More set-theory around

the weak Freese-Nation property

Sakaé Fuchino^{0}, Lajos Soukup^{1}

April 29, 1996, revised on May 24, 1996

to appear in ‘Fundamenta Mathematicae’

Abstract

We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s conjecture for ℵ_{ω}, we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals, a lot of ccc complete Boolean algebras of cardinality ≤ λ have the ℵ_{1}-Freese-Nation property provided that μ^{ℵ_{0}} = μ holds for every regular uncountable μ < λ and the very weak square principle holds for each cardinal ℵ_{0} < μ < λ of cofinality ω (Theorem 15). Finally we prove that there is no ℵ_{2}-Lusin gap if P(ω) has the ℵ_{1}-Freese Nation property (Theorem 17).

0 Institut für Mathematik II, Freie Universität Berlin, 14195 Berlin, Germany. e-mail:

fuchino@math.fu-berlin.de (present address: Department of Computer Sciences, Kitami Institute of Technology, Kitami Hokkaido 090 Japan. e-mail: fuchino@info.kitami-it.ac.jp).

1 Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary. e-mail:

soukup@math-inst.hu. The second author was supported by the Hungarian National Foundation for Scientific Research grant no. 16391.

1 Introduction

For a regular κ, a partial ordering P is said to have the κ Freese-Nation property (the κ-FN for short) if there is a mapping (κ-FN mapping) f : P → [P]^{<κ} such that for any p, q ∈ P if p ≤ q then there is r ∈ f(p) ∩ f(q) such that p ≤ r ≤ q.

Freese and Nation [5] used the ℵ_{0}-FN in a characterization of projective lattices and asked if this property alone already characterizes the projectiveness. L. Heindorf gave a negative answer to the question showing that the Boolean algebras with the ℵ_{0}-FN are exactly those which are openly generated. It is known that the class of openly generated Boolean algebras contains projective Boolean algebras as a proper subclass (see [8] — openly generated Boolean algebras are called ‘rcfiltered’ there). Heindorf and Shapiro [8] then studied the ℵ_{1}-FN which they called the weak Freese-Nation property and proved some elementary properties of the Boolean algebras with this property. Partial orderings with the κ-FN for arbitrary regular κ were further studied in Fuchino, Koppelberg and Shelah [6]. Koppelberg [10] gives some nice applications of the ℵ_{1}-FN.

In the following we shall quote some elementary facts from [6] which we need later. First of all, it can be readily seen that every small partial ordering has the κ-FN:

Lemma 1 ([6]) Every partial ordering P of cardinality ≤ κ has the κ-FN.

For a partial ordering P and a sub-ordering Q ⊆ P, we say that Q is a κ-sub-ordering of P and denote it with Q ≤_{κ} P if, for every p ∈ P, the set { q ∈ Q : q ≤ p } has a cofinal subset of cardinality < κ and the set { q ∈ Q : q ≥ p } has a coinitial subset of cardinality < κ.

Lemma 2 ([6]) Suppose that δ is a limit ordinal and (P_{α})_{α≤δ} a continuously increasing chain of partial orderings such that P_{α} ≤_{κ} P_{δ} for all α < δ. If P_{α} has the κ-FN for every α < δ, then P_{δ} also has the κ-FN.

For application of Lemma 2, it is enough to have P_{α} ≤_{κ} P_{δ} and the κ-FN of P_{α} for every α < δ such that either α is a successor or of cofinality ≥ κ: P_{α} ≤_{κ} P_{δ} for α < δ of cofinality < κ follows from this since such P_{α} can be represented as the union of < κ many κ-sub-orderings of P_{δ}. Hence by inductive application of Lemma 2, we can show that P_{α} satisfies the κ-FN for every α ≤ δ. Similarly, if δ is a cardinal > κ, then it is enough to have P_{α} <_{κ} P_{δ} and the κ-FN of P_{α} for every limit α < δ of cofinality ≥ κ.

Proposition 3 ([6]) For a regular κ and a partial ordering P, the following are equivalent:

1) P has the κ-FN;

2) For some, or equivalently, any sufficiently large χ, if M ≺ H_{χ} = (H_{χ}, ∈) is such that P ∈ M, κ ⊆ M and |M| = κ then P ∩ M ≤_{κ} P holds;

3) { C ∈ [P]^{κ} : C ≤_{κ} P } contains a club set.

Though Proposition 3,2) is quite useful to show that a partial ordering has the κ-FN, sometimes it is quite difficult to check Proposition 3, 2) as in the case of the ℵ_{1}-FN of P (ω) or [κ]^{<ω}: in these cases it is independent if Proposition 3, 2) holds. Applications like Corollary 11 in mind, we could think of another possible variant of Proposition 3, 2) in terms of the following weakening of the notion of internal approachability from [4]: for a regular κ and a sufficiently large χ, we shall call an elementary submodel M of H_{χ} V_{κ}-like if, either κ = ℵ_{0} and M is countable, or there is an increasing sequence (M_{α})_{α<κ} of elementary submodels of M of cardinality less than κ such that M_{α} ∈ M_{α+1} for all α < κ and M = ⋃_{α<κ} M_{α}.

In [6], a characterization of the κ-FN using V_{κ}-like elementary submodels in place of elementary submodels in Proposition 3, 2) was discussed. Unfortunately it appeared that some consequences of ¬0^{#} are necessary for the characterization (see “Added in Proof” in [6]). In this paper, we introduce a weakening of the very weak square principle from [3] — the principle □_{κ,μ}^{∗∗∗}. In section 2 we show the equivalence of □_{κ,μ}^{∗∗∗} with the existence of a matrix (M_{α,β})_{α<μ^{+},β<cf(μ)} — which we called (weak) (κ, μ)-dominating matrix — of elementary submodels of H(χ) for sufficiently large χ with certain properties. This fact is used in section 3 to show that □_{κ,λ}^{∗∗∗} together with a very weak version of the Singular Cardinals Hypothesis yields the characterization of partial orderings with the κ-FN in terms of V_{κ}-like elementary submodels (Theorem 10). ZFC or even ZFC + GCH is not enough for this characterization: in section 4, we show that, under Chang’s conjecture for ℵ_{ω}, there is a counter-example to the characterization. Together with Theorem 10, this counter-example also shows that □_{ℵ_{1},ℵ_{ω}}^{∗∗∗} is not a theorem in ZFC + GCH.

One of the most natural questions concerning the κ-FN would be if (P(ω), ⊆) has the ℵ_{1}-FN. It is easy to see that (P(ω), ⊆) has the ℵ_{1}-FN iff (P(ω)/fin, ⊆^{∗}) does (see [6]). See also Koppelberg [10] for some consequences of the ℵ_{1}-FN of P(ω)/fin. By Lemma 1, P(ω) has the ℵ_{1}-FN under CH. In section 5, we show that P(ω) and a lot of other ccc complete Boolean algebras still have the ℵ_{1}-FN in a model obtained by adding arbitrary number of Cohen reals to a model of, say, V = L. On the other hand it can happen very easily that P(ω) does not have the ℵ_{1}-FN. In [6], it was shown that this is the case when b > ℵ_{1} or, more generally, if there is a ⊆^{∗}-sequence of elements of P(ω) of order-type ≥ ω_{2}. In section 6, we show that the existence of an ℵ_{2}-Lusin gap can be another reason for failure of the ℵ_{1}-FN. At the moment the authors do not know if there are yet other reasons for failure of the ℵ_{1}-FN of P(ω):

Problem 1 Suppose that P(ω) does not have any increasing chain of length ≥ ℵ_{2} with respect to ⊆^{∗} and that there is no ℵ_{2}-Lusin gap. Does it follow that P(ω) has the ℵ_{1}-FN ?

Our notation is fairly standard. The following are possible deviations from the standard: for C ⊆ κ, we denote with (C)′ the set of limit points of C other than κ. For an ordinal α, Lim(α) = { β < α : β is a limit ordinal }. For a partial ordering P, cf(P) = min{ |X| : X ⊆ P, X is cofinal in P }. [λ]^{<κ} = { X ⊆ λ : |X| < κ } is often seen as the partial ordering ([λ]^{<κ}, ⊆). If Q is a sub-ordering of a partial ordering P and p ∈ P then Q ↑ p = { q ∈ Q : q ≥ p } and Q |` p = { q ∈ Q : q ≤ p }. Let I be an infinite set. Adopting the notation of [12], we denote with Fn(I, 2) the standard partial ordering for adding |I|-many Cohen reals, i.e. Fn(I, 2) is the partial ordering { p : p is a mapping with dom(p) ∈ [I]^{<ℵ_{0}} and range(p) ⊆ 2 } with the inverse inclusion.

2 Very weak square and dominating matrix

For a cardinal μ, the weak square principle for μ (notation: □_{μ}^{∗}) is the statement: there is a sequence (C_{α})_{α∈Lim(μ^{+})} such that for every α ∈ Lim(μ^{+})

w1) C_{α} ⊆ P(α) and |C_{α}| ≤ μ; w2) every C ∈ C_{α} is club in α and if cf(α) < μ then otp(C) < μ; w3) for every C ∈ C_{α} and δ ∈ (C)′, we have C ∩ δ ∈ C_{δ}.

Clearly we have □_{μ} → □_{μ}^{∗}. Jensen [9] proved that □_{μ}^{∗} is equivalent to the existence of a special Aronszajn tree on μ^{+}. Ben-David and Magidor [2] showed that the weak square principle for a singular μ is actually weaker than the square principle: they constructed a model of □_{ℵ_{ω}}^{∗} and ¬□_{ℵ_{ω}} starting from a model with a supercompact cardinal.

Foreman and Magidor considered in [3] the following principle which is, e.g. under GCH, a weakening of □^{∗} principle: for a cardinal μ, the very weak square principle for μ holds if there is a sequence (C_{α})_{α<μ^{+}} and a club D ⊆ μ^{+} such that for every α ∈ D

v1) C_{α} ⊆ α, C_{α} is unbounded in α; v2) for all bounded x ∈ [C_{α}]^{<ω_{1}}, there is β < α such that x = C_{β}.

In this paper, we shall use the following yet weaker variant of the very weak square principle. For a regular cardinal κ and μ > κ, let □_{κ,μ}^{∗∗∗} be 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) ≥ κ

y1) C_{α} ⊆ α, C_{α} is unbounded in α; y2) [α]^{<κ} ∩ { C_{α′} : α′ < α } dominates [C_{α}]^{<κ} (with respect to ⊆). Since y2) remains valid when C_{α}’s for α ∈ D are shrunk, we may replace y1) by

y1’) C_{α} ⊆ α, C_{α} is unbounded in α and otp(C_{α}) = cf(α).

A corresponding remark holds also for the sequence of the very weak square principle.

Lemma 4 a) The very weak square principle for μ implies □_{ω_{1},μ}^{∗∗∗}.

b) For a singular cardinal μ and a regular κ such that cf([λ]^{<κ}, ⊆) ≤ μ for every λ < μ, □_{μ}^{∗} implies □_{κ,μ}^{∗∗∗}.

Proof a) is clear. For b), let (C_{α})_{α∈Lim(μ^{+})} be a weak square sequence. Let C_{α} = { C_{α,β} : β < μ } for every α ∈ Lim(μ^{+}). By shrinking C_{α,β}’s if necessary, we may assume that |C_{α,β}| < μ for every α ∈ Lim(μ^{+}) and β < μ. Note that we need here the assumption that mu be singular. For α ∈ Lim(μ^{+}) and β < μ, let X_{α,β} be a cofinal subset of [C_{α,β}]^{<κ} of cardinality ≤ μ and let { C_{α} : α ∈ μ^{+} \ Lim(μ^{+}) } be an enumeration of ⋃{ X_{α,β} : α ∈ Lim(μ^{+}), β < μ }. For each α ∈ Lim(μ^{+}), let C_{α} = C_{α,0}. Let F : μ^{+} → μ^{+} be defined by F (ξ) = min{ γ < μ^{+} : X_{ξ,β} ⊆ { C_{α} : α < γ } for every β < μ }. Let D ⊆ μ^{+} be a club set closed with respect to F. Then (C_{α})_{α<μ^{+}} and D are as in the definition of □_{κ,μ}^{∗∗∗}. To see that y2) is satisfied, let α ∈ D be such that cf(α) ≥ κ and x ∈ [C_{α}]^{<κ}. By definition we have C_{α} = C_{α,0}. Hence there are α′ ∈ α ∩ Lim(μ^{+}) and β < λ such that x ∈ [C_{α′,β}]^{<κ}. Since α is closed with respect to F, there is some γ < α such that x ⊆ C_{γ} ∈ [C_{α′,β}]^{<κ}.

(Lemma 4)

□_{κ,μ}^{∗∗∗} has some influence on cardinal arithmetic of cardinals below μ:

Lemma 5 Suppose that κ is regular and μ is such that cf(μ) < κ. If □_{κ,μ}^{∗∗∗} holds, then we have cf([λ]^{<κ}, ⊆) < μ for every λ < μ.

Proof Let (C_{α})_{α<μ^{+}}, and D ⊆ μ^{+} be witnesses of □_{κ,μ}^{∗∗∗}. For λ < μ, let δ ∈ D be such that cf(δ) ≥ λ + κ. Then { C_{α} : α < δ } ∩ [δ]^{<κ} is cofinal in [C_{δ}]^{<κ}. Since |δ| ≤ μ it follows that cf([C_{δ}]^{<κ}, ⊆) ≤ μ. As the order type of C_{δ} is at least λ, it follows that ν = cf([λ]^{<κ}, ⊆) ≤ μ. But cf(ν) ≥ κ. Hence ν < μ. (Lemma 5)

Suppose now that κ is a regular cardinal and μ > κ is such that cf(μ) < κ. Let μ^{∗} = cf(μ). For a sufficiently large χ and x ∈ H(χ), let us call a sequence (M_{α,β})_{α<μ^{+},β<μ^{∗}} a (κ, μ)-dominating matrix over x — or just dominating matrix over x if it is clear from the context which κ and μ are meant — if the following conditions hold:

j1) M_{α,β} ≺ H(χ), x ∈ M_{α,β}, κ + 1 ⊆ M_{α,β} and |M_{α,β}| < μ for all α < μ^{+} and

β < μ^{∗}; j2) (M_{α,β})_{β<μ^{∗}} is an increasing sequence for each α < μ^{+} ; j3) if α < μ^{+} is such that cf(α) ≥ κ, then there is β^{∗} < μ^{∗} such that, for every

β^{∗} ≤ β < μ^{∗}, [M_{α,β}]^{<κ} ∩ M_{α,β} is cofinal in ([M_{α,β}]^{<κ}, ⊆) ;

For α < μ^{+}, let M_{α} = ⋃_{β<μ^{∗}} M_{α,β}. By j1) and j2), we have M_{α} ≺ H(χ).

j4) (M_{α})_{α<μ^{+}} is continuously increasing and μ^{+} ⊆ ⋃_{α<μ^{+}} M_{α}.

Foreman and Magidor [3] called a sequence of subsets of μ^{+} having some properties similar to those of the sequence (μ^{+} ∩ M_{α,β})_{α<μ^{+},β<μ^{∗}} for (M_{α,β})_{α<μ^{+},β<μ^{∗}} as above Jensen matrix. Our definition of dominating matrices and some ideas in the proofs in this section are inspired by their paper. Note that, in the case of 2^{<κ} = κ, j3) can be replaced by the following seemingly stronger property:

j3’) if α < μ^{+} is such that cf(α) ≥ κ, then there is β^{∗} < μ^{∗} such that, for every β^{∗} ≤ β < μ^{∗}, [M_{α,β}]^{<κ} ⊆ M_{α,β}.

This is simply because of the following observation:

Lemma 6 Suppose that 2^{<κ} = κ and M is an elementary submodel of H(χ) for some sufficiently large χ and κ ⊆ M. If [M]^{<κ} ∩ M is cofinal in [M]^{<κ}, then we have [M]^{<κ} ⊆ M.

Proof Let x ∈ [M]^{<κ}. We show that x ∈ M. By assumption there is y ∈ [M]^{<κ} ∩ M such that x ⊆ y. Let η = |P(y)|. Then η ∈ M and η ≤ κ. Let (y_{α})_{α<η} ∈ M be an enumeration of P(y). Then there is an α_{0} < η such that x = y_{α_{0}}. But since κ ⊆ M, we have α_{0} ∈ M and y_{α_{0}} ∈ M as well. (Lemma 6)

Note also that, if M ≺ H(χ) is V_{κ}-like, then [M]^{<κ} ∩ M is cofinal in [M]^{<κ}. Hence, under 2^{<κ} = κ, M ≺ H(χ) is V_{κ}-like if and only if |M| = κ and [M]^{<κ} ⊆ M.

In the following theorem, we show that □_{κ,μ}^{∗∗∗} together with a very weak version of the Singular Cardinals Hypothesis below μ implies the existence of a dominating matrix:

Theorem 7 Suppose that κ is a regular cardinal and μ > κ is such that cf(μ) < κ. If we have cf([λ]^{<κ}, ⊆) = λ for cofinally many λ < μ and □_{κ,μ}^{∗∗∗} holds, then, for any sufficiently large χ and x ∈ H(χ), there is a (κ, μ)-dominating matrix over x.

Proof Let μ^{∗} = cf(μ) and (μ_{β})_{β<μ^{∗}} be an increasing sequence of cardinals below μ such that μ_{0} > μ^{∗}, sup{ μ_{β} : β < μ^{∗} } = μ and cf([μ_{β}]^{<κ}, ⊆) = μ_{β} for every β < μ^{∗}. Let (C_{α})_{α∈μ^{+}} and D ⊆ μ^{+} be as in the definition of □_{κ,μ}^{∗∗∗}. Without loss of generality, we may assume that |C_{α}| ≤ cf(α) for all α < μ^{+}. We may also assume that α > μ for every α ∈ D.

In the following, we fix a well ordering ≤ on H(χ) and, when we talk about H(χ) as a structure, we mean H(χ) = (H(χ), ∈, ≤). X ⊆ H(χ) as a substructure of H(χ) is thus the structure (X, ∈ ∩X^{2}, ≤ ∩ X^{2}) — for notational simplicity we shall denote such a structure simply by (X, ∈, ≤).

Let N ∈ H(χ) be an elementary substructure of H(χ) such that N contains every thing needed below — in particular, we let μ^{+} ⊆ N and x, (C_{α})_{α<μ^{+}}, D, (μ_{α})_{α<μ^{∗}} ∈ N. Let (N_{ξ})_{ξ<κ} be an increasing sequence of elementary submodels of H(χ) such that

0) N_{0} = N ;

1) N_{ξ} ∈ H(χ) for every ξ < κ and

2) (N_{η})_{η≤ξ} ∈ N_{ξ+1} for every ξ < κ.

Now, for each ξ < κ, let

N_{ξ} = (N_{ξ}, ∈, ≤, R_{ξ}, κ, μ, μ^{∗}, D, (C_{α})_{α<μ^{+}}, (μ_{α})_{α<μ^{∗}}, x, η)_{η<ξ}

where R_{ξ} is the relation { (η, N_{η}) : η < ξ }. For X ⊆ μ^{+}, let us denote with sk_{ξ}(X) the Skolem-hull of X with respect to the built-in Skolem functions of N_{ξ} (induced from ≤). For ξ < ξ′ < κ, N_{ξ} is an element of N_{ξ′} by 2) and the Skolem functions of N_{ξ} are also elements of N_{ξ′}. In particular, we have sk_{ξ}(X) ⊆ sk_{ξ′}(X). It follows that sk(X) = ⋃_{ξ<κ} sk_{ξ}(X) is an elementary submodel of H(χ). Note also that, if X ⊆ μ^{+} is an element of sk_{ξ′}(Y) then, since sk_{ξ}(X) is definable in sk_{ξ′}(Y), we have sk_{ξ}(X) ∈ sk_{ξ′}(Y).

For the proof of the theorem, it is clearly enough to construct M_{α,β} with j1) — j4) for every α in the club set D and for every β < μ^{∗}. Let

M_{α,β} = sk(μ_{β} ∪ C_{α})

for α ∈ D and β < μ^{∗}. We show that (M_{α,β})_{α∈D,β<μ^{∗}} is as desired. It is clear that j1) and j2) hold. We need the following claim to show the other properties:

Claim 7.1 M_{α} = sk(α) for every α ∈ D.

⊢ “⊆” is clear since μ_{β} ∪ C_{α} ⊆ α for every α ∈ D and β < μ^{∗}. For “⊇”, it is enough to show that α ⊆ M_{α}. Let γ < α. By y1), there is γ_{1} ∈ C_{α} such that γ < γ_{1}. Let f ∈ M_{α,0} be a surjection from μ to γ_{1}, and let δ < μ be such that f(δ) = γ. Then we have γ = f(δ) ∈ M_{α,β^{∗}} ⊆ M_{α} for β^{∗} < μ^{∗} such that δ < μ_{β^{∗}}.

⊣ (Claim 7.1)

For j3), suppose that α ∈ D and cf(α) ≥ κ. Let β^{∗} be such that |C_{α}| < μ_{β^{∗}} and [α]^{<κ}∩{ C_{α′} : α′ < α, α′ ∈ M_{α,β^{∗}} } dominates [C_{α}]^{<κ}. The last property is possible by y2), Claim 7.1 and μ^{∗} < κ. We show that this β^{∗} is as needed in j3). Let β < μ^{∗} be such that β^{∗} ≤ β and suppose that x ∈ [M_{α,β}]^{<κ}. Then there are u ∈ [μ_{β}]^{<κ} and v ∈ [C_{α}]^{<κ} such that x ⊆ sk(u ∪ v). Since μ_{β} ∈ M_{α,β} and cf([μ_{β}]^{<κ}, ⊆) = μ_{β}, there is X ∈ M_{α,β} such that X ⊆ [μ_{β}]^{<κ}, |X| = μ_{β} and X is cofinal in ([μ_{β}]^{<κ}, ⊆). Since μ_{β} ⊆ M_{α,β}, it follows that X ⊆ M_{α,β}. Hence there is u′ ∈ M_{α,β} ∩ [μ_{β}]^{<κ} such that u ⊆ u′. On the other hand, by definition of β^{∗}, cf(α) ≥ κ, there is α′ ∈ α ∩ M_{α,β} such that C_{α′} ∈ [α]^{<κ} and v ⊆ C_{α′}. Thus, we have x ⊆ sk(u∪v) ⊆ sk(u′ ∪C_{α′}). By regularity of κ, there is ξ < κ such that x ⊆ sk_{ξ}(u′ ∪C_{α′}). But sk_{ξ}(u′ ∪C_{α′}) ∈ M_{α,β} and |sk_{ξ}(u′ ∪ C_{α′})| < κ.

j4) follows immediately from Claim 7.1. (Theorem 7)

Note that, in the proof above, the sequence (M_{α,β})_{α<μ^{+},β<μ^{∗}} satisfies also:

j5) for α < α′ < μ^{+} and β < μ^{∗}, there is β′ < μ^{∗} such that M_{α,β} ⊆ M_{α′,β′}.

[ Suppose that α, α′ ∈ D are such that α < α′ and β < μ^{∗}. By Claim 7.1, there is β′ < μ^{∗} such that β < β′, α ∈ M_{α′,β′} and otp(C_{α}) ≤ μ_{β′}. Then we have C_{α} ∈ M_{α′,β′} and C_{α} ⊆ M_{α′,β′}. Also μ_{β} ∈ M_{α′,β′} and μ_{β} ⊆ μ_{β′} ⊆ M_{α′,β′}. Hence it follows that M_{α,β} = sk(μ_{β} ∪ C_{α}) ⊆ M_{α′,β′}.]

Conversely, the existence of a (κ, μ)-dominating matrix (over some/any x) implies □_{κ,μ}^{∗∗∗}:

Theorem 8 Suppose that κ is a regular cardinal and μ > κ is such that cf(μ) < κ. If there exists a (κ, μ)-dominating matrix, then □_{κ,μ}^{∗∗∗} holds.

Proof Let μ^{∗} = cf(μ) and (M_{α,β})_{α<μ^{+},β<μ^{∗}} be a (κ, μ)-dominating matrix. Let M_{α} = ⋃_{β<μ^{∗}} M_{α,β} for each α < μ^{+}. For

X = ⋃{ [M_{α,β}]^{<κ} ∩ M_{α,β} : α < μ^{+}, β < μ^{∗} },

let { C_{α+1} : α < μ^{+} } be an enumeration of X. Let

D = { α < μ^{+} : M_{α} ∩ μ^{+} = α, { C_{α′+1} : α′ < α } ⊇ [M_{α^{†},β}]^{<κ} ∩ M_{α^{†},β} for every α^{†} < α, β < μ^{∗} }.

By j4), D is a club subset of μ^{+}. For α ∈ D with cf(α) ≥ κ, let C_{α} = M_{α,β_{α}} ∩ α where β_{α} < μ^{+} be such that M_{α,β_{α}} ∩α is cofinal in α (this is possible as M_{α}∩μ^{+} = α and μ^{∗} < κ) and that [M_{α,β_{α}}]^{<κ}∩M_{α,β_{α}} is cofinal in [M_{α,β_{α}}]^{<κ} (possible by j3) ). For α ∈ Lim(μ^{+}) \ { α ∈ D : cf(α) ≥ κ }, let C_{α} be anything, say C_{α} = ∅. We claim that (C_{α})_{α<μ^{+}} and D as above satisfy the conditions in the definition of □_{κ,μ}^{∗∗∗}: y1) is clear by definition of C_{α}’s. To show y2), let α ∈ D be such that cf(α) ≥ κ and x ∈ [C_{α}]^{<κ}. Then, by the choice of β_{α}, there is y ∈ [α]^{<κ} ∩ M_{α,β_{α}} such that x ⊆ y. By j4), there are α^{†} < α and β^{†} < μ^{∗} such that y ∈ M_{α^{†},β^{†}}. By definition of D, it follows that y = C_{α′+1} for some α′ < α. This shows that [α]^{<κ} ∩ { C_{α′} : α′ < α } dominates [C_{α}]^{<κ}. (Theorem 8)

Thus, by Theorem 7 and Theorem 8, if cf([λ]^{<κ}, ⊆) = λ for cofinally many λ < μ, □_{κ,μ}^{∗∗∗} is equivalent to the existence of a (κ, μ)-dominating matrix. The assumption “cf([λ]^{<κ}, ⊆) = λ for cofinally many λ < μ” cannot be removed from this equivalence theorem since □_{κ,μ}^{∗∗∗} implies this. However, using the following weakening of the notion of dominating matrix, we obtain a characterization of □_{κ,μ}^{∗∗∗} in ZFC: for a regular cardinal κ and μ > κ such that μ^{∗} = cf(μ) < κ, let us call a matrix (M_{α,β})_{α<μ^{+},β<μ^{∗}} of elementary submodels of H(χ) for a sufficiently large χ, a weak (κ, μ)-dominating matrix over x, if it satisfies j1), j2), j4) for M_{α} = ⋃_{β<μ^{∗}} M_{α,β}, α < μ^{+}, and

j3^{−}) if α < μ^{+} is such that cf(α) ≥ κ, then there is β^{∗} < μ^{∗} such that, for every β^{∗} ≤ β < μ^{∗}, [M_{α,β}]^{<κ} ∩ M_{α} is cofinal in ([M_{α,β}]^{<κ}, ⊆) .

Since μ^{∗} < κ, the condition above is equivalent to:

j3^{∗}) if α < μ^{+} is such that cf(α) ≥ κ, then there is β^{∗} < μ^{∗} such that, for every β^{∗} ≤ β < μ^{∗}, there is β′ < μ^{∗} such that [M_{α,β}]^{<κ} ∩ M_{α,β′} is cofinal in

([M_{α,β}]^{<κ}, ⊆) .

Theorem 9 Suppose that κ is a regular cardinal and μ > κ is such that μ^{∗} = cf(μ) < κ. Then □_{κ,μ}^{∗∗∗} holds if and only if there is a weak (κ, μ)-dominating matrix over some/any x.

Proof For the forward direction the proof is almost the same as that of Theorem 7. We let (μ_{β})_{β<μ^{∗}} here merely an increasing sequence of regular cardinals with the limit μ. Then (M_{α,β})_{α∈D,β<μ^{∗}} is constructed just as in the proof of Theorem 7. Lemma 5 is then used to see that j3^{−}) is satisfied by this matrix. For the converse, just the same proof as that of Theorem 8 will do. (Theorem 9)

Existence of a (weak) dominating-matrix is not a theorem in ZFC: we show in section 4 that the Chang’s conjecture for ℵ_{ω} together with 2^{ℵ_{ω}} = ℵ_{ω+1} implies that there is no (ℵ_{n}, ℵ_{ω})-dominating matrix for any n ≥ 1.

3 A characterization of the κ-Freese-Nation property

The following game over a partial ordering P was considered in [6, 7]. Let G^{κ}(P) be the following game played by Players I and II: in a play in G^{κ}(P), Players I and II choose subsets X_{α} and Y_{α} of P of cardinality less than κ alternately for α < κ such that

X_{0} ⊆ Y_{0} ⊆ X_{1} ⊆ Y_{1} ⊆ ⋯ ⊆ X_{α} ⊆ Y_{α} ⊆ ⋯ ⊆ X_{β} ⊆ Y_{β} ⊆ ⋯

for α ≤ β < κ. Thus a play in G^{κ}(P) looks like

Player I : X_{0}, X_{1}, …, X_{α}, …
Player II : Y_{0}, Y_{1}, …, Y_{α}, …

where α < κ. Player II wins the play if ⋃_{α<κ} X_{α} = ⋃_{α<κ} Y_{α} is a κ-sub-ordering of P. Let us call a strategy τ for Player II simple if, in τ, each Y_{α} is decided from the information of the set X_{α} ⊆ P alone (i.e. also independent of α).

Another notion we need here is the following generalization of V_{κ}-likeness. Let κ be regular and χ be sufficiently large. For D ⊆ { M ≺ H(χ) : |M| < κ }, we say that M ∈ [H(χ)]^{κ} is D-approachable if there is an increasing sequence (D_{α})_{α<κ} of elements of D such that

a) D_{α} ∪ { D_{α} } ⊆ D_{α+1} for every α < κ; and

b) M = ⋃_{α<κ} D_{α}. Clearly M ≺ H(χ) is V_{κ}-like if and only if M is D-approachable for D = { M ≺ H(χ) : |M| < κ }.

A slightly weaker version of the following theorem was announced in [6]:

Theorem 10 Let κ be a regular uncountable cardinal and κ ≤ λ. Suppose that i) ([μ]^{<κ}, ⊆) has a cofinal subset of cardinality μ for every μ such that κ < μ < λ and cf(μ) ≥ κ; and ii) □_{κ,μ}^{∗∗∗} holds for every μ such that κ ≤ μ < λ and cf(μ) < κ. Then, for a partial ordering P of cardinality ≤ λ, the following are equivalent:

1) P has the κ-FN ;

2) Player II has a simple winning strategy in G^{κ}(P) ;

3) for some, or equivalently any sufficiently large χ, and any V_{κ}-like M ≺ H(χ) with P, κ ∈ M, we have P ∩ M ≤_{κ} P ;

4) for some, or equivalently any sufficiently large χ, there is D ⊆ [H(χ)]^{<κ} such that D is cofinal in [H(χ)]^{<κ} and for any D-approachable M ⊆ H(χ), we have P ∩ M ≤_{κ} P.

Note that ¬0^{#} implies the conditions i) and ii). Also note that, for every λ < κ^{+ω}, the condition i) holds in ZFC. Hence the characterization above holds for partial orderings of cardinality ≤ κ^{+ω} without any additional assumptions.

Proof A proof of 1) ⇒ 2) ⇒ 3) is given in [6]. For 3) ⇒ 4), suppose that P satisfies 3). Then P together with D = { M ≺ H(χ) : |M| < κ, P, κ ∈ M } satisfies 4). The proof of 4) ⇒ 1) is done by induction on ν = |P | ≤ λ. If ν ≤ κ, then P has the κ-FN by Lemma 1. For ν > κ, let P and D be as in 4) and assume that 4) ⇒ 1) holds for every partial ordering of cardinality < ν. We need the following claims:

Claim 10.1 Let χ^{∗} be sufficiently large above χ. Suppose that M is an elementary submodel of H(χ^{∗}) such that P, H(χ), D ∈ M, κ + 1 ⊆ M and [M]^{<κ} ∩ M is cofinal in [M]^{<κ} with respect to ⊆. Then we have P ∩ M ≤_{κ} P.

⊢ Suppose not. then there is b ∈ P such that either

a) (P ∩ M) |` b has no cofinal subset of cardinality < κ; or

b) (P ∩ M) ↑ b has no coinitial subset of cardinality < κ.

To be definite, let us assume that we have the case a) — for the case b), just the same argument will do. We can construct an increasing sequence (N_{α})_{α<κ} of elements of D such that

c) N_{α} ∈ M and |N_{α}| < κ for α < κ (since κ + 1 ⊆ M, it follows that N_{α} ⊆ M) ;

d) N_{α} ∈ N_{α+1} for every α < κ;

e) (P ∩ N_{α}) |` b is not cofinal in (P ∩ N_{α+1}) |` b for every α < κ.

Then N = ⋃_{α<κ} N_{α} is D-approachable elementary submodel of H(χ) by c) and d). Hence, by 4), we have P ∩ M ≤_{κ} P. But, by e), (P ∩ N) |` b has no cofinal subset of cardinality < κ. This is a contradiction.

To see that the construction of N_{γ} is possible at a limit γ < κ, assume that N_{α}, α < γ have been constructed in accordance with c),d) and e). Let N′ = ⋃_{α<γ} N_{α}. By c), we have N′ ⊆ M and |N′| < κ. Since [M]^{<κ} ∩ M is cofinal in [M]^{<κ}, there is some N′′ ∈ M such that N′ ⊆ N′′ ⊆ H(χ) and |N′′| < κ. Hence by elementarity of M and by D ∈ M there is N′′′ ∈ M ∩ D such that N′′ ⊆ N′′′. Clearly, we may let N_{γ} = N′′′. For the construction at a successor step, assume that N_{α} have been chosen in accordance with c),d) and e). By assumption, there is c ∈ (P ∩ M) |` b such that there is no c′ ∈ (P ∩ N_{α}) |` b with c ≤ c′. By elementarity of M there is N^{∗} ∈ M ∩ D such that N_{α} ∪ { N_{α}, c } ⊆ N^{∗} and |N^{∗}| < κ. Then N_{α+1} = N^{∗} is as desired. ⊣ (Claim 10.1)

Claim 10.2 If Q ≤_{κ} P, then for every D-approachable M ≺ H(χ) with Q ∈ M we have Q ∩ M ≤_{κ} Q. In particular, such Q also satisfies the condition 4).

⊢ Let M ≺ H(χ) be D-approachable with Q ∈ M. By assumption, we have P ∩ M ≤_{κ} P. By elementarity of M and since Q ∈ M, we have Q ∩ M ≤_{κ} P ∩ M. It follows that Q ∩ M ≤_{κ} P and hence Q ∩ M ≤_{κ} Q. Now, let D_{0} = { M ∈ D : Q ∈ M }. Then it is clear that Q satisfies the condition 4) with D_{0} in place of D.

⊣ (Claim 10.2)

Now we are ready to prove the induction steps. Case I : ν is a limit cardinal or ν = μ^{+} with cf(μ) ≥ κ. Let ν^{∗} = cf ν. Then, by i), we can find an increasing sequence (M_{α})_{α<ν^{∗}} of elementary submodels of H(χ^{∗}) such that, for every α < ν^{∗}, |M_{α}| < ν and M_{α} satisfies the conditions in Claim 10.1; and P ⊆ ⋃_{α<ν^{∗}} M_{α}. By Claim 10.1, P ∩ M_{α} ≤_{κ} P for every α < ν^{∗}. For α < ν^{∗} let

P_{α} = \{ \begin{matrix}P ∩ M_{α}, & if α is a successor, \\ P ∩ (⋃_{β<α} M_{β}), & otherwise. \end{matrix}

Then (P_{α})_{α<ν^{∗}} is a continuously increasing sequence of sub-orderings of P such that |P_{α}| < ν for every α < ν^{∗} and P = ⋃_{α<ν^{∗}} P_{α}. We have also P_{α} ≤_{κ} P for every α < ν^{∗}: for a successor α < ν^{∗} this is clear. If a limit α < ν^{∗} has cofinality < κ then P_{α} can be represented as the union of an increasing sequence of < κ many κ-sub-ordering of P and hence P_{α} ≤_{κ} P. If α < ν^{∗} is a limit with cofinality ≥ κ, then P_{α} = P ∩ M where M = ⋃_{β<α} M_{β}. Now it is clear that M satisfies the conditions in Claim 10.1. Hence we again obtain that P_{α} = P ∩ M ≤_{κ} P. Now, by Claim 10.2, each of P_{α}, α < ν^{∗} satisfies the condition 4) and hence, by induction hypothesis, has the κ-FN. Thus, by Lemma 2, P also has the κ-FN.

Case II : ν = μ^{+} with cf(μ) < κ. Let μ^{∗} = cf(μ). Without loss of generality we may assume that the underlying set of P is ν. By Theorem 7, there is a (κ, μ)-dominating matrix (M_{α,β})_{α<ν,β<μ^{∗}} over (P, H(χ)). For α < ν and β < μ^{∗}, let P_{α,β} = P ∩M_{α,β} and P_{α} = ⋃_{β<μ^{∗}} P_{α,β}. By j4), the sequence (P_{α})_{α<ν} is continuously increasing and ⋃_{α<ν} P_{α} = P. |P_{α}| ≤ μ for every α < ν by j1).

Claim 10.3 P_{α} ≤_{κ} P for every α < ν.

⊢ For α < ν such that cf(α) ≥ κ, we have P_{α,β} ≤_{κ} P for every sufficiently large β < μ^{∗} by j3) and Claim 10.1. Since μ^{∗} < κ, it follows that P_{α} ≤_{κ} P. If cf(α) < κ, then, by the argument above, we have P_{α′} ≤_{κ} P for every α′ < α with cf(α′) ≥ κ. Since P_{α} can be represented as the union of < κ many of such P_{α′}’s, it follows again that P_{α} ≤_{κ} P. ⊣ (Claim 10.3)

Now, by Claim 10.2, each of P_{α}, α < ν satisfies the condition of 4). Hence, by induction hypothesis, they have the κ-FN. By Lemma 2, it follows that P also has the κ-FN. (Theorem 10)

Corollary 11 Suppose that κ and λ satisfy i), ii) in Theorem 10 and 2^{<κ} = κ. Then:

a) Every κ-cc complete Boolean algebra of cardinality ≤ λ has the κ-FN.

b) For any μ such that μ^{<κ} ≤ λ, the partial ordering ([μ]^{<κ}, ⊆) has the κ-FN.

Proof Let χ be sufficiently large. For a), let B be a κ-cc complete Boolean algebra. We show that B satisfies 3) in Theorem 10. Let M ≺ H(χ) be V_{κ}-like with B, κ ∈ M. By 2^{<κ} = κ, Lemma 6 and by the remark after the lemma, we have [M]^{<κ} ⊆ M. Hence B ∩ M is a complete subalgebra of B. It follows that B ∩ M ≤_{κ} M. For b), let M ≺ H(χ) be V_{κ}-like with λ, κ ∈ M. Then as above we have [M]^{<κ} ⊆ M. Hence, letting X = λ ∩ M, we have [λ]^{<κ} ∩ M = [X]^{<κ} ≤_{κ} [λ]^{<κ}.

(Corollary 11)

4 Chang’s Conjecture for ℵ_{ω}

Recall that (κ, λ) \overrightarrow{→} (μ, ν) is the following assertion:

For any structure A = (A, U,…) of countable signature with |A| = κ, U ⊆ A and |U| = λ, there is an elementary substructure A′ = (A′, U′,…) of A such that |A′| = μ and |U′| = ν.

In [13], a model of ZFC + GCH + Chang’s Conjecture for ℵ_{ω}, i.e. (ℵ_{ω+1}, ℵ_{ω}) \overrightarrow{→} (ℵ_{1}, ℵ_{0}), is constructed starting from a model with a cardinal having a property slightly stronger than huge. The following theorem together with Corollary 11 shows that the ℵ_{1}-FN of the partial ordering ([ℵ_{ω}]^{ℵ_{0}}, ⊆) is independent from ZFC (or even from ZFC + GCH).

Theorem 12 Suppose that (ℵ_{ω})^{ℵ_{0}} = ℵ_{ω+1} and (ℵ_{ω+1}, ℵ_{ω}) \overrightarrow{→} (ℵ_{1}, ℵ_{0}). Then ([ℵ_{ω}]^{ℵ_{0}}, ⊆) does not have the ℵ_{1}-FN.

Proof Assume to the contrary that there is an ℵ_{1}-FN mapping F : [ℵ_{ω}]^{ℵ_{0}} → [[ℵ_{ω}]^{ℵ_{0}}]^{ℵ_{0}}. Let us fix an enumeration (b_{α})_{α<ℵ_{ω+1}} of [ℵ_{ω}]^{ℵ_{0}} and consider the structure:

A = (ℵ_{ω+1}, ℵ_{ω}, ≤, E, f, g, h),

where

1) ≤ is the canonical ordering on ℵ_{ω+1};

2) E = { (α, β) : α ∈ ℵ_{ω}, β ∈ ℵ_{ω+1}, α ∈ b_{β} };

3) f : ℵ_{ω+1} × ℵ_{ω+1} → ℵ_{ω+1} is such that, for each α ∈ ℵ_{ω+1},

F (b_{α}) = { b_{f(α,n)} : n ∈ ω };

4) g : ℵ_{ω+1} ×ℵ_{ω+1} → ℵ_{ω} is such that, for each α ∈ ℵ_{ω+1}, g(α, ·) |` α is an injective mapping from α to ℵ_{ω}; and

5) h : ℵ_{ω+1}×ℵ_{ω+1}×ℵ_{ω+1} → ω+1 is such that for each α, β ∈ ℵ_{ω+1}, h(α, β, ·) |` (b_{α}∩ b_{β}) is injective.

Note that, by 5) and since ω is definable in A, we can express “b_{α} ∩ b_{β} is finite” as a formula φ(α, β) in the language of A. Now, applying the Chang’s conjecture for A with A = ℵ_{ω+1} and U = ℵ_{ω}, we obtain elementary substructure A′ = (A′, U′, ≤′, E′, f′, g′, h′) of A such that |A′| = ℵ_{1} and |U′| = ℵ_{0}.

Claim 12.1 otp(A′) = ω_{1}.

⊢ By 4) and elementarity of A′, every initial segment of A′ can be mapped into U′ injectively and hence countable. Since |A′| = ℵ_{1}, it follows that otp(A′) = ω_{1}.

⊣ (Claim 12.1)

Claim 12.2 For any α < ℵ_{ω+1}, there is γ < ℵ_{ω+1} such that b_{β} ∩ b_{γ} is finite for every β < α.

⊢ Since |α| ≤ ℵ_{ω}, we can find a partition (I_{n})_{n∈ω} of α such that |I_{n}| < ℵ_{ω} for every n < ω. For n < ω, let η_{n} = min(ℵ_{ω}\⋃{ b_{ξ} : ξ ∈ ⋃_{m≤n} I_{m} }). Let z = { η_{n} : n ∈ ω } and γ < ℵ_{ω+1} be such that b_{γ} = z. For any β < α, if β ∈ I_{m_{0}} for some m_{0} < ω, then we have b_{β} ∩ b_{γ} ⊆ { η_{n} : n < m_{0} }. Thus this γ is as desired. ⊣ (Claim 12.2)

Claim 12.3 For any countable I ⊆ A′, there is γ ∈ A′ such that b_{β} ∩ b_{γ} is finite for every β ∈ I.

⊢ By Claim 12.1, there is α ∈ A′ such that I ⊆ α. By elementarity of A′, the formula with the parameter α expressing the assertion of Claim 12.2 for this α holds in A′. Hence there is some γ ∈ A′ such that b_{β} ∩ b_{γ} is finite for every β ∈ A′ ∩ α.

⊣ (Claim 12.3)

Let

I = { ξ ∈ A′ : b_{ξ} ∈ F (U′) }.

Then I is countable. Hence, by Claim 12.3, there is γ ∈ A′ such that b_{β} ∩ b_{γ} is finite for every β ∈ I. As b_{γ} ⊆ U′ (this holds in virtue of h(γ, γ, ·)), there is b ∈ F (b_{γ}) ∩ F (U′) such that b_{γ} ⊆ b ⊆ U′. Let b = b_{ξ_{0}}. Then ξ_{0} ∈ I and |b_{γ} ∩ b_{ξ_{0}}| = |b_{γ}| = ℵ_{0}. This is a contradiction to the choice of γ. (Theorem 12)

We do not know if the assumption of Theorem 12 yields a counter example to Corollary 11, a) for κ = ℵ_{1}. Or, more generally:

Problem 2 Is there a model of ZFC + GCH where some ccc complete Boolean algebra does not have the ℵ_{1}-FN ?

Of course, we need the consistency strength of some large cardinal to obtain such a model by Corollary 11.

The following corollary slightly improves Theorem 4.1 in [3].

Corollary 13 Suppose that (ℵ_{ω})^{ℵ_{0}} = ℵ_{ω+1} and (ℵ_{ω+1}, ℵ_{ω}) \overrightarrow{→} (ℵ_{1}, ℵ_{0}). Then the equivalence of the assertions in Theorem 10 fails. Hence we have ¬□_{ℵ_{1},ℵ_{ω}}^{∗∗∗} under these assumptions.

Proof By Theorem 12 and Corollary 11, b). (Corollary 13)

Similarly we can prove ¬□_{ℵ_{n},ℵ_{ω}}^{∗∗∗} for every n ∈ ω under the assumption of 2^{ℵ_{ω}} = ℵ_{ω+1} and (ℵ_{ω+1}, ℵ_{ω}) \overrightarrow{→} (ℵ_{1}, ℵ_{0}).

5 Cohen models

Let V be our ground model and let G be a V -generic filter over P = Fn(τ, 2) for some τ. Suppose that B is a ccc complete Boolean algebra in V [G]. Without loss of generality we may assume that the underlying set of B is a set X in V. B is said to be tame if there is a P -name ˙≤ of partial ordering of B and a mapping t : X → [τ]^{ℵ_{0}} in V such that, for every p ∈ P and x, y ∈ X, if p ∥–_{P} “ x ˙≤ y ”, then p |` (t(x) ∪ t(y)) ∥–_{P} “ x ˙≤ y ”. A lot of ‘natural’ ccc complete Boolean algebras in V [G] are contained in the class of tame Boolean algebras:

Lemma 14 Let G be as above. Suppose that V [G] ⊧ “ B is a ccc complete Boolean algebra ” and either:

i) there is a Boolean algebra B′ in V such that B′ is dense subalgebra of B in V [G]; or ii) B is the completion of a Suslin forcing in V [G]. Then B is tame.

For Suslin forcing, see e.g. [1].

Theorem 15 Let P, G be as above and λ an infinite cardinal. Assume that, in V,

i) μ^{ℵ_{0}} = μ for every regular uncountable μ such that μ < λ; and ii) □_{ℵ_{1},μ}^{∗∗∗} holds for every μ such that ℵ_{0} < μ < λ and cf(μ) = ω. Then, for any tame ccc complete Boolean algebra B in V [G] of cardinality ≤ λ we have V [G] ⊧ “ B has the ℵ_{1}-FN ”.

Proof Let X, ˙≤ and t be as in the definition of tameness for B above. We may assume ∥–_{P} “ X is the underlying set of Ḃ ” and ∥–_{P} “ Ḃ is a ccc complete Boolean algebra ” where Ḃ is a P -name for B. Let χ be sufficiently large. The following is the key lemma to the proof:

Claim 15.1 Suppose that M ≺ H(χ) is such that τ, X, ˙≤, t ∈ M and [M]^{ℵ_{0}} ⊆ M. Let I = τ ∩ M, P′ = Fn(I, 2), G′ = G ∩ P′, X′ = X ∩ M and ˙≤′= ˙≤ ∩ M. Then a) ˙≤′ is a P′-name, ∥–_{P} “ B′ = (X′, ˙≤′) is a ccc complete Boolean algebra ”, B′ is tame (in V [G′]) and the infinite sum Σ^{B} in V [G] is an extension of the infinite sum Σ^{B′} in V [G′].

b) ∥–_{P} “ (X′, ˙≤′) ≤_{σ} (X, ˙≤) ”.

⊢ a): It is easy to see that ∥–_{P′} “ (X′, ˙≤′) is a Boolean algebra ” and ∥–_{P} “ (X′, ˙≤′) is a subalgebra of (X, ˙≤) ”. Since ∥–_{P} “ (X, ˙≤) has the ccc ”, it follows that ∥–_{P′} “ (X′, ˙≤′) has the ccc ”. By elementarity of M, it is also easy to see that t |` X′ witnesses the tameness of B′ in V [G′]. To see that ∥–_{P′} “ (X′, ˙≤′) is complete ” it is enough to see that every countable subset of B′ has its supremum in V [G′]. Let Ċ be a P′-name of countable subset of X′. Without loss of generality, we may assume that Ċ is countable and consists of sets of the form (p, x̌) where p ∈ P′ and x ∈ X′. Since [M]^{ℵ_{0}} ⊆ M, Ċ ∈ M. Clearly, we have M ⊧ “ Ċ is a P -name of countable subset of X ”. Hence, M ⊧ “ ∃p ∈ P ∃y ∈ X(p ∥–_{P} “ ΣĊ = y ”) ”. Let p ∈ P and y ∈ X be such elements of M. Then p ∈ P′ and y ∈ X′. By elementarity of M, we have p ∥–_{P} “ Σ^{B}Ċ = y ”. On the other hand, from M ⊧ “ p ∥–_{P} “ ΣĊ = y ” ” it follows that p ∥–_{P′} “ Σ^{B′} Ċ = y ”.

b): By assumption, for x ∈ X and y ∈ X′, we have y ≤ x in V [G], if and only if there is p ∈ G, dom(p) ⊆ t(x) ∪ t(y) such that p ∥–_{P} “ y ˙≤ x ”. For q ∈ G ∩ Fn(t(y), 2), the set U_{q} = { y ∈ X′ : ∃p ∈ G′ (p ∪ q ∥–_{P} “ y ˙≤ x ”) } is in V [G′]. Hence, by a), Σ^{B}U_{q} is an element in B′. Since X′ is closed under t, it follows that { Σ^{B}U_{q} : q ∈ G ∩ Fn(t(y), 2) } is cofinal in B′ |` y. ⊣ (Claim 15.1)

Now, let ν = |X|. Without loss of generality, we may assume that X = ν. We show by induction on ν that ∥–_{P} “ Ḃ has the ℵ_{1}-FN ”. For ν ≤ ℵ_{1} the assertion follows from Lemma 1 (applied in V^{P}). In the induction steps, we mimic the proof of Theorem 10. Let χ be sufficiently large.

Case I : ν is a limit cardinal or ν = μ^{+} for some μ with cf(μ) ≥ ω_{1}. By i), we can construct a continuously increasing sequence (M_{α})_{α<ν} of elementary submodels of H(χ) such that 0) τ, ν, ˙≤, t ∈ M_{0};

1) |M_{α}| < ν for every α < ν;

2) [M_{α+1}]^{ℵ_{0}} ⊆ M_{α+1} for every α < ν (note that it follows that the inclusion also holds for every limit < ν of cofinality ≥ ω_{1}); and

3) ν ⊆ ⋃_{α<ν} M_{α}.

For each α < ν, let X_{α} = X ∩ M_{α} and ˙≤^{α}= ˙≤ ∩M_{α} and let Ḃ_{α} be the P -name corresponding to (X_{α}, ˙≤^{α}). By Claim 15.1, we have ∥–_{P} “ Ḃ_{α} is ccc complete Boolean algebra and Ḃ_{α} ≤_{ℵ_{1}} Ḃ ”, for all α < ν such that either α is a successor or of cofinality ≥ ω_{1}. By induction hypothesis, we have ∥–_{P} “ Ḃ_{α} has the ℵ_{1}-FN ” for such α’s. Hence by Lemma 2 and the remark after the lemma (applied in V^{P}) it follows that ∥–_{P} “ Ḃ has the ℵ_{1}-FN ”.

Case II : ν = μ^{+} for a μ with cf(μ) = ω. By ii), there is an (ℵ_{1}, μ)-dominating matrix (M_{α,n})_{α<ν,n<ω} over (τ, ν, ˙≤, t). For α < ν, let M_{α} = ⋃_{n<ω} M_{α,n}. For α < ν and n < ω, let X_{α,n} = X ∩M_{α,n}, ˙≤^{α,n}= ˙≤ ∩M_{α,n} and Ḃ_{α,n} be the P -name corresponding to (X_{α,n}, ˙≤^{α,n}). Likewise, let X_{α} = X ∩ M_{α}, ˙≤^{α}= ˙≤ ∩M_{α} and Ḃ_{α} be the P -name corresponding to (X_{α}, ˙≤^{α}). Then we have X_{α} = ⋃_{n<ω} X_{α,n}, ˙≤^{α}= ⋃_{n<ω} ˙≤^{α,n} and ∥–_{P} “ Ḃ_{α} = ⋃_{n<ω} Ḃ_{α,n} ”. By Lemma 6 and i), j3’) holds for the dominating matrix. Hence, by Claim 15.1, we have ∥–_{P} “ Ḃ_{α,n} ≤_{ℵ_{1}} Ḃ_{α} and Ḃ_{α,n} is a ccc complete Boolean algebra ” for every α < ν with cf(α) > ω and every sufficiently large n < ω. By induction hypothesis, it follows that ∥–_{P} “ Ḃ_{α,n} has the ℵ_{1}-FN ” for such α and n. By Lemma 2 (applied in V^{P}) it follows that ∥–_{P} “ Ḃ_{α} has the ℵ_{1}-FN ” for every α < ν with cf(α) > ω. Hence again by Lemma 2 and the remark after that (applied in V^{P}) we obtain that ∥–_{P} “ Ḃ has the ℵ_{1}-FN ”. (Theorem 15)

Corollary 16 Suppose that V = L holds and let G be V -generic over P = Fn(τ, 2) for some τ. Then (among others) the following ccc complete Boolean algebras have the ℵ_{1}-FN: C_{κ} (≅ the completion of Fn(κ, 2)) for any κ; P(ω) (hence also P(ω)/fin); Borel(IR)/Null-sets.

In connection with Theorem 15, we would like to mention the following open problems:

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 ℵ_{1}-FN in V [G] ?

By Theorem 15, the answer to this question is positive if the number of added Cohen reals is less than ℵ_{ω}.

Problem 4 Does Theorem 15 hold also without the assumption of tameness ?

Or, more generally:

Problem 5 Are the following equivalent?

i) P(ω) has the ℵ_{1}-FN; ii) every ccc complete Boolean algebra has the ℵ_{1}-FN.

6 Lusin gap

For a regular κ, an almost disjoint family A ⊆ [ω]^{ℵ_{0}} is said to be a κ-Lusin gap if |A| = κ and for any x ∈ [ω]^{ℵ_{0}} either |{ a ∈ A : |a \ x| < ℵ_{0} }| < κ or |{ a ∈ A : |a ∩ x| < ℵ_{0} }| < κ.

Theorem 17 Assume that P(ω) has the ℵ_{1}-FN. Then there is no ℵ_{2}-Lusin gap.

Proof Let f : P(ω) → [P(ω)]^{ℵ_{0}} be an ℵ_{1}-FN mapping. We may assume that f(a) = f(b) = f(ω \ b) for all a, b ∈ P(ω) with a =^{∗} b. Thus x ⊆^{∗} y implies that there is z ∈ f(x) ∩ f(y) such that x ⊆^{∗} z ⊆^{∗} y and |x ∩ y| < ℵ_{0} implies that there is z ∈ f(x) ∩ f(y) such that x ⊆^{∗} z and |z ∩ y| < ℵ_{0}.

Suppose that A = { a_{α} : α < ω_{2} } is an almost disjoint family of subsets of ω. We show that A is not an ℵ_{2}-Lusin gap. Let χ be sufficiently large regular cardinal and consider the model H = (H(χ), ∈, ≤) where ≤ is any well-ordering on H. Let N be an elementary submodel of H such that A, f ∈ N, N ∩ω_{2} ∈ ω_{2} and cf(δ) = ω_{1} for δ = N ∩ ω_{2}. For α ∈ N we have |a_{α} ∩ a_{δ}| < ℵ_{0} and hence a_{α} ⊆^{∗} (ω \ a_{δ}). Thus there is b_{α} ∈ f(a_{α}) ∩ f(a_{δ}) such that a_{α} ⊆^{∗} b_{α} ⊆^{∗} (ω \ a_{δ}). Since f(a_{δ}) is countable and cf(δ) = ω_{1} there is b ∈ f(a_{δ}) such that I = { α < δ : b_{α} = b } is cofinal in δ. We show that b witnesses that A is not an ℵ_{2}-Lusin gap, i.e., J = { α < ω_{2} : a_{α} ⊆^{∗} b } and K = { α < ω_{2} : |b ∩ a_{α}| < ℵ_{0} } both have cardinality ℵ_{2}.

Claim 17.1 |J| = ℵ_{2}.

⊢ First note that b ∈ N since b ∈ f(a_{α}) for any α ∈ I ⊆ N. Hence we have J ∈ N and I ⊆ J. Since I is cofinal in N ∩ ω_{2}, we have N ⊧ “ J is cofinal in ω_{2} ”. By elementarity it follows that H ⊧ “ J is cofinal in ω_{2} ”. Hence J is really cofinal in ω_{2}. ⊣ (Claim 17.1)

Claim 17.2 |K| = ℵ_{2}.

⊢ Since b ∈ N it follows that K ∈ N. For β ∈ N ∩ ω_{2} = δ, we have H ⊧ “ δ ∈ K ∧ β < δ ”. Hence H ⊧ “ K ⊈ β ” and N ⊧ “ K ⊈ β ” by elementarity. It follows that N ⊧ “ K is unbounded in ω_{2} ”. Hence, again by elementarity, H ⊧ “ K is unbounded in ω_{2} ”. Thus K is really unbounded in ω_{2}. ⊣ (Claim 17.2)

(Theorem 17)

Corollary 18 b = ℵ_{1} or even the statement “ P(ω) does not contain any strictly increasing ⊆^{∗}-chain of length ω_{2}” does not imply that P(ω) has the ℵ_{1}-FN.

Proof Suppose that our ground model V satisfies the CH. Koppelberg and Shelah [11] proved that the forcing with Fn(ω_{2}, 2) can be represented as a two step iteration A∗Ḃ where ∥–_{A} “ P(ω) contains an ℵ_{2}-Lusin gap ”. Thus, by Theorem 17, we have ∥–_{A} “ P(ω) does not habe the ℵ_{1}-FN ”. On the other hand, we have ∥–_{A∗Ḃ} “ P(ω) does have the ℵ_{1}-FN ” by Theorem 15. Hence ∥–_{A∗Ḃ} “ there is no strictly increasing ⊆^{∗}-chain in P(ω) of length ω_{2} ”. It follows that ∥–_{A} “ there is no strictly increasing ⊆^{∗}-chain in P(ω) of length ω_{2} ”. (Corollary 18)

References

[1] T. Bartoszyński and H. Judah, Set Theory: on the structure of the real line, A K Peters, (1995), i–ix,1–546.

[2] S. Ben-David and M. Magidor, The weak □ is really weaker than full □, Journal of Symbolic Logic 51, (1986), 1029–1033.

[3] M. Foreman and M. Magidor, A very weak square principle, preprint.

[4] M. Foreman, M. Magidor and S. Shelah, Martin’s maximum, saturated ideals, and non-regular ultrafilters I, Annals of Mathematics. Second Series, 127 (1988), 1–47.

[5] R. Freese and J.B. Nation, Projective lattices, Pacific Journal of Mathematics, Vol. 75 (1978), 93–106.

[6] S. Fuchino, S. Koppelberg and S. Shelah, Partial orderings with the weak Freese-Nation property, to appear in Annals of Pure and Applied Logic.

[7] S. Fuchino, S. Koppelberg and S. Shelah, A game on partial orderings, to appear in Topology and its Applications.

[8] L. Heindorf and L.B. Shapiro, Nearly Projective Boolean Algebras, Springer Lecture Notes of Mathematics 1596, (1994), I–X, 1–202.

[9] R. Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, 4(3), (1972), 229–308.

[10] S. Koppelberg, Applications of σ-filtered Boolean algebras, preprint.

[11] S. Koppelberg and S. Shelah, Subalgebras of the Cohen algebra do not have to be Cohen, preprint.

[12] K. Kunen, Set Theory, North-Holland (1980), i–xvi, 1–313.

[13] J.-P. Levinski, M. Magidor and S. Shelah, On Chang’s conjecture for ℵ_{ω}, Israel Journal of Mathematics, 69, (1990), 161–172.