Almost-disjoint coding and strongly saturated ideals ^{∗}

Paul B. Larson^{†}

June 8, 2004

Abstract We show that Martin’s Axiom plus 𝔠 = ℵ_{2} implies that there is no

(ℵ_{2}, ℵ_{2}, ℵ_{0})-saturated σ-ideal on ω_{1}.

Given cardinals λ, κ and γ, a σ-ideal I on a set X is said to be (λ, κ, γ)- saturated if for every set {A_{α} : α < λ} ⊂ P(X) \ I there exists a set Y ∈ [λ]^{κ} such that for all Z ∈ [Y]^{γ}, ∩{A_{α} : α ∈ Z} ∉ I. Laver [6] was the first to show the consistency of an (ℵ_{2}, ℵ_{2}, ℵ_{0})-saturated ideal on ω_{1}, using a huge cardinal. Shelah [10] later showed that the nonstationary ideal on ω_{1} restricted to a given stationary set can be (ℵ_{2}, ℵ_{2}, ℵ_{0})-saturated, using a supercompact cardinal.

The cardinal characteristic 𝔞𝔭 is defined to be the least κ such that there is an almost disjoint family {e_{α} : α < κ} (i.e., each e_{α} is an infinite subset of ω, and for each distinct pair α, β < κ, e_{α} ∩ e_{β} is finite) and a set A ⊂ κ such that for no x ⊂ ω does it hold for all α < κ that α ∈ A if and only if e_{α} ∩ x is infinite (in [5] we called this 𝔮, but [1] shows that we shouldn’t have, as consistently every set of reals of cardinality 𝔞𝔭 is a Q-set). We let 𝔠 denote the cardinality of the continuum. It follows easily that 2^{γ} = 𝔠 for every infinite γ < 𝔞𝔭.

Given a cardinal γ, MA_{γ} is the variant of Martin’s Axiom that says that if P is a c.c.c. partial order and D_{α} (α < γ) are dense subsets of P, then there is a filter G ⊂ P such that G ∩ D_{α} is nonempty for each α < γ. It is a standard fact that MA_{γ} implies that 𝔞𝔭 > γ [4].

In this note, we show that the statement 𝔞𝔭 = 𝔠 = ℵ_{2} implies that there is no countably complete (ℵ_{2}, ℵ_{2}, ℵ_{0})-saturated σ-ideal on ω_{1}. This contradicts statements in [7, 8] to the effect that the axiom PFA (see [10]) had been shown to be consistent with the existence of a stationary subset of ω_{1} such that the nonstationary ideal restricted to this set is (ℵ_{2}, ℵ_{2}, ℵ_{0})-saturated. This situation is addressed in [9] and another corrigendum from Nyikos to appear.

For a fixed cardinal κ, an ideal on a set X is κ-dense if there is a subset A of P(X) \ I of cardinality κ such that every I-positive subset of X contains a member of A modulo I. It follows easily that every ℵ_{1}-dense σ-ideal is (ℵ_{2}, ℵ_{2}, ℵ_{0})-saturated. It is a classical fact due to Ulam that there is no ℵ_{0}-dense σ-ideal on ω_{1} (see, for instance, Lemma 10.13 of [3]). Taylor [11] showed that under MA_{ℵ_{1}} there is no ℵ_{1}-dense σ-ideal on ω_{1}. The proof of Theorem 18 in [2] shows that 𝔞𝔭 > ℵ_{1} suffices, i.e., that the following holds (a version of the argument appears also in [5]).

∗AMS 2000 Classification: Primary 03E50; Secondary 54D15

†The research in this paper was conducted with the support of of a FAPESP fellowship (Grant # 02/11551-3) at the University of São Paulo.

0.1 Fact. If 𝔞𝔭 > ℵ_{1}, then there is no ℵ_{1}-dense σ-ideal on ω_{1}.

For the rest of this note, we fix an almost disjoint family {e_{α} : α < ω_{1}}. For each n ∈ ω and each (possibly finite) σ ⊂ ω, we let E_{σ}^{n} = {α < ω_{1} | |e_{α}∩σ| ≥ n}, and we let F_{σ}^{n} = ω_{1} \ E_{σ}^{n}.

Lemma 0.2. Assume that 𝔞𝔭 > ℵ_{1}. Let I be a σ-ideal on ω_{1} and let β be a cardinal such that P(ω_{1})/I is not β-dense. Let {A_{α} : α < β} be a subset of P(ω_{1}) \ I. Then there exist an x ⊂ ω and an n ∈ ω such that

• F_{x}^{n} ∉ I

• for each α < β there exists an m ∈ ω such that E_{x∩m}^{n} ∩ A_{α} ∉ I.

Proof. Since P(ω_{1})/I is not β-dense, we may fix {B_{α} : α < β} ⊂ P(ω_{1}) \ I and D ∈ P(ω_{1}) \ I such that each B_{α} ⊂ A_{α} and each B_{α} ∩ D = ∅. Since 𝔞𝔭 > ℵ_{1}, there exists an x ⊂ ω such that e_{γ} ∩x is infinite for each γ ∈ ∪{B_{α} : α < β} and e_{γ} ∩ x is finite for each γ ∈ D. Since D ⊂ ∪{F_{x}^{n} : n < ω}, we may fix an n ∈ ω such that F_{x}^{n} ∉ I. Similarly, for each α < β, since B_{α} ⊂ ∪{E_{x∩m}^{n} : m < ω}, there is an m ∈ ω such that E_{x∩m}^{n} ∩ A_{α} ∉ I.

The following theorem shows that 𝔞𝔭 > ℵ_{1} implies that there is no σ-ideal I on ω_{1} which is (γ, γ, ℵ_{0})-saturated, where γ is the least cardinality of a dense subset of P(ω_{1})/I. In particular, if 𝔞𝔭 = 𝔠 = ℵ_{2}, then there is no (ℵ_{2}, ℵ_{2}, ℵ_{0})- saturated σ-ideal on ω_{1}.

Theorem 0.3. Assume that 𝔞𝔭 > ℵ_{1}, and let I be a σ-ideal on ω_{1}. Let γ be the least cardinal such that there exists a dense (modulo I) subset of P(ω_{1}) \ I of cardinality γ. Then there is a sequence ⟨D_{α} : α < γ⟩ of members of P(ω_{1}) \ I such that for every cofinal X ⊂ γ there exists a countable y ⊂ X such that ∩{D_{α} : α ∈ y} ∈ I.

Proof. Let {A_{α} : α < γ} enumerate a dense subset of P(ω_{1}) \ I modulo I. For each β < γ, apply Lemma 0.2 to {A_{α} : α < β}, obtaining x_{β}, n_{β} and D_{β} = F_{x_{β}}^{n_{β}} such that D_{β} ∉ I and such that for each α < β there exists an m ∈ ω such that A_{α} ∩ E_{x_{β}∩m}^{n_{β}} ∉ I.

Now let X ⊂ γ be cofinal. Let Z be the set of pairs (n, σ) (n ∈ ω, σ ⊂ ω finite) such that there exists a β ∈ X with E_{σ}^{n} ∩ D_{β} ∈ I. We claim that {E_{σ}^{n} : (n, σ) ∈ Z} is predense in P(ω_{1}) \ I, i.e., that for every α < γ there exist a β ∈ X, an n ∈ ω and a finite σ ⊂ ω such that E_{σ}^{n} ∩ D_{β} ∈ I and E_{σ}^{n} ∩ A_{α} ∉ I. To verify this, fix α < γ and let β be any member of X greater than α. Then D_{β} = F_{x_{β}}^{n_{β}} and there exists an m such that E_{x_{β}∩m}^{n_{β}} ∩A_{α} ∉ I, so β, n_{β} and x_{β} ∩m suffice for α.

Now, for each (n, σ) ∈ Z, choose β_{(n,σ)} ∈ X such that E_{σ}^{n}∩D_{β_{(n,σ)}} ∈ I. Then since {E_{σ}^{n} : (n, σ) ∈ Z} is predense in P(ω_{1})\I, ∩{D_{β_{(n,σ)}} : (n, σ) ∈ Z} ∈ I.

We do not know whether some forcing axiom implies that the nonstationary ideal on ω_{1} is not (ℵ_{2}, ℵ_{1}, ℵ_{0})-saturated. On the other hand, for all we know, some forcing axiom implies that the nonstationary ideal on ω_{1} is (ℵ_{2}, ℵ_{1}, ℵ_{0})- saturated.

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Department of Mathematics Miami University Oxford, Ohio 45056 USA larsonpb@muohio.edu