Iterated Forcing: a comparison between posets approach and Boolean algebraic approach.

approach and the Boolean algebraic approach

Pietro Salmaso April 14, 2015

1 Preliminary notions 7 1.1 Partially Ordered Sets and Boolean Algebras . . . 7 1.2 Homomorphisms and Embeddings . . . 10

2 Forcing 13

2.1 Boolean Forcing . . . 13 2.2 Poset forcing . . . 17 2.3 Equivalence . . . 20

3 Poset iterated forcing 27

3.1 Two step iteration . . . 27 3.2 α-stage iteration . . . 33

4 Boolean Forcing 39

4.1 Two step iteration with ∗ product . . . 39 4.2 Iteration Systems . . . 45

5 49

5.1 Boolean algebras and CBP forcing . . . 49 5.2 Classic and CBP forcing . . . 58

This thesis belongs to the research area of iterated forcing, which is a powerful tool used in independence proofs within set theory. Forcing has been introduced by Paul Cohen in 1963 in his proof of the independence of the continuum hypothesis from the axiomatic system of Zermelo-Fraenkel set theory.

Soon after the invention of forcing, several mathematicians started using and expanding this technique. One of the main generalization is iterated forcing.

While forcing consists of a extension of a model to another one once and for all, iterated forcing makes infinitely many extensions, thus giving additional power to the technique.

In this thesis we analyze two different approaches to iterated forcing: the poset iterated forcing and the Boolean iterated forcing. Our main aim is to prove that the second approach can simulate the first one.

The thesis is articulated as follows:

Chapter 1 contains the necessary algebraic background (posets, com-plete Boolean algebras, comcom-plete and regular embeddings, and related theorems.)

Chapter 2 presents both approaches to forcing following [7,8,9]. After giving the background definitions, it provides the basic theorems, with-out proofs. The equivalence between the two approach, in the basic, non

iterated, version is proved in this chapter.

Chapter 3 is devoted to present poset iterated forcing, following J. E. Baumgartner [2]. In the first part, we show with some detail the two-step iteration. In the second part, we present the general version.

Chapter 4, organized as the previous one, is dedicated to Boolean it-erated forcing, following M. Viale et al. [9]. As the preceeding chapter, we begin by considering the two-step iteration. In the second part, we present the general version.

Chapter 5 contains the full proof of a result which has been folklore in the community, but that lacks a rigorous proof in the current literature. Namely the fact that Boolean iterated forcing can simulate poset iterated forcing. We close the chapter with a wealth of conjectures, notably the possibility of the other direction of this simulation. After a closer look at this, it becomes clear that the proof of such an equivalence, if true, would be non-trivial at all.

Preliminary notions

We introduce here the necessary background from algebra. In particular we present definitions and we state the basic theorems which will be useful throughout the sequel. For a more complete introduction to these topic see e.g. [7,8,9]

1.1

Partially Ordered Sets and Boolean Algebras

Definition 1.1. A partially ordered set (shortly “poset”) is an ordered pair (P, ≤) where ≤ is a reflexive, transitive and antisymmetric binary relation over P .

When no ambiguity can arise, we omit the partial ordering ≤, and denote a poset (P, ≤) simply by its domain P .

We list here some basic definitions concerning posets.

Definition 1.2. Let (P, ≤) be a poset, let p, q ∈ P , and let A ⊆ P . • When we say that A is a subposet of P , we intend the poset

(A, ≤|_A×A).

• If P has a minimum, respectively a maximum, we denote it by 0P,

1P respectively.

• p is an atom of P if it is a minimal element in P , i.e., ∀r ∈ P (r ≤ p ⇒ r = p).

• P is atomless if no element of P is an atom.

• p and q are compatible in P (denoted by p k q) if there exists r ∈ P such that r ≤ p and r ≤ q.

• p and q are incompatible (denoted by p⊥q) if they are not compat-ible.

• A is a chain if it is totally ordered by ≤ |A×A.

• A is an antichain if its elements are pairwise incompatible.

• An antichain A ⊆ P is maximal in P if and only if no strict superset of A is an antichain in P .

• A is dense in P if ∀p ∈ P ∃a ∈ A a ≤ p.

• A is dense below p in P if ∀r ≤ p ∃a ∈ A, a ≤ r.

• A is upward directed if ∀p, q ∈ A ∃r ∈ P (p ≤ r ∧ q ≤ r), A is downward directed if ∀p, q ∈ A ∃r ∈ P (p ≥ r ∧ q ≥ r). • A is upward closed if ∀r ∈ A(p ≥ r ⇒ p ∈ A),

A is downward closed if ∀r ∈ A(p ≤ r ⇒ p ∈ A).

Remark 1.3. A subset A of a poset P is upward directed if and only if every finite subset of A has an upper bound in A. Similarly A is downward directed if and only if every finite subset of A has a lower bound in A.

Remark 1.4. Let Q be a poset, let P ⊆ Q be a dense subset of Q, and let D ⊆ P be a dense subset of P . Then D is a dense subset of Q. The following definition is fundamental for our purposes.

Definition 1.5. A poset (P, ≤) is separative if p 6≤ q ⇒ ∃r ∈ P (r ≤ p ∧ r⊥q).

Usually we assume without mention that all posets are separative, atom-less, and provided of a maximal element. if this is not the case, we men-tion explicitly the lack of any of these condimen-tions.

Definition 1.6. Let (P, ≤) be a poset. A nonempty set I ⊂ P is an ideal if it is upward directed and downward closed. A nonempty set F ⊂ P is a filter if it is downward directed and upward closed.

Following the general use, we call transitive model ZF C, a transitive1_setV

such that (V, ∈ |V ×V) |= ZF C.

Definition 1.7. Let M be a transitive model of ZF C and let (P, ≤) be a poset in M. A filter G ⊂ P is M-generic for P if D ∩ G ∩ M 6= ∅ for every set D ∈ M dense in P .

It is well known that if (P, ≤) is a separative atomless poset in a tran-sitive model M of ZF C, no filter in M is M-generic for P .

Definition 1.8. A poset (P, ≤) is a lattice if any two elements a, b have a unique supremum a ∨ b (least upper bound: join) and an unique infimum a ∧ b (greatest lower bound: meet).

Definition 1.9. Given a lattice L

• L is distributive if the operations of join and meet distribute over each other, i.e.,

∀a, b, c ∈ L ((a∧b)∨c = (a∨c)∧(b∨c) and (a∨b)∧c = (a∨b)∧(a∨c)) • L is bounded if it has a greatest element (1L) and a lowest element

(0_L).

• L is complemented if it is bounded and for every element a ∈ L there exists a unique complement ¬a such that a ∨ ¬a = 1L and

a ∧ ¬a = 0_L.

• B is a Boolean algebra if it is a distributive complemented lattice; • Moreover a Boolean algebra B is complete if every nonempty subset

A ⊆ B has a sumpremum and an infimum.

Remark 1.10. Notice that in every Boolean algebra one has: a ≤ b ⇐⇒ a ∧ b = a ⇐⇒ a ∨ b = b.

It is worthwhile to remark that, given a Boolean algebra B, the poset B \ {0B} is separative.

When B is a Boolean algebra, we say that a set X ⊆ B is dense, a filter or a generic ultrafilter if it is such a set in the poset B \ {0B}.

1.2

Homomorphisms and Embeddings

Definition 1.11. Given posets (P, ≤P), (Q, ≤Q), a function i : P → Q

is order preserving if

Furthermore an order preserving function i is • an immersion iff it is injective,

• a retraction if it is surjective,

• an homomorphism if it respects incompatibility between elements, i.e., for all a, b ∈ P a⊥b ⇒ i(a)⊥i(b),

• an isomorphism if i is bijective and i−1 _{is order preserving,}

• an embedding if it is an injective homomorphism that respects joints and meets whenever they exist,

• dense if i[P ] is dense in Q.

Definition 1.12. An embedding i : P → Q is a complete embedding if • i respects arbitrary suprema and infima whenever they exist ( i.e., for all E ⊆ P such that x := sup E exists in P , i(x) is the supre-mum of i[E] in P , and similarly for Y := inf E).

• There exists a map π : Q → P , the retraction associated to i such that

– π is order preserving, – π ◦ i is the identity on P , – For all q ∈ Q i(π(q)) ≥Q q.

Definition 1.13. Given Boolean algebras (B, ≤_{B), (C, ≤C}), a function i : B → C is a homomorphism of Boolean algebras iff

• i commutes with joints, meets and complements, i.e.,

∀p, q ∈ P i(p∧q) = i(p)∧i(q), i(p∨q) = i(p)∨i(q), and i(¬p) = ¬i(p), • i(1B) = 1C and i(0B) = 0C.

Moreover i is an embedding if it is an embedding of posets, i is a regular embedding if it is a complete embedding of posets.

Fact 1.14. Given two Boolean algebras (B, ≤_{B), (C, ≤C}), a homomor-phism of Boolean algebras i : B → C is a regular embedding iff it is injective and respects arbitrary suprema.

The proofs of the following theorem may be found in many books, for instance, in Chapter 7 of Jech’s book [7].

Theorem 1.15. Given a separative poset (P, ≤P), there exists a

com-plete Boolean algebra (RO(P ), ≤RO(P )) and a complete embedding of

posets iP : P → RO(P ) such that iP[P ] is dense in RO(P ) \ 0RO(P ).

Remark 1.16. We feel free to say that P is a subposet of RO(P ) when-ever it will simplify.

Forcing

We give here an introduction to the some basic forcing techniques, in particular we show the equivalence between Boolean and poset forcing. This will be instrumental to our results on iterated forcing. We follow the pattern of M. Viale et al. [9]. The reader interested to a deeper analysis of this topic could see e.g [7] or [8].

2.1

Boolean Forcing

Definition 2.1. Let V be a transitive model of ZF C, and let B be a Boolean algebra in V . We define inductively:

1. V0B = ∅.

2. Vα+1B : { ˙x : ˙x is a B − valued function, and dom( ˙x) ⊆ VαB }.

3. VλB = S_α<λVαB for λ limit ordinal.

4. VB ₌ S

α∈OrdVαB.

Elements of VB are usually called "B-names" or simply "names" and

denoted by dotted letters ( ˙x, ˙y, ...).

Definition 2.2. Given a transitive model of ZF C, and given a Boolean algebra B ∈ V , the rank of ˙x ∈ VB (denoted by rk( ˙x)) is

rk( ˙x) := min{α : α ∈ Ord and ˙x ∈ VB α+1}.

Definition 2.3. Given a transitive model of ZF C, and given a Boolean algebra B ∈ V , the forcing language on VB is a class language which

contains the usual language of set theory and a constant symbol for every name ˙a ∈ VB.

Definition 2.4. Let V be a transitive model of ZF C, and let B ∈ V be a complete Boolean algebra. Let ϕ(x, y) be one of the formulas x = y, x ⊆ y or x ∈ y. For names ˙x, ˙y ∈ VB define the Boolean value of

ϕ( ˙x, ˙y): _{Jϕ( ˙}x, ˙y)_{KB ∈ B, by recursion on the lexicographically ordered} pairs (rk( ˙x), rk( ˙y)).

• J ˙x ∈ ˙yK = supt∈dom( ˙y)(J ˙x = tK ∧ ˙y(t)), • J ˙x ⊆ ˙yK = inft∈dom( ˙x)(¬( ˙x(t)) ∨Jt ∈ ˙yK), • J ˙x = ˙yK = J ˙x ⊆ ˙yK ∧ J ˙y ⊆ ˙xK.

For any formula ψ in the forcing language, the Boolean value of ψ: JψKB

is defined by recursion on the complexity of the formula. • For the atomic formulas the definition is given above, • Jφ ∧ ψK := JφK ∧ JψK,

• J¬φK := ¬JφK.

• J∃x(φ(x)K) := sup{Jφ( ˙a)K : ˙a ∈ V

Remark 2.5. {_{Jφ( ˙}a)_{K :} ˙a ∈ VB_} is the subset of B that satisfies the

following formula

∃a(a ∈ VB _{∧ φ(a)).}

So the sup in the preceding definition is well posed.

Definition 2.6. Let V be a transitive model of ZF C, and let B ∈ V be a complete Boolean algebra. We define the canonical name ˇx of the element x ∈ V by recursion:

• ˇ∅ = ∅,

• ˇy = {(ˇx, 1B) : x ∈ y}.

We also define the canonical name for a V -generic filter as: ˙

G_B := {(ˇb, b) : b ∈ B} ∈ VB_.

We list without proof the following results. Theorem 2.7. (Łoś) [M. Viale et al. [9]]

Let V be a transitive model of ZF C, and let B be a complete Boolean algebra in V . Let G be any ultrafilter on B. For ˙a, ˙b ∈ VB we put

• ˙b =G _˙a iff

J ˙a = ˙bK ∈ G, • [˙b]G = { ˙c : ˙b =G ˙a},

• [˙b]G ∈G [ ˙a]G iff _J˙b ∈ ˙a_{K ∈ G,}

• VB_{/G = {[˙b]}

G : ˙b ∈ VB}.

Then

1. (VB_{/G, ∈}

2. (VB_{/G, ∈}

G) |= φ([˙b1]G, ...., [˙bn]G) iff _Jφ(˙b1, ...., ˙bn)_{K ∈ G .}

Definition 2.8. Let V be a transitive model of ZF C, let B ∈ V be a complete Boolean algebra in V , and let G be a V -generic filter for B. For any ˙b ∈ VB define

˙bG

:= { ˙aG : ∃p ∈ G ( ˙a, p) ∈ ˙b)}. Then define V [G] := {˙bG _{: ˙b ∈ VB}.}

Theorem 2.9. (Cohen’s forcing theorem) [M. Viale et al. [9]]

Let V be a transitive model of ZF C, let B ∈ V be a complete Boolean algebra, and let G be a V -generic filter for B. Then:

1. V [G] is isomorphic to VB_/G via the map ˙bG _{7→ [˙b]} G.

2. V [G] |= φ(˙bG

1 , ..., ˙bGn) iff Jφ(˙b1, ..., ˙bn)K ∈ G. 3. b ≤B Jφ(˙b1, ...., ˙bn)K iff V [G] |= φ(˙b

G

1, ...., ˙bGn) for all V -generic

fil-ters G such that b ∈ G. In this case we write b B φ(˙b1, ...., ˙bn).

Lemma 2.10. (Mixing) [M. Viale et al. [9]] Let B be a complete Boolean algebra and let {˙ba : a ∈ A} be a family of B-names indexed by an

antichain A ⊆ B. Then there exists ˙b ∈ VB such that

J˙b = ˙baK ≥ a for all a ∈ A.

Lemma 2.11. (Fullness) [M. Viale et al. [9]]

Let B be a complete Boolean algebra. For all ˙b1, ...., ˙bn ∈ VB and for any

formula φ(x, ˙b1, ..., ˙bn) there exists ˙b ∈ VB such that

J∃x φ(x,˙b1, ...., ˙bn)K = Jφ(˙b, ˙b1, ...., ˙bn)K.

2.2

Poset forcing

Definition 2.12. Let (V, ∈) be a transitive model of ZF C, and let P be a poset in V . We define: 1. VP 0 = ∅, 2. VP α+1 = P (VαP × P), 3. VP λ = S α<λV P α for limit λ, 4. VP ₌ S α∈OrdV P α .

Elements of VP _{are usually called "P -names" or simply "names", and}

denoted by a dot ( ˙x).

Notice that we use the same symbols in denoting B-names in the Boolean forcing and P -names in the poset forcing, since they play the same role in the construction of models.

Definition 2.13. Given a transitive model V of ZF C, and given a poset P ∈ V , the rank of ˙x ∈ VP _{(denoted by rk( ˙x)) is}

Definition 2.14.Given a transitive model V of ZF C, and given a poset P ∈ V, the forcing language on VP is a class language which contains the usual language of set theory and a constant symbol for every name

˙a ∈ VP.

Definition 2.15. Let V be a transitive model of ZF C, and let P ∈ V be a poset. We define by recursion the canonical name ˇx for the element x of V as follows:

• ˇ∅ = ∅,

• ˇy = {(ˇx, 1P) : x ∈ y}.

We also define the canonical name ˙G ∈ VP _{for a V -generic filter G by:}

˙

GP := {(ˇp, p) : p ∈ P }.

We refer to [7] for an explicit definition of the forcing relation .

Fact 2.16. For ˙x, ˙y ∈ VP _{p ˙x ∈ ˙y if and only if there exist q ≥ p,} ˙z ∈ VP such that ( ˙z, q) ∈ ˙y and p ˙x = ˙z.

Following [7](Chapter VII) we state without proof Theorem 2.17, , Theo-rem 2.19,Lemma 2.20, and Lemma 2.21 which are the poset counterparts of Theorem 2.7, Theorem 2.9,Lemma 2.10, and Lemma 2.11 of section 2.1.

Theorem 2.17. (Łoś)

Let V be a transitive model of ZF C, let P be a poset in V , and let G be a V -generic filter over P . For ˙a, ˙b ∈ VP _{we put}

• ˙b =G _˙a iff ∃p ∈ G p ˙a = ˙b,

• [˙b]G ∈G [ ˙a]G iff ∃p ∈ G p ˙b ∈ ˙a, • VP_{/G = {[˙b]} G : ˙b ∈ VP}. Then 1. (VP_{/G, ∈} G) is a model of ZF C, 2. (VP_{/G, ∈} G) |= φ([˙b1]G, ...., [˙bn]G) iff ∃p ∈ G p φ(˙b1, ...., ˙bn) .

Definition 2.18. Let V be a transitive model of ZF C, let P ∈ V be a poset in V , and let G be a V -generic filter on P . For any ˙b ∈ VP _we

define

˙bG _{:= { ˙a}G _{: ∃p ∈ G ( ˙a, p) ∈ ˙b}.}

V [G] = { ˙aG : ˙a ∈ VP}. Theorem 2.19. (Cohen’s forcing theorem)

Let V be a transitive model of ZF C, let P ∈ V be a poset, let ˙b1, ..., ˙bn ∈

VP, and let G be a V -generic filter for P . Then:

1. V [G] is isomorphic to VP_{/G via the map ˙b}G _{7→ [˙b]} G.

2. V [G] |= φ(˙bG

1 , ..., ˙bGn) iff ∃p ∈ G p φ(˙b1, ..., ˙bn).

3. p φ(˙b1, ...., ˙bn) iff V [G] |= φ(˙bG1, ...., ˙bGn) for all V -generic filters

G such that p ∈ G.

Lemma 2.20. (Mixing)

Let V be a transitive model of ZF C, let P be a poset in V , and let {˙ba : a ∈ A} be a family of P -names indexed by an antichain A of P .

Then there exists ˙b ∈ VP _{such that a ˙b = ˙b}

Lemma 2.21. (Fullness)

Let V be a transitive model of ZF C, and let P be a poset in V . For all formula φ(x, ˙b1, ...., ˙bn) and for all ˙b1, ...., ˙bn ∈ VP, there exists ˙b ∈ VP

such that

∀p ∈ P (p ∃xφ(x, ˙b1, ...., ˙bn) ⇔ p φ(˙b, ˙b1, ...., ˙bn).

2.3

Equivalence

Remark 2.22. Let M be a transitive model of ZF C, let Q be a poset in M, and let P be a subposet of Q. Then one can prove by transfinite induction that MP

α is a subset of MαQ for every ordinal α. Therefore

MP is a subclass of MQ.

In particular the class MP _{is a subclass of M}RO(P )_.

Let M be a transitive model of ZF C, and let P, Q ∈ M be such that P is a dense subset of the poset Q. Then

Lemma 2.23. 1. If G ⊂ Q is generic over M, then G ∩ P ⊂ P is generic over M.

2. Conversely, if H ⊂ P is generic over M, then

G := {q ∈ Q : ∃p ∈ H p ≤ q} ⊂ Q is generic over M. Proof. We begin by proving 1

G ∩ P is nonempty. In fact G is generic over M, P is dense in Q and P ∈ M, so, by definition of genericity, G ∩ P 6= ∅.

G ∩ P is upward closed. We have to prove that if p ∈ P and q < p is in G ∩ P , then p ∈ G ∩ P . Now q ∈ G, which is a filter in Q, so q < p ⇒ p ∈ G, and thus p ∈ G ∩ P.

G ∩ P is downward directed. We have to prove that if p, q ∈ G ∩ P , then there exists r ∈ G ∩ P such that r ≤ p and r ≤ q. In fact G is a filter, p, q ∈ G so ∃s ∈ G s ≤ p and s ≤ q.

Now we define the new set.

D := {x ∈ Q : (x ∈ P and x ≤ s) or (x⊥s)}. Then D ∈ M because it is defined from P, Q, s ∈ M. D is also dense in Q. In fact given t ∈ Q we have

t 6≤ s ⇒ ∃h ∈ Q h ≤ t and h⊥s because Q is separative, hence ⇒ h ≤ t and h ∈ D. On the other hand if t ≤ s, then

∃k ∈ P k ≤ t ≤ s ⇒ k ∈ P and k ≤ s ⇒ k ∈ D because P is dense.

Now D is dense, and G is M-generic, so G ∩ D 6= ∅, hence ∃r ∈ G ∩ D.

Since r, s ∈ G, they cannot be incompatible. By definition of D r ∈ D ∩ G and ¬(r⊥s) ⇒ r ≤ s and r ∈ G ∩ P

⇒ r ≤ p, r ≤ q and r ∈ G ∩ P. Thus we have proved that G ∩ P is a filter; it remains to prove

G ∩ P is M -generic. From Remark 1.4 a dense subset of a dense sub-set is dense. So, if D is a dense subsub-set of P , then D is also dense in Q. Thus G ∩ D 6= ∅, and we get

∅ 6= G ∩ D = G ∩ P ∩ D = (G ∩ P ) ∩ D

And so G ∩ P is M-generic over P . Now we prove 2.

G is nonempty. G ⊇ H 6= ∅ so G is nonempty.

G is upward closed. We have to prove that if p ∈ G and q > p, then q ∈ G. By definition of G

p ∈ G ⇒ ∃r ∈ H r ≤ p.

Thus r ∈ H and r < q, therefore q ∈ G.

G is downward directed. We have to prove that, if p, q ∈ G,, then there exists r ∈ G such that r ≤ p and r ≤ q.

p, q ∈ G ⇒∃h, k ∈ H h ≤ p, k ≤ q ⇒∃r ∈ H, r ≤ h, k ⇒≤ p, q.

By definition H ⊆ G, so r ∈ G, and G is a filter.

G is generic Let D be a dense subset of Q; we may define the "pro-jection" DP of D inside P as follows :

Now we prove that DP is dense in P : let q ∈ P

∃d ∈ D d ≤ q ⇒ ∃p ∈ P p ≤ d ≤ q ⇒ ∃p ∈ DP p ≤ q

The first clause follows from density of D in Q, the second one from density of P in Q, and the third one by definition of DP. So

H ∩ DP 6= ∅ ⇒∃h ∈ H ∩ DP ⇒ ∃d ∈ D d ≥ h,

⇒d ∈ G ∩ D.

The second implication follows by definition of DP and the third

one from definition of G.

The lemma above leads to a definition:

Definition 2.24. Let M be a transitive model of ZF C, let Q be a poset, let P be a dense subset of Q, and let H ⊂ P be an M-generic set; the extended M-generic set HQ is defined as follows:

HQ := {q ∈ Q : ∃p ∈ H p ≤ q}.

Remark that HQ_{is the unique generic filter F in Q such that F ∩P = H.}

So there is a one-to-one correspondence between generic sets in P and generic sets in Q.

Theorem 2.25. Let M be a transitive model of ZF C, let P, Q ∈ M be posets, let P ⊂ Q be a dense subset, let H ⊂ P be an M-generic set, and let HQ be the extended generic set given by Definition 2.24.

Since MP _{⊆ M}Q_{, the values ˙x}HQ

1 , ..., ˙xH

Q

1. For all ˙x1, ..., ˙xn ∈ MP and for all formulas φ in the language of

set theory

M [H] |= φ( ˙xH₁ , ..., ˙xH_n) ⇐⇒ M [HQ] |= φ( ˙xH₁ Q, ..., ˙xH_nQ). 2. Define the operator ˙x

e : MQ → MP _{by transfinite recursion as} ˙x e := {( ˙y e , p) : p ∈ P and ∃q ≥ p ( ˙y, q) ∈ ˙x}. Then ∀ ˙x ∈ MQ( ˙xHQ = ˙x e HQ_). Proof. 1. By Theorem 2.19 M [HQ] |= φ( ˙xH₁ Q, ..., ˙x_nHQ) ⇐⇒ ∃q ∈ HQ _{q φ( ˙x}1, ..., ˙xn), and M [H] |= φ( ˙xH₁ , ..., ˙xH_{n ) ⇐⇒ ∃p ∈ H p φ( ˙x}1, ..., ˙xn) By definition of HQ_{, we have} q ∈ HQ ⇒ ∃p ∈ P p ≤ q, p ≤ q and q φ( ˙x1, ..., ˙xn) ⇒ p φ( ˙x1, ..., ˙xn). So ∃q ∈ HQ _{q φ( ˙x}1, ..., ˙xn) ⇒ ∃p ∈ H p φ( ˙x1, ..., ˙xn).

On the other hand, H ⊆ HQ_{, so}

and point 1 is proved.

2. By definition of HQ _{it holds:}

q ∈ HQ ⇐⇒ ∃p ≤ q (p ∈ H) Therefore by definition of the operator ˙x

e ˙ yHQ ∈ ˙xHQ ⇔∃q ∈ HQ( ˙y, q) ∈ ˙x ⇔∃p ∈ H( ˙y e , p) ∈ ˙x e ⇔ ˙y e H _{∈ ˙x} e H_.

Proceeding by induction on the rank, for every ˙y such that rk( ˙y) < rk( ˙x), ˙y e HQ = ˙yHQ. Hence, ˙x e HQ = ˙xHQ by extensionality. Corollary 2.26. The correspondence

˙aH 7→ ˙aHQ

is an ∈-isomorphism between the transitive sets MH_{, and M}HQ

. There-fore they are equal.

The following remark will be useful in the next chapter:

Remark 2.27. Let M be a transitive model of ZF C, let P, Q ∈ M be posets, let i : P → Q be an isomorphism of posets, and let G ⊂ P be an M-generic filter. Then i[G] ⊂ Q is an M-generic filter, and the function i∗ _{: M}P _{→ M}Q defined by recursion on the rank as follows:

• i∗_{(∅) = ∅}

is an ∈-isomorphism.

Therefore we may identify M[G] and M[i[G])] as models of ZF C through the map

˙xG 7→ i∗( ˙x)i[G].

We conclude this section with an useful lemma whose proof can be found e.g. in the appendix of Chapter VII of Kunen’s book [8].

Lemma 2.28. Let B be a Boolean algebra in a transitive model M of ZF C, and let B+ _{:= B \ {0B}}, seen as a separative poset. Let G ⊂ B+

Poset iterated forcing

We introduce the poset iteration forcing. First we deal with a two step iteration, and then we illustrate the general case.

Throughout this chapter we follow the exposition of poset iterated forc-ing of J. E. Baumgartner [2].

3.1

Two step iteration

Remark 3.1. Given a transitive model V of ZF C, and given a poset P ∈ V, when we say that ˙Q ∈ VP is a name for a poset, we intend that 1P “ ˙Q is a separative atomless poset with a maximum”.

In particular, by Theorem 2.21, this implies that there exists a name ˙x ∈ VP such that

1P “ ˙x is the maximum of ˙Q”. (3.1)

We fix such a name of least rank, and we denote it by ˙1_Q˙, or shortly ˙1,

when it is clear from the context.

Definition 3.2. Let V be a transitive model of ZF C, let (P, ≤P) ∈ V

be a poset, and let ˙Q be a name in VP _{for a poset (i.e. 1}

P “ ˙Q is a

poset”). Then P ∗P O Q˙ is the class

{(p, ˙q) ∈ P × VP : 1P ˙q ∈ ˙Q}

equipped with the preorder relation ≤P ∗P OQ˙ defined by

(p, ˙q) ≤_{P ∗}

P OQ˙ (p

0_{, ˙q}0_{) ⇐⇒ p ≤}

P p0 and p ˙q ˙≤_Q˙q˙0.

Remark 3.3. The relation ≤_{P ∗}

P OQ˙ is not antisymmetric, in general.

Actually, there may exist names ˙q, ˙q0 _{such that ˙q 6= ˙q}0 _{and 1}

P ˙q, ˙q0 ∈

˙

Q, but p ˙q = ˙q0 for some p ∈ P . Then (p, ˙q) ≤_{P ∗}

P OQ˙ (p, ˙q

0_{) ≤}

P ∗P OQ˙ (p, ˙q),

but (p, ˙q) 6= (p, ˙q0_).

In order to avoid similar situations we define a suitable equivalence re-lation ≈P O:

(p, ˙q) ≈P O (p, ˙q0) ⇐⇒ p ˙q = ˙q0.

Lemma 3.4. Let V be a transitive model of ZF C, let P ∈ V be a poset, and let ˙Q be a name in VP _{for a poset.}

Then there exists a subset P ∗P O Q ⊆ P ∗˙ P O Q˙ such that

∀(p, ˙q) ∈ P ∗P O Q ∃!(p, ˙˙ q0) ∈ P ∗P O Q˙ s.t. (p, ˙q) ≈P O (p, ˙q0).

Such a set is called a regularized subset of P ∗P O Q˙.

Given a regularized subset P ∗P O Q˙, and given (p, ˙q) ∈ P ∗P O Q˙, its

regular equivalent (p, ˙qp) is the unique element of P ∗P O Q˙ such that

Proof. We define the set P ∗P O Q˙

^

:= {(p, ˙q) ∈ P ∗P O Q : rk( ˙˙ q) < rk( ˙Q)} ⊆ P × V_{rk( ˙}P _Q).

By Fact 2.16, for every ˙q such that 1P ˙q ∈ ˙Q, there exists ˙r such that

1P ˙r ∈ ˙Q, ˙r ≈P O q˙, and rk( ˙r) < rk( ˙Q).

Hence in P ∗P O Q˙

^

there are representatives for every equivalence class of the relation ≈P O.

We now use the axiom of choice and obtain the desired set P ∗P O Q˙ by

singling out an element from P ∗P O Q˙

^

for every equivalence class. Assumption 3.5. We assume without loss of generality that for every pair (p, ˙1_Q˙) ∈ P ∗P O Q˙, its regular equivalent (p, ˙1p) is (p, ˙1_Q˙) itself.

When it is clear from the context, we write shortly ∗ for ∗P O, and ≈ for

≈P O.

Lemma 3.6. Let V be a transitive model of ZF C, let P ∈ V be a poset, let ˙Q ∈ VP _{be such that 1}

P “ ˙Q is a poset”, and let P ∗ ˙Q be

a regularized subset of P ∗ ˙Q. Then P ∗ ˙Q, equipped with the induced order relation

≤_{P ∗ ˙Q}=≤_{P ∗ ˙Q} |_{P ∗ ˙Q×P ∗ ˙Q},

is a poset (separative, atomless, and with a maximum as stated in the first chapter).

Proof. The relation ≤P ∗ ˙Q is reflexive. Let (p, ˙q) ∈ P ∗ ˙Q. Then p ≤P

p by reflexivity of ≤P, and p ˙q ˙≤_Q˙q˙ because

p ≤ 1P “ ˙q ∈ ˙Q and ˙≤_Q˙ is a reflexive relation on ˙Q”.

The relation ≤_{P ∗ ˙Q} is antisymmetric. Let (p, ˙q), (p0, ˙q0) ∈ P ∗ ˙Q be such that (p, ˙q) ≤_{P ∗ ˙Q} (p0, ˙q0) ≤_{P ∗ ˙Q} (p, ˙q). Then • p = p0 _{by antisymmetry of ≤} P. • p = p0 ˙q ˙≤ ˙q0≤ ˙q,˙ hence p ˙q = ˙q0.

So (p, ˙q) ≈ (p0_{, ˙q}0_{). Hence (p, ˙q) = (p}0_{, ˙q}0_{) because P ∗ ˙Q is a}

regu-larized subset of P ∗ ˙Q.

The relation ≤_{P ∗ ˙Q} is transitive. Let (p, ˙q), (p0, ˙q0), (p00, ˙q00) ∈ P ∗ ˙Q be such that (p, ˙q) ≤P ∗ ˙Q≤P ∗ ˙Q (p 00_{, ˙q}00_{). Then} • p ≤P p00 by transitivity of ≤P • p ˙q ˙≤_Q˙q˙00 because p ≤ 1P “ ˙q, ˙q0, ˙q00 ∈ ˙Q, ˙≤_Q˙ is a transitive relation on ˙Q”, p ≤ p0 _˙q0 ≤ ˙q00_{, and p ˙q ≤ ˙q}0_. So p ˙q ≤ ˙q00_{, hence(p, ˙q) ≤} P ∗ ˙Q (p 00_{, ˙q}00_).

The poset (P ∗ ˙Q, ≤_{P ∗ ˙Q}) is separative. Let (p, ˙q), (p0, ˙q0) ∈ P ∗ ˙Qbe such that (p, ˙q) ≤ (p0_{, ˙q}0_{). We distinguish two cases:}

i) p 6≤ p0_{. Since P is a separative poset,}

∃p00 ∈ P : p00 ≤P p and p00⊥p0.

So

ii) p ≤ p0_{. In this case (p 6 ˙q ˙≤} ˙ Qq˙

0_{). Hence there exists p}00 _{≤ p such that}

p00 _{¬( ˙q ˙}≤_Q˙q˙0).

Now

p00 _{“ ˙}Q is a separative poset ”, hence p00 _{∃ ˙q}00 q˙00≤˙_Q˙q˙ and ˙q00⊥ ˙q0,

so (p00_{, ˙q}00_)⊥(p0_{, ˙q}0_).

The poset (P ∗ ˙Q, ≤_{P ∗ ˙Q}) is atomless. Given an ordered pair (p, ˙q) ∈ P ∗ ˙Q, there exists p0 < p since P is atomless. So the ordered pair (p0, ˙q) < (p, ˙q), which is not an atom.

The poset (P ∗ ˙Q, ≤_{P ∗ ˙Q}) has a maximum, namely the ordered pair (1P, ˙1_Q˙).

Remark 3.7. There are many different regularized subsets for every class P ∗ ˙Q; however all of them are pairwise isomorphic as posets. For this reason we consider fixed once for all one regularized subset P ∗ ˙Q. Forcing with P ∗ ˙Q accomplishes exactly the same task as forcing first with P and then with ˙Q. More precisely

Theorem 3.8. Let V be a transitive model of ZF C, let P ∈ V be a poset, let ˙Q ∈ VP _{be such that 1}

P “ ˙Q is a poset”. Then

1. Suppose G is P -generic over V and H is ˙QV [G]_{-generic over V [G].}

Put

J := {(p, ˙q) ∈ P ∗ ˙Q : p ∈ G and ˙qV [G] ∈ H}.

2. Suppose J is P ∗ ˙Q-generic over V . Then the set G := {p ∈ P : ∃ ˙q(p, ˙q) ∈ J }

is P -generic over V and the set

H := { ˙qV [G] : ∃p ∈ P s.t. (p, ˙q) ∈ J} is ˙QV [G]_{-generic over V [G].}

Proof. In both cases we check only the genericity condition 1. Suppose D is dense in P ∗ ˙Q. Define E in V [G] by putting

˙

qV [G] ∈ E ⇐⇒ ∃p ∈ G s.t. (p, ˙q) ∈ D. Let ˙E be a name for E.

If p ∈ P and 1P ˙q ∈ ˙Q, then there exists

(p0, ˙q0) ∈ D such that (p0, ˙q0) ≤ (p, ˙q), because D is dense. By definition, p0 _{“ ˙q}0 ∈ ˙E and ˙q0 ≤ ˙q”. Thus FE := {p ∈ P : p ∃q ≤ ˙q q ∈ ˙E}

is dense in P . Now we can choose ˙qV [G] _{∈ E ∩ H} and p ∈ G ∩ F E

by genericity of H and G, respectively. Hence p ˙q ∈ ˙E, and so (p, ˙q) ∈ J ∩ D.

2. If D is dense in P , then {(p, ˙q) ∈ P ∗ ˙Q : p ∈ D} is dense in P ∗ ˙Q. So there exists (p, ˙q) ∈ {(p, ˙q) ∈ P ∗ ˙Q : p ∈ D} ∩ J, in particular p ∈ D ∩ G, and so G is P -generic over V .

If E is dense in ˙QV [G] then there is a name ˙E ∈ VP such that

E = ˙EV [G] and 1P “ ˙E is dense in ˙Q”.

But then

F := {(p, ˙q) ∈ P ∗ ˙_{Q : p ˙q ∈ ˙}E} is dense in P ∗ ˙Q So there exists (p, ˙q) ∈ F ∩ J, in particular ˙qG _{∈ E ∩ H.}

3.2

α-stage iteration

The two stage iteration being so defined, to make a three stage iteration we can simply force with (P ∗ ˙Q) ∗ ˙R.

In order to uniformly iterate, it seems natural to consider instead of pairs ((p, ˙q), ˙r), sequences (p, ˙q, ˙r) of length three. This procedure leads ultimately to the following inductive definition, which works even for iterations into the transfinite.

Definition 3.9. [see J.E. Baumgartner [2]]

Let V be a transitive model of ZF C, and let α ≥ 1 be an ordinal 1. A partial ordering Pα ∈ V is an α-stage iteration iff Pα is a set of

1. If α = 1 then there is a partial ordering Q0 such that p ∈ P1 ⇐⇒ p(0) ∈ Q0 and p ≤ p0 ⇐⇒ p(0) ≤Q0 p 0 (0). In this case P1 ∼= Q0 2. If α = β + 1, β ≥ 1, then Pβ := {p|β : p ∈ Pα}

is a β-stage iteration and

∃ ˙Qβ : Pβ “ ˙Qβ is a poset”, and p ∈ Pα ⇔ p|β ∈ Pβ and Pβ p(β) ∈ ˙Qβ.

Moreover,

p ≤ p0 ⇔ p|β ≤Pβ p

0_|

β and p|β Pβ p(β) ≤Q˙β p

0_(β).

(Note that the ordering on Pβ is uniquely determined, by

induc-tion). Thus

Pα ∼= Pβ ∗ ˙Qβ

3. If α is a limit ordinal then ∀β ≤ α Pβ = {p|β : p ∈ Pα} is a

β-stage iteration, and

• ¯1 ∈ Pα, where ¯1(γ) = ˙1_Q˙_γ, the maximal element of ˙Qγ for all

γ < α,

• for all p, q ∈ Pα, p ≤ q ⇔ ∀β < α p|β ≤ q|β

• The following holds in Pα :

Where cβα(q)(γ) = q(γ) for γ < β and = ˙1_Q˙_γ elsewhere.

From now on we fix a transitive model V of ZF C, without mentioning it in every statement.

Definition 3.10. Let β ≥ 1 be an ordinal, the an iteration system hPαi_α<β is a sequence of α-stage iterations such that for all 1 ≤ η ≤

γ < β, Pη = {p|η : p ∈ Pγ}.

Definition 3.11. Let β ≥ 1 be an ordinal, and let P := hPαi_α<β be an

iteration system. A thread on P is a sequence hpαi_α<β such that for all

α < β, pα ∈ Pα, and for all 1 ≤ η ≤ γ < βpη = p|γ.

Definition 3.12. Let β ≥ 1, and let P := hPαi_α<β be an iteration

system.

• The inverse limit T (P) of P is the β-stage iteration Pβ = {p : p =

[

pα for some thread hpαi_α<β on P}.

• The direct limit C(P) of P (⊆ T (P)) is the β-stage iteration such that

p ∈ C(P) ⇔ ∃α < β∃q ∈ Pα p = cαβ(q).

• A limit for P is a poset L(Pλ) such that

1. C(Pλ) ⊆ L(Pλ) ⊆ T (Pλ)

2. L(Pλ) with the order relation induced by T (P) satisfies the

condition (3.2).

The support of p is the set

supp(p) := {α : p(α) 6= ˙1_Q˙}.

Definition 3.14. Let P be an iteration system of limit length λ, and let I a possibly improper ideal on P (λ).

I is a compatible ideal if and only if supp(c) ∈ I, for every c ∈ C(P) In this case, the I-limit TI(P) of P is the poset

TI(P) := {f ∈ T (P) : supp(f ) ∈ I},

equipped with the order relation induced by T (P).

Let P be an iteration system of limit length λ, and let I be an admissible ideal for P. The I-limit of P fulfills the condition (3.2), and so it is always a limit.

Remark 3.15.

• The compatibility of I with P is a necessary and sufficient condition for

C(P) ⊆ TI(P).

• The direct limit is the I-limit obtained by taking I := {S ⊂ λ : S is not cofinal in λ}.

• The inverse limit T (P) is the I-limit obtained by taking I = P (λ). • If I := {S ⊆ λ : |S| ≤ ℵ0}, then the I-limit is called the countable

• In general if I = {S ⊆ λ : |S| < κ} for some cardinal κ, then the I-limit is called the κ-support limit.

Definition 3.16. Let P be an iteration system of limit length λ, and let I be an ideal on P (λ) compatible with P. A downward closed I-limit is a poset L(P) such that C(P) ⊆ L(P) ⊆ TI(P), for every

f, g ∈ TI(P) \ C(P)

f ≤T (P) g and g ∈ L(P) ⇒ f ∈ L(P).

A downward closed I-limit respects the condition (3.2), and so is a limit. Definition 3.17 (M. Viale et al. [9). ] Let P be an iteration system of limit length λ. Define the revised countable support limit

RCS(P) := {p ∈ T (P) : p ∈ C(P) ∨ ∃α < λ(p(α) cf (ˇλ) = ˇω)} Remark 3.18. The RCS limit is a downward closed I-limit with I = P (λ).

Although not all limits may be downward closed I-limits, nevertheless most of the limit notions used by set theorists are in this category, and we will concentrate on this category in chapter 5.

Boolean Forcing

We introduce Boolean iterated forcing. Similarly to the preceding chap-ter, we first show the two step iteration process, and then we present the general technique.

Throughout this chapter we follow the presentation of M.Viale et al. [9].

4.1

Two step iteration with ∗ product

Let V be a universe, let B be a Boolean algebra in V , and let G be a generic ultrafilter on B. We can construct the Boolean valued universe VB, and the generic extension V [G]. Having at our disposal the new

universe V [G], we may consider a new Boolean algebra Q, and a new ultrafilter H. So we can obtain a new universe (V [G])[H], by using forcing.

The following question naturally arises: may we obtain this (V [G])[H] applying only one forcing process? The answer is yes. In order to define a suitable Boolean algebra, toghether a suitable ultrafilter, we introduce the following definition.

Definition 4.1.Let V be a transitive model of ZF C, let B be a complete Boolean algebra in V , and let ˙C be a B-name for a complete Boolean algebra1 In VB.

• Define the equivalence relation ≈CBA of B-names as follows:

˙a ≈CBA ˙b ⇔ _{J ˙}a = ˙b_KB = 1_B.

• Let ξ be an ordinal2 such that for every equivalence class [˙b] ≈CBA

with J˙b ∈ ˙CK = 1B, there exists a representative ˙b0 in VξB.

Then B ∗ ˙C is the set in V of all the equivalence classes in VB ξ of

B-names for elements of C.˙

• In order to obtain a Boolean algebra, we define the operations ∨ B∗C˙, ∧B∗C˙, and ¬B∗C˙ as follows: [ ˙d] ∨ B∗C˙ [ ˙e] = [ ˙f ] ⇔ Jd ∨˙ C ˙e = ˙fK = 1B; [ ˙d] ∧ B∗C˙ [ ˙e] = [ ˙f ] ⇔ Jd ∧˙ C ˙e = ˙fK = 1B; ¬ B∗C˙[ ˙d] = [ ˙e] ⇔ J ˙e = ¬C˙ ˙ d_{K = 1B}. Lemma 4.2. [M. Viale et al. [9]]

Let V be a transitive model of ZF C, let B ∈ V be a complete Boolean algebra, and let ˙C ∈ VB be a B-name for a complete Boolean algebra.

1_{i.e. 1}

B “ ˙C is a complete Boolean algebra ” 2_{The only purpose of considering ξ and V}

ξ is to obtain sets, and not, proper classes of B-names.

The existence of ξ follows from the fact that every equivalence class of names for elements of ˙C has members of rank less than that of ˙C itself.

Then for every b ∈ B there exists a unique [ ˙db] ∈ B ∗ ˙C such that r ˙ db = 1 z = b and rd˙b = 0 z

= ¬b, B ∗ ˙C is a complete Boolean algebra and the maps i_B∗C˙, π_B∗C˙ defined as

i B∗C˙ : B → B ∗ ˙C b 7→ [ ˙db]≈ π B∗C˙ : B ∗ ˙C → B [ ˙c]_≈ 7→ _{J ˙}c > 0_K are a regular embedding with its associated retraction.

Proof. The existence and uniqueness of ˙db for every b ∈ B is proved by

applying the Mixing Lemma (Lemma 2.20) to the maximal antichain {b, ¬b}, and the family of B-names {˙1, ˙0}.

We now verify that B ∗ ˙C is a Boolean algebra. Define the order relation in B ∗ ˙C by:

[ ˙c] ≤ [ ˙a] ⇐⇒ _{J ˙}c ∨ ˙a = ˙a_{K = 1B} ⇐⇒ _{J ˙}c ≤ ˙a_{K = 1B}. Clearly with this ordering B ∗ ˙C is a lattice.

It is also bounded by 1_B∗˙

C := [ ˙1C˙] and 0B∗C˙ := [ ˙0C˙].

For any [ ˙e] ∈ B∗ ˙C we have ¬_B∗C˙[ ˙e] = [¬_C˙( ˙e)], and [ ˙e]∧_B∗C˙ [¬_C˙ ˙e] = [ ˙0_C˙].

Hence

¬

B∗C˙[ ˙e] ∧B∗C˙ [ ˙e] = 0B∗C˙.

Similarly one obtains that [ ˙e] ∨_B∗C˙ ¬_B∗C˙[ ˙e] = 1_B∗C˙.

B ∗C; then˙ [ ˙a] ∧ B∗C˙ ([˙b] ∨B∗C˙ [ ˙c]) =[ ˙a] ∧B∗C˙ ([˙b ∨C˙ ˙c]) =[ ˙a ∧_C˙ (˙b ∨_C˙ ˙c)] =[( ˙a ∧˙ C ˙b) ∨C˙ ( ˙a ∧C˙ ˙c)] =[ ˙a ∧˙ C ˙b] ∨B∗C˙ [ ˙a ∧C˙ ˙c] =([ ˙a] ∧ B∗C˙ [˙b]) ∨B∗C˙ ([ ˙a] ∧B∗C˙ [ ˙c]).

The third equality follows from the fact that 1B “ ˙C is a distributive

lattice ”; the remaining one follows from the definition of ∧_B∗C˙ and ∨_B∗C˙.

The proof of the distributivity of ∨_B∗C˙ with respect to ∧_B∗C˙ is analogous.

So B ∗ ˙C is a Boolean algebra.

Now we prove that B ∗ ˙C is also complete: if {[ ˙dα] : α < δ} ⊆ B ∗ ˙C, let

˙c be such that r˙c = Wn ˙ dξ : ξ < δ oz = 1. Then [˙c] ≥ W n [ ˙dξ] : ξ < δ o since for all α < δ

r_ n ˙dξ : ξ < δ o ≥ ˙dα z = 1_B. Moreover if r˙a ≥ ˙dα z

= 1, for all α < δ, then ^ nr ˙a ≥ ˙dα z : α < δ o = 1,

thusJ ˙a ≥ ˙cK = 1, hence [ ˙a] ≥ [ ˙c], which gives that [˙c] = W n

[ ˙dα] : α < δo.

It remains to prove that i_B∗˙

C is a regular embedding and that πB∗C˙ is

its associated retraction. • Suppose that there exist

i

B∗C˙(b) = [ ˙c] = [ ˙d]. Then

b = _{J ˙}c = 1_{K ∧}rd = 1˙ z ≤ r˙c = ˙dz

¬b = _{J ˙}c = 0_{K ∧}rd = 0˙ z ≤ r˙c = ˙dz.

Hence 1 = b ∨ ¬b ≤r˙c = ˙dz and this implies [˙c] = [ ˙d].

• i_B∗C˙ preserves negation. Observe that ¬[ ˙db] = [ ˙d¬b]. In fact we have

that ¬rd˙b = 1 z = rd˙b = 0 z = r¬ ˙db = 1 z = ¬b and similarly ¬rd˙b = 0 z = r¬ ˙db = 0 z = b;

so, thanks to the uniqueness of ˙db, we obtain [¬ ˙db] = [ ˙d¬b].

There-fore

i

B∗C˙(¬b) = [ ˙d¬b] = ¬[ ˙db] = ¬iB∗C˙(b).

• i_B∗C˙ preserves joins. Consider {bα ∈ B : α < δ}. We have that

r_ ˙dbα = 0 z = ^ r ˙dbα = 0 z = ^(¬bα) = ¬( _ bα). We have also r_ ˙dbα = 1 z ≤ r_ ˙dbα > 0 z = _ r ˙dbα > 0 z = _ r ˙dbα = 1 z ≤ r_ ˙dbα = 1 z ; then rW ˙ dbα = 1 z = Wr ˙ dbα = 1 z = W bα. Hence i B∗C˙ _ bα  = h_ ˙dbα i = _[i B∗C˙(bα)] .

• i_B∗C˙ is regular. In fact, if i_B∗C˙(b) = i_B∗C˙(b0) = [ ˙d], then b0 =

r ˙

d = 1z = b.

• Finally we deal with the retraction π_B∗C˙. We have to show that

πi B∗ ˙C([ ˙c]) =J ˙c > 0K: by definition, we have πi B∗ ˙C([ ˙c]) = ^ {b ∈ B : i_B∗C˙(b) ≥ [ ˙c]}.

If b is such that i_B∗C˙(b) ≥ [ ˙c], then

r ˙ db ≥ ˙c z = 1 and we obtain b = rd˙b = 1 z = rd˙b > 0 z ≥ _{J ˙}c > 0_{K ∧}rd˙b ≥ ˙c z = _{J ˙}c > 0_{K ,}

so we have the first inequality πi

B∗ ˙C([ ˙c]) ≥ J ˙c > 0K .

In order to obtain the converse inequality, let i_B∗˙

C(J ˙c > 0K) = [d]˙. Then, on the one hand:

¬_{J ˙}c = 0_{K = J ˙}c > 0_{K =} rd = 1˙ z ≤r˙c ≤ ˙dz.

On the other hand:

J ˙c = 0K ≤ r

˙c ≤ ˙dz.

In particular, since ¬J ˙c = 0K ∨ J ˙c = 0K = 1B, we get

r

˙c ≤ ˙dz = 1_B, and thus [˙c] ≤ [ ˙d] = i_B∗C˙(_{J ˙}c > 0_{K), i.e.}

πi

and the proof is complete.

We omit the subscripts in i_B∗˙

C, πB∗C˙, whenever clear from the context.

We conclude this section by addressing briefly the case of three steps iterations in our framework.

Fact 4.3. Assume that V is a transitive model of ZF C, that B ∈ V is a complete Boolean algebra, that ˙C ∈ VB is a B-name for a complete

Boolean algebra and that ˙D ∈ VB∗C˙ is a B ∗ ˙C-name for a complete

Boolean algebra.

Let G be any ultrafilter on B and K be an ultrafilter on ˙CG_{. Put}

H = {c : [c]G ∈ K}.

Then

K = {[c]G : c ∈ H},

and ((B ∗ ˙C) ∗ ˙D/G)/K is isomorphic to (B ∗ ˙C) ∗ ˙D/H via the map

[[c]G]K 7→ [c]H.

4.2

Iteration Systems

Definition 4.4. Let δ be an ordinal, the sequence of complete Boolean algebras F = hBαi_α<δ is a Boolean iteration system of length δ iff for

all α ≤ β < δ there exists a regular embedding iBαBβ such that for all

α ≤ β ≤ γ < δ

• iBαBα is the identity on Bα

If F is compete Boolean iteration system of length δ, for γ < δ, the restricted iteration system F|γ as the γ-sequence hBαi_α<γ, equipped with

the same regular embeddings iBαBβ for all α ≤ β < γ.

We write shortly iαβ for i_BαBβ whenever no misunderstanding can arise.

Definition 4.5. Let λ be a limit ordinal and let F be a complete itera-tion system of length λ. Then:

• The inverse limit of the iteration is T (F ) := {f ∈ Y

α<λ

Bα∀α ∀β > α παβf (α) = f (β)}

• The direct limit is

C(F ) = {f ∈ T (F ) ∃α ∀β > αf (β) = iαβ(f (α))}

and its elements are called constant threads. The support supp(f) of a constant thread f is the least α such that iαβ ◦ f (α) = f (β)

for all β ≥ α.

• The revised countable support limit of F is

RCS(F ) = {f ∈ T (F ) : f ∈ C(F ) ∨ ∃α(f (α) cf (ˇλ) = ˇω)} • More generally L(F) is a limit for the iteration F iff C(F) ⊆

L(F ) ⊆ T (F ) and it respects (3.2) holds for L(F).

Remark 4.6. Given an iteration system F of limit length λ, we define on T (F) an order relation ≤T (F ) such that for every f, g ∈ T (F)

With the order relation induced by ≤T (F ), all limits of the iteration when

deprived of the constant thread 0 become separative atomless posets. Definition 4.7. Let F = hBαiα<λ be an iteration system of limit length

λ. For all α < λ, define the map ˜iαλ : Bα → C(F ) by 3

˜iαλ(b) 7→ hπαβ(b) : β < αi b hiαβ(b) : α ≤ β < λi , and the map ˜πλα : T (F ) → Bα by f 7→ f(α).

Let L(F) be a limit for the iteration F, and let j be a dense complete embedding of L(F) in RO(L(F)). Define the regular embedding

iαλ : Bα → RO(L(F )) by j ◦ ˜iαλ.

The following fact allows the extensions of iteration systems.

Fact 4.8. Given a limit L(F) of an iteration system F, the maps iαλ

of definition 1.13 are regular embeddings and their associate retractions πλα verify the equality παλ|j[L(F )] ◦ j = ˜παλ|L(F ).

3_{Given two sequences f, g, of length α and β, respectively, we denote here by f}

b g their con-catenation, i.e., the sequence of length α + β such that

∀γ < αf_{b g(γ ) = f (γ ) and} ∀γ < α + βf_{b g(α + γ ) = g(γ ).}

Equivalence between the two notions

of forcing

We introduce an iterated forcing technique which is equivalent to the Boolean iterated forcing. We prove that it simulates poset forcing and finally we discuss the issues which surround the other direction of the simulation.

Remark 5.1. Throughout this chapter, given a complete Boolean alge-bra B, we shall call "complete Boolean poset" B+ To simplify the

no-tation we denote B+

, C+ simply by the non blackboad bold letter B, C, .

Assumption 5.2. All limit steps in any poset iteration system are downward closed I-limits, unless explicitly otherwise specified.

5.1

Boolean algebras and CBP forcing

By comparing the definitions 1.12 1.13 we obtain the following

Remark 5.3. Let B and C be two complete Boolean algebras, then i : B → C

is a regular embedding iff

i|_B+_B+ → C+

is a complete embedding.

We will also need the following:

Lemma 5.4. Let B be a Boolean algebra, and ˙C be a name in VB for a

Boolean algebra. Then for every b ∈ B and for every names in VB _{˙c, ˙c}0

such that 1B ˙c, ˙c 0 _{∈ ˙}

C

b ˙c ≥ ˙c0 ⇐⇒ 1_{B ˙}db ∧ ˙c ≥ ˙db ∧ ˙c0.

Here ˙db is intended as in Lemma 4.2.

Proof. ⇐: Assume that

1_{B ˙}db ∧ ˙c ≥ ˙db ∧ ˙c0. Then b ˙db ∧ ˙c ≥ ˙db ∧ ˙c0. Now by definition of ˙db b ˙db = ˙1. So b ˙db ∧ ˙c = ˙1 ∧ ˙c = ˙c and b ˙db ∧ ˙c0 = ˙1 ∧ ˙c0 = ˙c0.

This proves

b ˙c ≥ ˙c0. ⇒: Let G be a generic ultrafilter on B.

If b ∈ G then ( ˙db ∧ ˙c)G = 1G ∧ ˙cG = ˙cG and ( ˙db ∧ ˙c0)G = 1G∧ ˙c0G = ˙c0G. So in V [G] it holds that ( ˙db ∧ ˙c)G ≥ ( ˙db ∧ ˙c0)G. If b /∈ G then ¬b ∈ G so ( ˙db ∧ ˙c)G = 0G∧ ˙cG = 0G and ( ˙db ∧ ˙c0)G = 0G∧ ˙c0G = 0G. Hence in V [G] it holds ˙db ∧ ˙c ≥ ˙db∧ ˙c0.

Therefore ˙db∧ ˙c ≥ ˙db∧ ˙c0 holds in V [G] for every generic ultrafilter

G. Then by the Forcing Theorem 2.9

1_{B ˙}db ∧ ˙c ≥ ˙db ∧ ˙c0.

Corollary 5.5. Let B a Boolean algebra and ˙C a name in VB for a

such that

1_{B ˙c}0, ˙c1 ∈ ˙C,

we have that

b ˙c0 = ˙c1 ⇐⇒ 1_B ˙db ∧ ˙c0 = ˙db ∧ ˙c1.

Theorem 5.6. Let B a complete Boolean algebra and ˙C a name for a complete Boolean algebra on VB. Then (B∗

cbaC) \ {0˙ _B∗C˙} is isomorphic

to B ∗poC˙ via the function1

j : B ∗ ˙_{C → B ∗ ˙C \ 0} B∗C˙ (b, ˙c) 7→ [ ˙db ∧ ˙c]≈ with inverse j−1 _{: (B ∗ ˙C) \ {0} → B ∗ ˙}C [˙b]≈ 7→ (_J˙b > 0_KB, ˙b J˙b>0KB)

Moreover let iB : B → B ∗ ˙C and iB : B → B ∗ ˙C are the standard

injections, then j ◦ iB = i_B|B

Proof. Let us decompose the proof in four steps: the first two will prove that j is a bijection with inverse function j−1_{, the third that j is an order}

isomorphism, the fourth will prove the commutativity of the diagram. j ◦ j−1 _{is the identity on B ∗ ˙C \ 0}

B∗C˙:

j(j−1([˙b]≈)) = j(J˙b > 0KB, ˙b_J˙b>0_KB) = [ ˙d_J˙b>0_KB ∧ ˙b_J˙b>0_KB]≈.

1_{Recall that ˙b}

J˙b>0KB is the unique name in ˙C such that

J˙b > 0KB ˙bJ˙b>0KB = ˙b and (J˙b > 0KB, ˙bJ˙b>0KB) ∈ B ∗ ˙C.

Now ¬J˙b > 0_{KB ˙b = 0 and ¬J}˙b > 0_{KB ˙}d

J˙b>0KB = 0. Hence

¬_{Jb > 0KB ˙}d

J˙b>0KB ∧ ˙bJ˙b>0KB = 0 ∧ ˙bJ˙b>0KB = 0 = ˙b.

On the other hand

J˙b > 0KB ˙b_J˙b>0_KB = ˙b by definition of (J˙b > 0_KB, ˙b J˙b>0KB). Moreover J˙b > 0KB ˙d_J˙b>0_KB = ˙1. Therefore J˙b > 0KB ˙d_J˙b>0_KB ∧ ˙b_J˙b>0_KB = ˙1 ∧ ˙b = ˙b. We conclude that 1_B = _J˙b > 0_KB ∨ ¬_J˙b > 0_{KB ˙}d J˙b>0KB ∧ ˙bJ˙b>0KB = ˙b.

This means that

[ ˙d

J˙b>0KB ∧ ˙bJ˙b>0KB]≈ = [˙b]≈.

j−1◦ j is the identity on B ∗ ˙C

j−1(j(b, ˙c)) = j−1([ ˙db ∧ ˙c]≈) = (_Jd˙b ∧ ˙c > 0_KB, ( ˙db ∧ ˙c)

Jd˙b∧ ˙c>0KB).

Now 1B ˙c > 0 since 1B ˙c ∈ ˙C, and ˙c is a name for a Boolean

algebra deprived of its 0, moreover Jd˙b > 0KB = b. We conclude

that

This shows that b is the first element of the ordered pair j−1_{(j(b, ˙c)).}

We now show thhat the second elements of the relevant ordered pairs are equal: b ˙db = ˙1 by definition of ˙db, hence

b ˙db ∧ ˙c = ˙1 ∧ ˙c = ˙c.

This implies (b, ˙db ∧ ˙c) ≈ (b, ˙c). Since (b, ˙c) ∈ B ∗ ˙C,

˙c = ( ˙db ∧ ˙c)b

giving the desired equality.

j and j−1 preserves orders Since j is a bijection we only need to prove that (b, ˙c) ≥ (b0, ˙c0) ⇐⇒ j((b, ˙c)) ≥ j((b0, ˙c0)). We observe that (b, ˙c) ≥ (b0, ˙c0) ⇔ ⇔ b ≥ b0 and b0 _{˙c ≥ ˙c}0 ⇔ 1B ˙db ≥ ˙db0 and b0 ˙c ≥ ˙c0. By Lemma 5.4 b0 _{˙c ≥ ˙c}0 iff 1B ˙db0 ∧ ˙c ≥ ˙d_b0 ∧ ˙c0.

Therefore b0 ≤_{J ˙}c ≥ ˙c0_{K ⇔} ⇔ _Jd˙b ≥ ˙db0 K ∧ Jd˙b0 ∧ ˙c ≥ ˙db0 ∧ ˙c 0 K = 1 ∧ 1 ⇔ _Jd˙b ∧ ˙c ≥ ˙db0 ∧ ˙c0 K = 1. j ◦ iB = iB|B We have that j(iB(b)) = j((b, ˙1)) = [ ˙db ∧ ˙1]≈ = [ ˙db]≈.

In the first two equalities we apply the definition of j and iB. The

third equality uses the fact that

1_{B ˙}db ∧ ˙1 = ˙db,

so ˙db ∧ ˙1 ≈ ˙db.

Corollary 5.7. Let B be a complete Boolean poset, and let ˙C ∈ VB be a name for a complete Boolean poset, then B ∗P O C˙ is a complete

boolean poset.

Let us now introduce an intermediate class of iterations, called CBP iterations.

Definition 5.8. Let γ be an ordinal. A CBP iteration of length γ is a sequence of complete Boolean posets B = hBαi_α<γ equipped with some

complete embeddings iB

αβ : Bα → Bβ for every α < β < γ such that

• For every α < γ − 1

– Bα+1 = Bα ∗P O C˙α for some name ˙Cα in VBα of a complete

– iB_{α α+1} is the standard embedding of Bα in Bα∗P O C˙α.

– For every β < α, iB_{β α+1} = iB_{α α+1}◦ iB_βα.

• The retraction παβ(bβ) = inf {b ∈ Bα : iαβ(b) ≥ bβ}.

• For every µ < γ limit ordinal,

– hbαi_α<µ such that bα ∈ Bα is a thread of length µ if πβα(bα) =

bβ for every β < α.

– Given bα ∈ Bα (α < µ), a constant thread of bα of length µ is

a thread hbβi_β<µ such that for all α < β < µ bβ = iαβ(bα).

– A set of threads of length µ that contains all constant threads of length µ, and respects (3.2) is a limit.

– Bµ = RO(L(hBαi_α<µ)) \ {0} for some limit L(hBαi_α<µ).

– For every α < µ iB_αµ(bα) is the constant thread of bα of length

µ.

Remark 5.9. Given a CBP iteration system hBαi_α<β, and iBαBγ its

standard complete embeddings, if we define Bα := RO(Bα), and i_BαBγ

such that

∀b ∈ Bα i_BαBγ(b) = iBαBγ(b),

i_BαBγ(0_Bα) = 0_Bγ.

Then hBαi_α<β is a Boolean iteration system forcing equivalent to hBαi_α<β.

Definition 5.10. Given a CBP iteration system hBαi_α<γ and given

b ∈ Bα for some α < γ, we define

Before giving the following important definition we provide a motivating example.

Example 5.11. The inverse limit of a Boolean iteration system in gen-eral is not a complete Boolean algebra.

Let F = hBαi_α<λ be a CBP iteration system, and assume that the

inverse limit is not a complete Boolean algebra.

Let b ∈ RO(T (F)) \ T (F). Then hπαλ(b) = bαi_α<λ is a thread belonging

to the inverse limit. Since b 6∈ T (F) the thread is strictly above b. We cannot exclude the possibility that there exists a thread c := hcαi_α<λ

such that

• cα ≤ bα for all α,

• c 6≤ b.

We want to rule out this pathological behaviour of threads. This moti-vates the following definition.

Definition 5.12 (S.). Let B be a CBP iteration of limit length λ and bλ := hbαi_α<λ ∈ T (B).

hbαi_α<λ is a poset iteration thread if there exists  < λ such that for

every limit ordinal  < γ < λ

bγ ∈ T (B|γ)

We call such a  a good point for bλ, and P IT (B) the set of all poset

iteration threads in T (B).

Remark 5.13. Sometimes the P IT limit coincides with the inverse limit, but not always: let B be a CBP iteration of limit length λ, if

λ = α + ω for some ordinal α, then α is a good point for all threads in T (B) and T (B) = P IT (B).

For the case λ of countable cofinality we can prove (though this will not be done in this thesis) that P IT (B) is a dense suborder of T (B).

The case for λ of uncountable cofinality remains open.

Definition 5.12, in virtue of the restrictions imposed on the limit threads will make possible the proofs that follow.

5.2

Classic and CBP forcing

Definition 5.14. Let Pβ be a β-stage iteration. as defined in 3.9. hBαi_α<β

is its equivalent CBP iteration if • hBαi_α<β is a CBP iteration.

• For every α < β, there exists a complete embedding jα : Pα → Bα

such that jα[Pα] is dense in Bα.

• For every p ∈ Pβ,

jβ(p) := hjα(p|α)i_α<β

is a thread in hBαiα<β.

In such a case, we will say that hjαiα<β is a coherent family of dense

embeddings.

Remark 5.15. Given an α-stage iteration Pβ, and a CBP iteration

system, we say that hjαi_α<β is a coherent family of dense embeddings if

and only if

2. For all α < β, jα is a complete embedding of posets.

3. jα[Pα] is dense in Bα.

4. For all α < β < λ, p ∈ Pα and q ∈ Pβ iBαBβ ◦ jα(p) = jβ(iαβ(p))

and πBαBβ ◦ jβ(q) = jα(q|α).

To have the latter equalities it is sufficient that:

(a) For all α + 1 < λ successor ordinal, for all sequences pb h ˙qi ∈ Pα+1, there exists an ordered pair (b, ˙c) ∈ Bα+1 such that

jα+1((pb h ˙qi)) = (b, ˙c) and b = jα(p).

(b) For every limit ordinal γ < λ, and for every thread p ∈ Pγ

jγ(p) = hjα(p|α)i_α<γ.

Remark 5.16. Let λ be a limit ordinal, let Pλ be a λ-stage iteration,

and let B be an CBP iterations system equivalent to Pλ via the family

of coherent embeddings hjαi_α<λ. Then for all p ∈ Pλ, and for all γ < λ,

jγ(p|γ) ∈ T (B|γ). Hence

jλ(p) ∈ P IT (B).

So

jλ[Pλ] ⊆ P IT (B).

For the reader’s convenience, in the statement and proof of the theorem below, we use as pedix of a function its domain.

This lemma will be useful to handle the proof of the successor step in the main theorem.

Lemma 5.17. Let us consider the following hypotheses H1 P be a poset,

H2 B be a complete Boolean poset, H3 jP : P → B be a dense embedding,

H4 ˙h : ˙Q → RO( ˙Q) be a P -name for a dense embedding of ˙Q in his Boolean completion deprived of the 0.

H5

j_{P ∗ ˙Q} : P ∗P O Q → B ∗˙ P O RO( ˙Q)

be the function defined as

j_{P ∗ ˙Q}((p, ˙q)) = (jP(p), ˙h( ˙q)).

H6 iP be the standard embedding of P in P ∗P O Q˙,

H7 iB be the standard embedding of B in B ∗P O RO( ˙Q).

Then jP ∗ ˙Q is a dense embedding and jP ∗P OQ˙ ◦ iP = iB ◦ jP.

P jP  iP _// P ∗P O Q˙ j P ∗P O ˙Q  B iB // B ∗P O RO( ˙Q)

Remark 5.18. The names in VP and the names in VB are two different classes.

However as stated in Corollary 2.26 we may identify a name ˙x in VP

with a correspondent j∗

P( ˙x) in VB such that V [G ∩ jP[P ]] = V [G], and

The fact below will be useful to prove Lemma 5.17

Fact 5.19. Let P, Q be two separative posets, 1P and 1Q be as in

Defi-nition 1.2, and j : P → Q be a dense complete embedding of posets. Then j(1P) = 1Q.

Proof. By contradiction, suppose

1Q 6= j(1P).

Since 1Q is the maximum of Q,

1Q ≥ j(1P).

This implies

¬(j(1P) ≥ 1Q).

Q is a separative poset, therefore

∃q ∈ Q q⊥j(1P).

Since j is a dense embedding, we have

∃p ∈ P j(p) ≤ q.

Given that j preserves orders by assumption, and 1P ≥ p, we can deduce

p ≤ 1P ⇒ j(p) ≤ j(1P)

Hence j(1P) and q are compatible, which is a contradiction.

Proof. of Theorem 5.17

1. jP ∗ ˙Q is an embedding of posets

(p, ˙q) ≤ (p0, ˙q0) means that p ≤ p0 and p ˙q ≤ ˙q0. By hypothesis jP(p) ≤ jP(p0). Since 1B ∀ ˙r, ˙r0 ∈ ˙Q ˙r ≤ ˙r0 ⇒ ˙h( ˙r) ≤ ˙h( ˙r0), we have j(p) ∀ ˙r, ˙r0 ∈ ˙Q ˙r ≤ ˙r0 ⇒ ˙h( ˙r) ≤ ˙h( ˙r0) p and also j(p) ˙q, ˙q0 ∈ ˙Q. Therefore j(p) ˙h( ˙q) ≤ ˙h( ˙q0). 2. the image of jP ∗ ˙Q is dense in B ∗ RO( ˙Q)

Let (b, ˙c) ∈ B ∗ RO( ˙Q). By hypothesis, we have

∃p ∈ P j(p) ≤ b.

Now

1B ˙h is a dense embedding of ˙Q in RO( ˙Q).

Hence 1B ∀ ˙d ∈ RO( ˙Q) ∃ ˙r ∈ ˙Q ˙h( ˙r) ≤ ˙d, and thus j(p) ∀ ˙d ∈ RO( ˙Q) ∃ ˙r ∈ ˙Q ˙h( ˙r) ≤ ˙d, and in particular j(p) ∃ ˙r ∈ ˙Q ˙h( ˙r) ≤ ˙c.

Therefore

(b, ˙c) ≥ (j(p), ˙h( ˙r)) = j_{P ∗ ˙Q}(p, ˙r). 3. The square is commutative

We have to prove that iB ◦ jP(p) = j_{P ∗ ˙Q}◦ iP(p), for every p ∈ P .

j_{P ∗ ˙Q}(iP(p)) =

= j_{P ∗ ˙Q}((p, ˙1_Q˙))

= (jP(p), ˙h( ˙1_Q˙))

= (jP(p), ˙1_{RO( ˙Q)})

= iB(jP(p)).

The first and the last equalities follow from the definition of iP and

iB, while the second equality follows from our definition of j_{P ∗ ˙Q}.

The third equality follows from Fact 5.19 applied to ˙h.

Corollary 5.20. Let P be a poset, and let ˙Q ∈ VP be a name for a poset. Then RO(P ) ∗P O RO( ˙Q) is the boolean completion of P ∗P O Q˙.

In the next lemma is useful the following definition

Definition 5.21. Let α ≤ β be ordinals, then α−β is the unique ordinal such that β + (α − β) = α.

Recall that the addition in the ordinals is not commutative in general, so (α − β) + β in general is not equal to α.

Lemma 5.22. Let B = hBαi_α<λ be a CBP iteration system of limit

iteration system and θλ : T (B) → Bβ ∗ T ( ˙Qβλ) such that θλ is an

isomorphism that commutes with the standard embeddings of Bβ in T (B)

and in Bβ ∗ T ( ˙Qβλ).

Moreover for every ordered pair (b, ˙q) ∈ Bβ ∗ T ( ˙Qβλ), it holds

supp(θ−1_λ ((b, ˙q))) = supp(b) ∪ tβ+1(supp( ˙q)), (5.1)

where

• supp( ˙q) is the set of successor ordinals δ + 1 such that 1Bβ ˙q( ˇδ + 1) 6= ( ˙q(ˇδ), ˙1)

• tβ+1(supp( ˙q)) is the translate of β + 1 of the set supp( ˙q), i.e.:

tβ+1(supp( ˙q)) = {α : ∃γ ∈ supp( ˙q), α = β + 1 + γ}

Proof. For every ordinal β < γ < λ, there exists a name for a complete Boolean poset ˙Qγ−(β+1) such that Bγ ∼= Bβ∗ ˙Qγ−(β+1). We also define φγ

as the isomorphism from Bγ to Bβ ∗ ˙Qγ−(β+1) and we define ψγ : Bγ →

VBβ such that φ

γ(b) = (πβγ(b), ψγ(b)).

• First we handle the successor case2:

Bβ ∗ ˙Qα+1−(β+1) ∼=Bα+1 ∼ =Bα ∗ ˙Cα ∼ =(Bβ ∗ ˙Qα−(β+1)) ∗ ˙Cα ∼ =Bβ ∗ ( ˙Qα−(β+1)∗ ˙Cα)

2_{In the third isomorhpism we are confusing ˙C}

α with φ∗βα( ˙Cα). An identification ever more

So by uniqueness of this decomposition, 1Bβ ˙Qα+1−(β+1)

∼

= ˙Qα−(β+1)∗ ˙Cα.

• Now we handle the limit case: let β < γ ≤ λ be a limit ordinal, and

Bγ = RO(L(hBαiα<γ))

for some limit L(hBαi_α<γ). Define the map

θβγ : L(hBαi_α<γ) → Bβ ∗ T (D ˙Qα E α<γ−(β+1)) letting θβγ(hbαi_α<γ) = (bβ, hψα(bα)i_β<α<γ)), where – D ˙Qα E

α<γ−(β+1) is a Bβ-name for the iteration system composed

by the ˙Qα,

– T (D ˙Qα

E

α<γ−(β+1))is a Bβ-name for the inverse limit of

D ˙_Q

α

E

α<γ−(β+1),

– hψα(bα)i_β<α<γ is a Bβ-name forced to be equal to the sequence

of Bβ-names hψα(bα)i_β<α<γ.

image seen as a suborder of Bβ ∗ T (D ˙Qα E α<γ−(β+1)). hbδi_δ<γ ≤ hb0δiδ<γ ⇔∀δ < γ (bδ ≤ b 0 δ) ⇔∀δ, β < δ < γ (bδ ≤ b0δ) ⇔∀δ, β < δ < γ (φβδ(bδ) ≤ φβδ(b0δ)) ⇔∀δ, β < δ < γ ((bβ, ψβδ(bδ)) ≤ (b0β, ψβδ(b0δ))) ⇔∀δ, β < δ < γ(bβ ≤ b0β and bβ (ψβδ(bδ) ≤ ψβδ(b0δ))) ⇔bβ ≤ b0β and ∀δ, β < δ < γ bβ (ψβδ(bδ) ≤ ψβδ(b0δ)) ⇔bβ ≤ b0β and bβ hψβδ(bδ)i_β<δ<γ ≤ hψβδ(b0δ)iβ<δ<γ ⇔(bβ, hψβδ(bδ)i_β<δ<γ) ≤ (b0β, hψβδ(b0δ)iβ<δ<γ) ⇔θβγ(hbδi_δ<γ) ≤ θβγ(hb0δiδ<γ).

Thus θβγ is an isomorphism with its image. Let

˙

X := {(hψβδ(b)i_β<δ<γ , 1Bβ) : b ∈ Bδ}.

Now we want to prove that 1Bβ forces ˙X to be a name for a limit

notion according to Definition 3.12 for D ˙Qα

E

α<γ−(β+1), and that

θβγ implements an isomorphism of L(hBαi_α<γ) with Bβ ∗ ˙X.

We first show that ˙X is a name for a limit notion: by the construc-tion made in the previous item, and by definiconstruc-tion of ψβγ, for all

α + 1 > β, it holds that

α + 1 ∈ supp(hbαi_α<γ) ⇔ α + 1 − (β + 1) ∈ supp(hbαi_β<α<γ).

i.e., θβγ maps a constant thread in an ordered pair in which the

second element is a name for a constant thread, and θβγ maps a

non constant thread in an ordered pair in which the second element is a name for a non constant thread.

1Bβ obviouvsly force ˙X to be included in the name of the inverse

limit, and by what we have said apropos of supports, it also forces the name for the direct limit to be included in ˙X.

So we have to check that 1Bβ forces ˙X to fulfill (3.2): let ˙q ∈ V

Bβ

be a name for an element in ˙Qα and ˙t ∈ VBβ be a name for a thread

in ˙X such that

1Bβ ˙q ≤ ˙t(α).

We must show that 3.2 is forced to hold for these Bβ-names: let ˙s

be a name for the constant thread for ˙q of length γ. θ−1

βγ(1Bβ, ˙s) is a

constant thread so, since L(hBαi_α<γ)satisfies (3.2), it is compatible

with θ−1

βγ(1Bβ, ˙t).

Recall that θ−1

βγ is an isomorphism, so this implies that (1Bβ, ˙t) is

compatible with (1Bβ, ˙s), in particular 1Bβ ˙t k ˙s, and we are

done.

Finally, to conclude the proof, we must show that 1Bβ ˙Qγ−(β+1)

∼

= RO( ˙X).

To this aim notice that by Lemma 5.17,