Analysis of Namba forcing's semiproperness reveals implications for saturated ideals and Prikry forcing.
We study some variations of Namba forcing Nm(κ,λ) over Pκλ and show that its semiproperness implies SSR([λ]ω,<κ). In particular, Prikry forcing at μ forces that Namba forcing Nm(μ+) is not semiproper. This shows that there are no semiproper saturated ideals in extensions by Prikry-type forcings. We also show that [λ+]μ+ cannot carry a λ+-saturated semiproper ideal if μ is singular with countable cofinality.
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Kenta Tsukuura (2025) studied this question.
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