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Universal domination and idealized forcing — arXiv:2608.18964

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.18964
License (as archived): CC0-1.0
Archived: 2026-10-18T14:00:00.619Z
SHA-256 of the archived text: 2404179e380fd7372934cce16e2a8c9cfd140c35fb3245ad543a55531520a40a
Stable URL of this page: https://metraipolis.com/documento/dc_7e7aa649d5bb

How to cite

Universal domination and idealized forcing — arXiv:2608.18964 — arXiv (en), https://arxiv.org/abs/2608.18964 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7e7aa649d5bb (Metraipolis archive dc_7e7aa649d5bb, sha256 2404179e380fd7372934cce16e2a8c9cfd140c35fb3245ad543a55531520a40a)

BibTeX

@misc{dc7e7aa649d5bb,
  title = {Universal domination and idealized forcing — arXiv:2608.18964},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.18964},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7e7aa649d5bb; sha256 2404179e380fd7372934cce16e2a8c9cfd140c35fb3245ad543a55531520a40a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.18964. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We introduce the universality property of a definable $\sigma$-ideal on a Polish space, which, on one hand, can serve as a benchmark for the properness of the associated idealized forcing of positive Borel sets ordered by inclusion, and, on the other hand, unifies many of the results that can be found in Zapletal's book. We show that under mild absoluteness assumptions, it implies properness, various dichotomy theorems, and closure under well-ordered unions in the Solovay model and under $\mathsf{AD}^+$, among other things. All major classes of proper idealized forcings studied in the book have this property. Further, we use this viewpoint to answer a question of Khomskii by showing that the naive idealized forcing for adding an eventually different real or a refining real is not proper below some condition. We also answer a question related to the definability of $\sigma$-ideals generated by Borel sets due to Kanovei, Sabok, and Zapletal.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-20 in math.LO).
Categorie arXiv: Logic (math.LO).
Pagina del lavoro: https://arxiv.org/abs/2608.18964 — 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 2026-10-18T13:58:45.839Z dalla pagina https://arxiv.org/catchup/math.LO/2026-08-20?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).