Small Embedding Characterizations for Large Cardinals

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.