Saturation of reduced products
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916103292190720 |
|---|---|
| author | De Bondt, Ben Farah, Ilijas Vignati, Alessandro |
| author_facet | De Bondt, Ben Farah, Ilijas Vignati, Alessandro |
| contents | We study reduced products $M=\prod_n M_n/\mathrm{Fin}$ of countable structures in a countable language associated with the Fréchet ideal. We prove that such $M$ is $2^{\aleph_0}$-saturated if its theory is stable and not $\aleph_2$-saturated otherwise (regardless of whether the Continuum Hypothesis holds). This implies that $M$ is isomorphic to an ultrapower (associated with an ultrafilter on $\mathbb N$) if its theory is stable, even if the CH fails. We also improve a result of Farah and Shelah and prove that there is a forcing extension in which such reduced product $M$ is isomorphic to an ultrapower if and only if the theory of $M$ is stable. All of these conclusions apply for reduced products associated with $F_σ$ ideals or more general layered ideals. We also prove that a reduced product associated with the asymptotic density zero ideal $\mathcal Z_0$, or any other analytic P-ideal that is not $F_σ$, is not even $\aleph_1$-saturated if its theory is unstable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Saturation of reduced products De Bondt, Ben Farah, Ilijas Vignati, Alessandro Logic 03C20, 03C50, 03C45 We study reduced products $M=\prod_n M_n/\mathrm{Fin}$ of countable structures in a countable language associated with the Fréchet ideal. We prove that such $M$ is $2^{\aleph_0}$-saturated if its theory is stable and not $\aleph_2$-saturated otherwise (regardless of whether the Continuum Hypothesis holds). This implies that $M$ is isomorphic to an ultrapower (associated with an ultrafilter on $\mathbb N$) if its theory is stable, even if the CH fails. We also improve a result of Farah and Shelah and prove that there is a forcing extension in which such reduced product $M$ is isomorphic to an ultrapower if and only if the theory of $M$ is stable. All of these conclusions apply for reduced products associated with $F_σ$ ideals or more general layered ideals. We also prove that a reduced product associated with the asymptotic density zero ideal $\mathcal Z_0$, or any other analytic P-ideal that is not $F_σ$, is not even $\aleph_1$-saturated if its theory is unstable. |
| title | Saturation of reduced products |
| topic | Logic 03C20, 03C50, 03C45 |
| url | https://arxiv.org/abs/2401.12539 |