COLLAPSING SUCCESSORS OF SINGULARS

JAMES CUMMINGS

Abstract. Let κ be a singular cardinal in V, and let W ⊇ V be a model such that κ_{V}^{+} = λ_{W}^{+} for some W -cardinal λ with W ⊧ cf(κ) ≠ cf(λ). We apply Shelah’s pcf theory to study this situation, and prove the following results.

1) W is not a κ^{+}-c.c generic extension of V.

2) There is no “good scale for κ” in V, so in particular weak forms of square must fail at κ.

3) If V ⊧ cf(κ) = ℵ_{0} then V ⊧ “κ is strong limit ⟹ 2^{κ} = κ^{+}”, and also ωκ ∩ W ⊋^{ω}κ ∩ V.

4) If κ = ℵ_{ω}^{V} then λ ≤ (2^{ℵ_{0}})_{V} .

1. Introduction

The results in this paper were motivated by the following natural question. Question 1: Can there exist a pair of inner models of ZFC, V and W say, such that V ⊆ W and ℵ_{ω+1}^{V} = ℵ_{2}^{W}? This question was raised by Bukovský and Copláková-Hartová [1] who observed (Theorem 1.7 of [1]) that if we have V and W as above then W ⊧ 2^{ℵ_{0}} ≥ ℵ_{2}, because V ⊧ ℵ_{ω}^{ℵ_{0}} ≥ ℵ_{ω+1}. They also note that if Question 1 has a positive answer then it is possible to violate CH by adding a real.

We can ask more general questions about the possibilities for collapsing successors of singulars; for example Question 2: Let V ⊧ “κ is singular”. Can there be W ⊇ V such that

W ⊧ “κ_{V}^{+} is a cardinal and cf(κ) ≠ cf(|κ|)”

The restriction “cf(κ) ≠ cf(|κ|)” is not unreasonable, because for example it is easy to collapse ℵ_{ω.2+1}^{V} to ℵ_{ω+1}^{V [G]} by forcing with Coll(ω, ℵ_{ω}).

Work of Shelah [10] shows that a positive solution to Question 1 or Question 2 would require the use of large cardinals.

Definition 1.1. Let κ be a cardinal, then AD_{κ} holds iff there exists a sequence ⟨A_{α} : α < κ^{+}⟩ such that

1. For each α < κ^{+}, A_{α} is an unbounded subset of κ.

2. For each β < κ^{+} there exists g : β ⟶ κ such that ⟨A_{α} \ g(α) : α < β⟩ is a sequence of mutually disjoint sets.

Shelah proved ([10], page 440) that

1. If κ is regular, or if κ is singular and □_{κ} holds, then AD_{κ} holds.

2. If V ⊧ AD_{κ}, W ⊇ V and W ⊧ “κ_{V}^{+} is a cardinal”, then W ⊧ cf(κ) = cf(|κ|).

1991 Mathematics Subject Classification. Primary 03E05; Secondary 03E35.

Key words and phrases. Squares, pcf, successors of singulars, good points.

Partially supported by Israel Science Foundation grant number 517/94.

In particular, if V ⊧ AD_{κ} + cf(κ) = ω then W ⊧ cf(|κ|) = ω, so that in W the cardinal κ_{V}^{+} must be ℵ_{1}^{W} or the successor of a cardinal of cofinality ω.

Remark 1.2. For κ singular the weak square principle □_{κ}^{∗} is enough to derive AD_{κ}. See [3] for a proof.

The connection with large cardinals arises because for κ singular the principle □_{κ}^{∗} holds unless there exist inner models of some very strong large cardinal axioms (at the level of Woodin cardinals). See [6], [7] and [8].

The results in this paper are more general than Shelah’s in the sense that they can be applied in some situations where V ⊧ ¬AD_{κ}. See [4] and [3] for a discussion of the implications between AD_{κ}, squares, scales and reflection principles.

The author would like to thank Matt Foreman and Menachem Magidor for many interesting and informative conversations about the connections between squares and pcf theory. Thanks also to Doug Burke for his incisive comments on the first version of this paper.

2. Pcf theory, good points and approachability

In this section we review some facts about pcf theory that will be used later. For a detailed development of pcf theory see [2] or [11]. All the facts quoted here are due to Shelah unless otherwise stated.

Definition 2.1. Let κ be a singular cardinal. A scale for κ (of length κ^{+}) is (κ⃗, f⃗) where

1. κ⃗ = ⟨κ_{i} : i < cf(κ)⟩ is a strictly increasing sequence of regular cardinals with sup_{i} κ_{i} = κ.

2. f⃗ = ⟨f_{α} : α < κ^{+}⟩ is a sequence such that (a) f_{α} ∈ ∏_{i} κ_{i}.

(b) f⃗ is strictly increasing and cofinal in (∏_{i} κ_{i}, <^{∗}) where <^{∗} is the eventual domination relation on ∏ κ_{i}.

Fact 2.2. If κ is singular then there exists a scale of length κ^{+} for κ.

Definition 2.3. Let (κ⃗, f⃗) be a scale of length κ^{+} for κ. α < κ^{+} is good (for f⃗) iff there exists A ⊆ α unbounded in α and i < cf(κ) such that whenever β, γ ∈ A with β < γ and j > i then f_{β}(j) < f_{γ}(j).

Remark 2.4. If cf(α) < cf(κ) then it is easy to see that α is good.

Remark 2.5. If cf(α) > cf(κ), α being good is equivalent to ⟨f_{β} : β < α⟩ having a least upper bound g such that cf(g(i)) = cf(α) for all large i.

Remark 2.6. The definition of good point actually makes sense for any κ⃗ and any increasing sequence from (∏_{i} κ_{i}, <^{∗}).

Fact 2.7. If ρ = cf(ρ) < κ and (κ⃗, f⃗) is a scale of length κ^{+} for κ then

{ α < κ^{+} : cf(α) = ρ and α is good }

is stationary in κ^{+}.

Since we will use some of the ideas later, we sketch a proof of this fact.

Proof. We prove it first when cf(κ) < ρ. Let C ⊆ κ^{+} be club. Fix θ some large regular cardinal. Build a increasing and continuous sequence ⟨N_{i} : i < ρ⟩ such that 1. N_{i} ≺ (H_{θ}, ∈, <_{θ}) where <_{θ} is some wellordering of H_{θ}.

2. C, κ⃗, f⃗ ∈ N_{0} and cf(κ) ⊆ N_{0}.

3. |N_{i}| < ρ.

4. ⟨N_{i} : i ≤ j⟩ ∈ N_{j+1} for all j < ρ (it will follow that i ⊆ N_{i} and hence that i < j ⟹ N_{i} ∈ N_{j}).

Notice that if i < j then sup(N_{i} ∩ κ^{+}) ∈ N_{j}, so that ⟨sup(N_{i} ∩ κ^{+}) : i < ρ⟩ is increasing and continuous. The limit of this sequence is sup(⋃_{i} N_{i} ∩ κ^{+}). It follows from elementarity that C is unbounded in N_{i} ∩ κ^{+}, so that sup(N_{i} ∩ κ^{+}) ∈ C.

Define χ_{i} ∈ ∏_{n} κ_{n} by setting χ_{i}(n) = 0 for κ_{n} < ρ, χ_{i}(n) = sup(N_{i} ∩ κ_{n}) for κ_{n} ≥ ρ. Since N_{i} ∈ N_{i+1} we see that χ_{i} ∈ N_{i+1}, so that there exists α_{i} ∈ N_{i+1} ∩κ^{+} such that α_{i} > sup(N_{i} ∩ κ^{+}) and χ_{i} <^{∗} f_{α_{i}}. Since cf(κ) ⊆ N_{i+1} we see that for all n < cf(κ) we have f_{α_{i}}(κ_{n}) ∈ N_{i+1} ∩ κ_{n}, in particular we have f_{α_{i}} <^{∗} χ_{i+1}.

Now for every i < ρ we may choose j < cf(κ) such that

k > j ⟹ χ_{i}(k) < f_{α_{i}}(k) < χ_{i+1}(k).

Since ρ > cf(κ) we may fix A ⊆ ρ unbounded and j < cf(κ) such that

∀i ∈ A ∀k > j χ_{i}(k) < f_{α_{i}}(k) < χ_{i+1}(k).

It follows that if i_{0}, i_{1} ∈ A with i_{0} < i_{1} then for k > j we have

f_{α_{i_{0}}} (k) < χ_{i_{0}+1}(k) ≤ χ_{i_{1}}(k) < f_{α_{i_{1}}} (k).

Now we know that sup(N_{i} ∩ κ^{+}) < α_{i} < sup(N_{i+1} ∩ κ^{+}), so this shows that sup_{i} α_{i} = sup_{i} sup(N_{i} ∩ κ^{+}) is a good point. This point is also a point of C with cofinality ρ, so we have shown the set of good points of cofinality ρ to be stationary. This concludes the proof when ρ > cf(κ).

To finish we observe that if A witnesses α to be a good point for f⃗, and β is a limit point of A, then A∩β witnesses that β is good for f⃗. Given ρ = cf(ρ) ≤ cf(κ) and a club set C, we first use the argument above to find α ∈ lim(C) with cf(α) = cf(κ)^{+} and α good and choose A witnessing this; then we choose β ∈ C a limit point of A with cofinality ρ, so that A ∩ β witnesses β is good.

Remark 2.8. It is easy to see that if (κ⃗, f⃗) and (κ⃗, g⃗) are two scales of length κ^{+} for κ in the same product ∏_{i} κ_{i}, then the set of good points for f⃗ is equal modulo clubs to the set of good points for g⃗.

Definition 2.9. A good scale for κ is a scale for κ in which, modulo the club filter, almost every point of cofinality greater than cf(κ) is good.

Now we look at a form of weak square (the approachability square AP_{κ}) introduced by Shelah in [9]. AP_{κ} is substantially weaker than □_{κ}^{∗}.

Definition 2.10. AP_{κ} holds iff there exists ⟨C_{α} : α < κ^{+}⟩ such that 1. C_{α} is a club subset of α of order type cf(α) for each limit α.

2. C_{γ+1} is a subset of κ^{+} for each successor ordinal γ + 1.

3. For a club of α < κ^{+}, ∀β < α ∃γ < α C_{α} ∩ β = C_{γ+1}.

We are interested in this principle mainly because of the following fact, which is Claim 4.4 in [4].

Fact 2.11. If AP_{κ} holds and (κ⃗, f⃗) is a scale then almost every point α with cf(α) > cf(κ) is good for f⃗. That is, every scale is good.

We also collect here a few more facts about good points.

Fact 2.12. Let cf(κ) = ω, and let κ be strong limit with 2^{κ} > κ^{+}. Then there exists a scale (κ⃗, f⃗) for κ such that every point of uncountable cofinality is good for f⃗.

Fact 2.13. If (κ⃗, f⃗) is a scale for ℵ_{ω} and α < ℵ_{ω+1}, cf(α) > 2^{ℵ_{0}} then α is good for f⃗.

3. The Main Lemma

In this section we will prove the key technical lemma of the paper, which connects the notion of “good point” with Questions 1 and 2 from the Introduction. We originally proved the lemma for points of cofinality λ, with the additional assumption that λ is regular in W; Burke pointed out that it holds in the more general form given here.

Lemma 3.1. Let V, W be inner models of set theory with V ⊆ W. Let κ be singular in V, and fix (κ⃗, f⃗) a scale of length κ^{+} for κ. Suppose that κ_{V}^{+} = λ_{W}^{+} where λ > ℵ_{0} and W ⊧ cf(λ) ≠ cf(κ). Then there exists η < λ such that for every δ ∈ REG^{W} ∩ (η, λ]

{ γ < κ_{V}^{+} : W ⊧ “cf(γ) = δ and γ is good for f⃗” }

is non-stationary in W.

Proof. We work in W. Since κ_{V}^{+} = κ_{W}^{+} we will just denote this cardinal by “κ^{+}”. Observe that μ =_{def} cf_{W} (κ) < λ because λ is the largest cardinal less than κ^{+}. cf_{W} (cf_{V} (κ)) = μ, so we choose C ⊆ cf_{V} (κ) a cofinal set of order type μ. |κ| = λ so we may write κ = ⋃_{i<λ} X_{i} where X⃗ is an increasing sequence of sets with |X_{i}| < λ for all i < λ.

Now for all α < κ^{+}, rge(f_{α} ↾ C) ⊆ κ = ⋃_{i} X_{i}. Since μ ≠ cf(λ), for every α < κ^{+} there is i < λ such that { n ∈ C : f_{α}(n) ∈ X_{i} } is unbounded in μ. Hence we can fix i < λ such that the set

B =_{def} { α < κ^{+} : { n ∈ C : f_{α}(n) ∈ X_{i} } is unbounded in μ }

is unbounded in κ^{+}.

Now let η = max{|X_{i}|, μ}, and fix δ ∈ REG^{W} with η < δ ≤ λ. Since B is unbounded it has a club set of limit points; we claim that every limit point of B with cofinality δ is not good for f⃗. To see this, suppose for a contradiction that γ is a limit point of B of cofinality δ and that γ is good. This means that there is A ⊆ γ unbounded and m < cf_{V} (κ) such that if α, β ∈ A, α < β and m < n < cf_{V} (κ) then f_{α}(n) < f_{β}(n).

Now we build an increasing sequence of ordinals ⟨γ_{j} : j < δ⟩ such that γ_{2j} ∈ A and γ_{2j+1} ∈ B. Notice that f_{γ_{2j}} <^{∗} f_{γ_{2j+1}} <^{∗} f_{γ_{2j+2}} for each j < δ. For each j let us choose n_{j} > m such that n_{j} ∈ C,

f_{γ_{2j}}(n_{j}) < f_{γ_{2j+1}}(n_{j}) < f_{γ_{2j+2}}(n_{j}),

and f_{γ_{2j+1}}(n_{j}) ∈ X_{i}.

Since δ > μ = |C| we can find Z ⊆ δ unbounded and a fixed n ∈ C such that for all j ∈ Z

f_{γ_{2j}}(n) < f_{γ_{2j+1}}(n) < f_{γ_{2j+2}}(n)

and f_{γ_{2j+1}}(n) ∈ X_{i}. Now if j, k are two elements of Z with j < k then

f_{γ_{2j+1}}(n) < f_{γ_{2j+2}}(n) ≤ f_{γ_{2k}}(n) < f_{γ_{2k+1}}(n).

This implies that |{ f_{γ_{2j+1}}(n) : j ∈ Z }| = |Z| = δ. However for each j ∈ Z we have f_{γ_{2j+1}}(n) ∈ X_{i}, and |X_{i}| < δ, which is a contradiction. So almost every point of cofinality δ is not good and we have proved the claim of the Lemma.

Remark 3.2. REG^{W} ∩(η, λ] is always non-empty. If λ is regular in W then λ will do, and λ is singular then λ is limit and we are guaranteed many suitable δ.

4. Exploiting the main lemma

Throughout this section our running assumptions are that V ⊆ W, κ_{V}^{+} = λ_{W}^{+} , V ⊧ “κ is a singular cardinal”, and W ⊧ cf(κ) ≠ cf(λ).

Theorem 1. W is not a κ^{+}-c.c. extension of V.

Proof. Fix (κ⃗, f⃗) a scale for κ. Applying Lemma 3.1, let δ ∈ REG^{W} be such that in W the set of good points with cofinality δ is non-stationary.

Let us define

S = { γ < κ^{+} : V ⊧ “cf(γ) = δ and γ is good for f⃗” }.

By Lemma 2.7, V ⊧ “S is stationary”. Since δ is regular in W we know that

V ⊧ “cf(γ) = δ and γ is good” ⟹ W ⊧ “cf(γ) = δ and γ is good”,

and so by the choice of δ we have W ⊧ “S is nonstationary”. Since any κ^{+}- c.c. forcing preserves stationary subsets of κ^{+}, W is not a κ^{+}-c.c. generic extension of V.

We originally proved the following result under the additional assumptions that λ is regular and V ⊧ cf(κ) < λ; Burke pointed out that it is true in the general form given here.

Theorem 2. V ⊧ “There is no good scale of length κ^{+} for κ”.

Proof. Let (κ⃗, f⃗) be a good scale for κ in V, and let C be a club in κ^{+} such that

V ⊧ “every point of C with cofinality greater than cf(κ) is good”.

Let B be the unbounded set constructed in the proof of Lemma 3.1, and let δ ∈ REG^{W} be such that every limit point of B with W -cofinality δ is not good and δ > cf_{W} (κ).

Choose γ ∈ C ∩ acc(B) such that W ⊧ cf(γ) = δ. By the choice of B and δ, γ cannot be good. On the other hand W ⊧ cf(γ) = δ > cf(κ) so that V ⊧ cf(γ) ≠ cf(κ); if V ⊧ cf(γ) < cf(κ) then γ is good by Remark 2.4 and if V ⊧ cf(γ) > cf(κ) then γ is good by the choice of C. Contradiction!

Theorem 3. Let V ⊧ cf(κ) = ω. Then 1. V ⊧ “κ is strong limit ⟹ 2^{κ} = κ^{+}”.

2. If λ = ℵ_{1}^{W} then ^{ω}κ ∩ W ⊋ ^{ω}κ ∩ V.

Proof. The first claim follows immediately from Fact 2.12 and Theorem 2. For the second claim, suppose for a contradiction that ^{ω}κ ∩ W = ^{ω}κ ∩ V and fix (in V) a scale (κ⃗, f⃗) of length κ^{+} for κ.

Now W ⊧ cf(κ_{n}) > ω for all n < ω and W ⊧ “f⃗ is cofinal in ∏_{n} κ_{n}”. This is enough for us to imitate the proof of Lemma 2.7 and to prove (by building in W an ℵ_{1}-chain of countable structures) that

W ⊧ “{ γ < κ^{+} : cf(γ) = ℵ_{1} and γ is good } is stationary”.

This contradicts Lemma 3.1.

Theorem 4. If κ = ℵ_{ω}^{V} then λ ≤ (2^{ℵ_{0}})_{V}.

Proof. Fix (κ⃗, f⃗) a scale of length κ^{+} for κ. Recall that by Fact 2.13 all points of cofinality greater than 2^{ℵ_{0}} are good for f⃗.

Suppose for a contradiction that λ > (2^{ℵ_{0}})_{V}. Using Lemma 3.1 find γ such that W ⊧ cf(γ) > (2^{ℵ_{0}})_{V}, and γ is not good for f⃗. Now V ⊧ cf(γ) > (2^{ℵ_{0}})_{V}, contradicting 2.13.

5. Concluding remarks

The problems discussed in this paper are connected with some other open problems in set theory.

5.1. Chang’s conjecture. The consistency of “(ℵ_{ω+1}, ℵ_{ω}) ↠ (ℵ_{2}, ℵ_{1})” is open. If this holds then S =_{def} { X ⊆ ℵ_{ω+1} : ot(X) = ℵ_{2} } is stationary, and supposing that there exists a Woodin cardinal δ we can force with Woodin’s stationary tower forcing [12] below the condition S. It is easy to see that S ⊩ ℵ_{ω+1}^{V} = ℵ_{2}^{V [G]}.

5.2. Good points. Let (κ⃗, f⃗) be a scale of length ℵ_{ω+1} for ℵ_{ω}. It is open whether the set of points { α < ℵ_{ω+1} : cf(α) > ℵ_{1} and α is not good } can ever be stationary. If this set cannot be stationary then Lemma 3.1 implies that ℵ_{ω+1} can only be collapsed to ℵ_{1} or ℵ_{2}.

This problem is also connected with “(ℵ_{ω+1}, ℵ_{ω}) ↠ (ℵ_{2}, ℵ_{1})”, because if this holds then by Claim 4.3 in [4] there is a stationary set of cofinality ℵ_{2} points which are not good. It is known [4] that if MM holds or (ℵ_{ω+1}, ℵ_{ω}) ↠ (ℵ_{1}, ℵ_{0}) then the set { α < ℵ_{ω+1} : cf(α) = ℵ_{1} and α is not good } is stationary. The statement “(ℵ_{ω+1}, ℵ_{ω}) ↠ (ℵ_{1}, ℵ_{0})” is known [5] to be consistent.

5.3. Saturated ideals. Burke and Matsubara have pointed out that the results in this paper can be used to show that the nonstationary ideal on P_{κ}λ is not saturated for certain values of κ and λ.

References

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Department of Mathematics 2-390, Massachusetts Institute of Technology, Cambridge MA 02139, USA

   E-mail address: cummings@math.mit.edu