Huge reflection

We study Structural Reflection beyond Vopenka'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 kappa < lambda 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 lambda, there is a structure B in C of rank kappa 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 omega, turn out to be very strong. Indeed, when restricted to l(1)-definable classes of structures they follow from the existence of I1-embeddings, while for more complicated classes of structures, e.g., sigma(2), 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. (C) 2022 Elsevier B.V. All rights reserved.