Analysis of HOD for Admissible Structures
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908273685299200 |
|---|---|
| author | Kruschewski, Jan Schlutzenberg, Farmer |
| author_facet | Kruschewski, Jan Schlutzenberg, Farmer |
| contents | Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in \mathbb{R}$ let $α_x$ be the least $β$ such that
\[
L_β[x] \models Σ_n \text{-KP} + \exists κ(``κ\text{ is inaccessible and }κ^+ \text{ exists}").
\]
We adapt the analysis of $\text{HOD}^{L[x,G]}$ as a strategy mouse to $L_{α_x}[x,G]$ for a cone of reals $x$. That is, we identify a mouse $\mathcal{M}^{\text{n-ad}}$ and define a class $H \subseteq L_{α_x}[x,G]$ as a natural analogue of $\text{HOD}^{L[x,G]} \subseteq L[x,G]$, and show that $H = M_\infty[Σ_0]$, where $M_\infty$ is an iterate of $\mathcal{M}^{\text{n-ad}}$ and $Σ_0$ a fragment of its iteration strategy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14458 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of HOD for Admissible Structures Kruschewski, Jan Schlutzenberg, Farmer Logic 03E45, 03E55, 03D60 Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in \mathbb{R}$ let $α_x$ be the least $β$ such that \[ L_β[x] \models Σ_n \text{-KP} + \exists κ(``κ\text{ is inaccessible and }κ^+ \text{ exists}"). \] We adapt the analysis of $\text{HOD}^{L[x,G]}$ as a strategy mouse to $L_{α_x}[x,G]$ for a cone of reals $x$. That is, we identify a mouse $\mathcal{M}^{\text{n-ad}}$ and define a class $H \subseteq L_{α_x}[x,G]$ as a natural analogue of $\text{HOD}^{L[x,G]} \subseteq L[x,G]$, and show that $H = M_\infty[Σ_0]$, where $M_\infty$ is an iterate of $\mathcal{M}^{\text{n-ad}}$ and $Σ_0$ a fragment of its iteration strategy. |
| title | Analysis of HOD for Admissible Structures |
| topic | Logic 03E45, 03E55, 03D60 |
| url | https://arxiv.org/abs/2503.14458 |