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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2106.01462 |
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| _version_ | 1866909057375272960 |
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| author | Bagaria, Joan Lücke, Philipp |
| author_facet | Bagaria, Joan Lücke, Philipp |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_01462 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Huge Reflection Bagaria, Joan Lücke, Philipp Logic 03E55, 03E65, 18A10, 18A15 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. |
| title | Huge Reflection |
| topic | Logic 03E55, 03E65, 18A10, 18A15 |
| url | https://arxiv.org/abs/2106.01462 |