Let J pcf

The present note is an answer to complains of E.Weitz on [Sh 371]. We present a corrected version of a part of chapter VIII of Cardinal Arithmetic. 1. J * [a] = b : b ⊆ b and for every inaccessible µ, we have µ > sup(b ∩ µ). 2. pcf * (a) = tcf(a/D) : D is an ultrafilter on a, D ∩ J * [a] = ∅. 3. If |a| < min(a), for µ ∈ pcf(a) let b a µ = b µ [a] be a subset of a such that J ≤µ [a] = J <µ [a] + b µ [a]. (Note that b a µ exists by [Sh:g, VIII 2.6], also a is a finite union of b µ [a]'s). 4. If |a| < min(a) let J pcf <λ [a] be the ideal of subsets of pcf(a) generated by {pcf(b µ [a]) : µ ∈ λ ∩ pcf(a)}. Let J pcf ≤λ [a] = J pcf <λ + [a]. pcf <λ [a] depends on a and λ only (and not on the choice of the b µ [a]'s). 2. If b ⊆ a then J pcf <λ [b] = P(b) ∩ J pcf <λ [a] and J * [b] = P(b) ∩ J * [a]. Proof. (1) Let b ′ µ [a] : µ ∈ pcf(a), b ′′ µ [a] : µ ∈ pcf(a) both be as in 1(3). So for each θ, b ′ θ [a] ⊆ b ′′ θ [a] ∪ ℓ<n b ′′ θ ℓ ′ θ [a]) ⊆ pcf(b ′′ θ [a]) ∪ ℓ<n pcf(b ′′ θ ℓ [a]), and each is in J pcf <λ [a] as defined by b ′′ σ [a] : σ ∈ pcf(a) (as θ ℓ < θ < λ). As this holds for every θ < λ, all generators of J pcf <λ [a] as defined by b ′ σ [a] : σ ∈ pcf(a) are in J pcf <λ [a] as defined by b ′′ σ [a] : σ ∈ pcf(a). As the situation is symmetric we finish. (2) Similar proof. The first phrase follows from part (1), and check the second. Lemma 3 ([Sh:g, VIII 3.2]). Suppose |a| + < min(a), a ⊆ b ∈ J * [pcf(a)], b / ∈ J =: J pcf <λ [a] and λ = max pcf(a). Then tcf(b/J) is λ. Proof. Remember that (by [Sh:g, VIII 2.6]) there …