4. Iterating subcomplete forcing 51
6.4 The extended Namba problem
π(α, B) = [
u∈Mα
πα(u∩B).
Thus ˜πmaps a subset of Hω1 ontoP(β). QED(Lemma 11)
6.4 The extended Namba problem
Shelah was the first to show that Namba forcing can be iterated without adding reals. If we iterate it out to a strongly inaccessibleκ, thenκbecomes the newω2
and arbitrarily large regular cardinals below κ becomeω-cofinal. However, many regular cardinals becomeω1-cofinal. The “extended Namba problem” asks whether, without adding reals, one can makeκbecomeω2while givingall of the regular car-dinals in the interval (ω1, κ) cofinality ω. This problem seemed so difficult that at one point we conjectured a provably negative answer in ZFC for all κ. Moti Gitik then disproved this conjecture by constructing a ZFC model in which the extended Namba problem had a positive solution for some κ. His model was a generic extension of a universe containing a supercompact cardinal. Following Gi-tik’s breakthrough we then obtained a positive solution in ZFC for all κ. It is impossible to give the full proof of that result in these notes, but we shall endeaver to give some account of the methods used. We may assume w.l.o.g. that GCH holds belowκ, since we may achieve this by a prior forcing in which all collapsed regular cardinals acquire a cofinality ≥ω2. If we then give the surviving regular cardinals in (ω1, κ) the cofinalityω, the collapsed ones will also becomeω-cofinal.
It is natural to try to solve this problem by an iterationhBi|i≤κi. We ask now what the initial steps of this iteration should look like. We follow the convention that B0 = 2. Thus B1 is the first stage which “does something”. We certainly expect it to giveω2 the cofinalityω without adding reals. By Lemma 11 it follows thatω3will be collapsed, soω3must acquire cofinality ω. But thenω4is collapsed etc. Thus every ωn must be collapsed with cofinality ω. By Lemma 11, it then
follows thatωω+1 is collapsed etc. This chain of implications does not break down Since eachBi is subcomplete, the inverse limitB∗ is also subcomplete. However, a bit of reflection shows thatB∗ is too small to do the job: At the limit stageωω1·ω
will be collapsed toω1. Hence by Lemma 11ω(ω1·ω)+1 will be collapsed and hence must acquire cofinalityω etc.
Proceeding in this fashion we see that ωω1·(ω+1) must be collapsed. Thus our limit algebra must be large, not containing any dense set of size less thanωω1(ω+1). At the same time it should have a dense subset of sizeωω1(ω+1) in order that the successor is preserved. It turns out that a limit with the requisite properties can be obtained by a construction rather like that of Example 2. We shall now sketch that construction, but a full verification of its properties is beyond the purview of these notes.
i | i < ωi of complete Boolean algebras in the sense of M with Bu
• Letπ:Mu→Mu,v cofinally. ThenhMu,v, πiis the liftup ofhMu, π0i.
We again set:
π:u⊳v iff hπ, u, vi ∈Π.
Thus hΓ,Πiis an approximation system which is related to hΓ0,Π0i exactly as in Chapter 6.2.
Again, letting M = LAβ be as above, and N = hHβ+, <, M, . . .i, we form the languageL onN containing only the core axioms.
Lemma 12 Lis consistent.
Proof. Let B∗ = the inverse limit of hBi | i < ωi. Then B∗ is subcomplete. Let B∗ be B∗-generic. We prove the consistency of L in V[B∗]. Let Bi = B∗∩Bi, B = S
i<ω
Bi. LetH =H(2β)+ in V. Letπ :H ≺H in V[B∗] s.t. H is countable and transitive. Let:
π(N , M , M0,hBi |i < ωi) =N, M, M0,hBi|i < ωi.
SetBi =π−1′′Bi fori < ω. Since we are working in V[B∗] we may assume that Bi isBi-generic overM fori < ω. ClearlyπtakesM0 toM0 cofinally. Moreover:
π↾M0:hM0, Bi⊳0hM0, Bi.
Now let hH,˜ πi˜ be the liftup of hH, π ↾M0i. Let: ˜π(M , N ,L) = ˜M ,N ,˜ L, where˜ π(L) =L. Since there is k: ˜H ≺H withk( ˜L) =L, it suffices to prove that ˜L is consistent. We claim:
Claim hHκ, Bimodels ˜L, where κ >2β is regular inV.
Proof. The only problematical case is: Let X ⊂M˜ be countable. There is u∈ Γ∩Hω1 s.t. u⊳hM , Bi˜ and X ⊂ rng(πu,hM ,Bi˜ ). Let Y ≺ H˜ be countable s.t.
rng(˜κ)∪X ⊂Y and whenever ∆∈Y is dense inBi (i < ω), then ∆∩B6=∅. Let:
π′ :H′ ∼
↔Y, π′(M0′, M′,hB′i|i < ωi) =M0,M ,˜ hBi|i < ωi.
Set: B′=π′−1′′Bi,π′′=π′↾M′. Claim π′′:hM′, B′i⊳hM , Bi.˜
Clearly: π′′↾ M0′ :hM0′, B′i⊳0hM0, Bi. Since π′′:M′ ≺M˜, it suffices to show that: Ifπ′′:M′→M∗cofinally, thenhM∗, π′′iis the liftup ofhM′, π′′↾M0′i– i.e.
that π′′ takes M′ γ′-cofinally to M∗ wheres γ′ = (ω1·ω)M′. This follows by the
usual argument. QED(Lemma 12)
The strong revisability lemma reads:
Lemma 13 For sufficiently largeθ >2βwe have: LetN∗=hHθ, M,P, <, . . .i. Let p conform to N∗ and set: N∗ =N∗(N∗, p) = hH, M ,P, <, . . .i. Let B ⊂ S
i<ω
BM
i
s.t. B∩BMi isBMi -generic overM for i < ω. Thenq=hhM , Bi, Fpi ∈P.
We must forego the proof of Lemma 13, since it is very long and involves prop-erties of the algebrasBi which we have not developed here.
An immediate corollary is:
Corollary 13.1 P is revisable, since revisability says that the above holds when B=BG for aG which isP-generic overN∗.
SinceLhas only the core axioms, it is then modest. But then we get:
Lemma 14 Pis subcomplete.
We sketch briefly the proof of Lemma 14, which is largely the same as before.
Letθbe big enough to verify the subcompleteness ofBi fori < ω. LetW =LAτ be a ZFC− model with Hθ ⊂W and θ < τ. Let π: W ≺W where W is countable and full. Letπ(θ,P, s) =θ,P, s.
Claim There is q∈Ps.t. ifG∋qisP-generic, there isσ∈V[G] with:
(a) σ:W ≺W (b) σ(θ,P, s) =θ,P, s
(c) CγW(rng(σ)) =CγW(rng(π)), whereγ=On∩M0= sup
i<ω
δ(Bi).
(d) σ′′G⊂G.
(Note γ≤δ(P), since otherwiseγwould not be collapsed.) Let Ω> θbe big enough to verify the strong revisability ofP. Set:
N∗=hHΩ, <, M, N,P, W, π, . . .i.
Let p conform to N∗. Set: N∗ = N∗(N∗, p) = hH′, M′, N′,P′, W′, π′, . . .i. Set:
B′
i = BM′
i (i < ω). Set θ′,P′, s′ =π′(θ,P, s). Setγ′ = π′(γ), where π(γ) = γ = On∩M0. Noting that W′ is countable and imitating the proof of Chapter 4, Theorem 2 we get:
Sublemma 14.1 There are σ′ and B′ ⊂ S
i<ω
B′i s.t. B′i =B′∩B′i is B′i-generic overW′ for i < ω and:
(a) σ′:W ≺W′ (b) σ′(θ,P, s) =θ′,P′, s′
(c) CγW′′(rng(σ′)) =CγW′′(rng(π′)) (d) σ′ ′′B⊂B′, whereB=BG.
To get this we successively define σ◦i, bi∈B′
is.t. wheneverBi′∋biisP′-generic over W′ andσ′i=σ◦iBi′, thenσi′ satisfies (a)–(c) and: σi′′′Bi⊂B′i(whereBi=B∩Bi).
We ensure hi(bi+1) =bi fori < ω. We then successively choose Bi′ ∋bi with: Bi′ isB′
i-generic over W′ and Bi′⊃Bℓ′ forℓ < i. We set: B′=S
i
Bi′ and letσ′ be the
’limit’ ofσ′i=σ◦iBi′ (i < ω) exactly as in the proof of Chapter 4, Theorem 2.
QED(Sublemma 14.1)
By the strong revisability lemma we have: q =hhM′, B′i, Fpi ∈P. Let G∋q be P-generic. Then πqG∪Fq extends uniquely to: σ∗: N∗ ≺N∗. Set σ=σ∗·σ′. It follows by a virtual repetition of previous proofs that σhas the desired properties.
QED(Lemma 14) collapsed, becoming the newω2. Hence we apply Example 2 at the next stage to collapse ̺(ω1) = the ω1-th successor of ̺to ω1. We continue in this fashion. We define an iteration hBi | i ≤ κiand a sequence h̺i | i ≤ κi as follows: ̺0 = ω1, B0= 2.
̺i+1 =̺(ωi 1) andBi+1 is constructed using Example 2 so as to collapse all regular τ ∈(ω1, ̺i+1) without collapsing̺+i+1. For limit λwe proceed as follows:
Case 1 λ has cofinality ω or has acquired cofinality ω at an earlier stage (i.e.
cf(λ)< λ∧cf(λ)6=ω1 inV).
By essentially the above construction we form a limit Bλ which collapses ̺λ = that we took the direct limit stationarily often belowλit follows that Bλ satisfies theλ-chain condition. Henceλis the newω2.
⋆ ⋆ ⋆ ⋆ ⋆
By induction on i we verify that Bi is subcomplete for i ≤ κ, using Chapter 4, Theorem 4 for Case 2 above. We stress, however, that in order to carry out the
induction we must also verify many other properties of theBi which have not been dealt with here. These include some strong symmetry properties.
Given that GCH holds belowκ, we can modify the above construction by making selective regularτ∈(ω1, κ) ω1-cofinal. The set of suchτ can be chosen arbitrarily in advance. Hence:
Theorem Let κbe inaccessible. Let GCH hold below κ. Let A⊂κ. There is a set of conditions P⊂Vκ s.t. wheneverGisP-generic, then in V[G]we have:
• κisω2.
• Ifτ ∈(ω1, κ)is regular inV, thencf(τ) =
ω1 if τ ∈A, ω if not.
Bibliography
[ASS] J. Barwise. Admissible Sets and Structures, Perspectives in Math. Logic Vol.
7, Springer Verlag, 1976.
[NA] H. Friedman, R. Jensen. A Note on Admissible Ordinals, in: The Syntax and Semantics of Infinitary Languages. Springer Lecture Notes in Math. Vol. 72, 1968.
[PR] R. Jensen, C. Karp. Primitive Recursive Set Functions, in Axiomatic Set The-ory, AMS Proceedings of Symposia in Pure Math. Vol. XIII, Part 1, 1971.
[PF] S. Shelah. Proper and Improper Forcing, Perspectives in Math. Logic, Springer Verlag, 1998.
[AS] R. Jensen: Admissible Sets*.
[LF] R. Jensen. L-Forcing*
[SPSC] R. Jensen. Subproper and Subcomplete Forcing*
[ENP] R. Jensen. The Extended Namba Problem*
[ITSC] R. Jensen. Iteration Theorems for Subcomplete and Related Forcings*
[DSP] R. Jensen. Dee-Subproper Forcing*
[FA] R. Jensen. Forcing Axioms Compatible with CH*
* These handwritten notes can be downloaded from
http://www.mathematik.hu-berlin/de/∼raesch/org/jensen.html (or enter ’Ronald B. Jensen’ in Google).
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