Small posets and the continuum

Matt Foreman  Paul B. Larson

April 12, 2004

Abstract

We present two proofs that the restriction of Martin’s Maximum to posets of cardinality ℵ_{2} implies that the continuum is ℵ_{2}. The techniques used in the proofs are all standard.

By now there eight or so constructions showing that the forcing axiom Martin’s Maximum implies that the continuum is ℵ_{2}. All of the standard constructions use posets of the size at least the continuum. Here we present two partial orders of cardinality ℵ_{2} which suffice. Specifically, we present two proofs of the following theorem.

Theorem 0.1. There exist partial orders P_{x} (x ⊂ ω) of cardinality ℵ_{2} and sets D_{α}^{x} (x ⊂ ω, α < ω_{1}) such that each D_{α}^{x} is a dense subset of the corresponding P_{x}, and such that, if for each x ⊂ ω there is a filter G_{x} ⊂ P_{x} intersecting each member of {D_{α}^{x} : α < ω_{1}}, then there is an injective function from P(ω) to ω_{2}.

The constructions we use here are standard, and so it it likely that this fact was previously known.

Given a cardinal κ and and ordinal α, we let C_{α}^{κ} denote the ordinals below α of cofinality κ. Both of our proofs use the following fact.

Lemma 0.2. Suppose that S is a stationary subset of ω_{1}, B is a stationary subset of C_{ω_{2}}^{ω}, and {g_{α} : α ∈ [ω_{1}, ω_{2})} is a set of functions such that each g_{α} is a bijection between ω_{1} and α. Let θ be a regular cardinal greater than or equal to ω_{2}. Then the set of countable Y ≺ H(θ) with Y ∩ ω_{1} ∈ S, sup(Y ∩ ω_{2}) ∈ B and

Y ∩ ω_{2} = g_{sup(Y ∩ω_{2})}[Y ∩ ω_{1}]

is stationary.

Proof. Letting θ̄ be (2^{|H(θ)|})^{+}, it suffices to find one countable Y ≺ H(θ̄) with Y ∩ ω_{1} ∈ S, sup(Y ∩ ω_{2}) ∈ B and Y ∩ ω_{2} = g_{sup(Y ∩ω_{2})}[Y ∩ ω_{1}]. Let Ȳ be an elementary submodel of H(θ̄) of cardinality ℵ_{1}, containing ω_{1}, such that sup(Ȳ ∩ ω_{2}) is in B. Let γ = sup(Ȳ ∩ ω_{2}). Let ⟨Y_{α} : α < ω_{1}⟩ be a continuous increasing chain of countable elementary submodels of Ȳ, with sup(Y_{0}∩ω_{2}) = γ. Then there is a club D ⊂ ω_{1} such that g_{γ}[α] = Y_{α} ∩ ω_{2} for all α ∈ D, and for any α ∈ S ∩ D, Y_{α} suffices.

We fix the following objects for both proofs. Let A be a subset of ω_{2} such that ω_{2}^{L[A]} = ω_{2}. Let {B_{i} : i ∈ ω} be a partition of the members of C_{ω_{2}}^{ω} into stationary sets. For each x ⊂ ω, let B_{x} = ∪{B_{i} : i ∈ x}, and let {S_{i}^{x} : i ∈ x} be a partition of ω_{1} into stationary sets.

Our first proof uses a trick due to Uri Abraham (see [1]).

First proof of Theorem 0.1. For each x ⊂ ω, we let P_{x} be the following two-step forcing. First we add ω_{2} many Cohen reals {c_{α} : α < ω_{2}}, with finite support. Call the generic for this forcing C. Then we force with Q_{x}, the set of countable continuous increasing sequences ⟨x_{α} : α < β⟩ in L[A][C] such that the following hold for each α < β:

• x_{α} is a countable subset of ω_{2},

• x_{α} ∩ ω_{1} is an ordinal, and sup(x_{α}) is in B_{i}, where i is the unique member of x such that x_{α} ∩ ω_{1} ∈ S_{i}.

The order on Q_{x} is extension. The first step of this forcing is c.c.c. and so preserves stationary subsets of ω_{1}. To see that Q_{x} does also, we work in V [C]. Let E be a stationary subset of ω_{1}, let τ be a Q_{x}-name for a club subset of ω_{1} and let A_{α} (α < ω_{1}) be maximal antichains in Q_{x} such that for each α < ω_{1} the members of A_{α} decide the α-th member of τ. Let θ be a regular cardinal greater than 2^{|Q_{x}|}, and let q be a condition in Q_{x}. Let i ∈ x be such that E ∩ S_{i} is stationary. Applying Lemma 0.2, fix a countable Y ≺ H(θ) with Q_{x}, q, τ and ⟨A_{α} : α < ω_{1}⟩ in Y, Y ∩ ω_{1} ∈ E ∩ S_{i} and sup(Y ∩ ω_{2}) ∈ B_{i}. Let X = Y ∩ H(ω_{2})^{L[A][C]}. Note that since L[A][C] ⊧ |H(ω_{2})| = ℵ_{2}, X ∈ L[A][C]. Let M_{X} be the transitive collapse of X, and let π: M_{x} → X be the inverse of the collapsing map. Let Q̄_{x} and q be the images of Q_{x} ∩ Y and q under the collapse of X, and for each α ∈ ω_{1} ∩ Y, let Ā_{α} be the image of A_{α} ∩ Y under this collapse.

The key point is that there is some γ < ω_{2} such that M_{X} and Q̄_{x} are in L[A][C↾γ] and that Ā_{α} (α ∈ Y ∩ ω_{1}) is in V [C↾γ]. Then Q̄_{x} is isomorphic to Cohen forcing, so the γ-th Cohen real, c_{γ}, codes a sequence ⟨a_{α} : α ∈ Y ∩ ω_{1}⟩ extending q̄ whose initial segments are members of Q̄_{x} eventually meeting each Ā_{α} (α ∈ Y ∩ ω_{1}). Then ⟨π[a_{α}] : α ∈ Y ∩ ω_{1}⟩^{⌢}{X ∩ ω_{2}} is a condition in Q_{x} forcing that Y ∩ ω_{1} is in τ. This shows that Q_{x} preserves stationary subset of ω_{1}.

Now if D_{α}^{x} are dense subsets of P_{x} deciding the members of the Q_{x}-generic, any filter contained in P_{x} meeting each D_{α}^{x} defines a bijection f_{x}: ω_{1} → γ_{x}, for some ordinal γ_{x} < ω_{2}, such that for club many α < ω_{1}, if (for some i < ω) α ∈ S_{i}^{x} then sup(f_{x}[α]) ∈ B_{i}. In particular, x is the set of i < ω such that the set of countable subset of γ_{x} with ordertype in B_{x} is stationary. The association x ↦ γ_{x} then is clearly injective.

Our second proof uses Stevo Todorcevic’s elementary submodel collapse (see [3]).

Second proof of Theorem 0.1. Now we let P_{x} be the partial order consisting of finite ϵ-chains p of elementary submodels of H(ω_{2})^{L[A]} which are elements of L[A] (and countable in L[A]) such that

• for each X ∈ p and i ∈ x, if X ∩ ω_{1} ∈ S_{i}^{x} then sup(X ∩ ω_{2}) is in B_{i},

• there exists in L[A] a countable, increasing continuous chain of countable elementary submodels of H(ω_{2})^{L[A]} containing all the members of p.

The order is containment.

To see that P_{x} preserves stationary subsets of ω_{1}, we apply Lemma 0.2. Let E be a subset of ω_{1}, and let τ be a P_{x}-name for a club subset of ω_{1}. Let p be a condition in P_{x}, and let θ = (2^{|P_{x}|})^{+}. In L[A], club many elementary submodels of H(ω_{2})^{L[A]} are of the form X ∩ H(ω_{2})^{L[A]} for some countable X ≺ H(θ)^{L[A]}. Let C denote the set of such elementary submodels of H(ω_{2})^{L[A]}, and let C^{∗} be the (club) set of γ < ω_{2} such that C ∩ [γ]^{ℵ_{0}} is club in [γ]^{ℵ_{0}}. Again we fix i ∈ ω such that E ∩ S_{i} is stationary. Modifying the proof of Lemma 0.2, slightly, we can find a countable elementary submodel Y of H(θ) such that p, τ, P_{x} ∈ Y, Y ∩ ω_{1} ∈ E ∩ S_{i}, sup(Y ∩ ω_{2}) ∈ B_{i} ∩ C^{∗}, Y ∩ ω_{2} ∈ L[A] and Y ∩ H(ω_{2})^{L[A]} ∈ C (call this set X). Then p′ = p ∪ {X} is a condition in P_{x}, and p′ forces that every antichain in Y will be met by a condition in the intersection of X and the generic filter. To see this, let q be a condition below p′ and let D be a dense subset of Q_{x} in Y. Then q ∩ X is a finite subset of X and thus a member of X. By elementarity, there is a condition r ∈ Y ∩ D below (q ∩ X). But r is a finite subset of Y ∩ L[A], so r is in L[A] and r is in X. Furthermore, r cup q is a finite ∈-chain of elementary submodels in L[A], so it is a condition.

Now P_{x} adds a continuous, increasing ∈-chain ⟨X_{α} : α < ω_{1}⟩ of elementary submodels of H(ω_{2})^{L[A]} such that, for each α < ω_{1} and each i ∈ x, if X_{α} ∩ ω_{1} ∈ S_{i}^{x} then sup(X_{α} ∩ ω_{2}) ∈ B_{i}. The rest of the proof then follows as before.

0.3 Question. Do either PFA or SPFA for posets of cardinality ℵ_{2} imply bounds on the continuum?

0.4 Question. Does MM for posets of cardinality ℵ_{1} imply a bound on the continuum?

References

[1] U. Abraham, S. Shelah, Forcing closed unbounded sets, J. Symbolic Logic 48 (1983), no. 3, 643–657

[2] M. Foreman, M. Magidor, S. Shelah, Martin’s Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annals of Mathematics, 127 (1988), 1-47

[3] S. Todorcevic, A note on the proper forcing axiom, Axiomatic set theory (Boulder, Colo., 1983), 209–218, Contemp. Math., 31, Amer. Math. Soc., Providence, RI, 1984