Further cardinal arithmetic

Abstract

We continue the investigations in the author's book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S ≤ℵ 0 (κ), ⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ, λ, θ + , 2) = pp(λ) in more cases even when cf(λ) = ℵ 0 (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6).

The PCF Theorem Revisited

“…(2) Note that our results are for µ = µ ℵ0 only; to remove this we need to improve the theorem on pp = cov (i.e. to prove cf(λ) = ℵ 0 < λ ⇒ pp(λ) = cov(λ, λ, ℵ 1 , 2) (or sup{pp(µ): cf µ = ℵ 0 < µ < λ} = cf(S ≤ℵ0 (λ), ⊆) (see [Sh-g], [Sh430,§1]), which seems to me a very serious open problem (see Analitic guide,14]). (4) We can approach 3.15 differently, by 3.20-3.23 below.…”

More on real-valued measurable cardinals and forcing with ideals

“…Then λ < j(κ). By [Sh430], then in a generic ultrapower N cov(λ, ℵ 1 , ℵ 1 , 2) < j(κ). However, in V λ + ≤ cov(λ, κ, ℵ 1 , 2) ≤ cov(λ, ℵ 1 ℵ 1 , 2)) N < j(κ).…”

On densities of box products

“…An additional property that we need to show in the present situation is that d <ℵ 1 (κ) cannot be below λ. But this follows by [Sh430,5.3,5.4] and the pcf structure of the models of [Git-Mag] or just directly using the correspondence established in Lemma 2.9 between basic clopen sets of κ 2 of V [G] and κ n 2 of M n . Since already κ 1 2 cannot have a dense set of cardinality less than λ because λ + 2 embeds it and GCH holds.…”