ℙ_{max} VARIATIONS FOR SEPARATING CLUB GUESSING PRINCIPLES

TETSUYA ISHIU AND PAUL B. LARSON

Abstract. In his book on ℙ_{max} [6], Woodin presents a collection of partial orders whose extensions satisfy strong club guessing principles on ω_{1}. In this paper we employ one of the techniques from this book to produce ℙ_{max} variations which separate various club guessing principles. The principle (+) and its variants are weak guessing principles which were first considered by the second author [3] while studying games of length ω_{1}. It was shown in [1] that the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence on ω_{1}. In this paper we give an alternate proof of the second of these results, using Woodin’s ℙ_{max} technology. We also present a variation which produces a model with a ladder system which weakly guesses each club subset of ω_{1} club often. We show that this model does not satisfy the Interval Hitting Principle, thus separating these statements. The main technique in this paper, in addition to the standard ℙ_{max} machinery, is the use of condensation principles to build suitable iterations.

In Chapter 8 of his book on ℙ_{max} [6], Woodin presents a collection of ℙ_{max} variations whose extensions satisfy strong club guessing principles on ω_{1}, along with the statement that the nonstationary ideal on ω_{1} (NS_{ω_{1}}) is saturated (see pages 499-500, for instance). In this paper we employ one of the techniques from that chapter to produce ℙ_{max} variations which separate various club guessing principles. The arguments and results in this paper are significantly simpler than the ones used there. The separation of club guessing principles is carried out via iterations; no local forcing arguments are used. We present these iteration arguments in full and outline the way in which they are incorporated in the standard ℙ_{max} machinery.

The principle (+) and its variants are weak guessing principles which were first considered by the second author [3] while studying games of length ω_{1}. It was shown in [1] that the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence on ω_{1}. In this paper we give an alternate proof of the second of these results, using Woodin’s ℙ_{max} technology. With the ℙ_{max} approach it is more natural to produce sequences which weakly guess clubs at club many points, so our model for (+) satisfies a strengthening of (+) for which the guessing happens club often. As always with ℙ_{max} variations, the continuum has cardinality ℵ_{2} in our models. This research was done at the same time as [1], though the results in that paper were proved first. As a warm-up we present a variation which produces a model with a ladder system which weakly guesses each club subset of ω_{1} club often. We show that this model does not satisfy the Interval Hitting Principle, thus separating these statements.

Date: June 15, 2010.

The first author is supported in part by NSF grant DMS-0700983. The second author is supported in part by NSF grants DMS-0401603 and DMS-0801009.

Section 10.2 of [4] outlines a general method for producing ℙ_{max} variations relative to a given Σ_{2} sentence for H(ℵ_{2}). The variations presented here do not literally fall inside this framework, but are similar in spirit. One difference is that the variations presented here use Woodin’s stationary tower in place of the Boolean algebra ℙ(ω_{1})/NS_{ω_{1}}, as explained in Section 1. Modulo this difference, the variation presented in Section 2 would fall inside the framework of [4], except that the sentence that it preserves, club weak club guessing, is Σ_{2} in H(ℵ_{2}) with a predicate for NS_{ω_{1}}, which can easily be added to the framework of [4]. The variation presented in Section 3 is still further removed, as the sentence it preserves, (+)_{ω}^{c}, is not in general expressible in H(ℵ_{2}).

The main technique in this paper, in addition to the standard ℙ_{max} machinery, is the use of condensation principles to build suitable iterations.

1. Condensation

Given a cardinal κ, we let H(κ) denote the collection of sets whose transitive closure has cardinality less than κ. Woodin [6, Definition 8.15] defines the strong condensation principle for H(κ) to be the statement that there is a function F: κ → H(κ) such that, for all X ≺ ⟨H(κ), F, ∈⟩, F_{X} = F ↾(Ord ∩ M_{X}), where F_{X} and M_{X} are the images of F and M under the transitive collapse of X. We will use a consequence of strong condensation for H(κ) (for any κ ≥ ℵ_{2}), which we will call weak condensation for H(ℵ_{2}). For our purposes, a club set of countable subsets of an uncountable set Z is the set of all countably infinite subsets of Z closed under a given finitary function on Z. We note that if C is such a club, and A is a set in Z, then {X ∈ C | A ∈ X} is club, and {X ∩ ω_{1} | X ∈ C} is a club subset of ω_{1} in the usual sense (see [2], for instance).

1.1 Definition. Weak condensation for H(ℵ_{2}) is the statement that there exist ν_{α} and N_{α} = {N_{β}^{α} : β < ν_{α}} (α < ω_{1}) such that for each α < ω_{1},

• ν_{α} ∈ ω_{1};

• for all β < ν_{α}, N_{β}^{α} is the transitive collapse of a countable elementary

submodel of H(ℵ_{2}) with ω_{1}^{N_{β}^{α}} = α;

• for all β < γ < ν_{α}, N_{β}^{α} ⊆ N_{γ}^{α};

and such that for club many countable elementary submodels X of H(ℵ_{2}), the transitive collapse of X is an element of N_{X∩ω_{1}}.

The club guessing principles which hold in the models in this article are not preserved by the usual forcings to make the nonstationary ideal on ω_{1} precipitous, as they guess club often. For this reason, we use Woodin’s stationary tower to generate the elementary embeddings which are used to define the order on our partial orders. We will be using the so-called countable tower ℚ_{<δ}, where δ is presumed to be a Woodin cardinal, see [2], for instance). Given a model M of ZFC and a Woodin cardinal δ of M, an iteration of (M, ℚ_{<δ}^{M}) (see [6, Definition 5.19]) consists of a family of models M_{α} (α ≤ ω_{1}) and a commuting system of elementary embeddings j_{α,β} (α ≤ β ≤ ω_{1}) such that

• M_{0} = M;
• each embedding j_{α,β} maps from M_{α} to M_{β};

• each j_{α,α+1} is a generic embedding derived from forcing over M_{α} with ℚ_{<j_{0,α}(δ)}^{M_{α}}, and M_{α+1} is the corresponding generic ultrapower (replaced with its Mostowski collapse if it is wellfounded);

• for each limit ordinal β ≤ ω_{1}, M_{β} is the direct limit of the models M_{α} (α < β) under the maps j_{α,γ} (α ≤ γ < β) and each map j_{α,β} (α < β) is induced by this directed system.

We often use j to indicate the embedding j_{0,ω_{1}} derived from an iteration, and sometimes refer to the iteration itself as j. If M is a countable transitive model of ZFC and δ is a Woodin cardinal of M, an iteration

j: (M, ℚ_{<δ}^{M}) → (M^{∗}, ℚ_{<j(δ)}^{M^{∗}})

is full if every member a of ℚ_{<j(δ)}^{M^{∗}} is stationary in M^{∗} (i.e., a = j_{α,ω_{1}}(a_{α}) for some α < ω_{1} and some

a_{α} ∈ ℚ_{<j_{0,α}(δ)}^{M_{α}},

and the set of β ∈ [α, ω_{1}) such that j_{α,β}(a_{α}) is in the generic filter at stage β is a stationary subset of ω_{1}). If M is a countable transitive model of ZFC and δ is a Woodin cardinal in M, we say that the pair (M, ℚ_{<δ}^{M}) is iterable if M^{∗} is wellfounded for every iteration

j: (M, ℚ_{<δ}^{M}) → (M^{∗}, ℚ_{<j(δ)}^{M^{∗}})

of (M, ℚ_{<δ}^{M}). If A is a set of reals, we say that (M, ℚ_{<δ}^{M}) is A-iterable if it is iterable and if A ∩ M^{∗} = j(A ∩ M) for every iteration j: (M, ℚ_{<δ}^{M}) → (M^{∗}, ℚ_{<j(δ)}^{M∗}) of (M, ℚ_{<δ}^{M}).

The proof of Theorem 8.42 from [6] shows that the conditions in the partial orders defined in this paper exist in suitable generality.

Theorem 1.2 (Woodin). If AD holds in L(ℝ) and A is a set of reals in L(ℝ) then there exist a countable transitive model M of ZFC and an ordinal δ ∈ M such that

• δ is a Woodin cardinal in M;

• (M, ℚ_{<δ}^{M}) is A-iterable;

• ⟨V_{ω+1} ∩ M, A ∩ M, ∈⟩ ≺ ⟨V_{ω+1}, A, ∈⟩;

• strong condensation holds for H(κ) in M, where κ is the least strongly inaccessible cardinal of M.

2. Club weak club guessing

This section is mostly a warm-up for the next section, in which our main result is proved. We prove one separation result here, mainly to illustrate our approach in a simpler setting. It is likely that other separation results can be proved in a similar fashion.

We let club weak club guessing denote the statement that there is a sequence ⟨a_{α} : α < ω_{1}⟩ such that each a_{α} is a cofinal subset of α of ordertype at most ω, and such that for every club C ⊆ ω_{1}, a_{α} ∩ C is infinite for club many α. In this section we present a ℙ_{max} variation for the existence of a club weak club guessing sequence. We drop the conventional naming system for variations of ℙ_{max} and call the partial order in this section ℙ_{0}.

2.1 Definition. The partial order ℙ_{0} consists of all pairs ⟨M, A, δ, X⟩ such that (1) M is a countable transitive model of ZFC;

(2) δ is a Woodin cardinal in M;

(3) (M, ℚ_{<δ}^{M}) is iterable;

(4) A is a club weak club guessing sequence in M;

(5) X is a set in M consisting of iterations j: (N, ℚ_{<κ}^{N}) → (N^{∗}, ℚ_{<j(κ)}^{N^{∗}}) which are full in M, for some ⟨N, B, κ, Y ⟩ ∈ ℙ_{0}, with j(B) = A and j(Y) ⊆ X, such that no two distinct members of X are iterations of the same pair.

The order on ℙ_{0} is as follows: ⟨M, A, δ, X⟩ < ⟨N, B, κ, Y ⟩ if there exists a full iteration j: (N, ℚ_{<κ}^{N}) → (N^{∗}, ℚ_{<j(κ)}^{N^{∗}}) in X.

The following statement was introduced by Kunen in unpublished work.

2.2 Definition. The Interval Hitting Principle (IHP) is the statement that there exists a set {b_{α} : α < ω_{1}} such that each b_{α} is a cofinal subset of α of ordertype at most ω and such that for every club C ⊆ ω_{1} there is a limit ordinal α < ω_{1} such that for all but finitely many β ∈ b_{α}, C ∩ [β, min(b_{α} \ (β + 1)) is nonempty.

We will show that, assuming that AD holds in L(ℝ), the ℙ_{0}-extension of L(ℝ) satisfies club weak club guessing but not IHP. The following lemma is the key step in proving each of these facts. The lemma shows, assuming weak condensation for H(ℵ_{2}), that it is possible to iterate a countable transitive model in such a way that a given club weak club guessing sequence in the countable model is mapped to such a sequence in V (i.e., in the model constructing the iteration) while simultaneously mapping a witness to IHP to a sequence which fails to witness IHP.

Lemma 2.3. Suppose that (M, ℚ_{<δ}^{M}) is an iterable pair, and that

A = {a_{α} : α < ω_{1}^{M}}

is a club weak club guessing sequence in M. Suppose that

B = {b_{α} : α < ω_{1}^{M}}

is a set in M such that each b_{α} is a cofinal subset of α of ordertype at most ω. Suppose that weak condensation holds for H(ℵ_{2}). Then there is an iteration j: (M, ℚ_{<δ}^{M}) → (M^{∗}, ℚ_{<j(δ)}^{M^{∗}}) such that j(A) witnesses club weak club guessing and the critical sequence of j shows that j(B) does not witness IHP.

Proof. We let the usual iteration construction determine cofinally many members of each generic filter, including the first member. This guarantees the genericity of each filter and fullness of the iteration.

For each α < ω_{1}, we let a_{α}^{∗} and b_{α}^{∗} be the unique members of j(A) and j(B) respectively which are cofinal subsets of α. Each a_{α}^{∗} and each b_{α}^{∗} are determined by j_{0,γ}, where γ < ω_{1} is minimal such that j_{0,γ}(ω_{1}^{M}) > α.

Let N_{α} = {N_{β}^{α} : β < ν_{α}} (α < ω_{1}) witness weak condensation for H(ℵ_{2}).

In order to ensure that the critical sequence of j shows that j(B) does not witness IHP, we include in the construction of G_{α} (for each limit α < ω_{1}) a stage for each β < α, where we ensure that the interval between some consecutive pair of elements of b_{ω_{1}^{Mα}}^{∗} above β will be disjoint from the critical sequence of j. At the stage for

each such β, some set d ∈ ℚ_{<j_{0,α}(δ)}^{M_{α}} has been chosen to put into G_{α}. Since each b_{η} is a cofinal subset of η of ordertype at most ω, there must be some n ∈ ω such that for cofinally many γ < ω_{1}^{M_{α}} there exist d_{γ} ≤ d such that for all Y ∈ d_{γ}, the n-th element of b_{Y ∩ω_{1}^{Mα}}^{∗} exists and is greater than γ. Let n_{0} be the least such n. There exist an ordinal of the form ξ + 1 < α and a

d′ ∈ ℚ_{<j_{0,ξ+1}(δ)}^{M_{ξ+1}}

such that j_{ξ+1,α}(d′) = d and ω_{1}^{M_{ξ}} > β. Then n_{0} satisfies the same definition in M_{ξ+1} with d′ in place of d, so there is a condition d′′ ≤ d′ in ℚ_{<j_{0,ξ+1}(δ)}^{M_{ξ+1}} such that the n_{0}-th and (n_{0} +1)-st elements of b_{Y ∩ω_{1}^{M_{ξ+1}}}^{∗} are the same values ζ and ζ′ (greater

than ω_{1}^{M_{ξ}} and less than ω_{1}^{M_{ξ+1}}) for all Y ∈ d′′. Then j_{ξ+1,α}(d′′) ≤ d forces that ζ and ζ′ are the n_{0}-th and (n_{0} + 1)-st elements of b_{ω_{1}^{Mα}}^{∗}, and no member of the critical sequence is in the interval [ζ, ζ′), as desired.

In order to guarantee that j(A) is a club weak club guessing sequence, the new part of the construction of each G_{α} includes a stage for each pair (β, C), where β < ω_{1}^{M_{α}} and C is a club subset of ω_{1}^{M_{α}} in some member of N_{ω_{1}^{Mα}}. At this stage,

some set d ∈ ℚ_{<j_{0,α}(δ)}^{M_{α}} has been chosen to put into G_{α}. If there is a d′ ≤ d forcing that γ ∈ a_{ω_{1}^{Mα}}^{∗} for some γ ∈ C \ β, then we choose such a d′ to put into G_{α} (and we say that we have met the challenge (β, C)). Otherwise, we do nothing at this stage.

Having completed the construction, suppose that C is a club subset of ω_{1}. Let X ≺ H(ℵ_{2}) be in the club corresponding to N_{α} (α < ω_{1}), with j and C in X. Let α = X ∩ ω_{1}, and note that ω_{1}^{M_{α}} = α. We will show that we met every challenge (β, C ∩ α) in the construction of G_{α}, which implies that a_{α} ∩ C is cofinal in α. The transitive collapse of X, call it N, is in N_{α}. Since j is full, every member of ℚ_{<j(δ)}^{M_{ω_{1}}} is stationary in V, so every member of ℚ_{<j_{0,α}(δ)}^{M_{α}} is stationary in N. Fix β < α and

let d ∈ ℚ_{<j_{0,α}(δ)}^{M_{α}} be the condition which was put into G_{α} just before the stage of the construction of G_{α} corresponding to (β, C ∩ α). The set f consisting of those γ < ω_{1}^{M_{α}} which are forced by some b′ ≤ b to be in a_{α}^{∗} is stationary in M_{α}. To see this, note that j_{0,α}(A) = {a_{γ}^{∗} : γ < α} is a club weak club guessing sequence in M_{α}. Let e be a club subset of α in M_{α}. Then there is in M_{α} a club set e′ ⊆ α \ (β + 1) such that e ∩ a_{γ}^{∗} is cofinal in γ for all γ ∈ e′. Let d_{0} be the set of Y ∈ d such that Y ∩ α ∈ e′. Then d_{0} ≤ d in ℚ_{<j_{0,α}(δ)}^{M_{α}}. There is a regressive function g on d_{0} such that each g(Y) is an element of a_{Y ∩α}^{∗} ∩ (e \ β). This function is constant on a condition d′ ≤ d in ℚ_{<j_{0,α}(δ)}^{M_{α}}. If γ is this constant value, then d′ forces that γ ∈ a_{α}^{∗} ∩ e, which shows that γ ∈ f ∩ e. Since M_{α} is stationarily correct in N, f is stationary in N, which means that (f ∩ C) \ β is nonempty. Thus we could and did meet the challenge at every stage of the construction of G_{α} corresponding to a pair (β, C ∩ α), for some β < α. It follows that a_{α}^{∗} ∩ C is cofinal in α. □

Assuming AD^{L(ℝ)}, the basic analysis of the ℙ_{0} extension requires only Lemma 2.3 in addition to standard ℙ_{max} arguments. Theorem 1.2 implies that ℙ_{0} conditions exist, and Theorem 1.2 and Lemma 2.3 together imply that every ℙ_{0} condition has a stronger condition below it (the version of this fact corresponding to ℙ_{1} is sketched after the proof of Lemma 3.3).

The ω-closure of ℙ_{0} is proved using the adaptation of Lemma 2.3 to limit structures (minus the part of the argument regarding IHP, which is not necessary).

Lemma 2.4. If AD^{L(ℝ)} holds, then every descending chain of ℙ_{0} conditions of length ω has a lower bound in ℙ_{0}.

Aside from notational complications, the iteration argument needed to prove Lemma 3.4 is essentially the same as the proof of Lemma 2.3. Briefly, given a descending sequence of conditions p_{i} = ⟨M_{i}, A_{i}, δ_{i}, X_{i}⟩, one forms a limit sequence ⟨M_{i}^{∗}, δ_{i}^{∗} : i < ω⟩ by composing the embeddings witnessing the order on these conditions. The images of the sets A_{i} under this composition are all the same set A^{∗}. One then applies Theorem 1.2 to find a suitable iterable pair (N, ℚ_{<κ}^{N}) with ⟨M_{i}^{∗}, δ_{i}^{∗} : i < ω⟩ ∈ H(ℵ_{1})^{N} and, working in N, constructs a full iteration j of ⟨M_{i}^{∗}, δ_{i}^{∗} : i < ω⟩ (in the corresponding sense) for which the image of A^{∗} is a club weak club guessing sequence in N. Again, the construction of j is just like the argument for Lemma 2.3. The condition ⟨N, j(A^{∗}), κ, Y ⟩ is then below each p_{i}, where Y is the union of the images of the sets X_{i} under the composition of the relevant embeddings.

Suppose that AD holds in L(ℝ), and let G ⊂ ℙ_{0} be an L(ℝ)-generic filter. For each p = ⟨M, A, δ, X⟩ ∈ G, if we consider all the stronger conditions ⟨N, B, κ, Y ⟩ in G, we see that the iterations of (M, ℚ_{<δ}^{M}) in these sets Y must be initial segments of one another, and must in fact all be proper initial segments of one iteration j_{p} of (M, ℚ_{<δ}^{M}) of length ω_{1}. We let P(ω_{1})_{G} denote the union of all sets of the form j_{p}(P(ω_{1})^{M}) for all p = ⟨M, A, δ, X⟩ ∈ G. The definition of the order on ℙ_{0} implies that j_{p}(A) is the same set for all p = ⟨M, A, δ, X⟩ ∈ G; we call this set A_{G}.

The following theorem gives the basic analysis of the ℙ_{0} extension. Again, the proof of the following theorem involves only the modification of the corresponding proof for ℙ_{max}, using the version of Lemma 2.3 for building a descending ω_{1}- sequence of conditions. We refer the reader to [6] for the definition of ψ_{AC}. For our purposes, the only relevant fact about ψ_{AC} is that we can deduce the Axiom of Choice from ψ_{AC} in the ℙ_{0} extension.

Theorem 2.5. Assume AD^{L(ℝ)}. Then the following hold in the ℙ_{0}-extension of L(ℝ).

(1) P(ω_{1}) = P(ω_{1})_{G};

(2) ω_{1}-DC;

(3) NS_{ω_{1}} is saturated;

(4) ψ_{AC};

(5) A_{G} is a club weak club guessing sequence;

(6) ¬ IHP.

Briefly, the proofs of the first two parts of Theorem 2.5 involve coding a given ℙ_{0}-name for a set of reals (below a given condition) with a set of reals B, and then applying Theorem 1.2 to obtain an B-iterable pair (M, ℚ_{<δ}^{M}) (this the the use of Biterability). Working inside M, one builds a descending ω_{1}-sequence of conditions which realizes ℵ_{1}-much of this name (from the perspective of M). The standard argument for doing this requires only choosing the first element of each generic filter correctly. The new steps from the proof of Lemma 2.3 can be worked in to preserve the selected club weak club guessing sequences. The last four parts of the theorem use the fact that P(ω_{1})_{G} = P(ω_{1}) in the ℙ_{0}-extension. As with the other parts of the theorem, the standard proofs of the third and fourth parts require only choosing the first element of each generic filter correctly, so the new steps can be worked in. That A_{G} witnesses club weak club guessing in the extension follows from P(ω_{1})_{G} = P(ω_{1}) and the fact that the selected club weak club guessing sequence A is preserved from one condition to the next. That IHP fails in the extension is due to Lemma 2.3, which in conjunction with Theorem 1.2 says that any IHP-sequence in any condition can be mapped to a sequence which fails to witness IHP in a stronger condition.

The Optimal Iteration Lemma [5] for the existence of a club weak club guessing sequence is the version of Lemma 2.3 obtained by replacing the hypothesis of weak condensation with the assumption of a club weak club guessing sequence (and removing the part concerning IHP). We do not know whether this lemma holds. As a result, we do not know if ℙ_{0} extensions are Π_{2}-maximal for H(ℵ_{2}) relative to the existence of a club weak club guessing sequences, or whether this form of Π_{2}-maximality is even possible. Similarly, we do not know if ℙ_{0} is homogeneous.

3. (+) and its variants

The original principle (+) asserts the existence of a stationary (i.e., intersecting each such club) family F consisting of countable elementary submodels of H(ℵ_{2}), with the property that whenever M, N are in F and M ∩ ω_{1} = N ∩ ω_{1}, all club subsets of ω_{1} in M intersect all club subsets of ω_{1} in N cofinally below M ∩ ω_{1}. This was reformulated by Justin Moore as the following statement: there exist ⟨f_{α} : α < ω_{1}⟩ such that each f_{α} is a (possibly empty) set of club subsets of α which pairwise have intersection cofinal in α, such that for every club subset C of ω_{1} there is an α with C ∩ α ∈ f_{α}. Here we consider a strengthening of this principle.

3.1 Definition. The principle (+)_{ω}^{c} asserts the existence of a sequence

⟨f_{α} : α < ω_{1}⟩

such that

• for each α < ω_{1},

– each element of f_{α} is a club subset of α;

– f_{α} is closed under finite intersections;

• for every club C ⊆ ω_{1}, {α < ω_{1} | C ∩ α ∈ f_{α}} contains a club.

Our goal in this section is to produce a ℙ_{max} variation in whose extension (+)_{ω}^{c} holds yet there is no club guessing sequence on ω_{1}.

3.2 Definition. The partial order ℙ_{1} consists of all pairs ⟨M, F, δ, X⟩ such that

(1) M is a countable transitive model of ZFC + CH;

(2) δ is a Woodin cardinal in M;

(3) (M, ℚ_{<δ}^{M}) is iterable;

(4) F = ⟨f_{α} : α < ω_{1}^{M}⟩ witnesses (+)_{ω}^{c} in M;

(5) X is a set in M, each element of X is an iteration j: (N, ℚ_{<κ}^{N}) → (N^{∗}, ℚ_{<j(κ)}^{N^{∗}})

such that • j is full in M,

• there exist H, Y such that ⟨N, H, κ, Y ⟩ ∈ ℙ_{1};

• j(Y) ⊆ X;

• letting j(H) = ⟨h_{α}^{∗} : α < ω_{1}^{N}⟩, h_{α}^{∗} ⊆ f_{α} for all α < ω_{1}^{M}; and no two distinct members of X are iterations of the same pair.

The order on ℙ_{1} is as follows: ⟨M, F, δ, X⟩ < ⟨N, H, κ, Y ⟩ if there exists an iteration j: (N, ℚ_{<κ}^{N}) → (N^{∗}, ℚ_{<j(κ)}^{N^{∗}}) in X.

As in the previous section, we use weak condensation to prove our iteration lemma.

Lemma 3.3. Suppose that • ⟨N_{β}^{α} : α < ω_{1}, β < ν_{α}⟩ witnesses weak condensation for H(ℵ_{2});

• (M, ℚ_{<δ}^{M}) is an iterable pair such that M ⊧ CH;

• ⟨f_{α} : α < ω_{1}^{M}⟩ witnesses (+)_{ω}^{c} in M;

• ⟨a_{α} : α < ω_{1}^{M}⟩ is such that for each limit α < ω_{1}, a_{α} is a cofinal subset of

α of ordertype at most ω. Then there exist a full iteration

⟨M_{α}, G_{β}, j_{α,γ} : β < ω_{1}, α ≤ γ ≤ ω_{1}⟩

of (M, ℚ_{<δ}^{M}) and a function e: ω_{1} → ω_{1} such that, letting

⟨f_{α}^{∗} : α < ω_{1}⟩ = j_{0,ω_{1}}(⟨f_{α} : α < ω_{1}^{M}⟩)

and

⟨a_{α}^{∗} : α < ω_{1}⟩ = j_{0,ω_{1}}(⟨a_{α} : α < ω_{1}^{M}⟩),
(1) for every α < ω_{1},

(a) e(α) ≤ ν_{α};

(b) the intersection of any member of f_{α}^{∗} with any club subset of α in ⋃_{β<e(α)} N_{β}^{α} is cofinal in α;

(c) if α is a limit ordinal, then a_{ω_{1}^{Mα}}^{∗} \ {ω_{1}^{M_{β}} : β < α} is infinite;

(2) for each club C ⊆ ω_{1}, the set of α such that C ∩ α ∈ ⋃_{β<e(α)} N_{β}^{α} contains a club.

Proof. We construct the iteration in the usual way, with the following modifications. We allow the ordinary construction to determine cofinally many members of each G_{β}, including the first one, and fill in the intervening steps ourselves. This guarantees the genericity of each filter and the fullness of the iteration.

To ensure conclusion (1c), when α is a limit ordinal we include a stage for each ξ < ω_{1}^{M_{α}}, as follows. When we come to this stage, we have some

b ∈ ℚ_{<j_{0,α}(δ)}^{M_{α}}

which we have decided to put into G_{α}. Let d = {ω_{1}^{M_{ξ}} : ξ < α}. Then d \ e is bounded in ω_{1}^{M_{α}} for each club e ∈ P(ω_{1})^{M_{α}}. Let

x = {ρ ∈ (ξ, ω_{1}^{M_{α}}) | {Y ∈ b | ρ ∈ a_{Y ∩ω_{1}^{Mα}}^{∗} } ∈ ℚ_{j_{0,α}(δ)}^{M_{α}}}.

This set is cofinal in ω_{1}^{M_{α}}, and its closure cannot be contained in d. Since d is closed, that means that x is not contained in d. Pick an ordinal ρ ∈ x \ d, and put

{Y ∈ b | ρ ∈ a_{Y ∩ω_{1}^{Mα}}^{∗} }

in G_{β}. This condition forces that ρ ∈ a_{ω_{1}^{Mα}}^{∗}. Collectively, then, these stages of the

construction ensure that a_{ω_{1}^{Mα}}^{∗} \ d is cofinal in ω_{1}^{M_{α}}.

It remains to describe the parts of the iteration which ensure the others parts of the conclusion of the lemma. Note that each f_{α}^{∗} will be of the form j_{β+1,ω_{1}}(f), for some f in M_{β+1}, where β is minimal such that ω_{1}^{M_{β+1}} > α. In fact, f_{α}^{∗} will be of the form

⋃_{γ∈[β+1,ω_{1})} j_{β+1,γ}(f).

We refer to each j_{β+1,γ}(f) as f_{α}^{∗} ∩ M_{γ}. This is a slight (but unambiguous) abuse of notation, as the sets of the form f_{α}^{∗} ∩M_{γ} will be determined once M_{γ} is determined, but f_{α}^{∗}, being in general an uncountable subset of P(α), will not be defined until the entire construction is completed.

While constructing G_{α}, we include a stage for each tuple (h, ξ, ζ, c) of the following type:

• ζ < ξ ≤ ω_{1}^{M_{α}};

• h is a function in M_{α} whose domain b_{h} is an element of ℚ_{<j_{0,α}(δ)}^{M_{α}} compatible

with every such element (i.e., a club of countable sets in V_{j_{0,α}(δ)}^{M_{α}});

• in the case ξ < ω_{1}^{M_{α}}, the codomain of h is f_{ξ}^{∗} ∩ M_{α};

• in the case ξ = ω_{1}^{M_{α}}, for all Y ∈ b_{h}, Y ∩ω_{1}^{M_{α}} ∈ ω_{1}^{M_{α}} and h(Y) ∈ f_{Y ∩ω_{1}}^{∗}∩M_{α};

• c is a club subset of ξ in N_{β}^{ω_{1}^{Mα}}, for some β < ν_{ω_{1}^{Mα}}.

When we come to the stage for a given (h, ξ, ζ, c), we have some b ∈ ℚ_{<j_{0,α}(δ)}^{M_{α}} which we have decided to put into G_{α}. Since b_{h} is a club, we may assume that b ≤ b_{h}. If possible, we find some γ ∈ (ζ, ξ) ∩ c such that

{Y ∈ b | γ ∈ h(Y ∩ ⋃ b_{h})}

is in ℚ_{<j_{0,δ}}^{M_{α}}, and we put this set in G_{α}. If there is no such δ, we do nothing at this stage.

Having completed the construction of the iteration, for each α < ω_{1}, let z_{α} be the set of β < ν_{α} such that there exists a countable elementary substructure X of H(ℵ_{2}) with the iteration ⟨M_{α}, G_{β}, j_{α,γ} : β < ω_{1}, α ≤ γ ≤ ω_{1}⟩ as a member, such that the transitive collapse of X is N_{β}^{α}. If z_{α} is empty, let e(α) = 0. If z_{α} has a maximal element β_{α}, let e(α) = β_{α} + 1. Otherwise, let e(α) be the supremum of z_{α}. In all cases, e(α) ≤ ν_{α}, and conclusions (1a) and (2) are satisfied.

It remains to check that conclusion (1b) is satisfied, i.e., that for each ξ < ω_{1}, the intersection of any member of f_{ξ}^{∗} with any member of ⋃_{β<e(ξ)} N_{β}^{ξ} is cofinal in

ξ. We need check only those ξ for which e(ξ) > 0. For these ξ, ω_{1}^{M_{ξ}} = ξ. Fix such a ξ, and fix a club subset c of ξ in ⋃_{β<e(ξ)} N_{β}^{ξ}. We show by induction on α ∈ (ξ, ω_{1}) that the intersection of any member of f_{ξ}^{∗} ∩ M_{α} with c is cofinal in ξ. Note that this is preserved automatically at limit stages.

First consider the case α = ξ + 1. When we reach the stage for a tuple of the form (h, ξ, ζ, c) in the construction of G_{ξ}, we have some b ∈ ℚ_{<j_{0,ξ}(δ)}^{M_{ξ}} which we have chosen to put into G_{ξ}. Since b_{h} and b are compatible, we may assume by shrinking b if necessary that b ≤ b_{h}. Consider the set E consisting of those γ ∈ (ζ, ω_{1}^{M_{ξ}}) for which there exists a condition b′ ≤ b such that γ ∈ h(Y ∩ ⋃ b_{h}) for all Y ∈ b′. Suppose towards a contradiction that E is nonstationary in M_{ξ}. Then there is a club D ∈ P(ω_{1})^{M_{ξ}} disjoint from E, and, since ⟨f_{β}^{∗} ∩ M_{ξ} : β < ξ⟩ witnesses (+)_{ω}^{c} in M_{ξ}, there is a club D′ ∈ P(ω_{1} \ (ζ + 2))^{M_{ξ}} such that for all ρ ∈ D′, D ∩ ρ ∈ f_{ρ}^{∗} ∩ M_{ξ}. Then there is a condition b_{0} ≤ b such that for all Y ∈ b_{0}, Y ∩ ω_{1}^{M_{ξ}} ∈ D′, and there is a regressive function on b_{0} which picks for each such Y an element of D ∩ h(Y ∩ ⋃ b_{h}) greater than ζ. Thinning b_{0} to make this regressive function constant gives a contradiction to the claim that D is disjoint from E. Then E is stationary in M_{ξ}, and also in N_{β}^{ξ} for all β < e(ξ), since M_{ξ} is stationarily correct in these models. Since E ∩ c is cofinal in ξ, we may choose γ ∈ E ∩ c ∩ (ζ, ξ) and b′ ≤ b such that b′ forces the element of f_{ξ}^{∗} ∩ M_{α} represented by h to intersect c above γ. Applying this argument for every tuple of the form (h, ξ, ζ, c) takes care of the case α = ξ + 1.

Now suppose that α > ξ and the induction hypothesis holds for all members of (ξ, α]. We show that it holds for α + 1. When we reach the stage for a tuple of the form (h, ξ, ζ, c) in the construction of G_{α}, we have some b ∈ ℚ_{<j_{0,α}(δ)}^{M_{α}} which we have

chosen to put into G_{α}. We may assume that b ≤ b_{h} and that Y ∩ ω_{1}^{M_{α}} ∈ ω_{1}^{M_{α}} \ ξ for all Y ∈ b. We need to see that there exists a γ ∈ (ζ, ξ) ∩ c such that the set of Y ∈ b such that γ ∈ h(Y ∩ ⋃ b_{h}) is stationary in M_{α}. Supposing that there is no γ as desired, then let d be the set of γ ∈ (ζ, ξ) such that b forces in M_{β} that γ is not in [h]_{G_{β}}. Then c\(ζ +1) ⊆ d. By our induction hypothesis, all elements of f_{ξ}^{∗} ∩M_{α} have cofinal intersection with c. So for each Y ∈ b there is a ρ(Y) ∈ d such that ρ(Y) ∈ h(α), and we get a contradiction again by thinning b to make this function constant. This completes the proof. □

The remainder of this section is similar to the end of the previous section. Assuming AD^{L(ℝ)}, the basic analysis of the ℙ_{1} extension requires only Lemma 3.3 in addition to standard ℙ_{max} arguments. Theorem 1.2 implies that ℙ_{1} conditions exist, and Theorem 1.2 and Lemma 3.3 together imply that every ℙ_{1} condition has a stronger condition below it. That is, if p = ⟨M, F, δ, X⟩ is a ℙ_{1} condition, Theorem 1.2 says that there is an iterable pair (N, ℚ_{<κ}^{N}) such that p ∈ H(ℵ_{1})^{N} and N satisfies weak condensation for H(ℵ_{2}). Letting ⟨N_{β}^{α} : α < ω_{1}^{N}, β < ν_{α}⟩ witnesses weak condensation for H(ℵ_{2}) in N, apply Lemma 3.3 in N to obtain a full iteration j of (M, ℚ_{<δ}^{M}) and a function e: ω_{1}^{N} → ω_{1}^{N} such that, letting j(F) = ⟨f_{α}^{∗} : α < ω_{1}⟩,

• for every α < ω_{1},

– e(α) ≤ ν_{α};

– the intersection of any member of f_{α}^{∗} with any club subset of α in

⋃_{β<e(α)} N_{β}^{α} is cofinal in α;

• for each club C ⊆ ω_{1}^{N} in N, the set of α such that C ∩ α ∈ ⋃_{β<e(α)} N_{β}^{α} contains a club.

Let Y = j(X) ∪ {j}. For each α < ω_{1}^{N}, let h_{α}^{0} be the collection of club subsets of α in ⋃_{β<e(α)} N_{β}^{α}, and let h_{α} be the set of all finite intersections of members of f_{α}^{∗} ∪ h_{α}^{0}. Let H = ⟨h_{α} : α < ω_{1}^{N}⟩. Then ⟨N, H, κ, Y ⟩ is a ℙ_{1} condition stronger than p.

The ω-closure of ℙ_{1} is proved using the adaptation of Lemma 3.3 to limit structures (minus the part of the argument regarding club guessing, which is not necessary).

Lemma 3.4. If AD^{L(ℝ)} holds, then every descending chain of ℙ_{1} conditions of length ω has a lower bound in ℙ_{1}.

Aside from notational complications, the iteration argument needed to prove Lemma 3.4 is essentially the same as the proof of Lemma 3.3. Briefly, given a descending sequence of conditions p_{i} = ⟨M_{i}, F_{i}, δ_{i}, X_{i}⟩, one forms a limit sequence ⟨M_{i}^{∗}, δ_{i}^{∗} : i < ω⟩ by composing the embeddings witnessing the order on these conditions. The images of the sequences F_{i} induce via coordinatewise union a sequence F^{∗} whose members are all closed under finite intersections. One then applies Theorem 1.2 to find a suitable iterable pair (N, ℚ_{<κ}^{N}) with

⟨M_{i}^{∗}, δ_{i}^{∗} : i < ω⟩ ∈ H(ℵ_{1})^{N}

and, working in N, constructs a full iteration j of ⟨M_{i}^{∗}, δ_{i}^{∗} : i < ω⟩ (in the corresponding sense) for which the image of F^{∗} satisfies the conclusion of Lemma 3.3. Again, the construction of j is just like the argument for Lemma 3.3. Then there is a set H as discussed above (after the proof of Lemma 3.3) such that the condition ⟨N, H, κ, Y ⟩ is below each p_{i}, where Y is the union of the images of the sets X_{i} under the composition of the relevant embeddings.

Suppose that AD holds in L(ℝ), and let G ⊂ ℙ_{1} be an L(ℝ)-generic filter. For each p = ⟨M, F, δ, X⟩ ∈ G, if we consider all the stronger conditions ⟨N, H, κ, Y ⟩ in G, we see that the iterations of (M, ℚ_{<δ}^{M}) in these sets Y must be initial segments of one another, and must in fact all be proper initial segments of one iteration j_{p} of (M, ℚ_{<δ}^{M}) of length ω_{1}. As before, we let P(ω_{1})_{G} denote the union of all sets of the form j_{p}(P(ω_{1})^{M}) for all p = ⟨M, A, δ, X⟩ ∈ G. We also get a sequence F_{G} = ⟨f_{α}^{G} : α < ω_{1}⟩ such that each f_{α}^{G} is the union of all sets of the form f_{α}^{∗}, where p = ⟨M, F, δ, X⟩ ∈ G, and j_{p}(F) = ⟨f_{α}^{∗} : α < ω_{1}⟩. The definition of the order on ℙ_{1} implies that each f_{α}^{G} is a collection of club subsets of α closed under finite intersections.

The following theorem gives the basic analysis of the ℙ_{1} extension. Again, the proof of the following theorem involves only the modification of the corresponding proof for ℙ_{max}, using the version of Lemma 3.3 for building a descending ω_{1}- sequence of conditions. Again, for our purposes, the only relevant fact about ψ_{AC} is that we can deduce the Axiom of Choice from ψ_{AC} in the ℙ_{1} extension.

Theorem 3.5. Assume AD^{L(ℝ)}. Then the following hold in the ℙ_{1}-extension of L(ℝ).

(1) P(ω_{1}) = P(ω_{1})_{G};

(2) ω_{1}-DC;

(3) NS_{ω_{1}} is saturated;

(4) ψ_{AC};

(5) F_{G} witnesses (+)_{ω}^{c};

(6) there is no club guessing sequence.

Again, the proofs of the first two parts of Theorem 3.5 involve coding a given ℙ_{1}-name for a set of reals (below a given condition) with a set of reals A, and then applying Theorem 1.2 to obtain an A-iterable pair (M, ℚ_{<δ}^{M}). Working inside M, one builds a descending ω_{1}-sequence of conditions which realizes ℵ_{1}-much of this name (from the perspective of M). The standard argument for doing this requires only choosing the first element of each generic filter correctly. The new steps from the proof of Lemma 3.3 can be worked in ensure that the image of the selected witness to (+)_{ω}^{c} can be extended to a witness to (+)_{ω}^{c} in the larger model. The last four parts of the theorem use the fact that P(ω_{1})_{G} = P(ω_{1}) in the ℙ_{1}-extension.

As with the other parts of the theorem, the standard proofs of the third and fourth parts require only choosing the first element of each generic filter correctly, so the new steps can be worked in. That F_{G} witnesses (+)_{ω}^{c} in the extension follows from P(ω_{1})_{G} = P(ω_{1}) and the fact that the members of the selected sequence F are extended from one condition to the next. That club guessing fails in the extension is due to Lemma 3.3, which in conjunction with Theorem 1.2 says that any club guessing sequence in any condition can be mapped to a sequence which fails to witness club guessing in a stronger condition.

Again, the Optimal Iteration Lemma for (+)_{ω}^{c} is the version of Lemma 3.3 obtained by replacing the hypothesis of weak condensation with (+)_{ω}^{c} itself (and removing the part concerning club guessing sequences). We do not know whether this lemma holds. As a result, we do not know if ℙ_{1} extensions are Π_{2}-maximal for H(ℵ_{2}) relative to (+)_{ω}^{c}, or whether this form of Π_{2}-maximality is even possible. Similarly, we do not know if ℙ_{1} is homogeneous.

References

[1] T. Ishiu, P. Larson, Some results about (+) proved by iterated forcing, in preparation.

[2] P. Larson, The Stationary Tower. Notes on a course by W. Hugh Woodin. University Lecture Series, Vol. 32. American Mathematical Society, Providence, RI. 2004.

[3] P. Larson, The canonical function game, Archive for Math. Logic 44 (2005) 7, 817-827

[4] P. Larson, Forcing over models of determinacy, in : The Handbook of Set Theory, Foreman, Kanamori, eds., Springer, 2010

[5] J. Zapletal, S. Shelah, Canonical models for ℵ_{1}-combinatorics, Ann. Appl. Logic 98 (1999) 1-3, 217-259

[6] W.H. Woodin,The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. DeGruyter Series in Logic and Its Applications, vol. 1, 1999

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