notions and connections to generalized tree properties.

Contribution

It provides the first embedding characterizations for subtle, ineffable, and λ-ineffable cardinals, and links internal large cardinals to generalized tree properties, offering new proofs.

Findings

Embedding characterizations for subtle, ineffable, and λ-ineffable cardinals.

Definition of internal large cardinals corresponding to generalized tree properties.

New proofs of existing results by Wei ext{ extquoteright}s and Viale and Wei ext{ extquoteright}s.

Abstract

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. In particular, we provide such embedding characterizations also for several large cardinal notions for which no embedding characterizations have been known so far, namely for subtle, for ineffable, and for $\lambda$-ineffable cardinals. As an application, which we will study in detail in a subsequent paper, we present the basic idea of our concept of internal large cardinals. We provide the definition of certain kinds of internally subtle, internally $\lambda$-ineffable and internally supercompact cardinals, and show that these correspond to generalized tree properties, that were investigated by Wei\ss\ in his [16] and [17], and by Viale and Wei\ss\ in [15]. In particular, this yields new proofs of Wei\ss 's results from [16] and [17], eliminating problems contained in the original proofs.