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Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing — arXiv:2609.25033

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.25033
License (as archived): CC0-1.0
Archived: 2026-10-08T16:34:10.804Z
SHA-256 of the archived text: 0e1bc9936050d76bfb32be59499f337942dbf60def0220dd70787a058c44e49b
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How to cite

Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing — arXiv:2609.25033 — arXiv (en), https://arxiv.org/abs/2609.25033 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ec560dd124e2 (Metraipolis archive dc_ec560dd124e2, sha256 0e1bc9936050d76bfb32be59499f337942dbf60def0220dd70787a058c44e49b)

BibTeX

@misc{dcec560dd124e2,
  title = {Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing — arXiv:2609.25033},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.25033},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ec560dd124e2; sha256 0e1bc9936050d76bfb32be59499f337942dbf60def0220dd70787a058c44e49b},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.25033. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

For every uncountable regular cardinal $\alpha$, $\mathbb S^{\ast}(\alpha)$ is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is $\sigma$-closed and, after adjoining a formal maximum, $\alpha$-strategically closed. For $\alpha\geq\omega_2$, every nonempty countable family of designated generic branches has a stationary and costationary common trace on the generic ladder-coordinate set $L_\alpha$, while no countable family of cofinal branches generates $L_\alpha$. These conclusions persist under a Kurepa-style level-size bound. In the unrestricted forcing, for every infinite cardinal $\mu<\alpha$ in the ground model, some level of the generic tree contains a copy of $({}^\mu 2)^V$. Consequently, the endpoint-corrected restriction family indexed by $\mathcal P_{\omega_2}\alpha$ is too wide, whereas the scaled restriction system indexed by $\mathcal P_\alpha\alpha$ has all levels of size less than $\alpha$ exactly when $\alpha$ is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of $L_\alpha$ relative to the scaled system is at least $\omega_1$. The low-cofinality empty-value convention also ensures that the set of domains of $L_\alpha$ contains no club in $\mathcal P_\alpha\alpha$. The unrestricted tree clause of the motivating presentation is retained, without asserting forcing equivalence.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-23 in math.LO).
Categorie arXiv: Logic (math.LO).
Pagina del lavoro: https://arxiv.org/abs/2609.25033 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T15:54:33.534Z dalla pagina https://arxiv.org/catchup/math.LO/2026-09-23?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).