Sticks and clubs

Sakaé Fuchino, Saharon Shelah and Lajos Soukup

November 18, 1996

Abstract

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin’s Axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω_{1} together with ¬CH and Martin’s Axiom for countable p.o.-sets.

1 Beating with sticks and clubs

In this paper, we study combinatorial principles known as ‘stick’ and ‘club’, and their diverse variants which are all weakenings of ◇. Hence some of the consequences of ◇ still hold under these principles. On the other hand, they are weak enough to be consistent with the negation of the continuum hypothesis or even with a weak version of Martin’s axiom in addition. See e.g. [2], [4], [10] for applications of these principles. We shall begin with introducing the principles and some cardinal numbers connected to them.

( |•) (read “stick”) is the following principle introduced in S. Broverman, J. Ginsburg, K. Kunen and F. Tall [2]:

( |•): There exists a sequence (x_{α})_{α<ω_{1}} of countable subsets of ω_{1} such that for any y ∈ [ω_{1}]^{ℵ_{1}} there exists α < ω_{1} such that x_{α} ⊆ y.

Of course the sequence (x_{α})_{α<ω_{1}} above is a bluff. What is essential here is that there exists an X ⊆ [ω_{1}]^{ℵ_{0}} of cardinality ℵ_{1} such that for any y ∈ [ω_{1}]^{ℵ_{1}} there is an x ∈ X with x ⊆ y. The formulation above is chosen here merely to make the connection to the principle (♣) introduced later, more apparent.

Note that ( |•) follows from CH.

The principle ( |•) suggests the following cardinal number:

|• = min{ | X | : X ⊆ [ω_{1}]^{ℵ_{0}}, ∀y ∈ [ω_{1}]^{ℵ_{1}} ∃x ∈ X x ⊆ y }.

We have ℵ_{1} ≤ |• ≤ 2^{ℵ_{0}} and ( |•) holds if and only if |• = ℵ_{1}. We also consider the following variants of |•:

|•′ = min{ κ : κ ≥ ℵ_{1}, there is an X ⊆ [κ]^{ℵ_{0}}
such that | X | = κ and ∀y ∈ [κ]^{ℵ_{1}} ∃x ∈ X x ⊆ y };
|•′′ = min{ κ : κ ≥ ℵ_{1}, there is an X ⊆ [κ]^{ℵ_{0}}
such that | X | = κ and ∀y ∈ [κ]^{κ} ∃x ∈ X x ⊆ y };
|•_{λ} = min{ | X | : X ⊆ [λ]^{ℵ_{0}}
such that ∀y ∈ [λ]^{ℵ_{1}} ∃x ∈ X x ⊆ y }.

We have ℵ_{1} ≤ |•′′ ≤ |•′ ≤ 2^{ℵ_{0}} and λ ≤ |•_{λ} ≤ λ^{ℵ_{0}}. ( |•) holds if and only if |• = |•′ = |•′′ = ℵ_{1}. Let us call X as in the definition of |• ( |•′, |•′′ and |•_{λ} respectively) a |• -set ( |•′-set, |•′′-set and |•_{λ}-set respectively).

Lemma 1.1

a) |• ≤ |•′.

b) If |• < ℵ_{ω_{1}} then |• = |•′. In particular, we have then |•′′ ≤ |•.

c) If λ ≤ λ′ then |•_{λ} ≤ |•_{λ′}.

d) |• ≤ |•_{|}_{•} ≤ |•′.

Proof a): Let X ⊆ [κ]^{ℵ_{0}} be a |•′-set of cardinality |•′. Then X_{0} = X ∩ [ω_{1}]^{ℵ_{0}} is a |• -set of cardinality ≤ |•′.

b): By a), it is enough to show |•′ ≤ |•. We show inductively that, for every uncountable κ ≤ |•,

(∗)_{κ} there exists an X_{κ} ⊆ [κ]^{ℵ_{0}} such that | X_{κ} | ≤ |• and ∀y ∈ [κ]^{ℵ_{1}} ∃x ∈ X_{κ} (x ⊆ y).

For κ = ℵ_{1} this is clear.

Assume that we have shown (∗)_{λ} for all λ < κ. If κ is a successor then by induction hypothesis, we can find X_{α} ⊆ [α]^{ℵ_{0}} for all α < κ such that | X_{α} | ≤ |• and ∀y ∈ [α]^{ℵ_{1}} ∃x ∈ X_{α} x ⊆ y. Let X_{κ} = ⋃_{α<κ} X_{α}. Then X_{κ} has the desired property: | X_{κ} | ≤ |• is clear. If y ∈ [κ]^{ℵ_{1}}, there is some α < κ such that y ∈ [α]^{ℵ_{1}}. Hence there is an x ∈ X_{α} ⊆ X such that x ⊆ y.

Suppose now that κ is a limit. By assumption, we have cof(κ) = ω. Let (κ_{n})_{n∈ω} be an increasing sequence of cardinals below κ such that κ = ⋃_{n∈ω} κ_{n}. For each n, let X_{κ_{n}} ⊆ [κ_{n}]^{ℵ_{0}} be as in (∗)_{κ_{n}} and let X_{κ} = ⋃_{n∈ω} X_{κ_{n}}. Then X_{κ} is as desired: clearly | X_{κ} | ≤ |•. If y ∈ [κ]^{ℵ_{1}} there is an n ∈ ω such that y ∩ κ_{n} is uncountable. Hence there exists an x ∈ X_{κ_{n}} ⊆ X_{κ} such that x ⊆ y ∩ κ_{n} ⊆ y.

In particular we have shown that (∗)_{|}^{•} holds and hence |•′ ≤ |•.

c): Similarly to a).

d): By a) and c), we have |• = |•_{ℵ_{1}} ≤ |•_{|}_{•} ≤ |•_{|}_{•}′ = |•′. (Lemma 1.1)

The question, whether |• < |•′ is consistent, turned out to be a very delicate one: the problem is connected with some natural weakenings of GCH whose status (i.e. whether they are theorems in ZFC) is still open. One of them implies that |• = |•′ (this is essentially stated in [14, 1.2, 1.2A] in the light of [13, 6.1 [D]]; for more see [15]) while the negation of the other implies that the inequality is consistent. In this paper, we shall treat the latter consistency proof (Proposition 3.4). In contrast, the consistency of the inequality |•′′ < |• can be shown without any such additional set-theoretic assumptions (Proposition 3.5).

The principle (♣) (‘club’), a strengthening of ( |•), was first formulated in Ostaszewski [10]. Let Lim(ω_{1}) = { γ < ω_{1} : γ is a limit }. For a stationary E ⊆ Lim(ω_{1}),

♣(E): There exists a sequence (x_{γ})_{γ∈E} of countable subsets of ω_{1} such that for every γ ∈ E, x_{γ} is a cofinal subset of γ with otp(x_{γ}) = ω and for every y ∈ [ω_{1}]^{ℵ_{1}} there is γ ∈ E such that x_{γ} ⊆ y.

Let us call (x_{γ})_{γ∈E} as above a ♣(E)-sequence. For E = Lim(ω_{1}) we shall simply write (♣) in place of ♣(Lim(ω_{1})). Clearly ( |•) follows from (♣). Unlike ( |•), (♣) does not follow from CH since (♣) + CH is known to be equivalent to ◇ (K. Devlin, see [10]). This equivalence holds also in the version argumented with a stationary E ⊆ Lim(ω_{1}).

Fact 1.2 For any stationary E ⊆ Lim(ω_{1}), ♣(E) + CH is equivalent to ◇(E).

Proof The proof in [10] argumented with E works. (Fact 1.2)

S. Shelah [11] proved the consistency of ¬CH + (♣) in a model obtained from a model of GCH by making the size of ℘(ω_{1}) to be ℵ_{3} by countable conditions and then collapsing ℵ_{1} to be countable. Soon after that, in an unpublished note, J. Baumgartner gave a model of ¬CH + ♣ where collapsing of cardinals is not involved: his model was obtained from a model of V = L by adding many Sacks reals by side by side product. I. Juhász then proved in an unpublished note that “¬CH + MA(countable) + (♣)” is consistent. Here MA(countable) stands for Martin’s axiom restricted to countable partial orderings. Later P. Komjáth [7] cited a remark by Baumgartner that Shelah’s model mentioned above also satisfies ¬CH + MA(countable) + (♣). In Section 3, we shall give yet another model of ¬CH + MA(countable) + (♣) in which collapsing of cardinals is not involved (Theorem 3.8). In section 5, we construct a model of ¬CH + MA(countable) + “♣(E) for every stationary E ⊆ Lim(ω_{1}) ” starting from a model of ZFC with an inaccessible cardinal (Theorem 5.6).

These results are rather optimal in the sense that a slight strengthening of MA(countable) implies the negation of (♣). Let MA(Cohen) denote Martin’s axiom restricted to the partial orderings of the form Fn(κ, 2) for some κ where, as in [8], Fn(κ, 2) is the p.o.-set for adding κ Cohen reals, i.e. the set of functions from some finite subset of κ to 2 ordered by reverse inclusion.

Fact 1.3 MA for the partial ordering Fn(ω_{1}, 2) implies |• = |•′ = 2^{ℵ_{0}}. Further, if MA(Cohen) holds, then we have also |•′′ = 2^{ℵ_{0}}.

Proof Both equations can be proved similarly. For the first equation, it is enough to show |• = 2^{ℵ_{0}} by Lemma 1.1. Suppose that X ⊆ [ω_{1}]^{ℵ_{0}} is of cardinality less than 2^{ℵ_{0}}. We show that X is not a |• -set. Let P = Fn(ω_{1}, 2). Then for each x ∈ X the set

D_{x} = { q ∈ Fn(ω_{1}, 2) : ∃α ∈ dom(q) ∩ x q(α) = 0 }

is dense in P. For each α < ω_{1},

E_{α} = { q ∈ Fn(ω_{1}, 2) : ∃β > α (β ∈ dom(q) ∧ q(b) = 1) }

is also a dense subset of P. Let D = { D_{x} : x ∈ X } ∪ { E_{α} : α < ω_{1} } and G be a D-generic filter over P. Then the uncountable set

Y = { α < ω_{1} : q(α) = 1 for some q ∈ G }

contains no x ∈ X as a subset. (Fact 1.3)

We shall see in Proposition 3.5 that MA for the partial ordering Fn(ω_{1}, 2) is not enough for the last assertion in Fact 1.3.

♣(E) is equivalent to the following seemingly much stronger statement. Let E ⊆ Lim(ω_{1}) be a stationary set.

♣^{†}(E): There exists a sequence (x_{γ})_{γ∈E} of countable subsets of ω_{1} such that for every γ ∈ E, x_{γ} is a cofinal subset of γ with otp(x_{γ}) = ω and for every X ∈ [ω_{1}]^{ℵ_{1}}, { α ∈ E : x_{α} ⊆ X } is stationary.

Fact 1.4 For any stationary E ⊆ Lim(ω_{1}), ♣(E) and ♣^{†}(E) are equivalent.

Proof Like Fact 1.2, an easy modification of the corresponding proof in [10] will work. Nevertheless we give here a proof for convenience of the reader.

Clearly it is enough to show ♣(E) ⇒ ♣^{†}(E). Suppose that (x_{γ})_{γ∈E} is a ♣(E)- sequence. We claim that (x_{γ})_{γ∈E} is then also a ♣^{†}(E)-sequence. Otherwise there would be a Y ∈ [ω_{1}]^{ℵ_{1}} and a club C ⊆ Lim(ω_{1}) such that x_{γ} ⊈ Y for every γ ∈ C ∩ E. By thinning out C if necessary, we may assume that Y ∩ α is cofinal in α for each α ∈ C. For α ∈ C, denoting by α^{+} the next element to α in C, let y_{α} ⊆ [α, α^{+})∩Y be a cofinal subset in α^{+} with otp(y_{α}) = ω. Now let Y′ = ⋃_{α∈C} y_{α}. Then Y′ ∈ [ω_{1}]^{ℵ_{1}} and Y′ ⊆ Y. We show that { γ ∈ E : x_{γ} ⊆ Y′ } = ∅ which is a contradiction: if γ ∈ E ∩ C then x_{γ} ⊈ Y′ follows from Y′ ⊆ Y. If γ ∈ E \ C then there is α ∈ C such that α < γ < α^{+}. By the choice of y_{α}, Y′ ∩ γ is not cofinal in γ. Hence again x_{γ} ⊈ Y′. (Fact 1.4)

Now, let us consider the following variants of the (♣)-principle:

(♣_{w}): There exists a sequence (x_{γ})_{γ∈Lim(ω_{1})} of countable subsets of ω_{1} such that for every γ ∈ Lim(ω_{1}), x_{γ} is cofinal subset of γ, otp(x_{γ}) = ω and for every y ∈ [ω_{1}]^{ℵ_{1}}, there is γ < ω_{1} such that x_{γ} \ y is finite.

(♣_{w^{2}}): There exists a sequence (x_{γ})_{γ∈Lim(ω_{1})} of countable subsets of ω_{1} such that for every γ ∈ Lim(ω_{1}), x_{γ} is cofinal subset of γ, otp(x_{γ}) = ω and for every y ∈ [ω_{1}]^{ℵ_{1}} { α < ω_{1} : x_{α} ∩ y is finite } ∪ { α < ω_{1} : x_{α} \ y is finite } is stationary in ω_{1}.

Clearly (♣) implies (♣_{w}). Similarly to Fact 1.4, we can prove the equivalence of (♣_{w}) with (♣_{w}^{†}) which is obtained from (♣_{w}) by replacing “there is an α < ω_{1} … ” with “there are stationary may α < ω_{1} … ”. Hence (♣_{w}) implies (♣_{w^{2}}). It is also easy to see that (♣_{w}) implies ( |•): if (x_{γ})_{γ∈Lim(ω_{1})} is a sequence as in the definition of (♣_{w}), then { x_{γ} \ u : γ ∈ Lim(ω_{1}), u ∈ [ω_{1}]^{<ℵ_{0}} } is a |• -set of cardinality ℵ_{1}. Džamonja and Shelah [3] gave a model of ¬CH + (♣_{w}) + ¬(♣). By the remark above this model also shows the consistency of non-equivalence of ( |•) and (♣) under ¬CH. In this paper we prove that (♣_{w^{2}}) is strictly weaker than (♣_{w}) by showing the consistency of ¬( |•) + (♣_{w^{2}}) (Corollary 3.12). The partial ordering used in Corollary 3.12 does not force MA(countable) hence the following problem remains open:

Problem 1.5 Is MA(countable) + ¬ ( |•) + (♣_{w^{2}}) consistent?

2 Pseudo product of partial orderings

In this section, we introduce a new kind of side-by-side product of p.o.’s which will be used in the next section to prove various consistency results. Let X be any set and (P_{i})_{i∈X} be a family of partial orderings. For p ∈ Π_{i∈X}P_{i} the support of p is defined by supp(p) = { i ∈ X : p(i) ≠ 1_{P_{i}} }. For a cardinal κ, let Π_{κ,i∈X}^{∗}P_{i} be the set

{ p ∈ Π_{i∈X}P_{i} : | supp(p) | < κ }

with the partial ordering

p ≤ q ⇔ p(i) ≤ q(i) for all i ∈ X and
\{ i ∈ X : p(i) {< \atop ≠} q(i) {< \atop ≠} 1_{P_{i}} \} is finite .

For κ = ℵ_{0} this is just a finite support product. We are mainly interested in the case where κ = ℵ_{1}. In this case we shall drop the subscript ℵ_{1} and write simply Π_{i∈X}^{∗}P_{i}. Further, if P_{i} = P for some partial ordering P for every x ∈ X, we shall write Π_{κ,X}^{∗}P (or even Π_{X}^{∗}P when κ = ℵ_{1}) to denote this partial ordering.

For p, q ∈ Π_{κ,i∈X}^{∗}P_{i} the relation p ≤ q can be represented as a combination of the two other distinct relations which we shall call horizontal and vertical, and denote by ≤_{h} and ≤_{v} respectively:

p ≤_{h} q ⇔ supp(p) ⊇ supp(q) and p|` supp(q) ⊆ q;
p ≤_{v} q ⇔ supp(p) = supp(q), p(i) ≤ q(i) for all i ∈ X and
\{ i ∈ X : p(i) {< \atop ≠} q(i) {< \atop ≠} 1_{P_{i}} \} is finite .

For p ∈ Π_{κ,i∈X}^{∗}P_{i} and Y ⊆ X let p⌈Y denote the element of Π_{κ,i∈X}^{∗}P_{i} defined by p⌈Y (i) = 1_{P_{i}} for every i ∈ X \ Y and p⌈Y (i) = p(i) for i ∈ Y.

The following is immediate from definition:

Lemma 2.1 For p, q ∈ Π_{κ,i∈X}^{∗}P_{i}, the following are equivalent:

a) p ≤ q;

b) There is an r ∈ Π_{κ,i∈X}^{∗}P_{i} such that p ≤_{h} r ≤_{v} q;

c) There is an s ∈ Π_{κ,i∈X}^{∗}P_{i} such that p ≤_{v} s ≤_{h} q.

Proof b) ⇒ a) and c) ⇒ a) are clear. For a) ⇒ b), let r = p⌈supp q; for a) ⇒ c), s = q|` supp(q) ∪ p|` (X \ supp(q)). (Lemma 2.1)

Lemma 2.2

1) If P_{i} has the property K for all i ∈ X then P = Π_{i∈X}^{∗}P_{i} preserves ℵ_{1}.

2) Suppose that λ ≤ κ. If P_{i} has the strong λ-cc (i.e. for every C ∈ [P_{i}]^{λ} there is pairwise compatible D ∈ [C]^{λ}), then P = Π_{κ,i∈X}^{∗}P_{i} preserves λ.

Proof This proof is a prototype of the arguments we are going to apply repeatedly.

1) and 2) can be proved similarly. For 1), assume that there would be p ∈ P and a P -name ḟ such that

*) p ∥–_{P} “ ḟ : (ω_{1})^{V} → ω and ḟ is 1-1 ”.

Then, let (p_{α})_{α<ω_{1}} and (q_{α})_{α<ω_{1}} be sequences of elements of P such that

a) p_{0} ≤ p and (p_{α})_{α<ω_{1}} is a descending sequence with respect to ≤_{h};

b) q_{α} ≤_{v} p_{α} and q_{α} decides ḟ(α) for all α < ω_{1};

c) p_{α}|` S_{α} = q_{α}|` S_{α} for every α < ω_{1} where

S_{α} = supp(q_{α}) \ (supp(p) ∪ ⋃_{β<α} supp(q_{β})).

For α < ω_{1} let d_{α} = ⋃_{β<α} supp(q_{β}). Then (d_{α})_{α<ω_{1}} is a continuously increasing sequence in [X]^{<ω_{1}}. Let u_{α} = { β ∈ supp(q_{α}) : q_{α}(β) ≠ p_{α}(β) } for α < ω_{1}. By b), u_{α} is finite and by c) we have u_{α} ⊆ d_{α}. Hence by Fodor’s lemma, there exists an uncountable (actually even stationary) Y ⊆ ω_{1} such that u_{α} = u^{∗} for all α ∈ Y, for some fixed u^{∗} ∈ [X]^{<ℵ_{0}}. Since Π_{i∈u^{∗}}P_{i} has the property K, there exists an uncountable Y′ ⊆ Y such that { q_{α}|` u^{∗} : α ∈ Y′ } is pairwise compatible. It follows that q_{α}, α ∈ Y′ are pairwise compatible. For each α ∈ Y′ there exists an n_{α} ∈ ω such that q_{α} ∥–_{P} “ n_{α} = ḟ(α) ” by b). By *), n_{α}, α ∈ Y′ must be pairwise distinct. But this is impossible as Y′ is uncountable.

For 2), essentially the same proof works with sequences of elements of P of length λ, using the Δ-system lemma argument in place of Fodor’s lemma.

(Lemma 2.2)

Lemma 2.3 If | P_{i} | ≤ 2^{<κ} for all i ∈ X, then Π_{κ,i∈X}^{∗}P_{i} has the (2^{<κ})^{+}-cc.

Proof By the usual Δ-system lemma argument. (Lemma 2.3)

Corollary 2.4 a) Under CH, if P_{i} satisfies the property K and | P_{i} | ≤ ℵ_{1} for every i ∈ X, then P = Π_{i∈X}^{∗}P_{i} preserves ℵ_{1} and has the ℵ_{2}-cc. In particular P preserves every cardinals.

b) Suppose that 2^{<κ} = κ. If P_{i} satisfies the strong λ-cc for every ℵ_{1} ≤ λ ≤ κ and | P_{i} | ≤ κ then Π_{κ,i∈X}^{∗}P_{i} preserves every cardinalities ≤ κ and has the κ^{+}-cc. In particular Π_{κ,i∈X}^{∗}P_{i} preserves every cardinals.

Proof By Lemmas 2.2, 2.3. (Corollary 2.4)

Lemma 2.5 For any Y ⊆ X and x ∈ X \ Y, we have

Π_{κ,i∈X}^{∗}P_{i} ≅ Π_{κ,i∈Y}^{∗} P_{i} × P_{x} × Π_{κ,i∈X\(Y ∪{x})}^{∗}P_{i}.

Proof The mapping from Π_{κ,i∈X}^{∗}P_{i} to Π_{κ,i∈Y}^{∗} P_{i} × P_{x} × Π_{κ,i∈X\(Y ∪{x})}^{∗}P_{i} defined by

p ↦ (p|` Y, p(x), p|` (X \ (Y ∪ {x})))

is an isomorphism. (Lemma 2.5)

In the following we mainly use the partial orderings of the form Fn(λ, 2) for some λ as P_{i} in Π_{κ,i∈X}^{∗}P_{i}. Note that Fn(λ, 2) has the property K and strong κ-cc in the sense above for every regular κ.

For a pseudo product of the form Π_{i∈X}^{∗}Fn(κ_{i}, 2), Lemma 2.2 can be still improved:

Theorem 2.6 (T. Miyamoto) For any set X, and sequence (κ_{i})_{i∈X}, the partial ordering P = Π_{i∈X}^{∗}Fn(κ_{i}, 2) satisfies the Axiom A.

Proof The sequence of partial orderings (≤_{n})_{n∈ω} defined by: p ≤_{0} q ⇔ p ≤ q and p ≤_{n} q ⇔ p ≤_{h} q for every n > 0 witnesses the Axiom A of P. We omit here the details of the proof since this assertion is never used in the following. The idea of the proof needed here is to be found in the proof of Lemmas 2.7 and 5.2.

(Theorem 2.6)

Lemma 2.7 Suppose that | P_{i} | ≤ κ for every i ∈ X and P = Π_{κ^{+},i∈X}^{∗}P_{i}. Then 1) If ẋ is a P -name with ∥–_{P} “ ẋ ∈ V ”, then for any p ∈ P there is q ∈ P such that q ≤_{h} p and

(†) for any r ≤ q, if r decides ẋ then r⌈supp(q) already decides ẋ.

2) Let G be P -generic. If u ∈ V [G] is a subset of V of cardinality < κ^{+}, then there is a ground model set X′ ⊆ X of cardinality ≤ κ (in the sense of V) such that u ∈ V [G ∩ (Π_{κ^{+},i∈X′}^{∗}P_{i})].

Proof 1): Let Φ : κ → κ × κ; α ↦ (φ_{1}(α), φ_{2}(α)) be a surjection such that φ_{1}(α) ≤ α for every α < κ. Let (p_{α})_{α<κ}, (p′_{α})_{α<κ} and (r_{α,β})_{α<κ,β<κ} be sequences of elements of P defined inductively by:

a) p_{0} = p; (p_{α})_{α<κ} is a descending sequence with respect to ≤_{h};

b) for a limit γ < κ, p_{γ} is such that supp(p_{γ}) = ⋃_{α<γ} supp(p_{α}) and, for i ∈ supp(p_{γ}), p_{γ}(i) = p_{α}(i) for some α < γ such that i ∈ supp(p_{α});

c) (r_{α,β})_{β<κ} is an enumeration of { r ∈ P : r ≤_{v} p_{α} };

d) let r = r_{φ_{1}(α),φ_{2}(α)} and

p′_{α} = r|` supp(r) ∪ p_{α}|` (X \ supp(r)).

If there is s ≤_{h} p′_{α} such that s decides ẋ, then let

p_{α+1} = p_{α}|` supp(p_{α}) ∪ s|` (X \ supp(p_{α})).

Otherwise let p_{α+1} = p_{α}.

Let q ∈ Π_{κ,i∈X}^{∗}P_{i} be defined by supp(q) = ⋃_{α<κ} supp(p_{α}) and, for i ∈ supp(q), q(i) = p_{α}(i) for some α < κ such that i ∈ supp(P_{α}). We show that this q is as desired: suppose that r ≤ q decides ẋ. Then there is some α < κ such that

r⌈supp(q) = p′_{α}|` supp(p′_{α}) ∪ q|` (X \ supp(p′_{α})).

By d), it follows that r⌈supp(q) ≤ r⌈supp(p_{α+1}) decides ẋ.

2): Let u̇ be a P -name for u and let ẋ_{α}, α < κ be P -names such that ∥–_{P} “ ẋ_{α} ∈ V ” for every α < κ and ∥–_{P} “ u = { ẋ_{α} : α < κ } ”. By 1), for each p ∈ P, we can build a sequence (p_{α})_{α<κ} of elements of P decreasing with respect to ≤_{h} such that p_{0} ≤_{h} p and

(†)_{α} for any r ≤ p_{α}, if r decides ẋ_{α}, then r⌈supp(p_{α}) already decides ẋ_{α}.

Let q ∈ P be defined by supp(q) = ⋃_{α<κ} supp(p_{α}) and, for i ∈ supp(q), q(i) = p_{α}(i) for some α < κ such that i ∈ supp(p_{α}). Then q satisfies:

(††) for any r ≤ q, if r decides ẋ_{α} for some α < κ, then r⌈supp(q) already decides ẋ_{α}.

The argument above shows that q’s with the property (††) are dense in P. Hence, by genericity, there is such q ∈ G. Clearly, G∩Π_{κ,i∈supp(q)}^{∗}P_{i} contains every information needed to construct u. (Lemma 2.7)

3 Consistency results

Proposition 3.1 (CH) For any infinite cardinal λ, let P = Π_{λ}^{∗}Fn(ω_{1}, 2). Then ∥–_{P} “ |• = λ ”.

Proof

Claim 3.1.1 ∥–_{P} “ |• ≥ λ ”.

⊢ If λ = ℵ_{1} this is clear. So assume that λ ≥ ℵ_{2}. For ξ < λ, let ḟ_{ξ} be the P -name of the generic function from ω_{1} to 2 added by the ξ-th copy of Fn(ω_{1}, 2) in P. Let G be a P -generic filter over V. In V [G] let X ⊆ [ω_{1}]^{ℵ_{0}} be such that | X | < λ. Then by ℵ_{2}-cc of P there exists ξ < λ such that X ∈ V [G′] for G′ = G ∩ Π_{λ\{ξ}}^{∗}Fn(ω_{1}, 2). Since (ḟ_{ξ})[G] is Fn(ω_{1}, 2)-generic over V [G′] by Lemma 2.5, we have x ⊈ ((ḟ_{ξ})[G])^{−1}{0} for every x ∈ X. ⊣ (Claim 3.1.1)

Claim 3.1.2 ∥–_{P} “ |• ≤ λ ”.

⊢ For u ∈ [λ]^{<ℵ_{0}}, let Ṗ_{u} be a P -name such that

∥–_{P} “ Ṗ_{u} = ([ω_{1}]^{ℵ_{0}})^{V [(ḟ_{ξ})_{ξ∈u}]} ”

where ḟ_{ξ} is as in the proof of the previous claim. Let Ṗ be a P -name such that

∥–_{P} “ Ṗ = ⋃{ Ṗ_{u} : u ∈ [λ]^{<ℵ_{0}} } ”.

For each u ∈ [λ]^{<ℵ_{0}}, (ḟ_{ξ}[G])_{ξ∈u} corresponds to a generic filter over Π_{u}^{∗}Fn(ω_{1}, 2) ≈ Fn(ω_{1}, 2). Hence, by CH, we have ∥–_{P} “ | Ṗ_{u} | = ℵ_{1} ”. It follows that ∥–_{P} “ | Ṗ | = λ ”. Thus it is enough to show that ∥–_{P} “ Ṗ is a |• -set ”.

Let p ∈ P and Ȧ be a P -name such that p ∥–_{P} “ Ȧ ∈ [ω_{1}]^{ℵ_{1}} ”. We show that there is an r ≤ p such that r ∥–_{P} “ ∃x ∈ Ṗ x ⊆ Ȧ ”.

Now we proceed as in the proof of Lemma 2.2. Let (p_{α})_{α<ω_{1}}, (q_{α})_{α<ω_{1}} be sequences of elements of P and (ξ_{α})_{α<ω_{1}} be a strictly increasing sequence of ordinals < ω_{1} such that

a) p_{0} ≤ p and (p_{α})_{α<ω_{1}} is a descending sequence with respect to ≤_{h};

b) q_{α} ≤_{v} p_{α} and q_{α} ∥–_{P} “ ξ_{α} ∈ Ȧ ” for all α < ω_{1};

c) p_{α}|` S_{α} = q_{α}|` S_{α} for every α < ω_{1} where

S_{α} = supp(q_{α}) \ (supp(p) ∪ ⋃_{β<α} supp(q_{β})).

For α < ω_{1} let u_{α} = { β ∈ supp(q_{α}) : q_{α}(β) ≠ p_{α}(β) }. As in the proof of Lemma 2.2, there exists u^{∗} ∈ [λ]^{<ℵ_{0}} such that S = { α ∈ ω_{1} : u_{α} = u^{∗} } is stationary. Now (q_{α}|` u)_{α∈S} is an infinite sequence of elements of P_{u^{∗}} = Π_{u^{∗}}Fn(ω_{1}, 2). Since P_{u^{∗}} satisfies the ccc, there exists an ε ∈ S and ζ < ω_{1} such that q_{ε}|` u^{∗} ∥–_{P_{u^{∗}}} “ { ξ ∈ S ∩ ζ : p_{ξ}|` u^{∗} ∈ Ġ } is infinite ”. Let r = q_{ε} ∪ p_{ζ} |` (supp(p_{ζ}) \ supp(p_{ε})). Let ḃ be a P -name such that

r ∥–_{P} “ ḃ = { ξ ∈ S ∩ ζ : q_{ξ}|` u^{∗} ∈ { p|` u^{∗} : p ∈ Ġ } } ”.

Let ẋ be a P -name such that r ∥–_{P} “ ẋ = { ξ_{α} : α ∈ ḃ } ”. Then r ∥–_{P} “ | ẋ | = ℵ_{0} ”. Since ḃ can be computed in V [(ḟ_{ξ}[G])_{ξ∈u^{∗}}] we have r ∥–_{P} “ ẋ ∈ Ṗ_{u^{∗}} ”. It is also clear by definition of ẋ that r ∥–_{P} “ ẋ ⊆ Ȧ ”. ⊣ (Claim 3.1.2)

(Proposition 3.1)

Proposition 3.1 shows that |• can be practically every thing. In particular we obtain:

Corollary 3.2 The assertion ‘cof( |•) = ω’ is consistent with ZFC.

Actually, Fn(λ, 2) forces almost the same situation:

Lemma 3.3 Suppose that λ is a cardinal such that μ^{ℵ_{0}} ≤ λ for every μ < λ. Then, for P = Fn(λ, 2), we have ∥–_{P} “ |• = λ ”.

Proof ∥–_{P} “ |• ≥ λ ” can be proved similarly to Claim 3.1.1. For ∥–_{P} “ |• ≤ λ ”, let G be a P -generic filter and let G_{α} = G ∩ Fn(α, 2) for α < λ. In V [G], let X = ⋃{ V [G_{α}] ∩ [ω_{1}]^{ℵ_{0}} : α < λ }. Then | X | = λ (here we need SCH in general). We show that X is a |• -set. For this, it is enough to show the following:

Claim 3.3.1 In V [G], if y ⊆ [ω_{1}]^{ℵ_{1}}, then there is α^{∗} < λ and infinite y′ ∈ V [G_{α^{∗}}] such that y′ ⊆ y.

⊢ In V, let ẏ be a P -name of y which is nice in the sense of [8]. For α < λ, let ẏ_{α} = ẏ ∩ { β̌ : β < ω_{1} } × Fn(α, 2). Then ∥–_{P} “ ẏ = ⋃_{α<λ} ẏ_{α} ”. Hence ∥–_{P} “ ∃α < λ ẏ_{α} is infinite ”. It follows that there is some α^{∗} < λ such that y′ = ẏ_{α^{∗}}[G] is infinite. Since ẏ_{α^{∗}} is an Fn(α^{∗}, 2)-name, ẏ_{α^{∗}}[G] ∈ V [G_{α^{∗}}]. Thus these α^{∗} and y′ are as desired. ⊣ (Claim 3.3.1)

(Lemma 3.3)

Proposition 3.4 (CH) Suppose that

(∗)_{λ,μ} There is a sequence (A_{i})_{i<μ} of elements of [λ]^{ℵ_{1}} such that | A_{i} ∩ A_{j} | < ℵ_{0} for every i, j < μ, i ≠ j

holds for some μ > λ ≥ 2^{ℵ_{0}}. Then there exists a partial ordering P such that

a) P preserves ℵ_{1} and and has the ℵ_{2}-cc;

b) ∥–_{P} “ |• = λ ” and

c) ∥–_{P} “ |•_{λ} ≥ μ ”. In particular, if (∗)_{λ,μ} is consistent with ZFC for some μ > λ ≥ 2^{ℵ_{0}}, then so is |• < |•′.

Remark. By [12, §6], (∗∗)_{μ} and (∗)_{λ,μ} for some λ < μ are equivalent, where

(∗∗)_{μ} there are finite a_{i} ⊆ Reg \ ℵ_{2} for i < ω_{1} such that, for any A ∈ [ω_{1}]^{ℵ_{0}}, max pcf(∪_{i∈A}a_{i}) ≥ μ.

For more see [15]. Proof Let P be as in Proposition 3.1. We claim that P is as desired: a) follows from Corollary 2.4 and b) from Proposition 3.1. For d), if X ⊆ [λ]^{ℵ_{0}} is a |•_{λ}-set then for each i < μ there is an x_{i} ∈ X such that x_{i} ⊆ A_{i}. Since A_{i}, i < μ are almost disjoint x_{i}, i < μ must be pairwise distinct.

The last assertion follows from Lemma 1.1, d). (Proposition 3.4) Now we show the consistency of the inequality |•′′ < |•:

Proposition 3.5 Assume 2^{ℵ_{1}} = ℵ_{2}. Then for any cardinal λ ≥ ℵ_{2} there exists a partial ordering P such that

a) P satisfies the ℵ_{3}-cc;

b) P preserves ℵ_{1} and ℵ_{2};

c) if λ^{ℵ_{0}} = λ in addition, then ∥–_{P} “ MA(Fn(ω_{1}, 2)) ”;

d) ∥–_{P} “ |• = λ ” and

e) ∥–_{P} “ |•′′ = ℵ_{2} ”.

Proof Without loss of generality let λ be regular and let P = Π_{ℵ_{2},λ}^{∗}Fn(ω_{1}, 2). Then a) and b) follow from Corollary 2.4. For c), note that ∥–_{P} “ 2^{ℵ_{0}} ” = λ under λ^{ℵ_{0}} = λ. Hence, by Lemma 2.7 and Lemma 2.5, we see easily that ∥–_{P} “ MA(Fn(ω_{1}, 2)) ”. An argument similar to the proof of of Proposition 3.1 shows that ∥–_{P} “ |• = λ ”. For e), we prove first the following:

Claim 3.5.1 Let X = [ℵ_{2}]^{ℵ_{0}}. Then we have ∥–_{P} “ X is a |•′′-set ”. In particular ∥–_{P} “ |•′′ ≤ ℵ_{2} ”.

⊢ Suppose that, for some p ∈ P and a P -name ẏ we have p ∥–_{P} “ ẏ ∈ [ω_{2}]^{ℵ_{2}} ”. Let ḟ be a P name such that p ∥–_{P} “ ḟ : ω_{2} → ẏ and ḟ is 1-1 ”. Let (p_{α})_{α<ω_{2}} and (q_{α})_{α<ω_{2}} be sequences of elements of P such that

f) p_{0} ≤ p and (p_{α})_{α<ω_{2}} is a descending sequence with respect to ≤_{h};

g) q_{α} ≤_{v} p_{α} and q_{α} decides ḟ(α) for all α < ω_{2};

h) p_{α}|` S_{α} = q_{α}|` S_{α} for every α < ω_{2} where

S_{α} = supp(q_{α}) \ (supp(p) ∪ ⋃_{β<α} supp(q_{β})).

For α < ω_{2}, let ξ_{α} ∈ ω_{2} be such that q_{α} ∥–_{P} “ ḟ(α) = ξ_{α} ”. Let u_{α} = { β ∈ supp(q_{α}) : q_{α}(β) ≠ p_{α}(β) } for α < ω_{2}. Just like in the proof of Lemma 2.4, we can find u^{∗} ∈ [λ]^{<ℵ_{0}} such that S = { α < ω_{2} : u_{α} = u^{∗} } is stationary in ω_{2}. Since | Fn(ω_{1}, 2) | = ℵ_{1}, there exists T ⊆ S of cardinality ℵ_{2} such that q_{α}|` u^{∗}, α ∈ T are all the same. Let α_{n}, n ∈ ω be ω elements of T and let q = ⋃_{n∈ω} q_{α_{n}} . Then q ≤ p and q ∥–_{P} “ { ξ_{α_{n}} : n ∈ ω } ⊆ ẏ ”. ⊣ (Claim 3.5.1)

Now by d), we have ∥–_{P} “ |•′′ > ℵ_{1} ”. Hence, by the claim above, it follows that ∥–_{P} “ |•′′ = ℵ_{2} ”. (Proposition 3.5)

Modifying the proofs of Propositions 3.1 and 3.5 slightly, we can also blow up the continuum while setting |• strictly between ℵ_{1} and 2^{ℵ_{0}}. For example:

Proposition 3.6 Assume CH and 2^{ℵ_{1}} = ℵ_{2}. Then for any cardinals λ, μ such that ℵ_{2} ≤ λ ≤ μ and μ^{ℵ_{1}} = μ, there exists a partial ordering P such that a) P satisfies the ℵ_{3}-cc;

b) P preserves ℵ_{1} and ℵ_{2};

c) ∥–_{P} “ MA(countable) ”;

d) ∥–_{P} “ |• = λ ”;

e) ∥–_{P} “ |•′′ = ℵ_{2} ” and

f) ∥–_{P} “ 2^{ℵ_{0}} = μ ”.

Proof For i < μ let

P_{i} = \{ \begin{matrix}Fn(ω_{1}, 2), & if i < λ, \\ Fn(ω, 2), & otherwise. \end{matrix}

Then P = Π_{ℵ_{2},i<μ}^{∗}P_{i} is as desired. e) can be proved by almost the same proof as that of Claim 3.5.1. a), b), c) can be shown just as in Proposition 3.5. Since P adds (at least) μ many Cohen reals over V and | P | = μ, f) follows from a). d) is proved similarly to Claims 3.1.1 and 3.1.2. For ∥–_{P} “ |• ≤ λ ” we need the following modification of Claim 3.1.2: let Ṗ be defined as in the proof of Claim 3.1.2. As there, we can show easily that ∥–_{P} “ | Ṗ | = λ ”. To show that ∥–_{P} “ Ṗ is a |• -set ”, let p ∈ P and Ȧ be a P -name such that p ∥–_{P} “ Ȧ ∈ [ω_{1}]^{ℵ_{1}} ”. Now let (p_{α})_{α<ω_{1}}, (q_{α})_{α<ω_{1}}, (ξ_{α})_{α<ω_{1}}, u^{∗} ∈ [μ]^{<ℵ_{0}} and S be just as in the proof of Claim 3.1.2. Let v^{∗} = u^{∗} \ λ. Since P_{v^{∗}} = Π_{i∈v^{∗}}P_{i} is countable, we may assume without loss of generality that q_{α}|` v^{∗}, α ∈ S are all the same. Now we can proceed just like in the proof of Claim 3.1.2 with u^{∗} replaced by u^{∗} \ v^{∗}.

The following Lemmas 3.7 and 3.9 show that, in spite of typographical similarity, Π_{λ}^{∗}Fn(ω_{1}, 2) and Π_{λ}^{∗}Fn(ω, 2) are quite different forcing notions: while the first one destroys (♣) or even ( |•) by Lemma 3.1, the second one not only preserves a (♣)-sequence in the ground model but also creates such a sequence generically.

Lemma 3.7 Let S = (x_{γ})_{γ∈E} be a ♣(E)-sequence for a stationary E ⊆ Lim(ω_{1}). Let P = Π_{κ}^{∗}Fn(ω, 2) for arbitrary κ. Then we have ∥–_{P} “ S is a ♣(E)-sequence ”.

Proof Let p ∈ P and Ȧ be a P -name such that p ∥–_{P} “ Ȧ ∈ [ω_{1}]^{ℵ_{1}} ”. We show that there is q ≤ p and γ ∈ E such that q ∥–_{P} “ x_{γ} ⊆ Ȧ ”. Let ḟ be a P -name such that p ∥–_{P} “ ḟ : ω_{1} → Ȧ and ḟ is 1-1 ”. Let (p_{α})_{α<ω_{1}} and (q_{α})_{α<ω_{1}} be sequence of elements of P satisfying the conditions a) – c) in the proof of Lemma 2.2. Also, let u_{α}, α < ω_{1} be as in the proof of Lemma 2.2. As there, we can find an uncountable Y ⊆ ω_{1} and u^{∗} ∈ [κ]^{<ℵ_{0}} such that u_{α} = u^{∗} for all α ∈ Y. Since Π_{u^{∗}}Fn(ω, 2) is countable we may assume that q_{α}|` u^{∗} are all the same for α ∈ Y. Now for each α ∈ Y let β_{α} be such that q_{α} ∥–_{P} “ ḟ(α) = β_{α} ” and let Z = { β_{α} : α ∈ Y }. Since q_{α}, α ∈ Y are pairwise compatible, β_{α}, α ∈ Y are pairwise distinct and so Z is uncountable. Note that Z is a ground model set. Hence there exists γ ∈ E such that x_{γ} ⊆ Z. Let q = ⋃_{α∈Y ∩γ} q_{α}. Then q ≤ p. Since sup{ β_{α} : α < γ } ≥ γ and ∥–_{P} “ { β_{α} : α < γ } is an initial segment of Z ”, we have q ∥–_{P} “ Z ∩γ ⊆ Ȧ ”. Hence q ∥–_{P} “ x_{γ} ⊆ Ȧ ”. (Lemma 3.7)

Theorem 3.8 “¬CH+ MA(countable) + there exists a constructible ♣ -sequence” is consistent.

Proof We can obtain a model of the statement by starting from a model of V = L and force with P = Π_{κ}^{∗}Fn(ω, 2) for a regular κ. By Corollary 2.4, every cardinal of V is preserved in V [G]. Since P adds κ many Cohen reals over V while | P | = κ and P has the ℵ_{2}-cc, we have V [G] ⊧ “ 2^{ℵ_{0}} = κ ”. By Lemma 2.5, V [G] ⊧ “ MA(countable) ”. By Lemma 3.7, the ◇-sequence in V remains a ♣ -sequence in V [G]. (Theorem 3.8)

In fact, we do not need a ♣-sequence in the ground model to get (♣) in the generic extension by Π_{κ}^{∗}Fn(ω, 2) :

Lemma 3.9 Let κ be uncountable and P = Π_{κ}^{∗}Fn(ω, 2). Then for any stationary E ⊆ Lim(ω_{1}) we have ∥–_{P} “ ♣(E) holds ”.

Proof For γ ∈ E let

f_{γ} : [γ, γ + ω) → γ

be a bijection and let

S_{γ} = { x ⊆ γ : x is a cofinal subset of γ, otp(x) = ω }.

For each x ∈ S_{γ} let p_{x} ∈ P be defined by

p_{x} = { (γ + n, { (0, i) }) : n ∈ ω, i ∈ 2, i = 1 ⇔ f_{γ}(γ + n) ∈ x }.

For distinct x, x′ ∈ S_{γ}, p_{x} and p_{x′} are incompatible. Hence, for each γ ∈ E, we can find a P -name ẋ_{γ} such that

∥–_{P} “ ẋ_{γ} is a cofinal subset of γ and otp(ẋ_{γ}) = ω ”

and

p_{x} ∥–_{P} “ ẋ_{γ} = x ” for each x ∈ S_{γ}.

We show that ∥–_{P} “ (ẋ_{γ})_{γ∈E} is a ♣(E)-sequence ”. For this, it is enough to show that, for any p ∈ P and a P -name Ȧ, if p ∥–_{P} “ Ȧ ∈ [ω_{1}]^{ℵ_{1}} ”, then there is q ≤ p and γ ∈ E such that q ∥–_{P} “ ẋ_{γ} ⊆ Ȧ ”. Let ḟ be such that

p ∥–_{P} “ ḟ : ω_{1} → Ȧ and ḟ is 1-1 ”.

Now let (p_{α})_{α<ω_{1}}, (q_{α})_{α<ω_{1}}, (u_{α})_{α∈ω_{1}}, Y and u^{∗} be as in the proof of Lemma 2.2. For each α ∈ Y let β_{α} be such that q_{α} ∥–_{P} “ ḟ(α) = β_{α} ” and let Z = { β_{α} : α ∈ Y }. Let

C = { γ ∈ Lim(ω_{1}) : ⋃_{α∈Y ∩γ}(supp(q_{α}) ∩ ω_{1}) ⊆ γ

and Z ∩ γ is unbounded in γ }.

Then C is closed unbounded in ω_{1} and hence there exists a γ^{∗} ∈ C ∩ E. Let q′ = ⋃_{α∈Y ∩γ^{∗}} q_{α}. Then we have q′ ≤ q and q′ ∥–_{P} “ Z ∩ γ^{∗} ⊆ Ȧ ”. Now let x ∈ S_{γ^{∗}} be such that x ⊆ Z ∩ γ^{∗}. Finally let q = q′ ∪ q_{x}. Then we have q ≤ p and q ∥–_{P} “ ẋ_{α} = x ⊆ Z ∩ γ^{∗} ⊆ Ȧ ”. (Lemma 3.9)

Note that E’s in Lemmas 3.7 and 3.9 are ground model sets. To force ♣(E) for every stationary E ⊆ Lim(ω_{1}) which may be also added generically, we need a sort of iteration described in the next section.

Toward the consistency of ¬(♣_{w}) + (♣_{w^{2}}), we consider first the following lemma which should be a well-known fact. Nevertheless, we include here a proof:

Lemma 3.10 Assume that there is a sequence (C_{β})_{β<κ} of elements of [ω_{1}]^{ℵ_{1}} such that | C_{β} ∩ C_{γ} | ≤ ℵ_{0} for all β < γ < κ. Then there exists a partial ordering P with the property K such that in V^{P} there is a sequence (B_{β})_{β<κ} of elements of [ω_{1}]^{ℵ_{1}} such that B_{β} ⊆ C_{β} and | B_{β} ∩ B_{γ} | < ℵ_{0} for all β < γ < κ.

Proof Let

P = { (D, f) : D ∈ [κ]^{<ℵ_{0}}, f : D → Fn(ω_{1}, 2),
f(δ) ∈ Fn(C_{δ}, 2) for all δ ∈ D }.

For (D, f), (D′, f′) ∈ P, let

(D′, f′) ≤ (D, f) ⇔ D ⊆ D′, f(δ) ⊆ f′(δ) for all δ ∈ D and

(f′(δ))^{−1}[{1}] \ (f(δ))^{−1}[{1}], δ ∈ D are pairwise disjoint.

By the usual Δ-system lemma argument, we can show that P has the property K. Since C_{β}, β < κ are pairwise disjoint modulo countable, the set

D_{β,δ} = { (D, f) ∈ P : β ∈ D, δ ∈ dom(f(β)) and
∃η > δ (η ∈ dom(f(β)) ∧ f(β)(η) = 1) }

is dense in P for every β < κ and δ < ω_{1}. Hence if G is a V -generic filter over P, then

B_{β} = { α < ω_{1} : f(β)(α) = 1 for some (D, f) ∈ G }

is cofinal in ω_{1} and hence uncountable. Also by the definition of ≤ on P, we have | B_{β} ∩ B_{γ} | < ℵ_{0} for every β < γ < κ. (Lemma 3.10) Note that if there is a sequence (B_{β})_{β<κ} as in Lemma 3.10 then by the argument in the proof of Proposition 3.4, we have |• ≥ κ. Lemma 3.11 There is a partial ordering Q with the property K such that ∥–_{Q} “ (♣_{w^{2}}) ”. Proof Let (Q_{α}, Ṙ_{α})_{α≤ω_{1}} be the finite support iteration of partial orderings with the property K such that for each γ ∈ Lim(ω_{1}), there is a Q_{γ} name U̇_{γ} such that Q_{γ} forces:

U̇_{γ} is an ultrafilter over γ, γ \ β ∈ U̇_{γ} for all β < γ, Ṙ_{α} is a p.o.-set with the property K and there is an Ṙ_{γ}-name ẋ_{γ} such that

∥–_{Ṙ_{γ}} “ ẋ_{γ} is a cofinal subset of γ of ordertype ω and | ẋ_{γ} \ a | < ℵ_{0} for all x ∈ U̇_{γ} ”.

For example, we can take the Mathias forcing for the ultrafilter U̇_{γ} as Ṙ_{γ}. For successor α < ω_{1} let ∥–_{Q_{α}} “ Ṙ_{α} = {1} ”.

Let Q = Q_{ω_{1}}. As (Q_{α}, Ṙ_{α})_{α≤ω_{1}} is a finite support iteration of property K p.o.s, Q satisfies also the property K (see e.g. [9]). Now let G be a V -generic filter over Q. In V [G], if X ∈ [ω_{1}]^{ℵ_{1}} then the set { α < ω_{1} : X ∩ α ∈ V [G_{α}] } contains a club subset C of Lim(ω_{1}). Let S_{0} = { α ∈ C : X ∩ α ∈ U̇_{α}[G] } and S_{1} = { α ∈ C : α \ X ∈ U̇_{α}[G] }. Since U̇_{α}[G] is an ultrafilter over α in V [G_{α}] for every α ∈ C, we have C = S_{0} ˙∪ S_{1}. We have | ẋ_{α}[G] \ X | < ℵ_{0} for α ∈ S_{0} and | ẋ_{α}[G] ∩ X | < ℵ_{0} for α ∈ S_{1}. Thus (ẋ_{α}[G])_{α∈Lim(ω_{1})} is a (♣_{w^{2}})-sequence in V [G].

Actually this proof shows that (ẋ_{α}[G])_{α∈Lim(ω_{1})} is even a (♣_{w^{2}})-sequence in the stronger sense that it satisfies the assertion of the definition of (♣_{w^{2}}) with “is stationary” replaced by “contains a club”. (Lemma 3.11) Corollary 3.12 There is a partial ordering R with property K such that ∥–_{R} “ |• ≥ ℵ_{2} but (♣_{w^{2}}) holds ”. In particular ¬ ( |•) + (♣_{w^{2}}) is consistent with ZFC. Further if CH holds then for any cardinal κ, there exists a cardinals preserving proper partial ordering R_{κ} such that ∥–_{R_{κ}} “ |• ≥ κ but (♣_{w^{2}}) holds ”. Proof Let R = P_{1} ∗ Ṗ_{2} where P_{1} is as P in Lemma 3.10 for κ = ℵ_{2} and Ṗ_{2} as Q in Lemma 3.11 in V^{P_{1}}.

For the second assertion, we let R_{κ} = Fn(κ, 2, ω_{1}) ∗ Ṗ_{1} ∗ Ṗ_{2}. Note that under CH, Fn(κ, 2, ω_{1}) is cardinals preserving and forces that 2^{ℵ_{1}} = κ. Hence there is a sequence (C_{β})_{β<κ} as in Lemma 3.10 in the generic extension. Thus in V^{Fn(κ,2,ω_{1})}, Ṗ_{1} can be taken as in Lemma 3.10 for our κ. Finally, in V^{Fn(κ,2,ω_{1})∗Ṗ_{1}} let Ṗ_{2} be as in Lemma 3.11. (Corollary 3.12)

4 CS^{∗}-iteration

In this section, we introduce an iterative construction of p.o.s which is closely related to the pseudo product we introduced in section 2. We adopt here the conventions of [5] on forcing. In particular, a p.o. (or forcing notion) P is a preordering with a greatest element 1_{P}. In the following, we just try to develop a minimal theory needed for Theorem 5.6. More general treatment of the iterations like the one described below should be found in [16].

We call a sequence of the form (P_{α}, Q̇_{α})_{α≤ε} a CS^{∗}-iteration if the following conditions hold for every α ≤ ε:

*0) P_{α} is a p.o. and, if α < ε, then Q̇_{α} is a P_{α} name such that ∥–_{P_{α}} “ Q̇_{α} is a

p.o. with a greatest element 1Q̇_{α} ”. *1)

P_{α} = { p : p is a function such that dom(p) ∈ [α]^{≤ℵ_{0}};
p|` β ∈ P_{β} for any β < α and,
if β ∈ dom(p) then ∥–_{P_{β}} “ p(β) ∈ Q̇_{β} ” }.

*2) For p, q ∈ P_{α}, p ≤_{P_{α}} q if and only if

i) for any β < α, p|` β ∥–_{P_{β}} “ p(β) ≤ q(β) ”;

ii) diff (p, q) = { β ∈ dom(p) ∩ dom(q) : p|` β ∥ –_{P_{β}} “ p(β) = q(β) ” } is

finite. We first show that such a sequence (P_{α}, Q̇_{α})_{α≤ε} is really an iteration in the

usual sense. In the following we assume always that (P_{α}, Q̇_{α})_{α≤ε} is a CS^{∗}-iteration as defined above.

Lemma 4.1 Suppose that α ≤ β ≤ ε. Then

0) if p ∈ P_{β}, then p|` α ∈ P_{α};

1) P_{α} ⊆ P_{β};

2) for p, q ∈ P_{α}, we have p ≤_{P_{α}} q ⇔ p ≤_{P_{β}} q;

3) for p, q ∈ P_{β}, if p ≤_{P_{β}} q then p|` α ≤_{P_{α}} q|` α.

Proof 1) can be proved by induction on β. Other assertions are clear from the definition of CS^{∗}-iteration. (Lemma 4.1)

Lemma 4.2 Suppose that α ≤ β ≤ ε and p, q ∈ P_{α}. Then p⊥_{P_{α}}q ⇔ p⊥_{P_{β}}q.

Proof Suppose that p and q are compatible in P_{α}, say r ≤_{P_{α}} p, q for some r ∈ P_{α}. Then r ∈ P_{β} by Lemma 4.1, 1) and r ≤_{P_{β}} p, q by Lemma 4.1, 2). Hence p and q are compatible in P_{β}.

Conversely, suppose that p and q are compatible in P_{β}, say s ≤_{P_{β}} p, q for some s ∈ P_{β}. Then we have s|` α ∈ P_{α} by Lemma 4.1, 0), s|` α ≤_{P_{α}} p|` α = p and s|` α ≤_{P_{α}} q|` α = q. Hence p and q are compatible in P_{α}. (Lemma 4.2)

Suppose that α ≤ β ≤ ε, p ∈ P_{β}. By Lemma 4.1, 0), we have p|` α ∈ P_{α}. For r ≤_{P_{α}} p|` α, let

p ^{⌢} r = p|` (dom(p) \ α) ∪ r.

For p, q ∈ P_{ε}, p ≤_{P_{ε}}^{h} q ⇔ p ≤_{P_{ε}} q and p|` dom(q) = q; p ≤_{P_{ε}}^{v} q ⇔ p ≤_{P_{ε}} q and dom(p) = dom(q) (h and v stand for ‘horizontal’ and ‘vertical’ respectively).

Lemma 4.3 1) Let α, β, p, r be as above. Then p ^{⌢} r ∈ P_{β} and p ^{⌢} r ≤_{P_{β}} r, p.

2) For p, q ∈ P_{ε}, r = q|` (dom(q) \ dom(p)) ∪ p is an element of P_{ε} and r ≤_{P_{ε}}^{h} p.

3) If p_{n} ∈ P_{ε} for n ∈ ω and p_{n+1} ≤_{P_{ε}}^{h} p_{n} for every n ∈ ω, then q = ⋃{ p_{n} : n ∈ ω } is an element of P_{ε} and q ≤_{P_{ε}}^{h} p_{n} for every n ∈ ω

Proof 1): By induction on β. If β = α then p ^{⌢} r = r ≤ p|` α = p. Suppose that we have shown the inequality for every β′ < β. Let p and r be as above. If β is a limit then we obtain easily p ^{⌢} r ∈ P_{β} and p ^{⌢} r ≤_{P_{β}} r, p by checking *1) and *2) of the definition of CS^{∗}-iteration. In particular, *2), ii) holds for the inequality p ^{⌢} r ≤_{P_{β}} r, p since diff (p ^{⌢} r, p) = diff (r, p|` α) and diff (p ^{⌢} r, r) = ∅. If β = γ + 1 for some γ ≥ α, then p|` γ ^{⌢} r ∈ P_{γ}, p|` γ ^{⌢} r ≤_{P_{γ}} r, p|` γ by induction hypothesis. If γ ∉ dom(p) then it follows p = p|` γ ∈ P_{β} and p ^{⌢} r ≤_{P_{β}} r, p. Otherwise (p ^{⌢} r)|` γ ∥–_{P_{γ}} “ p(γ) ≤Q̇_{γ} p(γ) ”. Hence again it follows that p ^{⌢} r ∈ P_{β} and p ^{⌢} r ≤_{P_{β}} r, p.

2) and 3) are trivial. (Lemma 4.3)

Lemma 4.4 Suppose that α ≤ β ≤ ε, p ∈ P_{α} and q ∈ P_{β}. If p and q are incompatible in P_{β} then p and q|` α are incompatible in P_{α}.

Proof Suppose that p and q|` α are compatible in P_{α}. Then there is r ∈ P_{α} such that r ≤_{P_{α}} p, q|` α. Let s = q ^{⌢} r. By Lemma 4.3, we have s ≤_{P_{β}} q, r. Hence p and q are compatible in P_{β}. (Lemma 4.4)

Lemma 4.5 Suppose that α ≤ β ≤ ε and that A is a maximal antichain in P_{α}. Then A is also a maximal antichain in P_{β}.

Proof By Lemma 4.1, 1), we have A ⊆ P_{β}. By Lemma 4.2, A is an antichain in P_{β}. Suppose that A were not a maximal antichain in P_{β}. Then there is some q ∈ P_{β} such that q is incompatible with each of p ∈ A. By Lemma 4.4, it follows that q|` α is incompatible with each of p|` α = p, p ∈ A. This is a contradiction to the assumption that A is a maximal antichain in P_{α}. (Lemma 4.5)

5 CS^{∗}-iteration of Cohen reals

In the rest, we consider the CS^{∗}-iteration (P_{α}, Q̇_{α})_{α≤κ} for a cardinal κ such that

∥–_{P_{α}} “ Q̇_{α} = Fn(ω, 2) ”

for every α < κ.

Lemma 5.1 Let p, q ∈ P_{κ} be such that p ≤ q. Then there is r ∈ P_{κ} such that r ≤ p and for any α ∈ diff (r, q), there is t ∈ Fn(ω, 2) such that r|` α ∥–_{P_{α}} “ r(α) = ť”.

Proof We define inductively a decreasing sequence (α_{n})_{n<ω} of ordinals and a decreasing sequence (p_{n})_{n∈ω} of elements of P_{κ} as follows: Let α_{0} = max diff (p, q). Choose p′_{0} ∈ P_{α_{0}} so that p′_{0} ≤ p|` α_{0} and that p′_{0} decides p(α_{0}). Let p_{0} = p ^{⌢} p′_{0}. If α_{n} and p_{n} have been chosen, let D_{n} = diff (p_{n}, q) ∩ α_{n}. If D_{n} = ∅ we are done. Otherwise, let α_{n+1} = max D_{n}. Choose p′_{n+1} ∈ P_{α_{n+1}} such that p′_{n+1} ≤ p_{n}|` α_{n+1} and p′_{n+1} decides p_{n}(α_{n+1}). Let p_{n+1} = p_{n} ^{⌢} p′_{n+1}. This process terminates after m steps for some m ∈ ω, since otherwise we would obtain an infinite decreasing sequence of ordinals. Clearly r = p_{m} is as desired. (Lemma 5.1)

Lemma 5.2 P_{κ} satisfies the axiom A.

Proof Let ≤_{n}, n ∈ ω be the relations on P_{κ} defined by p ≤_{n} q ⇔ p ≤_{P_{κ}}^{h} q for p, q ∈ P_{κ} and every n ∈ ω (in Ishiu [6] an axiom A p.o., for which the ≤_{n}’s can be taken to be all the same, is called uniformly axiom A). (≤_{n})_{n∈ω} has the fusion property by Lemma 4.3, 3). Hence it is enough to show the following:

Claim 5.2.1 For any p ∈ P_{κ} and maximal antichain D ⊆ P_{κ}, there is q ≤_{P_{κ}}^{h} p such that { r ∈ D : r is compatible with q } is countable.

⊢ Let Φ : ω → ω × ω; n ↦ (φ_{1}(n), φ_{2}(n)) be a surjection such that φ_{1}(n) < n for all n > 0 and, for any k, l ∈ ω, there are infinitely many n ∈ ω such that Φ(n) = (k, l). We construct inductively p_{k}, t_{k}, u_{k} ∈ P_{κ} and a sequence (s_{k,l})_{l∈ω} for k ∈ ω as follows: let p_{0} = p. If p_{k} has been chosen then let (s_{k,l})_{l∈ω} be an enumeration of Fn(dom(p_{k}), Fn(ω, 2)). If there are t ∈ D and u ∈ P_{κ} such that u ≤ t, p_{κ}, diff (u, p_{k}) = doms_{φ_{1}(k),φ_{2}(k)} and u|` diff (u, p_{k}) = s_{φ_{1}(k),φ_{2}(k)} (of course we identify here elements t of Fn(ω, 2) with corresponding P_{α}-name ť), then let t_{k} and u_{k} be such t and u and let p_{k+1} = p_{k} ∪ u|` (dom(u_{k}) \ dom(p_{k})). By Lemma 4.3, 2), we have p_{k+1} ∈ P. Otherwise let t_{k} = u_{k} = 1_{P_{κ}} and p_{k+1} = k_{k}.

Now, let q = ⋃_{k∈ω} p_{k}. Then by Lemma 4.3, 3), we have q ∈ P_{κ} and q ≤_{P_{κ}} p. We show that this q is as desired.

Suppose that t ∈ D is compatible with q. Then by Lemma 5.1, there is u ⊆_{P_{κ}} t, q such that u|` diff (q, r) has its values in Fn(ω, 2). Let n ∈ ω be such that diff (q, r) ⊆ q_{n} and k ≥ n be such that s_{φ_{1}(k),φ_{2}(k)} = u|` diff (q, r). Clearly t_{k} ∈ D by construction. We claim that t = t_{k}: otherwise t and t_{k} would be incompatible. Hence u_{k} and u should be incompatible. But this is a contradiction.

It follows that

{ r ∈ D : r is compatible with q } ⊆ { t_{k} : k ∈ ω }.

⊣ (Claim 5.2.1)

(Lemma 5.2)

In particular, P_{κ} is proper and hence the following covering property holds:

Corollary 5.3 Suppose that G is a P_{κ}-generic filter over V. Then for any a ∈ V [G] such that V [G] ⊧ “ a is a countable set of ordinals ”, there is a b ∈ V such that a ⊆ b and V ⊧ “ b is a countable set of ordinals ”.

Lemma 5.4 If κ is strongly inaccessible, then P_{κ} satisfies the κ-cc.

Proof Suppose that p_{β} ∈ P_{κ} for β < κ. We show that there are compatible conditions among them. Without loss of generality we may assume that { dom(p_{β}) : β < κ } is a Δ-system with the root x ∈ [κ]^{≤ℵ_{0}} Let α_{0} = sup{ γ + 1 : γ ∈ x }. Then α_{0} < κ and p_{β}|` x ∈ P_{α_{0}} for every β < κ. Since | P_{α} | < κ there are β, β′ < κ, β ≠ β′ such that p_{β}|` x = p_{β′}|` x. But then q = p_{β} ∪ p_{β′} ∈ P_{κ} and q ≤_{P_{κ}} p_{β}, p_{β′}.

(Lemma 5.4)

Lemma 5.5 Suppose that E ⊆ Lim(ω_{1}) is stationary. Then ∥–_{P_{κ}} “ ♣(E) ”.

Proof For each γ ∈ E let f_{γ} : [γ, γ + ω) → γ be a bijection and let

S_{γ} = { x ⊆ γ : x is a cofinal subset of γ, otp(x) = ω }.

For each x ∈ S_{γ}, let p_{x} ∈ P_{κ} be defined by

p_{x} = { (γ + n, q̇_{x,n}^{γ}) : n ∈ ω }

where q̇_{x,n}^{γ} is the standard P_{γ+n}-name for { (0, i) } with i ∈ 2 and i = 1 ⇔ f_{γ}(γ+n) ∈ x. For distinct x, x′ ∈ S_{γ}, p_{x} and p_{x′} are incompatible. Hence there is a P_{κ}-name ẋ_{γ} such that ∥–_{P_{κ}} “ ẋ_{γ} is a cofinal subset of γ with otp(ẋ_{γ}) = ω ” and p_{x} ∥–_{P_{κ}} “ ẋ_{γ} = x ” for every x ∈ S_{γ}.

We show that ∥–_{P_{κ}} “ (ẋ_{γ})_{γ∈E} is a ♣(E)-sequence ”. Suppose that p ∈ P_{κ} and Ȧ is a P_{κ}-name such that p ∥–_{P_{κ}} “ Ȧ ∈ [ω_{1}]^{ℵ_{1}} ”. We have to show that there is q ≤_{P_{κ}} p and γ ∈ E such that q ∥–_{P_{κ}} “ ẋ_{γ} ⊆ Ȧ ”.

Let ḟ be a P_{κ}-name such that p ∥–_{P_{κ}} “ ḟ : ω → Ȧ is 1-1 ”. Choose p_{α}, q_{α}, u_{α} for α < ω_{1} inductively such that a) p_{0} ≤_{P_{κ}} p and (p_{α})_{α<ω_{1}} is a decreasing sequence with respect to ≤_{P_{κ}}^{h} ;

b) q_{α} ≤_{P_{α}}^{v} p_{α} and q_{α} decides ḟ(α);

c) u_{α} = diff (q_{α}, p_{α}) ⊆ dom(p) ∪ ⋃_{β<α} dom(q_{β});

d) q_{α}|` u_{α} ∈ Fn(κ, Fn(ω, 2)).

The condition d) is possible because of Lemma 5.1. By Fodor’s lemma, there is Y ∈ [ω_{1}]^{ℵ_{1}} and r ∈ Fn(κ, Fn(ω, 2)) such that q_{α}|` u_{α} = r for every α ∈ Y. For each α ∈ Y, there is β_{α} ∈ ω_{1} such that q_{α} ∥–_{P_{κ}} “ ḟ(α) = β_{α} ” by b). Let Z = { β_{α} : α ∈ Y }. Let

C = { γ ∈ Lim(ω_{1}) : ⋃_{α∈Y ∩γ}(sup(q_{α}) ∩ ω_{1}) ⊆ γ

and Z ∩ γ is unbounded in γ }.

Then C is closed unbounded in ω_{1}. Since E was stationary, there exists a γ^{∗} ∈ C ∩ E. Let q′ = ⋃_{α∈Y ∩γ^{∗}} q_{α}. Then we have q′ ∥–_{P_{κ}} “ Z ∩ γ^{∗} ⊆ Ȧ ”. Now let x ∈ S_{γ^{∗}} be such that x ⊆ Z ∩ γ^{∗}. Finally let q = q′ ∪ q_{x}. Then we have q ≤_{P_{κ}}^{h} p and q ∥–_{P_{κ}} “ ẋ_{α} = x ⊆ Z ∩ γ^{∗} ⊆ Ȧ ”. (Lemma 5.5)

Let (P_{α}, Q̇_{α})_{α≤κ} be a CS^{∗}-iteration as above. For α < κ let P_{κ}/Ġ_{α} be a P_{α}- name such that ∥–_{P_{α}} “ P_{κ}/Ġ_{α} = { p ∈ P̌_{κ} : p|` α ∈ Ġ_{α} } with the ordering p ≤_{κ,α} q ⇔ p ≤_{P_{α}} q ”. As in [5], we can show that P_{κ} ≈ P_{α} ∗ P_{κ}/Ġ_{α}. Also, by Corollary 5.3, practically the same proof as in [5] shows that

∥–_{P_{α}} “ P_{κ}/Ġ_{α} is ≈ to a CS^{∗}-iteration of Fn(ω, 2) ”.

Now we are ready to prove the main theorem of this section:

Theorem 5.6 Suppose that ZFC + “there exists an inaccessible cardinal” is consistent. Then ZFC + ¬CH + MA(countable) + “♣(E) for every stationary E ⊆ Lim(ω_{1})” is consistent as well.

Proof Suppose that κ is strongly inaccessible. For P_{κ} as above, let G_{κ} be a P_{κ}- generic filter over V. We show that V [G_{κ}] models the assertions. Let E ⊆ Lim(ω_{1}) be a stationary set in V [G_{κ}]. Since P_{κ} has the κ-cc by Lemma 5.4, there is some α < κ such that E ∈ V [G_{α}] where G_{α} = G_{κ} ∩ P_{α}. Hence by the remark before this theorem, we may assume without loss of generality that E ∈ V. But then, by Lemma 5.5, we have V [G_{κ}] ⊧ “ ♣(E) ”.

Finally, we show that MA(countable) holds in V [G_{α}]. Let D be a family of dense subsets of Fn(ω, 2) in V [G_{κ}] of cardinality < κ. Again by the κ-cc of P_{κ}, we can find an α < κ such that D ∈ V [G_{α}]. Since we have

P_{κ} ≈ P_{α} ∗ Q̇_{α} ∗ P_{κ}/Ġ_{α+1} ,

the generic set over V [G_{α}] added by Q̇_{α}[G_{α}] = Fn(ω, 2) is D-generic over Fn(ω, 2) in V [G_{κ}]. (Theorem 5.6)

At the moment we — or more precisely the first and the third author — do not know if an inaccessible cardinal is really necessary in Theorem 5.6. As for CS-iteration, κ is collapsed to be of cardinality ℵ_{2} in the model above, since the continuum of each of the intermediate models is collapsed to ℵ_{1} in the following limit step of cofinality ≥ ω_{1}. Thus the following problem seems to be a rather hard one:

Problem 5.7 Is the combination MA(countable) + ♣(E) for every stationary E ⊆ Lim(ω_{1}) consistent with 2^{ℵ_{0}} > ℵ_{2} ?

Acknowledgments

The research of this paper began when the first author (S.F.) was at the Hebrew University of Jerusalem. He would like to thank The Israel Academy of Science and Humanities for enabling his stay there. He also would like to thank T. Miyamoto for some quite helpful remarks.

The second author (S.S.) was partially supported by the Deutsche Forschungsgemeinschaft(DFG) grant Ko 490/7–1. He also gratefully acknowledges partial support by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany). The present paper is the second author’s Publication No. 544.

The third author (L.S.) is partially supported by the Hungarian National Foundation for Scientific Research grant No. 16391 and the Deutsche Forschungsgemeinschaft (DFG) grant Ko 490/7–1.

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Authors’ addresses

Sakaé Fuchino Institut für Mathematik II, Freie Universität Berlin 14195 Berlin, Germany

fuchino@math.fu-berlin.de

Saharon Shelah Institute of Mathematics, The Hebrew University of Jerusalem 91904 Jerusalem, Israel and Department of Mathematics, Rutgers University New Brunswick, NJ 08854, USA

shelah@math.huji.ac.il

Lajos Soukup Mathematical Institute of the Hungarian Academy of Sciences

soukup@math-inst.hu