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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2502.11579 |
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| _version_ | 1866910831488270336 |
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| author | Tsukuura, Kenta |
| author_facet | Tsukuura, Kenta |
| contents | In this paper, we demonstrate that if, for every $κ$-complete fine filter $F$ over $\mathcal{P}_κλ$, the associated Namba forcing $\mathrm{Nm}(κ,λ,F)$ is semiproper, then $\square(μ,{<}\aleph_1)$ fails for all regular $μ\in [λ, 2^λ]$ under the certain cardinal arithmetic. In particular, this result establishes that the consistency strength of the semiproperness of $\mathrm{Nm}(\aleph_2,F)$ for every $\aleph_2$-complete filter $F$ over $\aleph_2$ exceeds the strength of infinitely many Woodin cardinals.
Minimal walk methods associated with a square sequece play a central role in this paper. These observations introduce two-cardinal walks with naive $C$-sequences and show that the existence of non-reflecting stationary subsets implies $\mathcal{P}_κλ\not\to [I_{κλ}^{+}]^{3}_λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11579 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Walks along a weak square sequence and the non-semiproperness of Namba forcings Tsukuura, Kenta Logic In this paper, we demonstrate that if, for every $κ$-complete fine filter $F$ over $\mathcal{P}_κλ$, the associated Namba forcing $\mathrm{Nm}(κ,λ,F)$ is semiproper, then $\square(μ,{<}\aleph_1)$ fails for all regular $μ\in [λ, 2^λ]$ under the certain cardinal arithmetic. In particular, this result establishes that the consistency strength of the semiproperness of $\mathrm{Nm}(\aleph_2,F)$ for every $\aleph_2$-complete filter $F$ over $\aleph_2$ exceeds the strength of infinitely many Woodin cardinals. Minimal walk methods associated with a square sequece play a central role in this paper. These observations introduce two-cardinal walks with naive $C$-sequences and show that the existence of non-reflecting stationary subsets implies $\mathcal{P}_κλ\not\to [I_{κλ}^{+}]^{3}_λ$. |
| title | Walks along a weak square sequence and the non-semiproperness of Namba forcings |
| topic | Logic |
| url | https://arxiv.org/abs/2502.11579 |