On forcing axioms and weakenings of the Axiom of Choice
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_ | 1866910834904530944 |
|---|---|
| author | Bomfim, Diego Lima Morgan, Charles da Silva, Samuel Gomes |
| author_facet | Bomfim, Diego Lima Morgan, Charles da Silva, Samuel Gomes |
| contents | We prove forcing axiom equivalents of two families of weakenings of the axiom of choice: a trichotomy principle for cardinals isolated by Lévy, ${\rm H\hskip0.05pt}_κ$, and ${\rm DC}_κ$, the principle of dependent choices generalized to cardinals $κ$, for regular cardinals $κ$. Using these equivalents we obtain new forcing axiom formulations of the axiom of choice.
A point of interest is that we use a new template for forcing axioms. For the class of forcings to which we asks that the axioms apply, we do not ask that they apply to all collections of dense sets of a certain cardinality, but rather only for each particular forcing to a specific family of dense sets of the cardinality in question. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10736 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On forcing axioms and weakenings of the Axiom of Choice Bomfim, Diego Lima Morgan, Charles da Silva, Samuel Gomes Logic 03E25, 03E57, 03E05 We prove forcing axiom equivalents of two families of weakenings of the axiom of choice: a trichotomy principle for cardinals isolated by Lévy, ${\rm H\hskip0.05pt}_κ$, and ${\rm DC}_κ$, the principle of dependent choices generalized to cardinals $κ$, for regular cardinals $κ$. Using these equivalents we obtain new forcing axiom formulations of the axiom of choice. A point of interest is that we use a new template for forcing axioms. For the class of forcings to which we asks that the axioms apply, we do not ask that they apply to all collections of dense sets of a certain cardinality, but rather only for each particular forcing to a specific family of dense sets of the cardinality in question. |
| title | On forcing axioms and weakenings of the Axiom of Choice |
| topic | Logic 03E25, 03E57, 03E05 |
| url | https://arxiv.org/abs/2404.10736 |