ISRAEL MATHEMATICAL CONFERENCE PROCEEDINGS Vol. 06, 1992

TOOLS FOR YOUR FORCING CONSTRUCTION

by MARTIN GOLDSTERN

Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel Current address: 2. Mathematisches Institut, FU Berlin, Arnimallee 3, 10000 Berlin 33, Germany E-mail address: goldstrn@bimacs.cs.biu.ac.il, goldstrn@math.fu-berlin.de

Abstract. A preservation theorem is a theorem of the form: “If ⟨P_{α}, Q_{α} : α < δ⟩ is an iteration of forcing notions, and every Q_{α} satisfies φ in V^{P_{α}}, then P_{δ} satisfies φ.” We give a simplified version of a general preservation theorem for countable support iteration due to Shelah. This version is particularly useful for problems dealing with sets of reals. We give several examples of applications, among them “countable support iteration of proper ω^{ω}-bounding forcing notions is ω^{ω}-bounding.” We also review the basic facts about countable support iteration and proper forcing, as well as Souslin proper forcing notions.

0 Introduction The main objective of this paper is to present a simplified version of Shelah’s preservation theorems. We hope that the reader will find here powerful tool for his/her forcing constructions, as well as good open problems in this abstract area of mathematical research: The study of (iterated) forcing.

Iterated forcing is a powerful tool for proving independence results. In an iterated forcing argument the “ground model” V_{0} is first extended to a model V_{1} using some forcing notion Q_{0}. Then V_{1} is extended to some universe V_{2}, using some forcing notion Q_{1} in V_{1}, etc. After ω many steps, a model V_{ω} containing ⋃_{n} V_{n} is constructed, and the iteration can continue—usually up to ω_{1}, ω_{2} or some large cardinal.

In an iterated forcing argument there two main points we have to take care of:

(1) In each extension by a single forcing notion Q_{i} we have to add certain generic objects that we want to have present in the final model.

(2) At no stage are we allowed to add certain objects that we do not

want to have in the final model. The argument given to deal with (1) usually depends heavily on the special

properties of the forcing notions Q_{i}. Similarly, dealing with (2) at a successor step i + 1 is done by arguments that are characteristic for the forcing notion Q_{i}.

A “preservation theorem” is a theorem that deals with problem (2) at limit stages, i.e. a theorem ensuring that no “unwanted” objects are introduced at limit stages whenever all the forcing notions Q_{i} that are used satisfy certain “niceness” conditions.

For example, the problem of preserving certain cardinals and cofinalities is an instance of (2). The earliest preservation theorem we are aware of is the following ([from 13])

The finite support limit of an iteration in which all iterands satisfy the countable chain condition also satisfies the countable chain condition (and thus preserves cofinalities and cardinals)

The corresponding theorem for countable support iteration is the following ([10]):

The countable support limit of an iteration in which all iterands are proper is itself proper (and thus does not collapse ℵ_{1}). Also, starting from CH, if the length of the iteration is ≤ ℵ_{2} and all iterands have size ≤ ℵ_{1}, then the limit satisfies the ℵ_{2}-cc, so all cardinals and cofinalities are preserved.

The following example shows that the question of what reals are introduced in limit stages of an iteration is usually nontrivial:

Example 1 0.1: Assume that V_{0} satisfies CH, Q_{0} adds no new reals to V_{0}, Q_{1} adds ℵ_{1}^{V_{0}} many new reals to V_{1}, Q_{2} adds ℵ_{2}^{V_{0}} many new reals to V_{2}, etc. Let P_{n} := Q_{0} ∗ Q_{1} ∗ ⋯ ∗ Q_{n−1}. So V_{n} is the generic extension of V_{0} by P_{n}.

Then in V_{ω} we will have at least ℵ_{ω} many new reals. But as the cofinality of the continuum must be uncountable, this means that in V_{ω} will have at least ℵ_{ω+1}^{V_{0}} many new reals. Since only ℵ_{ω} many of them appear in intermediate stages, “most” of these reals appear only in the limit stage. This shows that it is not trivial to keep control over which reals are added in a limit stages by controlling what reals are added in intermediate stages.

Example 2 0.2: Consider an iteration ⟨P_{n}, Q_{n} : n < ω⟩, where for all n, ∥−_{P_{n}}“Q_{n} is nontrivial, i.e., contains incompatible conditions.”

If we define V_{ω} as V^{P_{ω}}, where P_{ω} is the finite support limit of the iteration sequence ⟨P_{n}, Q_{n} : n < ω⟩, then (no matter what the forcing notions Q_{n} are), P_{ω} will add Cohen reals over the ground model. (Let ∥−_{n}“q_{n}^{0} ⊥_{Q_{n}} q_{n}^{1},” then the function f : ω → 2, defined by f(n) = 0 ⇔ q_{n}^{0} ∈ G(n) will be a Cohen real, where G(n) is the generic filter on Q_{n}.)

This example shows an inherent limitation of the method of finite support iteration: In limit stages of cofinality ω, Cohen reals are always added.

The general problem of whether Cohen reals can be added in a limit stage of a countable support iteration is open:

Assume P_{ω} is the countable support iteration of ⟨P_{n}, Q_{n} : n < ω⟩, where for all n we have:

∥−_{P_{n}}“There are no Cohen reals over V ”

Does this imply ∥−_{P_{ω}}“There are no Cohen reals over V ”?

At this moment it seems impossible to apply the preservation theorem we prove below to this problem directly. There are, however, stronger properties (of forcing notions) than “not adding Cohen reals”, that can be shown to be preserved under countable support iteration. The best known of them are “^{ω}ω-bounding” and “Laver property.” (see 6.3 and 6.25, below)

Remark 0.3: The same proof that shows that finite support iteration always adds Cohen reals in stage ω also shows an inherent limitation of countable support iteration: In stage ω_{1} (or indeed at any stage of cofinality ω_{1}) we add an “ω_{1}- Cohen set,” i.e., a generic filter for the forcing notion F n(ω_{1}, 2, ω_{1}) of all countable partial functions from ω_{1} to 2. At first generic filters for this forcing notion seem innocent for problems concerning sets of reals, as F n(ω_{1}, 2, ω_{1}) does not add any reals. However, a simple density argument shows that the continuum of the ground model is collapsed to ℵ_{1}. This makes it impossible to have 𝔠 > ℵ_{2} at any limit stage of uncountable cofinality during a countable support iteration.

For a specific example of where a preservation theorem can be used, consider the problem of proving

Con(Cov(N) + 𝔟 = 𝔠 + ¬Unif(N))

using iterated forcing.

Explanation:

Cov(N) means: the real line cannot be covered by less than 𝔠 (=continuum) many sets from N, the ideal of null (=measure zero) sets. 𝔟 = 𝔠 means: Whenever F ⊆ ^{ω}ω is a family of less than 𝔠 many functions, there exists a function g bounding every element of F, where “g bounds f” means ∃k ∀n ≥ k g(n) > f(n). ¬Unif(N) means that there is a set of reals of cardinality < 𝔠 that does not have measure zero.

There are two approaches to get a model satisfying the condition above:

(1) Start with a model where (some big fragment of) Martin’s axiom holds and 𝔠 is big. Then construct a short iteration of length κ < 𝔠. In each iteration stage i < κ, add a real r_{i} in such a way that at the end the set {r_{i} : i < κ} is nonmeasurable. Preserve enough of Martin’s axiom to ensure that in the final model, Cov(N) + 𝔟 = 𝔠 holds.

(2) Start with the constructible universe L (or some model satisfying enough of GCH) and construct a long iteration of length κ > ℵ_{1}. In each stage i < κ, add a function f_{i} and a real r_{i} such that f_{i} is not bounded by any function constructed so far, and r_{i} is not in any measure zero set constructed so far. Assuming that no new reals appear in stage κ, in the final model we will have 𝔠 = κ, the set {r_{i} : i < κ} will be a set of reals that cannot be covered by < κ many measure zero sets, and {f_{i} : i < κ} will witness 𝔟 = 𝔠. We have to take care not to cover the set of constructible reals by a measure zero set. This will ensure that there is a non-null set of size ℵ_{1} < 𝔠.

At the end of section 6, we will consider approach (2). Since a Cohen real makes the set of all old reals a measure zero set, we should not add Cohen reals during the iteration. So we cannot use finite support iteration.

r_{i} will be a random real. It is easy to see that any set that is not of measure zero in V will also not be of measure zero after adding a random real.

f_{i} will be a Laver real. [7] showed that Laver reals also do not make non null sets into null sets.

This two observations show that if in stage i, the set of constructible reals was not null, the also in stage i + 1 this property will hold. So it remains to show that in limit stages the constructible reals are not covered by a measure zero set. For this purpose we will use the iteration theorem proved below (5.14), with parameters given in application 2 (6.8, 6.12)

Notation for forcing 0.4: A forcing notion P = ⟨P, ≤_{P} , 1_{P} ⟩ is a set P equipped with a transitive reflexive relation ≤_{P} and a greatest element 1_{P}. ∥−_{P} denotes the forcing relation of P.

We interpret p ≥ q (or p ≥_{P} q) as “q extends p,” “q is stronger than p,” or “q has more information then p.” p is “incompatible” with q (p ⊥ q or p ⊥_{P} q) means ¬∃r ∈ P : p ≥ r & q ≥ r.

Note that p ⊥ q ⇔ p ∥−q ∉ G_{P}.

A set D ⊆ P is called dense if ∀p ∈ P ∃q ∈ D : q ≤ p. D is open iff ∀q ∈ D ∀r ≤ q : r ∈ D. D is an antichain if ∀p, q ∈ D : p ≠ q ⇒ p ⊥ q. D is predense if ∀p ∈ P ∃d ∈ D d ⊥̸ p. D is predense below p iff ∀q ≤ p ∃d ∈ D d ⊥̸ p.

D is a “filter” if any two elements of D are compatible with a witness in D, i.e. ∀p, q ∈ D ∃t ∈ D p ≥ r & q ≥ r.

If V is a model of set theory, G ⊆ P is called generic (over V) (for P), if G∩D ≠ 0 for all dense D ∈ V. We often omit mentioning P and/or V, if the context makes it clear what P and V should be.

The class of P -names is defined by ∈-recursion: \begin{matrix}x \\ ∼ \end{matrix} is a P -name iff every element of \begin{matrix}x \\ ∼ \end{matrix} is a pair of the form ⟨\begin{matrix}y \\ ∼ \end{matrix}, p⟩ where \begin{matrix}y \\ ∼ \end{matrix} is a P -name and p ∈ P.

To avoid having to work with a proper class, we introduce an equivalence relation ≡ on P -names by \begin{matrix}x \\ ∼ \end{matrix} ≡ \begin{matrix}y \\ ∼ \end{matrix} iff 1_{P} ∥−_{P} \begin{matrix}x \\ ∼ \end{matrix} = \begin{matrix}y \\ ∼ \end{matrix}, and choose a set of representatives from each class, say, those of least possible rank.

So for example, the set of P -names for functions from ω to ω is the set of all representatives \begin{matrix}x \\ ∼ \end{matrix} such that 1∥−\begin{matrix}x \\ ∼ \end{matrix} : ω → ω. (It can easily be seen that this is in fact a set.)

Usually we write (variables for) P -names with a tilde, as in \begin{matrix}x \\ ∼ \end{matrix}. For a P -name \begin{matrix}x \\ ∼ \end{matrix} and a generic G ⊆ P, we let \begin{matrix}x \\ ∼ \end{matrix}[G] be the evaluation of \begin{matrix}x \\ ∼ \end{matrix} by G:

\begin{matrix}x \\ ∼ \end{matrix}[G] := \begin{cases}y \\ ∼ \end{cases}[G] : ⟨\begin{matrix}y \\ ∼ \end{matrix}, p⟩ ∈ \begin{matrix}x \\ ∼ \end{matrix}, p ∈ G\}

But if M is a model of a large fragment of ZFC, then we let M[G] := \begin{cases}x \\ ∼ \end{cases}[G] : M ⊧ \begin{matrix}x \\ ∼ \end{matrix} is a P -name\}. Sometimes (if G is clear from the context) we use for the evaluation of a name by G the same variable as the one used for the name, but leaving out the tilde, i.e., if G is given, x abbreviates \begin{matrix}x \\ ∼ \end{matrix}[G], p abbreviates \begin{matrix}p \\ ∼ \end{matrix}[G], etc.

Similarly p_{n}^{1} is \begin{matrix}p \\ ∼ \end{matrix}_{n}^{1}[G], etc.)

For an element x of V, x̂ or (x) is the standard P -name for x, x̂ := {⟨ŷ, 1_{P} ⟩ : y ∈ x} but we usually write x for x̂.

For a forcing P, G_{P} is (depending on the context) either = {⟨p̂, p⟩ : p ∈ P }, the canonical name for the generic object (also called the generic filter) added by P, or a variable ranging over all V -generic filters G ⊆ P.

We will freely use the following “existential completeness lemma”, sometimes without explicitly mentioning it: Lemma 0.5: For any forcing P, any formula φ(x) there is a name \begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix} such that

1∥−∃x φ(x) ⇔ φ(\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix})

(See [8] for a proof.)

We will also need the following well-known definitions and facts about forcing: Definition 0.6: For p, q elements of a forcing notion P, we define p ≥^{∗} q iff for all r ≤ q, r is compatible with p.

Fact 0.7: p ≥^{∗} q iff q∥−_{P} p ∈ G_{P}. In particular, p ≥ q implies p ≥^{∗} q.

Most properties of ≤ are also shared by ≤^{∗}. E.g., p ⊥ q iff there is no condition r satisfying p ≥^{∗} r and q ≥^{∗} r. Also, if p∥−φ, and p ≥^{∗} q, then q∥−φ. Definition 0.8: If P, Q are forcing notions, we say that i : P → Q is a “dense* embedding”, iff

(1) ∀p_{1}, p_{2} ∈ P: p_{1} ≤_{P}^{∗} p_{2} iff i(p_{1}) ≤_{Q}^{∗} i(p_{2}). (“i is an embedding with respect to ≤^{∗}”)

(2) ∀q ∈ Q ∃p ∈ P : q ≥^{∗} i(p). (“i is dense*”) Note that i is not necessarily 1-1. If (1) and (2) hold with ≤ instead of ≤^{∗}, then

we say that i is a dense embedding.

Definition 0.9: If there exists a dense* embedding i : P → Q, we say that P and Q are equivalent, and we write P ≈ Q.

This is justified by the following fact: Fact 0.10: Assume i : P → Q is a dense* embedding. Then whenever G ⊆ P is generic, then the set H defined by

H := {q ∈ Q : ∃p ∈ G, i(p) ≤ q}

is generic for Q. Conversely, if a set H ⊆ Q is generic, then the set G, defined by

G := {p ∈ P : i(p) ∈ H}

is generic for P.

Moreover, in both cases V [G] = V [H], and there is a canonical translation function that maps P -names \begin{matrix}x \\ ∼ \end{matrix} to corresponding Q-names \begin{matrix}x \\ ∼ \end{matrix}′, and conversely. We will often identify names with their image under this translation function.

We have that p∥−_{P} φ(\begin{matrix}x \\ ∼ \end{matrix}) iff i(p)∥−_{Q}φ(\begin{matrix}x \\ ∼ \end{matrix}′).

The proof is a routine computation.

Finally, we define the concept of an “interpretation”: Definition 0.11: Assume Q is a forcing notion, \begin{matrix}f \\ ∼ \end{matrix}, is a Q-name of a functions in ωω, f^{∗} is a function in ^{ω}ω, ⟨p_{n} : n < ω⟩ an increasing sequence of conditions.

We say that ⟨p_{n} : n < ω⟩ interprets \begin{matrix}f \\ ∼ \end{matrix} as f^{∗}, if for all n, p_{n}∥−\begin{matrix}f \\ ∼ \end{matrix}↾n = f^{∗}↾n. We say that f^{∗} is an interpretation of \begin{matrix}f \\ ∼ \end{matrix} if there exists an increasing sequence

as above.

The main theorem (5.14) of this paper is a simplified version of a theorem of Shelah ([11, XVIII]). See also [10, V, VI], [12] and [7] for precursors. The proof presented here is a joint work of Judah and the author.

Theorem 8.4 has been proved for various instances by several people.

Souslin forcing and Souslin Proper forcing were introduced in [5].

Contents of this paper:

In section 1, we give a review of composition and iteration of forcing notions in a general context.

In section 2 we introduce finite support iteration and show that the countable chain condition is preserved in finite support limits.

In section 3 we explain the concept of properness, and give a simple proof of Shelah’s theorem “properness is preserved under countable support iteration.” This proof will serve as a basis for the proof of the preservation theorem in section 6.

In section 4 we continue the review of countable and finite support iteration, by considering the relationship between an intermediate model and the final model.

In section 5 we formulate and prove a general preservation for countable support iteration.

In section 6 we show how to apply the preservation theorem for countable support iteration by giving several examples of properties that can be preserved in this framework.

In section 7 we review the concept of “Souslin Proper forcing”, and we prove the corresponding preservation theorem.

In section 8 we formulate and prove a general preservation theorem for finite support iteration, and we give examples.

Those readers who are interested only in finite support iteration can skip section 3 and all references to countable support iteration in section 4 without loss of continuity. A dual remark applies to readers only interested in countable support iteration.

We conclude this introduction by mentioning some open problems concerning countable support iteration and proper forcing: Problem (Judah) 0.12: Is “MA(Axiom A) + projective measurability” equiconsistent with the existence of 2 weakly compact cardinals?

From [1] it essentially follows that MA(Axiom A) is equiconsistent with one weakly compact cardinal.

Problem (Judah) 0.13: Find a sequence ⟨⊏_{n}: n ∈ ω⟩, such that “Q preserves ⊏” iff Q does not add random reals.

The most outstanding problem concerning countable support iteration is the following: Problem (Judah-Shelah) 0.14: If ⟨P_{n}, Q_{n} : n < ω⟩ is a countable support iteration with countable support limit P_{ω}. Does

∀n : P_{n} does not add Cohen reals over V

imply that P_{ω} does not add Cohen reals over V?

(For a partial solution see [6].)

I want to thank Saharon Shelah for fruitful discussions about iterated forcing.

314  MARTIN GOLDSTERN  IMCP

1 Composition and iteration of forcing We briefly review a few facts about composition of forcing and iterated forcing.

If P is a forcing notion, and \begin{matrix}Q \\ ∼ \end{matrix} is a P -name of a forcing notion, then we can force in V^{P} with \begin{matrix}Q \\ ∼ \end{matrix} to obtain a new extension (V^{P})^{Q}. There is a single forcing notion P ∗ \begin{matrix}Q \\ ∼ \end{matrix} (the “composition” of P and \begin{matrix}Q \\ ∼ \end{matrix}) such that the extension V^{P ∗\begin{matrix}Q \\ ∼ \end{matrix}} is canonically isomorphic to (V^{P})\begin{matrix}Q \\ ∼ \end{matrix}. Definition 1.1: Assume that P is a forcing notion, and \begin{matrix}Q \\ ∼ \end{matrix} is a P -name for a forcing notion. Then we let P ∗ \begin{matrix}Q \\ ∼ \end{matrix} be the set of all pairs ⟨p, \begin{matrix}q \\ ∼ \end{matrix}⟩ such that p ∈ P and

p∥−_{Q}“\begin{matrix}q \\ ∼ \end{matrix} ∈ \begin{matrix}Q \\ ∼ \end{matrix}”.

We let ⟨p, q⟩ ≥_{P ∗\begin{matrix}Q \\ ∼ \end{matrix}} ⟨p′, q′⟩ iff p ≥ p′ and p′∥−_{P} “q ≥_{\begin{matrix}Q \\ ∼ \end{matrix}} q′”.

Remark 1.2: As defined, P ∗ \begin{matrix}Q \\ ∼ \end{matrix} is a proper class. However, as in [4], we choose for each class of equivalent P -names a representative. Then the official definition of P ∗ \begin{matrix}Q \\ ∼ \end{matrix} is: The set of all pairs ⟨p, \begin{matrix}q \\ ∼ \end{matrix}⟩, where p ∈ P, \begin{matrix}q \\ ∼ \end{matrix} a representative P -name, and p∥−\begin{matrix}q \\ ∼ \end{matrix} ∈ \begin{matrix}Q \\ ∼ \end{matrix}. Then P ∗ \begin{matrix}Q \\ ∼ \end{matrix} is a set.

A similar remark will apply to iteration of forcing.

([8] chooses a different way to avoid proper classes. However, this solutions causes anomalies that we want to avoid. See [8, Exercise VIII E2–E4] and [9].)

Fact 1.3: Assume that P, \begin{matrix}Q \\ ∼ \end{matrix} are as above. Then (see [8])

(1) If G ⊆ P is generic over V, H ⊆ \begin{matrix}Q \\ ∼ \end{matrix}[G] generic over V [G], then

G ∗ H := \{⟨p, \begin{matrix}q \\ ∼ \end{matrix}⟩ ∈ P ∗ \begin{matrix}Q \\ ∼ \end{matrix} : p ∈ G, \begin{matrix}q \\ ∼ \end{matrix}[G] ∈ H\}

is generic for P ∗ \begin{matrix}Q \\ ∼ \end{matrix} over V,and V [G ∗ H] = V [G][H].

(2) Conversely, if J ⊆ P ∗\begin{matrix}Q \\ ∼ \end{matrix} is generic over V, then G := \{p : ∃\begin{matrix}q \\ ∼ \end{matrix} ⟨p, \begin{matrix}q \\ ∼ \end{matrix}⟩ ∈ J\} is generic for P over V, H := \begin{cases}q \\ ∼ \end{cases}[G] : ∃p ∈ G, ⟨p, \begin{matrix}q \\ ∼ \end{matrix}⟩ ∈ J\} is generic for \begin{matrix}Q \\ ∼ \end{matrix} over V [G], and J = G ∗ H.

(3) Moreover, P ∗\begin{matrix}Q \\ ∼ \end{matrix}-names can be translated to P -names for Q-names, i.e., for every P ∗ \begin{matrix}Q \\ ∼ \end{matrix}-name \begin{matrix}x \\ ∼ \end{matrix} there is \begin{matrix}x \\ ∼ \end{matrix}′, a P -name for a \begin{matrix}Q \\ ∼ \end{matrix}-name such that whenever G, H are as above, then

\begin{matrix}x \\ ∼ \end{matrix}[G ∗ H] = (\begin{matrix}x \\ ∼ \end{matrix}′[G])[H]

Conversely, if \begin{matrix}x \\ ∼ \end{matrix}′ is P -name for a Q-name, we can find a

corresponding P ∗ \begin{matrix}Q \\ ∼ \end{matrix}-name \begin{matrix}x \\ ∼ \end{matrix}.
We often identify \begin{matrix}x \\ ∼ \end{matrix} and \begin{matrix}x \\ ∼ \end{matrix}′.

Definition 1.4: Let κ be ordinal, and let I be an ideal on κ containing all finite sets. (I is not necessarily a proper ideal.) By induction on ε < κ we will define what an I-supported iteration of length ε is, and what the I-supported limit of such an iteration is.

“finite support” means that we let I be the ideal of finite subsets of κ, and “countable support” means that we let I be the (possibly improper) ideal of all countable subsets of κ. In this context, “countable” is understood to include “finite”.

(1) Q̄^{ε} := ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ is an I-supported iteration iff for all α < ε,

P_{α} = lim_{I} ⟨P_{β}, \begin{matrix}Q \\ ∼ \end{matrix}_{β} : β < α⟩, and

∥−_{P_{α}}“Q_{α} = ⟨\begin{matrix}Q \\ ∼ \end{matrix}_{α}, ≤_{\begin{matrix}Q \\ ∼ \end{matrix}_{α}}, 1_{\begin{matrix}Q \\ ∼ \end{matrix}_{α}}⟩ is a forcing notion”.

(2)

(a) The underlying set of lim_{I} Q̄^{ε} is the set of all partial functions p with dom(p) ∈ I satisfying

∀β ∈ dom(p) : p↾β∥−_{β}p(β) ∈ \begin{matrix}Q \\ ∼ \end{matrix}_{β}

(b) For p, q ∈ lim_{I} ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ we define p ≥_{lim}_{I} Q̄^{ε} q iff

∀β ∈ dom(p) ∪ dom(q) : q↾β∥−p(β) ≥_{\begin{matrix}Q \\ ∼ \end{matrix}_{β}} q(β)

(where we agree to let p(β) = 1_{\begin{matrix}Q \\ ∼ \end{matrix}_{β}} for β ∉ dom(p).)
(c) 1_{lim}_{I} Q̄^{ε} = ∅.

Whenever we consider a I-supported iteration ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩, we automatically define P_{ε} to be the I-supported limit of this iteration. This allows us to avoid the more awkward notation ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{β} : α ≤ ε, β < ε⟩.

Example 1.5: P_{0} = \{∅\}. \begin{matrix}Q \\ ∼ \end{matrix}_{0} is a P_{0}-name for a forcing notion. Since P_{0} is the trivial forcing notion, G_{0} := P_{0} is a “generic filter”, and V = V [G_{0}]. We identify

\begin{matrix}Q \\ ∼ \end{matrix}_{0} with Q_{0} := \begin{matrix}Q \\ ∼ \end{matrix}_{0}[G_{0}].

P_{1} is the set of all partial functions from {∅} to Q_{0}, i.e., P_{1} is isomorphic to Q_{0}, similarly, P_{2} is isomorphic to Q_{0} ∗ \begin{matrix}Q \\ ∼ \end{matrix}_{1} by the map that sends p ∈ P_{2} to

⟨p(0), p(1)⟩ ∈ Q_{0} ∗ \begin{matrix}Q \\ ∼ \end{matrix}_{1}.

Remark 1.6: By expanding definitions it is easy to see that in general the forcing notion P_{α+1} := lim_{I} ⟨P_{β}, \begin{matrix}Q \\ ∼ \end{matrix}_{β} : β < α + 1⟩ is isomorphic to P_{α} ∗ \begin{matrix}Q \\ ∼ \end{matrix}_{α}, via the map that sends p ∈ P_{α+1} to ⟨p↾α, p(α)⟩.

The following two facts follow easily from the definitions. Fact 1.7: If ε is a limit ordinal, then

(1) p ∈ P_{ε} iff dom(p) ⊆ ε is in I, and ∀α < ε, p↾α ∈ P_{α}.

(2) For p, q ∈ P_{ε}, p ≤_{ε} q iff ∀α < ε, p↾α ≤_{α} q↾α. See also 1.19.

Fact 1.8: If ε = α + 1, then

(1) p ∈ P_{ε} iff p↾α ∈ P_{α}, and p↾α∥−_{α}p(α) ∈ Q_{α}. (Again, this includes the case that α ∉ dom(p), where we declare p(α) = 1_{Q_{α}}.)

(2) For p, q ∈ P_{ε}, p ≥_{ε} q iff p↾α ≥_{α} q↾α and q↾α∥−_{α}p(α) ≥_{Q_{α}} q(α).

(3) For p, q ∈ P_{ε}, p ≥_{ε}^{∗} q iff p↾α ≥_{α}^{∗} q↾α and q↾α∥−_{α}p(α) ≥_{Q_{α}}^{∗} q(α).

Fact 1.9: For all α ≤ ε:

(1) P_{α} ⊆ P_{ε}. In fact, P_{α} = {p ∈ P_{ε} : dom(p) ⊆ α}.

(2) If p, q ∈ P_{α}, then p ≥_{α} q iff p ≥_{ε} q.

(3) If p ∈ P_{α}, q ∈ P_{ε}, then p ≥_{ε} q iff p ≥_{α} q↾α. Proof: (1) and (2) follow immediately from the definition. For (3), note that for

β ∈ dom(q) − α, p(β) = 1_{\begin{matrix}Q \\ ∼ \end{matrix}_{β}}, so q↾β∥−p(β) ≥ q(β).

Fact and Notation 1.10: Let ⟨P_{α}, Q_{α} : α < ε⟩ be an iteration. If G_{α} ⊆ P_{α} is generic, and H ⊆ \begin{matrix}Q \\ ∼ \end{matrix}_{α}[G_{α}] is any set, we let

G_{α} ∗ H = {r ∈ P_{α+1} : r↾α ∈ G_{α}, r(α)[G_{α}] ∈ H}

Then: G_{α}∗H is generic iff G_{α} is generic and H ⊆ \begin{matrix}Q \\ ∼ \end{matrix}_{α}[G_{α}] is generic over V [G_{α}]. Conversely, writing G(α) for {q(α)[G_{α}] : q ∈ G_{α+1}}, we know that is a generic

filter on \begin{matrix}Q \\ ∼ \end{matrix}_{α}[G_{α}] over the model V [G_{α}], and G_{α+1} = G_{α} ∗ G(α).

Proof: This is just a restatement of 1.3, using 1.6.

Remark 1.11: The recursion theorem tells us that if F is a function (possibly a class) then for all ε there exists an I-supported iteration Q̄ = ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ such that:

If for all α < ε, F (Q̄↾α) is (defined and) a P_{α}-name for a forcing notion,

then for all α < ε ∥−_{α}\begin{matrix}Q \\ ∼ \end{matrix}_{α} = F (Q̄↾α)

For the following, let ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ be an I-supported iteration forcing iteration, and let P_{ε} be the I-supported limit of this iteration, and let α range over ε ∪ {ε}. Definition 1.12: Assume α ≤ ε. If p ∈ P_{ε}, r ∈ P_{α}, and p↾α ≥ r, then we let p ∧ r := r ∪ p↾[α, ε).

r ; p ; 0  α  ε ⟹  p ∧ r ; 0  α  ε

Fact 1.13: Assume p, r are as above. Then:

(1) p ∧ r ∈ P_{ε}

(2) p ∧ r ≤ p.

(3) If p_{1} ≥ p_{2}, then p_{1} ∧ r ≥ p_{2} ∧ r.

Proof: By induction on β we can show (simultaneously) (p ∧ r)↾β ∈ P_{β} and (p ∧ r)↾β ≤ p↾β:

If β ≤ α, we have (p ∧ r)↾β = r↾β ≤_{β} p↾β.

If β > α, β limit, then by induction hypothesis for all γ < β we have (p∧r)↾(γ + 1) ∈ P_{γ+1}, so (p ∧ r)↾γ∥−(p ∧ r)(γ) ∈ \begin{matrix}Q \\ ∼ \end{matrix}_{γ}. So by 1.7, (p ∧ r)↾β) ∈ P_{β}. Also by induction hypothesis for all γ < β we have (p ∧ r)↾γ ≤_{γ} p↾γ, so by 1.7, (p ∧ r) ≤_{β} p.

If β > α is a successor ordinal, say β = γ + 1, then (p ∧ r)↾γ ∈ P_{γ} by induction hypothesis, and (p ∧ r)↾γ∥−(p ∧ r)(γ) = p(γ) ∈ \begin{matrix}Q \\ ∼ \end{matrix}_{γ}, as (p ∧ r)↾γ ≤ p↾γ. So by 1.8, (p ∧ r)↾β ∈ P_{β}. Also by induction hypothesis have (p ∧ r)↾γ ≤_{γ} p↾γ, so by 1.8, (p ∧ r) ≤_{β} p.

(3): By induction, (p_{1} ∧ r)↾β ≥ (p_{2} ∧ r)↾β.

Fact 1.14: If α ≤ ε, p ∈ P_{α}, q ∈ P_{ε}, then p ⊥_{ε} q iff p ⊥_{α} q↾α.

Proof: By 1.9(3), if r ∈ P_{ε}, r ≤_{ε} p, q, then r↾α ≤_{α} p, q↾α. Conversely, if r ∈ P_{α}, r ≤_{α} p, q↾α, then r ∧ q ≤_{ε} p, q. Fact 1.15:

(1) If A ⊆ P_{α} is a maximal antichain, α ≤ ε, then A is also a maximal antichain in P_{ε}.

(2) If G_{ε} is generic for P_{ε}, then G_{α} := G_{ε} ∩ P_{α} is generic for P_{α}.

Proof of (1): A is an antichain in P_{ε}, by 1.14. Assume that r ∈ P_{ε}, r ⊥_{ε} a for all a ∈ A. Then again by 1.14, r↾α ⊥_{α} a for all a ∈ A, so A is not maximal in P_{α}.

(2) follows immediately from (1).

Notation 1.16: We write ≤_{α} for ≤_{P_{α}}, similarly ∥−_{α}, etc. We may write G_{α} for G_{P_{α}}, and G(α) for G_{Q_{α}}.

When we talk about a I-supported iteration ⟨P_{α}, Q_{α} : α < ε⟩, it is understood that P_{ε} is defined as the I-supported limit of this iteration.

If β ≤ α, and ⟨P_{α}, Q_{α} : α < ε⟩ is an iteration, then G_{β} always denotes G_{α} ∩ P_{β}. When we fix a ground model V = V_{0}, and consider an iteration ⟨P_{α}, Q_{α} : α < ε⟩ ∈

V_{0}, we write V_{α} for V [G_{α}]. Note that this conflicts with the notation that some authors use for the sets of rank < α.

Fact 1.17: Assume that λ is a limit ordinal. Then for a generic G_{λ} ⊆ P_{λ}, for all p ∈ P_{λ},

p ∈ G_{λ} ⇔ ∀α < λ p↾α ∈ G_{α}

Proof: Assume not, then there exists a condition q forcing this.

So q∥−_{α}∀β < λ p↾β ∈ G_{β} & p ∉ G_{λ}. For α ∈ dom(p) ∪ dom(q) we let q′(α) be a name such that

∥−_{α}q′(α) ≤ q(α) & (q′(α) ≤ p(α) ∨ q′(α) ⊥ p(α))

(We can get such q′(α) using 0.5) Now we claim that for all α, q′↾α∥−q′(α) ≤ p(α). Assume not, then there exists a generic filter G_{α} containing q′↾α such that p(α)[G_{α}] ⊥ q′(α)[G_{α}]. So we can extend G_{α} to a generic G_{α+1} := G_{α} ∗ G(α) containing q′↾α + 1 but not p↾α + 1. Now we can extend G_{α+1} to G_{ε}, containing q′ (hence q) but not p↾α + 1, a contradiction. Corollary 1.18: For p ∈ P_{λ}, we have p ∈ G_{λ} iff for all β < λ, p(β)[G_{β}] ∈ G(β).

Proof of the corollary: By induction on β < λ we can show p↾β ∈ G_{β}. Limit steps are handled by 1.17, and successor steps by 1.10. Corollary 1.19: If ε is a limit ordinal, then

For p, q ∈ P_{ε}, p ≥_{ε}^{∗} q iff ∀α < ε, p↾α ≥_{α}^{∗} q↾α.

Proof: If for all α < ε q↾α∥−_{α}p↾α ∈ G_{α}, then q∥−_{ε}p ∈ G_{ε}, by 1.17.

Conversely, if for some α < ε we have p↾α ≱^{∗} q↾α, then there is r ∈ P_{α}, r ≤ q↾α, r ⊥ p↾α. Letting q′ := q ∧ r we get q′∥−p ∉ G_{ε}. Since q ≥ q′, we do NOT have q∥−p ∈ G_{ε}, so q ≰^{∗} p.

The following fact shows that in finite support iteration of ccc forcing notions and in countable support iteration of proper forcing notions no new reals are added in limit steps of cofinality > ω. Lemma 1.20: Assume ⟨P_{α}, Q_{α} : α < ε⟩ is an iteration, and δ is a limit ordinal of cofinality > ω, and all there are no conditions in P_{δ} whose domain is unbounded in δ. (In particular, this will be true for finite or countable support iteration.) Then:

∥−_{δ}If cf(δ) > ω, then ^{ω}ω ∩ V [G_{δ}] = ⋃_{α<δ} ^{ω}ω ∩ V [G_{α}]

Proof: For any α < δ we define a P_{α}-name \begin{matrix}f \\ ∼ \end{matrix}_{α} satisfying the following:

∀n ∥−_{α}“If there is p ∈ G_{α}, j ∈ ω such that V ⊧ p∥−_{δ}\begin{matrix}f \\ ∼ \end{matrix}(ň) = ǰ, then \begin{matrix}f \\ ∼ \end{matrix}_{α}(n) = j

(Note that it is possible to define such a function, since any two p, p′ ∈ G_{α} must be compatible (in P_{δ}) hence cannot force two different values for \begin{matrix}f \\ ∼ \end{matrix}(n).)

We will show that ∥−_{δ}∃α\begin{matrix}f \\ ∼ \end{matrix} = \begin{matrix}f \\ ∼ \end{matrix}_{α}.

Work in V [G_{δ}]. For any n, let j_{n} := \begin{matrix}f \\ ∼ \end{matrix}[G_{δ}](n), and find a condition p_{n} ∈ G_{δ}

such that V ⊧ p_{n}∥−_{δ}\begin{matrix}f \\ ∼ \end{matrix}(n) = ǰ_{n}.

Since (in V [G_{δ}]) we have cf(δ) > ω, and the supports of the p_{n} are bounded in δ, we can find α < δ such that for all n, p_{n} ∈ P_{α}, and hence also p_{n} ∈ G_{α}. Clearly

\begin{matrix}f \\ ∼ \end{matrix}[G_{δ}] = \begin{matrix}f \\ ∼ \end{matrix}_{α}[G_{α}].

Vol. 06, 1992  TOOLS FOR YOUR FORCING CONSTRUCTION  319

2 Finite support iteration and ccc

A forcing notion P is said to satisfy the countable chain condition, if there is no uncountable set of pairwise incompatible conditions. Fact 2.1: “P has the ccc” is equivalent to the following statement:

Whenever \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} is a P -name, and ∥−“\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} is an ordinal”, then there
exists a countable set A of ordinals such that ∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ A.

(Equivalently, if ∥−\begin{matrix}x \\ ∼ \end{matrix} ∈ V for some P -name \begin{matrix}x \\ ∼ \end{matrix}, then there exists a countable set A in V such that ∥−\begin{matrix}x \\ ∼ \end{matrix} ∈ A.)

Proof: Assume that P has the ccc. Let C ⊆ P be a maximal set of conditions satisfying

(1) p ∈ C ⇒ ∃β p∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} = β̂.

(2) p_{1}, p_{2} ∈ C ⇒ p_{1} ⊥_{P} p_{2}. It is easy to see that C must be a maximal antichain in P (because the set of

conditions satisfying (1) is dense).

Let A := \{β : ∃p ∈ C : p∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} = β̂\}. Then A is countable because C is countable, and clearly ∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ A. [Proof: Every condition p ∈ C forces \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ A. A condition forcing \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∉ A cannot be compatible with any condition in C, which contradicts the above observation that C is a maximal antichain.]

Conversely, assume that ⟨p_{i} : i < ω_{1}⟩ is an antichain. Let \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} be a name of an ordinal such that

∥−∃i p_{i} ∈ G_{P} ⇒ p_{\begin{matrix}α \\ ∼ \end{matrix}} ∈ G_{P}

Assume, towards a contradiction, that there is a countable set A of ordinals such

that ∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ A. Let i ∈ ω_{1} − A. Then p_{i}∥−p_{i} ∈ G_{P}, but for all j ≠ i, p_{i}∥−p_{j} ∉ G.

So p_{i}∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} = i ∉ A, a contradiction.

The same proof also shows: P has the ccc, iff Whenever \begin{matrix}x \\ ∼ \end{matrix} is a P -name, and ∥−“\begin{matrix}x \\ ∼ \end{matrix} ∈ V ”, then there exists a

countable set A such that ∥−\begin{matrix}x \\ ∼ \end{matrix} ∈ A.

Corollary 2.2: If P has the ccc, and \begin{matrix}X \\ ∼ \end{matrix} is a name of a countable set ⊆ V, then there exists a countable set A in V such that ∥−\begin{matrix}X \\ ∼ \end{matrix} ⊆ A.

The next theorem can be phrased “ccc is preserved under composition of forcing notions.” Corollary 2.3: Assume P has the ccc, and ∥−_{P} “\begin{matrix}Q \\ ∼ \end{matrix} has the ccc.” Then P ∗ \begin{matrix}Q \\ ∼ \end{matrix} has the ccc.

Proof: Let \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} be a P ∗ \begin{matrix}Q \\ ∼ \end{matrix}-name of an ordinal. We consider \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} as a P -name for a \begin{matrix}Q \\ ∼ \end{matrix}-name of an ordinal. So since ∥−_{P} “\begin{matrix}Q \\ ∼ \end{matrix} has the ccc,” (and using the existential completeness lemma) we can find a P -name \begin{matrix}A \\ ∼ \end{matrix} such that

(1) ∥−_{P} “\begin{matrix}A \\ ∼ \end{matrix} is countable”

(2) ∥−_{P} “∥−_{Q}\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ \begin{matrix}A \\ ∼ \end{matrix}”.

By 2.2 we can find a countable set A′ in V such that

∥−_{P} A ⊆ A′.

So ∥−_{P} ∥−_{\begin{matrix}Q \\ ∼ \end{matrix}}\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ A′, hence ∥−_{P ∗\begin{matrix}Q \\ ∼ \end{matrix}}\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ A′.

The next theorem shows that the property of satisfying the countable chain condition is satisfied under finite support iteration. Theorem 2.4: Assume that ⟨P_{α}, Q_{α} : α < ε⟩ is a finite support iteration of forcing notions, and

∀α < ε : ∥−_{α}“Q_{α} has the ccc”

Then P_{ε} has the ccc.

The proof of this theorem proceeds by induction on ε. Successor steps are handled by 2.3 and 1.6.

As for limit steps, from the induction hypothesis we can conclude that

(∗) ∀α < ε P_{α} ⊧ ccc

We will show that (∗) implies that P_{ε} has the ccc. We need to use the following combinatorial fact (called the Δ-system lemma): Lemma 2.5: Assume that ⟨F_{i} : i < ω_{1}⟩ is a family of finite sets. Then there is a set S ⊆ ω_{1} of size ℵ_{1} such that the family ⟨F_{i} : i ∈ S⟩ is a Δ-system, i.e.

∃F finite ∀i, j ∈ S : i ≠ j ⇒ F_{i} ∩ F_{j} ⊆ F

F is called a “kernel”, “root” or “heart” of the Δ-system ⟨F_{i} : i ∈ S⟩.

See [8, II.1.5] for a proof.

Proof of 2.4, limit step:

Assume (toward a contradiction) that ⟨p_{i} : i < ω_{1}⟩ is an antichain of conditions in P_{ε}. Since ⟨dom(p_{i}) : i < ω_{1}⟩ is a family of finite sets, the Δ-lemma applies.

So we can find a set S ⊆ ω_{1} of size ℵ_{1} and a finite set F such that for all distinct i and j in S, dom(p_{i}) ∩ dom(p_{j}) ⊆ F. Let F ⊆ α < ε.

The family ⟨p_{i}↾α : i ∈ S⟩ is an uncountable familiy of conditions in P_{α}. By our assumption (∗), it cannot be an antichain. So there are i ≠ j in S and a condition r ∈ P_{α} such that p_{i} ≥ r and p_{j} ≥ r.

Now define p ∈ P_{ε} as follows:

p↾α = r.

∀γ ∈ dom(p_{i}) − α: p(γ) = p_{i}(γ).
∀γ ∈ dom(p_{j}) − α: p(γ) = p_{j}(γ).

Note that dom(p_{i}) ∩ dom(p_{j}) ⊆ F ⊆ α, so dom(p_{i}) − α and dom(p_{j}) − α are disjoint.

We leave it as an exercise to check that p is indeed a condition in P_{ε}, and p_{i} ≥ p, p_{j} ≥ p.

This is a contradiction, as ⟨p_{i} : i < ω_{1}⟩ was supposed to be an antichain.

3 Properness and countable support iteration

Definition 3.1: Assume that (N, ∈) is a model of a fragment of ZFC (which should be large enough to develop the general theory of forcing). Let P ∈ N be a forcing notion, and assume that for all p_{1}, p_{2} ∈ P ∩ N: N ⊧ p_{1} ⊥_{P} p_{2} ⇒ V ⊧ p_{1} ⊥_{P} p_{2}. We say that G is N-generic for P iff:

For all A ∈ N, if N ⊧ “A ⊆ P is a maximal antichain of P ”, then G ∩ N ∩ A ≠ ∅.

(So G is N-generic iff G ∩ N is N-generic.)

It is easy to see that we by replacing “maximal antichain” by “dense subset”, “dense open subset” or “predense subset”, we get an equivalent definition.

In fact, the following holds: Fact 3.2: Assume p ∈ P ∩ N, P ∈ N, p ≤ q ∈ P. Then the following are equivalent:

(1) For all D_{1} ∈ N: If N ⊧“D_{1} is predense below p”, then D_{1} ∩ N is predense below q.

(2) For all D_{2} ∈ N: If N ⊧“D_{2} is predense ⊆ P,” then q∥−D_{2} ∩ N ∩ G ≠ ∅.

(3) For all D_{3} ∈ N: If N ⊧“D_{3} is dense ⊆ P,” then q∥−D_{3}∩N ∩G ≠ ∅.

(4) For all D_{4} ∈ N: If N ⊧“D_{4} is a maximal antichain ⊆ P,” then q∥−D_{4} ∩ N ∩ G ≠ ∅.

(5) For all D_{5} ∈ N: If N ⊧“D_{5} is open dense ⊆ P,” then q∥−D_{5} ∩ N ∩ G ≠ ∅.

Proof: We will show \begin{aligned}& & & & (3) \\ & & & ↗ & & ↘ \\ (1) & → & (2) & & & & (5) & → (1). \\ & & & ↘ & & ↗ \\ & & & & (4) \end{aligned}

(2) ⇒ (4), (2) ⇒ (3), (3) ⇒ (5) are trivial.

(1) ⇒ (2): Let D_{2} be predense. Then D_{2} is predense below p, so D_{2} ∩ N is predense below q. Hence q∥−(D_{2} ∩ N) ∩ G ≠ ∅.

(4) ⇒ (5): Work in N. Assume that D_{5} is open dense. Let D_{4} be a maximal antichain ⊆ D_{5} (i.e., let D_{4} ⊆ D_{5} be an antichain such that there is proper superset D′_{4} ⊆ D_{5} that is also an antichain). It is enough to see that D_{4} is a maximal antichain in P (in N): Assume not, and let r be incompatible with every element of D_{4}, let s ≤ r be in D_{5}, then also s is incompatible with every element of D_{4}, contradicting the maximality of D_{4}.

(5) ⇒ (1): Let D_{1} ∈ N, N ⊧“D_{1} is predense below p.” The following takes place in N:

Let D_{5} := {r : either r ⊥ p, or ∃d ∈ D_{1} r ≤ d}. Then D_{5} is open. We claim that D_{5} is dense. To prove this, let s be any condition in P. If s ⊥ p, then s ∈ D_{1}.

Otherwise, there is a t ∈ P, s ≥ t, p ≥ t. t is compatible with some d ∈ D_{1}, so there is t′ extending d, t, s, p. So t ∈ D_{5}.

Hence N ⊧“D_{5} is dense”, so V ⊧ q∥−D_{5} ∩ N ∩ G ≠ ∅.

Finally we will to show that (in V), D_{1} ∩ N is predense below q. Let r ≤ q be incompatible with all elements of D_{1}∩N. Let r′ ≤ r, and r′∥−s ∈ D_{5}∩N ∩G. Then s ∈ D_{5} ∩ N. Either N ⊧ s ⊥ p, then s ⊥ p, which implies s ⊥ r′, a contradiction to r′∥−s ∈ G. Otherwise N ⊧ ∃d ∈ D_{1} s ≤ d. As r′ and s are compatible, r′ and d are compatible, so D_{1} ∩ N is indeed predense below q.

Definition 3.3: We say that q ∈ P is N-generic (or (N, P)-generic) iff

V ⊧ q∥−_{P} “G_{P} is N-generic”

(iff 3.2(2)–(5) hold).

Remark 3.4: If q is N-generic, and q′ ≤ q, then also q′ is N-generic.

Notation 3.5: For the following, we let χ be a “large enough” regular cardinal. “large enough” means that for all forcing notions P we consider, we have 𝔓(P) ∈ H(χ), i.e., the power set of P is hereditarily of size < χ. Since all forcing notions we consider will be hereditarily countable, or countable support iterations of length ≤ ℵ_{2} of hereditarily countable forcing notion, and all the universes we consider satisfy GCH except for possibly 2^{ℵ_{0}} = ℵ_{2}, we could choose χ := ℵ_{3}. To be on the safe side, we let

χ := ℶ_{ω}^{+}

(So also in every extension that we consider, χ = ℶ_{ω}^{+}).

We will consider countable elementary submodels of (H(χ), ∈).

(The notions we will define below will depend on χ, but a careful examination [which we will not carry out here] shows that this dependence is only apparent.)

Note that all essential properties of P are absolute between V and H(χ), for example

V ⊧ A is a maximal antichain of P ⇔ H(χ) ⊧ A is a maximal antichain of P

We also have the following fact: Fact 3.6: If ∥−\begin{matrix}x \\ ∼ \end{matrix} ∈ H(χ), then there is a name x̄ ; ∼ ∈ H(χ) such that ∥−\begin{matrix}x \\ ∼ \end{matrix} = x̄ ; ∼.

Proof: First note that for each element x of H(χ) there is λ < χ and a sequence ⟨x_{i} : i ≤ λ⟩ of sets in H(χ) such that for all i ≤ λ x_{i} ⊆ {x_{j} : j < i}, and x = x_{λ}.

Applying this fact in V [G] to \begin{matrix}x \\ ∼ \end{matrix}, we get a sequence ⟨\begin{matrix}x \\ ∼ \end{matrix}_{i} : i ≤ \begin{matrix}λ \\ ∼ \end{matrix}⟩. We can find an ordinal λ such that ∥−\begin{matrix}λ \\ ∼ \end{matrix} ≤ λ, so wlog we may assume ∥−\begin{matrix}λ \\ ∼ \end{matrix} = λ (since we can

define \begin{matrix}x \\ ∼ \end{matrix}_{i} = ∅ for \begin{matrix}λ \\ ∼ \end{matrix} < i < λ.

Now we define by induction x̄ ; ∼_{i} := \{⟨x̄ ; ∼_{j}, p⟩ : p∥−\begin{matrix}x \\ ∼ \end{matrix}_{j} ∈ \begin{matrix}x \\ ∼ \end{matrix}_{i}\}, and prove by induc-

tion ∥−x̄ ; ∼_{i} = \begin{matrix}x \\ ∼ \end{matrix}_{i}.

Definition 3.7: A forcing notion P is called proper, if for some x ∈ H(χ), for all countable models (N, ∈) ≺ (H(χ), ∈) that contain x and P, and for all p ∈ P ∩ N there exists a q ≤ p which N-generic.

Remark 3.8: The clause “for some x … ” above is not essential, but it makes some proofs slightly simpler. For example, when we consider an iterated forcing notion P_{ε} ∈ N, we could reconstruct the iteration ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ from P_{ε} (in N), but the above definition makes it unnecessary, as we can include “⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ ∈ N” into our assumption.

We leave as an exercise to the reader to show the following fact. (We will not use it in the rest of the paper). Note that this fact is independent of χ. Fact 3.9: P is proper iff player II has a winning strategy in the following infinite game (see also [2]):

In the first move, player I plays a condition p, and a maximal antichain A_{1}. Player II responds with a countable subset B_{1}^{1} ⊆ A_{1}.

In the n-th move (n > 1), player I plays a maximal antichain A_{n}, and player II plays countable sets B_{1}^{n} ⊆ A_{1}, …, B_{n}^{n} ⊆ A_{n}.

After ω many moves, player II wins iff there is a condition q ≤ p such that for all n, the set B_{n} := ⋃_{k≥n} B_{n}^{k} is predense below q.

[Sketch of proof: If player II has a winning strategy, and player I plays all maximal antichains of the model N, then all responses of player II will be subsets of N. Conversely, player II can, in each move generate a Skolem hull N_{n} (inside H(χ)) of everything played so far. At the end any condition generic for N := ⋃_{n} N_{n} will work. This definition of the strategy does not take place in H(χ), since H(χ) is needed as a parameter, but the resulting strategy itself IS in H(χ).] Fact 3.10: For N as above, q ∈ P is N-generic iff for all \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ N:

If N ⊧“\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} is a P -name for an ordinal”, then q∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ N

(Note that q∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ N really means q∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix}[G_{P}] ∈ N̂.)

Proof: Let A ∈ N, N ⊧“A is a maximal antichain.” Working in N, we can find a cardinal κ, a function f from κ onto A, and a P -name \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} of an ordinal such that

∥−_{P} f(\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix}) ∈ G_{P} ∩ A. Then if q∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ N, also q∥−f(\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix}) ∈ N, so q∥−N ∩ A ∩ G ≠ ∅.

Hence q satisfies 3.2(4).

Conversely, let \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ N be a P -name of an ordinal. Then the set D := \{r ∈ P : ∃β r∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} = β\} is dense. Let f be a function with domain D such that for all r ∈ D,

r∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} = f(r) . Then f ∈ N, so if q∥−∃r ∈ D ∩ N ∩ G, then also q∥−\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ N, as

∥−“r ∈ N ⇒ f(r) ∈ N.” Remark 3.11: A similar proof shows: q is N-generic iff

∀\begin{matrix}x \\ ∼ \end{matrix} ∈ N : If \begin{matrix}x \\ ∼ \end{matrix} is a P -name and ∥−\begin{matrix}x \\ ∼ \end{matrix} ∈ V, then q∥−\begin{matrix}x \\ ∼ \end{matrix} ∈ N

(or in other words, q∥−N[G] ∩ V = N).

Example 3.12: If P satisfies the countable chain condition, then every condition (or equivalently, the condition 1_{Q}) is N-generic, for any countable model N ≺ H(χ) containing P. Thus, any ccc forcing notion is proper.

Proof: Note that if A ∈ N is countable, then A ⊆ N, because A must be the range of a function f ∈ N that has domain ω. As f ∈ N, and for all n ∈ ω, n ∈ N, also f(n) ∈ N for all n, so A = rng(f) ⊆ N.

Let A ∈ N be a maximal antichain, then ∥−_{Q}A ∩ G ≠ ∅. So also 1_{Q}∥−_{Q}A ∩ N ∩ G = A ∩ G ≠ ∅. Hence 1_{Q} satisfies condition 3.2(4).

(Alternatively, we could use 2.1 to show that ccc ⇒ proper.)

Fact 3.13: If Q is proper, and A ∈ V [G_{Q}] a countable set of ordinals, then there is a countable set B ∈ V of ordinals such that A ⊆ B.

Proof: Let \begin{matrix}A \\ ∼ \end{matrix} be a name for A, and let ⟨\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix}_{n} : n ∈ ω⟩ ∈ V be a sequence of names for all ordinals in \begin{matrix}A \\ ∼ \end{matrix}. For each n, let A_{n} be a maximal antichain of conditions deciding \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix}_{n}. We will show that the set of conditions forcing “there exists a countable B in V covering A” is dense.

So fix a condition p. Let N ≺ H(χ) be a countable elementary model containing p, Q and ⟨A_{n} : n ∈ ω⟩ and let q ≤ p be N-generic.

Let B := ⋃_{n∈ω}{β : ∃r ∈ A_{n} ∩ N : r∥−α_{n} = β}. B is countable, and the genericity of q easily implies that q∥−\begin{matrix}A \\ ∼ \end{matrix} ⊆ B.

Corollary 3.14: If Q proper and cf(δ) > ω, then ∥−_{Q}cf(δ) > ω. In particular, ℵ_{1} is not collapsed.

Proof: Q cannot add a countable cofinal sequence in δ, by the previous fact. (Note that this proof works only for δ < χ, but for δ ≥ χ or indeed δ > |P | a much simpler argument shows that the cofinality of δ is preserved.)

Fact 3.15: If N ≺ H(χ), and Q ∈ N, then 1∥−N[G] ≺ H(χ)^{V [G]}.

Proof: (Remember that χ is quite large compared to Q.) We will use the Tarski-Vaught criterion. So it is enough to see for all names \begin{matrix}a \\ ∼ \end{matrix} in N:

(∗)  1_{Q}∥−_{Q}(∃x φ(x, \begin{matrix}a \\ ∼ \end{matrix}))^{H(χ)} ⇒ ∃x ∈ N[G] (φ(x, \begin{matrix}a \\ ∼ \end{matrix}))^{H(χ)}

By 0.5, there is a name \begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix} such that

(∗∗)  1_{Q}∥−_{Q}\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix} ∈ H(χ) & ((∃x φ(x, \begin{matrix}a \\ ∼ \end{matrix}))^{H(χ)} ⇒ φ(\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix}, \begin{matrix}a \\ ∼ \end{matrix}))^{H(χ)}.

As this last statement can be rewritten as a statement in H(χ) (thanks to 3.6), we can find such \begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix} in N.

Thus 1_{Q}∥−_{Q}\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix} ∈ N[G] & ((∃x φ(x, \begin{matrix}a \\ ∼ \end{matrix}))^{H(χ)} ⇒ φ(\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix}, \begin{matrix}a \\ ∼ \end{matrix})^{H(χ)}), which implies (∗).

Note that the same proof actually shows we even have (N[G], N[G]∩H(χ), ∈) ≺ (H(χ)^{V [G]}, H(χ)^{V} , ∈). So if we work with a proper forcing notion, by 3.11 we also get (N[G], N, ∈) ≺ (H(χ)^{V [G]}, H(χ)^{V} , ∈).

Lemma 3.16: Assume that Q_{0} ∗ \begin{matrix}Q \\ ∼ \end{matrix}_{1} is a composition of forcing notions. Then (q_{0}, \begin{matrix}q \\ ∼ \end{matrix}_{1}) is N-generic, iff q_{0} is N-generic for Q_{0}, and q_{0}∥−“\begin{matrix}q \\ ∼ \end{matrix}_{1} is N[G_{0}]-generic

for \begin{matrix}Q \\ ∼ \end{matrix}_{1}[G_{0}].”

Similarly, if we have an iteration ⟨P_{α}, Q_{α} : α < ε⟩, α ≤ ε, then q ∈ P_{α+1} is (P_{α+1}, N)-generic iff: q↾α is (P_{α}, N)-generic, and q↾α ∥−_{α}“q(α) is (Q_{α}, N[G_{α}])-generic.”

Proof: (Recall that we identify Q_{0} ∗ \begin{matrix}Q \\ ∼ \end{matrix}_{1}-names with Q_{0}-names for \begin{matrix}Q \\ ∼ \end{matrix}_{1}-names) We will use 3.10. For any name \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix} ∈ N of an ordinal we have

(q_{0}, \begin{matrix}q \\ ∼ \end{matrix}_{1})∥−_{Q_{0}∗\begin{matrix}Q \\ ∼ \end{matrix}_{1}} \begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix}[G_{Q_{0}∗\begin{matrix}Q \\ ∼ \end{matrix}_{1}}] ∈ N iff q_{0}∥−_{Q_{0}}(\begin{matrix}q \\ ∼ \end{matrix}_{1}∥−_{\begin{matrix}Q \\ ∼ \end{matrix}_{1}[G_{0}]}\begin{matrix}α \\ ∼ \end{matrix} \begin{matrix}α \\ ∼ \end{matrix}[G_{0}][G_{1}] ∈ N)

This shows that properness is preserved under composition of forcing notions. The following two lemmata will show that properness is also preserved under countable support iteration. Note that the first one does not mention properness or countable models—even the fact that the iteration has countable support plays no role. Preliminary Lemma 3.17: Let ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ be a countable support iteration.

Assume α_{1} ≤ α_{2} ≤ β ≤ ε, \begin{matrix}p \\ ∼ \end{matrix}_{1} is a P_{α_{1}}-name for a condition in P_{ε}. Let D be a dense open set of P_{β}.

Then 1_{P_{α_{2}}} ∥−_{α_{2}} ∃p_{2} φ(p_{2}), where φ(p_{2}) is the conjunction of the following clauses:

(1) p_{2} ∈ P̂_{β}, p_{2} ≤_{β} \begin{matrix}p \\ ∼ \end{matrix}_{1}.

(2) p_{2} ∈ D̂.

(3) If \begin{matrix}p \\ ∼ \end{matrix}_{1}↾α_{2} ∈ G_{α_{2}}, then p_{2}↾α_{2} ∈ G_{α_{2}}.

Remark 1: By the existential completeness lemma there is an α_{2}-name \begin{matrix}p \\ ∼ \end{matrix}_{2} for a condition in P_{β} such that ∥−_{α_{2}} φ(\begin{matrix}p \\ ∼ \end{matrix}_{2}).

Remark 2: The P_{α_{1}}-name \begin{matrix}p \\ ∼ \end{matrix}_{1} corresponds naturally to a P_{α_{2}}-name, which we also call \begin{matrix}p \\ ∼ \end{matrix}_{1}. In other words, we may wlog assume that α_{1} = α_{2}.

Proof of the lemma: Assume 1_{P_{α_{2}}} { \atop ∦−}_{α_{2}} ∃p_{2} φ(p_{2}), then there exists a condition r_{2} ∈ P_{α_{2}} such that

r_{2}∥−_{α_{2}} there is no p_{2} satisfying (1)–(3).

We may assume that r_{2} decides what \begin{matrix}p \\ ∼ \end{matrix}_{1} is, (i.e. r_{2} ∥−_{α_{2}}“\begin{matrix}p \\ ∼ \end{matrix}_{1} = p̂_{1}” for some

p_{1} ∈ V), and r_{2} also decides whether p_{1}↾α_{2} ∈ G_{α_{2}}.

Case 1: r_{2} ∥−_{α_{2}} p_{1}↾α_{2} ∉ G_{α_{2}}:

But then (3) is true for any p_{2}, so r_{2} ∥−_{α_{2}} there is no p_{2} satisfying (1)–(2).

which is a contradiction since D is dense open.

Case 2: r_{2} ∥−_{α_{2}} p_{1}↾α_{2} ∈ G_{α_{2}}. We may assume r_{2} ≤_{α_{2}} p_{1}↾α_{2}. Now let r := p_{1}∧r_{2} =

r_{2} ∪ p_{1}↾[α_{2}, β) ≤ p_{1}, and find p_{2} ∈ D, p_{2} ≤_{β} r. Then

p_{2}↾α_{2} ∥−_{α_{2}} p_{2} satisfies (1)–(3),

again a contradiction, because p_{2}↾α_{2} ≤ r_{2}.

Induction Lemma 3.18: Let Q̄ := ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ be a countable support iteration, and assume that for all α < ε, ∥−_{α}“\begin{matrix}Q \\ ∼ \end{matrix}_{α} is a proper forcing notion.” Let

N ≺ H(χ) be a countable model containing Q̄. Then:

For all β ∈ N ∩ ε:

For all α ∈ N ∩ β, all \begin{matrix}p \\ ∼ \end{matrix} ∈ N: Assume \begin{matrix}p \\ ∼ \end{matrix} is a P_{α}-name for a condition in P_{β}, and

(a) q ∈ P_{α}

(b) q is (P_{α}, N)-generic.

(c) q ∥−_{α} \begin{matrix}p \\ ∼ \end{matrix}↾α ∈ G_{α} ∩ N

Then there is a condition q^{+}:

(a)^{+} q^{+} ∈ P_{β}, q^{+}↾α = q (b)^{+} q^{+} is (P_{β}, N)-generic

(c)^{+} q^{+}∥−_{β}\begin{matrix}p \\ ∼ \end{matrix}↾β ∈ G_{β} ∩ N

The proof is by induction on β.

Successor step:

Let β = β′ + 1. Since we can first use the induction hypothesis on α, β′ to extend q to a condition q′ ∈ P_{β′} satisfying the appropriate version of (a)–(c), we may simplify the notation by assuming β = α + 1.

Since q ∥−_{α} N[G_{α}] ≺ H(χ)^{V [G_{α}]}, we also have

q ∥−_{α} “there is a (Q_{α}, N[G_{α}])-generic condition ≤ \begin{matrix}p \\ ∼ \end{matrix}(α)”

By “existential completeness”, there is a P_{α}-name q^{+}(α) for it. By 3.16, we are done.

Limit step:

Let β ∈ N be a limit ordinal, δ := sup(β ∩ N) = ⋃_{n} α_{n}, α = α_{0} < α_{1} < ⋯, α_{n} ∈ N. Let ⟨D_{n} : n ∈ ω⟩ enumerate all dense subsets of P_{β} that are in N.

First we will define a sequence ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n} : n ∈ ω⟩, \begin{matrix}p \\ ∼ \end{matrix}_{n} ∈ N, \begin{matrix}p \\ ∼ \end{matrix}_{0} = \begin{matrix}p \\ ∼ \end{matrix} such that the following will hold:

(0) \begin{matrix}p \\ ∼ \end{matrix}_{n} is a P_{α_{n}}-name for a condition in P_{β}

(1) ∥−_{α_{n+1}} \begin{matrix}p \\ ∼ \end{matrix}_{n+1} ≤ \begin{matrix}p \\ ∼ \end{matrix}_{n}

(2) ∥−_{α_{n+1}} \begin{matrix}p \\ ∼ \end{matrix}_{n+1} ∈ D_{n}.

(3) ∥−_{α_{n+1}} “If \begin{matrix}p \\ ∼ \end{matrix}_{n}↾α_{n+1} ∈ G_{α_{n+1}} then \begin{matrix}p \\ ∼ \end{matrix}_{n+1}↾α_{n+1} ∈ G_{α_{n+1}}”.

For each n we thus get a name \begin{matrix}p \\ ∼ \end{matrix}_{n} that is in N. For each n we can use the “preliminary lemma” (and remark 1 before its proof) in N to obtain \begin{matrix}p \\ ∼ \end{matrix}_{n+1}.

Now we define a sequence ⟨q_{n} : n ∈ ω⟩, q_{n} ∈ P_{α_{n}}, q = q_{0} ⊆ q_{1} ⊆ ⋯, q_{n+1}↾α_{n} = q_{n}, and q_{n} satisfies (a), (b), (c) (if we write q_{n} for q, p_{n} for p, α_{n} for α.)

q_{n+1} = q_{n}^{+} can be obtained by the induction hypothesis, applied to α_{n}, α_{n+1}, and \begin{matrix}p \\ ∼ \end{matrix}_{n}↾α_{n+1}. By (c)^{+} we know

q_{n}^{+}∥−\begin{matrix}p \\ ∼ \end{matrix}_{n}↾α_{n+1} ∈ G_{α_{n+1}} ∩ N

Hence by (3) we have

q_{n+1}∥−\begin{matrix}p \\ ∼ \end{matrix}_{n+1}↾α_{n+1} ∈ G_{α_{n+1}} ∩ N

Since q_{n+1}↾α_{n} = q_{n}, q = lim q_{n} exists and is ≤ q_{n} for all n.

We have to show that q∥−p ∈ G_{β} ∩ N and that q is generic. Let G_{β} be a generic filter containing q. We will write p_{n} for \begin{matrix}p \\ ∼ \end{matrix}_{n}[G_{α_{n}}]. (Note that p_{n} ∈ N, because q_{n} was N-generic and q_{n} ∈ G_{α_{n}}.)

Since q_{n} ≥ q ∈ G_{β}, we have p_{n}↾α_{n} ∈ G_{α_{n}} ∩ N and N ⊧ p_{n} ≤ p_{n−1} ≤ ⋯ ≤ p_{0}. Hence p↾α_{n} ∈ G_{α_{n}} ∩ N for all n, and so by 1.17, p↾δ ∈ G_{δ} ∩ N. As dom(p) ⊆ δ, p↾δ = p, so p ∈ G_{β}. Similarly, p_{n} ∈ G_{β} for all n.

Consider a dense set D_{n} ⊆ P_{β}. Since q_{n+1}∥−“\begin{matrix}p \\ ∼ \end{matrix}_{n+1} ∈ D_{n},” we have p_{n+1} ∈ G_{β} ∩ D_{n} ∩ N.

Hence q is generic.

Corollary 3.19: Let ⟨P_{α}, \begin{matrix}Q \\ ∼ \end{matrix}_{α} : α < ε⟩ be a countable support iteration of proper forcing notions Q_{α} (as in 3.18). Then P_{ε} (the countable support limit of this iteration) is proper.

Proof: Apply 3.18 with α = 0, β = ε.

4 Quotient forcing

Let ⟨P_{α}, Q_{α} : α < ε⟩ be either a countable support iteration of proper forcing notions, or a finite support iteration. Definition 4.1: We define P_{ε}/G_{α} to be a P_{α}-name for a forcing notion such that

∥−_{α}P_{ε}/G_{α} = {p ∈ P̂_{ε} : p↾α ∈ G_{α}}

For p, q ∈ P_{ε}/G_{α} we let p ≥_{P_{ε}/G_{α}} q iff p ≥_{P_{ε}} q.

Notation 4.2: We write ≤_{αε}, ∥−_{αε}, …, for ≤_{P_{ε}/G_{α}}, ∥−_{P_{ε}/G_{α}}, etc.

Fact 4.3: 1_{P_{α}} forces the following:

(1) P_{ε}/G_{α} ⊆ P̂_{ε}.

(2) If p, q ∈ P_{ε}/G_{α}, then p ≤_{αε} q ⇔ p ≤_{ε} q.

(3) If p, q ∈ P_{ε}/G_{α}, then p ⊥_{ε} q implies p ⊥_{αε} q. (However, note that p ⊥_{αε} q does not necessarily imply p ⊥_{ε} q!)

(4) If A ∈ V, A ⊆ P_{ε} is a maximal antichain, then A ∩ P_{ε}/G_{α} is a maximal antichain in P_{ε}/G_{α}.

(1) and (2) are true by definition.

For (3), note that any r ≤_{αε} p, q would also satisfy r ≤_{ε} p, q.

Proof of (4): By (3), ∥−_{α}“A∩P_{ε}/G_{α} is an antichain.” Assume that the conclusion is not true, then we can find a condition r ∈ P_{α} and a condition p ∈ P_{ε} such that

r∥−_{α}“p ∈ P_{ε}/G_{α}, but for all s ∈ A ∩ P_{ε}/G_{α} there is no t ∈ P_{ε}/G_{α} extending both p and s.”

Since r∥−“p↾α ∈ G_{α},” r and p↾α are compatible, so we may wlog assume r ≤ p↾α. Let p̄:= p ∧ r, then p̄ ≤ p and p̄↾α ≤ r. So

p̄↾α∥−_{α}“p̄ ∈ P_{ε}/G_{α}, and for all s ∈ A ∩ P_{ε}/G_{α} there is no t ∈ P_{ε}/G_{α} extending p̄ and s.”

Let s ∈ A be compatible with p̄, and let t ≤ s, p̄. Then t↾α∥−t ∈ P_{ε}/G_{α} & t ≤ s & t ≤ p̄, a contradiction, because t↾α ≤ p̄↾α. Fact 4.4: Assume G_{α} ⊆ P_{α} is generic over V, and G_{αε} ⊆ P_{ε}/G_{α} is generic over V [G_{α}]. Then

(1) G_{αε} ⊆ P_{ε} is a filter.

(2) G_{αε} is generic for P_{ε} over V.

(3) G_{αε} ⊇ G_{α}.

(4) G_{α} = G_{αε} ∩ P_{α}.

Proof: (1) G_{αε} ⊆ P_{ε} follows from P_{ε}/G_{α} ⊆ P_{ε}. Any two conditions p, q in G_{αε} have a common extension in G_{αε}, i.e., there is a r ∈ G_{αε}, p ≥_{αε} r, q ≥_{αε} r. Hence p ≥_{ε} r, q ≥_{ε} r.

(2): Let A ⊆ P_{ε} be a maximal antichain in V. Then in V_{α}, A ∩ P_{ε}/G_{α} is a maximal antichain by 4.3(3). So G_{αε} ∩ (A ∩ P_{ε}/G_{α}) and thus also G_{αε} ∩ A are nonempty.

Proof of (3): Assume p ∈ G_{α} but p ∉ G_{αε}. Then there is a condition q ∈ G_{αε} such that q ⊥_{αε} p. As p↾α, q↾α ∈ G_{α}, there is a condition r ∈ G_{α} such that r ≤ p↾α, q↾α. Then (q ∧ r)↾α = r ≤_{α} p, so q ∧ r ≤_{ε} p. We also have q ∧ r ≤ q, and q ∧ r ∈ P_{ε}/G_{α}. This contradicts q ⊥_{αε} p.

(4): “⊆” follows from (3), and if p ∈ G_{αε}∩P_{α}, then p ∈ P_{ε}/G_{α}, so p = p↾α ∈ G_{α}.

(Conversely, if G_{ε} ⊆ P_{ε} is generic over V, and G_{α} := G_{ε} ∩P_{α}, then G_{α} is generic for over V, and G_{ε} ⊆ P_{ε}/G_{α} is generic over V [G_{α}].)

So we can write G_{ε} for G_{αε}.

Fact 4.5: The map i : P_{ε} → P_{α} ∗ (P_{ε}/G_{α}), defined by

i(p) = ⟨p↾α, p̂⟩

is a dense embedding. Hence, forcing with P_{ε} amounts to the same as first forcing with P_{α}, and then with the “quotient forcing” P_{ε}/G_{α}.

Proof: Note that p↾α∥−_{α}“p ∈ P_{ε}/G_{α}”, so rng(i) ⊆ P_{α} ∗ P_{ε}/G_{α}. i preserves the ordering, as p ≥ q clearly implies q↾α∥−p̂ ≥ q̂.

To show that i is dense, let ⟨r, \begin{matrix}s \\ ∼ \end{matrix}⟩ ∈ P_{α} ∗ P_{ε}/G_{α}. So r∥−_{α}\begin{matrix}s \\ ∼ \end{matrix}↾α ∈ G_{α}. Since r∥−\begin{matrix}s \\ ∼ \end{matrix} ∈ P_{ε}, we can find a stronger condition r_{0} ∈ P_{α} and a condition s_{0} ∈ P_{ε} such that r_{0}∥−\begin{matrix}s \\ ∼ \end{matrix} = s_{0}. As r_{0}∥−_{α}s_{0}↾α ∈ G_{α}, r_{0} and s_{0}↾α are compatible, say r_{1} ≤ r_{0}, s_{0}↾α.

Let p := s_{0} ∧ r_{1}. Then p ∈ P_{ε}, p ≤ s_{0}, so ⟨r, \begin{matrix}s \\ ∼ \end{matrix}⟩ ≥ ⟨r_{0}, \begin{matrix}s \\ ∼ \end{matrix}⟩ ≥ ⟨r_{0}, ŝ_{0}⟩ ≥ ⟨r_{1}, ŝ_{0}⟩ ≥

⟨p↾α, p̂⟩.

How can we describe the forcing notion P_{ε}/G_{α} without explicitly mentioning P_{ε}? For example, comparing the forcing notions P_{α} and P_{α+1}, we have

P_{α+1} ≈ P_{α} ∗ Q_{α}

and also

P_{α+1} ≈ P_{α} ∗ (P_{α+1}/G_{α})

Similarly,

P_{α+2} ≈ P_{α} ∗ Q_{α} ∗ Q_{α+1}

and also

P_{α+2} ≈ P_{α} ∗ (P_{α+2}/G_{α})

This suggests that the forcing notions P_{α+β}/G_{α} are isomorphic to iterations of length β. Theorem 4.6 below shows that this is indeed the case. Theorem 4.6: Assume ⟨P_{α}, Q_{α} : α < ε⟩ is a countable support iteration of proper forcing notions, or a finite support iteration. Let α + β = ε. Then there exists a P_{α} name ⟨P̄_{γ}, Q̄_{γ} : γ < β⟩ of a countable/finite support iteration of length β, such that

∥−_{α}∀γ ≤ β : P̄_{β} ≈ P_{ε}/G_{α}

Proof: We will fix α, and proceed by induction on β. We will work in V [G_{α}], where G_{α} ⊆ P_{α} is generic over V.

By induction on β, we will (1) define a countable/finite support iteration ⟨P̄_{γ}, Q̄_{γ} : γ < β⟩

(2) define a map i_{αβ} from P_{α+β}/G_{α} into P̄_{β}

(3) prove that i_{αβ} is a dense* embedding. Case 1: β = 0

(1) ⟨P̄_{γ}, Q̄_{γ} : γ < β⟩ is the empty sequence, and P̄_{β} = P_{0} = {∅}

(2) i_{αβ} is the constant map.

(3) Any two conditions in P_{α+0}/G_{α} are compatible, as are any two

conditions in P̄_{0}. p_{1} ≤^{∗} p_{2} is true for all p_{1}, p_{2} ∈ P_{α}/G_{α}. Case 2: β = β′ + 1

(1) We have to define Q̄_{β′}.

In V, \begin{matrix}Q \\ ∼ \end{matrix}_{α+β′} is a P_{α+β′}-name for a forcing notion, which can be translated into a P_{α}-name for a P_{α+β′}/G_{α}-name. So in V [G_{α}], \begin{matrix}Q \\ ∼ \end{matrix}_{α+β′} is a P_{α+β′}/G_{α}-name, which by induction hypothesis can

be translated to a P̄_{β′}-name Q̄ ; ∼_{β′}.

(2) For p ∈ P_{α+β}/G_{α}, we define i_{αβ}(p) ∈ P̄_{β} by defining i_{αβ}(p)↾β′

and i_{αβ}(p)(β′):

i_{αβ}(p)↾β′ = i_{αβ}′(p↾(α + β′))

If p(α + β′) is undefined, we let i_{αβ}(p)(β′) be undefined. Otherwise, p(α + β′) is (in V [G_{α}]) a P_{α+β′}/G_{α}-name for an element of Q_{α+β′}, which can be translated to a P̄_{β′}-name for an element of Q̄_{β′}. This will be i_{αβ}(p)(β′). Or, more sloppily: i_{αβ}(p)(β′ = p(α + β′). Note that i_{αβ} extends i_{αβ}′.

(3) To see that i_{αβ} is an embedding, check that for p_{1}, p_{2} ∈ P_{α+β},

\begin{aligned}p_{1} ≥^{∗} p_{2} iff & p_{1}↾α+β′ ≥^{∗} p_{2}↾α+β′ & \\ & & p_{2}↾α+β′∥−p_{1}(α+β′) ≥^{∗} p_{2}(α+β′) \end{aligned}

which by induction hypothesis and using translation functions is equivalent to

\begin{aligned}p_{1} ≥^{∗} p_{2} iff & i_{αβ}(p_{1})↾β′ ≥^{∗} i_{αβ}(p_{2})↾β′ & \\ & & i_{αβ}(p_{2})↾β′∥−i_{αβ}(p_{1})(β′) ≥^{∗} i_{αβ}(p_{2})(β′) \end{aligned}

i.e., i_{αβ}(p_{1}) ≥^{∗} i_{αβ}(p_{2}). It is also easy to see that i_{αβ} is dense*.

Case 3: β is a limit ordinal.

(1) We define P̄_{β} as the (countable/finite support) limit.

(2) We let i_{αβ}(p) := ⋃_{γ<β} i_{αγ}.

(3) As before, we can show p ≥^{∗} q iff ∀γ < β, p↾α + γ ≥^{∗} q↾α + γ iff

∀γ < β i_{αβ}(p)↾γ ≤^{∗} etc. The crucial point is the density condition 0.8(2). So let p̄ ∈ P̄_{β}.

First we will deal with the case of a finite support iteration. So p̄ ∈ P̄_{γ}, for some γ < β. By induction hypothesis we can find p ∈ P_{α+γ}/G_{α} such that

i_{αγ}(p) ≤∗ ; P̄_{γ} p̄.

Then we also have p ∈ P_{α+β}/G_{α}, and i_{αβ}(p) = i_{αγ}(p) ≤P̄_{β} p̄.

Now assume that we are working with a countable support iteration of proper forcing notions. (If β has cofinality > ω in V [G_{α}], we can repeat the previous proof verbatim.)

In general, let p̄ ∈ P̄_{β}. In V [G_{α}], dom(p) is a countable set of ordinals, whichis not necessarily an element of V.

Here we can use the fact that P_{α} is proper: By 3.13, dom(p̄) is covered by a countable set of ordinals that itself lies in V. So there is a countable set F ∈ V such that dom(p̄) ⊆ {γ : α + γ ∈ F }.

Find a name p̄ ; ∼ ∈ V for p̄. From p̄ ; ∼ and F we can define a condition p ∈ P_{α+β}, p ∈ V as follows:

dom(p) = F.

If α + γ ∈ F, then let p(α + γ) be a name such that (in V)

∥− α+γ “If p̄ ; ∼(γ) ∈ Q_{α+γ}, then p(α + γ) = p̄ ; ∼(γ)

otherwise p(α + γ) = 1_{Q_{α+γ}}.

(We use the translation function here.) Because of F we know that p is indeed a function with countable domain. So

p ∈ P_{β}, and in fact ∥−_{α}p ∈ P_{β}/G_{α}.

Now we have to show that i_{αβ}(p) ≤ p̄. We have to prove by induction on γ that

∀γ ≤ β i_{αβ}(p)↾γ = i_{αβ}(p↾α + γ) ≤^{∗} p̄↾γ  (∗)

We leave this as an exercise to the reader (use 1.7 and 1.8).

5 A general preservation theorem for countable support iteration

Context 5.1: We will consider functions f from ω to ω. In applications, these functions may actually be from ω to HF (the hereditarily finite sets), or from ω^{<ω} to ω^{<ω}, etc. Since we can trivially (in a primitive recursive, absolute, …) way code such functions by functions in ^{ω}ω, all results from this section apply also to functions in ^{ω}HF, etc.

We fix a closed set C ⊆ ^{ω}ω. There is a tree T ⊆ ω^{<ω} such that

C = {f ∈ ^{ω}ω : ∀n f↾n ∈ T }.

When we work in a universe V_{1}, we write C for the set {f ∈ ^{ω}ω ∩ V_{1} : ∀n f↾n ∈ T }, i.e., we regard C not as a set per se, but as a formula defining a certain (closed) set.

Typical examples are C = ^{ω}ω, or C = {f ∈ ^{ω}ω : ∀nf(n) < H(n)} for some H ∈ ωω.

⟨⊏_{n}: n ∈ ω⟩ is an increasing sequence of two place relations on ^{ω}ω. In general, we do not assume that these relations are transitive.

⊏_{n} will always be given by an arithmetical definition. Again, we consider ⊏_{n} not as a set in itself, but rather as a definition of a certain arithmetical set, so if ⊏_{n} is defined in a universe V_{0}, f ⊏_{n} g makes sense even for f, g ∉ V_{0}. These definitions will be absolute between any two ∈-models, as the formula defining ⊏ is arithmetical.

(We do not require that the ⊏_{n} are uniformly arithmetical, i.e., each ⊏_{n} may be defined by a different formula. However, in all our applications there f ⊏_{n} g is expressed by a single formula φ(f, g, n).)

We let ⊏= ⋃_{n} ⊏_{n}.

A typical example is given by f ⊏_{n} g iff ∀k ≥ n f(n) < g(n). Then we have f ⊏ g iff f ≤^{∗} g. (Actually, this example is not so “typical” since here the relations ⊏_{n} are all transitive, which will not be true in general.)

Definition 5.2: We will only consider forcing notions that add a real, or at least introduce a new ω-sequence of ordinals. If p∥−\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix} : ω → Ord, we say that p decides

\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix}↾n, if for some t : n → Ord, p∥−\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix}↾n = t.

Note that if ∥−_{Q}\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix} : ω → Ord and ∥−_{Q}\begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix} ∉ V, then letting

E_{n} = \{p ∈ Q : p decides \begin{matrix}τ \\ ∼ \end{matrix} \begin{matrix}τ \\ ∼ \end{matrix}↾n\},

E_{n} is a dense open subset of Q and ⋂_{n} E_{n} is empty. Assumption 5.3: We assume that for all countable sets a ⊆ ^{ω}ω there is a g such that ∀f ∈ a ∩ C f ⊏ g.

We also assume that for every g and every n the set {f : f ⊏_{n} g} is closed.

Definition 5.4: We say that g (⊏, C)-covers) N if for all f ∈ N ∩ C we have f ⊏ g. (Usually we think of N as a countable elementary submodel of some H(χ).) When C and/or ⊏ are clear, we may just say “g covers N”.

We let χ be a “large enough” regular cardinal as in 3.5. We will consider countable elementary submodels of ⟨H(χ), ∈⟩.

(The notions we will define below will depend on χ, but a careful examination (which we do will not carry out here) shows that this dependence is only apparent.) Definition 5.5: We say that the forcing notion Q almost preserves ⊏, if

WHENEVER N ≺ (H(χ), ∈) is a countable model containing Q, C and ⊏, g covers N, p ∈ Q ∩ N,

THEN there exists an N-generic condition q stronger than p, such that q∥−“g covers N[G].”

Fact 5.6: If Q almost preserves ⊏, then ∥−_{Q}“∀f ∈ V [G] ∃g ∈ V f ⊏ g”. Proof: Assume not, so there is a condition p and a name \begin{matrix}f \\ ∼ \end{matrix} such that

p∥−there is no g ∈ V such that \begin{matrix}f \\ ∼ \end{matrix} ⊏ g.

Let N ≺ H(χ) be a countable model containing \begin{matrix}f \\ ∼ \end{matrix} and p, and let g (∈ V) cover N. Then we can find a condition q ≤ p forcing \begin{matrix}f \\ ∼ \end{matrix} ⊏ g, a contradiction.

Example 5.7: Let C = ^{ω}ω, and let (for f, g ∈ ^{ω}ω)

f ⊏_{n} g ⇔ ∀k ≥ n f(k) < g(k)

and f ⊏ g iff for some n, f ⊏_{n} g. All the ⊏_{n} are closed and transitive, and clearly for every countable set a ⊆ ^{ω}ω there is a g such that ∀f ∈ a f ⊏ g.

A forcing notion Q is called ^{ω}ω-bounding iff

∥−_{Q} ∀f ∈ ^{ω}ω ∩ V [G] ∃g ∈ ^{ω}ω ∩ V f ⊏ g

Fact: A proper forcing notion Q almost preserves ⊏ iff it is ^{ω}ω-bounding. (We will see later that we can omit the “almost” from this statement.)

Proof of the fact: If Q preserves ⊏, then Q is ^{ω}ω-bounding by 5.6. Conversely, assume that Q is ^{ω}ω-bounding, let N ≺ H(χ), and assume that g covers N. Then ∥−∀f ∈ N[G] ∃f′ ∈ V : f ⊏ f′, so any N-generic condition q forces ∀f ∈ N[G]∃f′ ∈ N : f ⊏ f′. Since for all f′ ∈ N, f′ ⊏ g (and ⊏ is absolute and transitive), for any N-generic q we have

q∥−∀f ∈ N[G] : f ⊏ g

So this definition 5.5 achieves exactly what we want. However, it is not clear whether this condition by itself is preserved under iteration. Before we give the definition we will actually use, we have recall the concept of “interpretation”: Definition 5.8: Assume Q is a forcing notion, \begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1} are Q-names of functions in C, f_{0}^{∗}, …, f_{k−1}^{∗} are functions in ^{ω}ω, ⟨p_{n} : n < ω⟩ an increasing sequence of conditions. We will write f̄ ; ∼ for ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}⟩ and f̄^{∗} for ⟨f_{0}^{∗}, …, f_{k−1}^{∗}⟩. A

sequence ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k}⟩ will be written as ⟨f̄ ; ∼, \begin{matrix}f \\ ∼ \end{matrix}_{k}⟩.

We say that ⟨p_{n} : n < ω⟩ interprets f̄ ; ∼ as f̄^{∗}, if for all i < k, for all n, p_{n}∥−\begin{matrix}f \\ ∼ \end{matrix}_{i}↾n = f_{i}^{∗}↾n.

(So when we say that ⟨p_{n} : n < ω⟩ interprets f̄ ; ∼ as f̄^{∗}, it is understood that the sequence ⟨p_{n} : n < ω⟩ is increasing, etc.)

Remark 5.9: If ⟨p_{n} : n < ω⟩ interprets \begin{matrix}f \\ ∼ \end{matrix} as f^{∗}, where \begin{matrix}f \\ ∼ \end{matrix} is a name for a function in C, then f^{∗} is a function in C.

Proof: C = {f : ∀n f↾n ∈ T } for some tree T. If f^{∗} ∉ C, then for some n f^{∗}↾n ∉ T, so p_{n}∥−\begin{matrix}f \\ ∼ \end{matrix}↾n ∉ T, a contradiction.

Definition 5.10: We say that the sequence ⟨p_{n} : n < ω⟩ ∈ ^{ω}Q is inconsistent, if there is no condition q such that ∀n q∥−p_{n} ∈ G or equivalently, ∥−_{Q}∃n : p_{n} ∉ G_{Q}. Note that if ⟨D_{Q,n} : n ∈ ω⟩ is a sequence of dense open sets with ⋂_{n} D_{Q,n} = ∅ and ∀n p_{n} ∈ D_{Q,n}, then ⟨p_{n} : n < ω⟩ is inconsistent.

(By our assumption in 5.2, for every forcing notion Q_{α} that we consider there is a sequence ⟨D_{Q,n} : n < ω⟩ as above.)

Definition 5.11: We say that the forcing notion Q preserves (⊏, C), if for some x:

WHENEVER N ≺ H(χ) is a countable model containing Q, x and ⊏, g covers N, p_{0} ∈ Q ∩ N, and whenever ⟨p_{n} : n < ω⟩ ∈ ^{ω}Q ∩ N (an increasing sequence of conditions) interprets f̄ ; ∼ ∈ N as f̄^{∗}, such that for all i < k f_{i}^{∗} ⊏_{n_{i}} g,

THEN there is a condition q ∈ Q such that

(a) q ≤ p.

(b) q is N-generic.

(c) q∥−∀f ∈ N[G] ∩ C f ⊏ g, i.e., q∥−“g covers N[G].”

(d) ∀i < k: q∥−\begin{matrix}f \\ ∼ \end{matrix}_{i} ⊏_{n_{i}} g.

Note that (a)–(c) just say that Q almost preserves ⊏. Also note that (c) is equivalent to

(c’) ∀\begin{matrix}f \\ ∼ \end{matrix} ∈ N: If ∥−_{Q}\begin{matrix}f \\ ∼ \end{matrix} ∈ C, then q∥−\begin{matrix}f \\ ∼ \end{matrix} ⊏ g.

When C is clear from the context (or irrelevant), we may say “Q preserves ⊏” instead of “Q preserves (⊏, C)”.

We call x = x_{Q} the “witness” for the statement “Q preserves ⊏.”

Lemma 5.12: If Q_{0} preserves ⊏, and ∥−_{Q_{0}}“\begin{matrix}Q \\ ∼ \end{matrix}_{1} preserves ⊏”, then Q_{0} ∗ \begin{matrix}Q \\ ∼ \end{matrix}_{1} preserves ⊏.

Idea of the proof: To use the assumptions on Q_{0} and Q_{1}, we have to find a Q_{0}-name f̄ ; ∼′ that “interpolates” between f̄ ; ∼ and f̄ ; ∼^{∗}, i.e., f̄ ; ∼^{∗} is an interpretation of

f̄ ; ∼′ which itself (in V^{Q_{0}}) is an interpretation of f̄ ; ∼.

Vol. 06, 1992  TOOLS FOR YOUR FORCING CONSTRUCTION  335

Proof: Let x_{0} witness that Q_{0} preserves ⊏, and ∥−“\begin{matrix}x \\ ∼ \end{matrix}_{1} witnesses that Q_{1} preserves ⊏”. x := ⟨x_{0}, \begin{matrix}x \\ ∼ \end{matrix}_{1}⟩ will witness that Q_{0} ∗ Q_{1} preserves ⊏.

Assume N, ⟨p_{n} : n < ω⟩, f̄ ; ∼, f̄^{∗}, n_{0}, …, n_{k−1} are as in 5.11, and g covers N. Each p_{n} is of the form ⟨p_{n}(0), p_{n}(1)⟩, where p_{n}(0) ∈ Q_{0} and p_{n}(0)∥−_{Q_{0}}p_{n}(1) ∈ \begin{matrix}Q \\ ∼ \end{matrix}_{1}.

We let p_{−1} := ⟨1_{Q_{0}}, 1_{\begin{matrix}Q \\ ∼ \end{matrix}_{1}}⟩.

Let G(0) ⊆ Q_{0} be generic, then we can define functions f′_{i} in V [G(0)] and conditions p′_{n}(1) in Q_{1} for i < k as follows:

If for all n ∈ ω p_{n}(0) ∈ G(0), then f′_{i} := f_{i}^{∗}, n^{∗} := ω, p′_{n}(1) := p_{n}(1). Otherwise, let n^{∗} := max{n ≥ −1 : p_{n}(0) ∈ G(0)}, and find a sequence ⟨p′_{n}(1) : n ∈ ω⟩ ∈ ^{ω}Q_{1} and functions f′_{i} for i < k such that p′_{0}(1) ≥ p_{n^{∗}}(1) and

⟨p′_{n}(1) : n < ω⟩ interprets f̄ ; ∼ as f̄′  (∗1)

(Note that (∗1) will be true also if n^{∗} = ω.) Then f′_{i}↾n^{∗} = f^{∗}↾n^{∗}: This is clear if n^{∗} = ω. Otherwise, p′_{0}(1)∥−_{Q_{1}}“\begin{matrix}f \\ ∼ \end{matrix}_{i}↾n^{∗} =

f_{i}^{∗}↾n^{∗},” so we also have p′_{n^{∗}}(1)∥−_{Q_{1}} “\begin{matrix}f \\ ∼ \end{matrix}_{i}↾n^{∗} = f_{i}^{∗}↾n^{∗}.”

Furthermore, p′_{n^{∗}}(1)∥− “\begin{matrix}f \\ ∼ \end{matrix}_{i}↾n^{∗} = f′_{i}↾n^{∗}”, so p′_{n^{∗}}(1)∥− “f′_{i}↾n^{∗} = f_{i}^{∗}↾n^{∗}.”

Coming back to V, we can find Q_{0}-names \begin{matrix}f \\ ∼ \end{matrix}′_{i}, \begin{matrix}p \\ ∼ \end{matrix}′_{n}(1), \begin{matrix}n \\ ∼ \end{matrix}^{∗} such that the above is forced. In particular,

p_{n}(0)∥−f′_{i}↾n = f_{i}^{∗}↾n,  (∗0)

as p_{n}(0)∥−\begin{matrix}n \\ ∼ \end{matrix}^{∗} ≥ n.

We can find these names \begin{matrix}f \\ ∼ \end{matrix}′_{i} and \begin{matrix}p \\ ∼ \end{matrix}′_{n} in N. Since Q_{0} preserves ⊏, by (∗0) there is a generic q(0) ∈ Q_{0} such that

(a_{0}) q(0) ≥ p_{0}(0). (b_{0}) q(0) is N-generic (c_{0}) q(0)∥−∀f ∈ N[G(0)] f ⊏ g.

(d_{0}) ∀i < k: q(0)∥−\begin{matrix}f \\ ∼ \end{matrix}′_{i} ⊏_{n_{i}} g.

Working again in V [G(0)], where q(0) ∈ G(0), we note that N[G(0)] is covered by g and n^{∗} ≥ 0, so since Q_{1} preserves ⊏ and by (∗1), we can find a condition q(1) ∈ Q_{1} such that

(a_{1}) q(1) ≤ p′_{0}(1) ≤ p_{0}(1). (b_{1}) q(1) is N[G(1)]-generic (c_{1}) q(1)∥−∀f ∈ N[G(0) ∗ G(1)] f ⊏ g.

(d_{1}) ∀i < k: q(1)∥−\begin{matrix}f \\ ∼ \end{matrix}_{i} ⊏_{n_{i}} g.

We can find a name \begin{matrix}q \\ ∼ \end{matrix}(1) such that the above is forced by q(0). Now we let q = ⟨q(0), \begin{matrix}q \\ ∼ \end{matrix}(1)⟩. Then 5.11(a)–(d) holds.

Recall that P_{β}/G_{α} = {p ∈ P_{β} : p↾α ∈ G_{α}}, and that the map i : P_{β} → P_{α} ∗ P_{β}/G_{α}, defined by i(p) = ⟨p↾α, p⟩ is a dense embedding. So to each P_{β} name \begin{matrix}f \\ ∼ \end{matrix} there is a naturally corresponding P_{α}-name for a P_{β}/G_{α}-name, which we also

call \begin{matrix}f \\ ∼ \end{matrix}.

For the following, assume that ⟨P_{α}, Q_{α} : α < ε⟩ is a countable support iteration ⊏-preserving forcing notions, i.e., for all α < ε,

P_{α}∥−“Q_{α} preserves ⊏(witnessed by \begin{matrix}x \\ ∼ \end{matrix}_{α}.”

Also assume that each Q_{α} adds a new sequence of ordinals.

Our goal is to show that P_{ε} preserves ⊏ (witnessed by x := ⟨\begin{matrix}x \\ ∼ \end{matrix}_{α} : α < ε⟩. So we fix a countable elementary model N containing all relevant information.

Induction Lemma 5.13: Let α ≤ β ≤ ε. Assume that ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n} : n < ω⟩ ∈ N is a sequence of P_{α}-names for conditions in P_{β}/G_{α} such that

(·) ∥−_{α}⟨\begin{matrix}p \\ ∼ \end{matrix}_{n} : n < ω⟩ interprets f̄ ; ∼ as f̄ ; ∼^{∗}

(where f̄ ; ∼ = ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}⟩, f̄ ; ∼^{∗} = ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}^{∗}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}^{∗}⟩, the \begin{matrix}f \\ ∼ \end{matrix}_{i} are P_{β}-names in N,

and the \begin{matrix}f \\ ∼ \end{matrix}_{i}^{∗} are P_{α}-names). Furthermore, assume that q ∈ P_{α}, and \begin{matrix}n \\ ∼ \end{matrix}_{0}, …, \begin{matrix}n \\ ∼ \end{matrix}_{k−1} are P_{α}-names for integers such that for some g:

(a) q∥−_{α}\begin{matrix}p \\ ∼ \end{matrix}_{0}↾α ∈ G_{α}. (This really follows from our assumption that ∥−_{α}\begin{matrix}p \\ ∼ \end{matrix}_{0} ∈

P_{β}/G_{α}.)

(b) q ∈ P_{α} is N-generic.

(c) q∥−∀f ∈ N[G_{α}] f ⊏ g.

(d) ∀i < k: q∥−\begin{matrix}f \\ ∼ \end{matrix}_{i}^{∗} ⊏_{\begin{matrix}n \\ ∼ \end{matrix}_{i}} g.

Then there exists q^{+} ∈ P_{β}, satisfying

(+) q^{+}↾α = q

(a)^{+} q^{+}∥−_{β}\begin{matrix}p \\ ∼ \end{matrix}_{0}↾β ∈ G_{β}.

(b)^{+} q^{+} ∈ P_{β} is N-generic (i.e., q^{+}∥−_{β} “G_{β} ∩ N is generic over N”.)

(c)^{+} q^{+}∥−∀f ∈ N[G_{β}] f ⊏ g.

(d)^{+} ∀i < k: q^{+}∥−\begin{matrix}f \\ ∼ \end{matrix}_{i} ⊏_{\begin{matrix}n \\ ∼ \end{matrix}_{i}} g.

The reason for considering ⊏-preserving forcing notions is the following corollary: Corollary 5.14: If ∀α ∥−_{P_{α}}“Q_{α} preserves ⊏”, then P_{ε} preserves ⊏.

Proof of the corollary: Use the induction lemma with with α = 0, β = ε, q = 1_{P_{0}}. We can find q^{+} ∈ P_{ε} satisfying (a)^{+}–(d)^{+}. Since q^{+}∥−p_{0} ∈ G_{ε}, there is a condition q^{++} ≤ q^{+}, q^{++} ≤ p_{0}. Then q^{++} will satisfy 5.11(a)–(d).

To prove the case “β limit” of the induction lemma, we will need the following lemma. (Note that, as in the preliminary lemma for the proof of preservation of properness, this lemma does not mention properness, ⊏-preservation, etc. — not even countable models.)

Preliminary lemma 5.15: Assume α_{1} < α_{2} ≤ β, ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{1} : n < ω⟩ is a sequence of P_{α_{1}}-names for conditions in P_{β}/G_{α_{1}}. Let D ⊆ P_{β} be a dense open set, j ∈ ω.

Assume ⟨f̄ ; ∼, \begin{matrix}f \\ ∼ \end{matrix}_{k}⟩ = ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k}⟩ are P_{β}-names, and f̄ ; ∼^{1} = ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}^{1}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}^{1}⟩ are

P_{α_{1}}-names, and

(1) ∥−_{α_{1}} ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{1} : n < ω⟩ interprets f̄ ; ∼ as f̄ ; ∼^{1}.

(2) ∥−_{α_{1}}⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{1}(α_{1}) : n < ω⟩ is inconsistent.

THEN there are: P_{α_{2}}-names ⟨f̄ ; ∼^{2}, \begin{matrix}f \\ ∼ \end{matrix}_{k}^{2}⟩ = ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}^{2}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k}^{2}⟩ and a sequence ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{2} : n < ω⟩

of P_{α_{2}}-names for conditions in P_{β}/G_{α_{2}} such that

(1)^{+} ∥−_{α_{2}}“⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{2} : n < ω⟩ interprets ⟨f̄ ; ∼, \begin{matrix}f \\ ∼ \end{matrix}_{k}⟩ as ⟨f̄ ; ∼^{2}, \begin{matrix}f \\ ∼ \end{matrix}_{k}^{2}⟩” and

∥−_{α_{2}}“\begin{matrix}p \\ ∼ \end{matrix}_{0}^{2} decides \begin{matrix}f \\ ∼ \end{matrix}_{0}↾j, …, \begin{matrix}f \\ ∼ \end{matrix}_{k}↾j.”

(2)^{+} ∥−_{α_{2}}⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{2}(α_{2}) : n < ω⟩ is inconsistent.

(3)^{+} ∥−_{α_{1}}⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{1}↾α_{2} : n < ω⟩ interprets f̄ ; ∼^{2} as f̄ ; ∼^{1}.

(4)^{+} ∥−_{α_{2}} If \begin{matrix}p \\ ∼ \end{matrix}_{n}^{1}↾α_{2} ∈ G_{α_{2}}, then \begin{matrix}p \\ ∼ \end{matrix}_{0}^{2} ≤ \begin{matrix}p \\ ∼ \end{matrix}_{n}^{1}.

(5)^{+} ∥−_{α_{2}} \begin{matrix}p \\ ∼ \end{matrix}_{0}^{2} ∈ D.

\begin{matrix}f̄ ; ∼^{1} & p_{n}^{1} & f̄ ; ∼ \\ α_{1} & α_{2} & β \end{matrix}  ⟹ \begin{aligned}& & f̄ ; ∼^{2}, \begin{matrix}f \\ ∼ \end{matrix}_{k}^{2} & p_{n}^{2} & f̄ ; ∼, \begin{matrix}f \\ ∼ \end{matrix}_{k} \\ f̄ ; ∼^{1} & p_{n}^{1}↾α & f̄ ; ∼^{2} \\ α_{1} & & α_{2} & & β \end{aligned}

Proof of the preliminary lemma:

We will work in V [G_{α_{2}}]. We write p_{n}^{1} for \begin{matrix}p \\ ∼ \end{matrix}_{n}^{1}[G_{α_{1}}]. Let p_{−1}^{1} := 1_{P_{β}}. Let n^{∗} be defined by

(i) n^{∗} := max\{n ≥ −1 : \begin{matrix}p \\ ∼ \end{matrix}_{n}^{1}[G_{α_{1}}]↾α_{2} ∈ G_{α_{2}}\}.

(By 5.15(2), not all p_{n}^{1}↾α_{2} can be in G_{α_{2}}.)

Let p_{−1}^{2} = p_{n^{∗}}^{1} ∈ P_{β}/G_{α_{2}}. By assumption on Q_{α}, there is a sequence ⟨E_{n} : n < ω⟩ of dense open subsets of Q_{α_{2}} such that ⋂_{n} E_{n} is empty. Let Ē_{n} := {r ∈ P_{β}/G_{α_{2}} : r(α_{2}) ∈ E_{n}}. Then Ē_{n} is a dense open subset of P_{β}/G_{α_{2}}, and ⋂_{n} Ē_{n} is empty.

For each n, let

D^{n} := D ∩ Ē_{n} ∩ \{r ∈ P_{β}/G_{α_{2}} : r decides \begin{matrix}f \\ ∼ \end{matrix}_{0}↾ max\{n, j\}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k}↾ max\{n, j\}\}.

Then D^{n} is open dense in P_{β}/G_{α_{2}}. By induction we can now find a sequence of conditions p_{n}^{2} in P_{β}/G_{α_{2}} such that for all n ∈ ω:

(ii) p_{n}^{2} in P_{β}/G_{α_{2}} (iii) p_{n−1}^{2} ≥ p_{n}^{2}. (So if p_{n}^{1}↾α_{2} ∈ G_{α_{2}}, then p_{n}^{1} ≥ p_{n^{∗}}^{1} = p_{−1}^{2} ≥ p_{n}^{2}.) (iv) p_{n}^{2} ∈ D^{n}. Since D^{n} ⊆ E_{n}, (iv) implies that ⟨p_{n}^{2}(α_{2}) : n < ω⟩ is inconsistent. By (iii) and (iv), there are functions f_{i}^{2} such that for all i ≤ k, all n:

(v) p_{n}^{2}∥−_{α_{2},β}\begin{matrix}f \\ ∼ \end{matrix}_{i}↾ max\{n, j\} = f_{i}^{2}↾ max\{n, j\}.

Coming back to V, we can find P_{α_{2}}-names \begin{matrix}p \\ ∼ \end{matrix}_{n}^{2}, \begin{matrix}f \\ ∼ \end{matrix}_{i}^{2}, \begin{matrix}n \\ ∼ \end{matrix}^{∗} such that (i)–(v) are forced by the empty condition of P_{α_{2}}.

Note that (v) implies that 5.15(1)^{+} will be satisfied, (iv) implies 5.15(2)^{+} and 5.15(5)^{+}, and (iii) implies 5.15(4)^{+}.

To show 5.15(3)^{+}, let G_{α_{1}} ⊆ P_{α_{1}} be any generic filter, n ∈ ω, i < k. We will

write p_{n}^{1} for \begin{matrix}p \\ ∼ \end{matrix}_{n}^{1}[G_{α_{1}}], f_{i}^{1} for \begin{matrix}f \\ ∼ \end{matrix}_{i}^{1}[G_{α_{1}}].

We claim that

V [G_{α_{1}}] ⊧ p_{n}^{1}↾α_{2}∥−_{α_{1},α_{2}} \begin{matrix}f \\ ∼ \end{matrix}_{i}^{2}↾n = f_{i}^{1}↾n

Proof of the claim: Let H ⊆ P_{α_{2}}/G_{α_{1}} be a V [G_{α_{1}}]-generic filter containing p_{n}^{1}↾α_{2}. Then H ∩ P_{α_{1}} = G_{α_{1}}, and H is generic for P_{α_{2}} over V, so we will write G_{α_{2}}

for H. Let f_{i}^{2} = \begin{matrix}f \\ ∼ \end{matrix}_{i}^{2}[G_{α_{2}}].

We have to check V [G_{α_{2}}] ⊧ f_{i}^{2}↾n = f_{i}^{1}↾n. It is enough to show that

V [G_{α_{2}}] ⊧ p_{n}^{2}∥−_{P_{β}/G_{α_{2}}} f_{i}^{2}↾n = \begin{matrix}f \\ ∼ \end{matrix}_{i}↾n = f_{i}^{1}↾n.

The first equality is clear by the definition of \begin{matrix}f \\ ∼ \end{matrix}_{i}^{2}. To prove the second, let G_{β} be

a V [G_{α_{2}}]-generic filter on P_{β}/G_{α_{2}} containing p_{n}^{2}. Again, G_{β} ⊇ G_{α_{2}} is also generic for P_{β} over V, and it contains p_{n}^{1}. (Remember that p_{n}^{1}↾α_{2} ∈ G_{α_{2}}, so by 5.15(4)^{+},

p_{n}^{2} ≥ p_{n}^{1}.) Hence V [G_{β}] ⊧ \begin{matrix}f \\ ∼ \end{matrix}_{i}[G_{β}]↾n = f_{i}^{1}↾n.

Proof of the induction lemma: We proceed by induction on β. The successor step is similar to the proof of 5.12: Assume ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n} : n ∈ ω⟩ is a sequence of P_{α}-names for conditions in P_{β+1}/G_{α}, in-

terpreting f̄ ; ∼ as f̄ ; ∼^{∗} in V [G_{α}]. (I.e., f̄ ; ∼^{∗} is a P_{α}-name, and f̄ ; ∼ is a P_{β+1}-name which we identify with the corresponding P_{α}-name for a P_{β+1}-name.)

Working in V_{β}, we can define f̄′, n^{∗}, ⟨p′_{n} : n ∈ ω⟩ such that the following hold: · n^{∗} = sup({n ∈ ω : p_{n}↾β ∈ G_{β}} ∪ {−1}) (also n^{∗} = ω is possible) · ⟨p′_{n} : n ∈ ω⟩ is an increasing sequence of conditions in Q_{β}, interpreting f̄ ; ∼ as f̄′.

· If n^{∗} ∈ ω, then p′_{0} ≤ p_{n^{∗}}(β).

· If n^{∗} = ω, then f̄′ = f̄ ; ∼^{∗}.

Coming back to V, we can find P_{β}-names \begin{matrix}n \\ ∼ \end{matrix}^{∗}, etc. Now ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}↾β : n ∈ ω⟩ is a

sequence of names for conditions in P_{β}/G_{α} forced to interpret f⃗ ; ∼′ as f̄ ; ∼^{∗}, and ⟨\begin{matrix}p \\ ∼ \end{matrix}′_{n} : n ∈ ω⟩ is a sequence of P_{β}-names for conditions in Q_{β} forced to interpret

f̄ ; ∼ as f⃗ ; ∼′. So we can use the induction hypothesis on α, β and the assumption on

Q_{β} to obtain q^{+}↾β and q^{+}(β), respectively.

This ends the proof of the successor case.

Let β be a limit ordinal.

Let ⟨D_{n} : n < ω⟩ enumerate all dense open subsets of P_{β} that are in N, where D_{0} = P_{β}. Let δ := sup(N ∩ β), δ = ⋃_{j} α_{j}, α = α_{0} < α_{1} < ⋯, ⟨α_{j} : j ∈ ω⟩ ∈ N. Fix P_{β}-names f̄ ; ∼ = ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}⟩ of functions.

Assume that

(1)_{0}  ∥−_{α}⟨p_{n} : n < ω⟩ ∈ ^{ω}(P_{β}/G_{α}) interprets f̄ ; ∼ as f̄ ; ∼^{∗}

and let for i < k \begin{matrix}n \\ ∼ \end{matrix}_{i} be names of integers such that ∥−_{α}\begin{matrix}f \\ ∼ \end{matrix}_{i} ⊏_{\begin{matrix}n \\ ∼ \end{matrix}_{i}} g.

First claim: wlog we may assume that

(2)_{0}  ∥−_{α}“⟨p_{n}(α) : n < ω⟩ ∈ ^{ω}Q_{α} is inconsistent”

[Proof of first claim: It is enough to find a sequence ⟨p′_{n} : n < ω⟩ satisfying (1)_{0} and (2)_{0} such that for all n, ∥−_{α}p′_{n} ≤^{∗} p_{n}.

To find this sequence, we work in V_{α}. If ⟨p_{n}(α) : n < ω⟩ is inconsistent, we let p′_{n} := p_{n}. Otherwise, let r_{0} ∈ Q_{α} be a condition such that for all n, r_{0} ≤^{∗} p_{n}(α), and let r_{0} ≤ r_{1} ≤ ⋯ be an increasing inconsistent sequence in Q_{α}. Now let p′_{n} := p_{n} ∧ r_{n}, i.e.,

p′_{n}(γ) = \{ \begin{matrix}r_{n} & if γ = α \\ p_{n}(γ) & if γ ≠ α \end{matrix}

Then ⟨p′_{n} : n < ω⟩ satisfies the requirements. This proves the first claim.]

Let ⟨\begin{matrix}f \\ ∼ \end{matrix}_{n} : n < ω⟩ enumerate all names \begin{matrix}f \\ ∼ \end{matrix} in N satisfying ∥−_{β}\begin{matrix}f \\ ∼ \end{matrix} ∈ C.

We will construct sequences ⟨⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{j} : n < ω⟩ : j < ω⟩ and ⟨⟨\begin{matrix}f \\ ∼ \end{matrix}_{i}^{j} : i < k + j⟩ : j < ω⟩ satisfying the following for all j ∈ ω:

(⋆) \begin{matrix}p \\ ∼ \end{matrix}_{n}^{j} is a P_{α_{j}}-name for an element of P_{β}/G_{α_{j}}

(∗) \begin{matrix}f \\ ∼ \end{matrix}_{i}^{j} a P_{α_{j}}-name for an element of ^{ω}ω

(0) ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{0} : n < ω⟩ = ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n} : n < ω⟩, ⟨\begin{matrix}f \\ ∼ \end{matrix}_{i}^{0} : i < k⟩ = ⟨\begin{matrix}f \\ ∼ \end{matrix}_{i} : i < k⟩.

(1)_{j+1} ∥−_{α_{j+1}} “⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{j+1} : n < ω⟩ ∈ ^{ω}(P_{β}/G_{α_{j+1}}) interprets ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k+j}⟩ as ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}^{j+1}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k+j}^{j+1}⟩, and \begin{matrix}p \\ ∼ \end{matrix}_{0}^{j} decides \begin{matrix}f \\ ∼ \end{matrix}_{0}↾j, …, \begin{matrix}f \\ ∼ \end{matrix}_{k+j}↾j”

(2)_{j+1} ∥−_{α_{j+1}}⟨p_{n}^{j+1}(α_{j+1}) : n < ω⟩ is inconsistent.

(3)_{j+1} ∥−_{α_{j}} ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{j}↾α_{j+1} : n < ω⟩ ∈ ^{ω}(P_{α_{j+1}}/G_{α_{j}}) interprets ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}^{j+1}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k+j−1}^{j+1}⟩

as ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}^{j}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k+j−1}^{j}⟩.

(4)_{j+1} ∥−_{α_{j+1}} If \begin{matrix}p \\ ∼ \end{matrix}_{0}^{j}↾α_{j+1} ∈ G_{α_{j+1}}, then \begin{matrix}p \\ ∼ \end{matrix}_{0}^{j+1} ≤ \begin{matrix}p \\ ∼ \end{matrix}_{0}^{j}.

(5)_{j+1} ∥−_{α_{j+1}} \begin{matrix}p \\ ∼ \end{matrix}_{0}^{j+1} ∈ D_{j+1}

Note that the statements (1)_{0} and (2)_{0} mentioned at the beginning of the proof are exactly (1)_{j+1} and (2)_{j+1} for j = −1.

We can obtain the sequences ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{j+1} : n < ω⟩ and ⟨\begin{matrix}f \\ ∼ \end{matrix}_{i}^{j+1} : i < k + j + 1⟩ from the

given sequences ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{j} : n < ω⟩, ⟨\begin{matrix}f \\ ∼ \end{matrix}_{i}^{j} : i < k + j + 1⟩ and D_{j} by applying the prelim-

inary lemma in N, so each ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n}^{j} : n ∈ ω⟩ will be in N.

Now we construct sequences ⟨q_{j} : j < ω⟩ (where each q_{j} is an N-generic condition in P_{α_{j}}) and ⟨\begin{matrix}n \\ ∼ \end{matrix}_{j} : j < ω⟩ (\begin{matrix}n \\ ∼ \end{matrix}_{j} a P_{α_{j}}-name for an integer) satisfying

(•) ∥−_{α_{j+1}}: If ∃n \begin{matrix}f \\ ∼ \end{matrix}_{k+j}^{j} ⊏_{n} g, then \begin{matrix}f \\ ∼ \end{matrix}_{k+j}^{j} ⊏_{\begin{matrix}n \\ ∼ \end{matrix}_{k+j}} g.

(+) q_{j+1}↾α_{j} = q_{j}.

(A) q_{j+1}∥−\begin{matrix}p \\ ∼ \end{matrix}_{0}^{j}↾α_{j+1} ∈ G_{α_{j+1}}.

(B) q_{j+1} ∈ P_{α_{j+1}} is N-generic. (C) q_{j+1}∥−∀f ∈ N[G_{α_{j+1}}] f ⊏ g.

(D) q_{j+1}∥−∀i < k + j + 1 : \begin{matrix}f \\ ∼ \end{matrix}_{i} ⊏_{\begin{matrix}n \\ ∼ \end{matrix}_{i}} g.

We let q_{0} = q. \begin{matrix}n \\ ∼ \end{matrix}_{0}, …, \begin{matrix}n \\ ∼ \end{matrix}_{k−1} are already defined. By assumption (a)–(d) of the induction lemma, (A)–(D) are now satisfied for j = −1.

Given q_{j} and \begin{matrix}n \\ ∼ \end{matrix}_{0}, …, \begin{matrix}n \\ ∼ \end{matrix}_{k+j−1}, we can easily find \begin{matrix}n \\ ∼ \end{matrix}_{j+k} by requirement (•). Now apply the induction assumption to the sequences ⟨p_{n}^{j} : n < ω⟩, ⟨\begin{matrix}f \\ ∼ \end{matrix}_{i} : i < k + j⟩,

⟨\begin{matrix}f \\ ∼ \end{matrix}_{i}^{j} : i < k + j⟩ to get q_{j+1}. This will show that q_{j+1} satisfies (+), (A), (B), (C). The induction assumption also implies that q_{j+1} will satisfy (D) for all i < k + j. Finally, (•) and (C) imply that (D) is also satisfied for i = k + j.

Note that q_{j+1}∥−\begin{matrix}p \\ ∼ \end{matrix}_{0}^{j}↾α_{j+1} ∈ G_{α_{j+1}}, so by (4),

q_{j+1}∥−\begin{matrix}p \\ ∼ \end{matrix}_{0}^{j+1} ≤ \begin{matrix}p \\ ∼ \end{matrix}_{0}^{j}

To conclude the proof of the Induction Lemma, let q = ⋃_{j} q_{j}. Then q ∈ P_{δ} ⊆ P_{β}. We have to check that this works.

Let G_{β} be any generic filter containing q. We write p_{n}^{j} for \begin{matrix}p \\ ∼ \end{matrix}_{n}^{j}[G_{α_{j}}], etc.

Clearly ⟨p_{0}^{j} : j < ω⟩ is an increasing sequence of conditions. First we claim that p_{0}^{j} ∈ D_{j} ∩ N. By (5), p_{0}^{j} ∈ D_{j}, and since q_{j} is generic, p_{0}^{j} = \begin{matrix}p \\ ∼ \end{matrix}_{0}^{j}[G_{α_{j}}] ∈ N.

Next we note that for all k, p_{0}^{k}↾α_{k} ∈ G_{α_{k}}. Since p_{0}^{j} ≥ p_{0}^{k} for j ≤ k, p_{0}^{j}↾α_{k} ∈ G_{α_{k}} for all k ≥ j, hence p_{0}^{j} ∈ G_{δ}. Since p_{0}^{j} ∈ N, dom(p_{0}^{j}) ⊆ δ. So p_{0}^{j} ∈ G_{β}. This shows G_{β} ∩ D_{j} ∩ N ≠ ∅. So q is generic.

For any i, f_{i}^{j}↾j = f_{i}↾j (by the second clause in (1)). So f_{i} = lim_{j<ω} f_{i}^{j}. Since for

all j, f_{i}^{j} ⊏_{n_{i}} g, and {f : f ⊏_{n_{i}} g} is closed, f_{i} ⊏_{n_{i}} g. So g covers N[G_{β}]. In particular, for i < k, f_{i} ⊏_{n_{i}} g, thus showing condition 5.11(d). So we also finished the limit case.

6 Applications

The general strategy for preserving a property in limit stages of a countable support iteration is as follows:

(1) Find a stronger property that can be written as Q∥−∀f ∈ V [G] ∃g ∈ V f ⊏ g for a Σ_{2}^{0} relation ⊏. Let ⊏= ⋃_{n} ⊏_{n}, where each ⊏_{n} is closed.

(2) Prove that all Q_{α} preserve ⟨⊏_{n}: n ∈ ω⟩. Then by the preservation theorem, for all α, P_{α}∥−∀f ∈ V [G_{α}] ∃g ∈ V f ⊏ g,

and hence P_{α} has the property we wanted to preserve. Fact 6.1: If for all g and all n, the set {f : f ⊏_{n} g} is closed and open, then:

Q almost preserves ⊏ ⇔ Q preserves ⊏

Proof: ⇐ is clear.

⇒: Assume that g covers the countable elementary model N, p̄:= ⟨p_{n} : n ∈ ω⟩ is an increasing sequence of conditions interpreting f̄ ; ∼ = (\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}) ∈ N as

f̄^{∗} = (f_{0}^{∗}, …, f_{k−1}^{∗}), and let f_{i}^{∗} ⊏_{n_{i}} g for all i < k.

Fix i < k. Since the set A_{i} := {f : f ⊏_{n_{i}} g} is open, there exists an integer n_{i}^{∗} such that [f_{i}^{∗}↾n_{i}^{∗}] ⊆ A_{i}.

Let n^{∗} := max{n_{i}^{∗} : i < k}. As Q almost preserves ⊏, we can find a generic condition q ≤ p_{n^{∗}}, satisfying 5.11(a)–(c).

For each i, we have q∥−\begin{matrix}f \\ ∼ \end{matrix}_{i}↾n^{∗} = f_{i}^{∗}↾n^{∗}, thus

q∥−\begin{matrix}f \\ ∼ \end{matrix}_{i} ∈ [f_{i}^{∗}↾n_{i}] ⊆ \{f : f ⊏_{n_{i}} g\}

so q also satisfies 5.11(d).

In general, “Q preserves ⊏” and “Q almost preserves ⊏” are not equivalent, as can be seen from the following example: Example 6.2: Let ⊏ = ⟨⊏_{n}: n ∈ ω⟩ be defined by f ⊏_{n} g ⇔ ∀k ≥ n : f(k) < g(k), assume that Q is ^{ω}ω-bounding (and thus preserves ⊏, see 6.5, below).

Let ⊏′_{n}=⊏_{n} for n > 0, and let f ⊏′_{0} g iff for all k, f(k) = 0, and ⊏′= ⋃_{n} ⊏′_{n}. Then for any f, g: f ⊏ g iff f ⊏′ g, hence for any model N:

g ⊏-covers N iff g ⊏′-covers N

342  MARTIN GOLDSTERN  IMCP

and thus: Q almost preserves ⊏ iff Q almost preserves ⊏′.

Claim: If Q adds reals, then Q does not preserve ⊏′. (But if Q is ^{ω}ω-bounding, then Q almost preserves ⊏′.)

Proof: Let \begin{matrix}x \\ ∼ \end{matrix} be a name such that ∥−_{Q}\begin{matrix}x \\ ∼ \end{matrix} ∈ ^{ω}ω & \begin{matrix}x \\ ∼ \end{matrix} ∉ V. Let p̄ := ⟨p_{n} : n ∈ ω⟩ be an increasing sequence of conditions interpreting \begin{matrix}x \\ ∼ \end{matrix}. Then

∥−_{Q}∃n : p_{n} ∉ G

Define a name \begin{matrix}f \\ ∼ \end{matrix} ∈ ^{ω}2 by requiring ∥−_{Q}“\begin{matrix}f \\ ∼ \end{matrix}(n) = 0 ⇔ p_{n} ∈ G_{Q}.” Let f^{∗} be the function with ∀nf^{∗}(n) = 0.

Then: f^{∗} ⊏′_{0} g, and ⟨p_{n} : n ∈ ω⟩ interprets \begin{matrix}f \\ ∼ \end{matrix} as f^{∗}, but ∥−_{Q}∃n\begin{matrix}f \\ ∼ \end{matrix}(n) ≠ 0, thus

∥−_{Q}¬(\begin{matrix}f \\ ∼ \end{matrix} ⊏′_{0} g).

Hence there is no condition satisfying 5.11(d).

Application 1: Preservation of ^{ω}ω-bounding

Definition 6.3: A forcing notion Q is called ^{ω}ω-bounding iff

∥−_{Q} ∀f ∈ ^{ω}ω ∩ V [G] ∃g ∈ ^{ω}ω ∩ V ∀n f(n) < g(n)

There is a natural way to translate this property into the framework of the “preservation theorem” in 5.11: Definition 6.4: We let (for f, g ∈ ^{ω}ω)

f ⊏_{n}^{bound} g ⇔ ∀k ≥ n f(k) < g(k)

This is a closed relation. Letting ⊏^{bound}= ⋃_{n} ⊏_{n}^{bound}, clearly Q is ^{ω}ω-bounding iff

∥−_{Q} ∀f ∈ ^{ω}ω ∩ V [G] ∃g ∈ ^{ω}ω ∩ V f ⊏^{bound} g

Lemma 6.5: A proper forcing notion Q preserves ⊏^{bound} iff it is ^{ω}ω-bounding.

Proof: Assume Q preserves ⊏^{bound}. Then for any name \begin{matrix}f \\ ∼ \end{matrix} and any condition p, let N be a model containing \begin{matrix}f \\ ∼ \end{matrix} and p. Let g cover N. We can find a condition q ≤ p forcing that \begin{matrix}f \\ ∼ \end{matrix} ⊏^{bound} g. Now g is in the ground model, so Q is ^{ω}ω-bounding.

Conversely, assume that Q is ^{ω}ω-bounding, and consider a model N and a sequence ⟨p_{n} : n < ω⟩ as in the hypothesis of 5.11. We also have names \begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}, and functions f_{i}^{∗} such that p_{n}∥−\begin{matrix}f \\ ∼ \end{matrix}_{i}↾n = f_{i}^{∗}↾n for all i < k.

First we note that any N-generic q will force

∀f ∈ N[G] ∃f′ ∈ N ∀n f(n) < f′(n)

Since any such f′ is eventually bounded by g, q∥−∀f ∈ N[G] f ⊏^{bound} g.

We still have to deal with our fixed names \begin{matrix}f \\ ∼ \end{matrix}_{i}. Assume f_{i}^{∗} ⊏_{n_{i}}^{bound} g. For each n we can find a condition p′_{n} ≤ p_{n} and functions f′_{i,n} ∈ N (i < k) such that

p′_{n}∥−∀k \begin{matrix}f \\ ∼ \end{matrix}_{i}(k) < f′_{i,n}(k). Let for all i < k f′_{i} be defined by

f′_{i}(k) = max{f′_{i,n}(k) : n ≤ k} + 1

We can find these sequences ⟨p′_{n} : n < ω⟩ and ⟨f′_{i,n} : i < k, n < ω⟩ in N, so there are n′_{i} such that f′_{i} ⊏_{n′_{i}}^{bound} g, n′_{i} ≥ n_{i}. Let n^{∗} ≥ n′_{i} for all i, and let q ≥ p′_{n^{∗}} be a generic condition. Then

\begin{matrix}q∥− & ∀k ≥ n^{∗} \begin{matrix}f \\ ∼ \end{matrix}_{i}(k) < f′_{i,n^{∗}}(k) < f′_{i}(k) < g(k) \\ q∥− & ∀k ∈ (n_{i}, n^{∗}) \begin{matrix}f \\ ∼ \end{matrix}_{i}(k) = f_{i}^{∗}(k) < g(k) \end{matrix}

so q∥−\begin{matrix}f \\ ∼ \end{matrix}_{i} ⊏_{n_{i}}^{bound} g.

Corollary 6.6: The countable support iteration of proper ^{ω}ω-bounding forcing notions is (proper and) ^{ω}ω-bounding.

Application 2: Preserving outer measure one

Let Ω be the set of clopen subsets of ^{ω}2. Ω is a countable set. We will consider functions f ∈ ^{ω}Ω and functions g ∈ ^{ω}2.

μ(A) is the Lebesgue measure of any measurable set A ⊆ ^{ω}2, and we let μ^{∗}(A) be the outer Lebesgue measure of any set A ⊆ ^{ω}2.

We let

C^{random} := {f ∈ ^{ω}Ω : ∀n ∈ ω μ(f(n)) ≤ 2^{−n}}

This is a closed set (in the product topology of ^{ω}Ω, where Ω is equipped with the discrete topology).

For f ∈ C^{random}, we let

A_{f} := ⋂_{n∈ω} ⋃_{k≥n} f(k)

Fact 6.7:

(1) If f ∈ C^{random}, then A_{f} ⊆ ^{ω}2 is a set of measure zero.

(2) If H ⊆ ^{ω}2 has measure zero, then there is f ∈ C^{random} such that H ⊆ A_{f}.

Proof: (1) follows from μ( ⋃_{k≥n} f(k)) ≤ 2^{−n} + 2^{−n−1} + ⋯ = 2^{−n+1}.

(2): Let Γ : ω × ω → ω be a monotone bijection. For each m, H can be covered by an open set of measure < 2^{−Γ(m,0)}, say

H ⊆ ⋃_{k∈ω}[s_{k}^{m}]  s_{k}^{m} ∈ 2^{<ω}  μ( ⋃_{k∈ω}[s_{k}^{m}]) ≤ 2^{−Γ(m,0)}

For each m, we can find a sequence 0 = k_{0}^{m} < k_{1}^{m} < ⋯ of integers satisfying

μ( ⋃_{k≥k_{j}^{m}} s_{k}^{m}) < 2^{−Γ(m,j)}

for all j ∈ ω.

Define f by

f(Γ(m, j)) = ⋃{[s_{k}^{m}] : k_{j}^{m} ≤ kk_{j+1}^{m}}

This is a finite union of basic clopen sets, hence clopen. Also,

μ(f(Γ(m, j))) ≤ μ( ⋃_{k≥k_{j}^{m}} s_{k}^{m}) ≤ 2^{−Γ(m,j)}

so f ∈ C^{random}.

Note that for all m, ⋃_{j∈ω} f(Γ(m, j)) = ⋃{[s_{k}^{m}] : k ∈ ω} ⊇ H. For all n ∈ ω there is m ∈ ω such that ∀j : Γ(m, j) > n, so

∀n ∃m H ⊆ ⋃_{j∈ω} f(Γ(m, j)) ⊆ ⋃_{k≥m} f(k)

so H ⊆ ⋂_{m∈ω} ⋃_{k≥m} f(k).

Definition 6.8: For f ∈ C^{random}, g ∈ ^{ω}2, n ∈ ω we let f ⊏_{n}^{random} g iff ∀k ≥ n g ∉ f(k).

This is a closed relation since (for fixed k), “g ∈ f(k)” is clopen. Fact 6.9: f ⊏^{random} g iff g ∉ A_{f}.

Fact 6.10: For a countable model N of ZF C:

g ⊏^{random}-covers N ⇔ g is random over N

Proof: Assume that g is random over N. Fix a function f ∈ C^{random}. Then as g is not an element of the null set A_{f}, we have f ⊏^{random} g.

Conversely, assume that ∀f ∈ N f ⊏^{random} g. Then for any measure zero set H in N we can find a sequence f ∈ C^{random} such that H ⊆ A_{f}. Since f ⊏^{random} g, g ∉ A_{f}, so g ∉ H. Fact 6.11: If Q preserves ⊏^{random}, then ∥−_{Q}μ^{∗}(V ∩^{ω}2) = 1.

Proof: Assume p∥−μ(V ∩^{ω}2) = 0. Then there is a name \begin{matrix}f \\ ∼ \end{matrix} such that p∥−\begin{matrix}f \\ ∼ \end{matrix} ∈

C^{random} & V ∩^{ω}2 ⊆ A_{\begin{matrix}f \\ ∼ \end{matrix}}. Take any countable elementary model N containing p and

\begin{matrix}f \\ ∼ \end{matrix}, and let g cover N. Then p∥−g ∈ A_{\begin{matrix}f \\ ∼ \end{matrix}}, but there is a condition q ≤ p, q∥−“g

covers N[G],” so q∥−g ∉ A_{\begin{matrix}f \\ ∼ \end{matrix}}, a contradiction.

This fact justifies the following definition: Definition 6.12: We say that a forcing notion Q “preserves outer measure one” iff Q preserves ⊏^{random}.

Vol. 06, 1992  TOOLS FOR YOUR FORCING CONSTRUCTION  345

In [7], the following property of forcing notions was considered: Definition 6.13: A forcing notion Q satisfies ∗_{4} if for every countable N ≺ H(χ),

if P ∈ N, ⟨p_{n} : n ∈ ω⟩ ∈ N, each p_{n} in P, and ⟨\begin{matrix}A \\ ∼ \end{matrix}_{n} : n ∈ ω⟩ ∈ N, each \begin{matrix}A \\ ∼ \end{matrix}_{n} a P -

name, for every n, p_{n}∥−“\begin{matrix}A \\ ∼ \end{matrix}_{n} is a Borel set and μ(\begin{matrix}A \\ ∼ \end{matrix}_{n}) < ε_{n}” and lim_{n→∞} ε_{n} = 0 and g ∈ ^{ω}2 is random over N then there exists q ∈ P such that

(i) q is (N, P)-generic.

(ii) q∥−“g is random over N[G_{Q}].”

(iii) there exists n such that q ≤ p_{n} and q∥−“x ∉ \begin{matrix}A \\ ∼ \end{matrix}_{n}.”

[7] also showed that Laver forcing has the property ∗_{4}. Lemma 6.14: If Q satisfies ∗_{4}, then Q “preserves outer measure one.”

Proof: Let ⟨p_{n} : n ∈ ω⟩, f̄ ; ∼ = ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}⟩ be as in 5.11, let N be a countable elementary model, and let g cover N. Then g is random over N.

Define Q-names \begin{matrix}A \\ ∼ \end{matrix}_{n} such that

∥−\begin{matrix}A \\ ∼ \end{matrix}_{n} = ⋃_{i<k} ⋃_{m≥n} \begin{matrix}f \\ ∼ \end{matrix}_{i}(m)

Then ∥−μ(A_{n}) ≤ ε_{n}, where ε_{n} := k · 2^{−n+1}.

Let ⟨p_{n} : n ∈ ω⟩ interpret f̄ ; ∼ as f̄^{∗}, and assume f_{i}^{∗} ⊏_{n_{i}}^{random} g. Let n^{∗} := max{n_{i} : i < k}. Applying ∗_{4} to the sequence ⟨p_{n} : n^{∗} ≤ n < ω⟩, we can find an n^{∗∗} ≥ n^{∗} and a generic condition q ≤ p_{n^{∗∗}} such that q∥−“g is random over N[G],” and q∥−g ∉ A_{n^{∗∗}}.

As g is random over N[G] iff g covers N[G], q satisfies 5.11(a)–(c).

For each i < k we have q∥−g ∉ ⋃_{m≥n^{∗∗}} \begin{matrix}f \\ ∼ \end{matrix}_{i}(m). So to get 5.11(d), it suffices to show

q∥−g ∉ ⋃_{n_{i}≤m<n^{∗∗}} \begin{matrix}f \\ ∼ \end{matrix}_{i}(m)

This follows easily from q ≤ p_{n^{∗∗}}, as

p_{n^{∗∗}}∥−g ∉ ⋃_{n_{i}≤m<n^{∗∗}} f_{i}^{∗}(m) = ⋃_{n_{i}≤m<n^{∗∗}} \begin{matrix}f \\ ∼ \end{matrix}_{i}(m)

Thus (by [7]) the Laver forcing “preserves outer measure one.” Lemma 6.15: The random real forcing notion “preserves outer measure one.”

To prove this lemma, we first show the following claim: Claim 6.16: Let Q be the random real forcing, and let \begin{matrix}A \\ ∼ \end{matrix} be a Q-name for a subset of IR, p ∈ Q, and assume that for some real c, p∥−μ^{∗}(\begin{matrix}A \\ ∼ \end{matrix}) < c. Then, letting

A(p) := \{x ∈ IR : p∥−x ∈ \begin{matrix}A \\ ∼ \end{matrix}\},

we have μ^{∗}(A(p)) ≤ c.

Proof of the claim: Let \begin{matrix}B \\ ∼ \end{matrix} be the name of a Borel set such that p∥−“\begin{matrix}A \\ ∼ \end{matrix} ⊆

\begin{matrix}B \\ ∼ \end{matrix} & μ(\begin{matrix}B \\ ∼ \end{matrix}) < c.

Notation: For any set C ⊆ IR^{2}, t ∈ IR, we let

\begin{matrix}C^{t} := & \{x ∈ IR : (x, t) ∈ C\} \\ C_{t} := & \{y ∈ IR : (t, y) ∈ C\} \end{matrix}

There is a Borel set C ⊆ IR^{2} such that p∥−\begin{matrix}B \\ ∼ \end{matrix} = C_{r}, where r is the canonical name of the random real. Wlog C ⊆ p × IR. In V we have for almost all x μ(C_{x}) ≤ c, so μ(C) ≤ c · μ(p).

Let D := {y : C^{y} =^{∗} p} = {y : p ⊆^{∗} C^{y}}. (We write X ⊆^{∗} Y iff μ(X − Y) = 0.) We claim D = B(p). Proof: We have y ∈ B(p) ⇔ p∥−y ∈ B ⇔ p∥−y ∈ C_{r} ⇔

p∥−(r, y) ∈ C ⇔ p∥−r ∈ C^{y} ⇔ μ(p − C^{y}) = 0.

Clearly μ(C) ≥ μ(D) · μ(p), so μ(D) ≤ μ(C) ; μ(p) ≤ c.

This ends the proof of 6.16.

Proof of 6.15:

We first show that Q almost preserves ⊏^{random} and in fact any condition in N is generic and forces “g is random over N[G],” if g is random over N.

Every condition is generic, because Q satisfies ccc. Now assume that { \atop ∦−} “g is random over N[G].” Then for some condition q, and some \begin{matrix}B \\ ∼ \end{matrix} ∈ N, ∥−_{Q}“μ(\begin{matrix}B \\ ∼ \end{matrix}) = 0” and q∥−_{Q}“g ∈ \begin{matrix}B \\ ∼ \end{matrix}.” Let C ∈ N such that ∥−“\begin{matrix}B \\ ∼ \end{matrix} = C_{r}.” So q∥−r ∈ C^{g}, which is impossible, since C is a set in N of measure zero, so C^{g} has measure zero. This shows that random real forcing almost preserves ⊏^{random}, C^{random}.

Assume f_{i}^{∗} ⊏_{n_{i}}^{random} g, p_{n}∥−\begin{matrix}f \\ ∼ \end{matrix}_{i}↾n = f_{i}^{∗}↾n.

In N we can define a sequence ⟨B_{n} : n < ω⟩ as follows:

B_{n} := \{x ∈ IR : p_{n}∥−x ∈ ⋃_{i<k} ( ⋃_{m≥n} \begin{matrix}f \\ ∼ \end{matrix}_{i}(m))\}

By 6.16,

μ^{∗}(B_{n}) ≤ k · ∑_{m≥n} 2^{−m} ≤ 2k ; 2^{n}

Since ⟨B_{n} : n ∈ ω⟩ ∈ N, and g is random over N, g ∉ ⋂_{n} B_{n}. So there exists a n^{∗} ≥ max(n_{0}, …, n_{k−1}) such that g ∉ B_{n}. There exists a condition q ≤ p_{n^{∗}},

q∥−∀i < k : g ∉ ⋃_{m≥n^{∗}} \begin{matrix}f \\ ∼ \end{matrix}_{i}(m)

Since for all i < k,

q∥−g ∉ ⋃_{n_{i}<m<n^{∗}} f_{i}^{∗}(m) = ⋃_{n_{i}<m<n^{∗}} \begin{matrix}f \\ ∼ \end{matrix}_{i}(m)

we have q∥−∀i < k \begin{matrix}f \\ ∼ \end{matrix}_{i} ⊏_{n_{i}}^{random} g.

A similar proof shows that random real forcing satisfies ∗_{4}, proving a conjecture of Miller.

Application 3: Preserving nonmeager sets

We consider functions f from ω^{<ω} to ω^{<ω}, and functions g from ω to ω. Fact 6.17:

(1) For any f : ω^{<ω} → ω^{<ω}, the set

A_{f} := {g : ∀n g↾n^{⌢}f(g↾n) ⊈ g}

is closed nowhere dense.

(2) Conversely, for every closed nowhere dense set H there is a function f such that H ⊆ A_{f}.

Proof: For every n, the set {g : g↾n^{⌢}f(g↾n) ⊆ g} = ⋃_{η∈^{n}ω}[η^{⌢}f(η)] is clopen, so A_{f} is closed. A_{f} cannot contain any basic interval [η], since A_{f} is disjoint from [η^{⌢}f(η)].

Conversely, let H be a closed nowhere dense set in ^{ω}ω. Then as H is closed, there is a tree T ⊆ ω^{<ω} such that H = {f ∈ ^{ω}ω : ∀n f↾n ∈ T }. As H is nowhere dense, for every η ∈ ω^{<ω} there is an extension η^{⌢}f(η) such that H ∩ [η^{⌢}f(η)] = ∅. Definition 6.18: We let C^{Cohen} be the set of all functions from ω^{<ω} to ω^{<ω}.

We define ⊏^{Cohen} by f ⊏_{k}^{Cohen} g iff f : ω^{<ω} → ω^{<ω}.

g : ω → ω.
∃n ≤ k : g↾n^{⌢}f(g↾n) ⊆ g.

Fact 6.19: f ⊏^{Cohen} g iff g ∉ A_{f}.

Fact 6.20: g ⊏^{Cohen}-covers N iff g is Cohen over N.

Proof: Recall that g ∈ ^{ω}2 is Cohen over a model, iff it not contained in any meager set coded in the model, iff it is not contained in any closed nowhere dense set coded in the model.

Assume that g is Cohen over N. Fix a function f ∈ C^{Cohen}. Then as g is Cohen, g is not an element of the meager set A_{f}, so there is an n such that f ⊏_{n}^{Cohen} g.

Conversely, assume that ∀f ∈ N f ⊏^{Cohen} g. Then for any closed nowhere dense set H in N we can find a sequence f ∈ C^{Cohen} such that H ⊆ A_{f}. Since there is some n such that f ⊏_{n}^{Cohen} g, g ∉ H. Fact 6.21: If Q preserves ⊏^{Cohen}, then Q∥−“V ∩^{ω}ω is not meager”.

Proof: Assume p∥−V ∩^{ω}2 is meager. Then there is a name \begin{matrix}f \\ ∼ \end{matrix} such that p∥−V ∩ ω2 ⊆ A_{\begin{matrix}f \\ ∼ \end{matrix}}. Take any countable elementary model N containing p and \begin{matrix}f \\ ∼ \end{matrix}, and let g

cover N. Then p∥−g ∈ A_{f}, a contradiction.

In [11, ch.18], a converse to this theorem is proved:

Theorem 6.22: Assume that (Q, ≤) is a Souslin proper forcing notion (see section 7), and

(∗) ∥−_{Q}V ∩^{ω}ω is not meager

and moreover, Q has property (∗) in any extension of V (by set forcing). Then Q preserves ⊏^{Cohen}.

Example 6.23: Cohen forcing preserves ⊏^{Cohen}.

Proof: Note that each relation ⊏_{n}^{Cohen} is clopen. By 6.1, it is enough to show that Cohen forcing almost preserves ⊏^{Cohen}.

We claim that if g covers N, then any condition forces “g covers N[G].” The proof is exactly the same as the proof in the previous section, with “measure zero” replaced by “meager”, “random” replaced by “Cohen”, etc.

Application 4: The Laver property

From now on until the end of this section, n will be a variable ranging over positive natural numbers. Definition 6.24: For a function h : ω → ω, an “h-cone” is a sequence ⟨A_{m} : m ∈ ω⟩ of finite subsets of ω with |A_{n}| ≤ h(n) for all n > 0.

If h, H : ω → ω, Ā an h-cone, we say that Ā is bounded by H if for all n > 0, A_{n} ⊆ H(n).

Q_{+} is the set of nonnegative rationals. For r ∈ Q_{+}, an r-cone is an h-cone where h(m) = ⌊2^{mr}⌋. (⌊x⌋ is the greatest integer ≤ x.)

A “cone” is a sequence ⟨A_{m} : m ∈ ω⟩ of finite subsets of ω with |A_{n}| ≤ 2^{n} (i.e., a 1-cone).

We say that Ā = ⟨A_{m} : m ∈ ω⟩ “covers” f ∈ ^{ω}ω if for all n > 0 f(n) ∈ A_{n}.

If f ∈ ^{ω}([ω]^{<ω}), we say that Ā covers f iff for all n > 0 f(n) ⊆ A_{n}.

For a function H ∈ ^{ω}ω, we write ∏ H for the set {f ∈ ^{ω}ω : ∀n > 0 f(n) < H(n)}. This is a closed subset of ^{ω}ω. Definition 6.25: A forcing notion Q is said to have the Laver Property iff for every H : ω → ω in V,

(Laver)_{H}  ∥−_{Q}“∀f ∈ ∏ H ∩ V [G] : ∃Ā ∈ V, Ā is a cone covering f

Note that if ∀n > 0 H(n) ≤ H′(n), then ∏ H ⊆ ∏ H′, so (Laver)_{H′} implies (Laver)_{H}. So without loss of generality we may restrict ourselves to some dominating family of functions H, e.g. all increasing functions in ^{ω}(ω − {0}). Fact 6.26: The following are equivalent for any two universes V_{0} ⊆ V_{1}:

(1) For all H ∈ ^{ω}ω ∩ V_{0}: If f ∈ ^{ω}ω ∩ V_{1} is bounded by H, then f is covered by some cone of V_{0}.

(2) For all H ∈ ^{ω}ω ∩ V_{0}, for all functions h_{0} ∈ ^{ω}ω ∩ V_{0} that diverge to infinity:

If f ∈ ^{ω}ω ∩ V_{1} is bounded by H, then f is covered by

some h_{0}-cone of V_{0}.

(3) For all H ∈ ^{ω}ω ∩ V_{0}: For all h_{0}, h_{1} ∈ ^{ω}ω ∩ V_{0}: If for all n,

h_{0}(n) ≤ h_{1}(n), and h_{0}(n) = o(h_{1}(n)), i.e., lim_{n→∞} h_{1}(n) ; h_{0}(n) = ∞,

then for all h_{0}-cones f ∈ V_{1}:

If f is bounded by H, then f is covered by some h_{1}-cone F ∈ V_{0}.

We will give the proof of this fact (a routine computation) below. Definition 6.27: If any/all of these conditions are satisfied, we say that V_{1} has the Laver property over V_{0}.

Note that by (1), a forcing notion Q has the Laver property iff

Q∥−V [G] has the Laver property over V

Notation 6.28: Fix some (recursive) map c from [ω]^{<ω} onto the set of rational numbers in [0,1). If c(x) = r we say “x codes r.”

Definition 6.29: Let

C^{H} := \{f ∈ ^{ω}([ω]^{<ω}) : If f(0) codes r_{0}, then {∀n > 0 : f(n) ⊆ H(n) \atop and |f(n)| ≤ 2^{nr_{0}}\}}

Definition 6.30: Let ⊏^{H} = ⟨⊏_{k}^{H}: k < ω⟩ be defined by f ⊏_{k}^{H} g iff g is a cone bounded by H. f(0) codes some r_{0}. f ∈ C^{H} (so ∀n > 0 |f(n)| ≤ 2^{nr_{0}}) ∀n > k f(n) ⊆ g(n).

We let ⊏^{H}= ⋃_{k} ⊏_{k}^{H}.

Clearly the set {f : f ⊏_{k}^{H} g} is closed for all g, and for any countable set a there is g such that ∀f ∈ a ∩ C^{H} f ⊏^{H} g. Definition 6.31: Let Q be a forcing notion. If H ∈ ^{ω}ω, 0 ≤ r < s, we write (Laver)_{H,r,s} for the statement:

(Laver)_{H,r,s}  ∥−_{Q} “Every r-cone f in V [G] that is bounded by H is covered by some s-cone F ∈ V ”

Thus, (Laver)_{H} from 6.25 is equivalent to (Laver)_{H,0,1}.

Lemma 6.32: Assume Q is a proper forcing notion. Then the following are equiavlent:

(1) Q satisfies the Laver property.

(2) For all H ∈ ^{ω}ω, all 0 ≤ r < s, Q satisfies (Laver)_{H,r,s}.

(3) For all H ∈ ^{ω}ω, all 0 ≤ r < 1, Q satisfies (Laver)_{H,r,1}.

(4) For all H ∈ ^{ω}ω, Q satisfies (Laver)_{H,0,1}.

(5) For all H, Q preserves ⊏^{H}.

(6) For all H, Q almost preserves ⊏^{H}.

Proof: We will show \begin{aligned}(1) & ⇒ & (2) & ⇒ & (3) & ⇒ & (4) & ⇒ & (1). \\ ⇓ & ⇑ \\ (5) & ⇒ & (6) \end{aligned}

(1) ⇒ (2) follows from the characterisation 6.26(3), using the functions h_{0}(n) = 2^{rn}, h_{1}(n) = 2^{sn}.

(2) ⇒ (3) ⇒ (4) ⇒ (1) is clear, and also (5) ⇒ (6) is clear.

(6) ⇒ (3) follows from 5.6. (Note that in (3) we can restrict ourselves to rational numbers r.)

So it remains to show (2) ⇒ (5). Assume that Q satisfies (Laver)_{H,r,s} for all 0 ≤ r < s. Let N be a countable

elementary model, H ∈ N, and assume that g covers N.

Claim: If G is generic over V, and G ∩ N is generic over N, then g covers N[G].

Proof of the claim: Let f ∈ C^{H} ∩ N[G], and let f(0) code r_{0}. Let t_{0} := (r_{0} + 1)/2.

So ∀n > 0 f(n) ⊆ H(n) and |f(n)| ≤ 2^{nr_{0}}. By (Laver)_{H,r_{0},t_{0}}, there is F ∈ V, ∀n > 0 |F (n)| ≤ 2^{nt_{0}} and f(n) ⊆ F (n) ⊆ H(n). Since G ∩ N is generic over N, we can find this F in N. We may assume that F (0) codes t_{0}. (So f ∈ C^{H} ∩ N.)

Since g covers N, there is a k such that F ⊏_{k}^{H} g_{s}. So for all n ≥ k, f(n) ⊆ F (n) ⊆ g_{s}(n). This ends the proof of the claim. So every generic condition will force 5.11(a)–(c). We still have to deal with condition (d) of 5.11. So assume that ⟨p_{n} : n < ω⟩

interprets ⟨\begin{matrix}f \\ ∼ \end{matrix}_{0}, …, \begin{matrix}f \\ ∼ \end{matrix}_{k−1}⟩ as ⟨f_{0}^{∗}, …, f_{k−1}^{∗}⟩, and assume ∀i < k f_{i}^{∗} ⊏_{n_{i}} g. Let

f_{i}^{∗}(0) code r_{i}, and let s_{i}, t_{i} be rationals such that r_{i} < s_{i} < t_{i} < 1.

By (Laver)_{H,r_{i},s_{i}}, for all n > 0 we can find a condition p′_{n} ≥ p_{n} such that for all i < k there exists a function F_{n,i} with F_{n,i}(m) ⊆ H(m) and |F_{n,i}(m)| ≤ 2^{ms_{i}} for all m, and

p′_{n}∥−∀m > 0 \begin{matrix}f \\ ∼ \end{matrix}_{i}(m) ⊆ F_{n,i}(m)

We can find these sequences ⟨p′_{n} : n < ω⟩ and ⟨F_{n,i} : n < ω, i < k⟩ in N. Now define for i < k F_{i} as follows: F_{i}(0) codes t_{i}, and for n > 0 let

F_{i}(n) = ⋃{F_{m,i}(n) : m ≤ 2^{n(t_{i}−s_{i})}}

Then F_{i}(n) ⊆ H(n), and |F_{i}(n)| ≤ 2^{nt_{i}}.

Furthermore, we have F_{m,i}(n) ⊆ F_{i}(n), if m ≤ 2^{n(t_{i}−s_{i})}. Let m_{0} be so large that

(i) For all n ≥ m_{0}, all i < k, 2^{n(t_{i}−s_{i})} > n.

(ii) For all i < k, F_{i} ⊏_{m_{0}}^{H} g. (iii) For all i < k, m_{0} > m_{i} (so f_{i}^{∗} ⊏_{m_{0}}^{H} g.) We now claim that for any generic q ≥ p′_{m_{0}},

q∥−∀i < k \begin{matrix}f \\ ∼ \end{matrix}_{i} ⊏_{n_{i}} g

Proof: Let G be a generic filter containing q, fix some i < k, and let f_{i} = \begin{matrix}f \\ ∼ \end{matrix}_{i}[G]. For n ∈ [n_{i}, m_{0}) we have f_{i}(n) = f_{i}^{∗}(n) because q ≤ p_{m_{0}}, and f_{i}^{∗}(n) ⊆ g(n)

because f_{i}^{∗} ⊏_{n_{i}}^{H} g.

For n ≥ m_{0}, we have f_{i}(n) ⊆ F_{m_{0},i}(n) because p_{m_{0}} ∈ G,

F_{m_{0},i}(n) ⊆ F_{i}(n) because m_{0} ≤ n ≤ 2^{n(t_{i}−s_{i})}, and
F_{i}(n) ⊆ g_{s_{i}}(n) because F_{i} ⊏_{m_{0}}^{H} g.

Hence for all n ≥ n_{i}, f_{i}(n) ⊆ g_{s_{i}}(n), so f_{i} ⊏_{n_{i}} g. Corollary 6.33: The Laver property is preserved under countable support iteration of proper forcing notions.

Proof of 6.26: Clearly (3) ⇒ (2) ⇒ (1). To show that (1) ⇒ (3), fix h_{0}, h_{1} in V_{0} and f in ∏_{n}[ω]^{<h_{0}(n)} ∩ V_{1}. We can find

a function ℓ ∈ ^{ω}ω ∩ V_{0} such that for all n ℓ(⌊log h_{1}(n) ; h_{0}(n)⌋) > n. (log is the logarithm to base 2.)

Let i be a bijection between HF (the hereditarily finite sets) and ω. Let H′(k) = 1 + max{i(η)(k) : |η| ≤ ℓ(k), ∀n η(n) < H(n)} and define f′ ∈ ^{ω}ω

by f′(k) = i(f↾ℓ(k)). Then H′ ∈ V_{0}, and f′(k) < H′(k). Using (1) on f′, we can get F′ ∈ V_{0}, such that ∀k f′(k) ∈ F′(k) & |F′(k)| ≤ 2^{k}. Clearly we may assume i(η) ∈ F′(k) ⇒ ∀n ∈ dom(η) |η(n)| ≤ h_{0}(n).

Let F (n) = ⋃{η(n) : i(η) ∈ F′(⌊ log(h_{1}(n)/h_{0}(n))⌋)}. So |F (n)| ≤ h_{0}(n) ×

h_{1}(n) ; h_{0}(n) ≤ h_{1}(n).

Now we claim that for all n, f(n) ⊆ F (n).

Proof: Fix n, and let k = ⌊ log h_{1}(n) ; h_{0}(n)⌋, η = f↾ℓ(k). Then ℓ(k) > n, so n ∈

dom(η).

Since i(η) = f′(k) ∈ F′(k), f(n) = η(n) ⊆ F (n).

Application 5: The Sacks Property

Definition 6.34: A forcing notion Q is said to have the Sacks Property iff

∥−_{Q}“∀f ∈ ^{ω}ω ∩ V [G] : ∃Ā ∈ V, Ā is a cone covering f

Fact and Definition 6.35: The following are equivalent for any two universes V_{0} ⊆ V_{1}:

(1) Every f ∈ V_{1} is covered by some cone of V_{0}.

(2) For all functions h_{0} ∈ ^{ω}ω ∩ V_{0} that diverge to infinity: Every f ∈ V_{1} is covered by some h_{0}-cone of V_{0}.

(3) For all h_{0}, h_{1} ∈ ^{ω}ω ∩ V_{0}: If for all n, h_{0}(n) ≤ h_{1}(n), and h_{0}(n) = o(h_{1}(n)), then for all h_{0}-cones f ∈ V_{1} there is an h_{1}-cone F ∈ V_{0} covering f.

If any/all these conditions are satisfied, we say that V_{1} has the Sacks property over V_{0}. We say that a forcing notion Q has the Sacks property iff

Q∥−V [G] has the Sacks property over V

Define H^{∞}(n) = ω for all n, then all proofs about the Laver property can be translated into proofs for the corresponding facts about the Sacks property, by letting H range over {H^{∞}} instead of over all increasing functions.

In particular we get

(1) Q has the Sacks property iff Q satisfies (Laver)_{H,r,s} (where H = H^{∞}) for all r < s iff Q preserves ⊏^{H^{∞}}.

(2) The Sacks property is preserved under countable support iteration

of proper forcing notions. (Alternatively, the fact that the Sacks property is preserved by countable support

iteration follows fron the fact that Q has the Sacks property iff Q has the Laver property and is ^{ω}ω-bounding.)

Application^{2} — An Example: We continue the example from the introduction. Recall that we are trying to build a model where every set of < 𝔠 many functions

is bounded, the set of reals cannot be covered by < 𝔠 many null sets, but there is a nonmeasurable set of size < 𝔠.

Starting in L we construct an countable support iteration ⟨P_{α}, Q_{α} : α < ω_{2}⟩ by requiring

(1) If α is even, then ∥−_{α}“Q_{α} = random real forcing”. Let r_{α} be the name of the random real added by Q_{α}.

(2) If α is odd, then ∥−_{α}Q_{α} = Laver real forcing. Let f_{α} be the

name of the Laver real added by Q_{α}. Then we have:

(1) If \begin{matrix}B \\ ∼ \end{matrix} is a P_{β}-name for a (code of a) Borel null set, and α > β is

even, then

∥−_{ε}r_{α} ∉ \begin{matrix}B \\ ∼ \end{matrix}

(2) If \begin{matrix}f \\ ∼ \end{matrix} is a P_{β}-name for a function in ^{ω}ω, and α > β is odd, then

∥−_{ε}\begin{matrix}f \\ ∼ \end{matrix} <^{∗} f_{α}

(Note that P_{ω_{2}} is proper and satisfies the ℵ_{2}-cc, so no cardinals are collapsed.)

By 1.20 we know that every real in V [G_{ω_{2}}] in fact appears in some V [G_{β}], for some β < ω_{2}. So in V_{ω_{2}}, no set of size ℵ_{1} of functions is unbounded, and no family of Borel null sets of size ℵ_{1} covers all reals numbers.

Hence V_{ω_{2}} ⊧ 𝔟 = 𝔡 + Cov(N).

To conclude this example, we claim that

V_{ω_{2}} ⊧ IR ∩ V_{0} is not of measure zero

To prove this claim it is by 6.11 enough to show that P_{ω_{2}} preserves ⊏^{random}. By the general preservation theorem it is enough to show that for all α < ω_{2} we get ∥−_{α}Q_{α} preserves ⊏^{random}.

But we remarked already that both random forcing and Laver forcing preserve ⊏^{random} (see 6.15 and [7]).

This finishes the application^{2}.

7. Souslin Proper forcing

We review the basic facts about iteration of Souslin proper forcing notions (from [5] and [3]).

Definition 7.1: Assume that (Q, ≤) is a forcing notion. We say that Q is a Souslin forcing notion iff Q, ≤, and the incompatibility relation ⊥_{Q} are analytic sets.

(Note that in general incompatibility is a Π_{1}^{1}-relation (if ≤ is analytic), so the demand on ⊥ really says that incompatibility is a Borel relation.)

Notation 7.2: Using a universal Σ_{1}^{1}-set, we can associate with each real d two Σ_{1}^{1} relations ≤_{d} and ⊥_{d} (subsets of IR × IR) such that every analytic pair ≤, ⊥ appears as some ≤_{d}, ⊥_{d}, and the relations x ≤_{d} y and x ⊥_{d} y are Σ_{1}^{1}.

Definition 7.3: We say that “d codes a Souslin forcing” iff

(1) Q_{d} := ⟨field(≤_{d}), ≤_{d}⟩ is a partial quasiorder. (We also write Q_{d}

for the underlying set field(≤_{d}))

(2) For all x, y ∈ Q_{d}: x ⊥_{d} y ⇔ ¬∃z ∈ Q_{d} : x ≥_{d} z & y ≥_{d} z, i.e., ⊥_{d} is equal to ⊥_{(Q_{d},≤_{d})}, the incompatibility relation for the forcing notion Q_{d}.

Remark 7.4: Clearly these are Π_{2}^{1} conditions on d.

Definition 7.5: Assume that d codes a Souslin forcing Q_{d}, and M is a transitive model model of ZFC* that contains d.

(1) We let Q_{d}^{M} be the Souslin forcing coded by d in M.

(2) If p ∈ Q_{d}^{M}, q ∈ Q_{d}, we say that q is (p, M)-generic, iff

q∥−_{Q_{d}}“G_{Q_{d}} ∩ M is Q_{d}^{M}-generic over M and contains p.”

(3) We say that “Q_{d} is a Souslin Proper forcing” or “d codes a Souslin Proper forcing” if d codes a Souslin forcing, and

(∗)

for all countable M as above, every p ∈ Q_{d}^{M} there exists a (p, M)-generic q ∈ Q_{d}

The property that ⊥ is analytic is sometimes not necessary. We say that (Q, ≤) is weakly Souslin, if ≤ (but not necessarily ⊥_{Q}) is analytic. Similarly, we say that (Q, ≤) is weakly Souslin proper if in addition, (3)(∗) holds.

Remark 7.6: (1) “d codes a Souslin proper forcing” is a Π_{3}^{1} statement about d. Hence (by Shoenfield’s absoluteness theorem) if it holds in V, it holds in every submodel that contains all countable ordinals.

(2) If (M, ∈) is a transitive model of a sufficiently large part of ZFC (M may be a class), and M ⊧“χ := ℶ_{ω}^{+} exists, and M_{0} := H(χ)^{M} is countable, then M_{0} is a countable model of ZFC*, and q is (p, M)-generic iff q is (p, M_{0})-generic. So for all practical purposes we can pretend that M is countable. (In particular this is true if ω_{1} is a inaccessible cardinal in M.)

Proof: (1) Every countable model M is isomorphic to some well-founded (ω, R). If x ∈ IR^{M}, we also write x for its image under this isomorphism.

It is enough to show that “d codes a Souslin Proper forcing notion” is a Π_{3}^{1} statement.

d codes a Souslin proper forcing iff for all R ⊆ ω × ω Either (ω, R) is not well-founded (i.e., there exists an R-descending sequence) or (ω, R) |̸= ZF C^{∗} (this is Δ_{1}^{1}) or d ∉ (ω, R) or for all p ∈ Q_{d}^{M} there is q ∈ Q_{d} such that for all r ≤ q (i) for all D such that (ω, R) ⊧ “D is open dense in Q_{d}”, there is an i, (ω, R) ⊧ i ∈ D, r ⊥̸ i

(ii) r ⊥̸ p.

((i) implies that q∥−G ∩ D ≠ ∅, and (ii) implies that q∥−p ∈ G.)

Proof of (2): M and M_{0} contain the same dense sets of Q_{d}. Context 7.7: In this whole section, ε will be an ordinal ≤ ω_{2}. S will be a countable subset of ω_{2} that is closed under immediate successors and predecessors, where the order type of S is in M. α and γ will stand for ordinals ≤ ε in S. M will be a countable transitive model of ZFC* (= a large enough fragment of ZFC), or an “essentially” countable model as in 7.6(2).

For α ∈ S, let α^{S} be the order type of α ∩ S.

d⃗ will be a sequence of length ε, and c⃗ will be a sequence of length ε^{S}, c⃗ ∈ M.

Definition 7.8: Given a sequence d⃗ = ⟨d_{α} : α < ε⟩ we define a countable support iteration ⟨P_{α}, Q_{α} : α < ε⟩, by letting Q_{α} be a P_{α}-name of Q_{d_{α}[G_{α}]} (if this is Souslin proper), i.e. P_{α} forces the following:

If d_{α}[G_{α}] is a code for a Souslin proper forcing, then Q_{α} = Q_{d_{α}}

otherwise Q_{α} = {∅} is the trivial forcing.

Notation and Remark 7.9: We write Q̄d⃗ for the iteration defined above, and we write P_{d⃗↾α} for P_{α}, i.e., P_{d⃗↾α} is the α-th iteration stage obtained from the definition d⃗, and Q_{α} = Q_{d_{α}} describes the successor extension. If d⃗ ∈ M ⊧ ZF C^{∗}, then P_{d⃗↾α}^{M}

is the α-th iteration stage, computed from the definition d⃗ in the model M.

We say “d⃗ codes a Souslin proper iteration”, if for all α, ∥−_{P_{d⃗↾α}}d_{α} codes a Souslin Proper forcing.

We will consider sequences d⃗ and c⃗, where |d⃗| = ε and |c⃗| = ε^{S}, c⃗ ∈ M.

Definition 7.10: Assume M, c⃗, d⃗, S, ε are as in 7.7.

By induction on α ∈ S ∩ ε, we will define a P_{α}-name G_{α}↾(S, M, c⃗↾α^{S}, d⃗↾α) (which we usually abbreviate to G′_{α} or G_{α}↾(S, M)), by requiring that 1_{P_{α}} forces the following:

G′_{α} ⊆ P_{c⃗↾α^{S}}^{M} and

• If α = β + 1, and (a) G′_{β} is (P_{β^{S}}^{M}, M)-generic

(b) d_{β}[G_{β}] = c_{β^{S}}[G′_{β}]

(c) d_{β}[G_{β}] codes a Souslin Proper forcing (in V [G_{β}])

then G′_{α} = G′_{β} ∗ (G(β) ∩ M[G′_{β}])

If α = β + 1 and (a)–(c) does not hold, then G′_{α} = ∅.

• If α is a limit, then for p ∈ P_{c⃗↾α^{S}}^{M} we let

p ∈ G′_{α} ⇔ ∀β ∈ S ∩ α p↾β^{S} ∈ G′_{β}

Definition 7.11: We call d⃗ and c⃗ “corresponding” sequences if for all α < ε,

∥−_{α}If G′_{α} is generic over M, then c_{α^{S}}[G′_{α}] = d_{α}[G_{α}]

Remark 7.12: We will only consider sequences d⃗ that code a Souslin proper iteration (see 7.9), and we will only consider corresponding c⃗.

Remarks and Notation 7.13:

(1) whenever the parameters M, S, c⃗, d⃗ are clear from the context we will write G′_{α} for G_{α}↾(S, M, c⃗↾α^{S}, d⃗↾α).

(2) If G_{α} is a P_{d⃗↾α}-generic filter, then we also write G′_{α} for the evaluation of the name G′_{α} by G_{α}, similarly for μ_{α}.

(3) Let M_{α} = M[G′_{α}].

(4) It might seem more natural to write G_{α}↾(S, M) as G′_{α^{S}} (= (G′)_{α^{S}}) instead of G′_{α} = (G_{α})′, but this would only complicate the notation.

(5) “G′_{α} is generic” means of course “generic for the forcing P_{c⃗↾α^{S}}^{M} over the model M”.

Remark 7.14: Assume c⃗ and d⃗ are as in 7.12, and M ⊧“\begin{matrix}p \\ ∼ \end{matrix} is a P_{c⃗↾γ^{S}}^{M}-name of a

condition in P M ; c⃗”.

We say that q is \begin{matrix}p \\ ∼ \end{matrix}-generic (or more precisely, (\begin{matrix}p \\ ∼ \end{matrix}, γ, M, S, \begin{matrix}c \\ ∼ \end{matrix}, \begin{matrix}d \\ ∼ \end{matrix})-generic) iff q forces:

(A) G′_{ε} is generic over M.

(B) \begin{matrix}p \\ ∼ \end{matrix}[G′_{γ}] ∈ G′_{ε}.

Theorem 7.15: Let d⃗, c⃗, S, M, γ, ε be as in 7.7. Assume that M ⊧ “\begin{matrix}p \\ ∼ \end{matrix} is a P_{c⃗↾γ^{S}}-name for a condition in Pc⃗” (so \begin{matrix}p \\ ∼ \end{matrix}↾γ is the name for its restriction to γ, and there is a canonical Pc⃗-name which we also call \begin{matrix}p \\ ∼ \end{matrix}).

Assume that q ∈ P_{d⃗↾γ} is (\begin{matrix}p \\ ∼ \end{matrix}↾γ, γ, M, S, c⃗↾γ^{S}, d⃗↾γ)-generic. Then there exists a

condition q^{+} ∈ Pd⃗ such that q^{+}↾γ = q and q^{+} is (\begin{matrix}p \\ ∼ \end{matrix}, γ, M, S, c⃗, d⃗)-generic.

Corollary 7.16: Assume M ⊧ p ∈ Pc⃗. Then there exists a (p, M)-generic condition q ∈ Pd⃗.

Proof of 7.15:

The proof is by induction on ε.

Successor step: Here is the only place where we explicitly use Souslin properness: let ε = α + 1.

Using the induction hypothesis on α, we get a (\begin{matrix}p \\ ∼ \end{matrix}↾α, γ)-generic condition q^{+}↾α ∈

P_{α}. To find q^{+}(α), we will work in V [G_{α}], where G_{α} is an arbitrary generic filter containing q^{+}↾α. Let d := d_{α}[G_{α}] (= c_{α^{S}}[G′_{α}], because c⃗ and d⃗ are corresponding sequences).

Since (a) G′_{α} ⊆ P_{c⃗↾α^{S}} is generic over M

(b) V [G_{α}] ⊧ d codes a Souslin proper forcing, and p(α)[G′_{α}] is in the Souslin proper forcing Q_{d}, by definition we can find an (p(α)[G′_{α}], M_{α})-generic condition q^{+}(α).

Coming back to V, we use the existential completeness lemma to get a name (which we also call q^{+}(α)) about which the above is forced by q^{+}↾α.

Clearly this construction ensures that q^{+} is generic, by 1.10.

Limit step: let ⟨α_{n} : n < ω⟩ be a cofinal sequence in ε ∩ S, α_{0} = γ. Let ⟨D_{n} : n ∈ ω⟩ enumerate all dense open subsets of P M ; c⃗ that are in M.

First we will define a sequence ⟨\begin{matrix}p \\ ∼ \end{matrix}_{n} : n ∈ ω⟩, \begin{matrix}p \\ ∼ \end{matrix}_{n} ∈ M, \begin{matrix}p \\ ∼ \end{matrix}_{0} = \begin{matrix}p \\ ∼ \end{matrix}, such that in M the following will hold:

(0) \begin{matrix}p \\ ∼ \end{matrix}_{n} is a P_{c⃗↾α_{n}^{S}}-name for a condition in Pc⃗

(1) ∥−_{α_{n+1}^{S}} \begin{matrix}p \\ ∼ \end{matrix}_{n+1} ≤ \begin{matrix}p \\ ∼ \end{matrix}_{n}

(2) ∥−_{α_{n+1}^{S}} \begin{matrix}p \\ ∼ \end{matrix}_{n+1} ∈ D_{n}.

(3) ∥−_{α_{n+1}^{S}} If \begin{matrix}p \\ ∼ \end{matrix}_{n}↾α_{n+1}^{S} ∈ G_{α_{n+1}^{S}} then \begin{matrix}p \\ ∼ \end{matrix}_{n+1}↾α_{n+1}^{S} ∈ G_{α_{n+1}^{S}}.

(Here, of course, G_{β} stands for the canonical name (in M) for the generic object of P_{c⃗↾β}^{M}, and ∥−_{β} is the forcing relation of P_{c⃗↾β}^{M} in M.)

For each n we thus get a name \begin{matrix}p \\ ∼ \end{matrix}_{n} that is in M. We use 3.17 and 0.5 (in M) to

obtain \begin{matrix}p \\ ∼ \end{matrix}_{n+1}.

Now we define a sequence ⟨q_{n} : n ∈ ω⟩, q_{n} ∈ P_{α_{n}}, q_{0} = q, such that for all n: (a) q_{n} ∈ P_{α_{n}}, n ≥ 1 ⇒ q_{n}↾α_{n−1} = q_{n−1}.

(b) q_{n} is generic for d⃗↾α, c⃗↾α^{S}, S, M.

(c) q_{n}∥−\begin{matrix}p \\ ∼ \end{matrix}_{n}↾α_{n}^{S} ∈ G′_{α_{n}}.

q_{n+1} = q_{n}^{+} can be obtained by the induction hypothesis, applied to α_{n}^{S}, α_{n+1}^{S},

and \begin{matrix}p \\ ∼ \end{matrix}_{n}↾α_{n+1}^{S}.

q_{n+1} = q_{n}^{+} can be obtained by the induction hypothesis, applied to α_{n}, α_{n+1}, and \begin{matrix}p \\ ∼ \end{matrix}_{n}↾α_{n+1}. By (c)^{+} we know

q_{n}^{+}∥−(\begin{matrix}p \\ ∼ \end{matrix}_{n})↾α_{n+1}^{S} ∈ G′_{α_{n+1}} ∩ M

Hence by (3) we have

q_{n+1}∥−(\begin{matrix}p \\ ∼ \end{matrix}_{n+1})↾α_{n+1}^{S} ∈ G′_{α_{n+1}} ∩ M

Since q_{n+1}↾α_{n} = q_{n}, q = lim q_{n} exists and is ≤ q_{n} for all n. We have to show that q∥−p ∈ G′_{ε} ∩ M and that q is generic. Let G_{ε} be a generic

filter containing q. We will write p_{n} for \begin{matrix}p \\ ∼ \end{matrix}_{n}[G′_{α_{n}}]. (Note that p_{n} ∈ M, because q_{n} was M-generic and q_{n} ∈ G_{α_{n}}.)

Since q_{n} ≥ q ∈ G_{ε}, we have p_{n}↾α_{n}^{S} ∈ G_{α_{n}} ∩ M and M ⊧ p_{n} ≤ p_{n−1} ≤ ⋯ ≤ p_{0}. Hence p↾α_{n}^{S} ∈ G′_{α_{n}} ∩ M for all n, and so by 1.17, p↾δ ∈ G_{δ} ∩ M. As dom(p) ⊆ δ,

p↾δ = p, so p ∈ G_{ε}. Similarly, p_{n} ∈ G_{ε} for all n.

Consider a dense set D_{n} ⊆ P_{ε}. Since q_{n+1}∥−“\begin{matrix}p \\ ∼ \end{matrix}_{n+1} ∈ D_{n},” we have p_{n+1} ∈ G_{ε} ∩ D_{n} ∩ M.

Hence q is generic.

Shelah has suggested the following modification of the definition of Souslin forcing: Definition 7.17: Assume (Q, ≤) is a forcing notion, Q ⊆ ^{ω}ω and epd_{Q} (“effectively predense”) is a relation on Q × [Q]^{ω}. (Q, ≤, epd) is called a Souslin^{+} forcing if

(1) ≤ is analytic, and epd is a Σ_{2}^{1} set.

(2) If epd(q, {p_{i} : i < ω}), then {p_{i} : i < ω} is predense below q. (Q, ≤, epd) is called Souslin^{+} proper iff in addition (3) Whenever (M, ∈) ⊧ ZF C is a countable model, Q ∈ M (i.e., the formulas defining ≤_{Q} and epd_{Q} have parameters in M), and p ∈ Q ∩ M, then there is a condition q ≤ p such that

(∗)

For all A ∈ M, if M ⊧ A is predense below p then epd(q, A ∩ M)

Definition 7.18: We call a condition q satisfying (∗) “(p, Q, M)-effectively generic”.

Fact 7.19: If q is (p, Q, M)-effectively generic, then q is (p, M)-generic.

Note that for Souslin forcing the relation “q is (Q, M)-generic” is in general only Π_{2}^{1}, whereas the relation (∗) above is Σ_{2}^{1}. Example 7.20: Sacks forcing is Souslin^{+} proper.

Proof:

Recall that Sacks forcing 𝕊 is the set of all perfect trees ⊆ 2^{<ω}. (The incompatibility relation for Sacks forcing is not analytic, because to find a condition r extending p and q, one has to intersect the trees p and q (this is an arithmetical computation) and then find a perfect subtree r ⊆ p ∩ q. The Cantor-Bendixson argument needed to obtain this tree r may take any countable number of steps, so an argument using the boundedness principle should show that “p ⊥ q” is not analytic.)

We let

epd(q, {p_{i} : i < ω}) ⇔ ∃F ⊆ q, F is a front, ∀η ∈ F ∃i p_{i} ≥ q^{[η]}.

(Recall that F ⊆ q is called a front, if F is an ⊆-antichain that meets every branch of q, and q^{[η]} = {ν ∈ q : ν ⊆ η ∨ η ⊆ ν}.)

Clearly epd(q, A) implies that A is predense below q.

To show that condition (3) is satisfied, we need a few definitions and a lemma about Sacks forcing 𝕊. Definition 7.21: For p ∈ 𝕊 we let

split(p) := {η ∈ p : η^{⌢}0 ∈ p & η^{⌢}1 ∈ p}
split_{n}(p) := {η ∈ split(p) : |{ν ∈ split(p) : ν ⊂ η}| = n}

stem(p) is the unique element of split_{0}(p), i.e., the first node in p at which splitting occurs.

If p ∈ 𝕊, D ⊆ 𝕊, we write p ∈^{∗} D iff there is a pure extension r of p such that r ∈ D. (r is a pure extension of p if stem(p) = r).

Lemma 7.22: Assume D ⊆ 𝕊 is an open dense set, and let p ∈ 𝕊, n ∈ ω. Then there is a condition q ≤_{n} p and a front F ⊆ q such that ∀η ∈ F, q^{[η]} ∈^{∗} D.

Proof: Consider the following game G(D, p, n): There are two players, who play ω many moves. We let A_{−1} := split_{n+1}(p).

In the nth move, player I plays η_{n} ∈ A_{n−1}, and player II responds by playing a set A_{n} ⊆ p of pairwise incompatible elements of ^{ω}2, |A_{n}| ≥ 2, and ∀ν ∈ A_{n}, ν ⊇ η.

Player I wins, if for some k, p^{[η_{k}]} ∈^{∗} D.

We claim that player II has no winning strategy. For assume that σ is a winning strategy for player II, then we can define

q := {η↾l : l ≤ |η|, η appears as some η_{n} in a play in which player II obeyed σ}

Clearly q is a tree. q is perfect: Assume η ∈ q. So there is an initial segment ⟨η_{0}, A_{0}, …, η_{k}, A_{k}⟩ of a play in which player II obeyed σ, and η ⊆ η_{k}. Let ν_{0}, ν_{1} be two incompatible elements of A_{k}, then both ν_{0} and ν_{1} are in q. Hence q ∈ 𝕊. It is also clear that q ≤_{n} p.

But if σ was a winning strategy for player II, then for all η ∈ q, q^{[η]} ∉^{∗} D, a contradiction.

The game is closed, so player I must have a winning strategy τ. Let q be defined as above, with “player II obeys σ” replaced by “player I obeys τ”.

Again q is perfect: Let η ∈ q, so η ⊆ η_{k}, where ⟨η_{0}, A_{0}, …, η_{k}⟩ is an initial segment of a play in which player I obeyed τ. There are 4 incompatible extensions ν_{0}, ν_{1}, ν_{2}, ν_{3} of η_{k}. Let

⟨η_{0}, A_{0}, …, η_{k}, {ν_{0}, ν_{1}}, η′⟩ and ⟨η_{0}, A_{0}, …, η_{k}, {ν_{2}, ν_{3}}, η′′⟩

be initial segments of plays according to τ. Then η′ ∈ {ν_{0}, ν_{1}}, η′′ ∈ {ν_{2}, ν_{3}}, so η′ and η′′ are incompatible extensions of η in q. Since τ is a winning strategy, we can find a front as required. Corollary 7.23: If D ⊆ 𝕊 is dense open, p ∈ 𝕊, n ∈ ω, then there is p′ ≤_{n} p and a front F ⊆ p′ such that for all η ∈ F, p′^{[η]} ∈ D.

Proof: FIrst we can get a condition q and a front F ⊆ q as in the previous lemma. We may assume that the front F is above the n-th splitting level, i.e., ∀η ∈ split_{n}(q) ∃ν ∈ F η ⊆ ν. Now thin out q at each η ∈ F to get into D, i.e., for all η ∈ F let q_{η} ≤ q^{[η]} be a condition with stem η, q_{η} ∈ D, and let p′ := ⋃_{η∈F} q_{η}.

Proof of 7.20: Let M ⊧ ZF C, p ∈ P ∩ M, and let ⟨D_{n} : n ∈ m⟩ enumerate all sets D such that M ⊧ D is dense open below p. Using 7.23 we can find a sequence ⟨(p_{n}, F_{n}) : n ∈ ω⟩, p = p_{0} ≥_{0} p_{1} ≥_{1} ⋯, p_{n}, F_{n} ∈ M, F_{n} ⊆ p_{n} a front, and ∀η ∈ F_{n} q_{n}^{[η]} ∈ D_{n}. Let q ≤ p_{n} for all n, then F_{n} ∩ q is a front in q, and for all η ∈ F_{n} ∩ q we have q^{[η]} ∈ D. Now it is easy to see that for all A ∈ M, if M ⊧ A predense below p, then A ∩ M is effectively predense below q. Remark 7.24: Similar arguments apply to many other forcing notions whose elements are perfect trees — Laver forcing, rational perfect sets, etc. A theorem analogous to 7.15 can be shown for Souslin^{+} proper forcing notions.

8 A simple preservation theorem for finite support iteration

Context 8.1: Similar to 5.1.

Assumption 8.2: We assume that for all f ∈ C the set {g : f ⊏_{n} g} is closed. (Often we even have that ⊏_{n} itself is closed, in some reasonable topology.)

Definition 8.3: We say that M is (⊏, ω)-closed if

∀A ⊆ M ∩ C (A countable) ∃f^{∗} ∈ M :
∀g : f^{∗} ⊏ g ⇒ ∀f ∈ A ∩ C f ⊏ g  (∗)

Remark: Note that (∗) is a Π_{1}^{1}-statement, hence absolute between any ∈-models of ZFC.

Theorem 8.4: Assume M ⊆ V is (⊏, ω)-closed. Let ⟨P_{α}, Q_{α} : α < ε⟩ (ε limit) be a finite support iteration of ccc forcing notions such that

∀α < δ ∥−_{α}∀g ∃f ∈ M ∩ C f ⊏̸ g

Then ∥−_{δ}∀g ∃f ∈ M f ⊏̸ g.

Proof: By 1.20, we may assume that cf(δ) = ω. Assume the conclusion is false, so there is a condition p_{0} ∈ P_{δ} and a P_{δ}-name \begin{matrix}g \\ ∼ \end{matrix} such that p_{0}∥−∀f ∈ M f ⊏ \begin{matrix}g \\ ∼ \end{matrix}.

Let δ_{0} < δ_{1} < ⋯ be a sequence of ordinals conerging to δ. For each n, let \begin{matrix}g \\ ∼ \end{matrix}_{n} be a P_{δ_{n}}-name such that

∥−_{δ_{n}} \begin{matrix}g \\ ∼ \end{matrix}_{n} interprets \begin{matrix}g \\ ∼ \end{matrix}  (considering \begin{matrix}g \\ ∼ \end{matrix} as a P_{δ_{n}}-name for a P_{δ}/G_{δ_{n}}-name)

and let \begin{matrix}f \\ ∼ \end{matrix}_{n} be a P_{δ_{n}}-name such that

∥−_{δ_{n}} \begin{matrix}f \\ ∼ \end{matrix}_{n} ∈ M & \begin{matrix}f \\ ∼ \end{matrix}_{n} ⊏̸ h

Since P_{δ} is ccc (even properness is sufficient here), there is a countable set F ∈ V,

F ⊆ C such that ∀n ∥−_{δ_{n}} \begin{matrix}f \\ ∼ \end{matrix}_{n} ∈ F.

By our assumption, M is (⊏, ω)-closed. So we can find a function f^{∗} ∈ M such that

∀h ∈ ^{ω}ω : f^{∗} ⊏ h ⇒ ∀f ∈ F ∩ C f ⊏ g

Since f^{∗} ∈ M, p_{0}∥−_{δ}f^{∗} ⊏ \begin{matrix}g \\ ∼ \end{matrix}. We can find a condition p_{1} ≤ p_{0} and an integer n such that p_{1}∥−f^{∗} ⊏_{n} \begin{matrix}g \\ ∼ \end{matrix}. Since p_{1} has finite support, there is an m ∈ ω such that p_{1} ∈ P_{δ_{m}}.

Now work in V [G_{δ_{m}}], where p_{1} ∈ G_{δ_{m}}. Clearly ∥−_{δ_{m},δ}f^{∗} ⊏_{n} \begin{matrix}g \\ ∼ \end{matrix}. Since f_{m} ⊏̸_{n} g_{m} (where f_{m} = \begin{matrix}f \\ ∼ \end{matrix}_{m}[G_{δ_{m}}], g_{m} = \begin{matrix}g \\ ∼ \end{matrix}_{m}[G_{δ_{m}}]), we also have f^{∗} ⊏̸_{m} g_{m}.

We consider the set {h : f^{∗} ⊏̸_{m} h}. By assumption on ⊏, this is an open set, containing g_{m}. So there is k ∈ ω such that

∀h : h↾k = g_{m}↾k ⇒ f^{∗} ⊏̸_{n} h

Let q ∈ P_{δ}/G_{δ_{m}} be a condition forcing \begin{matrix}g \\ ∼ \end{matrix}↾k = g_{m}↾k. Then q also forces f^{∗} ⊏̸_{n} \begin{matrix}g \\ ∼ \end{matrix}, a contradiction.

Example 1 8.5: Let

f ⊏_{n} g ⇔ ∀k ≥ n f(k) < g(k)

The “bounding” number 𝔟 is defined by

𝔟 := {min |B| : B ⊆ ^{ω}ω, B is unbounded}

where “B is unbounded” means that there is no function g such that for all f ∈ B we have f ⊏ g. We are interested in preserving an unbounded family. It is easy to construct (by induction) an unbounded family B that is well-ordered by ⊏ and has order type 𝔟. It is also easy to see that the cofinality of 𝔟 must be uncountable.

Hence B is (⊏, ω)-closed. So we get Fact 8.6: Assume ⟨P_{α}, Q_{α} : α < ε⟩ is a finite support iteration of ccc forcing notions such that for all α we have

∥−_{α} B is unbounded in ^{ω}ω ∩ V [G_{α}]

Then

∥−_{ε} B is unbounded in ^{ω}ω ∩ V [G_{ε}]

Proof: Apply 8.4.

In particular, we get: If no Q_{α} adds a dominating function, and V ⊧ 𝔟 = κ, then V [G_{ε}] ⊧ 𝔟 ≤ κ. Example 2 8.7: Let ⊏^{random} be defined as in 6.8. It was proved above that f ⊏ g iff g is not in the null set coded by f. So, letting M = V (which is clearly (⊏^{random}, ω)- closed, we get

Fact 8.8: If ⟨P_{α}, Q_{α} : α < ε⟩ is a finite support iteration of ccc forcing notions such that for all α we have

∥−_{α} there is no random real over V

then ∥−_{ε} there is no random real over V

362  MARTIN GOLDSTERN  IMCP

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