Huge Reflection

We study Structural Reflection beyond Vopěnka’s Principle, at the level of almost-huge cardinals and higher, up to rank-into-rank embeddings. We identify and classify new large cardinal notions in that region that correspond to some form of what we call Exact Structural Reflection (ESR). Namely, given cardinals κ < λ and a class C of structures of the same type, the corresponding instance of ESR asserts that for every structure A in C of rank λ, there is a structure B in C of rank κ and an elementary embedding of B into A. Inspired by the statement of Chang’s Conjecture, we also introduce and study sequential forms of ESR, which, in the case of sequences of length ω, turn out to be very strong. Indeed, when restricted to Π1-definable classes of structures they follow from the existence of I1-embeddings, while for more complicated classes of structures, e.g., Σ2, they are not known to be consistent. Thus, these principles unveil a new class of large cardinals that go beyond I1-embeddings, yet they may not fall into Kunen’s Inconsistency.