Cardinal invariants above the continuum

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 some consistency results about 𝔟(λ) and 𝔡(λ), which are natural generalisations of the cardinal invariants of the continuum 𝔟 and 𝔡. We also define invariants 𝔟_{cl}(λ) and 𝔡_{cl}(λ), and prove that almost always 𝔟(λ) = 𝔟_{cl}(λ) and 𝔡(λ) = 𝔡_{cl}(λ)

1 Introduction

The cardinal invariants of the continuum have been extensively studied. They are cardinals, typically between ω_{1} and 2^{ω}, whose values give structural information about ^{ω}ω. The survey paper [2] contains a wealth of information about these cardinals.

In this paper we study some natural generalisations to higher cardinals. Specifically, for λ regular, we define cardinals 𝔟(λ) and 𝔡(λ) which generalise the well-known invariants of the continuum 𝔟 and 𝔡.

∗Supported by a Postdoctoral Fellowship at the Hebrew University.

†Partially supported by the Basic Research Fund of the Israel Academy of Science.

Paper number 541.

For a fixed value of λ, we will prove that there are some simple constraints on the triple of cardinals (𝔟(λ), 𝔡(λ), 2^{λ}). We will also prove that any triple of cardinals obeying these constraints can be realised.

We will then prove that there is essentially no correlation between the values of the triple (𝔟(λ), 𝔡(λ), 2^{λ}) for different values of λ, except the obvious one that λ ⟼ 2^{λ} is non-decreasing. This generalises Easton’s celebrated theorem (see [3]) on the possible behaviours of λ ⟼ 2^{λ}; since his model was built using Cohen forcing, one can show that in that model 𝔟(λ) = λ^{+} and 𝔡(λ) = 2^{λ} for every λ.

𝔟(λ) and 𝔡(λ) are defined using the co-bounded filter on λ, and for λ > ω we can replace the co-bounded filter by the club filter to get invariants 𝔟_{cl}(λ) and 𝔡_{cl}(λ). We finish the paper by proving that these invariants are essentially the same as those defined using the co-bounded filter.

Some investigations have been made into generalising the other cardinal invariants of the continuum, for example in [7] Zapletal considers 𝔰(λ) which is a generalised version of the splitting number 𝔰. His work has a different flavour to ours, since getting 𝔰(λ) > λ^{+} needs large cardinals.

We are indebted to the referee and Jindrich Zapletal for pointing out a serious problem with the first version of this paper.

2 Definitions and elementary facts

It will be convenient to define the notions of “bounding number” and “dominating number” in quite a general setting. To avoid some trivialities, all partial orderings ℙ mentioned in this paper (with the exception of the notions of forcing) will be assumed to have the property that ∀p ∈ ℙ ∃q ∈ ℙ p <_{ℙ} q.

Definition 1: Let ℙ be a partial ordering. Then

• U ⊆ ℙ is unbounded if and only if ∀p ∈ ℙ ∃q ∈ U q ≰_{ℙ} p.

• D ⊆ ℙ is dominating if and only if ∀p ∈ ℙ ∃q ∈ D p ≤_{ℙ} q.

• 𝔟(ℙ) is the least cardinality of an unbounded subset of ℙ.

• 𝔡(ℙ) is the least cardinality of a dominating subset of ℙ.

The next lemma collects a few elementary facts about the cardinals 𝔟(ℙ) and 𝔡(ℙ).

Lemma 1: Let ℙ be a partial ordering, and suppose that β = 𝔟(ℙ) and δ = 𝔡(ℙ) are infinite. Then

β = cf(β) ≤ cf(δ) ≤ δ ≤ |ℙ|.

Proof: To show that β is regular, suppose for a contradiction that cf(β) < β. Let B be an unbounded family of cardinality β, and write B = ⋃_{α<cf(β)} B_{α} with |B_{α}| < β. For each α find p_{α} such that ∀p ∈ B_{α} p ≤ p_{α}, then find q such that ∀α < cf(β) p_{α} ≤ q. Then ∀p ∈ B p ≤ q, contradicting the assumption that B was unbounded.

Similarly, suppose that cf(δ) < β. Let D be dominating with cardinality δ and write D = ⋃_{α<cf(δ)} D_{α} where |D_{α}| < δ. For each α find p_{α} such that ∀p ∈ D_{α} p_{α} ≰ p, and then find q such that ∀α < cf(δ) p_{α} ≤ q. Then ∀p ∈ D q ≰ p, contradicting the assumption that D was dominating.

♦

The next result shows that we cannot hope to say much more.

Lemma 2: Let β and δ be infinite cardinals with β = cf(β) and δ^{<β} = δ. Define a partial ordering ℙ = ℙ(β, δ) in the following way; the underlying set is β × [δ]^{<β}, and (ρ, x) ≤ (σ, y) if and only if ρ ≤ σ and x ⊆ y.

Then 𝔟(ℙ) = β and 𝔡(ℙ) = δ.

Proof: Let B ⊆ ℙ be unbounded. If |B| < β then we can define

ρ = sup { σ | ∃y (σ, y) ∈ B } < β,
x = ⋃ { y | ∃σ (σ, y) ∈ B } ∈ [δ]^{<β}.

But then (ρ, x) is a bound for B, so |B| ≥ β and hence 𝔟(ℙ) ≥ β. On the other hand the set { (α, ∅) | α < β } is clearly unbounded, so that 𝔟(ℙ) = β.

Let D ⊆ ℙ be dominating. If |D| < δ then ⋃ { y | ∃σ (σ, y) ∈ D } ≠ δ, and this is impossible, so that 𝔡(ℙ) ≥ δ. On the other hand GCH holds and cf(δ) ≥ β, so that |ℙ| = β × δ^{<β} = δ. Hence 𝔡(ℙ) = δ.

♦

Definition 2: Let ℙ, ℚ be posets and f : ℙ ⟶ ℚ a function. f embeds ℙ cofinally into ℚ if and only if

• ∀p, p′ ∈ ℙ p ≤_{ℙ} p′ ⟺ f(p) ≤_{ℚ} f(p′).

• ∀q ∈ ℚ ∃p ∈ ℙ q ≤_{ℚ} f(p). That is, rge(f) is dominating.

Lemma 3: If f : ℙ ⟶ ℚ embeds ℙ cofinally into ℚ then 𝔟(ℙ) = 𝔟(ℚ) and 𝔡(ℙ) = 𝔡(ℚ).

Proof: Easy.

♦

Lemma 4: Let ℙ be any partial ordering. Then there is ℙ^{∗} ⊆ ℙ such that ℙ^{∗} is a dominating subset of ℙ and ℙ^{∗} is well-founded.

Proof: We enumerate ℙ^{∗} recursively. Suppose that we have already enumerated elements ⟨b_{α} : α < β⟩ into ℙ^{∗}. If { b_{α} | α < β } is dominating then we stop, otherwise we choose b_{β} so that b_{β} ≰ b_{α} for all α < β.

Clearly the construction stops and enumerates a dominating subset ℙ^{∗} of ℙ. To see that ℙ^{∗} is well-founded observe that b_{β} < b_{α} ⟹ β < α.

♦

Notice that the identity embeds ℙ^{∗} cofinally in ℙ, so 𝔟(ℙ^{∗}) = 𝔟(ℙ) and 𝔡(ℙ^{∗}) = 𝔡(ℙ).

We also need some information about the preservation of 𝔟(ℙ) and 𝔡(ℙ) by forcing.

Lemma 5: Let ℙ be a partial ordering with 𝔟(ℙ) = β, 𝔡(ℙ) = δ.

• Let V [G] be a generic extension of V such that every set of ordinals of size less than β in V [G] is covered by a set of size less than β in V. Then V [G] ⊨ 𝔟(ℙ) = β.

• Let V [G] be a generic extension of V such that every set of ordinals of size less than δ in V [G] is covered by a set of size less than δ in V. Then V [G] ⊨ 𝔡(ℙ) = δ.

Proof: We do the first part, the second is very similar. The hypothesis implies that β is a cardinal in V [G], and since “B is unbounded” is upwards absolute from V to V [G] it is clear that V [G] ⊨ 𝔟(ℙ) ≤ β. Suppose for a contradiction that we have C in V [G] unbounded with V [G] ⊨ |C| < β. By our hypothesis there is D ∈ V such that C ⊆ D and V ⊨ |D| < β, but now D is unbounded contradicting the definition of β.

♦

With these preliminaries out of the way, we can define the cardinals which will concern us in this paper.

Definition 3: Let λ be a regular cardinal.

1. If f, g ∈ ^{λ}λ then f <^{∗} g iff ∃α < λ ∀β > α f(β) < g(β).

2. 𝔟(λ) =_{def} 𝔟((^{λ}λ, <^{∗})).

3. 𝔡(λ) =_{def} 𝔡((^{λ}λ, <^{∗})).

These are defined by analogy with some “cardinal invariants of the continuum” (for a reference on cardinal invariants see [2]) known as 𝔟 and 𝔡. In our notation 𝔟 = 𝔟(ω) and 𝔡 = 𝔡(ω).

Lemma 6: If λ is regular then

• λ^{+} ≤ 𝔟(λ).

• 𝔟(λ) = cf(𝔟(λ)).

• 𝔟(λ) ≤ cf(𝔡(λ)).

• 𝔡(λ) ≤ 2^{λ}.

• cf(2^{λ}) > λ.

Proof: The first claim follows from the following trivial fact.

Fact 1: Let { f_{α} | α < λ } ⊆ ^{λ}λ. Then there is a function f ∈ ^{λ}λ such that ∀α < λ f_{α} <^{∗} f.

Proof: Define f(β) = sup { f_{γ}(β) + 1 | γ < β }. Then f(β) > f_{γ}(β) for γ < β < λ.

♦

The next three claims follow easily from our general results on 𝔟(ℙ) and 𝔡(ℙ), and the last is just König’s well-known theorem on cardinal exponentiation.

♦

We will prove that these are essentially the only restrictions provable in ZFC. One could view this as a refinement of Easton’s classical result (see [3]) on λ ⟼ 2^{λ}.

3 Hechler forcing

In this section we show how to force that certain posets can be cofinally embedded in (λ^{λ}, <^{∗}). This is a straightforward generalisation of Hechler’s work in [4], where he treats the case λ = ω.

We start with a brief review of our forcing notation. p ≤ q means that p is stronger than q, a κ-closed forcing notion is one in which every decreasing chain of length less than κ has a lower bound, and a κ-dense forcing notion is one in which every sequence of dense open sets of length less than κ has non-empty intersection.

If x is an ordered pair then x_{0} will denote the first component of x and x_{1} the second component.

Definition 4: Let λ be regular. 𝔻(λ) is the notion of forcing whose conditions are pairs (s, F) with s ∈ ^{<λ}λ and F ∈ ^{λ}λ, ordered as follows; (s, F) ≤ (t, F′) if and only if

1. dom(t) ≤ dom(s) and t = s ↾ dom(t).

2. s(α) ≥ F′(α) for dom(t) ≤ α < dom(s).

3. F (α) ≥ F′(α) for all α.

We will think of a generic filter G as adding a function f_{G} : λ ⟶ λ given by f_{G} = ⋃ { s | ∃F (s, F) ∈ G }. It is easy to see that

G = { (t, F) | t = f_{G} ↾ dom(t), dom(t) ≤ α ⟹ F (α) ≤ f_{G}(α) },

so that V [f_{G}] = V [G] and we can talk about functions from λ to λ being 𝔻(λ)-generic.

Lemma 7: Let λ^{<λ} = λ, and set ℙ = 𝔻(λ). Then

1. ℙ is λ-closed.

2. ℙ is λ^{+}-c.c.

3. If g : λ ⟶ λ is ℙ-generic over V then ∀f ∈ ^{λ}λ ∩ V f <^{∗} g.

Proof:

1. Let γ < λ and suppose that ⟨(t_{α}, F_{α}) : α < γ⟩ is a descending γsequence of conditions from ℙ. Defining t = ⋃ { t_{α} | α < γ } and F : β ⟼ sup { F_{α}(β) | α < γ }, it is easy to see that (t, F) is a lower bound for the sequence.

2. Observe that if (s, F) and (s, F′) are two conditions with the same first component then they are compatible, because if H : β ⟼ F (β)∪F′(β) the condition (s, H) is a common lower bound. There are only λ^{<λ} = λ possible first components, so that ℙ clearly has the λ^{+}-c.c.

3. Let f ∈ ^{λ}λ ∩ V, and let (t, F) be an arbitrary condition. Let us define F′ : β ⟼ (F (β) ∪ f(β)) + 1, then (t, F′) refines (t, F) and forces that f(α) < f_{G}(α) for all α ≥ dom(t).

♦

If μ = cf(μ) > λ = λ^{<λ} then it is straightforward to iterate 𝔻(λ) with < λ-support for μ steps and get a model where 𝔟(λ) = 𝔡(λ) = μ. Getting a model where 𝔟(λ) < 𝔡(λ) is a little harder, but we can do it by a “nonlinear iteration” which will embed a well chosen poset cofinally into (^{λ}λ, <^{∗}).

Theorem 1: Let λ = λ^{<λ}, and suppose that ℚ is any well-founded poset with 𝔟(ℚ) ≥ λ^{+}. Then there is a forcing 𝔻(λ, ℚ) such that

1. 𝔻(λ, ℚ) is λ-closed and λ^{+}-c.c.

2. V^{𝔻(λ,ℚ)} ⊨ ℚ can be cofinally embedded into (^{λ}λ, <^{∗})

3. If V ⊨ 𝔟(ℚ) = β then V^{𝔻(λ,ℚ)} ⊨ 𝔟(λ) = β.

4. If V ⊨ 𝔡(ℚ) = δ then V^{𝔻(λ,ℚ)} ⊨ 𝔡(λ) = δ.

Proof: We will define the conditions and ordering for 𝔻(λ, ℚ) by induction on ℚ. The idea is to iterate 𝔻(λ) “along ℚ” so as to get a cofinal embedding of ℚ into ^{λ}λ. It will be convenient to define a new poset ℚ^{+} which consists of ℚ together with a new element top which is greater than all the elements of ℚ.

We will define for each a ∈ ℚ^{+} a notion of forcing ℙ_{a}. If a ∈ ℚ^{+} then we will denote { c ∈ ℚ | c < a } by ℚ/a. It will follow from the definition that if c < a then ℙ_{c} is a complete subordering of ℙ_{a}, and that the map p ∈ ℙ_{a} ⟼ p ↾ ℚ/c is a projection from ℙ_{a} to ℙ_{c}.

Suppose that for all b <_{ℚ} a we have already defined ℙ_{b}.

1. p is a condition in ℙ_{a} if and only if

(a) p is a function, dom(p) ⊆ ℚ/a and |dom(p)| < λ.

(b) For all b ∈ dom(p), p(b) = (t, Ḟ) where t ∈ ^{<λ}λ and Ḟ is a ℙ_{b}-name for a member of ^{λ}λ.

2. If p, q ∈ ℙ_{a} then p ≤ q if and only if

(a) dom(q) ⊆ dom(p).

(b) For all b ∈ dom(q), if p(b) = (s, Ḣ) and q(b) = (t, İ) then

i. t = s ↾ dom(t).

ii. p ↾ (ℚ/b) ⊩_{ℙ_{b}} dom(t) ≤ α < dom(s) ⟹ s(α) > İ(α). iii. p ↾ (ℚ/b) ⊩_{ℙ_{b}} ∀α Ḣ(α) ≥ İ(α).

We define 𝔻(λ, ℚ) = ℙ_{top}, and verify that this forcing does what we claimed. The verification is broken up into a series of claims.

Claim 1: 𝔻(λ, ℚ) is λ-closed.

Proof: Let γ < λ and let ⟨p_{α} : α < γ⟩ be a descending γ-sequence of conditions. We will define a new condition p with dom(p) = ⋃_{α} dom(p_{α}). For each b ∈ dom(p_{α}) let p_{α}(b) = (t_{α}(b), Ḟ_{α}(b)).

Let p(b) = (t, Ḟ) where t = ⋃ { t_{α}(b) | b ∈ dom(p_{α}) } and Ḟ (b) is a ℙ_{b}- name for the pointwise supremum of { Ḟ_{α}(b) | b ∈ dom(p_{α}) }. Then it it is easy to check that p is a condition and is a lower bound for ⟨p_{α} : α < γ⟩

♦

Claim 2: 𝔻(λ, ℚ) is λ^{+}-c.c.

Proof: Let ⟨p_{α} : α < λ^{+}⟩ be a family of conditions. Since λ = λ^{<λ} we may assume that the domains form a Δ-system with root r. We may also assume that for b ∈ r, p_{α}(b) = (t_{b}, F_{α}(b)) where t_{b} is independent of α. It is now easy to see that any two conditions in the family are compatible.

♦

Claim 3: If c < a then ℙ_{c} is a complete subordering of ℙ_{a}, and the map p ⟼ p ↾ ℚ/c is a projection from ℙ_{a} to ℙ_{c}.

Proof: This is routine.

♦

If G is 𝔻(λ, ℚ)-generic, then for each a ∈ ℚ we can define f_{G}^{a} ∈ ^{λ}λ ∩ V [G] by f_{G}^{a} = ⋃ { t(a)_{0} | t ∈ G }. It is these functions that will give us a cofinal embedding of ℚ into (^{λ}λ ∩ V [G], <^{∗}), via the map a ⟼ f_{G}^{a}.

Claim 4: If a <_{ℚ} b then f_{G}^{a} <^{∗} f_{G}^{b}.

Proof: Let p be a condition and let Ḟ be the canonical ℙ_{b}-name for f_{G}^{a}. Refine p to q in the following way; q(c) = p(c) for c ≠ b, and if p(b) = (t, Ḣ) then q(b) = (t, İ) where İ names the pointwise maximum of Ḟ and Ḣ.

Then q forces that f_{G}^{b}(α) is greater than f_{G}^{a}(α) for α ≥ dom(t).

♦

Notice that by the same proof f_{G}^{b} dominates every function in V^{ℙ_{a}}.

Claim 5: If a ≮_{ℚ} b then f_{G}^{a} ≮^{∗} f_{G}^{b}.

Proof: If b <_{ℚ} a then we showed in the last claim that f_{G}^{b} <^{∗} f_{G}^{a}, so we may assume without loss of generality that b ≮_{ℚ} a.

Let p be a condition and let α < λ. Choose β large enough that {dom(p(a)_{0}), dom(p(b)_{0}), α} ⊆ β. Let p(b) = (t, Ḟ), and find q ∈ ℙ_{b} such that q ≤ p ↾ (ℚ/b) and q decides Ḟ ↾ (β + 1).

Let p_{1} be the condition such that p_{1}(c) = p(c) if c ≮_{ℚ} b and p_{1}(c) = q(c) if c <_{ℚ} b. Then p_{1} refines p and p_{1}(a) = p(a), p_{1}(b) = p(b).

Let p(a) = (s, Ḣ). Find r ∈ ℙ_{a} such that r ≤ p_{1} ↾ (ℚ/a) and r decides Ḣ ↾ (β + 1).

Let p_{2} be the condition such that p_{2}(c) = p_{1}(c) if c ≮_{ℚ} a and p_{2}(c) = r(c) if c <_{ℚ} a. Then p_{2} refines p_{1} and p_{2}(a) = p(a), p_{2}(b) = p(b).

Now it is easy to extend p_{2} to a condition which forces f_{G}^{a}(β) > f_{G}^{b}(β).

♦

Claim 6: The map a ⟼ f_{G}^{a} embeds ℚ cofinally into (^{λ}λ, <^{∗}) in the generic extension by 𝔻(λ, ℚ).

Proof: We have already checked that the map is order-preserving. It remains to be seen that its range is dominating.

Let G be 𝔻(λ, ℚ)-generic and let f ∈ ^{λ}λ ∩ V [G]. Then f = (ḟ)^{G} for some canonical name ḟ, and by the λ^{+}-c.c. we may assume that there is X ⊆ ℚ such that |X| = λ and ḟ only involves conditions p with dom(p) ⊆ X. Now 𝔟(ℚ) ≥ λ^{+} so that we can find a ∈ ℚ with X ⊆ ℚ/a.

This implies that ḟ is a ℙ_{a}-name for a function in ^{λ}λ, so that f <^{∗} f_{G}^{a} and we are done.

♦

Claim 7: If V ⊨ 𝔟(ℚ) = β then V^{𝔻(λ,ℚ)} ⊨ 𝔟(λ) = β.

Proof: Let G be 𝔻(λ, ℚ)-generic. By lemma 3 it will suffice to show that V [G] ⊨ 𝔟(ℚ) = β. This follows from lemma 5, the fact that 𝔻(λ, ℚ) is λ^{+}- c.c. and the assumption that 𝔟(ℚ) ≥ λ^{+}.

♦

Claim 8: If V ⊨ 𝔡(ℚ) = δ then V^{𝔻(λ,ℚ)} ⊨ 𝔡(λ) = δ.

Proof: Exactly like the last claim.

♦

This finishes the proof of Theorem 1.

♦

4 Controlling the invariants at a fixed cardinal

In this section we show how to force that the triple (𝔟(λ), 𝔡(λ), 2^{λ}) can be anything “reasonable” for a fixed value of λ.

Theorem 2: Let λ = λ^{<λ} and let GCH hold at all cardinals ρ ≥ λ. Let β, δ, μ be cardinals such that λ^{+} ≤ β = cf(β) ≤ cf(δ), δ ≤ μ and cf(μ) > λ.

Then there is a forcing 𝕄(λ, β, δ, μ) such that in the generic extension 𝔟(λ) = β, 𝔡(λ) = δ and 2^{λ} = μ.

Proof: In V define ℚ = ℙ(β, δ), as in lemma 2. We know that V ⊨ 𝔟(ℚ) = β and V ⊨ 𝔡(ℚ) = δ. Fix ℚ^{∗} a cofinal wellfounded subset of ℚ, and then define a new well-founded poset ℝ as follows.

Definition 5: The elements of ℝ are pairs (p, i) where either i = 0 and p ∈ μ or i = 1 and p ∈ ℚ^{∗}. (p, i) ≤ (q, j) iff i = j = 0 and p ≤ q in μ, i = j = 1 and p ≤ q in ℚ^{∗}, or i = 0 and j = 1.

Now we set 𝕄(λ, β, δ, μ) = 𝔻(λ, ℝ). It is routine to use the closure and chain condition to argue that 𝕄 makes 2^{λ} = μ. Since ℝ contains a cofinal copy of ℚ^{∗}, it is also easy to see that 𝕄 forces 𝔟(λ) = β and 𝔡(λ) = δ.

♦

5 A first attempt at the main theorem

We now aim to put together the basic modules as described in the previous section, so as to control the function λ ⟼ (𝔟(λ), 𝔡(λ), 2^{λ}) for all regular λ. A naive first attempt would be to imitate Easton’s construction from [3]; this almost works, and will lead us towards the right construction.

Let us briefly recall the statement and proof of Easton’s theorem on the behaviour of λ ⟼ 2^{λ}.

Lemma 8 (Easton’s lemma): If ℙ is κ-c.c. and ℚ is κ-closed then ℙ is κc.c. in V^{ℚ} and ℚ is κ-dense in V^{ℙ}. In particular^{<κ}ON ∩V^{ℙ×ℚ} = ^{<κ}ON ∩V^{ℙ}.

Theorem 3 (Easton’s theorem): Let F : REG ⟶ CARD be a class function such that cf(F (λ)) > λ and λ < μ ⟹ F (λ) ≤ F (μ). Let GCH hold. Then there is a class forcing ℙ which preserves cardinals and cofinalities, such that in the extension 2^{λ} = F (λ) for all regular λ.

Proof:[Sketch] The “basic module” is ℙ(λ) = Add(λ, F (λ)). ℙ is the “Easton product” of the ℙ(λ), to be more precise p ∈ ℙ iff

1. p is a function with dom(p) ⊆ REG and p(β) ∈ ℙ(β) for all β ∈ dom(p).

2. For all inaccessible γ, dom(p) ∩ γ is bounded in γ.

ℙ is ordered by pointwise refinement. There are certain complications arising from the fact that we are doing class forcing; we ignore them in this sketch.

If β is regular then we may factor ℙ as ℙ_{<β} × ℙ(β) × ℙ_{>β} in the obvious way. ℙ_{≥β} is always β-closed.

It follows from GCH and the Δ-system lemma that if γ is Mahlo or the successor of a regular cardinal then ℙ_{<γ} is γ-c.c. On the other hand, if γ is a non-Mahlo inaccessible or the successor of a singular cardinal, then ℙ_{<γ} is in general only γ^{+}-c.c.

In particular for γ regular ℙ_{≤γ} = ℙ_{<γ^{+}} is always γ^{+}-c.c. so that by Easton’s lemma ^{γ}ON ∩ V^{ℙ} = ^{γ}ON ∩ V^{ℙ_{≤γ}}. This implies that in the end we have only added F (γ) many subsets of γ.

It remains to be seen that cardinals and cofinalities are preserved. It will suffice to show that regular cardinals remain regular. If γ is Mahlo or the successor of a regular cardinal, then Easton’s lemma implies that <γON ∩ V^{ℙ} = ^{<γ}ON ∩ V^{ℙ_{<γ}}, and since γ is regular in V^{ℙ_{<γ}} (by γ-c.c.) γ is clearly regular in V^{ℙ}.

Now suppose that γ = μ^{+} for μ singular. If γ becomes singular in V^{ℙ} let its new cofinality be β, where we see that β < μ and β is regular in V. βON ∩ V^{ℙ} = ^{β}ON ∩ V^{ℙ_{≤β}}, so that γ will have cofinality β in V^{ℙ_{≤β}}. This is absurd as ℙ_{≤β} is β^{+}-c.c. and β^{+} < μ < γ. A very similar argument will work in case γ is a non-Mahlo inaccessible.

♦

Suppose that we replace Add(λ, F (λ)) by ℙ(λ) = 𝕄(λ, β(λ), δ(λ), μ(λ)), where λ ⟼ (β(λ), δ(λ), μ(λ)) is a function obeying the constraints given by Lemma 6. Let ℙ be the Easton product of the ℙ(λ). Then exactly as in the proof of Easton’s theorem it will follow that ℙ preserves cardinals and cofinalities, and that 2^{λ} = μ(λ) in V^{ℙ}.

Lemma 9: If λ is inaccessible or the successor of a regular cardinal then 𝔟(λ) = β(λ), 𝔡(λ) = δ(λ) and 2^{λ} = μ(λ) in V^{ℙ}.

Proof: For any λ, ^{λ}λ ∩ V^{ℙ} = ^{λ}λ ∩ V^{ℙ_{≤λ}}. 𝔟(λ) and 𝔡(λ) have the right values in V^{ℙ(λ)} by design, and these values are not changed by λ-c.c. forcing. So assuming ℙ_{<λ} is λ-c.c. those invariants have the right values in V^{ℙ_{≤λ}}, and hence in V^{ℙ}.

♦

We need some way of coping with the successors of singular cardinals and the non-Mahlo inaccessibles. Zapletal pointed out that at the first inaccessible in an Easton iteration we are certain to add many Cohen subsets, so that there really is a need to modify the construction.

6 Tail forcing

Easton’s forcing to control λ ⟼ 2^{λ} can be seen as a kind of iterated forcing in which we choose each iterand from the ground model, or equivalently as a kind of product forcing. Silver’s “Reverse Easton forcing” is an iteration in which the iterand at λ is defined in V^{ℙ_{λ}}. The “tail forcing” which we describe here is a sort of hybrid.

We follow the conventions of Baumgartner’s paper [1] in our treatment of iterated forcing, except that when have ℚ̇ ∈ V^{ℙ} and form ℙ∗ℚ̇ we reserve the right not to take all ℙ-names for members of ℚ (as long as we take enough names that the set of their denotations is forced to be dense). For example if ℚ ∈ V we will only take names q̂ for q ∈ ℚ, so ℙ ∗ ℚ̂ will just be ℙ × ℚ.

We will describe a kind of iteration which we call “Easton tail iteration” in which at successor stages we choose iterands from V, but at limit stage λ we choose ℚ̇_{λ} in a different way; possibly ℚ_{λ} ∉ V, but we will arrange things so that the generic G_{λ} factors at many places below λ and any final segment of G_{λ} essentially determines ℚ_{λ}. This idea comes from Magidor and Shelah’s paper [5].

We assume for simplicity that in the ground model all limit cardinals are singular or inaccessible. In the application that we intend this is no restriction, as the ground model will obey GCH.

Definition 6: A forcing iteration ℙ_{γ} with iterands ⟨ℚ̇_{β} : β + 1 < γ⟩ is an Easton tail iteration iff

1. The iteration has Easton support, that is to say a direct limit is taken at inaccessible limit stages and an inverse limit elsewhere.

2. ℚ̇_{β} = 0 unless β is a regular cardinal.

3. If β is the successor of a regular cardinal then ℚ_{β} ∈ V.

4. For all regular β, ℙ_{β+1} is β^{+}-c.c.

5. For λ a limit cardinal ℙ_{λ} is λ^{++}-c.c. if λ is singular, and λ^{+}-c.c. if λ is inaccessible.

6. For all regular α with α + 1 < γ there exists an iteration ℙ_{γ}^{α} dense in ℙ_{γ} such that ℙ_{α+1}^{α} = ℙ_{α+1}, and for β with α + 1 < β ≤ γ

(a) ℙ_{β}^{α} factors as ℙ_{α+1} × ℙ^{α} ↾ (α + 1, β).

(b) If β is inaccessible or the successor of a singular, and p ∈ ℙ_{γ}^{α}, then p(β) is a name depending only on ℙ^{α} ↾ (α + 1, β).

(c) ℙ^{α} ↾ (α + 1, β) is α^{+}-closed.

Clause 6 is of course the interesting one. It holds in a trivial way if ℙ_{γ} is just a product with Easton supports. Clauses 6a and 6b should really be read together, as the factorisation in 6a only makes sense because 6b already applies to β̄ < β, and conversely 6b only makes sense once we have the factorisation from 6a.

The following result shows that Easton tail iterations do not disturb the universe too much.

Lemma 10: Let ℙ_{γ} be an Easton tail iteration. Then

1. For all regular α < γ, ^{α}ON ∩ V^{ℙ_{γ}} = ^{α}ON ∩ V^{ℙ_{α+1}}.

2. ℙ_{γ} preserves all cardinals and cofinalities.

Proof: Exactly like Theorem 3.

♦

In the next section we will see how to define a non-trivial Easton tail iteration. If γ^{+} is the successor of a regular then it will suffice to choose ℚ_{γ^{+}} ∈ V as any γ^{+}-closed and γ^{++}-c.c. forcing. The interesting (difficult) stages are the ones where we have to cope with the other sorts of regular cardinal, here we will have to maintain the hypotheses on the chain condition and factorisation properties of the iteration. It turns out that slightly different strategies are appropriate for inaccessibles and successors of singulars.

7 The main theorem

Theorem 4: Let GCH hold. Let λ ⟼ (β(λ), δ(λ), μ(λ)) be a class function from REG to CARD^{3}, with λ^{+} ≤ β(λ) = cf(β(λ)) ≤ cf(δ(λ)) ≤ δ(λ) ≤ μ(λ) and cf(μ(λ)) > λ for all λ.

Then there exists a class forcing ℙ_{∞}, preserving all cardinals and cofinalities, such that in the generic extension 𝔟(λ) = β(λ), 𝔡(λ) = δ(λ) and 2^{λ} = μ(λ) for all λ.

Proof: We will define by induction on γ a sequence of Easton tail iterations ℙ_{γ}, and then let take a direct limit to get a class forcing ℙ_{∞}. The proof that ℙ_{∞} has the desired properties is exactly as in [3], so we will concentrate on defining the ℙ_{γ}. As we define the ℙ_{γ} we will also define dense subsets ℙ_{γ}^{α} intended to witness clause 6 in the definition of an Easton tail iteration.

Much of the combinatorics in this section is very similar to that in Section 3. Accordingly we have only sketched the proofs of some of the technical assertions about closure and chain conditions.

The easiest case to cope with is that where we are looking at the successor of a regular cardinal. So let γ be regular and assume that we have defined ℙ_{γ^{+}} (which is equivalent to ℙ_{γ+1} since we do trivial forcing at all points between γ and γ^{+}), and ℙ_{γ^{+}}^{α} (which is equivalent to ℙ_{γ+1}^{α}) for all α ≤ γ.

Definition 7: ℚ_{γ^{+}} = 𝕄(γ^{+}, β(γ^{+}), δ(γ^{+}), μ(γ^{+})) as defined in Section 4. ℙ_{γ^{+}+1} = ℙ_{γ^{+}} × ℚ_{γ^{+}}, and ℙ_{γ^{+}+1}^{α} = ℙ_{γ^{+}}^{α} × ℚ_{γ^{+}} for α ≤ γ.

It is now easy to check that this definition maintains the conditions for being an Easton tail iteration. Since ℙ_{γ^{+}} is γ^{+}-c.c. and we know that γ^{+}γ^{+} ∩ V^{ℙ_{∞}} = ^{γ^{+}}γ^{+} ∩ V^{ℙ_{γ^{+}+1}}, we will get the desired behaviour at γ^{+} in V^{ℙ_{∞}}.

Next we consider the case of a cardinal λ^{+}, where λ is singular. Suppose we have defined ℙ_{λ^{+}} (that is ℙ_{λ}) and ℙ_{λ^{+}}^{α} appropriately. Let ℝ be a well-founded poset of cardinality μ(λ) with 𝔟(ℝ) = β(λ) and 𝔡(ℝ) = δ(λ), as defined in Section 4. Let ℝ^{+} be ℝ with the addition of a maximal element top. We will define ℙ_{λ^{+}} ∗ ℚ̇_{a} by induction on a ∈ ℝ^{+}, and then set ℙ_{λ^{+}+1} = ℙ_{λ^{+}} ∗ ℚ̇_{top}. In the induction we will maintain the hypothesis that, for each α < λ, ℙ_{λ^{+}}^{α} ∗ ℚ̇_{a} can be factored as ℙ_{α+1} × (ℙ^{α} ↾ (α + 1, λ^{+}) ∗ ℚ̇_{a}). Let us now fix a, and suppose that we have defined ℙ_{λ^{+}} ∗ ℚ̇_{b} for all b below a in ℝ^{+}.

Definition 8: Let b < a, and let τ̇ be a for a function from λ^{+} to λ^{+}. Then τ̇ is symmetric iff for all α < λ, whenever G_{0} × G_{1} and G′_{0} × G_{1} are two generics for ℙ_{α+1} × (ℙ^{α} ↾ (α + 1, λ^{+}) ∗ ℚ_{b}), then τ̇^{G_{0}×G_{1}} = τ̇^{G′_{0}×G_{1}}.

Of course the (technically illegal) quantification over generic objects in this definition can be removed using the truth lemma, to see that the collection of symmetric names really is a set in V.

Definition 9: (p, q) is a condition in ℙ_{λ^{+}} ∗ ℚ̇_{a} iff

1. p ∈ ℙ_{λ^{+}}.

2. q is a function, dom(q) ⊆ ℝ^{+}/a and |dom(q)| ≤ λ.

3. For each b ∈ dom(q), q(b) is a pair (s, Ḟ) where s ∈ ^{<λ^{+}}λ^{+} and Ḟ is a symmetric ℙ_{λ^{+}} ∗ ℚ̇_{b}-name for a function from λ^{+} to λ^{+}.

Definition 10: Let (p, q) and (p′, q′) be conditions in ℙ_{λ^{+}} ∗ℚ̇_{a}. (p′, q′) refines (p, q) iff

1. p′ refines p in ℙ_{λ^{+}}.

2. dom(q) ⊆ dom(q′).

3. For each b ∈ dom(q), if we let q(b) = (s, F) and q(b′) = (s′, F′), then

(a) s′ extends s.

(b) If α ∈ lh(s′) − lh(s) then (p′, q′ ↾ (ℝ^{+}/b)) ⊩ s′(α) ≥ F (α).

(c) For all α, (p′, q′ ↾ (ℝ^{+}/b)) ⊩ F′(α) ≥ F (α).

Definition 11: We define ℙ_{λ^{+}+1} as ℙ_{λ^{+}} ∗ℚ̇_{top}. If α < λ then we define ℙ_{λ^{+}+1}^{α} as ℙ_{λ^{+}}^{α} ∗ ℚ̇_{top}.

It is now routine to check that this definition satisfies the chain condition and factorisation demands from Definition 6. The chain condition argument works because 2^{λ} = λ^{+}, and any incompatibility in ℚ_{top} is caused by a disagreement in the first coordinate at some b ∈ ℝ. The factorisation condition (clause 6a) holds because symmetric names can be computed using any final segment of the ℙ_{λ}-generic, and the closure condition (clause 6c) follows from the fact that the canonical name for the pointwise sup of a series of functions with symmetric names is itself a symmetric name.

Now we check that we have achieved the desired effect on the values of 𝔟(λ^{+}) and 𝔡(λ^{+}).

Lemma 11: The function added by ℙ_{λ^{+}} ∗ ℚ̇_{top} at b ∈ ℝ^{+} eventually dominates all functions in ^{λ^{+}}λ^{+} ∩ V^{ℙ_{λ^{+}}∗ℚ̇_{b}}.

Proof: It suffices to show that if Ḟ is a ℙ_{λ^{+}} ∗ ℚ̇_{b}-name for a function from λ^{+} to λ^{+} then there is a symmetric name Ḟ′ such that ⊩ ∀β F (β) ≤ F′(β).

For each regular α < λ we factor ℙ_{λ^{+}} ∗ℚ̇_{b} as ℙ_{α+1}×(ℙ^{α} ↾ (α+1, λ^{+})∗ℚ̇_{b}). In V^{ℙ^{α}↾(α+1,λ^{+})∗ℚ̇_{b}} we may treat Ḟ as a ℙ_{α+1}-name and define

G_{α}(β) = sup({ γ | ∃p ∈ ℙ_{α+1} p ⊩ Ḟ (β) = γ }).

Notice that we may also treat G_{α} as a ℙ_{λ^{+}} ∗ ℚ̇_{b}-name, and that if α < ᾱ then ⊩ G_{α} ≤ Gᾱ. Now let F′ be the canonical name for a function such that ∀β F′(β) = sup_{α<λ} G_{α}(β), then it is easy to see that F′ is a symmetric name for a function from λ^{+} to λ^{+} and that ⊩ ∀β F (β) ≤ F′(β).

♦

Lemma 12: In V^{ℙ_{λ^{+}+1}} there is a copy of ℝ embedded cofinally into ^{λ^{+}}λ^{+}.

Proof: Let ḟ name a function from λ^{+} to λ^{+} in V^{ℙ_{λ^{+}+1}}. As ℙ_{λ^{+}+1} has the λ^{++}-c.c. we may assume that ḟ only depends on λ^{+} many coordinates in ℝ, and hence (since 𝔟(ℝ) = β(λ^{+}) ≥ λ^{++}) that ḟ is a ℙ_{λ^{+}} ∗ ℚ̇_{b}-name for some b ∈ ℝ. By the preceding lemma the function which is added at coordinate b will dominate ḟ.

♦

It remains to be seen what we should do for λ inaccessible. The construction is very similar to that for successors of singulars, with the important difference that we need to work with a larger class of names for functions in order to guarantee that we dominate everything that we ought to. This in turn leads to a slight complication in the definition of ℙ^{α}.

Suppose that we have defined ℙ_{λ} and ℙ_{λ}^{α} appropriately. Let ℝ be a poset with the appropriate properties (|ℝ| = μ(λ), 𝔟(ℝ) = β(λ), 𝔡(ℝ) = δ(λ)) and let ℝ^{+} be ℝ with a maximal element called top adjoined. As before we define ℙ_{λ} ∗ ℚ̇_{a} by induction on a ∈ ℝ^{+}. In the induction we will maintain the hypothesis that for each α < λ there is a dense subset of ℙ_{λ}^{α} ∗ ℚ̇_{a} which factorises as ℙ_{α+1} × (ℙ^{α} ↾ (α + 1, λ) ∗ ℚ̇_{a}^{α}). Let us fix a, and suppose that we have defined everything for all b below a. Definition 12: (p, (μ, q)) is a condition in ℙ_{λ} ∗ ℚ̇_{a} iff

1. p ∈ ℙ_{λ}.

2. μ < λ, μ is regular.

3. q is a function, dom(q) ⊆ ℝ^{+}/a and |dom(q)| < λ.

4. For each b ∈ dom(q), q(b) is a pair (s, Ḟ) where s ∈ ^{<λ^{+}}λ^{+} and Ḟ is a

ℙ^{μ} ↾ (μ + 1, λ) ∗ ℚ̇_{b}^{μ}-name for a function from λ^{+} to λ^{+}. Definition 13: Let (p, (μ, q)) and (p′, (μ′, q′)) be conditions in ℙ_{λ^{+}} ∗ ℚ̇_{a}. (p′, (μ′, q′)) refines (p, (μ, q)) iff

1. p′ refines p in ℙ_{λ^{+}}.

2. dom(q) ⊆ dom(q′).

3. μ′ ≥ μ.

4. For each b ∈ dom(q), if we let q(b) = (s, F) and q(b′) = (s′, F′), then (a) s′ extends s.

(b) If α ∈ lh(s′) − lh(s) then (p′, q′ ↾ (ℝ^{+}/b)) ⊩ s′(α) ≥ F (α).

(c) For all α, (p′, q′ ↾ (ℝ^{+}/b)) ⊩ F′(α) ≥ F (α). Notice that since any ℙ_{λ}∗ℚ̇_{b}-generic induces a ℙ^{μ} ↾ (μ+1, λ)∗ℚ̇_{b}^{μ}-generic,

there is a natural interpretation of any ℙ^{μ} ↾ (μ+1, λ)∗ℚ̇_{b}^{μ}-name as a ℙ_{λ} ∗ℚ̇_{b}- name. We are using this fact implicitly when we define the ordering on the conditions. We need to maintain the hypothesis on factorising the forcing, so we make the following definition.

Definition 14: Let μ be regular with μ < λ. ℙ_{μ+1} × (ℙ^{μ} ↾ (μ + 1, λ) ∗ ℚ̇_{a}^{μ}) is defined as the set of (p_{0}, (p_{1}, (ν, q))) such that p_{0} ∈ ℙ_{μ+1}, p_{1} ∈ ℙ^{μ} ↾ (μ + 1, λ) and (p_{0} ⌢ p_{1}, (ν, q)) ∈ ℙ_{λ} ∗ ℚ̇_{a} with ν ≥ μ.

The key point here is that the factorisation makes sense, because for such a condition q(b) depends only on ℙ^{μ} ↾ (μ + 1, λ) ∗ ℚ̇_{b}^{μ}.

Definition 15: We define ℙ_{λ+1} as ℙ_{λ} ∗ ℚ̇_{top}, and ℙ_{λ+1}^{μ} as ℙ_{λ} ∗ ℚ̇_{top}^{μ},

As in the case of a successor of a singular, it is straightforward to see that we have satisfied the chain condition and factorisation conditions. To finish the proof we need to check that the forcing at λ has achieved the right effect, which will be clear exactly as in the singular case when we have proved the following lemma.

Lemma 13: The function added by ℙ_{λ} ∗ ℚ̇_{top} at b ∈ ℝ eventually dominates all functions in ^{λ}λ ∩ V^{ℙ_{λ}∗ℚ̇_{b}}.

Proof: We do a density argument. Suppose that ḟ names a function in λλ ∩ V^{ℙ_{λ}∗ℚ̇_{b}}, and let (p, (μ, q)) be a condition in ℙ_{λ} ∗ ℚ̇_{top}. We factor ℙ_{λ} ∗ ℚ̇_{b} as ℙ_{μ+1} × (ℙ^{μ} ↾ (μ + 1, λ) ∗ ℚ̇_{b}^{μ}), and use the fact that ℙ_{μ+1} has μ^{+}-c.c. in V^{ℙ^{μ}↾(μ+1,λ)∗ℚ̇_{b}^{μ}} to find a ℙ^{μ} ↾ (μ + 1, λ) ∗ ℚ̇_{b}^{μ}-name ġ such that ⊩ ∀β ḟ(β) ≤ ġ(β).

Now we can refine (p, (μ, q)) in the natural way by strengthening the second component of q(b) to dominate ġ. This gives a condition which forces that the function added at b will eventually dominate ḟ.

♦

This concludes the proof of Theorem 4.

♦

8 Variations

In this appendix we discuss the invariants that arise if we work with the club filter in place of the co-bounded filter. It turns out that this does not make too much difference. All the results here are due to Shelah. Definition 16: Let λ be regular.

1. Let f, g ∈ ^{λ}λ. f<_{cl}g iff there is C ⊆ λ closed and unbounded in λ such that α ∈ C ⟹ f(α) < g(α).

2. 𝔟_{cl}(λ) =_{def} 𝔟((^{λ}λ, <_{cl})).

3. 𝔡_{cl}(λ) =_{def} 𝔡((^{λ}λ, <_{cl})). Theorem 5: 𝔡_{cl}(λ) ≤ 𝔡(λ) ≤ 𝔡_{cl}(λ)^{ω}. Proof: If a family of functions is dominating with respect to <^{∗} it is dominating with respect to <_{cl}, so that 𝔡_{cl}(λ) ≤ 𝔡(λ).

For the converse, let us fix D ⊆ ^{λ}λ such that D is dominating with respect to <_{cl} and |D| = 𝔡_{cl}(λ). We may assume that every function in D is increasing (replace each f ∈ D by f^{∗} : γ ⟼ ⋃_{α≤γ} f(α)).

Let g_{0} ∈ ^{λ}λ. Define by induction f_{n}, g_{n}, and C_{n} such that

1. f_{n} ∈ D.

2. C_{n} is club in λ, and α ∈ C_{n} ⟹ g_{n}(α) < f_{n}(α).

3. C_{n+1} ⊆ C_{n}.

4. g_{n} is increasing.

5. g_{n+1}(β) > g_{n}(β) for all β.

6. g_{n+1}(β) > f_{n}(min(C_{n} − (β + 1))) for all β.

Now let α = min(⋂_{n<ω} C_{n}). We will prove that g_{0}(γ) ≤ ⋃_{n} f_{n}(γ) for γ > α.

Fix some γ > α. For each n we know that C_{n} ∩γ ≠ ∅, so that if we define γ_{n} = sup(C_{n} ∩ (γ + 1)) then γ_{n} is the largest point of C_{n} less than or equal to γ. Notice that min(C_{n} − (γ + 1)) = min(C_{n} − (γ_{n} + 1)).

Since C_{n+1} ⊆ C_{n}, γ_{n+1} ≤ γ_{n}, so that for all sufficiently large n (say n ≥ N) we have γ_{n} = γ̄ for some fixed γ̄. We claim that g_{0}(γ) ≤ f_{N+1}(γ), which we will prove by building a chain of inequalities. Let us define

δ = min(C_{N} − (γ + 1)) = min(C_{N} − (γ̄ + 1)).

Then

g_{0}(γ) ≤ g_{N}(γ) ≤ g_{N}(δ) < f_{N}(δ) < g_{N+1}(γ̄) < f_{N+1}(γ̄) ≤ f_{N+1}(γ),

where the key point is that γ̄ ∈ C_{N+1} and hence g_{N+1}(γ̄) < f_{N+1}(γ̄).

Now it is easy to manufacture a family of size 𝔡_{cl}(λ)^{ω} which is dominating with respect to <^{∗}, so that 𝔡(λ) ≤ 𝔡_{cl}(λ)^{ω}.

♦

Theorem 6: 𝔟_{cl}(λ) = 𝔟(λ).

Proof: If a family of functions is unbounded with respect to <_{cl} it is unbounded with respect to <^{∗}, so that 𝔟(λ) ≤ 𝔟_{cl}(λ).

Suppose for a contradiction that 𝔟(λ) < 𝔟_{cl}(λ), and fix U_{0} ⊆ ^{λ}λ such that |U_{0}| = 𝔟(λ) and U_{0} is unbounded with respect to <^{∗}. We may assume without loss of generality that every function in U_{0} is increasing. We perform an inductive construction in ω steps, whose aim is to produce a bound for U_{0} with respect to <^{∗}.

By assumption U_{0} is bounded with respect to <_{cl}, so choose g_{0} which bounds it modulo the club filter. Choose also club sets { C_{f}^{0} | f ∈ U_{0} } such that α ∈ C_{f}^{0} ⟹ f(α) < g_{0}(α). For each f ∈ U_{0} define another function f^{[0]} by f^{[0]} : β ⟼ f(min(C_{f}^{0} − (β + 1)))

For n ≥ 1 define U_{n} = U_{n−1} ∪ { f^{[n−1]} | f ∈ U_{n−1} }. By induction it will follow that |U_{n}| = 𝔟(λ), so that we may choose g_{n} such that g_{n}(β) > g_{n−1}(β) for all β and g_{n} bounds U_{n} modulo clubs. We choose clubs { C_{f}^{n} | f ∈ U_{n} } such that

1. α ∈ C_{f}^{n} ⟹ f(α) < g_{n}(α).

2. If f ∈ U_{n−1}, then C_{f}^{n} ⊆ C_{f}^{n−1}.

3. If n ≥ 2 and f ∈ U_{n−2} then C_{f}^{n} ⊆ C_{f^{[n−2]}}^{n−1}, where this makes sense because in this case f^{[n−2]} ∈ U_{n−1}.

For each f ∈ U_{n} we define f^{[n]} : β ⟼ f(min(C_{f}^{n} − (β + 1))) to finish round n of the inductive construction.

Now we claim that the pointwise sup of the sequence ⟨g_{n} : n < ω⟩ is an upper bound for U_{0} with respect to <^{∗}. Let us fix f ∈ U_{0}, and then let α = min(⋂ C_{f}^{n}).

We will now give a very similar argument to that of Theorem 5. Fix γ > α. We define γ_{n} = sup(C_{f}^{n} ∩ (γ + 1)) and observe that γ_{n+1} ≤ γ_{n}, so we may find N and γ̄ such that n ≥ N ⟹ γ_{n} = γ̄.

Let δ = min(C_{F}^{N} − (γ + 1)) = min(C_{F}^{N} − (γ̄ + 1)). We now get a chain of inequalities

f(γ) ≤ f(δ) = f^{[N]}(γ̄) < g_{N+1}(γ̄) ≤ g_{N+1}(γ).

This time the key point is that γ̄ ∈ C_{f}^{N+2} ⊆ C_{f^{[N]}}^{N+1}, so that f^{[N]}(γ̄) < g_{N+1}(γ̄).

We have proved that γ > α ⟹ f(γ) ≤ ⋃_{n} g_{n}(γ), so that every function f ∈ U_{0} is bounded on a final segment of λ by γ ⟼ ⋃_{n} g_{n}(γ). This contradicts the choice of U_{0} as unbounded with respect to <^{∗}, so we are done.

♦

It is natural to ask whether the first result can be improved to show that 𝔟_{cl}(λ) = 𝔟(λ). This can be done for λ sufficiently large, at the cost of using a powerful result from Shelah’s paper [6].

Definition 17: Let θ = cf(θ) < μ.

1. P^{θ}(μ) = { X ⊆ μ | |X| = θ }.

2. μ^{[θ]} is the least cardinality of a family P ⊆ P^{θ}(μ) such that

∀A ∈ P^{θ}(μ) ∃B ⊆ P (|B| < θ ∧ A ⊆ ⋃ B).

One of the main results of [6] is that ZFC proves a weak form of the GCH.

Theorem 7: Let μ > ℶ_{ω}. Then μ^{[θ]} = μ for all sufficiently large θ < ℶ_{ω}.

It is easy to see that if P ⊆ P^{θ}μ is such that

∀A ∈ P^{θ}(μ) ∃B ⊆ P (|B| < θ ∧ A ⊆ ⋃ B),

then ∀A ∈ P^{θ}(μ) ∃C ∈ P |A ∩ C| = θ. This is all we use in what follows, and in fact we could get away with ∀A ∈ P^{θ}(μ) ∃C ∈ P |A ∩ C| = ℵ_{0}.

Theorem 8: Let λ = cf(λ) > ℶ_{ω}. Then 𝔡(λ) = 𝔡_{cl}(λ).

Proof: Let μ = 𝔡_{cl}(λ). Then μ > ℶ_{ω}, so that we may apply Theorem 7 to find a regular θ < ℶ_{ω} such that μ^{[θ]} = μ. let us fix P ⊆ P^{θ}(μ) such that |P | = μ and ∀A ∈ P^{θ}(μ) ∃C ∈ P |A ∩ C| = θ.

Now let D ⊆ ^{λ}λ be such that |D| = μ and D is dominating in (^{λ}λ, <_{cl}). We may suppose that D consists of increasing functions. Enumerate D as ⟨h_{α} : α < θ⟩, and then define h_{A} : γ ⟼ ⋃_{α∈A} h_{α}(γ) for each A ∈ P. Since θ < ℶ_{ω} ≤ λ, h_{A} ∈ ^{λ}λ. We will prove that { h_{A} | A ∈ P } is dominating in (^{λ}λ, <^{∗}).

We will do a version of the construction from Theorem 5. Let g_{0} ∈ ^{λ}λ. Define by induction f_{α}, g_{α}, and C_{α} for α < θ, with the following properties.

1. f_{α} ∈ D.

2. C_{α} is club in λ, and β ∈ C_{α} ⟹ g_{α}(β) < f_{α}(β).

3. α < ᾱ ⟹ Cᾱ ⊆ C_{α}.

4. g_{α} is increasing.

5. If α < ᾱ, then gᾱ(β) > g_{α}(β) for all β.

6. If α < ᾱ, then gᾱ(β) > f_{α}(min(C_{α} − (β + 1))) for all β.

This is easy, because θ < λ. By the choice of P we may find a set A ∈ P such that |{ h_{β} | β ∈ A } ∩ { f_{α} | α < θ }| = θ. Enumerate the first ω many α such that f_{α} ∈ { h_{β} | β ∈ A } as ⟨α_{n} : n < ω⟩.

We may now repeat the proof of Theorem 5 with f_{α_{n}}, g_{α_{n}} and C_{α_{n}} in place of f_{n}, g_{n} and C_{n}. We find that for all sufficiently large γ we have g_{0}(γ) ≤ g_{α_{0}}(γ) ≤ ⋃_{n} f_{α_{n}}(γ). By the definition of h_{A} and the fact that { f_{α_{n}} | n < ω } ⊆ { h_{β} | β ∈ A }, ⋃_{n} f_{α_{n}}(γ) ≤ h_{A}(γ) for all γ, so that g_{0} <^{∗} h_{A}. This shows { h_{A} | A ∈ P } to be dominating, so we are done.

♦

We do not know whether it can ever be the case that 𝔡_{cl}(λ) < 𝔡(λ). This is connected with some open and apparently difficult questions in pcf theory.

References

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[2] E. K. van Douwen, The integers and topology, Handbook of Set-Theoretic Topology, (K. Kunen and J. E. Vaughan, editors), North-Holland, Amsterdam, 1984, pp. 111–167.

[3] W. Easton, Powers of regular cardinals, Annals of Mathematical Logic, vol. 1 (1964), pp. 139–178.

[4] S. Hechler, On the existence of certain cofinal subsets of ω^{ω}, Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, vol. 13 part II, American Mathematical Society, Providence, Rhode Island, 1974, pp. 155–173.

[5] M. Magidor and S. Shelah, When does almost free imply free?, Journal of the American Mathematical Society (to appear).

[6] S. Shelah, The Generalised Continuum Hypothesis revisited, Israel Journal of Mathematics (to appear).

[7] J. Zapletal, Splitting number and the core model, (to appear).