From Internal to External: Classical Models of ZF + PP + $\neg$AC
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908783052062720 |
|---|---|
| author | Gilson, Frank |
| author_facet | Gilson, Frank |
| contents | Goal. We analyze when the Partition Principle ($\mathsf{PP}$) holds without $\mathsf{AC}$ in models arising from a free finite $H$-action on Cantor space, and reconcile two standard routes to such models. Approach. Route I proceeds via a Boolean-valued presentation $\mathrm{Sh}(\mathbb{B})$ and symmetric names; Route II uses direct forcing with $\mathrm{Fn}(\mathbb{N}\times H,2)$ and finite-support automorphisms. We prove a unification theorem identifying the resulting symmetric submodels and develop a Local-to-Global Embedding Principle (LEP) for hereditarily symmetric names. Results. We prove external $\mathsf{PP}$ in the symmetric model $N$ built via Route II. From LEP we obtain that $\mathsf{PP}$ holds in the symmetric model, hence $N\models \mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$. Along the way, we unify the Route I/Route II presentations functorially. Limitations. Our proof exploits the countable-support stratification of $\mathrm{Fn}(\mathbb{N}\times H,2)$; extending the LEP/gluing to uncountable presentations (e.g. $\mathrm{Fn}(κ\times H,2)$ for $κ>ω$) or, more generally, to $κ$-directed families of finite supports remains open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09764 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Internal to External: Classical Models of ZF + PP + $\neg$AC Gilson, Frank Logic 03E25, 03E40, 03G30 Goal. We analyze when the Partition Principle ($\mathsf{PP}$) holds without $\mathsf{AC}$ in models arising from a free finite $H$-action on Cantor space, and reconcile two standard routes to such models. Approach. Route I proceeds via a Boolean-valued presentation $\mathrm{Sh}(\mathbb{B})$ and symmetric names; Route II uses direct forcing with $\mathrm{Fn}(\mathbb{N}\times H,2)$ and finite-support automorphisms. We prove a unification theorem identifying the resulting symmetric submodels and develop a Local-to-Global Embedding Principle (LEP) for hereditarily symmetric names. Results. We prove external $\mathsf{PP}$ in the symmetric model $N$ built via Route II. From LEP we obtain that $\mathsf{PP}$ holds in the symmetric model, hence $N\models \mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$. Along the way, we unify the Route I/Route II presentations functorially. Limitations. Our proof exploits the countable-support stratification of $\mathrm{Fn}(\mathbb{N}\times H,2)$; extending the LEP/gluing to uncountable presentations (e.g. $\mathrm{Fn}(κ\times H,2)$ for $κ>ω$) or, more generally, to $κ$-directed families of finite supports remains open. |
| title | From Internal to External: Classical Models of ZF + PP + $\neg$AC |
| topic | Logic 03E25, 03E40, 03G30 |
| url | https://arxiv.org/abs/2511.09764 |