μ-Clubs OF P_κ(λ) : Paradise on earth

2023/08/28 by Matet, Pierre
#03E04 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2308.14773

Abstract

If V = L, and μ, κ and λ are three infinite cardinals with μ= \rm cf (μ) < κ= \rm cf(κ) ≤ λ, then, as shown in \citeHeaven, the μ-club filters on P_κ(λ) and P_κ(λ^{< κ}) are isomorphic if and only if \rm cf (λ) \not= μ. Now in L, λ^{< κ} equals u (κ, λ) (the least size of a cofinal subset in (P_κ(λ), ⊆)) equals λ if \rm cf (λ) ≥ κ, and λ⁺ otherwise. We show that, in ZFC, there are many triples (μ, κ, λ) for which (u (κ, λ) > λ and) the μ-club filters on P_κ(λ) and P_κ(u (κ, λ)) are isomorphic.

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