Fitxers
Tipus de document
ArticleVersió
Versió publicadaData de publicació
Llicència de publicació
Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/217380
Huge Reflection
Títol de la revista
Autors
Director/Tutor
ISSN de la revista
Títol del volum
Recurs relacionat
Resum
We 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.
Matèries (anglès)
Citació
Citació
LÜCKE, Philipp and BAGARIA, Joan. Huge Reflection. Annals of Pure and Applied Logic. 2023. Vol. 174, num. 1. ISSN 0168-0072. [consulted: 4 of June of 2026]. Available at: https://hdl.handle.net/2445/217380