The $\mathsf{HOD}$ Hypothesis and a supercompact cardinal
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2018
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908569529483264 |
|---|---|
| author | Cheng, Yong |
| author_facet | Cheng, Yong |
| contents | In this paper, we prove that: if $κ$ is supercompact and the $\mathsf{HOD}$ Hypothesis holds, then there is a proper class of regular cardinals in $V_κ$ which are measurable in $\mathsf{HOD}$. Woodin also proved this result. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the $\mathsf{HOD}$ Hypothesis and supercompact cardinals, large cardinals in $\mathsf{V}$ are reflected to be large cardinals in $\mathsf{HOD}$ in a local way, and reveals the huge difference between $\mathsf{HOD}$-supercompact cardinals and supercompact cardinals under the $\mathsf{HOD}$ Hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1801_10420 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The $\mathsf{HOD}$ Hypothesis and a supercompact cardinal Cheng, Yong Logic 03E55, 03E99 In this paper, we prove that: if $κ$ is supercompact and the $\mathsf{HOD}$ Hypothesis holds, then there is a proper class of regular cardinals in $V_κ$ which are measurable in $\mathsf{HOD}$. Woodin also proved this result. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the $\mathsf{HOD}$ Hypothesis and supercompact cardinals, large cardinals in $\mathsf{V}$ are reflected to be large cardinals in $\mathsf{HOD}$ in a local way, and reveals the huge difference between $\mathsf{HOD}$-supercompact cardinals and supercompact cardinals under the $\mathsf{HOD}$ Hypothesis. |
| title | The $\mathsf{HOD}$ Hypothesis and a supercompact cardinal |
| topic | Logic 03E55, 03E99 |
| url | https://arxiv.org/abs/1801.10420 |