Huge Reflection

TL;DR

This paper introduces and classifies new large cardinal notions called Exact Structural Reflection (ESR), extending the understanding of structural reflection principles beyond known large cardinal axioms, and explores their consistency and strength.

Contribution

It defines ESR principles at high levels of the large cardinal hierarchy, introduces sequential forms inspired by Chang's Conjecture, and analyzes their consistency and implications.

Findings

ESR principles extend large cardinal hierarchy beyond Vopěnka's Principle.

Sequential ESR forms are very strong, related to I1-embeddings.

Some ESR principles are consistent, others remain open questions.

Abstract

We study Structural Reflection beyond Vop\v{e}nka'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 $\kappa<\lambda$ 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 $\lambda$, 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 $\mathrm{ESR}$, 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 $I1$-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.