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 ($\mathrm{ESR}$). Namely, given cardinals $κ<λ$ and a class $\mathcal{C}$ of structures of the same type, the corresponding instance of $\mathrm{ESR}$ asserts that for every structure $A$ in $\mathcal{C}$ of rank $λ$, there is a structure $B$ in $\mathcal{C}$ 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 $\mathrm{ESR}$, which, in the case of sequences of length $ω$, turn out to be very strong. Indeed, when restricted to $Π_1$-definable classes of structures they follow from the existence of $I1$-embeddings, while for more complicated classes of structures, e.g., $Σ_2$, 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.

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