Young Set Theory Workshop 2010, Raach, Austria.

The Proper Forcing Axiom: a tutorial

Justin Tatch Moore^{1} Notes taken by Giorgio Venturi

In these notes we will present an exposition of the Proper Forcing Axiom (PFA). We will first discuss examples of the consequences of PFA. We will then present two proper partial orders which are used to force two combinatorial principles which follow from PFA: The P-Ideal Dichotomy (PID) and Todorcevic’s formulation of the Open Coloring Axiom. On one hand, these posets and the proofs of their properness are quite typical of direct applications of PFA. On the other hand, these principles already capture a large number of the consequences of PFA and do not use any terminology or technical tools from the theory of forcing.

1 Consequences of PFA

We start with the definition of PFA.

Definition 1.1. PFA holds if, given a proper poset ℚ and a collection D of dense subsets of ℚ, with |D| ≤ ℵ_{1}, then there is a filter G ⊆ ℚ such that G ∩ D ≠ ∅ for all D ∈ D.

While we will give a definition of proper later on, it is sufficient for the moment to say that properness is a weakening of the countable chain condition (c.c.c.). In particular, PFA is a strengthening of MA(ℵ_{1}).

The following two theorems predated PFA and were important in its formulation.

Theorem 1.2. (Solovay, Tennenbaum [15]) Souslin Hypothesis is consistent with ZFC.

Theorem 1.3. (Baumgartner [4]) The following statement is consistent with ZFC: every pair of ℵ_{1}-dense^{2} subsets of ℝ are isomorphic.

1The research of Justin Moore was supported in part by NSF grant DMS-0757507. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

2A set X ⊆ ℝ is said to be κ-dense, if every interval meets X in κ points.

Martin observed that Solovay and Tennenbaum’s proof of the consistency of Souslin’s Hypothesis could be adapted to prove the consistency of a stronger statement, now known as MA(ℵ_{1}). The model which Baumgartner constructed to establish the above theorem is, like Solovay and Tennenbaum’s model, obtained by a finite support iteration of forcings which satisfy the c.c.c.. Unlike Souslin’s Hypothesis, however, the conclusion of Baumgartner’s theorem did not apparently follow from MA(ℵ_{1}) (this was later confirmed by work of Abraham and Shelah [2]).

Let us now consider two principles which we will later demonstrate are consequences of PFA. Recall that an ideal I ⊆ [S]^{ω} is a P-ideal if

• I contains every finite subsets of S,

• for every family {X_{n} : n ∈ ω} ⊆ I there is an X ∈ I such that X_{n} is contained in X modulo a finite set for every n ∈ ω.

Definition 1.4. (P-Ideal Dichotomy (PID)) If S is a set and I ⊆ [S]^{ω} is a P-ideal, then either

• there is an uncountable Z ⊆ S such that [Z]^{ω} ⊆ I, or

• S = ⋃_{n∈ω} S_{n} such that no infinite subset of any S_{n} is in I.

Definition 1.5. (Open Coloring Axiom (OCA)) If G is an open graph on a separable metric space X, then either

• there is an uncountable Z ⊆ X such that [Z]^{2} ⊆ G, or

• X = ⋃_{n∈ω} X_{n} such that [X_{n}]^{2} ∩ G = ∅, for every n ∈ ω.

Remark 1.6. The formulation of this principle is due to Todorcevic and is based on the different formulations of OCA presented in [1].

We will now turn to some of the consequences of these principles.

Theorem 1.7. (Todorcevic [21]) PFA implies that if X is a Banach space of density ℵ_{1}, then X has a quotient with a basis of length ω_{1}.

In fact for Todorcevic’s result, the conjunction of PID and MA(ℵ_{1}) is sufficient to derive the desired conclusion.

Theorem 1.8. (Farah [6]) OCA implies that if H is a separable infinite dimensional Hilbert space, then all automorphisms of B(H)/K(H) are inner.

Theorem 1.9. (Moore [12]) PFA implies that every uncountable linear order contains an isomorphic copy of one of the following: X, ω_{1}, −ω_{1}, C, −C. Here X is a set of reals of cardinality ℵ_{1} and C is a Countryman line.

Theorem 1.10. (Moore [13]) PFA implies that η_{C} is universal for the Aronszajn lines. Here η_{C} is the direct limit of the finite lexicographic products of the form C × (−C) × … × (±C).

Theorem 1.11. (Martinez [8]) PFA implies that the Aronszajn lines are well quasi ordered by embeddability.

Theorem 1.12. (Todorcevic, Veličković [5] [23]) PFA implies that 2^{ℵ_{0}} = ℵ_{2}.

Theorem 1.13. (Todorcevic [16]) PFA implies that for every regular κ > ℵ_{1}, □(κ) fails.

Theorem 1.14. (Viale [26]) PFA implies 2^{μ} = μ^{+} whenever μ is a singular strong limit cardinal.

The two last theorems are consequences of PID [20] [27].

Theorem 1.15. (Todorcevic [17]) OCA implies that every (κ, λ^{∗})-gap in ω^{ω}/fin is of the form κ = λ = ω_{1} or min(κ, λ) = ω and max(κ, λ) ≥ ω_{2}.

2 Some technical background

Both when defining properness and when building proper forcings, it will be convenient to work with set models of a sufficiently strong fragment of ZFC. The most appropriate fragment in the present setting is ZFC with the power set axiom omitted. Models of this theory are provided by the structures (H(θ), ∈), where θ is an uncountable regular cardinal and H(θ) is the collection of all sets of hereditary cardinality less than θ. Note in particular that if A is in H(θ), then P(A) ⊆ H(θ) and if A and B are in H(θ), then so is A × B. In particular, H(θ) ⊧ |A| ≤ |B| if and only if |A| ≤ |B|. Since we will frequently be working with elementary substructures of some fixed H(θ), it will be useful to also fix a well ordering ◁ of H(θ). This provides a nice set of Skolem functions for H(θ).

In what follows, we will only be interested in countable elementary submodels of H(θ) and will write M ≺ H(θ) to mean that M is a countable elementary submodel of H(θ). We will say that θ is sufficiently large with respect to X if P(X) is in H(θ). A suitable model for X is an M ≺ H(θ) where X is in M and θ is sufficiently large for X.

Definition 2.1. If ℚ is a forcing notion and M is a suitable model for ℚ, then a condition q ∈ ℚ is (M, ℚ)-generic if whenever r ≤ q and D ⊆ ℚ is dense and in M, there is a s ∈ D ∩ M, such that s and r are compatible (i.e. D ∩ M is predense below q).

The above definition is equivalent to saying that

q ⊩ Ġ ∩ M̌ is M̌-generic

where Ġ is the ℚ-name for the generic filter.

Definition 2.2. A forcing notion ℚ is proper if whenever M is a suitable model for ℚ and q is in ℚ ∩ M, q has an extension which is (M, ℚ)-generic.

The following are useful facts about countable elementary submodels.

Fact 2.3. If X is definable in H(θ) from parameters in M ≺ H(θ), then X ∈ M.

Fact 2.4. If X ∈ M ≺ H(θ), then X is countable if and only if X ⊆ M

Proof. Only the reverse implication requires argument. Observe that since elements of ω are definable, ω ⊆ M. If X is countable, then, by elementarity, M will contain a function from X into ω. Since ω is contained in X, its preimage under this function is as well and hence X ⊆ M.

Fact 2.5. If M ≺ H(θ), then M ∩ ω_{1} is a countable ordinal.

Proof. If α ∈ M ∩ ω_{1}, then α ⊆ M by the previous fact. Hence M ∩ ω_{1} is a transitive set of ordinals and therefore an ordinal.

3 Forcings associated to OCA and PID

We will now define posets associated to OCA and to PID and prove that they are proper.

3.1 The poset for OCA

Suppose that G is an open graph on X. Let ℚ_{X,G} be the poset consisting of all the pairs q = (H_{q}, N_{q}) such that:

1. H_{q} ⊆ X is a finite clique in G;

2. N_{q} is an increasing finite ∈-chain of elementary submodels of H(2^{ℵ_{0}}^{+}) which each contain (X, G); 3. if x ≠ y are in H_{q}, then there is an N in N_{q} such that |N ∩ {x, y}| = 1;

4. if N is in N_{q} and E ⊆ X is in N with [E]^{2} ∩ G = ∅, then E ∩ H_{q} ⊆ N.

Define the order on ℚ_{X,G} by p ≤ q if H_{q} ⊆ H_{p} and N_{q} ⊆ N_{p}. Notice that, by condition 3, N_{q} induces an order on H_{q} corresponding to the number of elements of N_{q} which do not contain a given element of H_{q}. It will be convenient to let x_{q}(i) denote the i^{th} element of H_{q} in this enumeration.

Remark 3.1. If CH holds, then can modify the forcing ℚ_{X,G} (and ℚ_{X,G}^{∗} below) as follows. Let N be a continuous ∈-chain of length ω_{1} of countable elementary submodels of H(2^{ℵ_{0}}^{+}), each containing the relevant objects. Define, e.g., ℚ_{X,G} to be all H_{q} such that (H_{q}, N_{q}) is in ℚ_{X,G} whenever N_{q} is a finite subset of N. One can verify that this modified forcing in fact satisfies the countable chain condition. The role of CH here can be explained as follows: if X is a separable metric space and if M and N are two countable elementary submodels of (H(2^{ℵ_{0}}^{+}), ∈, X) such that M∩ω_{1} = N∩ω_{1}, then CH implies that M∩H(ℵ_{1}) = N∩H(ℵ_{1}) and hence M and N have the same intersection with the closed subsets of X. Notice that in condition 4, the set E can be assumed to be closed since if [E]^{2} ∩ G is empty, the same is true for the closure of E.

3.2 The poset for PID

Let I be a P-ideal on a set S and suppose that θ is a regular cardinal such that I is in H(θ). For each countable subset X of I, let I_{M} be an element of I such that I ⊆_{∗} I_{X} for every I in X. Here I ⊆_{∗} J means that the set I \ J is finite. If M is a countable elementary submodel of H(θ), will let I_{M} denote I_{M∩I}. Define I^{⊥} to be the collection of all subsets of S which have finite intersection with every element of I.

Let ℚ_{I} be the poset consisting of all the pairs q = (Z_{q}, N_{q}) such that

1. Z_{q} ⊆ S is finite;

2. N_{q} is an increasing finite ∈-chain of elementary submodels of H(θ) which each contain I;

3. if x, y ∈ S, then there is an N in N_{q} such that |N ∩ {x, y}| = 1;

4. if N is in N_{q} and X is in N ∩ I^{⊥}, then X ∩ Z_{q} ⊆ N.

Define the order on ℚ_{I} by:

1. p ≤ q if Z_{q} ⊆ Z_{p}; 2. N_{q} ⊆ N_{p};

3. (Z_{p} \ Z_{q}) ∩ N ⊆ I_{N}, whenever N is in N ∈ N_{q}.

4 Verification of properness

The above examples of forcings all have a common from. For instance, their elements consist of pairs q = (X_{q}, N_{q}) where X_{q} is a finite approximation of some desired object and N_{q} is a finite ∈-chain of countable elementary submodels of some suitably chosen H(θ). In order to verify properness of such forcings, the general strategy is to argue that if M is a suitable model for ℚ and M ∩ H(θ) is in N_{q}, then q is (M, ℚ)-generic. The forcing ℚ is defined in such a way that it is trivial to verify that if q_{0} is in M ∩ ℚ, then

q = (X_{q_{0}}, N_{q_{0}} ∪ {M ∩ H(θ)})

a condition in ℚ which extends q_{0}.

In order to verify that M ∩ H(θ) ∈ N_{q} implies the (M, ℚ)-genericity of q, one usually argues that:

(*): if D ⊆ ℚ is in M and q is in D, then q is compatible with an element of D ∩ M.

Notice that this implies genericity: if D ⊆ ℚ is dense and in M, then we can first extend q to an element r of D and then appeal to (*) to find a condition s in D ∩ M which is compatible with r and hence with q.

We now show that ℚ_{X,G} and ℚ_{I} are proper and argue that ℚ_{A,B} can be viewed as a special case of a modified version of ℚ_{X,G}.

Theorem 4.1. ℚ_{X,G} is proper.

Proof. Suppose that M is suitable for ℚ_{X,G}, r is in ℚ_{X,G}, and N = M ∩H(2^{ℵ_{0}}^{+}) is in N_{r}. It is sufficient to verify that r is (M, ℚ_{X,G})-generic and this will be done by verifying (*). To this end, let D ⊆ ℚ_{X,G} be in M and contain r. Fix disjoint open sets U_{i} (i < n) which are in N such that x_{r}(i) is in U_{i} and U_{i} × U_{j} ⊆ G whenever i ≠ j < n.

By replacing D with a subset if necessary, we may assume that if s is in D, then

• s ≤ r_{0},

• for some N_{s} ∈ N_{s}, r_{0} = (H_{s} ∩ N_{s}, N_{s} ∩ N_{s}), • |H_{s}| = |H_{r}| = n, and

• x_{s}(i) ∈ U_{i} for all i < n.

Let r_{0} = r ∩ M and note that r_{0} ∈ N.

To find a condition s in D ∩ M compatible with r we have to check that [H_{r} ∪ H_{s}]^{2} ⊆ G. By construction of the set D, we just need to show that

{x_{r}(i), x_{s}(j)} ∈ G for i = j,

because for i ≠ j, U_{i} and U_{j} already witness the fact that {x_{r}(i), x_{s}(j)} ∈ G. The following is now the key lemma.

Lemma 4.2. Suppose that:

• N ≺ H(2^{ℵ_{0}}^{+}) with (X, G) ∈ N;

• A ⊆ X^{n} is in N;

• x̄ ∈ cl(A) and x̄ ↾ n − 1 ∈ M;

• for every E ⊆ X in M with [E]^{2} ∩ G = ∅, x(n − 1) is not in cl(E).

Then there is a basic open U ⊆ X not containing x(n − 1) such that

{x(n − 1), y} ∈ G for every y ∈ U,
x̄ ↾ n − 1 ∈ cl({ȳ ↾ (n − 1) : ȳ ∈ A ∧ y(n − 1) ∈ U}).

Proof. Let {W_{i} : i ∈ ω} be a neighborhood base for x̄ ↾ (n − 1) which is in M and define

E_{0} = ⋂_{i} cl({ȳ(n − 1) : ȳ ∈ A and ȳ ↾ (n − 1) ∈ W_{i}}).

Notice that x(n − 1) ∈ E_{0}. Observe that E_{0} is in M since it is definable from parameters in M. Similarly,

E = {z ∈ E_{0} : ∀y ∈ E_{0} {z, y} ∉ G}

is in M. By the fourth condition in the definition of ℚ_{X,G}, x(n − 1) ∉ E. Pick then y ∈ E_{0} such that {x(n − 1), y} ∈ G. Since G is open, we can find a basic open neighborhood U of y which does not contain x such that {x(n−1)}×U ⊆ G.

We now must show that

x̄ ↾ n − 1 ∈ cl({ȳ ↾ (n − 1) : ȳ ∈ A ∧ y(n − 1) ∈ U}).

Let j ∈ ω be given. Since U is open and y is in E_{0}, there is a z̄ in A such that z̄ ↾ (n − 1) is in W_{j} and z(n − 1) is in U. Since {W_{i} : i ∈ ω} is a neighborhood base at x̄ ↾ (n − 1) and j was arbitrary, we have the desired conclusion.

We are now ready to complete the proof that ℚ_{X,G} is proper. Set A = {x̄_{s} : s ∈ D} and, using the above lemma, inductively construct a sequence of sets A_{i} (i ≤ n) in N such that:

• A_{0} = A and A_{i} ⊆ A_{i−1} if i > 0;

• x_{r} ↾ i is an accumulation point of {y ↾ i : y ∈ A_{i}};

• if i < j < n and y is in A_{i}, then {x(j), y(j)} ∈ G.

Now if s is in N and x̄_{s} is in A_{n}, it follows that s is compatible with r.

Finally we turn to the proof that the PID forcing is proper.

Lemma 4.3. ℚ_{I} is proper.

Proof. Let M be a suitable model for ℚ_{I} and suppose that r is in ℚ_{I} with M ∩ H(θ) in N_{r}. Define r_{0} = (Z_{r} ∩ M, N_{r} ∩ M) and let n = |Z_{r} \ M|. We will verify that (*) holds. To this end, suppose that D ⊆ ℚ_{I} with D in M and r in D. By replacing D with a subset if necessary, we may assume that if s is in D then:

• there is an N_{s} in N_{s} such that r_{0} = (Z_{s} ∩ N_{s}, N_{s} ∩ N_{s});

• |Z_{s} \ N_{s}| = n.

By definition of extension in ℚ_{I}, our goal will be to find an s ∈ D ∩M such that Z_{s}\N_{s} ⊆ I_{P}, for each P ∈ N_{r}\M. This is sufficient, since then (Z_{s}∪Z_{r}, N_{s}∪N_{r}) is in ℚ_{I} and is a common extension of r and s.

Define T = {t_{s} : t_{s} = Z_{s} \ N_{s} and s ∈ D}.

Claim 4.4. There is a T′ ⊆ T which is J^{+}-splitting, where J is the σ-ideal generated by I^{⊥}.

Proof. If U ⊆ S^{n}, define ∂U to be the set of all u in U such that there is no k < n with

{v(k) : (v ∈ U) ∧ (u ↾ k = v ↾ k)} ∈ J .

By definition, U is J^{+}-splitting if ∂U = U and U is non empty. Observe that for any U ⊆ S^{n}, if u is in ∂^{i+1}U \ ∂^{i}U, then the k which witnesses this must be less than n − i. In particular, ∂^{n+1}T = ∂^{n}T. Observe that T′ = ∂^{n}T is in N and contains t_{r} = Z_{r} \ N.

Now we inductively build σ_{0}, σ_{1}, … σ_{n} ∈ M such that σ_{0} = ∅ and each σ_{i} is the initial part of some s ∈ T′ with σ_{i+1} having σ_{i} as an initial part whenever i < n. Given σ_{i} consider

{x ∈ S : σ_{i}^{⌢}x has a extension in T′}.

This set is in M, since it is definable from parameters which are in M. By elementarity of M and the definition of J, the above set contains a countably infinite subset H in M that is in I. In particular, H ⊆_{∗} I_{P} for all P ∈ N_{r}. Hence there is an element x ∈ H ∩ ⋂_{P ∈N_{r}} I_{P} such that σ_{i+1} = σ_{i}^{⌢}x has an extension in T′. Finally, at stage n, σ_{n} = σ = Z_{s} \ N_{s} ⊆ ⋂_{P ∈N_{r}} I_{P}, for some s ∈ D. Such an s is now compatible with r. We have therefore established that ℚ_{I} is proper.

5 Density arguments

While the main difficulty in proving implications such as PFA implies PID lies in the verification that the relevant forcing is proper, some additional argument is usually required to show that certain sets are dense. In the case of PID and OCA, this verification is straightforward. In order to illustrate the argument, we will go through the remainder of the proof that PFA implies PID.

Theorem 5.1. (PFA) PFA implies PID.

Proof. Suppose that I ⊆ [S]^{ω} is a P-ideal. If S can be covered by countably many sets in I^{⊥}, then there is nothing to show. Suppose that this is not the case and let M be a suitable model for ℚ_{I} and x ∈ X be outside of every element of I^{⊥} ∩ M. Define q = ({x}, {M ∩ H(θ)}). We have established that q is (M, ℚ_{I})-generic and therefore q forces that

Ż = ⋃_{p∈Ġ} Z_{p}

is uncountable. Also observe that q forces that every countable subset of S is contained in N for some N in Ṅ = ⋃_{p∈Ġ}. This is easily seen for countable sets in V and holds for sets in V [G] as well since ℚ_{I} is proper. Furthermore, q forces that if N is in Ṅ, then N ∩ Ż ⊆^{∗} I_{N} and hence is in I.

Let ḟ be a ℚ_{I}-name for an injection from ω_{1} into Ż and let ġ be a ℚ_{I}-name for a function from ω_{1} into I such that q forces ∀δ < ω_{1}(ḟ′′δ ⊆ ġ(δ)).

If we let D_{ξ} be the set of all p such that p decides ḟ(ξ) and ġ(ξ), then any filter G meeting D_{ξ} for all ξ < ω_{1} must satisfy that Z = Ż[G] is an uncountable set such that every countable subset of Z is in I.

6 Some application of PID

We now present some consequences of PID.

Definition 6.1. An ideal K on a set S is countably generated in X if there is a countable family {K_{n} : n ∈ ω} ⊆ X such that for all K ∈ K, there is an n such that K ⊆ K_{n}. We will say that K is countably generated if it is countably generated in K.

In what follows we will use the notational convention that K ↾ X = K∩P(X).

Theorem 6.2. PID implies that if K is an ideal on S that is not countably generated in K^{⊥⊥}, then there is an X ⊆ S of size less or equal to ℵ_{1} such that K ↾ X is not countably generated in K^{⊥⊥}.

Proof. Define I = K^{⊥} and assume that, for every countable X ⊆ S, K ↾ X is countably generated in K^{⊥⊥}.

Claim 6.3. I is a P-ideal

Proof. Let I_{n} ∈ I, for n ∈ ω and set X = ⋃_{n} I_{n}. Since X ⊆ S is countable, K ↾ X is countably generated in K^{⊥⊥}, and so fix a family {K_{i} : i ∈ ω} ⊆ K^{⊥⊥} which witnesses this. Without loss of generality we can also suppose that K_{i} ⊆ K_{i+1}.

Now define

I_{∗} = ⋃_{i}(I_{i} \ K_{i}).

To see that this set is in I, it is sufficient to show that it has finite intersection with K_{n} for each n < ω. For a given n, the intersection of I_{∗} with K_{n} is contained in ⋃_{i<n} I_{i}, a set which is in I and has finite intersection with K_{n}. Hence I_{∗} is in I as desired. Moreover I_{n} ⊆_{∗} I_{∗}, since I_{n} \ I_{∗} ⊆ K_{n} ∩ I_{n}, which is finite.

We can now apply PID. If S = ⋃_{n} S_{n} such that, for all n, S_{n} ∈ I^{⊥}, then {S_{n} : n ∈ ω} witnesses that K is countably generated in K^{⊥⊥}.

On the other hand, if Z ⊆ S is uncountable and [Z]^{ω} ⊆ I, then Z ∩ K is finite, for every K ∈ K, because if Z ∩ K, were infinite, then there would be a countable Y ⊆ Z ∩ K, but then Y ∈ [Z]^{ω} ⊆ I and hence Y ∩ K would be finite. Z witnesses that K ↾ Z is not countably generated in K^{⊥⊥}, because there is no family {K_{n}}_{n∈ω} in K^{⊥⊥}, such that, for all K ∈ K, there is an n such that K ⊆ K_{n}.

Corollary 6.4. PID implies SCH.

Proof. We can state SCH in the following way: for every strong limit cardinal μ, with cof(μ) < μ, 2^{μ} = μ^{+}. Suppose that SCH fails and let μ be the least witness to this. By Silver’s theorem, cof(μ) = ω. Fix an increasing sequence {μ_{n}}_{n∈ω} that converges to μ. For each β < μ^{+}, find K_{β,n} ⊆ β such that

• |K_{β,n}| ≤ μ_{n},

• ⋃_{n} K_{β,n} = β,

• ∀β < β′, for every m there exists an n such that K_{β,m} ⊆ K_{β′,n}

The key point here is that if X ⊆ μ^{+} is countable, then there is a β such that for every β′ > β, the ideal on X generated by {K_{β,n} ∩ X : n ∈ ω} equals the ideal on X generated by {K_{β′,n} : n ∈ ω}. This follows from the fact that 2^{2^{ℵ_{1}}} < μ^{+}.

The above fact implies that, if K is the ideal on μ^{+} generated by {K_{β}, n : β < μ^{+}, n ∈ ω}, then K ↾ X is countably generated ∀X ∈ [μ^{+}]^{ℵ_{1}}.

So, by Theorem 6.2 and by PID, K is countably generated in K^{⊥⊥}. In particular there is a Z ⊆ μ^{+} cofinal in μ^{+} such that all countable subsets of Z are contained in K_{β,n}, for some β and n.

Thus |Z^{ω}| = μ^{+} = |μ^{ω}| and so 2^{μ} = μ^{+}, since any subset X ⊆ μ can be seen as X = ⋃_{n∈ω} X ∩ μ_{n}.

Corollary 6.5. PID the failure of □(κ), for every regular κ > ℵ_{1}.

Proof. We just sketch the proof. Recall that □(κ) is the following principle: there is a sequence ⟨C_{α} : α < κ⟩ such that:

• C_{α} ⊆ α is club in α;

• if α is a limit point of C_{β}, then C_{α} = C_{β} ∩ α,

• there is no club C ⊆ κ such that, for all limit points α of C, C ∩ α = C_{α}.

Now, for α < β < κ inductively define

ϱ_{2}(α, β) = 1 + ϱ_{2}(α, min(C_{β} \ α)),
ϱ_{2}(α, α) = 0.

Thus ϱ_{2}(α, β) is the length of the walk from β to α.

We define, for β < κ,

K_{β,n} = {α < β : ϱ_{2}(α, β) ≤ n}.

It can be verified that ϱ_{2} has the following properties:

• if β < β′, then there is an n such that |ϱ_{2}(α, β) − ϱ_{2}(α, β′)| ≤ n for all α < β;

• if X ⊆ κ is unbounded, then there is a β such that {ϱ_{2}(α, β) : α ∈ X ∩ β} is infinite.

It follows that the ideal K generated by {K_{β,n} : (β < κ) ∧ (n < ω)} is not countably generated in K^{⊥⊥} and yet K ↾ β is countably generated in K^{⊥⊥} for all β < κ.

References

[1] Uri Abraham, Matatyahu Rubin, and Saharon Shelah. On the consistency of some partition theorems for continuous colorings, and the structure of ℵ_{1}-dense real order types. Ann. Pure Appl. Logic, 29(2):123–206, 1985.

[2] Uri Avraham and Saharon Shelah. Martin’s axiom does not imply that every two ℵ_{1}-dense sets of reals are isomorphic. Israel J. Math., 38(1-2):161–176, 1981.

[3] B. Balcar, T. Jech, and T. Pazák. Complete CCC Boolean algebras, the order sequential topology, and a problem of von Neumann. Bull. London Math. Soc., 37(6):885–898, 2005.

[4] James E. Baumgartner. All ℵ_{1}-dense sets of reals can be isomorphic. Fund. Math., 79(2):101–106, 1973.

[5] M. Bekkali. Topics in Set Theory. Springer-Verlag, Berlin, 1991. Lebesgue measurability, large cardinals, forcing axioms, ρ-functions, Notes on lectures by Stevo Todorčević.

[6] Ilijas Farah. All automorphisms of the Calkin algebra are inner. ArXiv preprint 0705.3085v9 (9/29/2009).

[7] —. Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Mem. Amer. Math. Soc., 148(702), 2000.

[8] Carlos Martinez. It is consistent that the Aronszajn lines are well quasiordered. preprint, 2009.

[9] Justin Tatch Moore. The Proper Forcing Axiom. Proceedings of the 2010 meeting of the ICM, Hyderabad, India.

[10] —. Open colorings, the continuum and the second uncountable cardinal. Proc. Amer. Math. Soc., 130(9):2753–2759, 2002.

[11] —. Set mapping reflection. J. Math. Log., 5(1):87–97, 2005.

[12] —. A five element basis for the uncountable linear orders. Ann. of Math. (2), 163(2):669–688, 2006.

[13] —. A universal Aronszajn line. Math. Res. Lett., 16(1):121–131, 2009.

[14] Saharon Shelah. Proper and Improper Forcing. Springer-Verlag, Berlin, second edition, 1998.

[15] Robert Solovay and S. Tennenbaum. Iterated Cohen extensions and Souslin’s problem. Ann. of Math., 94:201–245, 1971.

[16] Stevo Todorcevic. A note on the proper forcing axiom. In Axiomatic set theory (Boulder, Colo., 1983), volume 31 of Contemp. Math., pages 209– 218. Amer. Math. Soc., Providence, RI, 1984.

[17] —. Partition Problems In Topology. Amer. Math. Soc., 1989.

[18] —. A classification of transitive relations on ω_{1}. Proc. London Math. Soc. (3), 73(3):501–533, 1996.

[19] —. Basis problems in combinatorial set theory. Doc. Math., Extra Vol. II:43–52, 1998.

[20] —. A dichotomy for P-ideals of countable sets. Fund. Math., 166(3):251– 267, 2000.

[21] —. Biorthogonal systems and quotient spaces via Baire category methods. Math. Ann., 335(3):687–715, 2006.

[22] —. Walks on ordinals and their characteristics, volume 263 of Progress in Mathematics. Birkhäuser, 2007.

[23] Boban Veličković. Forcing axioms and stationary sets. Adv. Math., 94(2):256–284, 1992.

[24] —. ccc forcing and splitting reals. Israel J. Math., 147:209–220, 2005.

[25] —. OCA and automorphisms of (ω)/fin. Topology Appl., 49(1):1–13, 1993.

[26] Matteo Viale. The proper forcing axiom and the singular cardinal hypothesis. Jour. Symb. Logic, 71(2):473–479, 2006.

[27] —. A family of covering properties. Math. Res. Lett., 15(2):221–238, 2008. 14