Non‐saturation of the non‐stationary ideal onP_κ(λ) with λ of countable cofinality

Abstract

MSC (2010) 03E05Given a regular uncountable cardinal κ and a cardinal λ > κ of cofinality ω, we show that the restriction of the non-stationary ideal on Pκ (λ) to the set of all a with cf(sup(a ∩ κ)) = ω is not λ + + -saturated and even not 2 (λ < κ ) -saturated in case 2 λ = λ + . We actually prove the stronger result that there is Q ⊆ NG + κ ,λ with |Q| = λ + + such that A ∩ B is a non-cofinal subset of Pκ (λ) for any two distinct members A, B of Q, where NG κ ,λ denotes the game ideal on Pκ (λ). We also remark that for κ > ω1 , adding λ + 3 Cohen subsets of ω1 to L makes NG κ ,λ λ + 3 -saturated.

Non-saturation of the nonstationary ideal on P κ (λ) in case κ ≤ cf (λ) < λ

“…Lemma 4.4 (Shelah) Suppose κ ≤ cf(λ) < λ. Then the following hold : [7]) If pp(λ) = λ ++ , then κ, cf(λ), λ, λ ++ holds.…”