Lücke, PhilippBagaria, Joan2025-01-132025-01-132023-010168-0072https://hdl.handle.net/2445/217380We study Structural Reflection beyond Vopěnka's Principle, at the level of almosthuge 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 $\mathcal{C}$ of structures of the same type, the corresponding instance of ESR asserts that for every structure $A$ in $\mathcal{C}$ of rank $\lambda$, there is a structure $B$ in $\mathcal{C}$ 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 $\Pi_1$-definable classes of structures they follow from the existence of $I 1$-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.32 p.application/pdfengcc by (c) Joan Bagaria et al., 2023http://creativecommons.org/licenses/by/3.0/es/Nombres cardinalsTeoria de conjuntsCategories (Matemàtica)Cardinal numbersSet theoryCategories (Mathematics)info:eu-repo/semantics/article7443302025-01-13info:eu-repo/semantics/openAccess