Some independence results on reflection

James Cummings Hebrew University of Jerusalem

              cummings@math.huji.ac.il ^{∗}

Saharon Shelah Hebrew University of Jerusalem

               shelah@math.huji.ac.il ^{†}

February 25, 1998

Abstract We prove that there is a certain degree of independence between

stationary reflection phenomena at different cofinalities.

1 Introduction

Recall that a stationary subset S of a regular cardinal κ is said to reflect at α < κ if cf(α) > ω and S ∩ α is stationary in α. Stationary reflection phenomena have been extensively studied by set theorists, see for example [7].

Definition 1.1 Let κ = cf(κ) < λ = cf(λ). T_{κ}^{λ} =_{def} { α < λ | cf(α) = κ }. If m < n < ω then S_{m}^{n} =_{def} { α < ℵ_{n} | cf(α) = ℵ_{m} }.

∗Supported by grant number 517/94 of the Israel Science Foundation.

†Research supported by the Israel Science Foundation, administered by The Israel Academy of Sciences and Humanities. Publication number 596.

Baumgartner proved in [1] that if κ is weakly compact, GCH holds, and ω < δ = cf(δ) < κ then forcing with the Levy collapse Coll(δ, < κ) gives a model where for all ρ < δ and all stationary T ⊆ T_{ρ}^{κ} the stationarity of T reflects to some α ∈ T_{δ}^{κ}. In this last result all the cofinalities ρ < δ are on the same footing; we will build models where reflection holds for some cofinalities but fails badly for others.

We introduce a more compact terminology for talking about reflection.

Definition 1.2 Let κ = cf(κ) < λ = cf(λ) < μ = cf(μ).

1. Ref(μ, λ, κ) holds iff for every stationary S ⊆ T_{κ}^{μ} there is a α ∈ T_{λ}^{μ} with S ∩ α stationary in α.

2. Dnr(μ, λ, κ) (Dense Non-Reflection) holds iff for every stationary S ⊆ T_{κ}^{μ} there is a stationary T ⊆ S such that for no α ∈ T_{λ}^{μ} is T ∩ α stationary in α.

We will use a variation on an idea from Dzamonja and Shelah’s paper [4].

Definition 1.3 Let κ = cf(κ) < λ = cf(λ) < μ = cf(μ). Snr(μ, λ, κ) (Strong Non-Reflection) holds iff there is F : T_{κ}^{μ} ⟶ λ such that for all α ∈ T_{λ}^{μ} there is C ⊆ α closed and unbounded in α with F ↾ C ∩ T_{κ}^{μ} strictly increasing.

As the name suggests, Snr(μ, λ, κ) is a strong failure of reflection. It is easy to see that if Jensen’s Global □ principle holds then Snr(μ, λ, κ) holds for all κ < λ < μ; in some sense the strong non-reflection principle captures exactly that part of □ which is useful for building non-reflecting stationary sets.

Lemma 1.4 Snr(μ, λ, κ) ⟹ Dnr(μ, λ, κ).

Proof: Let S ⊆ T_{κ}^{μ} be stationary, and let F : T_{κ}^{μ} ⟶ λ witness the strong non-reflection. Let T ⊆ S be stationary such that F ↾ T is constant. Let α ∈ T_{λ}^{μ} and let C be a club in α on which F is strictly increasing, then C meets T at most once and hence T is non-stationary in α.

♦

We will prove the following results in the course of this paper. Theorem 3.5: If the existence of a weakly compact cardinal is consistent, then Ref(ℵ_{3}, ℵ_{2}, ℵ_{0}) + Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}) is consistent. Theorem 3.6: If the existence of a measurable cardinal is consistent, then Ref(ℵ_{3}, ℵ_{2}, ℵ_{0}) + Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}) + Snr(ℵ_{3}, ℵ_{1}, ℵ_{0}) is consistent. Theorem 3.7: If the existence of a supercompact cardinal with a measurable above is consistent, then Ref(ℵ_{3}, ℵ_{2}, ℵ_{0}) + Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}) + Ref(ℵ_{3}, ℵ_{1}, ℵ_{0}) is consistent. Theorem 4.1: If Snr(μ, λ, κ) and κ < κ^{∗} < λ then Snr(μ, λ, κ^{∗}). Theorem 4.2: Let κ < λ < μ < ν. Then Ref(ν, μ, λ) + Ref(ν, λ, κ) ⟹ Ref(ν, μ, κ) and Ref(ν, μ, κ) + Ref(μ, λ, κ) ⟹ Ref(ν, λ, κ). Theorem 5.10: If the existence of a weakly compact cardinal is consistent, then Ref(ℵ_{3}, ℵ_{2}, ℵ_{1}) + Dnr(ℵ_{3}, ℵ_{2}, ℵ_{0}) is consistent.

Theorems 3.5, 3.6 and 3.7 were proved by the first author. Theorems 4.1 and 5.10 were proved by the second author, answering questions put to him by the first author. Theorem 4.2 was noticed by the first author (but has probably been observed many times). We would like to thank the anonymous referee for their very thorough reading of the first version of this paper.

2 Preliminaries

We will use the idea of strategic closure of a partial ordering (introduced by Gray in [5]).

Definition 2.1 Let ℙ be a partial ordering, and let η be an ordinal.

1. The game G(ℙ, η) is played by two players I and II, who take turns to play elements p_{α} of ℙ for 0 < α < η, with player I playing at odd stages and player II at even stages (NB limit ordinals are even). The rules of the game are that the sequence that is played must be decreasing (not necessarily strictly decreasing), the first player who cannot make a move loses, and player II wins if play proceeds for η stages.

2. ℙ is η-strategically closed iff player II has a winning strategy in G(ℙ, η).

3. ℙ is < η-strategically closed iff for all ζ < η ℙ is ζ-strategically closed.

Strategic closure has some of the nice features of the standard notion of closure. For example a (δ + 1)-strategically closed partial ordering will add no δ-sequences, and the property of being (δ + 1)-strategically closed is preserved by forcing with ≤ δ-support (see [3] for more information on this subject). We will need to know that under some circumstances we can preserve a stationary set by forcing with a poset that has a sufficient degree of strategic closure.

The following is well-known.

Lemma 2.2 Let GCH hold, let λ = cf(λ) and κ = λ^{+}. Let δ = cf(δ) ≤ λ, and suppose that ℙ is (δ +1)-strategically closed and S is a stationary subset of T_{δ}^{κ}. Then S is still stationary in V^{ℙ}.

Proof: Let p ⊩ “Ċ is club in κ”. Build ⟨X_{α} : α < κ⟩ a continuous increasing chain of elementary substructures of some large H_{θ} such that everything relevant is in X_{0}, |X_{α}| = λ, ^{<λ}X_{α} ⊆ X_{α+1}. We make the remark here that this would not be possible for λ singular, and indeed the theorem can fail in that case (see [8] for details).

Now find some limit γ such that X_{γ} ∩ κ ∈ S, clearly cf(γ) = δ and so ^{<δ}X_{γ} ⊆ X_{γ}. Let β =_{def} X_{γ} ∩ κ and fix ⟨β_{i} : i < δ⟩ cofinal in β. Since ℙ, δ ∈ X_{0} ⊆ X_{γ} we can find in X_{γ} a winning strategy σ for the game G(ℙ, δ + 1).

Now we build a sequence ⟨p_{α} : α ≤ δ⟩ such that

1. For each even β, p_{β} = σ(p⃗ ↾ β).

2. For β < δ, p_{β} ∈ ℙ ∩ X_{γ}.

3. For each i < δ, there is η such that β_{i} < η < β and p_{2i+1} ⊩ η̂ ∈ Ċ.

We can keep going because X_{γ} ≺ H_{θ}, ^{<δ}X_{γ} ⊆ X_{γ} and σ is a winning strategy. At the end of the construction p_{δ} is a refinement of p_{0} which forces that β is a limit point of Ċ, and we are done.

♦

3 Some consistency results

In this section we prove (starting from a weakly compact cardinal) the consistency of ZF C+Ref(ℵ_{3}, ℵ_{2}, ℵ_{0})+Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}). We also show that together with this we can have either Ref(ℵ_{3}, ℵ_{1}, ℵ_{0}) or Snr(ℵ_{3}, ℵ_{1}, ℵ_{0}).

We begin by defining a forcing ℙ_{Snr} to enforce Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}).

Definition 3.1 p is a condition in ℙ_{Snr} iff p is a function from a bounded subset of S_{1}^{3} to ℵ_{2}, and for every γ ∈ S_{2}^{3} with γ ≤ sup(dom(p)) there is C club in γ such that p ↾ C ∩ S_{1}^{3} is strictly increasing. ℙ_{Snr} is ordered by extension.

It is easy to see that ℙ_{Snr} is ω_{2}-closed, and in fact that it is ω_{2}-directed closed.

Lemma 3.2 ℙ_{Snr} is (ω_{2} + 1)-strategically closed.

Proof: We describe a winning strategy for player II in G(ℙ_{Snr}, ω_{2} + 1). Suppose that p_{α} is the condition played at move α. Let β be an even ordinal, then at stage β II will play as follows.

Define q_{β} =_{def} ⋃_{α<β} p_{α}, ρ_{β} =_{def} dom(q_{β}), and then let II play as follows: p_{β} = q_{β} unless β is limit and cf(β) = ℵ_{1}, in which case p_{β} = q_{β} ∪ {(ρ_{β}, β)}.

The strategy succeeds because when play reaches stage ω_{2}, { ρ_{β} | β < ω_{2} } is a club witnessing that p_{ω_{2}} is a condition.

♦

This shows that ℙ_{Snr} preserves cardinals and cofinalities up to ℵ_{3}, from which it follows that V^{ℙ_{Snr}} ⊨ Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}). If GCH holds then |ℙ_{Snr}| = ℵ_{3}, so ℙ_{Snr} has the ℵ_{4}-c.c. and all cardinals are preserved.

Now we define in V^{ℙ_{Snr}} a forcing ℚ. This will enable us to embed ℙ_{Snr} into the Levy collapse Coll(ω_{2}, ω_{3}) in a particularly nice way.

Definition 3.3 In V^{ℙ_{Snr}} let F : S_{1}^{3} ⟶ ℵ_{2} be the function added by ℙ_{Snr}. q ∈ ℚ iff q is a closed bounded subset of ℵ_{3}, the order type of q is less than ℵ_{2}, and F ↾ lim(q) ∩ S_{1}^{3} is strictly increasing.

The aim of ℚ is to add a club of order type ω_{2} on which F is increasing. It is clear that ℚ is countably closed and collapses ω_{3}.

Lemma 3.4 If GCH holds, ℙ_{Snr} ∗ ℚ̇ is equivalent to Coll(ℵ_{2}, ℵ_{3}).

Proof: Since ℙ_{Snr} ∗ ℚ̇ has cardinality ℵ_{3} and collapses ℵ_{3}, it will suffice to show that it has an ℵ_{2}-closed dense subset. To see this look at those conditions (p, c) where c ∈ V, and max(c) = sup(dom(p)). It is easy to see that this set is dense and ℵ_{2}-closed.

♦

Theorem 3.5 Let κ be weakly compact, let GCH hold. Define a two-step iteration by ℙ_{0} =_{def} Coll(ω_{2}, < κ) and ℙ_{1} =_{def} (ℙ_{Snr})_{V^{ℙ_{0}}}. Then V^{ℙ_{0}∗ℙ_{1}} ⊨ Ref(ℵ_{3}, ℵ_{2}, ℵ_{0}) + Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}).

Proof: We will first give the proof for the case when κ is measurable and then show how to modify it for the case when κ is just weakly compact. Assuming that κ is measurable, let j : V ⟶ M be an elementary embedding into a transitive inner model with critical point κ, where ^{κ}M ⊆ M. Notice that by elementarity and the closure of M, j(ℙ_{0}) = Coll(ω_{2}, < j(κ))_{M} = Coll(ω_{2}, < j(κ))_{V}.

Let G be ℙ_{0}-generic over V and let H be ℙ_{1}-generic over V [G]. We already know that V [G ∗ H] ⊨ Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}), so let us assume that in V [G ∗ H] we have S a stationary subset of S_{0}^{3}. To prove that S reflects we will build a generic embedding with domain V [G ∗ H] extending j. Notice that since |ℙ_{0} ∗ ℙ_{1}| = κ we can prove (by looking at canonical names) that V [G ∗ H] ⊨ ^{κ}M[G ∗ H] ⊆ M[G ∗ H], so in particular we have S ∈ M[G ∗ H].

We start by forcing with ℚ over V [G ∗ H] to get a generic object I. H ∗ I is generic over V [G] for (ℙ ∗ ℚ)_{V [G]} which is equivalent to Coll(ω_{2}, κ), so we can regard G ∗ H ∗ I as being generic for Coll(ω_{2}, ≤ κ). Now let J be Coll(ω_{2}, [κ, j(κ)))-generic over V [G∗H ∗I], then G∗H ∗I ∗J is j(ℙ_{0})-generic over V (so a fortiori over M) and j“G ⊆ G ∗ H ∗ I ∗ J so that we can lift to get j : V [G] ⟶ M[G ∗ H ∗ I ∗ J].

It remains to lift j onto V [G ∗ H], for which we need to force a generic K for j(ℙ_{1}) with the property that j“H ⊆ K. We will get K by constructing a master condition in j(ℙ_{0}) (that is, a condition refining all the conditions in j“H) and forcing below that master condition. A natural candidate for a master condition is F =_{def} ⋃ j“H, where it is easily seen (since crit(j) = κ and j ↾ ℙ_{1} = id) that F is the generic function from κ to ℵ_{2} added by H.

The models V [G] and M[G ∗ H ∗ I ∗ J] agree in their computations of T_{ℵ_{1}}^{κ} and T_{ℵ_{2}}^{κ}, so F is increasing on a club at all the relevant points below κ. Now κ = (ℵ_{3})_{V [G]} is an ordinal of cofinality ℵ_{2} in M[G ∗ H ∗ I ∗ J], but there is no problem here because I has introduced a club in κ on which F is increasing. Hence F is a condition in j(ℙ_{1}) and we can force to get K ⊇ j“H as desired.

We claim that S is still stationary in M[G ∗ H ∗ I ∗ J ∗ K]. I is generic for countably closed forcing, J is generic for ℵ_{2}-closed forcing and K is generic for ℵ_{2}-closed forcing so that S (being a set of cofinality ω ordinals) remains stationary. Now we argue as usual that since j(S) ∩ κ = S and cf(κ) = ω_{2} in M[G∗H ∗I ∗J ∗K], there must exist α < κ in V [G∗H] such that cf(α) = ω_{2} and S ∩ α is stationary in α.

We promised at the start of this proof that we would show how to weaken the assumptions on κ from measurability to weak compactness. We will actually sketch two arguments, based on two well-known characterisations of weak compactness. See Hauser’s paper [6] for detailed accounts of some similar arguments.

1. κ is weakly compact iff for every A ⊆ V_{κ} and every Π_{1}^{1} formula ϕ, V_{κ} ⊨ ϕ(A) ⟹ ∃α V_{α} ⊨ ϕ(A ∩ V_{α}).

2. κ is weakly compact iff κ is strongly inaccessible and for every transitive M such that |M| = κ, ^{<κ}M ⊆ M and M models enough set theory there is a transitive set N and an elementary embedding j : M ⟶ N with crit(j) = κ.

Argument 1: ℙ_{0} ∗ ℙ_{1} ⊆ V_{κ}, and so if Ṡ is a name for a stationary subset of κ we can represent it by S^{∗} = { (p, α) | p ⊩ α̂ ∈ Ṡ } ⊆ V_{κ}. The fact that S is forced to be stationary can be written as a Π_{1}^{1} sentence (the universal secondorder quantification is over names for clubs). Using the first characterisation given above we can find inaccessible α < κ such that S^{∗} ∩ V_{α} is a Coll(ω_{2}, < α) ∗ (ℙ_{Snr})_{V^{Coll(ω_{2},<α)}} name for a stationary set.

Now the argument is just like the one from a measurable, only with α playing the role of κ and κ replacing j(κ). We see that S has an initial segment S ∩α which is stationary in a certain intermediate generic extension, and just need to check that S ∩ α remains stationary and that α becomes a point of cofinality ℵ_{2}. This is routine. Argument 2: Given a name S for a stationary subset of S_{0}^{3}, build S and ℙ_{0} ∗ ℙ_{1} into an appropriate model M of size κ. Get j as in the second characterisation as above. Repeat (mutatis mutandis) the argument from a measurable.

♦

Having proved Theorem 3.5, it is natural to ask whether there is any connection between reflection to points of cofinality ω_{2} and reflection to points of cofinality ω_{1}. The following results provide a partial (negative) answer.

Theorem 3.6 Con(Ref(ℵ_{3}, ℵ_{2}, ℵ_{0}) + Snr(ℵ_{3}, ℵ_{2}, ℵ_{1}) + Snr(ℵ_{3}, ℵ_{1}, ℵ_{0})) follows from the consistency of a measurable cardinal.

Proof: Let κ be measurable. Without loss of generality GCH holds (as we can move to the inner model L[μ]). We will sketch a proof that we can force to get Snr(κ, ℵ_{1}, ℵ_{0}) without destroying the measurability of κ. A more detailed argument for a very similar result is given in [3] (alternatively one can argue that because □ holds in L[μ], Snr(κ, ℵ_{1}, ℵ_{0}) is already true in that model). Let j : V ⟶ M be the ultrapower map associated with some normal measure U on M.

We will do a reverse Easton iteration of length κ + 2, forcing at every regular cardinal α ≤ κ^{+} with ℚ_{α}, where ℚ_{α} is the natural forcing to add a witness to Snr(α, ℵ_{1}, ℵ_{0}) by initial segments. An easy induction shows that ℚ_{α} is < α-strategically closed, the point being that the witnesses added below α can be used to produce strategies in the game played on ℚ_{α}. The argument is exactly parallel to the proof of Lemma 6 in [3].

Let us break up the generic as G ∗ g ∗ h, where G is ℙ_{κ}-generic, g is ℚ_{κ}-generic and h is ℚ_{κ^{+}}-generic. The key point is that j(ℙ_{κ})/G ∗ g ∗ h is κ^{+}-strategically closed in V [G ∗ g ∗ h], so that by GCH we can build H ∈ V [G ∗ g ∗ h] which is j(ℙ_{κ})/G ∗ g ∗ h-generic over M[G ∗ g ∗ h]. Since j“G ⊆ G ∗ g ∗ h ∗ H, we can lift j to get j : V [G] ⟶ M[G ∗ g ∗ h ∗ H]. It is easy to see that ⋃ j“g(= ⋃ g) will serve as a master condition, so using GCH again we may find g^{+} ∈ V [G ∗ g ∗ h] such that j“g ⊆ g^{+} and g^{+} is j(ℙ_{κ})-generic over M[G ∗ g ∗ h ∗ H]. Finally we claim that j“h generates a j(ℙ_{κ^{+}})-generic filter over M[G ∗ g ∗ h ∗ H ∗ g^{+}], because ℙ_{κ^{+}} is distributive enough and M[G ∗ g ∗ h ∗ H ∗ g^{+}] = { j(F)(κ) | dom(F) = κ, F ∈ V [G ∗ g] }. Hence in V [G ∗ g ∗ h] we can lift j onto V [G ∗ g ∗ h], so the measurability of κ is preserved.

Now we just repeat the construction (from a measurable) of Theorem 3.5, and claim that in the final model Snr(ℵ_{3}, ℵ_{1}, ℵ_{0}) holds. The point is that if F : T_{ℵ_{0}}^{κ} ⟶ ω_{1} witnesses the truth of Snr(κ, ℵ_{1}, ℵ_{0}), then in the final model F witnesses Snr(ℵ_{3}, ℵ_{1}, ℵ_{0}) because the forcing from Theorem 1 does not change T_{ℵ_{0}}^{κ} or T_{ℵ_{1}}^{κ}.

♦

We can also go to the opposite extreme.

Theorem 3.7 Let κ be λ-supercompact, where λ > κ and λ is measurable. Let GCH hold. Then Ref(ℵ_{3}, ℵ_{2}, ℵ_{0})+Snr(ℵ_{3}, ℵ_{2}, ℵ_{1})+Ref(ℵ_{3}, ℵ_{1}, ℵ_{0}) holds in some forcing extension.

Proof: We will start by forcing with ℙ =_{def} Coll(ω_{1}, < κ), after which κ is ω_{2} and λ is still measurable. Then we will do the construction of Theorem 3.5, that is we force with (Coll(κ, < λ) ∗ ℙ_{Snr})_{V^{ℙ}}. Let ℙ_{0} = Coll(κ, < λ)_{V^{ℙ}} and ℙ_{1} = (ℙ_{Snr})_{V^{ℙ∗ℙ_{0}}}.

We need to check that Ref(ℵ_{3}, ℵ_{1}, ℵ_{0}) holds in the final model. To see this fix j : V ⟶ M such that crit(j) = κ, j(κ) > λ and ^{λ}M ⊆ M. Let G, H and I be the generics for ℙ, ℙ_{0} and ℙ_{1} respectively.

Since ℙ_{0}∗ℙ_{1} is countably closed and has size λ, we may find an embedding i : ℙ∗ℙ_{0}∗ℙ_{1} ⟶ j(ℙ) such that i ↾ ℙ = id and j(ℙ)/i“(ℙ∗ℙ_{0}∗ℙ_{1}) is countably closed. Let J be j(ℙ)/i“(ℙ ∗ ℙ_{0} ∗ ℙ_{1})-generic, then we can lift j in the usual way to get j : V [G] ⟶ M[G ∗ H ∗ I ∗ J]. Since ℙ_{0} ∗ ℙ_{1} is a κ-directed-closed forcing notion of size λ, we may find a lower bound for j“(H ∗I) in j(ℙ_{0} ∗ℙ_{1}) and use it as a master condition, forcing K such that j“(H ∗ I) ⊆ K and lifting j to j : V [G ∗ H ∗ I] ⟶ M[G ∗ H ∗ I ∗ J ∗ K].

Now suppose that in V [G ∗ H ∗ I] we have T ⊆ S_{0}^{3} a stationary set. If μ = ⋃ j“λ then we may argue as usual that T ∈ M[G ∗ H ∗ I] and that M[G∗H∗I] ⊨ j(T) ∩ μ is stationary in μ. Since J∗K is generic for countably closed forcing and j(T)∩μ ⊆ T_{ℵ_{0}}^{μ} this will still be true in M[G∗H ∗I ∗J ∗K], and since cf(μ) = ℵ_{1} in this last model we have by elementarity that T reflects to some point in S_{1}^{3}.

♦

4 Some ZFC results

In the light of the results from the last section it is natural to ask about the consistency of Ref(ℵ_{3}, ℵ_{2}, ℵ_{1})+Snr(ℵ_{3}, ℵ_{2}, ℵ_{0}). The first author showed by a rather indirect proof that this is impossible, and the second author observed that there is a simple reason for this.

Theorem 4.1 If Snr(μ, λ, κ) and κ < κ^{∗} = cf(κ^{∗}) < λ then Snr(μ, λ, κ^{∗}).

Proof: Let F : T_{κ}^{μ} ⟶ λ witness Snr(μ, λ, κ). Define F^{∗} : T_{κ^{∗}}^{μ} ⟶ λ by

F^{∗} : σ ⟼ min { ⋃_{α∈C∩T_{κ}^{μ}} F (α) | C club in σ }.

We claim that F^{∗} witnesses Snr(μ, λ, κ^{∗}). To see this, let δ ∈ T_{λ}^{μ} and fix D club in δ such that F ↾ D ∩T_{κ}^{μ} is strictly increasing. Let σ ∈ lim(D)∩T_{κ^{∗}}^{μ}, and observe that F^{∗}(σ) = ⋃_{α∈D∩T_{κ}^{μ}} F (α), because for any club C in σ we have

⋃_{α∈C∩T_{κ}^{μ}} F (α) ≥ ⋃_{α∈C∩D∩T_{κ}^{μ}} F (α) = ⋃_{α∈D∩T_{κ}^{μ}} F (α).

It follows that F^{∗} ↾ lim(D) ∩ T_{κ^{∗}}^{μ} is strictly increasing, because if σ_{0}, σ_{1} ∈ lim(D) ∩ T_{κ^{∗}}^{μ} with σ_{0} < σ_{1} and β is any point in lim(D) ∩ T_{κ}^{μ} ∩ (σ_{0}, σ_{1}) then F^{∗}(σ_{0}) ≤ F (β) < F^{∗}(σ_{1}).

♦

We also take the opportunity to record some other easy remarks, which put limits on the extent of the independence between different forms of reflection.

Theorem 4.2 Let κ < λ < μ < ν be regular. Then

1. Ref(ν, μ, λ) + Ref(ν, λ, κ) ⟹ Ref(ν, μ, κ).

2. Ref(ν, μ, κ) + Ref(μ, λ, κ) ⟹ Ref(ν, λ, κ).

Proof: For the first claim: let S ⊆ T_{κ}^{ν} be stationary, and let us define T = { α ∈ T_{λ}^{ν} | S ∩ α is stationary }. We claim that T is stationary in ν. To see this suppose C is club in ν and disjoint from T, and consider C ∩ S; this set must reflect at some γ ∈ T_{λ}^{ν}, but then on the one hand S ∩γ is stationary (so γ is in T) while on the other hand C is unbounded in γ (so γ ∈ C), contradicting the assumption that C and T are disjoint.

Now let δ ∈ T_{μ}^{ν} be such that T ∩ δ is stationary in δ. We claim that S reflects at δ. For if D is club in δ then there is γ ∈ T ∩ lim(D) by the stationarity of T ∩ δ, and now since D ∩ γ is club in γ and S ∩ γ is stationary in γ there is β ∈ D ∩ S. This proves the first claim.

For the second claim: S ⊆ T_{κ}^{ν} be stationary, and let δ ∈ T_{μ}^{ν} be such that S ∩ δ is stationary in δ. Let f : μ ⟶ δ be continuous increasing and cofinal in δ, and let S^{∗} = { α < μ | f(α) ∈ S }. Then S^{∗} is stationary in μ, and we can find β ∈ T_{λ}^{μ} such that S^{∗} ∩ β is stationary in β. Now f(β) ∈ T_{λ}^{ν} and S ∩ f(β) is stationary in f(β). This proves the second claim.

♦

5 More consistency results

From the results in the previous section we saw in particular that we cannot have Ref(ℵ_{3}, ℵ_{2}, ℵ_{1}) + Snr(ℵ_{3}, ℵ_{2}, ℵ_{0}). In this section we will see that Ref(ℵ_{3}, ℵ_{2}, ℵ_{1}) + Dnr(ℵ_{3}, ℵ_{2}, ℵ_{0}) is consistent.

We will need some technical definitions and facts before we can start the main proof.

Definition 5.1 Let S be a stationary subset of ℵ_{3}. We define notions of forcing ℙ(S), ℚ(S) and ℝ(S).

1. c is a condition in ℙ(S) iff c is a closed bounded subset of ℵ_{3} such that c ∩ S = ∅.

2. d is a condition in ℚ(S) iff d is a function with dom(d) < ℵ_{3}, d : dom(d) ⟶ 2, d(γ) = 1 ⟹ γ ∈ S, and for all α ≤ dom(d) if cf(α) > ω then there is C ⊆ α closed unbounded in α such that γ ∈ C ⟹ d(γ) = 0.

3. e is a condition in ℝ(S) iff e is a closed bounded subset of ℵ_{3} such that for every point α ∈ lim(e) with cf(α) > ω the set S∩α is non-stationary in α.

In each case the conditions are ordered by end-extension.

The aims of these various forcings are respectively to kill the stationarity of S (ℙ(S)), to add a non-reflecting stationary subset of S (ℚ(S)), and to make S non-reflecting on a closed unbounded set of points (ℝ(S)). Notice that for some choices of S the definitions of ℙ(S) and ℝ(S) may not behave very well, for example if S = ω_{3} then ℙ(S) is empty and ℝ(S) only contains conditions of countable order type. Notice also that ℚ(S) and ℝ(S) are countably closed.

Lemma 5.2 Let GCH hold. Let S ⊆ S_{0}^{3} and suppose that there is a club C of ω_{3} such that S ∩ α is non-stationary for all α ∈ C with cf(α) > ω. Let T be a stationary subset of S_{1}^{3}. Then ℙ(S) adds no ω_{2}-sequences of ordinals, and also preserves the stationarity of T.

Proof: First we prove that ℙ(S) is ω_{2}-distributive. Let ⟨D_{α} : α < ω_{2}⟩ be a sequence of dense sets in ℙ(S), and let c ∈ ℙ(S) be a condition. Fix some large regular cardinal θ and let c, C, D⃗, S ∈ X ≺ H_{θ} where |X| = ω_{2}, ω_{1}X ⊆ X. Let γ be the ordinal X ∩ ω_{3}, then cf(γ) = ω_{2} by the closure of X. By elementarity it follows that C is unbounded in γ, so that γ ∈ C and S ∩ γ is non-stationary.

Fix B ⊆ γ closed unbounded in γ such that B ∩ S = ∅ and B has order type ω_{2}. Now we build a chain of conditions c_{α} ∈ ℙ(S) ∩ X for α < ω_{2} such that c_{0} ≤ c, c_{2β+1} ∈ D_{β} and max(c_{2β}) ∈ B; we can continue at each limit stage because B is disjoint from S and X is sufficiently closed. Finally we let d =_{def} ⋃_{α<ω_{2}} c_{α} ∪ {γ}, then d ≤ c and d ∈ D_{β} for all β < ω_{2}. This shows that ℙ(S) is ω_{2}-distributive.

The argument for the preservation of stationarity is similar. Let Ė be a ℙ(S)-name for a closed unbounded subset of ω_{3} and let c ∈ ℙ(S) be a condition. This time build X such that c, C, Ė, S ∈ X ≺ H_{θ} where |X| = ω_{1}, ωX ⊆ X and δ =_{def} sup(X ∩ω_{3}) ∈ T. Again δ ∈ C, so S ∩δ is nonstationary. Choose B ⊆ δ closed unbounded of order type ω_{1} with B ∩S = ∅ and build a chain of conditions c_{α} ∈ ℙ(S) ∩ X such that c_{0} ≤ c, max(c_{2β}) ∈ B and c_{2β+1} forces that Ė ∩ (max(c_{2β}), max(c_{2β+1})) ≠ ∅. Finally if d =_{def} ⋃_{α<ω_{1}} c_{α} ∪ {δ} then d ≤ c and d ⊩ δ̂ ∈ T̂ ∩ Ė.

♦

Now we describe a certain kind of forcing iteration. It will transpire that all iterations of this type are ω_{4}-c.c. and (ω_{2} + 1)-strategically closed, so that in particular all cardinalities and cofinalities are preserved.

Definition 5.3 Fix F : ω_{4} × ω_{3} ⟶ ω_{4} such that for all β < ω_{4} the map i ⟼ F (β, i) is a surjection from ω_{3} onto β.

ℙ_{β} is a nice iteration iff

1. β ≤ ω_{4}.

2. ℙ_{β} is an iteration of length β with ≤ ω_{2}-supports.

3. ℚ_{0} = {0}.

4. ℚ_{2γ+1} is ℚ(Ṡ_{γ})_{V^{P_{2γ+1}}} where Ṡ_{γ} is some ℙ_{2γ+1}-name for a stationary subset of S_{0}^{3}. Let S_{γ}^{∗} be the non-reflecting stationary subset of S_{γ}

which is added by ℚ_{2γ+1}.

5. ℚ_{2γ} is ℝ(Ṙ_{γ}) where R_{γ} is the diagonal union of ⟨S_{F (γ,i)}^{∗} : i < ω_{3}⟩. That is R_{γ} = { δ ∈ S_{0}^{3} | ∃i < δ δ ∈ S_{F (γ,i)}^{∗} }.

It is clear that an initial segment of a nice iteration is nice. Also every final segment of a nice iteration is countably closed, so that all the sets S_{γ}^{∗} remain stationary throughout the iteration. The following remark will be useful later.

Lemma 5.4 Let γ < δ < ω_{4}. Then R_{γ} − R_{δ} is non-stationary.

Proof: Let C be the closed and unbounded set of i < ω_{3} such that

{ F (γ, j) | j < i } = { F (δ, j) | j < i } ∩ γ.

Let i ∈ C ∩R_{γ}. Then for some j < i we have i ∈ S_{F (γ,j)}^{∗}, and by the definition of C there is k < i such that F (γ, j) = F (δ, k). So i ∈ S_{F (δ,k)}^{∗}, i ∈ R_{δ}, and we have proved that C ∩ R_{γ} ⊆ R_{δ}.

♦

We define a certain subset of ℙ_{β}, which we call ℙ_{β}^{∗}.

Definition 5.5 If ℙ_{β} is a nice iteration then ℙ_{β}^{∗} is the set of conditions p ∈ ℙ_{β} such that

1. p(γ) ∈ V̂ (that is, p(γ) is a canonical name for an object in V) for all γ ∈ dom(p).

2. There is an ordinal ρ(p) such that (a) 2δ ∈ dom(p) ⟹ max(p(2δ)) = ρ(p).

(b) 2δ + 1 ∈ dom(p) ⟹ dom(p(2δ + 1)) = ρ(p) + 1.

(c) 2δ + 1 ∈ dom(p) ⟹ p(2δ + 1)(ρ(p)) = 0.

3. If 2δ ∈ dom(p) then ∀i < ρ(p) 2F (δ, i) + 1 ∈ dom(p).

4. If 2δ + 1 ∈ dom(p) then p ↾ 2δ + 1 decides S_{δ} ∩ ρ(p).

Lemma 5.6 If p ∈ ℙ_{β}^{∗} and 2δ ∈ dom(p) then p ↾ 2δ forces that ρ(p) ∉ R_{δ}.

Proof: Let ρ = ρ(p). By the definition of ℙ_{β}^{∗}, we see that 2F (δ, i) + 1 ∈ dom(p) and p(2F (δ, i) + 1)(ρ) = 0 for all i < ρ. This means that p ⊩ ρ ∉ Ṡ_{F (δ,i)}^{∗} for all i < ρ, which is precisely to say p ↾ 2δ ⊩ ρ ∉ Ṙ_{δ}.

♦

Lemma 5.7 Let ℙ_{β} be a nice iteration. Let δ ≤ ω_{2} be a limit ordinal and let ⟨p_{γ} : γ < δ⟩ be a decreasing sequence of conditions from ℙ_{β}^{∗} such that ⟨ρ(p_{γ}) : γ < δ⟩ is continuous and increasing. Define q by setting dom(q) = ⋃_{γ<δ} dom(p_{γ}), ρ = ⋃_{γ<δ} ρ(q_{γ}), q(2ϵ) = ⋃ { p_{γ}(2ϵ) | 2ϵ ∈ dom(p_{γ}) } ∪ {ρ}, q(2ϵ + 1) = ⋃ { p_{γ}(2ϵ + 1) | 2ϵ + 1 ∈ dom(p_{γ}) } ∪ {(ρ, 0)}.

Then q ∈ ℙ_{β}^{∗} and ρ(q) = ρ.

Proof: Clearly it is enough to show that q ∈ ℙ_{β}. Most of this is routine; the key points are that q ↾ 2ϵ forces that R_{ϵ} ∩ ρ is non-stationary, and that q ↾ 2ϵ + 1 forces that S_{ϵ}^{∗} ∩ ρ is non-stationary.

For the first point, observe that for all sufficiently large γ < δ we have 2ϵ ∈ dom(p_{γ}), so that by the last lemma p_{γ} ↾ 2ϵ ⊩ ρ(p_{γ}) ∉ R_{ϵ}; since ⟨ρ(p_{γ}) : γ < δ⟩ is continuous and q ↾ 2ϵ refines p_{γ} ↾ 2ϵ, this implies that q ↾ 2ϵ forces that R_{ϵ} is not stationary in ρ.

Similarly, 2ϵ + 1 ∈ dom(p_{γ}) for all large γ, so that p_{γ}(2ϵ + 1)(ρ(p_{γ})) = 0 for all large γ. It follows immediately that q ↾ (2ϵ + 1) forces that S_{ϵ}^{∗} ∩ ρ is non-stationary.

♦

Lemma 5.8 ℙ_{β}^{∗} is dense in ℙ_{β}, and ℙ_{β} is (ω_{2} + 1)-strategically closed.

Proof: The proof is by induction on β. We prove first that ℙ_{β}^{∗} is dense. β = 0: there is nothing to do. β = 2α + 2: fix p ∈ ℙ_{β}. Since ℙ_{2α+1} is strategically closed and ℙ_{2α+1}^{∗} is dense we may find q_{1} ∈ ℙ_{2α+1}^{∗} such that q_{1} ≤ p ↾ (2α + 1), q_{1} decides p(2α + 1), and ρ(q_{1}) > dom(p(2α + 1)).

Now we build a decreasing ω-sequence q⃗ of elements of ℙ_{2α+1}^{∗} such that ρ(q_{n}) is increasing and q_{n+1} decides Ṡ_{α} ∩ ρ(q_{n}); at each stage we use the strategic closure of ℙ_{2α+1} and the fact that ℙ_{2α+1}^{∗} is a dense subset. After ω steps we define q as follows; let ρ = ⋃ ρ(q_{n}), dom(q) = ⋃_{n} dom(q_{n})∪{2α+1}, and

1. For γ < α, q(2γ + 1) = ⋃_{n} q_{n}(2γ + 1) ∪ {(ρ, 0)}.

2. For γ ≤ α, q(2γ) = ⋃_{n} q_{n}(2γ) ∪ {ρ}.

3. q(2α + 1)(i) = p(2α + 1)(i) if i ∈ dom(p(2α + 1)), and 0 otherwise.

By the last lemma, q ↾ 2α + 1 ∈ ℙ_{2α+1}^{∗}. It is routine to check that q ∈ ℙ_{2α+2}^{∗} and ρ(q) = ρ. β = 2α + 1: this is exactly like the last case, except that now we demand ∀i < ρ(q_{n}) 2F (α, i) + 1 ∈ dom(q_{n+1}). β is limit, cf(β) = ω_{1}: Fix ⟨β_{i} : i < ω_{1}⟩ which is continuous increasing and cofinal in β. Let p ∈ ℙ_{β}. Find q_{0} ∈ ℙ_{β_{0}}^{∗} such that q_{0} ≤ p ↾ β_{0} and set p_{0} = q_{0} ⌢ p ↾ [β_{0}, β).

Now we define q_{i} and p_{i} by induction for i ≤ ω_{1}.

1. Choose q_{i+1} ≤ p_{i} ↾ β_{i+1} with q_{i+1} ∈ ℙ_{β_{i+1}}^{∗}, and then define p_{i+1} = q_{i+1} ⌢ p ↾ [β_{i+1}, β).

2. For i limit let ρ_{i} = ⋃_{j<i} ρ(q_{j}), q_{i}(2γ + 1) = ⋃_{j<i} q_{j}(2γ + 1) ∪ {(ρ_{i}, 0)}, q_{i}(2γ) = ⋃_{j<i} q_{j}(2γ) ∪ {ρ_{i}}. Then let p_{i} = q_{i} ⌢ p ↾ [β_{i}, β_{i+1}).

For i < ω_{1} it is easy to see that p_{i}, q_{i} are conditions. We claim that q = q_{ω_{1}} ∈ ℙ_{β}^{∗}. The only subtle point is to see that q ∈ ℙ_{β}. Let 2δ + 1 ∈ dom(q). Then for all large i we know 2δ +1 < β_{i}, 2δ +1 ∈ dom(q_{i}), so that in particular q_{i}(2δ + 1)(ρ_{i}) = 0 for all large i. This means that q(2δ + 1) is the characteristic function of a set which does not reflect at ρ, so is a legitimate condition in ℚ_{2δ+1}. Similarly if 2δ ∈ dom(q) then for all large i we see that q_{i} ↾ 2δ ⊩ ρ_{i} ∉ R_{δ}, so that q ↾ 2δ forces that the stationarity of R_{δ} does not reflect at ρ. β is a limit, cf(β) = ω or ω_{2}: similar to the cofinality ω_{1} case. cf(β) = ω_{3}: easy because ℙ_{β} is the direct limit of the sequence ⟨ℙ_{γ} : γ < β⟩.

This concludes the proof that ℙ_{β}^{∗} is dense. It is now easy to see that ℙ_{β} is (ω_{2} + 1)-strategically closed; the strategy for player II is simply to play into the dense set ℙ_{β}^{∗} at every successor stage, and to play a lower bound constructed as in Lemma 5.7 at each limit stage.

♦

Lemma 5.9 Let ℙ_{2γ} be a nice iteration of length less than ω_{4}. Then ℙ_{2γ} ∗ ℙ(Ṙ_{γ}) is (ω_{2} + 1)-strategically closed.

Proof: This is just like the last lemma.

♦

Notice that the effect of forcing with ℙ(R_{γ}) is to destroy the stationarity of all the sets S_{δ}^{∗} for δ < γ. We are now ready to prove the main result of this section.

Theorem 5.10 If the existence of a weakly compact cardinal is consistent, then Ref(ℵ_{3}, ℵ_{2}, ℵ_{1}) + Dnr(ℵ_{3}, ℵ_{2}, ℵ_{0}) is consistent.

Proof: As in the proof of Theorem 3.5, we will first give a proof assuming the consistency of a measurable cardinal and then show how to weaken the assumption to the consistency of a weakly compact cardinal. We will need a form of “diamond” principle.

Lemma 5.11 If κ is measurable and GCH holds, then in some forcing extension

1. κ is measurable.

2. There exists a sequence ⟨S_{α} : α < κ⟩ such that S_{α} ⊆ α for all α, and for all S ⊆ κ there is a normal measure U on κ such that if j_{U} : V ⟶ M_{U} ≃ Ult(V, U) is the associated elementary embedding then j_{U}(S⃗)_{κ} = S (or equivalently, { α | S ∩ α = S_{α} } ∈ U.

Proof:[Lemma 5.11] The proof is quite standard. For a similar construction given in more detail see [2].

Fix j : V ⟶ M the ultrapower map associated with some normal measure on κ. Let ℚ_{α} be the forcing whose conditions are sequences ⟨T_{β} : β < γ⟩ where γ < α and T_{β} ⊆ β for all β, ordered by end-extension. (This is really the same as the Cohen forcing Add(α, 1)). Let ℙ_{κ+1} be a Reverse Easton iteration of length κ + 1, where we force with (ℚ_{α})_{V^{ℙα}} at each inaccessible α ≤ κ.

Let G_{κ} be ℙ_{κ}-generic over V and let g be ℚ_{κ}-generic over V [G_{κ}]. We will prove that the sequence given by S_{α} = g(α) will work, by producing an appropriate U for each S ∈ V [G_{κ}][g] with S ⊆ κ. Let us fix such an S.

By GCH and the fact that j(ℙ_{κ})/G_{κ} ∗ g is κ^{+}-closed in V [G_{κ}][g], we may build H ∈ V [G_{κ}][g] which is j(ℙ_{κ})/G_{κ} ∗g-generic over M[G_{κ}][g]. Now for the key point: we define a condition q ∈ ℚ_{j(κ)} by setting q(α) = g(α) for α < κ and q(κ) = S. Then we build h ∋ q which is ℚ_{j(κ)}-generic over M[G_{κ}][g][H], using GCH and the κ^{+}-closure of ℚ_{j(κ)} in V [G_{κ}][g].

To finish we define j : V [G_{κ}][g] ⟶ M[G_{κ}][g][H][h] by j : τ̇^{G_{κ}∗g} ⟼ j(τ̇)^{G_{κ}∗g∗H∗h}, where this map is well-defined and elementary because j“(G_{κ} ∗ g) ⊆ G_{κ} ∗ g ∗ H ∗ h. The extended j is still an ultrapower by a normal measure (U say) because M[G_{κ}][g][H][h] = { j(F)(κ) | F ∈ V [G_{κ}][g] }. It is clear from the definition of this map j that j(g)(κ) = h(κ) = q(κ) = S, so the model V [G_{κ}][g] is as required.

This concludes the proof of Lemma 5.11.

♦

Fixing some reasonable coding of members of H_{α^{+}} by subsets of α, we may write the diamond property in the following equivalent form: there is a sequence ⟨x_{α} : α < κ⟩ such that for every x ∈ H_{κ^{+}} there exists U such that j_{U}(x⃗)_{κ} = x. Henceforth we will assume that we have fixed a sequence x⃗ with this property.

Now we describe a certain Reverse Easton forcing iteration of length κ+1. It will be clear after the iteration is defined that it is ℵ_{2}-strategically closed, so that in particular ℵ_{1} and ℵ_{2} are preserved. At stage α < κ we will force with ℚ_{α}, where ℚ_{α} is trivial forcing unless

1. α is inaccessible.

2. V^{ℙ_{α}} ⊨ α = ℵ_{3}, α_{V}^{+} = ℵ_{4}

3. x_{α} is a ℙ_{α}-name for a nice iteration ℙ_{2δ+1}^{α} of some length 2δ + 1 < α^{+}.

In this last case ℚ_{α} is defined to be ℙ_{2δ+1}^{α} ∗ ℙ(R_{δ}^{α}) ∗ Coll(ℵ_{2}, α).

At stage κ (which will be (ℵ_{3})_{V^{ℙκ}}), we will do a nice iteration ℚ_{κ} of length κ^{+}, with some book-keeping designed to guarantee that for every stationary S ⊆ S_{0}^{3} in the final model there exists T ⊆ S a non-reflecting stationary subset. So by design Dnr(ℵ_{3}, ℵ_{2}, ℵ_{0}) holds in the final model (and in fact so does Dnr(ℵ_{3}, ℵ_{1}, ℵ_{0})). It remains to be seen that Ref(ℵ_{3}, ℵ_{2}, ℵ_{1}) is true.

Let T ⊆ S_{1}^{3} be a stationary subset of S_{1}^{3} in the final model. Since ℚ_{κ} has the κ^{+}-c.c. we may assume that T is the generic extension by ℙ_{κ} ∗ (ℚ_{κ} ↾ (2δ + 1)) for some δ < κ^{+}. Let \overline{ℚ} = ℚ_{κ} ↾ (2δ + 1). Using the diamond property of x⃗ and the definition of the forcing iteration we may find U such that

j_{U}(ℙ_{κ}) = ℙ_{κ} ∗ \overline{ℚ} ∗ ℙ(R_{δ}) ∗ ℝ_{κ+1,j_{U} (κ)}

where ℝ_{κ+1,j_{U} (κ)} is the iteration above κ. To save on notation, denote j_{U} by j.

Now if M ≃ Ult(V, U) is the target model of j, then we may assume by the usual arguments that T ∈ M^{ℙ_{κ}∗\overline{ℚ}}. Notice that the last step in the iteration \overline{ℚ} was to force with ℝ(R_{δ}), that is to add a club of points at which the stationarity of R_{δ} fails to reflect. Applying Lemma 5.2 we see that T is still stationary in the extension by ℙ_{κ} ∗ \overline{ℚ} ∗ ℙ(R_{δ}). Since ℝ_{κ+1,j(κ)} is ℵ_{2}-strategically closed, T will remain stationary in the extension by j(ℙ_{κ}) (although of course κ will collapse to become some ordinal of cofinality ℵ_{2}).

To finish the proof we will build a generic embedding from V^{ℙ_{κ}∗\overline{ℚ}} to M^{j(ℙ_{κ}∗\overline{ℚ})}. It is easy to get j : V^{ℙ_{κ}} ⟶ M^{j(ℙ_{κ})}, what is needed is a master condition for \overline{ℚ} and j. Since κ is ℵ_{3} in V^{ℙ_{κ}} and ℚ_{κ} is an iteration with ≤ ℵ_{2}- supports, it is clear what the condition should be, we just need to check that it works.

Definition 5.12 Define q by setting dom(q) = j“(2δ + 1), q(j(2γ + 1)) = f_{γ} ∪ {(κ, 0)}, q(2γ) = C_{γ} ∪ {κ}, where f_{γ} : κ ⟶ 2 is the function added by \overline{ℚ} at stage 2γ + 1 and C_{γ} is the club added at stage 2γ.

We claim that q is a condition in j(\overline{ℚ}). To see this we should first check that f_{γ} is the characteristic function of a non-stationary subset of κ; this holds because at stage κ in the forcing j(ℙ_{κ}) we forced with ℙ(R_{δ}) and made S_{γ}^{∗} non-stationary for all γ < δ. We should also check that R_{γ} is non-stationary in γ, and again this is easy by Lemma 5.4 and the fact that R_{δ} has been made non-stationary.

Forcing with j(\overline{ℚ}) adds no bounded subsets of j(κ), so that clearly T is still stationary and cf(κ) is still ℵ_{2} in the model M^{j(ℙ_{κ}∗\overline{ℚ})}. By the familiar reflection argument, there exists α ∈ S_{2}^{3} such that T ∩ α is stationary in the model V^{ℙ_{κ}∗\overline{ℚ}}. T ∩ α will still be stationary in V^{ℙ_{κ}∗ℚ_{κ}}, because the rest of the iteration ℚ_{κ} does not add any bounded subsets of ℵ_{3}. We have proved that Ref(ℵ_{3}, ℵ_{2}, ℵ_{1}) holds in V^{ℙ_{κ}∗ℚ_{κ}}, which finishes the proof of Theorem 5.10 using a measurable cardinal.

It remains to be seen that we can replace the measurable cardinal by a weakly compact cardinal. To do this we will use the following result of Jensen.

Fact 5.13 Let V = L and let κ be weakly compact. Then there exists a sequence ⟨S_{α} : α < κ⟩ with S_{α} ⊆ α for all α, such that for all S ⊆ κ and all Π_{1}^{1} formulae ϕ(X) with one free second-order variable

V_{κ} ⊨ ϕ(S) ⟹ (∃α < κ S_{α} = S ∩ V_{α}, V_{α} ⊨ ϕ(S_{α})).

Using this fact we can argue exactly as in Argument 1 at the end of the proof of Theorem 3.5. This concludes the proof of Theorem 5.10.

♦

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