On the weak Freese-Nation property of complete Boolean algebras

Sakaé Fuchino, Stefan Geschke, Saharon Shelah, Lajos Soukup

April 6, 2000

Abstract

The following results are proved:

(a) In a model obtained by adding ℵ_{2} Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property.

(b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH.

(c) If a weak form of □_{μ} and cof([μ]^{ℵ_{0}}, ⊆) = μ^{+} hold for each μ > cf(μ) = ω, then the weak Freese-Nation property of ⟨P(ω), ⊆⟩ is equivalent to the weak Freese-Nation property of any of ℂ(κ) or ℝ(κ) for uncountable κ.

(d) Modulo the consistency of (ℵ_{ω+1}, ℵ_{ω}) \overrightarrow{→} (ℵ_{1}, ℵ_{0}), it is consistent with GCH that ℂ(ℵ_{ω}) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ_{ω} Cohen reals destroys the weak Freese-Nation property of ⟨P(ω), ⊆⟩.

These results solve all of the problems listed in Fuchino-Soukup [5] and some other problems posed by S. Geschke.

∗The first author was partially supported by Grant-in-Aid for Scientific Research (C) No.

10640099 of the Ministry of Education, Science, Sports and Culture, Japan.

Some of the results here, in particular earlier versions of the results in Section 5, were included in the second author’s Ph.D. thesis [7].

This paper is [FGShS:712] of the third author’s publications list. His research is supported by “The Israel Science Foundation”.

The fourth author was partially supported by Grant-in-Aid for JSPS Fellows No. 98259 of the Ministry of Education, Science, Sports and Culture, Japan, and by Hungarian National Foundation for Scientific Research grant no. 25745.

Section 6 of the present paper was worked out during the conference ”Algebra and Discrete Mathematics” in Hattingen, Germany 1999 which was organized by the third author and attended by all of the other authors.

1 Introduction

A quasi-ordering ⟨P, ≤⟩ is said to have the weak Freese-Nation property if there is a mapping f : P → [P]^{≤ℵ_{0}} such that:

For any p, q ∈ P with p ≤ q there is r ∈ f(p) ∩ f(q) such that p ≤ r ≤ q.

A mapping f as above is called a weak Freese-Nation mapping on P.

The weak Freese-Nation property was introduced in Chapter 4 of [8] as a weakening of a notion of almost freeness of Boolean algebras. The property was further studied in [4] and [5].

In [4], it is shown that ⟨P(ω_{1}), ⊆⟩ does not have the weak Freese-Nation property. If a complete Boolean algebra B does not have the c.c.c., then ⟨P(ω_{1}), ⊆⟩ can be completely embedded into B. Hence, in this case, B cannot have the weak Freese-Nation property (see Proposition 1.2 (1)).

It is easily seen that every quasi-ordering of cardinality ≤ ℵ_{1} has the weak Freese-Nation property (see e.g. [4]). It follows that, under CH, ⟨P(ω), ⊆⟩ has the weak Freese-Nation property.

Similarly to the situation with P(ω_{1}) and non-c.c.c. complete Boolean algebras, if ⟨P(ω), ⊆⟩ does not have the weak Freese-Nation property, then no infinite c.c.c. complete Boolean algebra (and hence no infinite complete Boolean algebra at all) can have the weak Freese-Nation property.

To simplify the formulation of some of the results below, let us say that a model of set-theory is neat if cof([μ]^{ω}, ⊆) = μ^{+} for each μ > cf(μ) = ω and the very weak variant of □_{μ} introduced in [5] (see the end of section 4) holds.

In [4] and [5], it was shown that if CH holds, then every c.c.c. complete Boolean algebra of size < ℵ_{ω} has the weak Freese-Nation property; and in a neat model, CH implies that every c.c.c. complete Boolean algebra has the weak Freese-Nation property. The following questions remained unanswered in [5]:

Question 1 ([5, Problem 5]) Are the following equivalent?

(a) ⟨P(ω), ⊆⟩ has the weak Freese-Nation property;

(b) every c.c.c. complete Boolean algebra has the weak Freese-Nation property.

Question 2 ([5, Problem 2]) Does ZFC + GCH imply that every c.c.c. complete Boolean algebra has the weak Freese-Nation property ?

The final version of the paper was then prepared during the Workshop on Set-Theoretical Topology 1999 at Erdős Center, Budapest Hungary 1999.

The authors would like to thank I. Juhász, S. Koppelberg and the anonymous referee for pointing out some errors in earlier versions of the paper.

We give here negative answers to these questions: see Corollary 3.4 for question 1 and Theorem 4.2 for question 2. By the result in [5] already mentioned above, we need the consistency strength of some large cardinal to give a negative answer to Question 2. Indeed, the ground model V in the negative solution to this question is obtained by starting from a model of ZFC with a supercompact cardinal.

In [4] it was shown that if CH holds, then adding less than ℵ_{ω} many Cohen reals preserves the weak Freese-Nation property of ⟨P(ω), ⊆⟩. By [5], in the generic extension obtained by adding any number of Cohen reals to a neat model satisfying CH, not only ⟨P(ω), ⊆⟩ but every tame c.c.c. complete Boolean algebra has the weak Freese-Nation property. Here, letting P = Fn(τ, 2) (= the standard p.o. for adding τ Cohen reals), a Boolean algebra B in a P -generic extension is said to be tame, if there is a P -name ˙≤ of a partial ordering of B and a mapping t : B → [τ]^{ℵ_{0}} in V such that, for every p ∈ P and x, y ∈ B, if p ∥–_{P} “ x ˙≤ y ”, then p|` (t(x) ∪ t(y)) ∥–_{P} “ x ˙≤ y ” (we assume here without loss of generality that B is chosen so that its underlying set is a ground model set).

These results suggest the following questions posed in [5]:

Question 3 ([5, Problem 3]) Assume that V [G] is a model obtained by adding Cohen reals to a model of ZFC + CH. Is it true that P(ω) has the weak Freese-Nation property in V [G] ?

Question 4 ([5, Problem 4]) Assume that V [G] is a model obtained by adding ℵ_{2} Cohen reals to a model of ZFC + CH. Is it true that every c.c.c. complete Boolean algebra (not just the tame ones) has the weak Freese-Nation property in V [G]?

The results of the present paper answer these questions in the negative: see Theorem 6.1 for question 3 and Corollary 3.3 for question 4.

By the result in [5] already mentioned above, we need the consistency strength of some large cardinal for a negative solution of Question 3. The ground model V in the negative solution of this problem given in Theorem 6.1 is obtained by starting from a model of ZFC with a large cardinal slightly stronger than a huge cardinal.

After the negative solution of Problem 5 in [5] (listed here as Question 1), the following question still remains:

Problem 1 For which Boolean algebras B, is the weak Freese-Nation property of B equivalent with the weak Freese-Nation property of ⟨P(ω), ⊆⟩?

The following trivial lemma can also be seen in connection with this problem:

Lemma 1.1 The following are equivalent:

(a) ⟨P(ω), ⊆⟩ has the weak Freese-Nation property;

(b) ⟨P(ω), ⊆^{∗}⟩ has the weak Freese-Nation property;

(c) ⟨P(ω)/fin, ⊆^{∗}⟩ has the weak Freese-Nation property;

(d) ⟨^{ω}ω, ≤⟩ has the weak Freese-Nation property;

(e) ⟨^{ω}ω, ≤^{∗}⟩ has the weak Freese-Nation property.

(f) ⟨^{ω}ℝ, ≤⟩ has the weak Freese-Nation property.

S. Koppelberg [10] pointed out that the weak Freese-Nation property of the Cohen algebra ℂ(ω) is equivalent to the weak Freese-Nation property of ⟨P(ω), ⊆⟩. In the present paper, we show that it is also equivalent to the weak Freese-Nation property of the measure algebra ℝ(ω) (Proposition 5.1) and moreover, in a neat model, also to the weak Freese-Nation property of ℂ(κ) and/or ℝ(κ) for any κ ≥ ℵ_{0} (Corollary 5.4). Here, we denote by ℂ(κ) and ℝ(κ) the c.c.c. complete Boolean algebras Borel(^{κ}2)/meager(^{κ}2) and Borel(^{κ}2)/null(^{κ}2) respectively. We show that some extra set-theoretic assumptions are really necessary in Corollary 5.4 by constructing a model of GCH and the negation of weak Freese-Nation property of ℂ(ℵ_{ω}) starting from a model of GCH and Chang’s conjecture for ℵ_{ω}.

Assume that ⟨P_{α}, Q̇_{α} : α < ω_{2}⟩ is a finite support iteration such that the forcing with Q̇_{α} just adds a real to V^{P_{α}}. Then, as S. Geschke showed in [7] , if this iteration preserves the weak Freese-Nation property of P(ω), then in many cases, we can conclude that for stationarily many α the partially ordered set Q̇_{α} just adds one Cohen real. On the other hand, in any model obtained by adding ≥ ℵ_{2} Cohen reals, there is a c.c.c. complete Boolean algebra B without the weak Freese-Nation property (Corollary 3.3). So there is no easy way to blow up the continuum while preserving the weak Freese-Nation property of all c.c.c. complete Boolean algebras. This suggests that the following question is a quite reasonable one:

Problem 2 Does CH follow from the assumption that every c.c.c. complete Boolean algebra has the weak Freese-Nation property?

If b > ℵ_{1} or if there is an ℵ_{2}-Luzin-gap, then ⟨P(ω), ⊆⟩ does not have the weak Freese-Nation property (see [4] and [5]). The following question (([5, Problem 1])) was raised against this background:

Suppose that P(ω) does not have any increasing chain of length ≥ ω_{2} with respect to ⊆^{∗} and that there is no ℵ_{2}-Luzin gap. Does it follow that P(ω) has the weak Freese-Nation property ?

This can be solved negatively using results from [2] and [7]: Let V be a model of CH and V [G] its generic extension by adding many random reals side by side.

Using results from [2] we see that, in V [G], there are neither increasing ω_{2} chains in P(ω) with respect to ⊆^{∗} nor ℵ_{2}-Luzin gaps. On the other hand S. Geschke [7] showed that, in V [G], ⟨P(ω), ⊆⟩ does not have the weak Freese-Nation property.

Consequences of the weak Freese-Nation property of ⟨P(ω), ⊆⟩ were studied in [10] and [6]. In the latter paper it was shown that a set-theoretic universe with the weak Freese-Nation property of ⟨P(ω), ⊆⟩ looks quite similar to a Cohen model. In particular, under the weak Freese-Nation property of ⟨P(ω), ⊆⟩, all cardinal invariants which appear in [1] take the same value as in a Cohen model with the same size of 2^{ℵ_{0}}.

Problem 3 Find a combinatorial (Π_{1}^{1}) characterization of the weak Freese-Nation property of P(ω).

The weak Freese-Nation property of a quasi-ordering ⟨P, ≤⟩ is actually a property of the corresponding partial ordering ⟨\overline{P} , ≤⟩ obtained as the quotient structure of ⟨P, ≤⟩ with respect to the equivalence relation “x ≤ y ∧ y ≤ x”: ⟨P, ≤⟩ has the weak Freese-Nation property if and only if ⟨\overline{P} , ≤⟩ does (the first author thanks David Fremlin for pointing out this fact).

The following criteria of the weak Freese-Nation property are used in the later sections. A partial ordering Q is said to be a retract of a partial ordering P if there are order preserving mappings i : Q → P and j : P → Q such that j ∘ i = id_{Q}.

Q is said to be a σ-subordering of P (notation: Q ≤_{σ} P) if, for every p ∈ P, Q|` p = {q ∈ Q : q ≤ p} has a countable cofinal subset and Q ↑ p = {q ∈ Q : q ≥ p} has a countable coinitial subset. Note that if C is a complete subalgebra of a complete Boolean algebra B (notation: C ≤_{c} B) or a countable union of complete subalgebras of B, then it follows that C ≤_{σ} B.

Proposition 1.2 (1) (Lemma 2.7 in [4]) If Q is a retract of P and P has the weak Freese-Nation property then Q has the weak Freese-Nation property.

(2) Suppose that Q is a complete Boolean algebra and there is a strictly orderpreserving embedding f of Q into P (i.e. f preserves ordering and incomparability). If P has the weak Freese-Nation property then Q also has the weak Freese-Nation property.

(3) (Lemma 2.3 (a) in [4]) If Q ≤_{σ} P and P has the weak Freese-Nation property, then Q also has the weak Freese-Nation property.

(4) (Lemma 2.6 in [4]) If P_{α}, α < δ is an increasing sequence of partial orderings with the weak Freese-Nation property such that P_{α} ≤_{σ} P_{α+1} for every α < δ and P_{γ} = ⋃_{α<γ} P_{α} for all γ < δ with cf(γ) > ω, then P = ⋃_{α<δ} P_{α} also has the weak Freese-Nation property.

Proposition 1.2 (2) follows easily form Proposition 1.2 (1): the mapping g : P → Q defined by g(p) = ∑{q ∈ Q : f(q) ≤ p} for p ∈ P witnesses that Q is a retract of P.

2 The partial ordering P_{S}

In this section we introduce a construction of partial orderings P_{S} and Boolean algebras B_{S} which will be used in the following Section 3 and 4. For S ⊆ κ and an indexed family S = ⟨S_{α} : α ∈ S⟩ of subsets of κ, let

P_{S} = {x_{i} : i ∈ κ} ∪ {y_{α} : α ∈ S}

where the x_{i}’s and y_{α}’s are pairwise distinct, and let ≤_{S} be the partial ordering on P_{S} defined by

p ≤_{S} q ⇔ p = q or
p = x_{i} and q = y_{α} for some i ∈ κ and α ∈ S with i ∈ S_{α} .

Let B_{S} be the Boolean algebra generated freely from P_{S} except the relation ≤_{S}. Note that the identity map on P_{S} canonically induces a strictly order-preserving embedding of P_{S} into B_{S}.

Proposition 2.1 Suppose that cf(κ) ≥ ω_{2}, S ⊆ κ is stationary such that S ⊆ {α < κ : cf(α) ≥ ω_{1}} and S = {S_{α} : α ∈ S} is such that S_{α} is a cofinal subset of α for each α ∈ S. If P_{S} is embedded into a partial ordering P by a strictly order-preserving mapping then P does not have the weak Freese-Nation property. In particular, B_{S} and its completion do not have the weak Freese-Nation property.

Proof Without loss of generality, we may assume that P_{S} is a subordering of P. Assume to the contrary that there is a weak Freese-Nation mapping f : P → [P]^{≤ℵ_{0}}. Let

\begin{aligned}C = \{ξ < κ : & ∀η < ξ ∀p ∈ f(x_{η}) \\ & [ ∃α ∈ S [ x_{η} ≤ p ≤ y_{α}] → ∃α′ ∈ S ∩ ξ [ x_{η} ≤ p ≤ y_{α′}] ] \}. \end{aligned}

Then C is a club subset of κ. Let α ∈ C ∩ S and let

A = {p ∈ f(y_{α}) : ∃η ∈ S_{α} [ p ∈ f(x_{η}) ∧ x_{η} ≤ p ≤ y_{α}] }.

Since α ∈ C, for each p ∈ A there is α_{p} < α such that p ≤ y_{α_{p}}. Let α^{∗} = sup{α_{p} : p ∈ A}. Since A is countable we have α^{∗} < α. Let β ∈ S_{α} \ α^{∗}. Since x_{β} ≤ y_{α}, there is a p ∈ A such that x_{β} ≤ p ≤ y_{α}. Hence x_{β} ≤ y_{α_{p}}. But this is impossible since α_{p} < α^{∗} ≤ β. (Proposition 2.1)

3 Cohen models

Consider the following principle:

(∗∗) There is a sequence ⟨S_{α} : α ∈ Lim(ω_{2})⟩ such that each S_{α} is a cofinal subset of α and for any pairwise disjoint ⟨x_{β} : β < ω_{1}⟩ with x_{β} ∈ [ω_{2}]^{<ℵ_{0}} for β < ω_{1}, there are β_{0} < β_{1} < ω_{1} such that x_{β_{0}} ∩ S_{α} = ∅ for all α ∈ x_{β_{1}} ∩ Lim(ω_{2}) and that x_{β_{1}} ∩ S_{α} = ∅ for all α ∈ x_{β_{0}} ∩ Lim(ω_{2}) .

Proposition 3.1 Let P = Fn(ω_{2}, 2). Then ∥–_{P} “ (∗∗) ”.

Proof Without loss of generality we may assume P = Fn(⋃_{α∈Lim(ω_{2})} α × {α}, 2). For α ∈ Lim(ω_{2}), let Ṡ_{α} be a P -name such that ∥–_{P} “ Ṡ_{α} = {β ∈ α : ġ(β, α) = 1} ” where ġ is the canonical name for the generic function. By genericity, ∥–_{P} “ Ṡ_{α} is cofinal in α ” for every α ∈ Lim(ω_{2}). Let Ṡ be a P -name such that ∥–_{P} “ Ṡ = ⟨Ṡ_{α} : α ∈ Lim(ω_{2})⟩ ”.

To show that Ṡ is forced to satisfy the property in (∗∗), let ⟨ẋ_{β} : β < ω_{1}⟩ be a P -name of a sequence of pairwise disjoint finite subsets of ω_{2}. For each β < ω_{1}, let p_{β} and x_{β} ∈ [ω_{2}]^{<ℵ_{0}} be such that p_{β} ∥–_{P} “ ẋ_{β} = x_{β} ”. By thinning out the index set ω_{1}, we may assume without loss of generality that dom(p_{β}), β < ω_{1} form a Δ-system with the root d and p_{β}|` d, β < ω_{1} are all equal to the same p ∈ P. Since p_{β}, β < ω_{1} are then pairwise compatible, x_{β}, β < ω_{1} are pairwise disjoint. Further, we may assume that s_{β}, β < ω_{1} form a Δ-system with the root s where s_{β} = {γ : ⟨γ, α⟩ ∈ dom(p_{β}) for some α ∈ Lim(ω_{2})}. Note vthat x_{β}, β < ω_{1} are pairwise disjoint since p_{β}’s are pairwise compatible.

Let β_{0} < β_{1} < ω_{1} be such that x_{β_{0}} ∩ s = ∅, x_{β_{1}} ∩ s = ∅, x_{β_{0}} ∩ s_{β_{1}} = ∅ and x_{β_{1}} ∩ s_{β_{0}} = ∅. To take such β_{0} and β_{1}, first fix a β_{0} such that x_{β_{0}} ∩ s = ∅. This is possible since s is finite and x_{β}’s are pairwise disjoint. By the same argument we can find infinitely many β’s such that x_{β} ∩ (s ∪ s_{β_{0}}) = ∅. Now for such β’s, since s_{β} \ s are pairwise disjoint, there are infinitely may β’s among them with x_{β_{0}} ∩ s_{β} = ∅. Let β_{1} be one of such β’s.

Let

\begin{aligned}p^{∗} = & p_{β_{0}} ∪ p_{β_{1}} ∪ \{⟨⟨β, α⟩, 0⟩ : β ∈ x_{β_{0}}, α ∈ x_{β_{1}} ∩ Lim(ω_{2})\} \\ & ∪\{⟨⟨β, α⟩, 0⟩ : β ∈ x_{β_{1}}, α ∈ x_{β_{0}} ∩ Lim(ω_{2})\} \end{aligned}

Then p^{∗} ∈ P. Clearly p^{∗} ∥–_{P} “ ẋ_{β_{0}} ∩ Ṡ_{α} = ∅ ” for all α ∈ ẋ_{β_{1}} ∩ Lim(ω_{2}) and p^{∗} ∥–_{P} “ ẋ_{β_{1}} ∩ Ṡ_{α} = ∅ ” for all α ∈ ẋ_{β_{0}} ∩ Lim(ω_{2}). (Proposition 3.1)

Proposition 3.2 Suppose that ⟨S_{α} : α ∈ Lim(ω_{2})⟩ is as in (∗∗). Let S = {α < ω_{2} : cf(α) = ω_{1}} and S = ⟨S_{α} : α ∈ S⟩. Then B_{S} satisfies the c.c.c.

Proof Otherwise we can find I_{α} ∈ [ω_{2}]^{<ℵ_{0}}, J_{α} ∈ [S]^{<ℵ_{0}} for α < ω_{1} and t(α, i), u(α, ξ) ∈ {+1, −1} for each i ∈ I_{α}, ξ ∈ J_{α} and α < ω_{1} such that

z_{α} = ∏_{i∈I_{α}} t(α, i) x_{i} · ∏_{ξ∈J_{α}} u(α, ξ) y_{ξ}, α < ω_{1}

form a pairwise disjoint family of elements of B_{S}^{+}.

By a Δ-system argument, we may assume that I_{α} ∪ J_{α}, α < ω_{1} are pairwise disjoint. Applying (∗∗) to ⟨I_{α} ∪ J_{α} : α < ω_{1}⟩, we find β_{0} < β_{1} < ω_{1} such that I_{β_{0}} ∩ S_{ξ} = ∅ for all ξ ∈ J_{β_{1}} and that I_{β_{1}} ∩ S_{ξ} = ∅ for all ξ ∈ J_{β_{0}} . By definition of B_{S}, it follows that z_{β_{0}} · z_{β_{1}} ≠ 0 . This is a contradiction. (Proposition 3.2)

Theorem 3.3 In a Cohen model (i.e. any model obtained by adding ≥ ℵ_{2} Cohen reals) there is a c.c.c. complete Boolean algebra B of density ℵ_{2} without the weak Freese-Nation property.

Proof By Proposition 3.1, (∗∗) holds in a Cohen model. Hence B_{S} as in Proposition 3.2 satisfies the c.c.c. By Proposition 2.1, the completion of B_{S} does not have the weak Freese-Nation property. (Theorem 3.3)

Corollary 3.4 The weak Freese-Nation property of ⟨P(ω), ⊆⟩ does not imply the weak Freese-Nation property of all c.c.c. complete Boolean algebras.

Proof If we start from a model of CH and add ℵ_{2} Cohen reals, then ⟨P(ω), ⊆⟩ has the weak Freese-Nation property in the resulting model (see e.g. [5]). On the other hand, by Theorem 3.3, there is a c.c.c. complete Boolean algebra without the weak Freese-Nation property in such a model. (Corollary 3.4)

Under CH, every c.c.c. complete Boolean algebra of size ℵ_{2} has the weak Freese-Nation property ([4]). Hence it follows from the result above that CH implies the negation of the principle (∗∗). This can be also seen directly as follows:

Proposition 3.5 CH implies ¬(∗∗).

Proof Let ⟨S_{α} : α ∈ Lim(ω_{2})⟩ be any sequence such that each S_{α} is a cofinal subset of α for α ∈ Lim(ω_{2}). To show that ⟨S_{α} : α ∈ Lim(ω_{2})⟩ is not as in (∗∗), let χ be sufficiently large and let M ≺ H(χ) be such that | M | = ℵ_{1}; ⟨S_{α} : α ∈ Lim(ω_{2})⟩ ∈ M; ω_{1} ⊆ M; ω_{2} ∩ M ∈ ω_{2} and, letting γ = ω_{2} ∩ M, cf(γ) = ω_{1}. By CH — and since ω_{1} ⊆ M and cf(γ) = ω_{1}, we have [γ]^{ℵ_{0}} ⊆ M.

Now, by induction, choose distinct α_{β}^{0}, α_{β}^{1} < γ for β < ω_{1} such that (1) α_{β}^{0} ∈ S_{γ}, and (2) {α_{ξ}^{0} : ξ < β} ⊆ S_{α_{β}^{1}} for all β < ω_{1}. (2) is possible: since {α_{ξ}^{0} : ξ < β} ⊆ S_{γ} and {α_{ξ}^{0} : ξ < β} ∈ M, we have

M ⊧ ∃ν < ω_{2} [ sup{α_{ξ}^{1} : ξ < β} < ν ∧ {α_{ξ}^{0} : ξ < β} ⊆ S_{ν} ]

by elementarity. Let x_{β} = {α_{β}^{0}, α_{β}^{1}} for β < ω_{1}. Then there are no β_{0} < β_{1} < ω_{1} such that x_{β_{0}} ∩ S_{α} = ∅ for all α ∈ x_{β_{1}} ∩ Lim(ω_{2}). (Proposition 3.5)

4 The Weak Freese-Nation property of c.c.c. complete Boolean algebras under GCH

In [5] it is proved that, assuming CH and a weak form of the square principle at singular cardinals of cofinality ω, every c.c.c. complete Boolean algebra has the weak Freese-Nation property. In this section we show that even GCH does not suffice for this result.

Hajnal, Juhász and Shelah [9] showed that, starting from a model with a supercompact cardinal, a model of GCH and the following principle can be constructed:

(∗∗∗) There are a stationary S ⊆ {α < ω_{ω+1} : cf(α) = ω_{1}} and a family S = ⟨S_{α} : α ∈ S⟩ such that each S_{α} is a cofinal subset of α of ordertype ω_{1} and that, for all distinct α, β ∈ S, S_{α} ∩ S_{β} is finite.

Proposition 4.1 Suppose that S = ⟨S_{α} : α ∈ S⟩ is as in (∗∗∗). Then B_{S} satisfies the c.c.c.

Proof Otherwise we can find I_{α} ∈ [ω_{ω+1}]^{<ℵ_{0}}, J_{α} ∈ [S]^{<ℵ_{0}}, α < ω_{1} and t(α, i), u(α, ξ) ∈ {+1, −1} for each α < ω_{1}, i ∈ I_{α} and ξ ∈ J_{α} such that

z_{α} = ∏_{i∈I_{α}} t(α, i) x_{i} · ∏_{ξ∈J_{α}} u(α, ξ) y_{ξ}, α < ω_{1}

form a pairwise disjoint family of elements of B_{S}^{+}.

By a Δ-system argument, we may assume that I_{α} ∪ J_{α}, α < ω_{1} are pairwise disjoint and each I_{α} has the same size, say n.

For α < β < ω_{1}, since z_{α} ·z_{β} = 0, either (I) there is η ∈ J_{α} such that I_{β} ∩S_{η} ≠ ∅ or else (II) there is ξ ∈ J_{β} such that I_{α} ∩ S_{ξ} ≠ ∅. If (I) holds then let us say that ⟨α, β⟩ is of type (I).

Now, one of the following two cases holds. We show that both of them lead to a contradiction.

Case I. There is an infinite subset S of ω such that for every β ∈ ω_{1} \ ω, {k ∈ S : ⟨k, β⟩ is of type (I)} is cofinite in S. In this case, by thinning out the index set ω_{1}, we may assume that, for any k ∈ ω and β ∈ ω_{1} \ ω, ⟨k, β⟩ is of type (I). For all β ∈ ω_{1} \ω, since | I_{β} | = n, there are 0 ≤ i^{0}(β) < i^{1}(β) < n+1 such that I_{β}^{∗} = I_{β} ∩ S_{α^{0}(β)} ∩ S_{α^{1}(β)} ≠ ∅ for some α^{k}(β) ∈ J_{i^{k}(β)} for k = 0, 1 by the pigeonhole principle. Hence we can find an infinite X ⊆ ω_{1} \ ω, 0 ≤ i^{0} < i^{1} < n + 1 and α^{0}, α^{1} ∈ ω_{1}, α^{0} ≠ α^{1} such that α^{0}(β) = α^{0} and α^{1}(β) = α^{1} for all β ∈ X. But then ⋃_{β∈X} I_{β}^{∗} ⊆ S_{i^{0}} ∩ S_{i^{1}}. Since the set on the left side is infinite as a disjoint union of pairwise disjoint non-empty sets, this is a contradiction to (∗∗∗).

Case II. For any infinite subset S ⊆ ω, there is β ∈ ω_{1} \ ω such that for infinitely many k ∈ S, ⟨k, β⟩ is not of type (I). In this case, by thinning out the first ω elements of the index set ω_{1}, we may assume that for each k ∈ ω, there is ξ(k) ∈ J_{ω} such that I_{k} ∩S_{ξ(k)} ≠ ∅. Note that J_{ω} is finite. So by thinning out further the first ω elements of the index set ω_{1}, we may assume that there is ξ_{0} ∈ J_{ω} such that I_{k} ∩ S_{ξ_{0}} ≠ ∅ for every k < ω. Similarly we may also assume that there are ξ_{i} ∈ J_{ω+i} for 1 ≤ i ≤ n such that I_{k} ∩S_{ξ_{i}} ≠ ∅ for every k < ω. Then for each k < ω, we can find i^{0}(k) < i^{1}(k) ≤ n such that I_{k}^{∗} = I_{k}∩S_{ξ_{i^{0}(k)}}∩S_{ξ_{i^{1}(k)}} ≠ ∅ by the pigeonhole principle. Since there are only n(n − 1)/2 possibilities of i^{0}(k) < i^{1}(k) ≤ n, there are i^{0} < i^{1} ≤ n and an infinite set X ⊆ ω such that for every k ∈ X, i^{0}(k) = i^{0} and i^{1}(k) = i^{1}. It follows that S_{ξ_{i^{0}}} ∩ S_{ξ_{i^{1}}} ⊇ ⋃_{k∈X} I_{k}^{∗}. As S_{ξ_{i^{0}}} ∩ S_{ξ_{i^{1}}} is finite, while ⋃_{k∈X} I_{k}^{∗} is infinite as a union of infinitely many pairwise disjoint non-empty sets, this is a contradiction. (Proposition 4.1)

Theorem 4.2 It is consistent with GCH (modulo the consistency strength of a supercompact cardinal) that there is a c.c.c. complete Boolean algebra of size ℵ_{ω+1} without the weak Freese-Nation property.

Proof Let S be a family as in (∗∗∗). By Proposition 4.1, B_{S} satisfies the c.c.c. Hence the completion B of B_{S} is a c.c.c. complete Boolean algebra of size ℵ_{ω+1}. By Proposition 2.1, B does not have the weak Freese-Nation property. (Theorem 4.2)

The following weak form of the square principle was introduced in [5]. □_{ℵ_{1},μ}^{∗∗∗} is the following assertion: there exists a sequence ⟨C_{α} : α < μ^{+}⟩ and a club set D ⊆ μ^{+} such that for α ∈ D with cf(α) ≥ ω_{1}

(y1) C_{α} ⊆ α, C_{α} is unbounded in α; (y2) [α]^{<ω_{1}} ∩ {C_{α′} : α′ < α} dominates [C_{α}]^{<ω_{1}} (with respect to ⊆).

It can be easily seen that □_{ℵ_{1},μ}^{∗∗∗} follows from the very weak square principle for μ by Foreman and Magidor [3] (see [5]).

Proposition 4.3 □_{ℵ_{1},μ}^{∗∗∗} together with CH implies the negation of (∗∗∗).

Proof In [5] it is proved that under CH and □_{ℵ_{1},ℵ_{ω}}^{∗∗∗}, every c.c.c. complete Boolean algebra of cardinality ℵ_{ω+1} has the weak Freese-Nation property. Hence the assertion follows from the proof of Theorem 4.2. (Proposition 4.3)

By Proposition 4.3, we see that the consistency strength of some large cardinal is involved in GCH + (∗∗∗).

5 The weak Freese-Nation property of c.c.c. complete Boolean algebras under very weak square principles

In this section, we turn to the question for which c.c.c. complete Boolean algebras B the weak Freese-Nation property of B is equivalent to the weak Freese-Nation property of ⟨P(ω), ⊆⟩. Lemma 1.1 was already an easy observation in ZFC in this direction. S. Koppelberg pointed out in [10] that the Cohen algebra ℂ(ω) is such a Boolean algebra.

Note that ⟨P(ω), ⊆⟩ can be embedded in every complete Boolean algebra. Hence it follows from Proposition 1.2 (2) that, if ⟨P(ω), ⊆⟩ does not have the weak Freese-Nation property then no complete Boolean algebra can have the weak Freese-Nation property.

Proposition 5.1 The measure algebra ℝ(ω) has the weak Freese-Nation property if and only if ⟨P(ω), ⊆⟩ has the weak Freese-Nation property.

Proof By the remark above and by Proposition 1.2 (2) and Lemma 1.1, it is enough to find a strictly order-preserving embedding of ℝ(ω) into ⟨^{ω}ℝ, ≤⟩. We may replace ω by the countable set Clop(^{ω}2) where Clop(^{ω}2) denotes the clopen sets of the Cantor space ^{ω}2.

We define e : ℝ(ω) → ^{Clop(^{ω}2)}ℝ by taking e(a)(c) = μ(a ∩ c) for c ∈ Clop(^{ω}2). Then clearly e is an order-preserving map of ℝ(ω) into ^{Clop(^{ω}2)}ℝ.

To show that e is a strictly order-preserving, assume that μ(a \ b) > 0. Then there is c ∈ Clop(^{ω}2) such that μ((a \ b) ∩ c) > μ(c)/2. Then e(a)(c) > μ(c)/2 > e(b)(c), so e(a) ≰ e(b). (Proposition 5.1)

In a similar way, we can also show that ℝ(ω) is a retract of P(ω) as a partially ordered set though it is known that ℝ(ω) is not a retract of P(ω) as a Boolean algebra: the mapping e : ℝ(ω) → P(Clop(^{ω}2) × ℚ) defined by e(c) = {⟨a, q⟩ : a ∈ Clop(^{ω}2), q < μ(a ∩ c)} is easily seen to be a strictly order preserving embedding.

In general, the weak Freese-Nation property of ℂ(κ) or that of ℝ(κ) for arbitrary κ ≥ ω is not equivalent with the weak Freese-Nation property of ⟨P(ω), ⊆⟩. In the next section we shall give a model where ⟨P(ω), ⊆⟩ has the weak Freese-Nation property while ℂ(ℵ_{ω}) (and hence also ℝ(ℵ_{ω})) does not.

However the equivalence does hold if κ < ℵ_{ω} or some consequences of ¬0^{#} are available. To prove this, we need the following instance of Theorem 7 in [5]:

Theorem 5.2 (Theorem 7 in [5] for κ = ℵ_{1}) Suppose that μ > cf(μ) = ω, cf([λ]^{ℵ_{0}}, ⊆) = λ for cofinally many λ < μ and □_{ℵ_{1},μ}^{∗∗∗} holds. Then for any sufficiently large regular χ and x ∈ H(χ), there is a matrix ⟨M_{α,i} : α < μ^{+}, i < ω⟩ such that

(1) M_{α,i} ≺ H(χ), x ∈ M_{α,i}, ω_{1} ⊆ M_{α,i} and | M_{α,i} | < μ for all α < μ^{+} and i < ω;

(2) ⟨M_{α,i} : i < ω⟩ is an increasing sequence for each α < μ^{+};

(3) If α < μ^{+} and cf(α) ≥ ω_{1}, then there is an i^{∗} < ω such that for every i^{∗} ≤ i < ω, [M_{α,i}]^{ℵ_{0}} ∩ M_{α,i} is cofinal in ⟨[M_{α,i}]^{ℵ_{0}}, ⊆⟩;

(4) Let M_{α} = ⋃_{i<ω} M_{α,i} for α < μ^{+}. Then M_{α} ≺ H(χ) (by (1) and (2)). Moreover ⟨M_{α} : α < μ^{+}⟩ is continuously increasing and μ^{+} ⊆ ⋃_{α<μ^{+}} M_{α}.

For a complete Boolean algebra B and X ⊆ B, let us denote by ⟨X⟩_{B}^{cm} the complete subalgebra of B generated completely by X. The following theorem shows that in a neat model (in the sence of Section 1) if every countably generated c.c.c. Boolean algebra has the weak Freese-Nation property, then every c.c.c. Boolean algebra (without the restriction on the size of its set of generators) has the weak Freese-Nation property.

Theorem 5.3 Let λ be a cardinal. Suppose that for every μ < λ with μ > cf(μ) = ω, we have:

(i) cf([μ]^{ℵ_{0}}, ⊆) = μ^{+}; (ii) □_{ℵ_{1},μ}^{∗∗∗}. Then for any c.c.c. complete Boolean algebra B with a set of complete generators of size < λ, B has the weak Freese-Nation property if and only if every complete subalgebra of B generated completely by a countable subset of B has the weak Freese-Nation property.

Proof The “only if” part of the theorem follows from Proposition 1.2 (3). The “if” part of the theorem is proved by induction on the minimal cardinality of a subset X of B completely generating B.

If X is countable, then there is nothing to prove. Let μ = | X | < λ and suppose that we have the theorem for any c.c.c. complete Boolean algebra with a set of complete generators of size < μ.

If cf(μ) > ω, let X = {x_{α} : α < μ} and B_{β} = ⟨{x_{α} : α < β}⟩_{B}^{cm} for β < μ. B_{β} is a complete subalgebra of B and hence B_{β} ≤_{σ} B for all β < μ. By induction hypothesis, every B_{β}, β < μ has the weak Freese-Nation property. By the c.c.c. of B, we have B_{γ} = ⋃_{β<γ} B_{β} for all limit γ < μ with cf(γ) > ω and B = ⋃_{β<μ} B_{β}. By Proposition 1.2 (4), it follows that B has the weak Freese-Nation property.

If cf(μ) = ω, then there is ⟨M_{α,i} : α < μ^{+}, i < ω⟩ as in the previous theorem for x = ⟨B, X⟩.

For α < μ^{+} and i < ω, let B_{α,i} = ⟨B ∩ M_{α,i}⟩_{B}^{cm} and B_{α} = ⋃_{i<ω} B_{α,i}.

Claim 5.3.1 For every α < μ^{+}, B_{α} has the weak Freese-Nation property and B_{α} ≤_{σ} B.

⊢ For every i < ω, B_{α,i} has the weak Freese-Nation property by induction hypothesis. Since B_{α,i} is a complete subalgebra of B for every i < ω it follows that B_{α} = ⋃_{i<ω} B_{α,i} ≤_{σ} B. Also by Proposition 1.2 (4) it follows that B_{α} has the weak Freese-Nation property. ⊣ (Claim 5.3.1)

Claim 5.3.2 If γ < μ^{+} and cf(γ) > ω, then B_{γ} = ⋃_{α<γ} B_{α}.

⊢ Suppose that a ∈ B_{γ}. Then, by the c.c.c. of B, there is an i < ω and s ∈ [B ∩ M_{γ,i}]^{ℵ_{0}} such that a ∈ ⟨s⟩_{B}^{cm}. By (3) in Theorem 5.2, we may assume that s ∈ M_{γ,i}. By (4), there is α < γ and j < ω such that s ∈ M_{α,j}. It follows that s ⊆ M_{α,j} and hence a ∈ B_{α,j} ⊆ B_{α}. ⊣ (Claim 5.3.2)

Claim 5.3.3 B = ⋃_{α<μ^{+}} B_{α}.

⊢ By the last statement of (4) in Theorem 5.2 and (i), [X]^{ℵ_{0}} ∩⋃_{α<μ^{+}} M_{α} is cofinal in ⟨[X]^{ℵ_{0}}, ⊆⟩.

Suppose now that a ∈ B. Then by the c.c.c. of B, there is a countable s ∈ [X]^{ℵ_{0}} such that a ∈ ⟨s⟩_{B}^{cm}. By the remark above, we may assume that s ∈ ⋃_{α<μ^{+}} M_{α}, say s ∈ M_{α^{∗},i^{∗}} for some α^{∗} < μ^{+} and i^{∗} < ω. Then s ⊆ B ∩ M_{α^{∗},i^{∗}} and hence a ∈ B_{α^{∗},i^{∗}} ⊆ B_{α^{∗}}. ⊣ (Claim 5.3.3)

Now by Theorem 1.2 (4) and the claims above, it follows that B has the weak Freese-Nation property. (Theorem 5.3)

Corollary 5.4 Let λ be as in Theorem 5.3 and ω ≤ κ < λ. Then the following are equivalent:

(a) ⟨P(ω), ⊆⟩ has the weak Freese-Nation property;

(b) ℂ(κ) has the weak Freese-Nation property;

(c) ℝ(κ) has the weak Freese-Nation property.

Proof (1) ⇒ (0) and (2) ⇒ (0) follow from Proposition 1.2 (3).

For (0) ⇒ (2) assume that ⟨P(ω), ⊆⟩ has the weak Freese-Nation property. Since ℝ(κ) has the set Clop(^{κ}2) of complete generators of cardinality κ, it is enough by Theorem 5.3 to show that every countably generated subalgebra of ℝ(κ) has the weak Freese-Nation property. Let A be such an algebra. Then there is X ∈ [κ]^{ℵ_{0}} such that A is a complete subalgebra of ℝ(X). Since ℝ(X) has the weak Freese-Nation property by Proposition 5.1, it follows by Proposition 1.2 (3) that A also has the weak Freese-Nation property.

The proof of (0) ⇒ (1) is similar. (Corollary 5.4) Note that the conditions on λ in Theorem 5.3 hold vacuously for λ = ℵ_{ω}. Hence

we obtain the following as a special case of the corollary above:

Corollary 5.5 The following are equivalent (in ZFC):

(a) ⟨P(ω), ⊆⟩ has the weak Freese-Nation property;

(b) ℂ(ℵ_{n}) has the weak Freese-Nation property for some/all n ∈ ω;

(c) ℝ(ℵ_{n}) has the weak Freese-Nation property for some/all n ∈ ω.

6 Chang’s Conjecture

In this section, we give the negative answer to Problem 3 mentioned in the introduction (Theorem 6.1). We also show that Corollary 5.5 in the last section is an optimal result in ZFC (Theorem 6.2).

Theorem 6.1 Suppose that V_{0} is a transitive model of ZFC such that

V_{0} ⊧ GCH + (ℵ_{ω+1}, ℵ_{ω}) \overrightarrow{→} (ℵ_{1}, ℵ_{0}) .

Let P be a c.c.c. partial ordering in V_{0} of cardinality ℵ_{1} adding a dominating real. Let η ∈ ^{ω}ω be a dominating real over V_{0} generically added by P and let V_{1} = V_{0}[η]. Note that V_{1} ⊧ GCH. In V_{1} let Q = Fn(ℵ_{ω}, ω) and let Q̇ be a corresponding P -name. Then we have:

V_{1} ⊧ ∥–_{Q} “ ⟨P(ω), ⊆⟩ does not have the weak Freese-Nation property ”.

Proof In the proofs of this and the next theorem, we shall denote by a dotted symbol a name of an element in a generic extension. By the same symbol without the dot, we denote the corresponding element in a fixed generic extension. Without further mention, we shall identify P ∗ Q̇ names with the corresponding Q-name in V_{1} and vice versa.

Now toward a contradiction, assume that there is a Q-name Ḟ in V_{1} such that

(⊗) V_{1} ⊧ ∥–_{Q} “ Ḟ is a weak Freese-Nation mapping on ⟨^{ω}ω, ≤^{∗}⟩ ”.

Let φ̇ be a P ∗Q̇-name of the function ℵ_{ω} → ω generically added by Q over V_{1}. Let V_{2} = V_{1}[φ]. By GCH, we can find in V_{0} a scale ⟨f_{α} : α < ℵ_{ω+1}⟩ in ⟨∏_{n∈ω} ℵ_{n}, ≤^{∗}⟩. Without loss of generality, we may assume that for every α < ℵ_{ω+1} and n ∈ ω, f_{α}(n) ∈ ℵ_{n} \ ℵ_{n−1} (where we set ℵ_{−1} = ∅). For each α ∈ ℵ_{ω+1}, let g_{α} : ω → ω be defined by

g_{α} = φ ∘ f_{α} .

Let χ be sufficiently large and let N ≺ ⟨H(χ), ∈⟩ be such that N contains everything we need in the course of the proof — in particular ⟨f_{α} : α < ℵ_{ω+1}⟩ ∈ N, | ℵ_{ω} ∩ N | = ℵ_{0}, and otp(ℵ_{ω+1} ∩ N) = ω_{1}. The last two conditions are possible by the assumption of V_{0} ⊧ (ℵ_{ω+1}, ℵ_{ω}) \overrightarrow{→} (ℵ_{1}, ℵ_{0}).

In V_{0}, let {ξ_{n,k} : k ∈ ω} be an enumeration of (ℵ_{n} \ ℵ_{n−1}) ∩ N for each n ∈ ω. Here again, we use the convention that ℵ_{−1} = ∅. Let ḣ^{∗} be a P ∗ Q̇-name of an element of ^{ω}ω such that

∥–_{P ∗Q̇} “ ḣ^{∗}(n) = max{φ̇(ξ_{n,k}) : k ≤ η̇(n)} for all n ∈ ω ”.

Claim 6.1.1 For every α ∈ ℵ_{ω+1} ∩ N, ∥–_{P ∗Q̇} “ ġ_{α} ≤^{∗} ḣ^{∗} ”.

⊢ Since α ∈ N we have f_{α} ∈ N. Let e_{α} : ω → ω be the function in V_{0} such that f_{α}(n) = ξ_{n,e_{α}(n)} for all n ∈ ω. Since η is dominating, there is n^{∗} ∈ ω such that

V_{1} ⊧ e_{α}|` (ω \ n^{∗}) ≤ (η|` ω \ n^{∗}).

By definition of ḣ^{∗}, it follows that

V_{2} ⊧ g_{α}(n) = φ ∘ f_{α}(n) = φ(ξ_{n,e_{α}(n)}) ≤ h^{∗}(n)

for all n ≥ n^{∗}. ⊣ (Claim 6.1.1)

Let N_{0} = N, N_{1} = N_{0}[η] and N_{2} = N_{1}[φ]. Then we have N_{2} ≺ H(χ)[η][φ]. Let ḣ_{l} ∈ N_{0}, l ∈ ω be P ∗ Q̇-names such that

∥–_{P ∗Q̇} “ {ḣ_{l} : l ∈ ω} = Ḟ (ḣ^{∗}) ∩ Ṅ_{2} ”.

For l ∈ ω, Let S_{l} ∈ [ℵ_{ω}]^{ℵ_{0}} ∩ N_{0} be such that, regarding ḣ_{l} as a P -name of Q̇-name,

V_{1} ⊧ ∥–_{P} “ ḣ_{l} is a Fn(S_{l}, ω)-name ”.

This is possible since P has the c.c.c., ∥–_{P} “ Q̇ has the c.c.c. ” and by the elementarity of N_{0}. For each l ∈ ω, let s_{l} ∈ ∏_{n∈ω} ℵ_{n} ∩ N_{0} be defined (in N_{0}) by s_{l}(0) = 0 and

s_{l}(m) = sup S_{l} ∩ ℵ_{m}

for m ∈ ω \ {0}. Since ⟨f_{α} : α < ℵ_{ω+1}⟩ was taken to be a scale on ⟨∏_{n∈ω} ℵ_{n}, ≤^{∗}⟩, there is α_{l} ∈ ℵ_{ω+1}∩N_{0} for each l ∈ ω such that s_{l} ≤^{∗} f_{α_{l}}. Since otp(ℵ_{ω+1}∩N) = ω_{1}, we can find an α^{∗} ∈ ℵ_{ω+1} ∩ N_{0} such that sup{α_{l} : l ∈ ω} ≤ α^{∗}.

Now, by the choice of ḣ_{l}, l ∈ ω, the following claim together with Claim 6.1.1 contradicts to (⊗) and hence proves the theorem:

Claim 6.1.2 V_{1} ⊧ ∥–_{Q} “ ġ_{α^{∗}} ≰^{∗}ḣ_{l} for all l ∈ ω ”.

⊢ Assume to the contrary that, in V_{1}, we have

q ∥–_{Q} “ ġ_{α^{∗}}|` (ω \ k) ≤ ḣ_{l}|` (ω \ k) ”

for some q ∈ Q and k, l ∈ ω. We may assume that

s_{l}|` (ω \ k) < f_{α^{∗}}|` (ω \ k)

and sup(dom(q)) < f_{α^{∗}}(m) for all m ∈ ω \ k as well. Working further in V_{1}, let m^{∗} = k + 1 and let q′ ≤ q be such that

q′ ∥–_{Q} “ ḣ_{l}(m^{∗}) = j^{∗} ”

for some j^{∗} ∈ ω. Then q′′ = q ∪ (q′|` S_{l}) also forces the same statement. Since f_{α^{∗}}(m^{∗}) ∈ ℵ_{m^{∗}} \ (s_{l}(m^{∗}) ∪ dom(q)), we have

f_{α^{∗}}(m^{∗}) ∉ dom(q′′).

Hence

q^{∗} = q′′ ∪ {⟨f_{α^{∗}}(m^{∗}), j^{∗} + 1⟩}

is an element of Q and q^{∗} ≤ q′′ ≤ q. But

q^{∗} ∥–_{Q} “ ġ_{α^{∗}}(m^{∗}) = φ̇ ∘ f_{α^{∗}}(m^{∗}) = j^{∗} + 1 > j^{∗} = ḣ_{l}(m^{∗}). ”

This is a contradiction. ⊣ (Claim 6.1.2) (Theorem 6.1)

A similar but slightly simpler proof yields the following:

Theorem 6.2 Suppose that V_{0}, P, η, η̇, V_{1} are as in the previous theorem. Then

V_{1} ⊧ ℂ(ℵ_{ω}) does not have the weak Freese-Nation property.

Proof Suppose to the contrary that F ∈ V_{1} is a weak Freese-Nation mapping on ℂ(ℵ_{ω}). Let X = {x_{ξ} : ξ < ℵ_{ω}} be the canonical free subset of ℂ(ℵ_{ω}) completely generating the whole ℂ(ℵ_{ω}). We have X ∈ V_{0}.

Let ⟨f_{α} : α < ℵ_{ω+1}⟩ be as in the proof of the previous theorem. For n ∈ ω, let c_{n} = x_{n} · − ∑_{m<n} x_{m}. Note that {c_{n} : n ∈ ω} is a partition of ℂ(ℵ_{ω}). For each α < ℵ_{ω+1} and n ∈ ω, let

b_{α,n} = ∑_{m>n}(x_{f_{α}(m)} · c_{m}).

Let χ be sufficiently large and let N ≺ ⟨H(χ), ∈⟩ be such that N contains everything we need in the course of the proof, | ℵ_{ω} ∩ N | = ℵ_{0}, and otp(ℵ_{ω+1} ∩ N) = ω_{1}. The last two conditions are possible because of V_{0} ⊧ (ℵ_{ω+1}, ℵ_{ω}) \overrightarrow{→} (ℵ_{1}, ℵ_{0}). In V_{0}, let {ξ_{n,k} : k ∈ ω} be an enumeration of (ℵ_{n} \ ℵ_{n−1}) ∩ N for each n ∈ ω. In V_{1} = V_{0}[η] let

b^{∗} = ∑_{n∈ω}(c_{n} · ∑_{l≤η(n)}x_{ξ_{n,l}}) .

Claim 6.2.1 For every α ∈ ℵ_{ω+1} ∩ N, there is n_{α} ∈ ω such that b_{α,n_{α}} ≤ b^{∗}.

⊢ In V_{0}, let t_{α} ∈ ^{ω}ω be such that f_{α}(n) = ξ_{n,t_{α}(n)} for all n ∈ ω. Note that t_{α} is well-defined by f_{α} ∈ N. Since η is dominating, there is an n_{α} ∈ ω such that t_{α}|` (ω \ n_{α}) ≤ η|` (ω \ n_{α}). By definition of b_{α,n} and b^{∗}, it follows that b_{α,n_{α}} ≤ b^{∗}.

⊣ (Claim 6.2.1)

Now, let N_{1} = N[η]. Note that we have N_{1} ≺ H(χ)^{V_{1}}. Let ⟨d_{l} : l ∈ ω⟩ be an enumeration of F (b^{∗}) ∩ ℂ(ℵ_{ω})|` b^{∗} ∩ N_{1} and ḋ_{l}, l ∈ ω be corresponding P -names. Since P is proper, we can choose these names so that {ḋ_{l} : l ∈ ω} ⊆ N.

By c.c.c. of P and elementarity of N, we can find S_{l} ∈ [ℵ_{ω}]^{ℵ_{0}} ∩ N for each l ∈ ω such that

∥–_{P} “ ḋ_{l} ∈ ⟨{x_{ξ} : ξ ∈ S_{l}}⟩_{ℂ(ℵ_{ω})}^{cm} ”.

For l ∈ ω, let s_{l} ∈ ∏_{n∈ω} ℵ_{n} ∩ N be defined by s_{l}(0) = 0 and

s_{l}(m) = sup ℵ_{m} ∩ S_{l}

for m ∈ ω \ {0}. Since ⟨f_{α} : α < ℵ_{ω+1}⟩ was taken to be a scale on ⟨∏_{n∈ω} ℵ_{n}, ≤^{∗}⟩, there is α_{l}^{∗} ∈ ℵ_{ω+1} ∩ N for each l ∈ ω such that s_{l} <^{∗} f_{α_{l}^{∗}}. By otp(ℵ_{ω+1} ∩ N) = ω_{1}, we can find an α^{∗} ∈ ℵ_{ω+1} ∩ N such that α_{l}^{∗} < α^{∗} for all l ∈ ω.

Since b_{α^{∗},n_{α^{∗}}} ∈ N_{1} and b_{α^{∗},n_{α^{∗}}} ≤ b^{∗} by Claim 6.2.1, the following claim gives the desired contradiction:

Claim 6.2.2 In V_{1}, b_{α^{∗},n_{α^{∗}}} ≰ d_{l} for all l ∈ ω, .

⊢ For l ∈ ω, let m ∈ ω be such that (1) n_{α^{∗}} < m, and

(2) s_{l}(m) < f_{α^{∗}}(m). Note that we have c_{m} ≰ d_{l}, since d_{l} ≤ b^{∗} and c_{m} ≰ b^{∗} by definition of b^{∗}. By (2), f_{α^{∗}}(m) ∉ S_{l} (note that we also have ω < f_{α^{∗}}(m)). Hence x_{f_{α^{∗}}(m)} · c_{m} ≰ d_{l}. But x_{f_{α^{∗}}(m)} · c_{m} ≤ b_{α^{∗},n_{α^{∗}}} by (1) and by the definition of b_{α^{∗},n_{α^{∗}}}. It follows that b_{α^{∗},n_{α^{∗}}} ≰ d_{l}. ⊣ (Claim 6.2.2)

(Theorem 6.2)

References

[1] A. Blass, Combinatorial cardinal characteristics of the continuum, to appear in Handbook of Set-Theory.

[2] J. Brendle and S. Fuchino: Coloring ordinals by reals, preprint.

[3] M. Foreman and M. Magidor, A very weak square principle. J. Symbolic Logic 62, no. 1(1997), 175–196.

[4] S. Fuchino, S. Koppelberg and S. Shelah, Partial orderings with the weak Freese-Nation property, Annals of Pure and Applied Logic, 80 (1996), 35–54.

[5] S. Fuchino and L. Soukup, More set-theory around the weak Freese-Nation property, Fundamenta Matematicae 154 (1997), 159–176.

[6] S. Fuchino, S. Geschke and L. Soukup, On the weak Freese-Nation property of P(ω), to appear in Archive for Mathematical Logic.

[7] S. Geschke, On σ-filtered Boolean algebras, Dissertation, Berlin (1999).

[8] L. Heindorf and L.B. Shapiro, Nearly Projective Boolean Algebras, Springer Lecture Notes of Mathematics 1596, (1994).

[9] A. Hajnal, I. Juhász and S. Shelah, Splitting strongly almost disjoint families, Transactions of the American Mathematical Society 295 No.1 (1986), 369-397.

[10] S. Koppelberg, Applications of σ-filtered Boolean algebras, in: Advances in Algebra and Model Theory, eds.: M. Droste and R. Göbel, Gordon and Breach sc. publ. (1997), 199–213.

Authors’ addresses

Sakaé Fuchino Kitami Institute of Technology, Kitami, Japan.

fuchino@math.cs.kitami-it.ac.jp

Stefan Geschke II. Mathematisches Institut, Freie Universität Berlin, Germany.

geschke@math.cs.kitami-it.ac.jp

Saharon Shelah Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel. Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, U.S.A.

shelah@math.huji.ac.il

Lajos Soukup Institute of Mathematics, Hungarian Academy of Science, Budapest, Hungary.

soukup@math.cs.kitami-it.ac.jp