Donder’s Version of Revised Countable Support

arXiv:math/9207204v1 [math.LO] 30 Jul 1992Donder’s Version of Revised Countable SupportUlrich Fuchs∗September 6, 2018AbstractShelah introduced the revised countable support (RCS) iteration toiterate semiproperness [Shelah]. This was an endpoint in the search foran iteration of a weak condition, still implying that ℵ1 is preserved.

Itwas one of the key tools in the proof of the relative consistency of Martin’sMaximum. Dieter Donder found a better manageable approach to thisiteration, which is presented here.

More iterations of semiproperness areformulated in [Schlindwein] and [Miyamoto].TerminologyAn iteration is a well-ordered commuting family of complete embeddings ofcomplete Boolean algebras. From now on let B = ⟨Bγ|γ ∈λ⟩be an iteration,which hence means ∀α < β < λBα ⋖Bβ.

We will formulate in this sectionstatements and definitions of notions for this special case only, but they all haveformulations and generalizations for the general case.If α < λ we have ⊩α (B/G is iteration). (A filter G in Bα defines quotientsBβ/G (β > α) and embeddings Bβ/G −→Bγ/G (γ > β > α) even if thealgebras in B are not complete.

The system is called B/G).Of course the family of the canonical projections hα : S B −→Bα, b 7→V{c ∈Bα|b ⩽c} does also commute. (i.e.

α < β =⇒hα ◦hβ = hα).A thread in B is a f ∈Q B with ∀α ⩽β < λf(α) = hα(f(β)).If f is a thread, we can define an iteration B↾f as ⟨Bα↾f(α)|α < λ⟩togetherwith the complete embeddings Bα↾f(α) −→Bβ↾f(β), b 7→b ∧f(β).T (B) is the set of all threads, which is canonically componentwise partiallyordered.Let c : S B −→T (B) be defined by b 7→⟨hα(b)|α < λ⟩. The range of c isthe set C(B) of all eventually constant threads.∗Freie Universit¨at Berlin, e-mail: FuX@Math.FU-Berlin.De1

Let C(B) ⊂F ⊂T (B). If∀α < λ∀f ∈F∀b ∈Bαb ⩽f(α) =⇒f and c(b) are compatible in F(1)then F is separative and the mappings Bα −→F, b 7→c(b) are complete embed-dings.

Hence there is a complete Boolean algebra B(F) and a dense embeddingd : F −→B(F) such that d ◦c ⊂id i.e.⌢BB (F) is an iteration.C(B) and T (B) satisfy (1). Let Dir(B) = B(C(B)) and Inv(B) = B(T (B)).Factor Property: Let E = Inv(B) and D = Dir(B).

For all α < λ⊩αE/G ∼= Inv(B/G)andD/G ∼= Dir(B/G).Fact: (proof in [Jech], Lemma 36.5, page 460).If λ > ω is regular and{α | Bα = Dir(B↾α)} is stationary in λ and each Bα satisfies the λ-antichain-condition, then so does Dir(B).Revised countable support iterating semiproper-nessLet us assume familiarity with the following facts about the iteration ofsemiproperness, more or less proved by simultaneously playing semipropergames in different generic extensions:Theorem 11. Let P be semiproper and ⊩P ( ˙Q is semiproper), then P ∗Q is semiproper.2.

Let B = ⟨Bn|n ∈ω⟩be some iteration with semiproper B0 and ∀n⊩n(Bn+1/G is semiproper). Then Inv(B) is semiproper.3.

Let λ > ω be regular and B = ⟨Bα|α ∈λ⟩be an iteration with: for allα < λ Bα is semiproper and∀β ∈(α, λ)⊩α (Bβ/G is semiproper)cof(α) = ω=⇒Bα = Inv(B↾α)If λ = ω1 or Dir(B) satisfies the λ-antichain-condition, then Dir(B) will besemiproper.Definition 2 We call a thread f ∈T (B) short iff∃α < λf(α) ⊩α (cof(λ) = ω).S(B), the set of all short or eventually constant threads, satisfies (1). DefineRlim(B) = B(S(B)).2

We remark that Rlim satsfies the factor property. Now our RCS-iteration, thatmeans taking Rlim at limit stages, of semiproperness will work:Theorem 3 Let B = ⟨Bα|α ⩽λ⟩be a RCS-iteration satisfyingB0 is semiproper∀α < λ⊩α+1 (|Bα| ⩽ℵ1)⊩α (Bα+1/G is semiproper).Then Bλ is semiproper.Prove by induction on β ⩽λ that ∀α < β⊩α (Bβ/G is semiproper).

In thelimit case use the factor property of Rlim and the following lemma:Lemma 4 Let B = ⟨Bα|α < λ⟩be RCS-iteration, λ limit ordinal such that:∀α < λBα is semiproper⊩α+1 (|Bα| ⩽ℵ1)∀β ∈(α, λ)⊩α (Bβ/G is semiproper).Then Rlim(B) is semiproper.Proof of lemma 4: Let B = Rlim(B) and d : S(B) −→B be dense. Let usshow thatb B↾b is semiproperis dense in B.

So let f ∈S(B) and we haveto find a g ∈S(B) below f such that B↾d(g) is semiproper.• first case: ∃α < λf(α) ̸⊩α (cof(λ) > ω).Hence there is a thread g below f and an α < λ such that g(α) ⊩α(cof(λ) = ω). We have B↾d(g) ∼= Inv(B↾g) (every thread below g is short).In a generic extension over Bα↾g(α) we apply (up to isomorphism) 1.2 andget⊩Bα↾g(α) (Inv(B↾g/G) is semiproper),the factor property for the inverse limit gives B↾d(g) is semiproper.• second case: ∀α < λ f(α) ⊩α (cof(λ) > ω).Hence below f there is no short thread.

We distinguish two more cases:– ∃α < λf(α) ̸⊩α (cof(λ) > ℵ1).Hence there is an α < λ and an eventually constant thread g belowf such that g(α) ⊩α (cof(λ) = ω1). Now proceed as in the first case,using 1.3 instead of 1.2 and the factor property for the direct limitto get B↾d(g) ∼= Dir(B↾g) is semiproper.3

– it remains: ∀α < λ f(α) ⊩α (cof(λ) > ℵ1).So ∀α < λ|Bα| < λ, and λ is regular. (Otherwise the cofinal-ity would have been collapsed).Let S = {α < λ| cof(α) = ℵ1}.We have ∀α ∈S⊩α (cof(α) = ℵ1) and therefore Bα↾f(α) ∼=Dir((B↾f)↾α).

Since S is stationary in λ, Dir(B↾f) satisfies the λ-antichain-condition. (apply the Fact of the previous section).Now we can apply 1.3 and get the semiproperness of B↾d(f) ∼=Dir(B↾f).References[Jech]Thomas Jech.

Set Theory. Academic Press, 1978.

[Miyamoto]Tadatoshi Miyamoto. A Note on Iterated Forcing.

Handwrittennotes. [Shelah]Saharon Shelah.

Proper Forcing. Volume 940, Springer-Verlag,Lecture Notes in Mathematics edition, 1982.

[Schlindwein] Chaz Schlindwein. Simplified RCS iterations.

Abstract bye-prints.4

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