- 1 6 Not Collapsing Cardinals ≤ Κ in ( < Κ ) – Support Iterations

We deal with the problem of preserving various versions of completeness in (< κ)–support iterations of forcing notions, generalizing the case " S–complete proper is preserved by CS iterations for a stationary co-stationary S ⊆ ω 1 ". We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal κ and a stationary set S ⊆ κ with fat complement we can have uniformization for A δ : δ ∈ S , A δ ⊆ δ = sup A δ , cf(δ) = otp(A δ) and a stationary non-reflecting set S ⊆ S. Section 0: Introduction We put this work in a context and state our aim. –0.1 Background: Abelian groups –0.2 Background: forcing [We define (< κ)–support iteration.] –0.3 Notation CASE A Here we deal with Case A, say κ = λ + , cf(λ) = λ, λ = λ <λ. Section A.1: Complete forcing notions We define various variants of completeness and related games; the most important are the strong S–completeness and real (S 0 , ˆ S 1 , D)–completeness. We prove that the strong S–completeness is preserved in (< κ)–support iterations (A.1.13) Section A.2: Examples We look at guessing clubs ¯ C = C δ : δ ∈ S. If [α ∈ nacc(C δ) ⇒ cf(α) < λ] we give a forcing notion (in our context) which adds a club C of κ such that C ∩ nacc(C δ) is bounded in δ for all δ ∈ S. (Later, using a preservation theorem, we will get the consistency of " no such ¯ C guesses clubs " .) Then we deal with uniformization (i.e., Pr S) and the (closely related) being Whithead. Section A.3: The iteration theorem We deal extensively with (standard) trees of conditions, their projections and inverse limits. The aim is to build a (P γ , N)–generic condition forcing G ˜ γ ∩ N , and the trees of conditions are approximations to it. The main result in the preservation theorem for our case (A.3.7). Section A.4: The Axiom We formulate a Forcing Axiom relevant for our case and we state its consistency.