The *-variation of the Banach-Mazur game and forcing axioms
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2015
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| _version_ | 1866911486839881728 |
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| author | Yoshinobu, Yasuo |
| author_facet | Yoshinobu, Yasuo |
| contents | We introduce a property of posets which strengthens (ω_1+1)-strategic closedness. This property is defined using a variation of the Banach-Mazur game on posets, where the first player chooses a countable set of conditions instead of a single condition at each turn. We prove PFA is preserved under any forcing over a poset with this property. As an application we reproduce a proof of Magidor's theorem about the consistency of PFA with some weak variations of the square principles. We also argue how different this property is from (ω_1+1)-operational closedness, which we introduced in our previous work, by observing which portions of MA^+(ω_1-closed) are preserved or destroyed under forcing over posets with either property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1503_06703 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | The *-variation of the Banach-Mazur game and forcing axioms Yoshinobu, Yasuo Logic 03E57 (primary), 03E35 (secondary) We introduce a property of posets which strengthens (ω_1+1)-strategic closedness. This property is defined using a variation of the Banach-Mazur game on posets, where the first player chooses a countable set of conditions instead of a single condition at each turn. We prove PFA is preserved under any forcing over a poset with this property. As an application we reproduce a proof of Magidor's theorem about the consistency of PFA with some weak variations of the square principles. We also argue how different this property is from (ω_1+1)-operational closedness, which we introduced in our previous work, by observing which portions of MA^+(ω_1-closed) are preserved or destroyed under forcing over posets with either property. |
| title | The *-variation of the Banach-Mazur game and forcing axioms |
| topic | Logic 03E57 (primary), 03E35 (secondary) |
| url | https://arxiv.org/abs/1503.06703 |