Addendum to “ Maximal Chains In

This note is intended as a supplement and clarification to the proof of Theorem 3.3 of [1]; namely, it is consistent that b = ℵ 1 yet for every ultrafilter U on ω there is a ≤ * chain {f ξ : ξ ∈ ω 2 } such that {f ξ /U : ξ ∈ ω 2 } is cofinal in ω/U. The general outline of the the proof remains the same. In other words, a ground model is taken which satisfies 2 ℵ0 = ℵ 1 and in which there is a ♦ ω2 sequence {D ξ : ξ ∈ ω 2 } such that for every X ⊆ ω 2 there is a stationary set of ordinals, µ, such that cof(µ) = ω 1 and such that X ∩ µ = D µ. Actually, a coding will be used to associate with subsets of ω 2 , names for subsets on ω in certain partial orders. The details of this coding will be ignored except to state that c(D η) will denote the coded set and that if P ω2 = lim{P ξ : ξ ∈ ω 2 } is the finite support iteration of ccc partial orders of size no greater than ω 1 and 1 Pω 2 " X ⊆ [ω] ℵ 0 " then there is a stationary set S X ⊆ ω 2 , consisting of ordinals of uncountable cofinality, such that 1 P ξ " X ↾ P ξ = c(D ξ) " for each ξ ∈ S X. Here X ↾ P ξ denotes the P ξ-name obtained by considering only those parts of X that mention conditions in P ξ ; to be more precise here would requires providing the details of a specific development of names in the theory of forcing, and so this will not be done.