On Indestructible Strongly Guessing Models
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915079091388416 |
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| author | Mohammadpour, Rahman Velickovic, Boban |
| author_facet | Mohammadpour, Rahman Velickovic, Boban |
| contents | In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(ω_3,ω_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $ω_2$ and $ω_3$. In this paper we formulate a strengthening of ${\rm GM}^+(ω_3,ω_1)$ that we call ${\rm SGM}^+(ω_3,ω_1)$. We also prove, modulo the consistency of two supercompact cardinals, that ${\rm SGM}^+(ω_3,ω_1)$ is consistent with ZFC. In addition to all the consequences of ${\rm GM}^+(ω_3,ω_1)$, the principle ${\rm SGM}^+(ω_3,ω_1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $ω_2$ either adds a real or collapses some cardinal. This gives a partial answer to a question of Abraham \cite{AvrahamPhD} and extends a previous result of Todorčević \cite{Todorcevic82} in this direction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_17458 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Indestructible Strongly Guessing Models Mohammadpour, Rahman Velickovic, Boban Logic 03E35, 03E55, 03E57, 03E65 In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(ω_3,ω_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $ω_2$ and $ω_3$. In this paper we formulate a strengthening of ${\rm GM}^+(ω_3,ω_1)$ that we call ${\rm SGM}^+(ω_3,ω_1)$. We also prove, modulo the consistency of two supercompact cardinals, that ${\rm SGM}^+(ω_3,ω_1)$ is consistent with ZFC. In addition to all the consequences of ${\rm GM}^+(ω_3,ω_1)$, the principle ${\rm SGM}^+(ω_3,ω_1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $ω_2$ either adds a real or collapses some cardinal. This gives a partial answer to a question of Abraham \cite{AvrahamPhD} and extends a previous result of Todorčević \cite{Todorcevic82} in this direction. |
| title | On Indestructible Strongly Guessing Models |
| topic | Logic 03E35, 03E55, 03E57, 03E65 |
| url | https://arxiv.org/abs/2303.17458 |