Bounding by canonical functions, with CH^{∗}^{†}

Paul Larson^{‡}  Saharon Shelah^{§}

December 10, 2003

Abstract We show that the members of a certain class of semi-proper iterations do not add countable sets of ordinals. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from ω_{1} to ω_{1} is bounded on a club by a canonical function for an ordinal less than ω_{2}.

1 Introduction

Given an ordinal γ, a function f: ω_{1} → Ord is a canonical function for γ if the empty condition (i.e., ω_{1}) in the forcing P(ω_{1})/NS_{ω_{1}} forces that j(f)(ω_{1}^{V}) = γ, where j is the elementary embedding induced by the generic. For each α < ω_{1}, the constant function with value α is the canonical function for α. For α ∈ [ω_{1}, ω_{2}), a canonical function f for α is obtained by taking a bijection g: ω_{1} → α and letting f(β) be the ordertype of g[β]. In this paper we let Bounding denote the statement that every function from ω_{1} to ω_{1} is bounded on a club subset of ω_{1} by a canonical function for an ordinal less than ω_{2}. It is fairly easy to see that if the nonstationary ideal on ω_{1} (NS_{ω_{1}}) is saturated, then Bounding holds. The second author has shown [12] that given the existence of a Woodin cardinal there is a semi-proper forcing making NS_{ω_{1}} saturated, and it has been known for some time that there is a simpler forcing making Bounding hold from a weaker large cardinal hypothesis. The most quotable result in this paper is that this standard forcing to make every function from ω_{1} to ω_{1} bounded by a canonical function is (ω, ∞)-distributive (i.e., it does not add ω-sequences of ordinals), and so this statement is consistent with the Continuum Hypothesis, even in the presence of large cardinals. This is in contrast with saturation, as Woodin [15] has shown that if NS_{ω_{1}} is saturated and sufficiently large cardinals exist, then there is a definable counterexample to CH. We give a more general theorem stating that the members of a certain class of semi-proper iterations are (ω, ∞)-distributive. This class includes the standard forcing to make Bounding hold, and is general enough to show that a generalization of Bounding for certain sets of reals is also consistent with CH, answering a question in [15].

∗This research was conducted while both authors were in residence at the Mittag-Leffler Institute. We thank the Institute for its hospitality.

†MSC 2000: 03E35, 03E50, 03E55. Keywords: Iterated Forcing, Canonical Functions, Continuum Hypothesis.

‡The research of the first author was supported in part by the NSERC grants of Juris Steprāns, Paul Szeptycki and Franklin D. Tall, and the Centre de Recerca Matemàtica of the Institut d’Estudis Catalan.

§The research of the second author was supported by the Israel Science Foundation, founded by the Israel Academy of Sciences. Publication number 746.

The key construction used in the proof generalizes the notion of α-properness from Chapter V of [12] to semi-proper forcing. Briefly, a forcing is α-semi-proper if for any ∈-chain of countable elementary submodels of length α, there is a condition which is simultaneously semi-generic for each model in the sequence. The problem in applying the method to show that a given improper iteration is (ω, ∞)-distributive is that for a given model N in the sequence, N[G] ∩ κ can be a proper superset of N ∩ κ, where κ is the length of the iteration and G is generic for some initial segment, so that new steps appear. For the forcings in this paper, however, we have a good understanding of how to enlarge each such N, as well as how to produce the appropriate tower of models to overcome this.

This can be generalized further, getting the consistency of certain forcing axioms, using ideas from [12], Chapters V and VIII, and [14]. The reader is referred to [11] for more on this topic and on RCS in particular.

Interest in this question derives also from the study of Woodin’s ℙ_{max} forcing [15], which produces a model in which CH fails and all forceable Π_{2} sentences for H(ω_{2}) hold simultaneously. It is not known whether all such Π_{2} sentences forceably consistent with CH can hold together with CH. The generalized form of bounding in this paper is a candidate for showing that this is impossible. Candidates for the other half of the incompatibility appear in [6, 13] and in Section 10.6 of [15].

2 Skolem Hulls

Given a structure M with a predicate <^{∗} for a wellordering of the domain of M, and a subset X of the domain of M, we let Sk_{(M,∈,<^{∗})}(X) denote the Skolem hull of X in (M, ∈, <^{∗}). If X is a countable elementary substructure of some H(χ) and η is an ordinal in X, then we let D_{η}^{X} be the set of all a ∈ [η]^{<ω} such that f(a) ∈ X for all f: [η]^{|a|} → ω_{1} in X. The point is that if <^{∗} is a wellordering of H(χ), X ≺ (H(χ), ∈, <^{∗}) is countable, and z is a subset of some ordinal η ∈ X, then

Sk_{(H(χ),∈,<^{∗})}(X ∪ z) = {f(a) : a ∈ [z]^{<ω} ∧ f ∈ H(χ)^{([η]^{<ω})} ∩ X}

and so if z is such that [z]^{<ω} ⊂ D_{η}^{X},

Sk_{(H(χ),∈,<^{∗})}(X ∪ z) ∩ ω_{1} = X ∩ ω_{1}.

Note that if η < η′ are ordinals in an elementary submodel X, then

D_{η}^{X} = D_{η′}^{X} ∩ [η]^{<ω}.

In order to verify that our forcings satisfy the semi-properness condition we require, we need to repeatedly apply a certain simultaneous-extendibility property. The following lemma will be useful in this regard.

Lemma 2.1. Say that M is a structure and <^{∗} is a wellordering of the domain of M. Let X be a countable elementary substructure of (M, ∈, <^{∗}). Let η be an ordinal in X and let z_{0}, z_{1} be countable subsets of η such that [z_{0}]^{<ω} ⊂ D_{η}^{X} and [z_{1}]^{<ω} ⊂ D_{η}^{X′}, where X′ = Sk_{(M,∈,<^{∗})}(X ∪ z_{0}). Then [z_{0} ∪ z_{1}]^{<ω} ⊂ D_{η}^{X}.

Proof. Let a ∈ [z_{0}]^{<ω} and b ∈ [z_{1}]^{<ω}, and let f: [η]^{|a|+|b|} → ω_{1}. Then f_{a}: [η]^{|b|} → ω_{1}, defined by letting f_{a}(y) = f(a ∪ y), is a function in

X′ = Sk_{(M,∈,<^{∗})}(X ∪ z_{0}),

so f(a ∪ b) ∈ X′ ∩ ω_{1} = X ∩ ω_{1}.

Say that X, Y are countable elementary submodels of some (H(χ), ∈, ≤_{χ}) with X ∈ Y. Let η be an ordinal in X and let γ ∈ Y ∩ η. Then even though Sk_{(H(χ),∈,≤_{χ})}(X ∪ {γ}) is not directly definable in Y, the set itself is in Y, since it is equivalent to {f(γ) | f ∈ X ∧ f: η → H(χ)}.

2.2 Remark. Similarly, many of the arguments in this paper prove facts about sequences of elementary submodels by induction on the length of the sequence. Of course if X, Y are countable elementary submodels of (H(χ), ∈, ≤_{χ}) with X ∈ Y, Y does not see that X is an elementary submodel, and so the induction hypothesis cannot be applied directly in Y. However, the statement about whether an object exists with a certain relation to X, a semi-generic extending a certain condition, say, is formalizable in Y, and so if one exists in H(χ) then one exists in Y.

3 A class of (ω, ∞)-distributive iterations

Each step of the iterations we are considering is a forcing which shoots a continuous increasing sequence of length ω_{1} through a given stationary set of countable sets of ordinals. Under certain assumptions on the stationary set, such forcings are a typical example of improper forcings which preserve stationary subsets of ω_{1}, and they have been well studied in recent years (see, for example, [3]). For the iterations in this paper, we require that these stationary sets be definable from sets from the ground model with the help of functions from ω_{1} to ω_{1} added by initial segments of the iteration.

3.1 Definition. Let ⟨λ_{ρ} : ρ < κ⟩ be a continuous increasing sequence of ordinals, and let A = ⟨A_{β}^{ρ} : ρ < κ, β < ω_{1}⟩ be such that each A_{β}^{ρ} ⊂ [λ_{ρ+1}]^{<ω_{1}}. Given ρ < κ and f: ω_{1} → ω_{1}, let Q_{ρ,f} be the forcing whose conditions are countable, continuous, increasing sequences ⟨x_{β} : β ≤ γ⟩ such that for each β ≤ γ, x_{β} ∩ ω_{1} ∈ ω_{1} and x_{β} ∈ A_{f(x_{β}∩ω_{1})}^{ρ}, ordered by extension. Then an A-iteration is a structure Q̄ = ⟨P_{ρ}, Q_{∼}_{ρ}, f_{∼}_{ρ} : ρ < κ⟩ such that (a) ⟨P_{ρ}, Q_{∼}_{ρ} : ρ < κ⟩ is a Revised Countable Support iteration,

(b) each f_{∼}_{ρ} is a P_{ρ}-name for a function from ω_{1} to ω_{1},

(c) for all ρ < κ, 1_{P_{ρ}}⊩Q_{∼}_{ρ} = Q_{ρ,f_{∼}_{ρ}}.

The stationary sets in our iterations must also satisfy certain extendibility conditions with respect to the countable elementary submodels of a sufficiently large initial segment of the universe.

Theorem 3.2. Let ⟨λ_{ρ} : ρ < κ⟩ be a continuous increasing sequence of strong limit cardinals with supremum κ. Fix a regular cardinal χ > (2^{κ})^{+}, and let ≤_{χ} be a wellordering of H(χ). Let A = ⟨A_{β}^{ρ} : ρ < κ, β < ω_{1}⟩ be such that each A_{β}^{ρ} ⊂ [λ_{ρ+1}]^{<ω_{1}}, and let Q̄ = ⟨P_{ρ}, Q_{∼}_{ρ}, f_{∼}_{ρ} : ρ < κ⟩ be an A-iteration, such that the following hold.

1. For all ρ < κ, β < ω_{1}, if E ∈ A_{β}^{ρ} and E′ ∈ [λ_{ρ+1}]^{<ω_{1}} with E ⊂ E′ then E′ ∈ A_{β}^{ρ},

2. For all ρ < κ, 1_{P_{ρ}} forces that for all countable X ≺ (H(χ)^{V^{Pρ}} , ∈, ≤_{χ}) with λ_{ρ+1}, Q̄ ∈ X, and for all β < ω_{1} there exists a countable z ⊂ λ_{ρ+1} such that [z]^{<ω} is a subset of

⋂{D_{λ_{ρ+1}}^{Z} : Z ≺ (H(χ)^{V^{Pρ}} , ∈, ≤_{χ}) ∧ λ_{ρ+1}, Q̄ ∈ Z ∈ X}

and, letting Y = Sk_{(H(χ)^{V ^{Pρ}} ,∈,≤_{χ})}(X ∪ z),

(i) X ∩ ω_{1} = Y ∩ ω_{1},

(ii) Y ∩ λ_{ρ+1} ∈ A_{β}^{ρ}.

3. For all countable X ≺ (H(χ), ∈, ≤_{χ}) with Q̄ ∈ X, and for all ρ ∈ X ∩ κ, β < ω_{1} there exists a countable Y ≺ (H(χ), ∈, ≤_{χ}) such that

(i) X ⊂ Y

(ii) X ∩ V_{λ_{ρ}+2} = Y ∩ V_{λ_{ρ}+2},

(iii) Y ∩ λ_{ρ+1} ∈ A_{β}^{ρ}.

Let P be the Revised Countable Support limit of ⟨P_{ρ}, Q_{∼}_{ρ} : ρ < κ⟩. Then P is (ω, ∞)-distributive.

3.3 Remark. By standard RCS arguments, if in the statement of Theorem 3.2 we assume in addition that κ is strongly inaccessible, then P is κ-c.c. (see Theorem 4.6).

Conditions 2 and 3 of Theorem 3.2 could easily be subsumed into one condition; indeed, in our applications we verify both conditions at the same time. The conditions correspond to separate parts of the proof, however. Condition 2 is needed to show that each successor step of the iteration is α-semi-proper for all countable α (Theorem 4.13), and could in fact be replaced by this requirement, though this would mean more work in applying the theorem. Condition 3 is used to construct the systems of models which we use to show that no countable sets of ordinals are added (Lemma 4.20).

Since in the end we show that P does not add ω-sequences, these are actually Countable Support iterations. Nonetheless, at this time we do not have a proof of (the corresponding version of) Theorem 3.2 which avoids RCS.

For the rest of this paper, sets denoted by κ, ⟨λ_{ρ} : ρ < κ⟩, χ, ≤_{χ}, Q_{ρ,f}, Q̄, P, A and ⟨A_{β}^{ρ} : ρ < κ, β < ω_{1}⟩ are supposed to have the properties given in the hypotheses of Theorem 3.2. This policy will be modified in two ways. First, in the application sections, we will add extra properties for these terms. On the other hand, in proving the main theorem we sometimes state our lemmas more generally in terms of certain properties of these objects, temporarily forgetting the others. We hope that it is clear when we are doing this, so that there will be no confusion.

4 Proof of Theorem 3.2

The proof of Theorem 3.2 breaks into two largely disjoint parts, which come together only in Lemma 4.21. In the first part, we define α-semi-proper forcing, where α is a countable ordinal, and show that for each countable ordinal α, α-semi-properness is preserved under RCS iterations. In Theorem 4.13 we use Condition 2 from Theorem 3.2 to show that each successor step from one of these iterations is forced to be α-semi-proper for all countable α, which then carries over to the entire iteration. In the second part of the proof we use the end-extension property given in Condition 3 of Theorem 3.2 to find a system of elementary submodels suitable for proving (ω, ∞)-distributivity, which is shown in Lemmas 4.17, 4.18, 4.20 and 4.21. By Theorem 4.7, the iteration is κ-c.c.

4.1 Revised Countable Support

The reader is referred to [12, 11] for the definition of Revised Countable Support and its basic analysis. Alternate presentations of RCS can be found in [2, 8]. We will show that the arguments here follow in all three versions. The only facts about RCS that we need outside of this section are Theorems 4.1, 4.7 and 4.11, and Property 4.3. These are (essentially - see Remark 4.4) proved in [8, 9] for the version of RCS presented in [8]. We present proofs of these facts that work for the versions of RCS in [12] and [2].

The following two theorems hold for all presentations of RCS. The first gives an important property of RCS which distinguishes it from Countable Support, that names for conditions are essentially conditions themselves.

Theorem 4.1. Say that ⟨P_{α}, Q_{∼}_{α} : α < κ⟩ is an RCS iteration with RCS limit P. Fix p ∈ P and γ < κ. Suppose that A is a maximal antichain in P_{γ} below p↾γ and f : A → P is a function such that for each a ∈ A f(a) ≤ p and f(a)↾γ = a. Then there is a condition p′ ≤ p such that p′ ≤ p, p↾γ = p′↾γ and each a ∈ A forces that p′↾[γ, κ) = f(a)↾[γ, κ).

Theorem 4.2 follows from Theorem 4.1 and Corollary 2.8 from Chapter X of [12] (and the corresponding theorem from [2]).

Theorem 4.2. Suppose that ⟨P_{γ}, Q_{∼}_{γ} : γ < κ⟩ is an RCS iteration with RCS limit P_{κ} such that each Q_{∼}_{γ} is forced to be semi-proper and each Q_{∼}_{γ} forces the corresponding P_{γ} to have cardinality ℵ_{1}. Then the following hold.

1. P_{κ} is semi-proper.

2. For all regular cardinals χ > 2^{|P_{κ}|}, for all countable X ≺ H(χ) with P_{κ} ∈ X, for all γ < δ in X ∩ (κ + 1), if

(a) p ∈ P_{κ},

(b) q is an (X, P_{γ})-semi-generic condition in P_{γ} below p↾γ,

(c) q forces “there exists r in P_{κ} ∩ X such that p↾[γ, κ) = r↾[γ, κ),”

then there exists an (X, P_{δ})-semi-generic condition q′ ∈ P_{δ} such that q′↾γ = q and q′ ≤ p↾δ.

In [2], RCS is characterized by the following property.

4.3 Property. If ⟨P_{α} : α < κ⟩ is an RCS iteration with RCS limit P and p ∈ P, then for all q ≤ p in P there exist γ < κ, r ∈ P_{γ} such that r ≤ q↾γ and either

r⊩cof(κ̌) = ω or r⊩∀α ∈ (γ̌, κ̌) p(α) = 1_{Q_{∼}_{α}}.

If ⟨P_{α} : α < κ⟩ is an RCS iteration with RCS limit P then for each condition p in P there is an associated P -name supp(p) for the support of p, the set of α < κ such that

p(α) ≠ 1_{Q_{∼}_{α}},

as decided by the P -generic filter. One can easily prove by induction on κ that Property 4.3 implies that for each p in P supp(p) is forced to be countable.

4.4 Remark. Property 4.3 is shown in [8] for the version of RCS in [8], for the special case where κ = ω_{1} (so the first possibility for r cannot hold), en route to (essentially) proving Theorem 4.6. Together these two facts give Property 4.3 for this version of RCS, since in the remaining case (where |P_{α}| < cof(κ) for all α < κ) a maximal antichain deciding the supremum of supp(p) for some condition p must have maximal restriction in some initial segement of the iteration.

In [12], we have the following property (see Definition 1.1 and Claim 1.3 (1) of [12], Chapter X).

4.5 Property. If ⟨P_{α} : α < κ⟩ is an RCS iteration of limit length with RCS limit P and p ∈ P, then there exist P -names τ_{i} (i < ω) for ordinals less than κ such that

1_{P} ⊩supp(p̌) = {τ_{i} : i < ω}

and such that for each i < ω and each q ≤ p in P there exist γ < κ, r ∈ P_{γ+1} such that r ≤ q↾(γ + 1) and r⊩τ_{i} = γ̌.

If we assume that in addition each Q_{∼}_{α} forces that the cardinality of the corresponding P_{α} is ℵ_{1}, then Property 4.5 implies Property 4.3. We need only check the limit case. Fix p and q, let {τ_{i} : i < ω} be the sequence given by Property 4.5 with respect to p, and assume that there is no pair γ, r such that

• γ < κ,

• r ∈ P_{γ},

• r ≤ q↾γ,

• r⊩cof(κ̌) = ω.

If some P_{α} has cardinality greater than the cofinality of κ, then (assuming that q is in the generic filter), κ will have cofinality ω_{1} in the extension by P_{α+1}. Then, working in the P_{α+1}-extension, let X be a countable elementary substructure of a large enough H(χ) with P, p, q, {τ_{i} : i < ω} ∈ X. Define ζ = ⋃(X ∩ κ) and let q′ ≤ q↾[α + 1, κ) be (X, P/P_{α+1})-semi-generic. Then q′ forces that supp(p) will be contained in ζ. This means that for each i < ω every condition q′′ ≤ q′ in P/P_{α+1} will be compatible with an r ∈ P_{ζ}/P_{α+1} forcing that τ_{i} < ζ. But then q′↾ζ will also have this property, and thus q′↾ζ also forces that supp(p) ⊂ ζ. Therefore, there exists a condition r ∈ P_{γ} for some γ < κ such that r↾(α + 1) forces that γ will have the properties of ζ as above, and that r↾[α + 1, γ) will satisfy the properties of q′↾ζ. Such an r suffices.

On the other hand, if cof(k) > |P_{α}| for all α, then there is a sequence of pairs (γ_{i}, r_{i}) (i < ω) such that

• the γ_{i}’s are increasing,

• each r_{i} ∈ P_{γ_{i}} forces a bound below κ on τ_{i},

• for all i < j, r_{j}↾γ_{i} = r_{i},

• each r_{i} ≤ p↾γ_{i}.

Then the limit of the r_{i}’s is the desired condition. To find γ_{i+1}, r_{i+1}, find a maximimal antichain A in P_{γ_{i}} below r_{i} such that for each a ∈ A there is a γ_{a} < κ and an r_{a} ∈ P_{γ_{a}} such that

• r_{a}↾γ_{i} = a,

• r_{a} ≤ p,

• r_{a}⊩τ_{i} = γ̌_{a}.

Then apply Theorem 4.1 to A and the function a ↦ r_{a} to find r_{i+1}, and let γ_{i+1} = sup{γ_{a} : a ∈ A}, which must be below κ since cof(κ) > |P_{γ_{i}}|.

Similar considerations give the following (see also Lemma 36.5 of [5]). The point is that the RCS limit of an iteration of cofinality ω_{1} is just the direct limit, and the ordinals of cofinality ω_{1} are stationary below cof(κ) as below. Then for any maximal antichain A in P as below there is some γ < κ of cofinality ω_{1} such that A ∩ P_{γ} is a maximal antichain in P_{γ}. But then |A| ≤ |P_{γ}|.

Theorem 4.6. Say that ⟨P_{γ}, Q_{∼}_{γ} : γ < κ⟩ is an RCS iteration with RCS limit P such that cof(κ) > |P_{γ}| for all γ < κ, and such that each Q_{∼}_{γ} forces the corresponding P_{γ} to have cardinality ℵ_{1}. Then P is cof(κ)-c.c.

Applying Theorems 4.2 and 4.6, we have reduced the proof of Theorem 3.2 to showing that each P_{β} (β < κ) is (ω, ∞)-distributive. Theorem 4.7. Let ⟨P_{β}, Q_{∼}_{β} : β < κ⟩ be an RCS iteration of strongly inaccessible length with RCS limit P_{κ} such that

• each 1_{P_{β}} forces the corresponding Q_{∼}_{β} to be semi-proper,

• each Q_{∼}_{β} makes the corresponding P_{β} have cardinality ℵ_{1},

• each |P_{β}| < κ,

• each P_{β} is (ω, ∞)-distributive. Then for all ρ < β ≤ κ, P_{β}/P_{ρ} is semi-proper and κ-c.c. Therefore P_{κ} is (ω, ∞)-distributive.

4.2 Semi-generics for sequences

The following are generalizations of ideas from Chapters V, X and XII of [12]. 4.8 Definition. Let α be a countable ordinal.

1. The set SEQ_{α}(χ) consists of all N̄ = ⟨N_{β} : β < α⟩ such that (a) each N_{β} is a countable elementary substructure of (H(χ), ∈, ≤_{χ}),

(b) for each γ < α, N̄↾γ = ⟨N_{β} : β < γ⟩ ∈ N_{γ}.

2. Let N̄ ∈ SEQ_{α}(χ) and let P be a forcing construction in N_{0}. A condition

p ∈ P is (N̄, P)-semi-generic if

p⊩Ň_{δ}[G_{∼}_{P}] ∩ ω_{1} = Ň_{δ} ∩ ω_{1}

for all δ < α, where G_{∼}_{P} is a P -name for the generic filter. The condition p is (N̄↾[γ, β], P)-semi-generic if this holds for all δ ∈ [γ, β].

3. A forcing construction P is α-semi-proper if for every N̄ ∈ SEQ_{α}(χ) with P ∈ N_{0} and for all p ∈ P ∩ N_{0} there is an (N̄, P)-semi generic q ∈ P such that p ≥ q.

Given N̄ ∈ SEQ_{α}(χ) and P ∈ N_{0}, if G ⊂ P is a generic filter then N̄[G] is the sequence ⟨N_{β}[G] : β < α⟩. Both parts of the following lemma are immediate. Note that in the second part there is no need to find a p ∈ G_{∼}_{P} which is (N̄, P)- semi-generic. Lemma 4.9. Let α be a countable ordinal and fix N̄ ∈ SEQ_{α}(χ).

1. Let P be an α-semi-proper forcing in N_{0}, and let Q_{∼} be a P -name in N_{0} for an α-semi-proper forcing. Let q ∈ P be (N̄, P)-semi-generic and let p_{∼} be a P -name for a condition in (⋃ N̄[G_{∼}_{P}])∩Q_{∼}. Then there is a P -name r_{∼} for a condition in Q_{∼} such that 1_{P} ⊩r_{∼} ≤ p_{∼} and (q, r_{∼}) is (N̄, P ∗ Q)-semi-generic.

2. 1⊩_{P} ˇ N̄[G_{∼}_{P}] ∈ SEQα̌(χ̌)^{V [G_{∼}_{P}]}.

Lemma 4.10. Let α be a countable ordinal. Say that P_{κ} is the RCS limit of an RCS iteration ⟨P_{ρ}, Q_{∼}_{ρ} : ρ < κ⟩ in a in a set forcing extension V^{∗} such that each 1_{P_{ρ}} forces the corresponding Q_{∼}_{ρ} to be semi-proper, and each Q_{∼}_{ρ} forces P_{ρ} to have cardinality ℵ_{1}. Then the following hold in V^{∗}.

1. P_{κ} is α-semi-proper.

2. Fix β ≤ γ ≤ κ, and fix N̄ ∈ SEQ_{α}(χ) with P_{κ} and β, γ ∈ ⋃ N̄. Let p ∈ P_{κ}, q ∈ P_{β} and δ < α be such that

• β ∈ N_{δ},

• p is (N̄↾δ, P_{κ})-semi-generic,

• q is (N̄↾[δ, α), P_{β})-semi-generic,

• p↾β ≥ q,

• q forces that for some r ∈ P_{κ} ∩ N_{δ}, p↾[β, κ) = r↾[β, κ).

Let δ^{∗} ∈ [δ, α) be such that γ ∈ N_{δ^{∗}}. Then there exist q^{∗} ∈ P_{γ} and p^{∗} ∈ P_{κ} such that

• p^{∗} is (N̄↾δ^{∗}, P_{κ})-semi-generic,

• q^{∗} is (N̄↾[δ^{∗}, α), P_{γ})-semi-generic,

• q^{∗} ≤ p^{∗}↾γ,

• p^{∗} ≤ p,

• q^{∗}↾β = q,

• q^{∗} forces that for some r ∈ P_{κ} ∩ N_{δ^{∗}}, p^{∗}↾[γ, κ) = r↾[γ, κ).

Proof. We prove part 2 by induction on α, working in V^{∗}. The first part follows immediately. We fix the notation that G_{ρ} is the generic filter for P_{ρ}, for ρ < κ. The case α = 1 is given by Theorem 4.2. For the case where α = α′ + 1, there are two subcases. If δ^{∗} < α′, then we may assume by the induction hypothesis that δ^{∗} = δ and that there is a (N̄↾[δ^{∗}, α′), P_{γ})-semi-generic q′ ∈ P_{γ} such that q′↾β = q and q′ ≤ p↾γ. Then since q forces that N_{α′}[G_{β}] will be elementary in H(χ)^{V^{∗}[G_{β}]}, where G_{β} is the generic filter for P_{β}, we can replace q′ with a q′′ with the additional property that q forces that q′′↾[β, γ) will be equal to r↾[β, γ) for some r ∈ N_{α′} ∩ P_{κ}. By part 2 of Theorem 4.2, then, there is a q^{∗} ∈ P_{γ} such that q^{∗}↾β = q, q^{∗} ≤ q′′ and q^{∗} is (N_{α′}, P_{γ})-semi-generic. Such a q^{∗} suffices.

For the subcase δ^{∗} = α′, by the induction hypothesis there is a (N̄↾α′, P_{κ})- semi-generic q′ ∈ P_{γ} such that q′↾β = q and q′ ≤ p. Then since q forces that N_{α′}[G_{β}] will be elementary in H(χ)^{V^{∗}[G_{β}]}, where G_{β} is the generic filter for P_{β}, we can replace q′ with a q′′ with the additional property that q forces that q′′↾[β, κ) will be equal to r↾[β, κ) for some r ∈ N_{α′} ∩ P_{κ}, and we can let p^{∗} = q′′. Then as in the previous paragraph, by part 2 of Theorem 4.2 there is a q^{∗} as desired.

The case where α is a limit and κ = κ′+1 is similar, now fixing α and inducting on κ. By the induction hypothesis for α we may assume that δ = δ^{∗} and by the induction hypothesis for κ we may assume that there is a (N̄↾[δ^{∗}, α), P_{κ′})- semi-generic q′ ∈ P_{κ′} such that q′↾β = q and q′ ≤ p↾κ′. Since Q_{∼}_{κ′} is forced to be α-semi-proper, by Theorem 4.1 (or part 1 of Lemma 4.9) there is a (N̄, P_{κ})- semi-generic q^{∗} below p such that q^{∗}↾κ = q′.

For the case where α and κ are both limits there are two subcases (in each of which we may assume that δ = δ^{∗}). If κ has cofinality ω or ω_{1}, or if |P_{ρ}| < cof(κ) for each ρ < κ, then we fix an increasing sequence ⟨ξ_{i} : i < ω⟩ cofinal in α, with ξ_{0} = δ. If the cofinality of κ is countable, fix an increasing sequence of ordinals ⟨β_{i} : i < ω⟩ ∈ N_{0} cofinal in κ, with β_{0} = β. Otherwise, let β_{i} be any ordinal in N_{ξ_{i}} greater than sup(⋃ N̄↾ξ_{i} ∩ κ), again with β_{0} = β. Let p_{0} = p and let q_{0} = q. Alternately choose conditions p_{i+1}, q_{i+1} (i < ω), such that

1. each p_{i+1} is a condition in P_{κ},

2. each q_{i+1} is a condition in P_{β_{i+1}},

3. each q_{i} ≤ p_{i}↾β_{i},

4. each p_{i+1} ≤ p_{i},

5. for all i < j < ω, q_{j}↾β_{i} = q_{i},

6. each p_{i+1} is (N̄↾ξ_{i+1}, P_{κ})-semi-generic,

7. each q_{i+1} is (N̄↾[ξ_{i+1}, α), P_{β_{i+1}})-semi-generic,

8. each q_{i} forces that for some condition r ∈ P_{κ} ∩ N_{ξ_{i}}, r↾[β_{i}, κ) = p_{i}↾[β_{i}, κ).

For the case where no condition in any P_{η} makes cof(κ) ≤ ω_{1}, we modify condition 8 as follows:

8a. each q_{i} forces that for some condition r ∈ P_{κ} ∩ N_{ξ_{i}} such that for some γ < κ 1_{P_{κ}}⊩supp(ř) ⊂ γ̌, r↾[β_{i}, κ) = p_{i}[β_{i}, κ).

That such conditions exist is immediate by the induction hypothesis (8a follows from Property 4.3). Then the limit of the q_{i}’s (call it q^{∗}) will be the desired (N̄, P_{κ})-semi-generic, as long as it is below each p_{i}. This fact follows from the fact that q^{∗} forces that {β_{i} : i < ω} will be cofinal in ⋃{supp(p_{i}) : i < ω}. This is clear if cof(κ) = ω, and if cof(κ) = ω_{1} it follows from the fact that each N_{ξ_{i}} ∩ κ will be cofinal in N_{ξ_{i}}[G_{β_{i}}] ∩ κ since q_{i} is (N_{ξ_{i}}, P_{β_{i}})-semi-generic. For the remaining case it follows from condition 8a.

If |P_{ρ}| ≥ cof(κ) for some ρ < κ, then we may assume that β > ρ and so cof(κ) ≤ ω_{1} in the P_{β}-extension. Then we may apply the previous argument in the P_{β}-extension, along with Theorem 4.1.

Theorem 4.11. Let α be a countable ordinal. Say that P_{κ} is the RCS limit of an RCS iteration ⟨P_{β}, Q_{∼}_{β} : β < κ⟩ such that each P_{β+1} = P_{β} ∗ Q_{∼}_{β} forces |P_{β}| ≤ ℵ_{1} and each 1_{P_{β}} forces that Q_{∼}_{β} is α-semi-proper. Then for all β < γ ≤ κ, P_{γ}/P_{β} is α-semi-proper for every α < ω_{1}.

To show that each Q_{ρ,f} is α-semi-proper for all α < ω_{1}, we show that we can extend sequences of models in a suitable way.

Lemma 4.12. Assume that λ < χ, Q̄, and ⟨A_{β} : β < ω_{1}⟩ ⊂ [λ]^{<ω_{1}} are such that

• for all β < ω_{1}, E ∈ A_{β}, if E ⊂ E′ then E′ ∈ A_{β},

• for all countable X ≺ (H(χ), ∈, ≤_{χ}) with λ, Q̄ ∈ X, and for all β < ω_{1} there exists a countable z ⊂ λ such that [z]^{<ω} is a subset of

⋂{D_{λ}^{Z} : Z ≺ (H(χ), ∈, ≤_{χ}) ∧ Q̄, λ ∈ Z ∈ X}

and, letting Y = Sk_{(H(χ),∈,≤_{χ})}(X ∪ z),

(i) X ∩ ω_{1} = Y ∩ ω_{1},

(ii) Y ∩ λ ∈ A_{β}.

Let N̄ ∈ SEQ_{ξ}(χ) (for some ξ < ω_{1}) with Q̄, λ ∈ N_{0}, and let g: ω_{1} → ω_{1} be a function such that for each η < ξ g↾(∪{N_{η′} ∩ ω_{1} : η′ < η} + 1) ∈ N_{η}. Then there exists N̄′ ∈ SEQ_{ξ}(χ) such that

(a) each N_{η} ⊂ N′_{η},

(b) each N′_{η} ∩ ω_{1} = N_{η} ∩ ω_{1},

(c) each N′_{η} ∩ λ ∈ A_{g(N′_{η}∩ω_{1})},

(d) letting N′ = ⋃{N′_{η} : η < ξ}, N′ ∩ λ ∈ A_{g(N′∩ω_{1})}.

Proof. By induction on ξ. If N̄ has a last model, then we are done by the extension assumption on countable elementary submodels of H(χ). For the limit case, applying the induction hypothesis we may assume that every condition already holds, except part (d). Let

N = ⋃{N_{η} : η < ξ}.

Applying the end-extension assumption, let z be a countable subset of λ such that

[z]^{<ω} ⊂ ⋂{D_{λ}^{Z} : Z ≺ (H(χ), ∈, ≤_{χ}) ∧ λ, Q̄ ∈ Z ∈ N}

and such that, letting N′ = Sk_{(H(χ),∈,≤_{χ})}(N ∪ z), we have

1. N ⊂ N′, 2. N ∩ ω_{1} = N′ ∩ ω_{1},

3. N′ ∩ λ ∈ A_{g(N′∩ω_{1})}.

Let h: ω → z be a bijection, and fix an increasing sequence ⟨ζ_{i} : i < ω⟩ cofinal in ξ with ζ_{0} = 0. Now let each

N′_{η} = Sk_{(H(χ),∈,≤_{χ})}(N_{η} ∪ h[i_{η}]),

where i_{η} is the largest i such that ζ_{i} ≤ η. Since these are all finite extensions, each initial sequence of the new sequence is an element of the later models. By the choice of z, each N_{η} ∩ ω_{1} = N′_{η} ∩ ω_{1}, and so N̄′ is as desired.

Theorem 4.13. If P_{ρ} is as in Theorem 3.2 and f: ω_{1} → ω_{1} is a function added by P_{ρ}, then Q_{ρ,f} is α-semi-proper in the extension by P_{ρ} for all α < ω_{1}.

Proof. By induction on α, using the Lemma 4.12 and working in the extension by P_{ρ}. Fix N̄ ∈ SEQ_{α}(χ) and p ∈ Q_{ρ,f} with p, Q̄, λ ∈ N_{0}, and let N̄′ be as in Lemma 4.12, with respect to ⟨A_{β}^{ρ} : β < ω_{1}⟩. If N̄′ has a final model (i.e., if α is a successor), then we can choose a (N̄′↾(α − 1), Q_{ρ,f})-semi-generic p′ ∈ N′_{α−1} extending p by the induction hypothesis (see Remark 2.2). Then since

N′_{α−1} ∩ λ_{ρ+1} ∈ A_{f(N′_{α−1}∩ω_{1})}^{ρ},

any N′_{α−1}-generic for Q_{ρ,f} extending p′ will suffice as the desired condition. For the case where α is a limit, we let N̄ and N̄′ be as in the proof of Lemma 4.12, and fix an increasing sequence ξ_{i} (i < ω) of ordinals cofinal in α. Then letting p be p_{0} we can successively pick conditions p_{i} (i < ω) such that each p_{i+1} is a (N̄′↾ξ_{i+1}, Q_{ρ,f})-semi-generic in N′_{ξ_{i+1}} extending p_{i}, and such that the last member of each p_{i+1} contains N′_{ξ_{i}} ∩ λ_{ρ+1}. By Condition (d) from Lemma 4.12, the limit of the p_{i}’s (adjoined by their union) will be a condition in Q_{ρ,f}, and it will be (N̄, Q_{ρ,f})-semi-generic since it extends an (N̄↾β, Q_{ρ,f})-semi-generic for each β < α.

4.3 Systems of models and conditions

To show that each initial segment P_{β} (β < κ) is (ω, ∞)-distributive, we choose a suitable pair (M, N̄) where M is a countable elementary submodel of H(χ) and N̄ ∈ SEQ_{o.t.(M∩β)}(χ), and find an (N̄, P_{β})-semi-generic condition in P_{β} which extends an M-generic filter. As defined below, a (Q̄, ρ, β)-system is a partial construction of such an object (minus the M-genericity requirement), where ρ is the length of the initial segment for the iteration for which the desired condition has been constructed, and β is the target length.

4.14 Definition. Given ordinals ρ ≤ β ≤ κ, we say that (M, N̄, p, q) is a (Q̄, ρ, β)-system if (letting δ = o.t.(M ∩ ρ) and γ = o.t.(M ∩ β)):

1. M ≺ (H(χ), ∈, ≤_{χ}), 2. Q̄, ρ, β belong to M,

3. N̄ ∈ SEQ_{γ}(χ) and Q̄ ∈ N_{0},

4. q ∈ P_{ρ} and q is (N̄↾[δ, γ), P_{ρ})-semi-generic,

5. for all ρ′ ∈ ρ ∩ M, q↾ρ′ is (N_{o.t.(M∩ρ′)}, P_{ρ′})-semi-generic,

6. for all η ∈ M ∩ β,

(a) for all ν ∈ N_{o.t(M∩η)} ∩ ω_{1}, M ∩ λ_{η+1} ∈ A_{ν}^{η},

(b) M ∩ V_{λ_{η}+2} ∈ N_{o.t.(M∩η)},

7. p ∈ P_{β}, p↾ρ ≥ q and for all η ∈ [ρ, β) ∩ M, p↾η ∈ N_{o.t.(M∩η)}.

4.15 Remark. Note that if β = ρ + 1 and (M, N̄, p, q) is a (Q̄, ρ, β)-system, then (M, N̄, p, q) is also a (Q̄, β, β)-system, as (the second part of) Condition 4 becomes vacuous in the second case, and Condition 4 in the first case completes Condition 5 in the second.

4.16 Definition. Let M ≺ (H(χ), ∈, ≤_{χ}) and β < κ with P_{β} ∈ M. A filter g ⊂ P_{β} ∩ M is M-generic if for all dense sets D ⊂ P_{β} in M the intersection of g and D is nonempty. A condition p ∈ M is a potential-M-generic for P_{β} if there exists a g ⊂ P_{β} ∩ M M-generic for P_{β} such that for all ρ ∈ M ∩ β, 1_{P_{ρ}} forces that if

M ∩ λ_{ρ+1} ∈ A_{f_{∼}_{ρ}(M∩ω_{1})}^{ρ}

then p(ρ) = g(ρ)^{⌢}⟨∪g(ρ)⟩, and if for all ρ ∉ M ∩ β, p(ρ) = 1_{Q_{∼}_{ρ}}.

Note that if g ⊂ P_{β} is M-generic and p is in g then since supp(p) is a P_{β}-name for a countable subset of β the support of p as determined by g is contained in M ∩ β. We need to see that potential-M-generics exist in suitable generality.

Lemma 4.17. For any β ≤ κ, if (M, N̄, p, ∅) is a (Q̄, 0, β)-system with p ∈ M, then there is a potential-M-generic p̄ ≤ p such that p̄↾η ∈ N_{o.t.(η∩M)} for all η ∈ M ∩ β.

Proof. Given g ⊂ P ∩M M-generic for P_{β}, one can easily build a corresponding potential-M-generic p, letting p(ρ) be the empty condition when

M ∩ λ_{ρ+1} ∉ A_{f_{∼}_{ρ}(M∩ω_{1})}^{ρ}.

The point then is just to find an M-generic filter g ⊂ M ∩ P_{β} with p ∈ g, such that g ∩ P_{η} ∈ N_{o.t.(η∩M)} for all η ∈ M ∩ β.

Inducting primarily on β ∈ M ∩ κ, and secondarily on γ ∈ M ∩ β, we show that if p^{∗} ∈ P_{β} ∩ M and g^{∗} ∈ N_{o.t.(M∩γ)} is an M-generic filter for P_{γ} such that p^{∗}↾γ ∈ g^{∗} and g^{∗} ∩ P_{γ′} ∈ N_{o.t.(M∩γ′)} for all γ′ ∈ M ∩ γ, then there exists an M-generic g ⊂ P_{β} such that g ∩ P_{γ} = g^{∗}, p^{∗} ∈ g and g ∩ P_{η} ∈ N_{o.t.(M∩η)} for all η ∈ M ∩ β. Note that for each γ ∈ M ∩ β, since

P(P(λ_{γ})) ∩ M ∈ N_{o.t.(M∩γ)}

(and |P_{γ}| ≤ 2^{λ_{γ}}), the M-genericity of g^{∗} can be verified in N_{o.t.(M∩γ)} (M itself is not in any member of N̄, so this is not automatic).

The base case β = 0 is trivial. For the successor step, we may assume that γ + 1 = β Then since for the successor case there is no restriction on the last coordinate of the M-generic filter (other than M-genericity), g^{∗} can be extended in any fashion to an M-generic for P_{β}.

For the limit, fix an increasing sequence η_{i} (i < ω) cofinal in β ∩ M, and let p = p_{0}. Now in ω many steps alternately pick

• M-generic g_{i} ⊂ P_{η_{i}} in N_{o.t.(M∩η_{i})} (applying the induction hypothesis plus the elementarity of N_{o.t.(M∩η_{i})} plus the fact that N̄↾o.t.(M ∩ η_{i}) ∈ N_{o.t.(M∩η_{i})}, as in Remark 2.2) with p_{i}↾η_{i} ∈ g_{i} such that

– g_{i} ∩ P_{η} ∈ N_{o.t.(M∩η)} for all η ∈ M ∩ η_{i},

– g_{i} ∩ P_{η_{j}} = g_{j} for all j < i,

• p_{i+1} ≤ p_{i} in M meeting the ith dense set in M for P_{β} such that p_{i+1}↾η_{i} ∈ g_{i} (such a p_{i+1} exists because g_{i} is M-generic for P_{η_{i}}).

Since each p_{i+1} ∈ M, its initial segments are automatically in the corresponding N_{ξ}’s. Then {p_{i} : i < ω} generates an M-generic filter g for P_{β}, and for each η_{i}, g ∩ P_{η_{i}} = g_{i}. Then by the induction hypothesis, g ∩ P_{η} ∈ N_{o.t.(M∩η)} for all η ∈ M ∩ β, since for all i < ω with η_{i} ≥ η, g ∩ P_{η} = g_{i} ∩ P_{η}.

Lemma 4.18. Fix β ≤ κ. If for all p_{0} ∈ P_{β} there is a (Q̄, β, β)-system (M, N̄, p, q) with p_{0} ∈ M and p ≤ p_{0} a potential-M-generic, then the forcing P_{β} is (ω, ∞)-distributive.

Proof. Towards a contradiction, fix the least β for which the lemma fails, and let p_{0} be a condition in P_{β} forcing that P_{β} adds a new ω-sequence of ordinals. Let (M, N̄, p, q) be a (Q̄, β, β)-system with p_{0} ∈ M and p ≤ p_{0} a potential-M-generic, as given by the hypothesis of the lemma. Then there exists a P_{β}-name τ in M such that p_{0} forces that τ will be a new ω-sequence of ordinals. Let g ⊂ P_{β} ∩ M be an M-generic filter witnessing that p is a potential-M-generic. We wish to see that q is below each member of g. Then we will be done, as for each integer i there is a member of g intersecting the antichain in P_{β} determining the ith member of τ. So we will show by induction on ρ ∈ (β + 1) ∩ M that q↾ρ ≤ p′↾ρ for all p′ ∈ g (simultaneously). The cases where ρ = 0 or ρ is a limit and M ∩ ρ is cofinal in ρ are clear.

For the successor step from ρ to ρ+1, f_{∼}_{ρ}(M ∩ω_{1}) is a P_{ρ}-name in N_{o.t.(M∩ρ)} for a countable ordinal, so by Conditions 5 and 6 of Definition 4.14, q↾ρ forces that

M ∩ λ_{ρ+1} ∈ A_{f_{∼}_{ρ}(M∩ω_{1})}^{ρ}.

Then q↾ρ forces that p(ρ) = g(ρ)^{⌢}(∪g(ρ)). By Condition 7 of Definition 4.14, q↾(ρ + 1) ≤ p↾(ρ + 1), so q↾ρ forces that q(ρ) extends p(ρ). Now fix p′ ∈ g. Since q↾ρ forces that p(ρ) extends p′(ρ), we have that q↾(ρ + 1) ≤ p′↾(ρ + 1).

Lastly, for the case where ρ is a limit and M ∩ ρ is not cofinal in ρ, note that ρ has uncountable cofinality, and also that we have shown that each P_{ρ′}, ρ′ < ρ is (ω, ∞)-distributive. By Property 4.3 then, densely many conditions in P_{ρ} (and therefore g ∩ P_{ρ}) are conditions in some P_{ρ′}, ρ′ < ρ, and so densely many conditions in g ∩ P_{ρ} are in some P_{ρ′} with ρ′ ∈ ρ ∩ M.

Given a countable X ≺ (H(χ), ∈, ≤_{χ}) with λ ∈ X, and given η < λ, we say that Y ≺ (H(χ), ∈, ≤_{χ}) is a minimal (η, λ)-extension of X if the following hold.

1. X ∩ η = Y ∩ η.

2. Y = Sk_{(H(χ),∈,≤_{χ})}(X ∪ A), for some A ⊂ λ.

The fact about these extensions that we will use is given in the following lemma.

Lemma 4.19. Let X ≺ (H(χ), ∈, ≤_{χ}), and let λ < γ be ordinals in X with γ a regular cardinal. Let Y be a minimal (η, λ)-extension of X for some η < λ. Then X ∩ γ is cofinal in Y ∩ γ.

Proof. Since γ is regular, each f: [λ]^{<ω} → γ has bounded range below γ. If f is in X then this bound exists in X.

Lemma 4.20. For any set x ∈ H(χ) and any ordinal β ≤ κ there is a (Q̄, 0, β)- system (M, N̄, 1_{P_{β}}, ∅) with x ∈ M.

Proof. Let M_{0} ≺ (H(χ), ∈, ≤_{χ}) be countable with {Q̄, β, x} ∈ M. For some ζ ≤ ω_{1} we build ⟨M_{ξ} : ξ ≤ ζ⟩, ⟨γ_{ξ} : ξ ≤ ζ⟩ and ⟨N_{ξ} : ξ < ζ⟩ satisfying the following conditions.

1. Each M_{ξ} is a countable elementary submodel of (H(χ), ∈, ≤_{χ}).

2. Each N_{ξ} is a countable elementary submodel of (H(χ), ∈, ≤_{χ}).

3. Each γ_{ξ} is the ξth ordinal in M_{ξ} (0 being the 0th ordinal).

4. Q̄ ∈ N_{0}.

5. For all ξ < ζ, ⟨N_{η} : η < ξ⟩ ∈ N_{ξ}.

6. For all ξ < ζ, M ∩ V_{λ_{ξ}+2} ∈ N_{ξ}.

7. For all ξ < ζ and for all ν ∈ N_{ξ} ∩ ω_{1}, M_{ξ+1} ∩ λ_{γ_{ξ}+1} ∈ A_{ν}^{γ_{ξ}}.

8. For each ξ < ζ, M_{ξ+1} is a minimal (2^{2^{λγ_{ξ}}} , λ_{γ_{ξ}+1})-extension of M_{ξ},

9. If ξ is a limit ordinal, then M_{ξ} = ⋃{M_{η} : η < ξ}.

10. If there exists a ξ < ω_{1} such that γ_{ξ} = β, then ζ is the least such ξ; otherwise ζ = ω_{1}.

Given M_{ξ}, by applying Condition 3 in the statement of Theorem 3.2 repeatedly, once for each A_{ν}^{γ_{ξ}} with ν = N_{ξ} ∩ ω_{1}, we can choose each M_{ξ+1} to meet Conditions 7 and 8. While the model Y resulting from this repeated application may not be a minimal (2^{2^{λγ_{ξ}}} , λ_{γ_{ξ}+1})-extension of M_{ξ},

Y^{∗} = Sk_{(H(χ),∈,≤_{χ})}(M_{ξ} ∪ (Y ∩ λ_{γ_{ξ}+1}))

will be, as Y ∩ V_{λ_{γ_{ξ}+1}} = Y^{∗} ∩ V_{λ_{γ_{ξ}+1}}, so we can take this Y^{∗} as our M_{ξ+1}. The rest of the construction is straightforward.

We claim that for each ξ ≤ ζ the sequence ⟨γ_{η} : η ≤ ξ⟩ lists the first ξ + 1 ordinals of M_{ξ} in increasing order. This follows by induction on ξ. It is clear when ξ = 0, and when ξ = ξ′ + 1 it follows from Condition 8 and the fact that

2^{2^{λγ_{ξ′}}} > γ_{ξ′}

(so in this case γ_{ξ} = γ_{ξ′} + 1). If ξ is a limit ordinal, we have from the fact that M_{ξ} = ⋃{M_{η} : η < ξ} and the induction hypothesis that ⟨γ_{η} : η < ξ⟩ lists the first ξ ordinals of M_{ξ} in increasing order. Then the definition of γ_{ξ} finishes the proof of the claim.

Next we claim that γ_{ξ} = β for some ξ < ω_{1}, and so ζ < ω_{1}. Given this we are done, letting M = M_{ζ} and N̄ = ⟨N_{ξ} : ξ < ζ⟩. All the conditions of Definition 4.14 are satisfied trivially, aside from Condition 6. Condition 6 is satisfied since for each η ∈ M_{ζ} ∩ β, there is some ξ < ζ such that η = γ_{ξ} (and so ξ = o.t.(M ∩ η)). Then N_{ξ} and M_{ξ} were chosen to satisfy Condition 6 of Definition 4.14 by Conditions 6 and 7 of the construction, and this relationship was preserved for all later M_{ξ} by Condition 8.

Assume to the contrary that ζ = ω_{1}. If γ_{ω_{1}} = β, then since every ordinal in M_{ω_{1}} ∩ β is equal to some γ_{ξ}, there is a limit ordinal ξ < ω_{1} such that

{γ_{η} : η < ξ} = M_{ξ} ∩ β.

But then γ_{ξ} = β, contradicting ζ = ω_{1}. So we may assume that γ_{ω_{1}} < β. We will show that the cofinality of M_{ω_{1}} ∩ γ_{ω_{1}} is countable, which is a contradiction since ⟨γ_{ξ} : ξ < ω_{1}⟩ is increasing and cofinal in it. If γ_{ω_{1}} ≤ λ_{γ_{ξ}} for some γ_{ξ} with γ_{ω_{1}} ∈ M_{ξ}, then M ∩ γ_{ω_{1}} = M_{ξ} ∩ γ_{ω_{1}}, which is countable. If not, γ_{ω_{1}} = λ_{γ_{ω_{1}}} . If γ_{ω_{1}} is singular, let ρ be the cofinality of γ_{ω_{1}}. Then ρ < λ_{γ_{ξ}} for some γ_{ξ} with {ρ, γ_{ω_{1}}} ∈ M_{ξ}, and there exists a cofinal map f: ρ → γ_{ω_{1}} in M_{ξ}. Since ρ ∩ M_{ω_{1}} = ρ ∩ M_{ξ}, f[M_{ξ} ∩ ρ] is a countable set cofinal in M_{ω_{1}} ∩ γ_{ω_{1}}. The last remaining case is that γ_{ω_{1}} is a regular limit cardinal. Let ξ be least with γ_{ω_{1}} ∈ M_{ξ}, and fix a cofinal sequence in M_{ξ} ∩γ_{ω_{1}}. By Lemma 4.19 and Condition 8 of the construction, this sequence is cofinal in M_{ω_{1}} ∩ γ_{ω_{1}}.

We finish by applying the following lemma to the case ρ = 0, γ = β, from Lemma 4.20. By Lemma 4.18, then, we are done.

Lemma 4.21. Fix β ≤ κ. Let (M, N̄, p, q) be a (Q̄, ρ, β)-system with ρ ≤ γ ≤ β in M, and assume that p is a potential-M-generic. Then there exists a condition q′ ∈ P_{γ} such that 1. q′↾ρ = q,

2. (M, N̄, p, q′) is a (Q̄, γ, β)-system.

Proof. We first note that the lemma follows from the restricted version of the lemma where γ = β. To see this, fix M, N̄, p, q, ρ, γ, β as given by the hypothesis of the unrestricted version, and note that the restricted version then gives a q^{∗} ∈ P_{γ} such that q^{∗}↾ρ = q and (M, N̄↾o.t.(M ∩γ), p↾γ, q^{∗}) is a (Q̄, γ, γ)-system. Now, if G_{ρ} ⊂ P_{ρ} is V -generic with q ∈ G_{ρ}, then M[G_{ρ}] ∩ Ord = M ∩ Ord (since p is a potential-M-generic and q ≤ p) and the following hold.

• ∀α < o.t.(β ∩ M) N_{α}[G_{ρ}] ∩ ω_{1} = N_{α} ∩ ω_{1},

• N̄[G_{ρ}] = ⟨N_{α}[G_{ρ}] : α < o.t.(M ∩ β)⟩ ∈ SEQ_{o.t.(M∩β)}(χ)^{V [G_{ρ}]}.

Furthermore, in V [G_{ρ}],

• q^{∗}↾[ρ, γ) ∈ P_{γ}/P_{ρ}

• ∀γ′ ∈ (γ \ ρ) ∩ M, q^{∗}↾[ρ, γ′) is (N_{o.t.(M∩γ′)}[G_{ρ}], P_{γ′}/P_{ρ})-semi-generic,

• p↾[ρ, γ) ≥ q^{∗}↾[ρ, γ).

Applying the elementarity of N_{o.t.(M∩γ)}[G_{ρ}] in H(χ)^{V [G_{ρ}]}, we see that there exists a condition q̄ ∈ N_{o.t.(M∩γ)}[G_{ρ}] ∩ P_{γ}/P_{ρ} satisfying these conditions. By part 2 of Lemma 4.10, then, there is a condition q′ ∈ P_{γ} such that q′↾ρ = q and q forces that q′↾[ρ, γ) is a

(N̄[G_{ρ}]↾[o.t.(M ∩ γ), o.t.(M ∩ β)), P_{γ}/P_{ρ})-semi-generic

condition extending such a q̄. This q′ suffices. We have Conditions 5 and 7 of Definition 4.14 by the properites listed above for q^{∗}↾[ρ, γ), Condition 4 by our extension, and the others by the assumptions of the lemma.

The restricted version of the lemma follows by induction on o.t.((β \ρ)∩M). Note that our induction hypothesis entitles us (once we have fixed β and ρ) to assume that the unrestricted version holds whenever γ < β.

Now, when ρ = β there is nothing to show. The argument for increasing o.t.((β \ ρ) ∩ M) by one follows from the case where β = ρ + 1, and this case is also trivial (see Remark 4.15).

The only remaining case then is when ρ < β and β = γ is a limit ordinal. Fix an increasing sequence η_{i} (i < ω) cofinal in M ∩ β with η_{0} = ρ, and succesively apply the induction hypothesis (in V, as opposed to some submodel). That is, let q_{0} = q, and given q_{i}, let q_{i+1} ∈ P_{η_{i+1}} be such that q_{i+1}↾η_{i} = q_{i} and (M, N̄, p, q_{i+1}) is a (Q̄, η_{i+1}, β)-system. Then each q_{i} is (N̄↾[o.t.(M∩η_{i}), o.t.(M∩ β)), P_{η_{i}})-semi-generic, and the limit of the q_{i}’s is the desired q′. Condition 4 of Definition 4.14 is then easily satisfied (the second half being vacuous), and Conditions 5 and 7, being local properties, are satisfied by the induction hypothesis.

5 Applications

5.1 Bounding

The following is a a minor modification of a standard fact.

Theorem 5.1. Let λ be a measurable cardinal, let θ be a regular cardinal such that V_{λ+2} ⊂ H(θ), and let ≤^{∗} be a wellordering of H(θ). Then for every countable ordinal β and every X ≺ (H(θ), ∈, ≤^{∗}), there is a z ∈ [λ]^{β} such that

(∗∗) [z]^{<ω} ⊂ ⋂{D_{λ}^{Z} : Z ≺ (H(θ), ∈, ≤^{∗}) ∧ Z ∈ X},

and such that, letting

Y = Sk_{(H(θ),∈,≤^{∗})}(X ∪ z),

X ∩ λ is a proper initial segment of Y ∩ λ.

Proof. It suffices to prove the theorem for the case β = 1, since then we may repeat the construction any countable ordinal number of times. To see that (∗∗) is preserved, note that if after repeating the construction some countable number of times we have the λ-end-extension X′ of X and we have added γ_{0}, …, γ_{n} (possibly among other ordinals) to z and Z ≺ (H(θ), ∈, ≤^{∗}) is in X, then Z′ = Sk_{(H(θ),∈,≤^{∗})}(Z ∪ {γ_{0}, …, γ_{n}}) ∈ X′. Further, if γ_{n+1} ∈ D_{λ}^{Z′}, then by Lemma 2.1, {γ_{0}, …, γ_{n+1}} ∈ D_{λ}^{Z}.

Now, to prove the theorem for the case β = 1, let μ be the ≤^{∗}-least normal measure on λ, and fix X as in the statement of the theorem. Let γ be any member of ⋂(X ∩ μ), and let

Y = Sk_{(H(θ),∈,≤^{∗})}(X ∪ {γ}).

The key point is that if f: λ → λ is a regressive function in any elementary submodel of (H(θ), ∈, ≤^{∗}), then f is constant on a set A ∈ μ, and since μ is ≤^{∗}-least, μ and therefore A and the constant value are also in the model. Along with the fact that ⋂(X ∩ μ) ⊂ ⋂(Z ∩ μ) for any Z ≺ (H(θ), ∈, ≤^{∗}) with Z ∈ X, this shows that γ satisfies (∗∗). To see that Y ∩ λ is a proper end-extension of X ∩ λ, note that γ ∉ X, and since any ordinal η ∈ Y ∩ λ is of the form f(γ) for some f: λ → λ in X, if η ∈ Y ∩ γ then any corresponding f is regressive on a set in μ, and so η ∈ X, by the key point.

Putting together Theorems 3.2 and 5.1 we have the following.

Corollary 5.2. Say that there exists a strongly inaccessible limit of measurable cardinals. Then there is a semi-proper forcing extension in which Bounding holds, along with the Continuum Hypothesis.

Proof. Let ⟨λ_{ρ} : ρ < κ⟩ be a continuous increasing sequence of cardinals with supremum κ strongly inaccessible such that each λ_{ρ+1} is a measurable cardinal. Fix a regular cardinal χ > (2^{κ})^{+}, and let ≤_{χ} be a wellordering of H(χ). Let

A = ⟨A_{β}^{ρ} ⊂ [λ_{ρ+1}]^{<ω_{1}} : ρ < κ, β < ω_{1}⟩

be such that each A_{β}^{ρ} is the set of countable subsets of λ_{ρ+1} of ordertype greater than β. Then by Theorem 5.1, Conditions 1-3 of Theorem 3.2 are satisfied (Condition 2 by the fact that small forcing preserves measurable cardinals), so any forcing P as in the statement of Theorem 3.2 is semi-proper and (ω, ∞)- distributive. We may further require that for every function f added by some initial stage of the iteration there is a ρ < κ such that f = f_{∼}_{ρ}. Each Q_{ρ,f} forces that |λ_{ρ+1}| = ℵ_{1} and that any canonical function for λ_{ρ+1} dominates f on a club, so P forces Bounding. Further, since 2^{ω} < κ in V and P forces κ = ω_{2}, P forces CH.

Deiser and Donder have recently shown [1] that Bounding is equiconsistent with a strongly inaccessible limit of measurable cardinals.

5.2 Suslin Bounding

The previous application can be generalized to show that the following statement can be forced without adding reals, answering a question in [15].

5.3 Definition. ([4]) Suppose that A ⊂ ℝ. Then A is universally Baire if for any compact Hausdorff space X and any continuous function π: X → ℝ, the set {a ∈ X | π(a) ∈ A} has the property of Baire in X.

5.4 Definition. Suslin Bounding is the following statement. Suppose that A ⊂ ℝ is universally Baire, and fix a function f: ω_{1} → A. Then there is a tree T on ω × ω_{1} such that A = p[T] and such that {α < ω_{1} | f(α) ∈ p[T ↾(ω × α)]} contains a club.

Theorem 5.5. Bounding is equivalent to Suslin Bounding for Π_{1}^{1} sets.

Proof. For the forward direction, we adapt the proof that Π_{1}^{1} sets are projections of trees on ω × ω_{1} (see [10]). Let S be a tree on ω × ω, let B = p[T] and let A = ω^{ω} \ B. Recall that for all x ∈ A,

S_{x} = {σ ∈ ω^{<ω} | (x↾|σ|, σ) ∈ S}

is wellfounded. For each x ∈ A, let β_{x} be the least ordinal (necessarily countable) such that there exists a function r_{x}: ω^{<ω} → β_{x} with the property that if σ, σ′ ∈ S_{x} are such that σ is a proper initial segment of σ′, then r_{x}(σ) > r_{x}(σ′). Fix f: ω_{1} → A, and for each α < ω_{1} let h(α) = β_{f(α)}. Now fix γ < ω_{2} and a bijection g: ω_{1} → β such that for every α in a fixed club C ⊂ ω_{1}, o.t.(g[α]) > h(α). Let σ: ω → ω^{<ω} be a bijection such that |σ(i)| ≤ i + 1 for all i < ω. Let T be the tree on ω × ω_{1} consisting of all pairs (ρ, τ) such that if i, j < |ρ| (= |τ|) and σ(i) is a proper initial segment of σ(j) with

(ρ↾|σ(i)|, σ(i)), (ρ↾|σ(j)|, σ(j))

both in S, then g(τ(i)) > g(τ(j)). Then p[T] = A, and, for each x ∈ A, if δ_{x} is the least (again, necessarily countable) ordinal δ such that o.t.(g[δ]) ≥ β_{x} and i_{x} is an order preserving embedding of β_{x} into g[δ_{x}], then (x, g^{−1} ∘ i_{x} ∘ r_{x} ∘ σ) is a path through T ↾(ω × δ_{x}). Since δ_{f(α)} ≤ α for all α ∈ C, this shows that T satisfies the definition of Suslin Bounding for f.

For the other direction, let W ⊂ ω^{ω} be the set of functions coding wellordings of ω under some fixed coding with the property that for each g ∈ ω^{ω} and n ∈ ω, g↾(n + 1) codes how n + 1 compares with each m < n + 1. Each function from ω_{1} to ω_{1} induces a corresponding function from ω_{1} into W. Let f: ω_{1} → W be such a function, and let T be the tree given by Suslin Bounding, with C the witnessing club set. We have a partial, not necessarily transitive, order ≤_{o} on the sequences in T, where σ_{0} ≤_{o} σ_{1} if one extends the other, and, σ_{i} being the longer one, the first cordinate of the last element of σ_{i} codes that |σ_{0}| < |σ_{1}| in the corresponding ordering. Let ≤_{t} be the least transitive ordering containing ≤_{o}. Then ≤_{t} is a wellfounded partial order. Seeing this requries checking that the resulting order is antireflexive and wellfounded. To see antireflexivity, assume that

τ ≤_{o} σ_{0} ≤_{o} … ≤_{o} σ_{n} ≤_{o} τ

is the shortest possible counterexample. First note that σ_{0} and σ_{n} cannot be comparable in ≤_{o} (or identical) since then there would be a shorter counterexample, removing the τ’s and placing either σ_{n} and the beginning of the sequence or σ_{0} at the end if they are unequal. Therefore σ_{0} and σ_{n} must be incompatible extensions of τ, and σ_{1} must be comparable with τ and distinct from σ_{n}. But then depending on whether σ_{1} ≥_{o} τ or τ ≥_{o} σ_{1}, there is a shorter counterexample, removing either σ_{0} or σ_{2} … σ_{n} from the original sequence. To see wellfoundedness, let σ_{i} (i < ω) be a descending sequence in ≤_{o}. First note that if some τ ∈ T has infinitely many extensions in the sequence, then all but finitely many of them must be extensions of a fixed immediate successor of τ, since otherwise initial segments of τ are visited infinitely often by the sequence, which is impossible (using antireflexivity of ≤_{t}), there being only finitely many of them. But the empty sequence has infinitely many extension in the sequence, which means that we can build an infinite chain though T all of whose members have this property. We claim that infinitely many members of this chain must be in the sequence, which gives a contradiction since the chain codes a wellordering. To see the claim, fix a member τ_{0} of the chain, and an arbitrary integer n. Then there is some σ_{i}, i > n, extending τ_{0}, and a member τ_{1} of the chain with length greater than |σ_{i}|. Let σ_{j} be an extension of τ_{1}, for some j > i. Then if σ_{i} is not on the chain, there must be some k in the interval (i, j) such that σ_{k} is an initial segment of τ_{1}. Since n was arbitrary, the claim follows.

Now extend ≤_{t} to a wellordering ≤_{T} of T, and let γ be the length of ≤_{T}. Let h: ω_{1} → T be a bijection, and define g: ω_{1} → ω_{1} by letting g(α) be the ordertype of ≤_{T} restricted to h[α]. Then g is a canonical function for γ. Furthermore, for a club C′ ⊂ C of α < ω_{1}, T ↾(ω × α) = h[α]. For these α, g(α) is greater than the ordertype of every wellordering in the projection of T ↾(ω × α), and thus greater than f(α).

5.6 Remark. Note that Suslin Bounding for Π_{1}^{1} sets and Suslin Bounding for Π_{∼}_{1}^{1} sets are identical.

Instead of working directly with the definition of universal Baireness, we will consider the equivalent (in the presence of large cardinals) form given by Theorem 5.8. Given a set X, we let m(X) denote the set of countably complete ultrafilters on X. Recall that a sequence of measures ⟨μ_{i} : i < ω⟩ such that each μ_{i} concentrates on κ^{i} is a countably complete tower if for any sequence ⟨A_{i} : i < ω⟩ such that each A_{i} ∈ μ_{i} there is a sequence z ∈ κ^{ω} such that each z↾i ∈ A_{i}. See [7, 15] for more detail.

5.7 Definition. Suppose that κ is a nonzero ordinal and that T is a tree on ω ×κ. Then T is δ-homogeneous if there is a partial function π: ω^{<ω} → m(κ^{<ω}) such that

1. if s ∈ dom(π) then π(s) is a δ-complete measure on κ^{|s|} and π(s)(T_{s}) = 1, where T_{s} = {t ∈ κ^{|s|} : (s, t) ∈ T },

2. for all x ∈ ω^{ω}, x ∈ p[T] if and only if

(a) {x↾k : k ∈ ω} ⊂ dom(π),

(b) ⟨π(x↾k) : k ∈ ω⟩ is a countably complete tower.

A set A ⊂ ℝ is δ-homogeneously Suslin if A = p[T] for some δ-homogeneous tree T. A is ^{∞}-homogeneously Suslin if it is δ-homogeneously Suslin for arbitrarily large δ.

Theorem 5.8. ([4]) Suppose that there is a proper class of Woodin cardinals and that A ⊂ ℝ. Then the following are equivalent.

1. A is universally Baire.

2. A is ^{∞}-homogeneously Suslin.

We use the following fact to ensure that our forcing iteration considers all universally Baire sets.

Lemma 5.9. Let P be an (ω, ∞)-distributive partial order, and let G ⊂ P be V - generic. Then in V [G], for all A ⊂ ℝ and all γ > (2^{|P |})^{+}, A is γ-homogeneously Suslin if and only if A ∈ V and A is γ-homogeneously Suslin in V.

Proof. This follows from the following standard facts about measures, where V [G] is an extension by a forcing P such that (2^{|P |})^{+} < γ ≤ κ.

1. For every γ-complete measure U on κ^{<ω} in V [G], U ∩ V ∈ V. (Otherwise, densely often in P there is a set in V whose membership in the measure is undecided; by genericity then there will be a subset of the measure of size ≤ |P | with empty intersection.)

2. For every γ-complete measure on κ^{<ω} in V [G], every positive set contains a positive set in V. (For each P -name for a positive set and each condition in P, consider the set of sequences p forces into the positive set.)

3. Every γ-complete measure on κ^{<ω} in V extends to one in V [G]. (All sets containing positive sets from the ground model.)

For the forward direction, let T be a γ-homogeneous tree on ω × κ in V [G] such that p[T] = A, for some γ > (2^{|P |})^{+}. Let π: ω^{<ω} → m(κ^{<ω}) witness that T is γ-homogeneous. Each π(σ) extends a measure in V on κ^{<ω}, and since P is (ω, ∞)-distributive, the corresponding function π′ taking each σ to the restriction of π(σ) to V exists in V. For each x ∈ ℝ \ p[T], let ⟨A_{k}^{x} : k < ω⟩ be a witness to the fact that ⟨π(x↾k) : k < ω⟩ is not countably complete. For each s ∈ ω^{<ω}, let

B_{s} = ⋂{A_{|s|}^{x} | x ∈ ℝ \ p[T] ∧ s ⊂ x}.

Since every positive set for each π(x↾k) contains one from V, and since P is (ω, ∞)-distributive, we can assume by shrinking if necessary that ⟨B_{s} | s ∈ ω^{<ω}⟩ is in V. Now let T′ ∈ V be the set of pairs s, t such that t ∈ B_{s}. Since the measures are all γ-complete, each B_{s} is positive for π(s). Then T′ is γhomogeneous (with π′ as a witness) with the same projection as T.

For the other direction, assume that T (on ω × κ) and π in V witness that A is γ-homogeneously Suslin. Extend the π(σ)’s to V [G]-measures, inducing a function π′: ω^{<ω} → m(κ^{<ω}). Since each positive set in V [G] contains one in V, and since no ω-sequences of ordinals have been added by P, for each x ∈ ω^{ω} the countable completeness of the corresponding tower is not changed by P. Since no countable sets of ordinals have been added the projection of T is the same, so T (along with π′) witnesses in V [G] that A is γ-homogeneously Suslin.

Let T be a tree on ω × κ and let f: ω_{1} → p[T]. Our one-step forcing R_{f,T} is the set of continuous increasing sequences ⟨x_{α} ∈ [κ]^{<ω_{1}} : α ≤ β⟩ of countable length such that for all α ≤ β, x_{α} ∩ ω_{1} ∈ ω_{1} and f(x_{α} ∩ ω_{1}) ∈ p[T ↾(ω × x_{α})], ordered by extension.

Given a δ^{+}-homogeneous set of reals, we use the measures witnessing homogeneity to suitably expand countable elementary substructures of H(χ).

Lemma 5.10. Fix δ ≥ ω_{1}, and let T be a δ^{+}-homogeneous tree on ω × κ, witnessed by π: ω^{<ω} → m(κ^{<ω}). Let χ > 2^{κ} be a regular cardinal with ≤_{χ} a wellordering of H(χ), and let X ≺ (H(χ), ∈, ≤_{χ}) with T, π ∈ X. Then for any countable a ⊂ p[T] there exists a countable z ⊂ κ such that letting Y = Sk_{(H(χ),∈,≤_{χ})}(X ∪ z) we have that

1. X ∩ δ = Y ∩ δ,

2. a ⊂ p[T ↾(ω × (Y ∩ κ))].

3. [z]^{<ω} ⊂ ⋂{D_{κ}^{Z} : Z ≺ (H(χ), ∈, ≤_{χ}) ∧ {T, π} ∈ Z ∈ X}.

Proof. Fix T, π, X and a = {a_{i} : i < ω}. By Lemma 2.1, it suffices to show that we can deal with a_{0}, as we can just repeat the process ω times. For each k < ω, let A_{k} = ⋂(X ∩ π(a_{0}↾k)). Then by the definition of δ^{+}-homogeneity (i.e., countable completeness) there exists z ∈ (κ \ δ)^{ω} such that for all k < ω, (a_{0}↾k, z↾k) ∈ T and z↾k ∈ A_{k}. Now z is as desired, since by the δ^{+}-completeness of each π(a_{0}↾k), if k ∈ ω and h: κ^{k} → δ then h is constant on a set C_{h} ∈ π(a_{0}↾k), and so this constant value must be an element of any elementary submodel of (H(χ), ∈, ≤_{χ}) with h and π as members. Further, if h is in X, then z↾k ∈ C_{h}, so h(z↾k) is the corresponding constant value.

Given Y as in Lemma 5.10 where f(Y ∩ ω_{1}) ∈ a, the union of any Y -generic for R_{f,T}, adjoined by its union, is a condition.

To get the consistency of Suslin Bounding from Theorem 3.2, we start from a proper class of Woodin cardinals, and let κ be a strongly inaccessible cardinal such that every ^{∞}-homogeneously Suslin set of reals is ^{∞}-homogeneously Suslin in V_{κ}. The assumption of the Woodin cardinals is just to make ^{∞}-homogeneous Suslinity equal to universal Baireness. Let F: ω_{1} → ℝ be a wellordering of the reals. Let our bookkeeping for P be such that each pair (A, f), where A is a universally Baire set (from the ground model) and f: ω_{1} → A is added by some initial segment of the iteration, is associated to some ρ greater than the stage at which f was added, such that A is the projection of a |P(P(λ_{ρ}))|^{+}- homogeneously Suslin tree T ∈ V_{κ} on ω×κ for some κ ≤ λ_{ρ+1}. For this ρ, we let each A_{β}^{ρ} be the set of countable x ⊂ λ_{ρ+1} such that F (β) ∈ p[T ↾(ω × (x ∩ κ))], so that Q_{ρ,(F^{−1}∘f)} = R_{f,T}. By Lemma 5.10, these sets also satisfy Conditions 1-3 of Theorem 3.2. This scheme then gives the following corollary.

Corollary 5.11. Suppose that there is a proper class of Woodin cardinals and let κ strongly inaccessible be such that every ^{∞}-homogeneously Suslin set of reals is ^{∞}-homogeneously Suslin in V_{κ}. Then there is a semi-proper (ω, ∞)- distributive forcing of size κ in whose extension the Continuum Hypothesis and Suslin Bounding hold.

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Department of Mathematics and Statistics Miami University Oxford, Ohio 45056 USA

larsonpb@muohio.edu

Institute of Mathematics Department of Mathematics Hebrew University Rutgers University 91904 Jerusalem New Brunswick, NJ 08903 Israel USA

shelah@math.huji.ac.il