Distributivity and minimality in perfect tree forcings for singular cardinals
Abstract: Dobrinen, Hathaway and Prikry studied a forcing ℙκ consisting of perfect trees of height λ and width κ where κ is a singular λ-strong limit of cofinality λ. They showed that if κ is singular of countable cofinality, then ℙκ is minimal for ω-sequences assuming that κ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption.
Prikry proved that ℙκ is (ω, ν)-distributive for all ν < κ given a singular ω-strong limit cardinal κ of countable cofinality, and Dobrinen et al. asked whether this result generalizes if κ has uncountable cofinality. We answer their question in the negative by showing that ℙκ is not (λ, 2)-distributive if κ is a λ-strong limit of uncountable cofinality λ and we obtain the same result for a range of similar forcings, including one that Dobrinen et al. consider that consists of pre-perfect trees. We also show that ℙκ in particular is not (ω, ·, λ+)-distributive under these assumptions.
While developing these ideas, we address natural questions regarding minimality and collapses of cardinals
- Location
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Deutsche Nationalbibliothek Frankfurt am Main
- Extent
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Online-Ressource
- Language
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Englisch
- Notes
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Israel journal of mathematics. - 261, 2 (2024) , 549-588, ISSN: 1565-8511
- Event
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Veröffentlichung
- (where)
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Freiburg
- (who)
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Universität
- (when)
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2024
- Creator
- DOI
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10.1007/s11856-024-2607-z
- URN
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urn:nbn:de:bsz:25-freidok-2473189
- Rights
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Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
- Last update
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14.08.2025, 11:00 AM CEST
Data provider
Deutsche Nationalbibliothek. If you have any questions about the object, please contact the data provider.
Associated
- Levine, Maxwell
- Mildenberger, Heike
- Universität
Time of origin
- 2024
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