aa r X i v : . [ m a t h . L O ] J a n Some Forcing Techniques:Ultrapowers, templates, and submodels
J¨org Brendle ∗ Graduate School of System InformaticsKobe UniversityRokko-dai 1-1, Nada-kuKobe 657-8501, Japanemail: [email protected]
January 28, 2021
Abstract
This is an expository paper about several sophisticated forcing techniques closed related to standard finitesupport iterations of ccc partial orders. We focus on the four topics of ultrapowers of forcing notions,iterations along templates, Boolean ultrapowers of forcing notions, and restrictions of forcing notions toelementary submodels.
Introduction
The method of finite support iteration (fsi) of ccc forcing, originally developed by Solovay and Tennenbaum toprove the consistency of Suslin’s hypothesis [ST], has since been used for a plethora of independence proofs,both in set theory proper and in other areas of mathematics like topology or algebra. One recurring themehas been its use for independence proofs about cardinal invariants of the continuum, that is, cardinal numbersdescribing the combinatorial structure of the Baire space ω ω or Cantor space 2 ω and typically taking valuesbetween the first uncountable cardinal ℵ and the size c of the continuum. Examples of such cardinal invariantsare the unbounding number b and the dominating number d , the least size of a family F of functions in ω ω such that no single function dominates modulo finite all functions in F (such that all functions are dominatedmodulo finite by a member of F , respectively) The fact that the continuum can be made arbitrarily large addsto the versatility of the fsi method.However, there are situations when an fsi cannot be used (or when it is not known whether it can be used).In such cases, a countable support iteration of proper forcing [Sh2] may be appropriate. This method, though,makes the continuum of size at most ℵ and therefore is of no use for distinguishing three or more cardinalinvariants. Therefore, a number of intricate and sophisticated methods, which are to some extent modificationsof fsi, have been developed for solving specific problems, and later used for further results about cardinalinvariants. The purpose of the present survey paper is to introduce four such methods with the hope of makingthem more accessible to researchers in the field. Specifically, we shall discuss • ultrapowers of partial orders (Section 1), • iterations along templates (Section 2), • Boolean ultrapowers of partial orders (Section 3), and • restrictions of partial orders to elementary submodels (Section 4).The first two methods were introduced by Shelah [Sh4] to prove the consistency of d < a , first using a measurablecardinal and then in ZFC. Here a is the almost disjointness number , that is, the least size of an infinite maximalalmost disjoint (mad) family of infinite sets of natural numbers, one of the most important cardinal invariants ∗ Partially supported by Grant-in-Aid for Scientific Research (C) 18K03398, Japan Society for the Promotion of Science.
1f the continuum. The third was introduced in two papers, one, still unpublished, by Raghavan and Shelah [RS]dealing with the consistency of d ( λ ) < a ( λ ) where λ is regular uncountable, and another, by Goldstern, Kellner,and Shelah [GKS] showing the consistency of Cicho´n’s maximum, the statement that all cardinal invariants inCicho´n’s diagram can be simultaneously distinct. Both use large cardinal assumptions. The forth, then, wasfirst used by Goldstern, Kellner, Mej´ıa, and Shelah [GKMS1] to prove the consistency of Cicho´n’s maximum inZFC.For each of the four topics, we first describe the technique and prove a number of basic results in one or twosubsections. In the next subsection, we present one proof obtained by the respective method in full detail andin the final subsection, we provide an overview of results obtained by the same method.It should be noted that while only the template technique inherently is an fsi-style method, apart from thementioned work by Raghavan and Shelah, the others so far have been used in an fsi context. For ultrapowersthis seems to have to do with the fact that the rather complicated limit construction in the iteration only worksin this case, but for the other two methods, Boolean ultrapowers and submodels, we expect more applications tohigher cardinal invariants and, thus, e.g., to forcing notions which are < λ -closed and λ + -cc for some uncountableregular λ . We assume basic knowledge about forcing theory (see [Je] and [Ku]), as well as some practice with cardinalinvariants and their interplay with forcing (see [BJ], [Bl], and [Ha]).We often freely switch between the partial order (p.o.) language and the complete Boolean algebra (cBa)language when dealing with forcing. We use P < ◦ Q to denote that a p.o. P completely embeds into a p.o. Q ,and P ⋆ ˙ Q is the two-step iteration of P with ˙ Q . “lim dir” denotes the direct limit of a directed system of forcingnotions.Let P and Q be complete Boolean algebras (cBas) with P ⊆ Q . Then the projection mapping h QP : Q → P is defined by h QP ( q ) = Q { p ∈ P : q ≤ Q p } for q ∈ Q . Notice that in this context, P < ◦ Q is equivalent to sayingthat for all p ∈ P and q ∈ Q , if p is compatible with h QP ( q ) in P then p is compatible with q in Q .A forcing notion ( P , ≤ ) is said to be Suslin ccc if P is ccc and P ⊆ ω ω , ≤ ⊆ ( ω ω ) , as well as ⊥ ⊆ ( ω ω ) are Σ sets. See [BJ, Section 3.6] for basic properties about Suslin ccc forcing. The following two basic lemmas arecrucial and we therefore include the short proofs. Lemma 1. If P is Suslin ccc, “ A ⊆ P is a maximal antichain” is a Π statement and therefore absolute betweenmodels of ZFC.Proof. Let A = { x n : n ∈ ω } ⊆ P . “ A is a maximal antichain” iff • ∀ n = m ∀ y ( y / ∈ P ∨ y x n ∨ y x m ) • ∀ y ( y / ∈ P ∨ ∃ n ¬ ( y ⊥ x n ))Both formulas are Π , and therefore Σ -absoluteness applies. Lemma 2.
Assume P , P ′ are partial orders with P < ◦ P ′ . Also let Q be a Suslin ccc forcing. Then P ⋆ ˙ Q < ◦ P ′ ⋆ ˙ Q where the first ˙ Q is the P -name for (the interpretation of the code of ) Q in V P and the second the corresponding P ′ -name.Proof. Assume B = { ( p α , ˙ q α ) : α < κ } is a maximal antichain in P ⋆ ˙ Q . We need to show B is still maximal in P ′ ⋆ ˙ Q .Assume ( p ′ , ˙ q ′ ) ∈ P ′ ⋆ ˙ Q . Let G ′ be a P ′ -generic filter over V with p ′ ∈ G ′ . Then G := G ′ ∩ P is P -generic over V and, in V [ G ], { ˙ q α [ G ] : p α ∈ G } is a maximal antichain in ˙ Q [ G ]. (This means in particular that { α : p α ∈ G } is at most countable.) By the previous lemma, this is still a maximal antichain in ˙ Q [ G ′ ] in V [ G ′ ]. Therefore, in V [ G ′ ], there is α with p α ∈ G such that ˙ q α [ G ] and ˙ q ′ [ G ′ ] are compatible. Hence there are p ′′ ≤ p α , p ′ in P ′ anda P ′ -name ˙ q ′′ for an element of ˙ Q such that ( p ′′ , ˙ q ′′ ) ≤ ( p α , ˙ q α ) , ( p ′ , ˙ q ′ ), as required.In Sections 1 and 2, we need the following basic notion from [Br3]. Let P ∧ < ◦ P j < ◦ P ∨ , j ∈ { , } , becBa’s. We say projections in the diagram P ∧ h P i i = P P P ∨ (cid:0)(cid:0)❅❅ (cid:0)(cid:0)❅❅ re correct if h ∨ ( p ) = h ∧ ( p ) for all p ∈ P iff h ∨ ( p ) = h ∧ ( p ) for all p ∈ P iff whenever h ∧ ( p ) = h ∧ ( p ) then p and p are compatible in P ∨ . Notice this implies (but is not equivalent to) P ∧ = P ∩ P .A typical example for a diagram with correct projections is given by letting P ∧ = { , } and P ∨ the usualproduct forcing, that is, the completion of ( P \ { } ) × ( P \ { } ). Another important example is obtainedby letting P ∧ < ◦ P be arbitrary forcing notions and putting P := P ∧ ⋆ ˙ Q and P ∨ := P ⋆ ˙ Q , where Q isa Suslin ccc forcing notion. In both cases, correctness is straightforward. More on correctness can be foundin [Br5].Correctness can be used to show complete embeddability between direct limits. Lemma 3.
Let K be a directed index set. Assume ( P k : k ∈ K ) and ( Q k : k ∈ K ) are systems of cBa’s suchthat P k < ◦ P ℓ , Q k < ◦ Q ℓ , and P k < ◦ Q k for any k ≤ ℓ . Assume further projections in all diagrams of the form P k Q k P ℓ Q ℓ (cid:0)(cid:0)❅❅ (cid:0)(cid:0)❅❅ are correct for k ≤ ℓ . Then P := lim dir k ∈ K P k completely embeds into Q := lim dir k ∈ K Q k . Furthermore,correctness is preserved in the sense projections in all diagrams of the form P k Q k PQ (cid:0)(cid:0)❅❅ (cid:0)(cid:0)❅❅ for k ∈ K are correct.Proof. Let A ⊆ S k ∈ K P k be a maximal antichain in P . We have to show A is still maximal in Q . Choose q ∈ Q .Then q ∈ Q k for some k ∈ K . By maximality of A there is p ∈ A such that h Q k P k ( q ) is compatible with p in P and thus with h PP k ( p ) in P k . Find ℓ ≥ k such that p ∈ P ℓ . By correctness, p and q are compatible in Q ℓ andthus in Q , as required. Preservation of correctness is straightforward. Assume P is a ccc partial order and κ is a measurable cardinal as witnessed by the κ -complete ultrafilter D .Then the ultrapower P κ / D is again a ccc partial order and P completely embeds into P κ / D so that we mayview P κ / D as a two-step iteration of P and some remainder forcing (see Subsection 1.1 for details). P κ / D sharesmany properties with P and some objects added by P will actually be preserved by the ultrapower, while onthe other hand, if P forces that a has size at least κ , then P κ / D destroys all mad families of the intermediateextension V P . This simple and ingenious observation, due to Shelah, forms the basis of his consistency proofsof d < a and u < a [Sh4].Typically, the ultrapower operation is applied to iterations P = h P γ : γ ≤ µ i , and ultrapowers of iterationsare again iterations. When iterating the process of taking such ultrapowers the question arises what to doin limit steps. One option is a direct limit (this has been used e.g. in Theorem 19), but often embeddingthe iterations obtained by taking ultrapowers into a larger iteration is necessary (e.g. for d < a and u < a ).Technical details of this are discussed in Subsection 1.2.In Subsection 1.3 we present a complete proof of Shelah’s consistency of d < a from a measurable (Theo-rem 14), and in Subsection 1.4 we discuss further results obtained by the ultrapower method. Our expositionfollows to some extent our earlier [Br4]. Let κ be a measurable cardinal and let D be a κ -complete ultrafilter on κ . For a p.o. P and f ∈ P κ ,[ f ] = f / D = { g ∈ P κ : { α < κ : f ( α ) = g ( α ) } ∈ D} is the equivalence class of f modulo D . The ultrapower P κ / D of P consists of all such equivalence classes. It ispartially ordered by [ f ] ≤ [ g ] iff { α < κ : f ( α ) ≤ g ( α ) } ∈ D
3s usual, we identify p ∈ P with the class [ f ] of the constant function f ( α ) = p , α < κ , and thus construe P asa subset of P κ / D . Lemma 4. If P is κ -cc then P < ◦ P κ / D .Proof. Fix ν < κ , and let A = { p γ : γ < ν } be a maximal antichain in P . Given arbitrary f ∈ P κ , for all α < κ there is γ < ν such that f ( α ) and p γ are compatible. By κ -completeness of D , there is γ < ν such that { α < κ : f ( α ) and p γ are compatible } belongs to D . Hence [ f ] is compatible with p γ in P κ / D , and A is still amaximal antichain in P κ / D .Notice that the converse holds as well. If P is not κ -cc, then there is a maximal antichain { p γ : γ < µ } ⊆ P with µ ≥ κ , and [ f ] given by f ( α ) = p α for α < κ is an element of P κ / D incompatible with all p γ . Hence P does not completely embed into P κ / D . Lemma 5. If P is ν -cc for some ν < κ then so is P κ / D .Proof. Take arbitrary elements f γ ∈ P κ , γ < ν . By the ν -cc of P , for each α < κ , there are γ < δ < ν such that f γ ( α ) and f δ ( α ) are compatible. By κ -completeness of D , there are γ < δ < ν such that { α < κ : f γ ( α ) and f δ ( α ) are compatible } belongs to D . Thus [ f γ ] and [ f δ ] are compatible. Therefore every antichain of P κ / D hassize less than ν .On the other hand, if P is not ν -cc for any ν < κ , then P κ / D is not κ -cc. Indeed, let A α = { p α,γ : γ < ν α } be a maximal antichain in P of size ν α ≥ | α | for each α < κ , and fix p ∈ P arbitrarily. Define f γ ∈ P κ for γ < κ by f γ ( α ) = (cid:26) p α,γ if ν α > γp otherwiseIt is easy to see that { [ f γ ] : γ < κ } is an antichain in P κ / D .For the remainder of this section, we assume P is ccc. Therefore P < ◦ P κ / D and P κ / D is ccc by the twoprevious lemmata.We next describe the relationship between P -names and P κ / D -names for real numbers. First notice thatgiven κ many maximal antichains { p αn : n ∈ ω } , α < κ , in P , letting f n ( α ) = p αn for α < κ we obtain a maximalantichain { [ f n ] : n ∈ ω } in P κ / D . Furthermore all maximal antichains of P κ / D are of this form. Now recallthat a P -name ˙ x for a real in ω ω is given by maximal antichains { p n,i : n ∈ ω } and numbers { k n,i : n ∈ ω } , i ∈ ω , such that p n,i (cid:13) P ˙ x ( i ) = k n,i . Therefore, a P κ / D -name ˙ y for a real is given by maximal antichains { p αn,i : n ∈ ω } and numbers { k n,i : n ∈ ω } , i ∈ ω and α < κ , such that, letting f n,i ( α ) = p αn,i ,[ f n,i ] (cid:13) P κ / D ˙ y ( i ) = k n,i . Since { p αn,i : n ∈ ω } and { k n,i : n ∈ ω } , i ∈ ω , determine a P -name ˙ x α for a real, we may think if ˙ y as the average or mean of the ˙ x α and write ˙ y = h ˙ x α : α < κ i / D . Notice that every P κ / D -name for a real is of thisform. Lemma 6.
Let P be ccc. Assume ˙ A is a P -name for a mad family of size at least κ . Then P κ / D forces that ˙ A is not maximal. In particular, if P forces a ≥ κ , then no a.d. family of V P is maximal in V P κ / D .Proof. Assume ˙ A = { ˙ A γ : γ < ν } where ν ≥ κ and all ˙ A γ are P -names for infinite subsets of ω . Then˙ A = h ˙ A α : α < κ i / D is a P κ / D -name for an infinite subset of ω by the preceding discussion.Fix γ < ν . Since for all α < κ with α = γ , (cid:13) P | ˙ A α ∩ ˙ A γ | < ℵ , we see that { α < κ : (cid:13) P | ˙ A α ∩ ˙ A γ | < ℵ } belongs to D . Therefore (cid:13) P κ / D | ˙ A ∩ ˙ A γ | < ℵ because ˙ A is the average of the ˙ A α . Hence ˙ A witnesses that P κ / D forces non-maximality of ˙ A .4 emma 7. Let P be ccc, µ = κ regular, and assume P adjoins a scale { ˙ d β : β < µ } . Then P κ / D forces that { ˙ d β : β < µ } is still a scale. In particular, if P κ / D ∼ = P ⋆ ˙ Q , then P forces that ˙ Q is an ω ω -bounding forcingnotion.Proof. Let ˙ y be a P κ / D -name for a real in ω ω . By the preceding discussion, there are f n,i ∈ P κ and k n,i ∈ ω , n, i ∈ ω , such that ˙ y is determined by { [ f n,i ] : n ∈ ω } and { k n,i : n ∈ ω } , i ∈ ω . Letting p αn,i = f n,i ( α ), { p αn,i : n ∈ ω } and { k n,i : n ∈ ω } , i ∈ ω , determine P -names ˙ x α for reals, and ˙ y = h ˙ x α : α < κ i / D .By ccc-ness of P , for each α < κ there is β α < µ such that (cid:13) P ˙ x α ≤ ∗ ˙ d β α . Since µ = κ are both regular, we obtain β < µ such that { α < κ : β α ≤ β } ∈ D (use the κ -completeness of D in case µ < κ ). Therefore (cid:13) P κ / D ˙ y ≤ ∗ ˙ d β because ˙ y is the average of the ˙ x α . Hence { ˙ d β : β < µ } remains a scale in the P κ / D -extension. STRATEGY.
These two simple lemmas provide us with a scenario for proving the consistency of d < a . Let κ < µ < λ be regular cardinals. Force b = d = µ with a ccc p.o. P . Then keep taking ultrapowers of P for λ many steps. By Lemma 7, b = d = µ should be preserved while, by the ZFC-inequality b ≤ a , Lemma 6 tells usthat mad families of size less than λ will be destroyed so that a = λ in the final model. The problem, however,is what to do in limit steps of the procedure of iteratively taking ultrapowers. To get a handle on this, we shalllook at ultrapowers of iterations and iterations of ultrapowers in the next subsection. We start with the discussion of two-step iterations.
Lemma 8.
Assume P < ◦ Q are cBa’s. Then P κ / D < ◦ Q κ / D and, in fact, the projection mapping is given by h Q κ / D P κ / D ([ f ]) = h h QP ( f ( α )) : α < κ i / D for f ∈ Q κ . Furthermore, projections in the diagram h P , Q , P κ / D , Q κ / Di are correct.Proof. Let g ∈ P κ and notice that[ g ] ≥ h Q κ / D P κ / D ([ f ]) ⇐⇒ [ g ] ≥ [ f ] ⇐⇒ { α < κ : g ( α ) ≥ f ( α ) } ∈ D⇐⇒ { α < κ : g ( α ) ≥ h QP ( f ( α )) } ∈ D⇐⇒ [ g ] ≥ h h QP ( f ( α )) : α < κ i / D This shows equality. Thus [ g ] ⊥ h Q κ / D P κ / D ([ f ]) ⇐⇒ [ g ] ⊥h h QP ( f ( α )) : α < κ i / D⇐⇒ { α < κ : g ( α ) ⊥ h QP ( f ( α )) } ∈ D⇐⇒ { α < κ : g ( α ) ⊥ f ( α ) } ∈ D⇐⇒ [ g ] ⊥ [ f ]and we obtain complete embeddability. To see correctness, let q ∈ Q , p := h QP ( q ), g ( α ) = q , and f ( α ) = p forall α . Then h Q κ / D P κ / D ( q ) = h Q κ / D P κ / D ([ g ]) = h h QP ( g ( α )) : α < κ i / D = h f ( α ) : α < κ i / D = [ f ] = p as required. Lemma 9.
Let P be a p.o. and let Q be a Suslin ccc forcing notion. Then ( P ⋆ ˙ Q ) κ / D ∼ = P κ / D ⋆ ˙ Q . Note that the first ˙ Q is a P -name while the second is a P κ / D -name. Proof.
Taking ( p α , ˙ x α ) ∈ P ⋆ ˙ Q , α < κ , we see that an arbitrary condition in ( P ⋆ ˙ Q ) κ / D is of the form h ( p α , ˙ x α ) : α < κ i / D . Therefore, letting f ∈ P κ be defined by f ( α ) = p α and setting ˙ y = h ˙ x α : α < κ i / D , wesee that we can identify h ( p α , ˙ x α ) : α < κ i / D with the condition ([ f ] , ˙ y ) ∈ P κ / D ⋆ ˙ D . This obviously defines anisomorphism between (dense subsets of) the two partial orders.5e now move to transfinite iterations. Let µ be an ordinal. Say a sequence of cBas P = h P γ : γ ≤ µ i is an iteration (see also [Br4]) if P γ < ◦ P δ for γ < δ . Note that, for technical reasons which will become obvious lateron (see Lemma 11), we do not require that P δ is any kind of limit of P γ , γ < δ , for limit ordinals δ . For γ < δ ,let h δγ : P δ → P γ be the projection. The support of p ∈ P µ is defined bysupp( p ) = { δ : there is no γ < δ such that h µδ ( p ) = h µγ ( p ) } Note that δ + 1 ∈ supp( p ) iff h µδ +1 ( p ) < h µδ ( p ). Similarly, for limit ordinals δ , δ ∈ supp( p ) iff h µδ ( p ) < h µγ ( p ) forall γ < δ .An iteration P has finite supports if supp( p ) is finite for all p ∈ P µ . While this is not the usual definitionof a finite support iteration (fsi) the following simple lemma implies that an iteration with finite supports isequivalent to an fsi (we leave the details of this to the reader). Lemma 10.
Assume P = h P γ : γ ≤ µ i has finite supports. Let δ ≤ µ be a limit ordinal. Then lim dir γ<δ P γ < ◦ P δ where, as usual, lim dir denotes the direct limit of forcing notions.Proof. Let p ∈ P δ . If δ / ∈ supp( p ), then p = h µδ ( p ) = h µγ ( p ) for some γ < δ . Therefore p ∈ P γ ⊆ S γ<δ P γ =lim dir γ<δ P γ , and there is nothing to prove.If δ ∈ supp( p ), then p = h µδ ( p ) < h µγ ( p ) for all γ < δ . However, since supports are finite, there is γ < δ suchthat h µγ ( p ) = h µγ ( p ) for all γ with γ ≤ γ < δ . Set p = h µγ ( p ). So p ∈ P γ ⊆ S γ<δ P γ = lim dir γ<δ P γ . Weclaim p is a reduction of p to lim dir γ<δ P γ . Indeed, suppose q ≤ p belongs to lim dir γ<δ P γ . Then there is γ with γ ≤ γ < δ such that q ∈ P γ . Since h µγ ( p ) = p ≥ q in P γ , p and q are compatible in P δ (with commonextension p · q ), as required.We next investigate ultrapowers of iterations. Lemma 11.
Let P = h P γ : γ ≤ µ i be an iteration. Then P κ / D = h ( P γ ) κ / D : γ ≤ µ i also is an iteration.Moreover, if P has finite supports then so does P κ / D .Proof. The first part is obvious by Lemma 8. So assume that P has finite supports. Let f ∈ ( P µ ) κ . Thensupp( f ( α )) is finite for all α , say supp( f ( α )) = { γ α < γ α < ... < γ αn α − } for all α . By completeness of D thereis n ∈ ω such that { α < κ : n α = n } belongs to D . For each i < n there is γ i ≤ µ such that • either { α < κ : γ αi = γ i } ∈ D • or { α < κ : γ αi < γ i } ∈ D while { α < κ : γ αi ≤ δ } / ∈ D for all δ < γ .In the latter case we necessarily have cf( γ i ) = κ by the κ -completeness of D .We claim that supp([ f ]) = { γ i : i < n } and so is finite as required. (Therefore, | supp([ f ]) | ≤ n , but equalitydoes not necessarily hold because the γ i are not necessarily distinct.)Indeed, γ ∈ supp([ f ]) ⇐⇒ ∀ δ < γ ( h µγ ([ f ]) < h µδ ([ f ])) ⇐⇒ ∀ δ < γ ( { α < κ : h µγ ( f ( α )) < h µδ ( f ( α )) } ∈ D ) (by Lemma 8) ⇐⇒ either { α < κ : γ ∈ supp( f ( α )) } ∈ D (first case)or cf( γ ) = κ and ∀ δ < γ ( { α < κ : ( δ, γ ) ∩ supp( f ( α )) = ∅} ∈ D ) (second case) ⇐⇒ γ = γ i for some i < n We note that even if P δ = lim dir γ<δ P γ for all limits δ , this is not necessarily the case for ( P δ ) κ / D . Indeed,the proof above shows that for δ with cf( δ ) = κ , lim dir γ<δ ( P γ ) κ / D will be a strict complete suborder of( P δ ) κ / D . This is the reason for our notion of iteration with finite supports .We want to iterate the procedure of taking ultrapowers – and thus obtain iterations of ultrapowers. To thisend, we need to explain what to do in limit steps. The following lemma (Lemma 7 of [Br4]) is a special case ofthe amalgamated limit from [Br5]. Lemma 12.
Let µ and λ be limit ordinals. Assume P α = h P αγ : γ ≤ µ i , α < λ , are iterations such that P αγ < ◦ P βγ for α < β < λ and γ ≤ µ . Also assume h P λγ : γ < µ i is an iteration such that P αγ < ◦ P λγ for α < λ and γ < µ .Furthermore, assume that for all α < β ≤ λ and γ < δ ≤ µ with ( β, δ ) = ( λ, µ ) , the projections in the diagram h P αγ , P βγ , P αδ , P βδ i are correct.Then there is a p.o. P λµ such that P λ = h P λγ : γ ≤ µ i is an iteration and P αµ < ◦ P λµ for all α < λ . Moreover,correctness is preserved in the sense that the projections in the diagram h P αγ , P λγ , P αµ , P λµ i are correct as well.Assume also all P α , α < λ , and h P λγ : γ < µ i have finite supports. Then so does P λ . roof. Elements of P λµ are formal products of the form p · q where p ∈ P αµ for some α < λ , q ∈ P λγ for some γ < µ , and h γ,λγ,α ( q ) = h µ,αγ,α ( p ). In this case we say that the pair α, γ witnesses p · q ∈ P λµ . For formal products p · q with witness α, γ and p · q with witness β, δ , where β ≥ α and δ ≥ γ , we define the partial order ≤ on P λµ by p · q ≤ p · q if p ≤ p in P βµ and q ≤ q in P λδ .We first verify that if a pair α, γ witnesses p · q ∈ P λµ , then the pair β, δ is also a witness for any β ≥ α and δ ≥ γ . For indeed, by correctness, we have h γ,λγ,β ( q ) ≤ h γ,λγ,α ( q ) = h µ,αγ,α ( p ) = h µ,βγ,β ( p ). Thus, letting p ′ = p · h γ,λγ,β ( q ) ∈ P βµ , we see h µ,βγ,β ( p ′ ) = h γ,λγ,β ( q ) so that the formal product p ′ · q belongs to P λµ as witnessed bythe pair β, γ . Obviously, p ′ · q ≤ p · q , and it is easy to see that any p ′′ · q ′′ ≤ p · q is compatible with p ′ · q sothat p ′ · q and p · q in fact describe the same condition. By symmetry we can also replace γ by δ .This fact provides us with an easier description of the ordering: given formal products p · q and p · q in P λµ , we may assume they have the same witness α, γ , and we let p · q ≤ p · q if p ≤ p in P αµ and q ≤ q in P λγ .Next we prove completeness of the embeddings. By symmetry it suffices to show that P αµ < ◦ P λµ ( P λγ < ◦ P λµ is analogous). Fix p · q ∈ P λµ with witness β, γ . By the preceding discussion we may assume that β ≥ α .Let p = h µ,βµ,α ( p ) ∈ P αµ . We need to show that any p ≤ p in the partial order P αµ is compatible with p · q .Let p ′ = p · p ∈ P βµ . Then clearly p ′ ≤ p in P βµ and h µ,βγ,β ( p ′ ) ≤ h µ,βγ,β ( p ) = h γ,λγ,β ( q ) in P βγ . Thus, letting q ′ = q · h µ,βγ,β ( p ′ ) ∈ P λγ , we get that h γ,λγ,β ( q ′ ) = h µ,βγ,β ( p ′ ) and p ′ · q ′ ≤ p · q as required.Preservation of correctness is obvious by the definition of P λµ as the collection of formal products.The above arguments show in fact that h µ,λµ,β ( p · q ) = p · h γ,λγ,β ( q ) for any β ≥ α where the pair α, γ witnesses p · q ∈ P λµ . Similarly h µ,λδ,λ ( p · q ) = h µ,αδ,α ( p ) · q for any δ ≥ γ where α, γ witnesses p · q ∈ P λµ . Thus, if p · q ∈ P λµ , thensupp( p · q ) = supp( p ) ∪ supp( q ) so that P λ has finite supports if h P λγ : γ < µ i and all the P α , α < λ , do.We next need to argue that the limit construction of the previous lemma preserves the ccc. This is far fromobvious. In view of later applications (Theorem 14 and Subsection 1.4) we do this in the special situation when P α +1 is the ultrapower of P α and the iterands in the P α are sufficiently simple.We say a forcing notion ( P , ≤ ) is Suslin σ -linked if it is Suslin ccc and P = S n P n where all P n are linked (thatis, any two elements of P n have a common extension) Σ sets. This implies “ P n is linked” is a Π statementand therefore absolute. (Indeed, linkedness of P n is equivalent to “ ∀ x, y ( x / ∈ P n ∨ y / ∈ P n ∨ ¬ ( x ⊥ y ))”.) Lemma 13.
Let µ and λ be limit ordinals. Let Q = S n Q n be a Suslin σ -linked forcing notion. Assume P α = h P αγ : γ ≤ µ i are iterations with finite support, α < λ , such that • P = { , } and P δ = lim dir γ<δ P γ for limit δ , • P αγ +1 = P αγ ⋆ ˙ Q for all α and γ , • P α +1 γ = ( P αγ ) κ / D for all α and γ , • P βγ is built according to Lemma 12 for limit β and γ .Then all P αγ satisfy property K (and thus are ccc). Let us note that by Lemma 9, there is no conflict between the second and third clauses in the assumption.
Proof.
By recursion on α ≤ λ , we define I α , D αγ , γ ≤ µ , and s α , such that1. the I α are linear orders, I = µ , and I α ⊆ I β for α ≤ β ,2. the D αγ are dense subsets of P αγ and D αγ ⊆ D βδ for α ≤ β and γ ≤ δ ,3. the s α are functions with domain D αµ and s α ⊆ s β for α ≤ β ,4. if p ∈ D αγ , γ ≤ µ , then s α ( p ) : I α → ω is a finite partial function with dom( s α ( p )) ∩ µ = { δ : δ + 1 ∈ supp( p ) } ,5. if p ∈ D αγ , γ ≤ µ , δ < γ with δ + 1 ∈ supp( p ), then h γ,αδ,α ( p ) (cid:13) p ( δ ) ∈ ˙ Q s α ( p )( δ ) ,6. if p, q ∈ D αγ , γ ≤ µ , δ < γ , s α ( p ) and s α ( q ) agree on their common domain, and r ≤ h γ,αδ,α ( p ) , h γ,αδ,α ( q ) in P αδ , then there is r ≤ p, q in P αγ such that h γ,αδ,α ( r ) = r .7ote that for δ = 0, the last clause means that p and q in P αγ are compatible if s α ( p ) and s α ( q ) agree on theircommon domain. By the ∆-system lemma, it is then immediate that P αγ has property K and the ccc follows.So it suffices to carry out the recursion.Basic step α = 0. Let I := µ . Define D γ and s with the required properties by recursion on γ ≤ µ .If γ = 0 there is nothing to do.Assume γ = δ + 1 is successor, and D δ and s ↾ D δ have been defined. Let D γ = D δ ∪ { p = ( p , ˙ x ) ∈ P γ : p ∈ D δ and p (cid:13) ˙ x ∈ ˙ Q n for some n } . Clearly, D γ is dense in P γ . Next, for such p = ( p , ˙ x ) ∈ D γ , notice that δ + 1 ∈ supp( p ), let dom( s ( p )) =dom( s ( p )) ∪ { δ } and define s ( p ) such that s ( p ) ⊆ s ( p ) and s ( p )( δ ) = n where p (cid:13) ˙ x ∈ ˙ Q n . Then (5) issatisfied and we need to show (6).If p = ( p , ˙ x ) , q = ( q , ˙ y ) ∈ D γ , s ( p ) and s ( q ) agree on their common domain, and r ≤ p , q , then r (cid:13) ˙ x, ˙ y ∈ ˙ Q n so that r forces ˙ x, ˙ y to be compatible and there is a name ˙ z such that r (cid:13) ˙ z ≤ ˙ x, ˙ y . Letting r = ( r , ˙ z ), we see that r is as required.Finally assume γ is a limit ordinal. Since P γ = lim dir δ<γ P δ , D γ = S δ<γ D δ is dense in P γ , and properties(3) through (6) hold vacuously.Successor step α = β + 1. Then P αγ = ( P βγ ) κ / D for all γ ≤ µ . We let I α = ( I β ) κ / D . Clearly, I α is linearlyordered with [ v ] ≤ [ w ] if { ξ < κ : v ( ξ ) ≤ w ( ξ ) } ∈ D for v, w : κ → I β , and I β ⊆ I α . Next let D αγ = { [ f ] ∈ P αγ : f : κ → P βγ and { ξ < κ : f ( ξ ) ∈ D βγ } ∈ D} . Clearly D αγ is dense and D βδ ⊆ D αγ for δ ≤ γ . For such f , and for all ξ with f ( ξ ) ∈ D βγ , dom( s β ( f ( ξ ))) ⊆ I β isfinite, say dom( s β ( f ( ξ ))) = { i ξ < ... < i ξm ξ − } . By ω -completeness of D , there is m such that { ξ < κ : m ξ = m } belongs to D . Define v j : κ → I β by v j ( ξ ) = (cid:26) i ξj if this is defined0 otherwisefor j < m . (Note that the first case occurs D -almost everywhere.) Then { [ v ] < ... < [ v m − ] } ⊆ I α and we letdom( s α ([ f ])) = { [ v j ] : j < m } . Applying once more ω -completeness of D , we see that there are n j , j < m ,such that { ξ < κ : s β ( f ( ξ ))( i ξj ) = n j } ∈ D . Thus we let s α ([ f ])([ v j ]) = n j for j < m . Clearly s β ⊆ s α .To see (5), if δ < γ and s α ([ f ])( δ ) = n , then { ξ < κ : s β ( f ( ξ ))( δ ) = n } ∈ D . Also, h γ,αδ +1 ,α ([ f ]) = h h γ,βδ +1 ,β ( f ( ξ )) : ξ < κ i / D = h ( h γ,βδ,β ( f ( ξ )) , ˙ x ξ ) : ξ < κ i / D = ( h γ,αδ,α ([ f ]) , ˙ y ) ∈ ( P βδ ) κ / D ⋆ ˙ Q where ˙ y = h ˙ x ξ : ξ < κ i / D (see Lemmata 8 and 9). By induction hypothesis (5) we know that { ξ < κ : h γ,βδ,β ( f ( ξ )) (cid:13) ˙ x ξ ∈ ˙ Q n } ∈ D . Therefore h γ,αδ,α ([ f ]) (cid:13) ˙ y ∈ ˙ Q n .To prove (6), assume δ < γ , [ f ] , [ g ] ∈ D αγ , s α ([ f ]) and s α ([ g ]) agree on their common domain, and [ h ] ≤ h γ,αδ,α ([ f ]) , h γ,αδ,α ([ g ]) in P αδ . Let { [ v ] < ... < [ v n − ] } list the common domain of s α ([ f ]) and s α ([ g ]). Then { ξ < κ : V ( ξ ) = { v ( ξ ) < ... < v n − ( ξ ) } = dom( s β ( f ( ξ ))) ∩ dom( s β ( g ( ξ ))) and s β ( f ( ξ )) ↾ V ( ξ ) = s β ( g ( ξ )) ↾ V ( ξ ) } belongs to D . Also, the set { ξ < κ : h ( ξ ) ≤ h γ,βδ,β ( f ( ξ )) , h γ,βδ,β ( g ( ξ )) } belongs to D . For ξ which belong toboth sets we find, by induction hypothesis (6), h ( ξ ) ∈ P βγ with h ( ξ ) ≤ f ( ξ ) , g ( ξ ) and h γ,βδ,β ( h ( ξ )) = h ( ξ ). So[ h ] ≤ [ f ] , [ g ] and h γ,αδ,α ([ h ]) = [ h ], as required.Limit step α . Let I α = S β<α I β , equipped with the obvious ordering. As in the basic step, we define D αγ and s α by recursion on γ ≤ µ .The cases γ = 0 and γ = δ + 1 are identical to the basic step. The only difference is that, this time, D αγ must contain all D βγ , and that s α must extend all s β , for β < α .So assume γ is a limit ordinal, and D αδ and s α ↾ D αδ have been defined for δ < γ . Since supports are finite(see also Lemmas 11 and 12), we know that ¯ P αγ < ◦ P αγ where ¯ P αγ := lim dir δ<γ P αδ , by Lemma 10. By the proof ofLemma 12, elements of P αγ are formal products p · ¯ p with p ∈ P βγ for some β < α , ¯ p ∈ ¯ P αγ , and h ¯ γ,α ¯ γ,β (¯ p ) = h γ,β ¯ γ,β ( p )(where we use ¯ γ as an index for the direct limit ¯ P jγ of the P jδ , δ < γ , which completely embeds into P jγ , where j = α, β ). By strengthening p and ¯ p , if necessary, we may assume p ∈ D βγ . By further strengthening ¯ p , we may8ssume ¯ p ∈ ¯ D αγ := S δ<γ D αγ . In general, we will then only have h ¯ γ,α ¯ γ,β (¯ p ) ≤ h γ,β ¯ γ,β ( p ), but this does not concern usbecause the collection of formal products satisfying this weaker condition is obviously forcing equivalent withthe original P αγ . Hence, if we let D αγ consist of formal products p · ¯ p with p ∈ D βγ for some β < α , ¯ p ∈ ¯ D αγ , and h ¯ γ,α ¯ γ,β (¯ p ) ≤ h γ,β ¯ γ,β ( p ), then D αγ is dense in P αγ . Clearly ¯ D αγ ⊆ D αγ and D βγ ⊆ D αγ for β < α . For such p · ¯ p ∈ D αγ , wedefine s α ( p · ¯ p ) by dom( s α ( p · ¯ p )) = dom( s α (¯ p )) ∪ { i ∈ dom( s β ( p )) : i ≥ δ for all δ < γ } and s α ( p · ¯ p )( i ) = (cid:26) s α (¯ p )( i ) for i ∈ dom( s α (¯ p )) s β ( p )( i ) for i ∈ dom( s β ( p )) with i > δ for all δ < γ. (5) is immediate by induction hypothesis (5) for ¯ p .To prove (6), let δ < γ , p · ¯ p, q · ¯ q ∈ D αγ , s α ( p · ¯ p ) and s α ( q · ¯ q ) agree on their common domain, and r ≤ h γ,αδ,α ( p · ¯ p ) , h γ,αδ,α ( q · ¯ q ) in P αδ . It is immediate that h γ,αδ,α ( p · ¯ p ) = h ¯ γ,αδ,α (¯ p ) and similarly for q · ¯ q . Thus, byinduction hypothesis (6), there is ¯ r ≤ ¯ p, ¯ q in ¯ P αγ such that h ¯ γ,αδ,α (¯ r ) = r . Let β p , β q < α be such that p ∈ D β p γ and q ∈ D β q γ . Without loss of generality β p ≤ β q , and we let β := β q . By correctness h γ,β ¯ γ,β ( p ) = h γ,β p ¯ γ,β p ( p ). Sowe know that h ¯ γ,α ¯ γ,β (¯ r ) ≤ h ¯ γ,α ¯ γ,β (¯ p ) ≤ h ¯ γ,α ¯ γ,β p (¯ p ) ≤ h γ,β p ¯ γ,β p ( p ) = h γ,β ¯ γ,β ( p ) and h ¯ γ,α ¯ γ,β (¯ r ) ≤ h ¯ γ,α ¯ γ,β (¯ q ) ≤ h γ,β ¯ γ,β ( q ). Thus byinduction hypothesis (6) there is r ≤ p, q in P βγ such that h γ,β ¯ γ,β ( r ) = h ¯ γ,α ¯ γ,β (¯ r ). Hence h γ,αδ,α ( r · ¯ r ) = h ¯ γ,αδ,α (¯ r ) = r ,and r · ¯ r ≤ p · ¯ p, q · ¯ q is as required. d < a from a measurable cardinal Recall that
Hechler forcing D consists of ( s, x ) ∈ ω <ω × ω ω with s ⊆ x ordered by ( t, y ) ≤ ( s, x ) if t ⊇ s and y ≥ x everywhere. D is a Suslin σ -centered forcing adding a dominating real. In particular it is Suslin σ -linked.We outline the proof of: Theorem 14 (Shelah [Sh4]) . Assume ZFC + “there is a measurable cardinal” is consistent. Then so is d < a .More explicitly, if GCH holds, κ is measurable and λ > µ > κ are regular, then there is a ccc forcing extensionsatisfying b = d = µ and a = c = λ .Proof. As usual, let D be the κ -complete ultrafilter on κ . By recursion on α ≤ λ we construct iterations (withfinite supports) P α = h P αγ : γ ≤ µ i such that1. P = { , } and P δ = lim dir γ<δ P γ for limit δ ,2. P αγ +1 = P αγ ⋆ ˙ D for all α and γ ,3. P α +1 γ = ( P αγ ) κ / D for all α and γ ,4. P βδ is built according to Lemma 12 for limit β and δ ,5. if α ≤ β and γ ≤ δ , then P αγ < ◦ P βδ ,6. projections in all diagrams h P αγ , P αδ , P βγ , P βδ i , α < β and γ < δ , are correct.For β = 0 construct P according to the first two clauses, that is, P is a µ -stage finite support iteration ofHechler forcing. If β = α + 1, P β is the ultrapower of P α , see Lemma 11. Thus (3) is satisfied, and (5) and (6)follow from Lemmas 4 and 8. Furthermore, (2) also holds for by definition, the induction hypothesis for (2),and Lemma 9 we have P βγ +1 = ( P αγ +1 ) κ / D = ( P αγ ⋆ ˙ D ) κ / D = ( P αγ ) κ / D ⋆ ˙ D = P βγ ⋆ ˙ D . Finally, for limit β , we define P βδ for successor δ = γ + 1 by (2), and for limit δ , by (4). (5) and (6) follow fromLemma 2 and the comment after the definition of correctness (in Subsection 0.1) in the first case, and fromLemma 12, in the second case. This completes the recursive construction.By Lemma 13 all P αγ are ccc. In particular so is P := P λµ .We next note that | P | = λ . Indeed, by induction on α , we have that max {| α | , µ } ≤ | P αµ | ≤ max {| α | , µ } κ forall α ≤ λ : | P µ | = µ = µ κ is obvious. For successor, the formula follows from | P α +1 µ | ≤ | P αµ | κ , and, for limit,from | P βµ | = P α<β | P αµ | . In particular, a standard argument with nice names gives c ≤ λ .9he last iteration P λ = h P λγ : γ ≤ µ i cofinally often adds a dominating real (more explicitly, P λγ +1 adds aHechler real dominating the reals of the P λγ -extension), and therefore b = d = µ holds in the extension. (In fact,the sequence of µ Hechler reals is already added by P , see also Lemma 7).To see a ≥ λ , we use Lemma 6: assume A is an a.d. family of size ν for some ν < λ in the generic extension.If ν < µ then A is not maximal because b ≤ a in ZFC. If ν ≥ µ > κ , by the ccc and the regularity of λ , thereis an α < λ such that ˙ A is a P αµ -name. By Lemma 6, we then see that P α +1 µ = ( P αµ ) κ / D forces that ˙ A is notmaximal. This completes the proof. We collect a number of results obtained by related methods.In his original work [Sh4], Shelah also obtains the consistency of u < a by basically the same method.Recall here that, given a nonprincipal ultrafilter U on ω , B ⊆ U is a base of U if for every A ∈ U there is B ∈ B such that B ⊆ A . If we only require that B ⊆ [ ω ] ω , B is called a π -base . The character χ ( U ) ( π -character πχ ( U ), respectively) of U is the least size of a base ( π -base, resp.) of U . We define u := min { χ ( U ) : U is a nonprincipal ultrafilter on ω } , the ultrafilter number . Theorem 15 (Shelah [Sh4]) . Assume ZFC + “there is a measurable cardinal” is consistent. Then so is u < a .More explicitly, if GCH holds, κ is measurable and λ > µ > κ are regular, then there is a ccc forcing extensionsatisfying u = b = d = µ and a = c = λ . This is proved by replacing Hechler forcing D with Laver forcing L U with an ultrafilter U in the iteratedforcing construction of the proof of Theorem 14. For details see also [Br4].In two subsequent papers ([Sh5], [Sh6]), Shelah used this construction for obtaining several results aboutnon-convexity of the character spectrum Spec( χ ) = { χ ( U ) : U is a nonprincipal ultrafilter on ω } . Theorem 16 (Shelah [Sh5]) . Assume ZFC + “there is a measurable cardinal” is consistent. Then so is “
Spec( χ ) is not convex”. More explicitly, if GCH holds, κ is measurable and λ > κ > µ are uncountable regular, thenthere is a ccc forcing extension satisfying u = b = d = µ , c = λ , { µ, λ } ⊆ Spec( χ ) and κ / ∈ Spec( χ ) . Using two measurables, this can be combined with Theorem 15 [Sh5, Theorem 1.1]. Furthermore, there isan extension to a variant of the π -character spectrum [Sh5, Theorem 2.5]. Again, details can be found as wellin [Br4]. Theorem 17 (Shelah [Sh6]) . Given two disjoint sets Θ and Θ of regular cardinals such that θ <θ = θ forall θ ∈ Θ and all members of Θ are measurable cardinals, there is a partial order forcing Θ ⊆ Spec( χ ) and Θ ∩ Spec( χ ) = ∅ . This is a generalization of Theorem 16, obtained by a product of a ccc forcing with iterated ultrapowers (toguarantee the measurables in Θ are not characters) and a forcing not adding reals but adjoining ultrafilters ofcharacter in Θ . Theorem 18 (Shelah [Sh6]) . Assuming the consistency of infinitely many strongly compact cardinals, it isconsistent that for any A ⊆ ω \ { } , {ℵ n : n ∈ A } is the set of characters below ℵ ω . This is proved by combining the previous theorem with a product of Levy collapses. Some of these resultsanswer questions originally addressed in [BS].A family
S ⊆ [ ω ] ω is a splitting family if for all A ∈ [ ω ] ω there is B ∈ S with | A ∩ B | = | A \ B | = ω (we say B splits A ). The splitting number s is the smallest cardinality of a splitting family. Iterating theultrapower construction as in Subsection 1.2 in a matrix and also destroying splitting families while preservingan unbounded family added by the first iteration, but taking direct limits in the limit step (this makes thecomplex Lemmas 12 and 13 unnecessary) one obtains: Theorem 19 (Brendle and Fischer [BF]) . Assume GCH holds, κ is measurable and λ > µ > κ are regularcardinals. Then there is a ccc forcing extension in which b = µ < s = a = c = λ . This generalizes an old result of Shelah [Sh1] who proved that b = ℵ < s = a = c = ℵ is consistent with acountable support iteration of proper forcing. 10 Templates
In standard well-ordered iterations h P α , ˙ Q α : α < µ i where µ is an ordinal, the initial segments of the iteration P α and the iterands ˙ Q α are handed down by the same recursion, with the latter being P α -names, one defines P α +1 = P α ⋆ ˙ Q α , and then one has to specify what to do in the limit step. However, while wellfoundednessis a necessity for recursively defining the initial segments in many cases, this is not so for the iterands, andthey may well be indexed by a non-wellfounded structure. This is what iterations along templates do. Theindex set of their iterands is an arbitrary linear order L , and their initial segments are produced by specifyinga wellfounded subfamily I of P ( L ) and then recursively defining the segment P ↾ A for A ∈ I . Since we need toconsider the same iterand Q x for x ∈ L over various initial extensions, definability of such iterands is crucial,and iteration along templates can be seen as a generalization of finite support iteration of Suslin ccc forcing.See Subsection 2.1 for details (the exposition follows largely [Br3]).As an application, we provide a complete proof of Shelah’s proof of the consistency of d < a in ZFC [Sh4] inSubsection 2.2 (Theorem 29), following roughly [Br1]. Subsection 2.3 presents further results obtained by thetemplate method. Let h L, ≤ L i be a linear order. For x ∈ L let L x = { y ∈ L : y < L x } be the initial segment determined by x . Definition 20. A template is a pair ( L, ¯ I = {I x : x ∈ L } ) such that h L, ≤ L i is a linear order, I x ⊆ P ( L x ) forall x ∈ L , and1. I x contains all singletons and is closed under unions and intersections,2. I x ⊆ I y for x < L y ,3. I := S x ∈ L I x ∪ { L } is wellfounded with respect to inclusion, as witnessed by the depth function Dp I : I →
On.If A ⊆ L and x ∈ L , we define I x ↾ A := { B ∩ A : B ∈ I x } , the trace of I x on A and let ¯ I ↾ A = {I x ↾ A : x ∈ A } . L is meant to be an index set for the iterands of an iteration while I describes in a sense the support. Thewellfoundedness condition 3 is crucial because it allows for a recursive definition of the iteration. By clause 1,a real r y added at stage y will be generic over a real r x added at stage x for x < L y while by clause 2, if r x isgeneric over some initial stage of the iteration then so is r y . Conditions 1 and 2 together imply that I is alsoclosed under intersections and unions.Note that ( A, ¯ I ↾ A ) is a template as well and, if B ∈ I and C = B ∩ A ∈ I ↾ A , then Dp I ↾ A ( C ) ≤ Dp I ( B ).Thus, with every A ⊆ L we can associate its depth , Dp( A ) := Dp I ↾ A ( A ). Dp has the following properties: Lemma 21.
1. If B ∈ I ↾ A then for any C ⊆ B , C ∈ I ↾ B iff C ∈ I ↾ A .2. If B ∈ I ↾ A , then Dp I ↾ A ( B ) = Dp( B ) .3. If B ⊆ A then Dp( B ) ≤ Dp( A ) and if additionally B ∈ I ↾ A and B ( A then Dp( B ) < Dp( A ) .4. If B = B ∪ { x } ⊆ A , x / ∈ B , and B , B ∈ I ↾ A then Dp( B ) = Dp( B ) + 1 .Proof. (1) If C ∈ I ↾ A then there is D ∈ I with C = A ∩ D . Thus C = B ∩ D and C ∈ I ↾ B follows.On the other hand if C ∈ I ↾ B then there is D ∈ I with C = B ∩ D . Also there is D ′ ∈ I with B = A ∩ D ′ .Thus C = A ∩ ( D ′ ∩ D ) and therefore C ∈ I ↾ A .(2) is immediate by (1), and (3) is obvious.(4) Dp( B ) ≥ Dp( B )+ 1 follows by (3), and Dp( B ) ≤ Dp( B )+ 1 is shown by induction on Dp( B ). SupposeDp( B ) = α . Let C ( B with C ∈ I ↾ B and x ∈ C . Put C = C ∩ B = C \ { x } . Then C ∈ I ↾ B and C ( B ,and therefore Dp( C ) < α by (3). Since also C , C ∈ I ↾ A , by induction hypothesis Dp( C ) ≤ Dp( C ) + 1 ≤ α .Thus Dp( B ) ≤ α + 1.Call a Suslin ccc forcing notion Q correctness-preserving if for every diagram h P i i with correct projections,the projections in the diagram h P i ⋆ ˙ Q i are correct as well. Definition and Theorem 22.
Assume ( L, ¯ I ) is a template. Also assume ( Q x : x ∈ L ) is a family ofcorrectness-preserving Suslin ccc forcing notions whose definition lies in the ground model. By recursion-induction on Dp( A ) , A ⊆ L , . we define the partial order P ↾ A ,2. we prove that P ↾ D ⊆ P ↾ A and ≤ P ↾ D ⊆≤ P ↾ A for D ⊆ A ,3. we prove that P ↾ A is transitive,4. we describe how P ↾ A is obtained from P ↾ B where B ( A with B ∈ I ↾ A (so Dp( B ) < Dp( A ) ),5. we prove that P ↾ D< ◦ P ↾ A for D ⊆ A ,6. we prove that for D ⊆ L with Dp( D ) ≤ Dp( A ) , we have P ↾ ( A ∩ D ) = P ↾ A ∩ P ↾ D ,7. we prove correctness in the sense that if A ′ , D ( A , and D ′ = A ′ ∩ D , then projections in the diagram h P ↾ D ′ , P ↾ A ′ , P ↾ D, P ↾ A i are correct. P ↾ L is called iteration along a template .Definition and Proof. (1) First consider the case Dp( A ) = 0. This is equivalent to A = ∅ . Let P ↾ ∅ = {∅} .So assume Dp( A ) >
0. Let P ↾ A consist of all finite partial functions p with domain contained in A and suchthat, putting x = max(dom( p )), there is B ∈ I x ↾ A (so Dp( B ) < Dp( A )) such that p ↾ ( A ∩ L x ) ∈ P ↾ B and p ( x )is a P ↾ B -name for a condition in ˙ Q x (where we construe ˙ Q x as a P ↾ B -name as well).The ordering on P ↾ A is given as follows. q ≤ P ↾ A p if dom( q ) ⊇ dom( p ) and, putting x = max(dom( q )), thereis B ∈ I x ↾ A such that q ↾ ( A ∩ L x ) ∈ P ↾ B and • either x / ∈ dom( p ), p ∈ P ↾ B , and q ↾ ( A ∩ L x ) ≤ P ↾ B p , • or x ∈ dom( p ), p ↾ ( A ∩ L x ) ∈ P ↾ B , q ↾ ( A ∩ L x ) ≤ P ↾ B p ↾ ( A ∩ L x ), and p ( x ) and q ( x ) are P ↾ B -names forconditions in ˙ Q x such that q ↾ ( A ∩ L x ) (cid:13) P ↾ B q ( x ) ≤ ˙ Q x p ( x ).Concerning the first alternative here, note that it is easy to see that given p ∈ P ↾ A and x > L max(dom( p ))with x ∈ A , there is B ∈ I x ↾ A such that p ∈ P ↾ B . (Indeed, let y = max(dom( p )). Then there is B ∈ I y ↾ A such that p ↾ ( A ∩ L y ) ∈ P ↾ B and p ( y ) is a P ↾ B -name for a condition on ˙ Q y . Now B = B ∪ { y } ∈ I x ↾ A and p ∈ P ↾ B .)(2) Let D ⊆ A and p ∈ P ↾ D . Also let x = max(dom( p )). There is E ∈ I x ↾ D such that p ↾ ( D ∩ L x ) ∈ P ↾ E and p ( x ) is a P ↾ E -name for a condition in ˙ Q x . Let B ∈ I x ↾ A be such that E = B ∩ D . Then E ⊆ B and, byinduction hypothesis (2), p ↾ ( D ∩ L x ) = p ↾ ( A ∩ L x ) ∈ P ↾ B . By induction hypothesis (5), p ( x ) is a P ↾ B -name aswell. Therefore p ∈ P ↾ A .The inclusion for the order is proved similarly.(3) We use completeness of the embeddings (induction hypothesis (5)) and closure of the template underunions.Assume r ≤ P ↾ A q ≤ P ↾ A p . Let y and x be the maximal elements of dom( r ) and dom( q ), respectively. Thereare B y ∈ I y ↾ A and B x ∈ I x ↾ A witnessing the order relationship. In particular, r ↾ ( A ∩ L y ) , q ↾ ( A ∩ L y ) ∈ P ↾ B y , q ↾ ( A ∩ L x ) , p ↾ ( A ∩ L x ) ∈ P ↾ B x , and r ↾ ( A ∩ L y ) ≤ P ↾ B y q ↾ ( A ∩ L y ), q ↾ ( A ∩ L x ) ≤ P ↾ B x p ↾ ( A ∩ L x ). Let B = B y ∪ B x ∈ I y ↾ A . We check that B witnesses r ≤ P ↾ A p .If x < y , we have x ∈ B , (2) gives us q, p ↾ ( A ∩ L x ) ∈ P ↾ B and therefore also p ∈ P ↾ B , and r ↾ ( A ∩ L y ) ≤ P ↾ B q ≤ P ↾ B p . By induction hypothesis (3), r ↾ ( A ∩ L y ) ≤ P ↾ B p , and r ≤ P ↾ A p follows.If x = y , r ↾ ( A ∩ L y ) ≤ P ↾ B q ↾ ( A ∩ L y ) ≤ P ↾ B p ↾ ( A ∩ L y ), and thus by induction hypothesis (3), r ↾ ( A ∩ L y ) ≤ P ↾ B p ↾ ( A ∩ L y ), so we are done if x / ∈ dom( p ). Assume x ∈ dom( p ). Then r ( y ) and q ( y ) are P ↾ B y -names, and q ( x )and p ( x ) are P ↾ B x -names. By induction hypothesis (5), they are all P ↾ B -names and r ↾ ( A ∩ L y ) (cid:13) P ↾ B r ( y ) ≤ q ( y ) ≤ p ( y ), as required.(4) We consider several cases.Case 1. There is x = max( A ) such that A := A ∩ L x = A \ { x } ∈ I x ↾ A . Then P ↾ A is easily seen to be thestandard two-step iteration P ↾ A ⋆ ˙ Q x where ˙ Q x is a P ↾ A -name, for if p ∈ P ↾ A , then p ↾ A ∈ P ↾ A and p ( x ) isa P ↾ A -name for a condition in ˙ Q x . In particular, P ↾ A < ◦ P ↾ A .Case 2a. There is x = max( A ), but A = A ∩ L x / ∈ I x ↾ A . Let p ∈ P ↾ A . There is B ∈ I x ↾ A such that p ↾ A ∈ P ↾ B and either x / ∈ dom( p ) or p ( x ) is a P ↾ B -name. Note B ( A . Let B := B ∪ { x } ( A . ClearlyDp( B ) < Dp( A ). (This holds because Dp( B ) = Dp( B ) + 1 and Dp( A ) ≥ Dp( B ) + ω by parts 3 and 4 ofLemma 21.) Also B = B ∩ L x ∈ I x ↾ B and thus p ∈ P ↾ B . Since I x ↾ A is closed under unions, the collection of B ( A with B ∩ L x ∈ I x ↾ A is directed. Also note that by induction hypothesis (5), if B ⊆ B ′ are of this form,then P ↾ B< ◦ P ↾ B ′ . Therefore P ↾ A is the direct limit of the P ↾ B with B ( A and B ∩ L x ∈ I x ↾ A .12ase 2b. A has no maximum. Let p ∈ P ↾ A be a condition. There is x > max(dom( p )) with x ∈ A andtherefore, as remarked earlier, there is B ∈ I x ↾ A such that p ∈ P ↾ B . Clearly Dp( B ) < Dp( A ). The collectionof B ∈ I ↾ A with B ∈ I x ↾ A for some x ∈ A is directed and therefore, using again induction hypothesis (5), wesee that P ↾ A is the direct limit of the P ↾ B for such B .(5) Let D ⊆ A . We split the proof into cases according to (4) for A .Case 1. Let D := D ∩ A = D ∩ L x ∈ I x ↾ D . Since Dp( A ) < Dp( A ), we may use the induction hypothesis(5) and see P ↾ D < ◦ P ↾ A . We know already P ↾ A ∼ = P ↾ A ⋆ ˙ Q x . If x / ∈ D , then D = D , and P ↾ D< ◦ P ↾ A < ◦ P ↾ A follows. If x ∈ D , then P ↾ D ∼ = P ↾ D ⋆ ˙ Q x where ˙ Q x is a P ↾ D -name. Since Q x is Suslin ccc, P ↾ D< ◦ P ↾ A followsfrom Lemma 2.Case 2a. Assume first D = D ∩ L x ∈ I x ↾ D . So there is B ∈ I x ↾ A such that D = D ∩ B . Put B := B ∪ { x } ( A . Then D ⊆ B and P ↾ D< ◦ P ↾ B< ◦ P ↾ A where the first < ◦ is by induction hypothesis (5)(because Dp( B ) < Dp( A )) and the second, by Case 2a of (4) above.So assume D / ∈ I x ↾ D . Suppose first that x ∈ D . By Case 2a of (4) applied to D instead of A , P ↾ D is thedirect limit of the P ↾ E where E ( D with E ∩ L x ∈ I x ↾ D . Each such E is of the form D ∩ B where B ( A and B ∩ L x ∈ I x ↾ A . Conversely, any D ∩ B is such an E . Using the inductive hypothesis for correctness (7),we see that projections in all diagrams of the form h P ↾ ( D ∩ B ) , P ↾ B, P ( D ∩ B ′ ) , P ↾ B ′ i , where B ⊆ B ′ ( A with B ∩ L x , B ′ ∩ L x ∈ I x ↾ A , are correct. By Lemma 3, this means, however, that the direct limit of the P ↾ E completely embeds into the direct limit of the P ↾ B , as required.Suppose finally that x / ∈ D . Then D = D and, since D / ∈ I x ↾ D , we must be in Case 2 for D and,depending on whether D has a maximum or not, we are either in Case 2a or Case 2b. In the first case, by (4),if y = max( D ), then P ↾ D is the direct limit of P ↾ E where E ( D with E ∩ L y ∈ I y ↾ D . In the second case,again by (4), P ↾ D is the direct limit of P ↾ E where E ( D with E ∈ I y ↾ D for some y ∈ D . In either case, such E belongs to I x ↾ D (though not all E ∈ I x ↾ D are necessarily of this form). Since P ↾ E ⊆ P ↾ D by (2) and thecollection of E ∈ I x ↾ D is directed, P ↾ D must in fact be the direct limit of P ↾ E where E ∈ I x ↾ D . We note againthat such E agree with the sets of the form D ∩ B where B ( A and B ∩ L x ∈ I x ↾ A . By correctness and theinductive hypothesis for (7), we can apply Lemma 3 and see that P ↾ D< ◦ P ↾ A .Case 2b. If D ∈ I x ↾ D for some x ∈ A , we are done because then D ⊆ B for some B ∈ I x ↾ A , and P ↾ D< ◦ P ↾ B< ◦ P ↾ A by induction hypothesis (5) and Case 2b of (4) above.So assume D / ∈ I x ↾ D for any x ∈ A . Again, we must be in Case 2 for D and, as in the last paragraphof Case 2a in (5), we see that P ↾ D is the direct limit of P ↾ E where E ∈ I x ↾ D for some x ∈ A . Using againLemma 3, we conclude that P ↾ D< ◦ P ↾ A .(6) P ↾ ( A ∩ D ) ⊆ P ↾ A ∩ P ↾ D is immediate by part (2). So assume p ∈ P ↾ A ∩ P ↾ D . Let x = max(dom( p )).There are B ∈ I x ↾ A and E ∈ I x ↾ D such that p ↾ L x ∈ P ↾ B ∩ P ↾ E and p ( x ) is both a P ↾ B -name and a P ↾ E -name,and thus a P ↾ B ∩ P ↾ E -name. (To see the latter simply note that since p ( x ) is a name for a real, being a P ↾ B -namemeans that all Boolean values [[ p ( x )( i ) = j ]] belong to P ↾ B , and similarly for P ↾ E . Hence the Boolean valuesmust belong to P ↾ B ∩ P ↾ E .) Since Dp( B ) < Dp( A ) and Dp( E ) < Dp( D ) ≤ Dp( A ), we may apply the inductionhypothesis (6) and get P ↾ ( B ∩ E ) = P ↾ B ∩ P ↾ E . Note that B ∩ E ∈ I x ↾ ( A ∩ D ). Therefore p ∈ P ↾ ( A ∩ D ), asrequired.(7) Again we split into cases according to (4) for A .Case 1. x = max( A ), A = A ∩ L x ∈ I x ↾ A , and P ↾ A = P ↾ A ⋆ ˙ Q x . If x / ∈ A ′ , we get A ′ ⊆ A and thus P ↾ A ′ < ◦ P ↾ A , and correctness follows from induction hypothesis (7). Similarly if x / ∈ D . So we may assume x ∈ A ′ ∩ D = D ′ . Then let A ′ = A ′ ∩ L x , D = D ∩ L x , D ′ = D ′ ∩ L x , apply the induction hypothesis (7) tothe diagram h P ↾ D ′ , P ↾ A ′ , P ↾ D , P ↾ A i and use that Q x is correctness-preserving. (This is the only place wherethis assumption is needed.)Case 2a. x = max( A ), A = A ∩ L x / ∈ I x ↾ A , and P ↾ A is the direct limit of the P ↾ B where B ( A and B ∩ L x ∈ I x ↾ A . Let p ∈ P ↾ A ′ ⊆ P ↾ A . We need to show that the projections agree, that is, that h A ′ D ′ ( p ) = h AD ( p ).First assume that D = D ∩ L x ∈ I x ↾ D . Then, by the discussion in (5) (Case 2a), D ⊆ B for a B as aboveand, enlarging B if necessary, we may assume p ∈ P ↾ B . Let B ′ = A ′ ∩ B . By (6) we know that p ∈ P ↾ B ′ .By Dp( B ) < Dp( A ) and induction hypothesis (7), h A ′ D ′ ( p ) = h B ′ D ′ ( p ) = h BD ( p ) = h AD ( p ), as required. If A ′ = A ′ ∩ L x ∈ I x ↾ A ′ , then, by the symmetry of correctness, the same argument works.So assume D / ∈ I x ↾ D and A ′ / ∈ I x ↾ A ′ . By the discussion in (5) (Case 2a), we know that P ↾ D ( P ↾ A ′ ,respectively) is the direct limit of P ↾ ( D ∩ B ) ( P ↾ ( A ′ ∩ B ), resp.) where B is as above. Again fix such B suchthat p ∈ P ↾ B . (6) gives us p ∈ P ↾ ( A ′ ∩ B ). Using Dp( B ) < Dp( A ) and the induction hypothesis (7), we seethat h A ′ ∩ B,D ′ ∩ B ( p ) = h B,D ∩ B ( p ). Again by induction hypothesis (7), we have that h B ,D ∩ B ( p ) = h B,D ∩ B ( p )for any B ⊆ B ( A with B ∩ L x ∈ I x ↾ A . Since P ↾ A and P ↾ D are the direct limits of such P ↾ B and P ↾ ( D ∩ B ), respectively, h AD ( p ) = h B,D ∩ B ( p ) follows (see Lemma 3). In case D ′ = D ′ ∩ L x ∈ I x ↾ D ′ , argue as13n the previous paragraph to see that we may assume D ′ ⊆ B and thus obtain h A ′ D ′ ( p ) = h A ′ ∩ B,D ′ ( p ), while if D ′ / ∈ I x ↾ D ′ , h A ′ D ′ ( p ) = h A ′ ∩ B,D ′ ∩ B ( p ) follows as in the previous sentence. In either case, h A ′ D ′ ( p ) = h AD ( p ),and we are done.Case 2b. Depending on whether D ∈ I x ↾ D or A ′ ∈ I x ↾ A ′ for some x ∈ A , we repeat the previous argument,referring to Case 2b of (5).While the definition of the iteration along a template looks complicated, clause (4) should be seen as sayingthat such iterations are recursively built up using the two simple operations of two-step iteration and directlimit – as are standard finite support iterations (fsi). Note in this context that an fsi is the special case where L = µ is an ordinal and I α = { β ∪ F : β ≤ α and F ∈ [ α ] <ω } for α < µ . Lemma 23.
Assume ( L, ¯ I ) is a template, and the Q x , x ∈ L , are correctness-preserving Suslin σ -linked partialorders coded in the ground model, Q x = S n Q x,n . Then, for any A ⊆ L , P ↾ A is a ccc p.o.Proof. We argue in three steps.Step 1. By induction on Dp( A ), we show that given p ∈ P ↾ A , there is q ≤ P ↾ A p such that for all x ∈ dom( q )there are B ∈ I x ↾ A and n = n q,x such that q ↾ ( A ∩ L x ) ∈ P ↾ B and q ↾ ( A ∩ L x ) (cid:13) P ↾ B q ( x ) ∈ ˙ Q x,n .Indeed, let p ∈ P ↾ A . Also let x = max(dom( p )). There is B ∈ I x ↾ A such that p ↾ ( A ∩ L x ) ∈ P ↾ B and p ( x )is a P ↾ B -name for a condition in ˙ Q x . Thus we may find r ∈ P ↾ B and n ∈ ω with r ≤ P ↾ B p ↾ ( A ∩ L x ) and suchthat r (cid:13) P ↾ B p ( x ) ∈ ˙ Q x,n . Since Dp( B ) < Dp( A ), there is q ∈ P ↾ B with q ≤ P ↾ B r satisfying the inductionhypothesis. Let q ∈ P ↾ A be such that dom( q ) = dom( q ) ∪ { x } , q ↾ ( A ∩ L x ) = q and q ( x ) = p ( x ). Then q is asrequired.Step 2. Assume p, q ∈ P ↾ A are as in Step 1, that is, the n p,x and n q,x exist for all x ∈ dom( p ) and x ∈ dom( q ),respectively. Also suppose that n p,x = n q,x for all x ∈ dom( p ) ∪ dom( q ). Then p and q are compatible.This is proved by building a common extension by recursion on dom( p ) ∪ dom( q ). For x = min(dom( p ) ∪ dom( q )), r x ∈ P ↾ ( A ∩ L x ) is the trivial condition. Assume r x ∈ P ↾ ( A ∩ L x ) has been produced for some x ∈ dom( p ) ∪ dom( q ). Let y be the successor of x in dom( p ) ∪ dom( q ) or let y = ∞ if x = max(dom( p ) ∪ dom( q )). In thelatter case also let L ∞ = L . If x ∈ dom( p ) \ dom( q ), let r y ∈ P ↾ ( A ∩ L y ) be such that dom( r y ) = dom( r x ) ∪ { x } , r y ↾ ( A ∩ L x ) = r x , and r y ( x ) = p ( x ). If x ∈ dom( q ) \ dom( p ), define r y analogously. If x ∈ dom( p ) ∩ dom( q ),find r y ↾ ( A ∩ L x ) ≤ r x and r y ( x ) such that r y ↾ ( A ∩ L x ) (cid:13) P ↾ ( A ∩ L x ) r y ( x ) ≤ p ( x ) , q ( x ). This is possible because n p,x = n q,x . Letting r = r ∞ , we see that r is a common extension of p and q .Step 3. ccc-ness now follows by a straightforward ∆-system argument. Lemma 24.
Let ( L, ¯ I ) be a template. Also assume the Q x are as in the previous lemma. Let A ⊆ L .1. If p ∈ P ↾ A , then there is a countable C ⊆ A such that p ∈ P ↾ C .2. If ˙ f is a P ↾ A -name for a real, then there is a countable C ⊆ A such that ˙ f is a P ↾ C -name.Proof. This is proved by a simultaneous induction on Dp( A ).(1) Assume p ∈ P ↾ A . Let x = max(dom( p )). There is B ∈ I x ↾ A such that p ↾ ( A ∩ L x ) ∈ P ↾ B and p ( x ) is a P ↾ B -name. By induction hypothesis (1), there is a countable C ⊆ B such that p ↾ ( A ∩ L x ) ∈ P ↾ C . By inductionhypothesis (2), since p ( x ) is a name for a real, there is a countable C ⊆ B such that p ( x ) is a P ↾ C -name. Let C = C ∪ C ∪ { x } . Then C is countable and p ∈ P ↾ C .(2) Assume ˙ f is a P ↾ A -name. For i ∈ ω , let { p n,i : n ∈ ω } be a maximal antichain of conditions deciding˙ f ( i ). This uses the ccc-ness proved in the previous lemma. By part (1), there are countable C n,i ⊆ A such that p n,i ∈ P ↾ C n,i . Let C = S n,i C n,i . Then ˙ f is a P ↾ C -name. Corollary 25.
Let ( L, ¯ I ) be a template. Also assume the Q x are as in Lemma 23. Then P ↾ L is the direct limitof the P ↾ A where A ⊆ L is countable.Proof. By the previous lemma, P ↾ L = S { P ↾ A : A ⊆ L is countable } . Since the collection of countable subsetsof L is directed, P ↾ L = lim dir { P ↾ A : A ⊆ L is countable } follows.An easy consequence of this is for example that the limit of an fsi of Suslin ccc partial orders can berepresented as the direct limit of its countable fragments. More explicitly, if h P α , ˙ Q α : α < µ i is such aniteration, then P µ = lim dir { P A : A ⊆ µ is countable } where P A is obtained by only iterating the ˙ Q α with α ∈ A . When iterating along a wellorder, ccc-ness is preserved, so the σ -linkedness of Lemma 23 is not needed. .2 The consistency of d < a in ZFC For showing the consistency of d < a in ZFC, the ultrapower argument is replaced by an isomorphism-of-namesargument. Recall the following folklore result (see e.g. [Br1, Proposition 3.1] for a proof). Proposition 26.
Assume CH, and let λ = λ ω be a cardinal. In the forcing extension obtained by adding λ Cohen reals, every mad family has either size ℵ or size c = λ . STRATEGY.
The point of the proof of Proposition 26 is that using CH and a ∆-system argument, if ℵ ≤ κ < λ , and { ˙ A α : α < κ } is a name for an a.d. family, then one can produce another name ˙ A κ isomorphic to ℵ many ˙ A α and such that ˙ A κ is a.d. from all ˙ A α . By global homogeneity of Cohen forcing, the isomorphismproducing ˙ A κ comes from an automorphism of the whole forcing, but this is more than what is needed. Toobtain the consistency of d < a , it suffices to build a partial order forcing b = d = ℵ and having sufficient localhomogeneity to allow for the construction of ˙ A κ . This is exactly what the template method achieves. Madfamilies of size ℵ are ruled out in this scenario by b = ℵ . Lemma 27.
Hechler forcing is a correctness-preserving Suslin ccc forcing notion: assume projections in h P i i are correct. Then so are projections in h P i ⋆ ˙ D i .Proof. We may assume without loss of generality that P ∧ = { , } . Let q = ( p , ( s , ˙ f )) ∈ P ⋆ ˙ D and fix( s, f ) ≤ h , ∧ ( q ). We may suppose that s ⊇ s . Then, given any t ⊇ s such that t dominates f on its domain,there is p ′ ≤ p forcing t ≥ ˙ f on its domain. (This is so because ( t, g ) ≤ ( s, f ) with g ↾ [ | t | , ∞ ) = f ↾ [ | t | , ∞ ) iscompatible with q .)Now assume q = ( p , ( s , ˙ f )) ∈ P ⋆ ˙ D extends ( s, f ). Thus s ⊇ s and s dominates f on its domain. By theprevious paragraph find p ′ ≤ p forcing s ≥ ˙ f on its domain. By correctness, p and p ′ are compatible in P ∨ and, clearly, the common extension p · p ′ forces that ( s , ˙ f ) and ( s , ˙ f ) are compatible. Thus ( s, f ) ≤ h ∨ , ( q )in P ⋆ ˙ D .We now introduce the template for the proof of the main theorem (Theorem 29).Let µ and λ be cardinals. As usual, λ ∗ denotes (a disjoint copy of) λ with the reverse ordering. Elementsof λ will be called positive , and members of λ ∗ are negative . Choose a partition λ ∗ = S α<ω S α such that each S α is coinitial in λ ∗ . Define L = L ( µ, λ ) as follows. Elements of L are non-empty finite sequences x such that x (0) ∈ µ and x ( n ) ∈ λ ∗ ∪ λ for n >
0. The order is naturally given by x < y if • either x ( y and y ( | x | ) ∈ λ , • or y ( x and x ( | y | ) ∈ λ ∗ , • or x (0) < y (0), • or, letting n := min { m : x ( m ) = y ( m ) } > x ( n ) < y ( n ) in the natural ordering of λ ∗ ∪ λ .It is immediate that this is indeed a linear ordering. We identify sequences of length one with their range sothat µ ⊆ L is cofinal. Say x ∈ L is relevant if | x | ≥ x ( n ) is negative for odd n and positive for even n , x ( | x | − < ω , and whenever n < m are even such that x ( n ) , x ( m ) < ω , then there are β < α such that x ( n − ∈ S α and x ( m − ∈ S β . For relevant x , set J x = [ x ↾ ( | x | − , x ), the interval of nodes between x ↾ ( | x | −
1) and x in the order of L . Notice that if x < y are relevant, then either J x ∩ J y = ∅ or J x ( J y (inwhich case we also have | y | ≤ | x | , x ↾ ( | y | −
1) = y ↾ ( | y | −
1) and x ( | y | − ≤ y ( | y | − x ∈ L , let I x consist of finite unions of L α , where α ≤ x and α ∈ µ , of J y , where y ≤ x is relevant, andof singletons. Lemma 28. ( L, ¯ I = {I x : x ∈ L } ) is a template.Proof. By definition I x contains singletons, is closed under unions, and I x ⊆ I y for x ≤ y . Closure underintersections follows easily from the discussion immediately preceding the definition of I x . Hence it suffices toshow that I := S x ∈ L I x ∪ { L } is wellfounded.Assume A n , n ∈ ω , is a decreasing chain from I . Let α n be such that L α n occurs in A n as a component. The α n must be decreasing and therefore eventually constant. This means it suffices to consider the J x componentsof the A n and we may as well assume without loss of generality that A = J x hi , and that there is a finitelybranching tree T ⊆ ω <ω such that A n = S σ ∈ T ∩ ω n J x σ ∪ F n where the F n ⊆ L are finite, such that σ ⊆ τ implies J x τ ⊆ J x σ , and such that the J x σ , σ ∈ T ∩ ω n , are pairwise disjoint. Now note that if f ∈ [ T ] is a branch, then15he sequence { x f ↾ n : n ∈ ω } must eventually stabilize. Indeed, if | x f ↾ n | → ∞ , then { α : x f ↾ n ( | x f ↾ n | − ∈ S α for some n } would constitute a decreasing sequence of ordinals, by the definition of “relevant”, a contradiction.Therefore | x f ↾ n | is eventually constant. But then the decreasing sequence x f ↾ n ( | x f ↾ n | −
1) must be eventuallyconstant as well, and so must be x f ↾ n . Since T is a finitely branching tree this means that the total number of x σ is finite which in turn implies that the sequence of the A n eventually stabilizes.Note that, ordered by inclusion, L is a tree of countable height. Countable subtrees A, B ⊆ L are called isomorphic if there is a bijection ϕ = ϕ A,B : A → B such that for all x, y ∈ A and all n ∈ ω , • | ϕ ( x ) | = | x | , • ϕ ( x ) ↾ n = ϕ ( x ↾ n ), • x < y iff ϕ ( x ) < ϕ ( y ), • x ( n ) is positive iff ϕ ( x )( n ) is positive, • Q x = Q ϕ ( x ) , and • ϕ maps I ↾ A to I ↾ B .Since the trace of I on any countable set is countable, there are at most c many isomorphism types of trees.Note that, in view of the last two clauses, if A and B are isomorphic, then so are P ↾ A and P ↾ B , for the partialorder only depends on the structure of the template and on the iterands. If only the first four clauses hold, wecall the trees weakly isomorphic . Theorem 29 (Shelah [Sh4]) . Assume CH. Let λ > µ > ℵ be regular cardinals with λ ω = λ . Then there is accc forcing extension satisfying b = d = µ and a = c = λ .Proof. Take the template ( L, ¯ I ) introduced above. Let P = P ↾ L be the iteration of Hechler forcing along thistemplate, that is, Q x = D for all x ∈ L in Definition and Theorem 22. Using the description of P ↾ A as atwo-step iteration or direct limit in (4) of the latter, it is easy to prove by induction on Dp( A ), for A ⊆ L , that P ↾ A has size | A | ω and that there are | A | ω many P ↾ A -names for reals. Thus | P | = λ ω = λ and P forces c ≤ λ .Also, letting ˙ d α , α < µ , be the P -name of the Hechler generic added at stage α , we see that the ˙ d α forma scale of length µ . Indeed, if α < β , then since ˙ d β is generic over P ↾ L β and α ∈ L β , ˙ d β dominates ˙ d α .Furthermore, if ˙ x is an arbitrary name for a real, by Lemma 24, there is a countable A ⊆ L such that ˙ x is a P ↾ A -name. Choosing α such that A ⊆ L α and recalling L α ∈ I , we see that ˙ d α dominates ˙ x . Thus b = d = µ follows.We are left with showing a ≥ λ . Since b ≤ a in ZFC, we already know a ≥ µ . Thus let ˙ A be a name foran almost disjoint family of size < λ and ≥ µ , say ˙ A = { ˙ A α : α < κ } where κ ≥ ω · B α ⊆ L such that the ˙ A α are P ↾ B α -names.More explicitly, letting { p αn,i : n ∈ ω } , i ∈ ω , be maximal antichains and { k αn,i ∈ { , } : i, n ∈ ω } be such that p αn,i (cid:13) i ∈ ˙ A α iff k αn,i = 1 and p αn,i (cid:13) i / ∈ ˙ A α iff k αn,i = 0, we have { p αn,i : i, n ∈ ω } ⊆ P ↾ B α . We may also assumeall B α ’s are trees. Letting B := S α<κ B α we see that | B | < λ . By CH and the ∆-system lemma we may alsoassume that { B α : α < ω } forms a ∆-system with root R and that • ϕ α,β : B α → B β is an isomorphism of trees (as defined above) fixing R , • the induced isomorphism ψ α,β : P ↾ B α → P ↾ B β maps p αn,i to p βn,i , • there are numbers k n,i such that k αn,i = k n,i for all α < ω , • there is some θ < ω such that whenever α < ω , x ∈ B α , j odd, and x ( j ) ∈ λ ∗ , then x ( j ) ∈ S θ for some θ < θ .Note that the second and third clauses immediately imply that ψ α,β also maps the name ˙ A α to ˙ A β .For α < ω , write B α = { x αs : s ∈ T } where T ⊆ ( ω ∗ ∪ ω ) <ω is the canonical tree weakly isomorphicto any B α . This means in particular that | s | = | x αs | , that s ( n ) is positive iff x αs ( n ) is positive, and that ϕ α,β ( x αs ) = x βs . Let S ⊆ T be the subtree corresponding to the root R , that is, s ∈ S iff x αs ∈ R for any16 < ω . So, for α = β , x αs = x βs iff s ∈ S . List the immediate successors of S in T as { t n : n ≥ } , i.e., { t n : n ≥ } = { t ∈ T \ S : t ↾ ( | t | − ∈ S } . For α < β < ω define F ( { α, β } ) = n if either t n ( | t n | − ∈ ω and x αt n ( | t n | − > x βt n ( | t n | − t n ( | t n | − ∈ ω ∗ and x αt n ( | t n | − < x βt n ( | t n | − n exists and is minimal with this property0 otherwiseNote that, by wellfoundedness of the ordinals, for every n ≥
1, any subset of ω homogeneous in color n mustbe finite. Hence, by the Erd˝os-Rado Theorem, we obtain a subset of size ω homogeneous in color 0 and mayas well assume that ω itself is 0-homogeneous. Using further pruning arguments, we may additionally supposethat if s ∈ S and ζ, ξ ∈ ω ∗ ∪ ω with s ˆ ζ, s ˆ ξ ∈ T \ S (so s ˆ ζ = t n , s ˆ ξ = t m , for some n = m ≥ α < β < ω , • if ζ is positive, then x αs ˆ ζ ( | s | ) < x βs ˆ ζ ( | s | ), all x αs ˆ ζ ( | s | ) are larger than ω , and if ζ < ξ then – either x βs ˆ ζ ( | s | ) < x αs ˆ ξ ( | s | ) (this is the case when sup α<ω x αs ˆ ζ ( | s | ) < sup α<ω x αs ˆ ξ ( | s | )), – or x αs ˆ ξ ( | s | ) < x βs ˆ ζ ( | s | ) (this is the case when sup α<ω x αs ˆ ζ ( | s | ) = sup α<ω x αs ˆ ξ ( | s | )), • if ζ is negative, then x αs ˆ ζ ( | s | ) > x βs ˆ ζ ( | s | ), and if ζ > ξ then – either x βs ˆ ζ ( | s | ) > x αs ˆ ξ ( | s | ) (this is the case when inf α<ω x αs ˆ ζ ( | s | ) > inf α<ω x αs ˆ ξ ( | s | )), – or x αs ˆ ξ ( | s | ) > x βs ˆ ζ ( | s | ) (this is the case when inf α<ω x αs ˆ ζ ( | s | ) = inf α<ω x αs ˆ ξ ( | s | )).Define x κs ∈ L by recursion on the length of s ∈ T , as follows. If s ∈ S , then let x κs = x αs for any α < ω (inparticular, | x κs | = | x αs | = | s | ). If s ∈ S and s ˆ ζ / ∈ S , we will have | x κs ˆ ζ | = | s ˆ ζ | + 2. First let x κs ˆ ζ ( | s | ) be thelimit of the x αs ˆ ζ ( | s | ) (so it is either the sup or the inf, depending on whether ζ is positive or negative). Nextfind γ < λ with γ > ω and γ ∗ ∈ S θ , such that for all s and ζ , • if x κs ˆ ζ ( | s | ) = sup α<ω x αs ˆ ζ ( | s | ), then for all y ∈ B with y ↾ ( | s | + 1) = x κs ˆ ζ ↾ ( | s | + 1), we have y ( | s | + 1) > γ ∗ , • if x κs ˆ ζ ( | s | ) = inf α<ω x αs ˆ ζ ( | s | ), then for all y ∈ B with y ↾ ( | s | + 1) = x κs ˆ ζ ↾ ( | s | + 1), we have y ( | s | + 1) < γ .It is clear that such a γ exists because λ > | B | is regular. In the first case, let x κs ˆ ζ ( | s | + 1) = γ ∗ , and in thesecond case, x κs ˆ ζ ( | s | + 1) = γ . To complete the definition of x κs ˆ ζ define x κs ˆ ζ ( | s | + 2) = (cid:26) x s ˆ ζ ( | s | ) if | s | > ξ + 2 n + 1 if | s | = 0 and ζ = ξ + n with ξ limitFinally, for the remaining t ∈ T , stipulate again that | x κt | = | t | + 2, find s ( t with s ∈ S maximal, put x κt ↾ ( | s | + 3) = x κs ˆ t ( | s | ) and x κt ( j + 2) = x t ( j ) for j > | s | .Let B κ = { x κs : s ∈ T } . Notice that B κ , though very tree-like, is not a tree like the B α ’s. For α < ω define ϕ α,κ : B α → B κ by ϕ α,κ ( x αs ) = x κs for s ∈ T . We proceed to show that ϕ α,κ maps I ↾ B α to I ↾ B κ andthat therefore P ↾ B α and P ↾ B κ are isomorphic by the induced map ψ α,κ . It suffices to consider the case α = 0.Clearly, ϕ = ϕ ,κ is order-preserving.First fix β and consider L β . Note that there is β ≤ β such that ϕ ( L β ∩ B ) = L β ∩ B κ = L β ∩ B κ . Forany s ∈ T with x s ∈ L β yet x κs / ∈ L β , we must have x κs (0) > β ≥ x s (0) ≥ β and x κs (0) = sup α<ω x αs (0). Inparticular, for all such s , x κs (0) must have the same value, say γ . Also x κs (1) = γ ∗ and x κs (2) = ξ + 2 n + 1 < ω where s (0) = ξ + n with ξ limit. If, for some s ∈ T , x s (0) = β , let η = ξ + 2 n + 1 where s (0) = ξ + n with ξ limit. If there is no such s and ξ + n = sup { s (0) + 1 : x s (0) < β } , ξ limit, let η = ξ + 2 n . Then we see that L β ∩ B is mapped to ( L β ∪ J x ) ∩ B κ via ϕ , where | x | = 3, x (0) = γ , x (1) = γ ∗ , and x (2) = η (note that this x is indeed relevant).Next assume x is relevant and consider J x . Assume that J x ∩ B = ∅ . Then there must be s ∈ T suchthat | s | = | x | − x s = x ↾ ( | x | − s ∈ S , we have x κs = x s and J x ∩ B is mapped to J x ∩ B κ via ϕ because, by construction, we must have y ∈ R for any y ∈ B with | y | = | x | , y ↾ ( | x | −
1) = x s and y ( | x | − ≤ x ( | x | − < ω . In case s ∈ T \ S , let j < | s | be maximal with s ↾ j ∈ S . Define y by | y | = | x | + 2, y ↾ ( | y | −
1) = x κs and y ( | y | −
1) = x ( | x | −
1) and note that J x ∩ B gets mapped to J y ∩ B κ via ϕ provided wecan show that y is relevant. In case j >
0, this follows because whenever x s ( j ) > ω where j ≥ j is even then17lso x κs ( j + 2) = x s ( j ) > ω , and, if j is even, we additionally have x κs ( j ) = sup α<ω x αs ( j ) > ω while, if j is odd, we additionally have x κs ( j + 1) = γ > ω . In case j = 0 this is true because x κs (1) ∈ S θ and θ islarger than all the θ for which x κs ( j ) ∈ S θ where j > J x ∩ B κ = ∅ then there is a relevant y such that ϕ ( J y ∩ B ) = J x ∩ B κ .As mentioned already this means that ψ α,κ is an isomorphism of P ↾ B α and P ↾ B κ , and we can define ˙ A κ as the ψ α,κ -image of ˙ A α (where α < ω is arbitrary). More explicitly, p κn,i = ψ α,κ ( p αn,i ), and p κn,i (cid:13) i ∈ ˙ A κ iff k n,i = 1 and p κn,i (cid:13) i / ∈ ˙ A κ iff k n,i = 0.By construction, it is then also clear that if β < κ is arbitrary, we can find α < ω such that B α ∪ B β and B κ ∪ B β are order isomorphic via the mapping ϕ ′ fixing nodes of B β and sending the x αs to the corresponding x κs via ϕ α,κ (in fact, this is true for all but countably many α ). Also ϕ ′ maps I ↾ ( B α ∪ B β ) to I ↾ ( B κ ∪ B β ).Thus the induced map ψ ′ : P ↾ ( B α ∪ B β ) → P ↾ ( B κ ∪ B β ) is an isomorphism fixing the name ˙ A β and mappingthe name ˙ A α to the name ˙ A κ . Since P ↾ ( B α ∪ B β ) forces that ˙ A α ∩ ˙ A β is finite, P ↾ ( B κ ∪ B β ) forces that ˙ A κ ∩ ˙ A β is finite. As this is true for any β , ˙ A is not maximal, and the proof is complete.Note that the template framework is in a sense more general than the framework of Subsection 1.2, for theproof of Theorem 14 in Subsection 1.3 can also be done using templates. See [Br1, Section 2] for a discussionof this. By modifying the template of Theorem 29, Shelah also proved a may be a singular cardinal of uncountablecofinality. Theorem 30 (Shelah [Sh4]) . Assume GCH. Let µ > ℵ be regular and λ > µ singular of uncountable cofinality.Then there is a ccc forcing extension satisfying b = d = µ and a = c = λ . Embedding Hechler’s forcing for adding a mad family of size ℵ ω into the template framework, the authorobtained a model in which a = ℵ ω has countable cofinality. Theorem 31 (Brendle [Br2]) . Assume CH and let λ be a singular cardinal of countable cofinality. Then thereis a ccc forcing extension satisfying a = λ . In particular, a = ℵ ω is consistent. A subgroup G of Sym( ω ) is called cofinitary if any non-identity member of G fixes only finitely manynumbers. The cardinal invariant a g , the minimal size of a maximal cofinitary group, is a relative of a , andsimilar results about it can be proved with the same techniques. For example: Theorem 32 (Fischer and T¨ornquist [FT]) . Assume CH and let λ be a singular cardinal of countable cofinality.Then there is a ccc forcing extension satisfying a g = λ . In particular, a g = ℵ ω is consistent. More recently, the template technique has been used to obtain several consistency results about the -independence number i in [BHKLS] (see there for a definition), e.g.: Theorem 33 (Brendle, Halbeisen, Klausner, Lischka, and Shelah [BHKLS]) . Assume CH and let λ be a singularcardinal of countable cofinality. Then there is a ccc forcing extension satisfying i = λ . In particular, i = ℵ ω is consistent. The interest of these three results stems from the fact that for most cardinal invariants of the continuum, itis known that they must have uncountable cofinality, and apart from the above and some more variations of a ,the only cardinal which is known to consistently have countable cofinality is cov ( N ) [Sh3]. It is open whetherthe independence number i or the reaping number r (see [Bl] for definitions) can have countable cofinality.Replacing Hechler forcing in the template framework by other Suslin ccc forcings, one obtains a numberof related consistency results about the order relationship of cardinal invariants of the continuum. See [Br1,Section 4] for details. For example: Theorem 34 (Brendle [Br1]) . Assume CH. Let λ > µ > ℵ be regular cardinals with λ ω = λ . Then there is accc forcing extension satisfying add ( N ) = cof ( N ) = µ and a = c = λ . In all the template models discussed so far s = ℵ (this is so because iterations of Suslin ccc forcing notionskeep s small, see [BJ, Theorem 3.6.21]), and the question arose as to whether one could also increase s in thetemplate framework. Incorporating the ultrapower construction from Section 1, Mej´ıa introduced iterations ofnon-definable ccc partial orders along templates and proved:18 heorem 35 (Mej´ıa [Me]) . Assume GCH and let θ < κ < µ < λ be uncountable regular cardinals with κ measurable. Then there is a ccc p.o. forcing s = θ , b = d = µ , and a = c = λ . The large cardinal assumption in fact can be removed:
Theorem 36 (Fischer and Mej´ıa [FM]) . Assume GCH and let θ < µ < λ be uncountable regular cardinals.Then there is a ccc p.o. forcing s = θ , b = d = µ , and a = c = λ . As remarked in Subsection 1.4 (see Theorem 15), Shelah also used the technique of iterating ultrapowersof ccc forcing notions to obtain the consistency of ℵ < u < a , assuming the consistency of a measurablecardinal. It is not known whether this can be done in ZFC alone. The problem is that while non-definableccc forcings of the type L U can be incorporated into the matrix-like framework discussed in Subsection 1.2, itis not clear how to do this with the more complex template framework in Subsection 2.1. However, using acountable support iteration of proper forcing, Guzm´an and Kalajdzievski [GK] recently proved the consistencyof u = ℵ < a = c = ℵ . On the other hand, whether d = ℵ < a is consistent is a famous old open problem ofRoitman’s from the seventies. Assume P is a ccc partial order, κ is a strongly compact cardinal, and B is a κ + -cc and < κ -distributive cBa.Given a κ -complete ultrafilter D on B we may form the Boolean ultrapower Ult D ( P , B ). This is again a ccc partialorder, P completely embeds into Ult D ( P , B ), and much of the basic theory is very similar to the ultrapowers ofSection 1 (see Subsection 3.1). In particular Boolean ultrapowers of iterations are again iterations. Since thereis considerable freedom in selecting both the cBa B and the ultrafilter D , this method turns out to be morepowerful and there is a lot of control as to what can be achieved by just taking the Boolean ultrapower once.Accordingly, all results obtained with this method (Theorems 50 through 56) are obtained by finitely manyBoolean ultrapowers, and sophisticated limit constructions as in Subsection 1.2 become unnecessary.In Subsection 3.3 we present a proof of the result, due to Goldstern, Kellner, and Shelah [GKS], saying thatit is consistent that all cardinal invariants in Cicho´n’s diagram simultaneously assume distinct values, assumingthe consistency of four strongly compact cardinals (Theorem 50). For this, the combinatorial properties COB and
EUB and their behavior under Boolean ultrapowers is central (see Subsection 3.2). Further results usingBoolean ultrapowers can be found in Subsection 3.4.
Assume κ is a strongly compact cardinal. Let B be a κ + -cc and < κ -distributive cBa. Then every κ -completefilter on B can be extended to a κ -complete ultrafilter. Let D be a κ -complete ultrafilter on B . For a p.o. P define F = F ( P , B ) = { f : dom( f ) is a maximal antichain in B , ran( f ) ⊆ P } For f, g ∈ F , the
Boolean value of f = g is defined by[[ f = g ]] = _ { b ∈ B : ∃ a f ∈ dom( f ) , a g ∈ dom( g ) ( b ≤ a f , a g and f ( a f ) = g ( a g )) } Similarly we define Boolean values of other statements, e.g. [[ f ≤ g ]] etc. For f ∈ F ,[ f ] = f / D = { g ∈ F : [[ f = g ]] ∈ D} is the equivalence class of f modulo D . The Boolean ultrapower
Ult D ( P , B ) consists of all such equivalenceclasses. It is partially ordered by [ f ] ≤ [ g ] iff [[ f ≤ g ]] ∈ D As in the discussion of ultrapowers in Section 1, we identify p ∈ P with the class [ f ] of the constant function f ( ) = p and think of P as a subset of Ult D ( P , B ). Lemma 37. If P is κ -cc then P < ◦ Ult D ( P , B ) .Proof. Like the proof of Lemma 4.
Lemma 38. If P is ν -cc for some ν < κ then so is Ult D ( P , B ) . roof. This is like the proof of Lemma 5, but we provide the argument for the sake of completeness. Let f γ ∈ Ult D ( P , B ), γ < ν . By < κ -distributivity of B , the maximal antichains dom( f γ ) have a common refinement A and we may assume dom( f γ ) = A for all γ < ν . By the ν -cc of P , for all a ∈ A there are γ < δ < ν such that f γ ( a ) and f δ ( a ) are compatible. By κ -completeness of D , there are γ < δ such that W { a ∈ A : f γ ( a ) and f δ ( a )are compatible } belongs to D . Thus [ f γ ] and [ f δ ] are compatible, as required.For the remainder of this section, assume P is ccc. Then so is Ult D ( P , B ) and P completely embeds intoUlt D ( P , B ). As in Section 1, we obtain a natural description of Ult D ( P , B )-names for reals in terms of P -namesfor reals. Let A be a maximal antichain in B , and let { p an : n ∈ ω } , a ∈ A , be | A | many maximal antichains in P . Defining f n : A → P by f n ( a ) = p an for a ∈ A we obtain a maximal antichain { [ f n ] : n ∈ ω } in Ult D ( P , B ).Furthermore, by distributivity of B , all maximal antichains of Ult D ( P , B ) are of this form. Next assume wehave | A | many P -names ˙ x a for reals in ω ω , a ∈ A , given by maximal antichains { p an,i : n ∈ ω } and numbers { k an,i : n ∈ ω } , i ∈ ω and a ∈ A , such that p an,i (cid:13) P ˙ x a ( i ) = k an,i . Then, letting f n,i ( a ) = p an,i and defining k n,i to be the unique ℓ such that W { a ∈ A : k an,i = ℓ } ∈ D , we obtainan Ult D ( P , B )-name ˙ y for a real given by [ f n,i ] (cid:13) Ult D ( P , B ) ˙ y ( i ) = k n,i . This is the average or mean of the ˙ x a , and we will usually write ˙ y = h ˙ x a : a ∈ A i / D . Using again thedistributivity of B we see that every Ult D ( P , B )-name for a real is of this form. Lemma 39.
Assume P < ◦ Q . Then Ult D ( P , B ) < ◦ Ult D ( Q , B ) .Proof. Like the proof of Lemma 8.The following result is not needed, but we include it to show that much of the theory can be developed likefor ultrapowers (Section 1).
Lemma 40.
Let P be a p.o. and let Q be a Suslin ccc forcing notion. Then Ult D ( P ⋆ ˙ Q , B ) ∼ = Ult D ( P , B ) ⋆ ˙ Q .Proof. Like the proof of Lemma 9.
Lemma 41.
Let P = h P γ : γ ≤ µ i be an iteration. Then Ult D ( P , B ) = h Ult D ( P γ , B ) : γ ≤ µ i also is aniteration. Moreover, if P has finite supports then so does Ult D ( P , B ) .Proof. Like the proof of Lemma 11.In the main result of this section (Theorem 50), we will apply the Boolean ultrapower operation (finitelyoften) to an iteration. By the previous lemma, the result is again an iteration, though this is not really relevantfor us.
COB and
EUB
We introduce and present the basic properties of two principles,
COB and
EUB , which are important forpreservation of cardinal invariants. They will be used again in Section 4.Suppose we have a binary Borel relation R on the Baire space ω ω (or the Cantor space 2 ω ) such that • for all x ∈ ω ω there is y ∈ ω ω with xRy , • for all y ∈ ω ω there is x ∈ ω ω with ¬ ( xRy ).If xRy holds, we say that y R -dominates x , and if ¬ ( xRy ), x is R -unbounded over y . We associate two cardinalswith this relation R , the unbounding number b ( R ) := min {| F | : F ⊆ ω ω is not R -dominated by a single y ∈ ω ω } and the dominating number d ( R ) := min {| F | : F ⊆ ω ω and all x ∈ ω ω are R -dominated by a member of F }
20 typical example is when R = ≤ ∗ , the eventual domination ordering : say x ≤ n y if for all k ≥ n , x ( k ) ≤ y ( k )holds. ≤ ∗ = S n ≤ n , and b ( ≤ ∗ ) = b ( d ( ≤ ∗ ) = d , repsectively) is the usual unbounding (dominating, resp.)number. We shall see more examples shortly.Given such a relation R , a ccc partial order P , and cardinals λ ≤ ν with λ regular, we say P forces a < λ -directed R -cone of bounds of size ν , COB ( R, P , λ, ν ) in symbols, if there are a < λ -directed partial order h S, ≤i of size ν and P -names ( ˙ z s : s ∈ S ) for reals such that for every P -name ˙ y for a real there is s ∈ S suchthat for all t ≥ s , (cid:13) P ˙ yR ˙ z t The connection between
COB and the values of the cardinals b ( R ) and d ( R ) in the forcing extension is givenby: Lemma 42.
COB ( R, P , λ, ν ) implies that b ( R ) ≥ λ and d ( R ) ≤ ν in the P -generic extension.Proof. Clearly the ( ˙ z s : s ∈ S ) form a witness for d ( R ) in the extension. On the other hand, by < λ -directedness,any ˙ F ⊆ ω ω of size < λ will be bounded.We next discuss the relationship between COB for a partial order and its ultrapower.
Lemma 43.
Assume λ ≤ ν are regular and COB ( R, P , λ, ν ) .1. If κ < λ or ν < κ , then COB ( R, Ult D ( P , B ) , λ, ν ) .2. If λ < κ and κ ≤ ν , then COB ( R, Ult D ( P , B ) , λ, max( ν, µ ) κ ) where µ = | B | .Proof. (1) This is similar to Lemma 7, but we sketch the argument. Let ( ˙ z s : s ∈ S ) be < λ -directed and cofinalof size ν in the P -generic extension. We show this property is preserved in the Ult D ( P , B )-generic extension. Tosee this, let ˙ y = h ˙ x a : a ∈ A i / D be a Ult D ( P , B )-name for a real in ω ω . For each a ∈ A , find s ( a ) ∈ S such thatfor all t ≥ s ( a ), (cid:13) P ˙ x a R ˙ z t If κ < λ , directedness of S gives us s ∈ S bigger than all s ( a ). If ν < κ , from the completeness of D we obtain s ∈ S such that W { a ∈ A : s ( a ) = s } ∈ D . In either case, for all t ≥ s , (cid:13) Ult D ( P , B ) ˙ yR ˙ z t follows easily.(2) Again let ( ˙ z s : s ∈ S ) be < λ -directed and cofinal of size ν in the P -generic extension. Letting U =Ult D ( S, B ) we easily see that U is < λ -directed of size ≤ max( ν, µ ) κ . For [ u ] ∈ U , let ˙ y [ u ] = h ˙ z u ( a ) : a ∈ A i / D where A ⊆ B is a maximal antichain and u : A → S . We claim that ( ˙ y [ u ] : [ u ] ∈ U ) is cofinal in the Ult D ( P , B )-generic extension. Taking ˙ x = h ˙ x a : a ∈ A i / D arbitrarily, there is u : A → S such that for all a ∈ A and all t ≥ u ( a ), (cid:13) P ˙ x a R ˙ z t In particular, if [ v ] ∈ U with [ v ] ≥ [ u ] (i.e. [[ v ≥ u ]] ∈ D ), we see that W { a ∈ A : (cid:13) P x a R ˙ z v ( a ) } ∈ D and therefore (cid:13) Ult D ( P , B ) ˙ xR ˙ y [ v ] as required.Given a Borel relation R , a ccc partial order P , and a limit ordinal ν , we say P forces an eventually R -unbounded sequence of length ν , EUB ( R, P , ν ) in symbols, if there are P -names ( ˙ x α : α < ν ) for reals such thatfor all P -names ˙ y for reals there is α < ν such that for all β ≥ α , (cid:13) P ¬ ( ˙ x β R ˙ y )(i.e. ˙ y does not R -dominate ˙ x β ).Note that for every Borel relation R on the Baire space, we have the dual relation , R ⊥ , given by xR ⊥ y ⇐⇒ ¬ ( yRx )It is well-known and easy to see that b ( R ⊥ ) = d ( R ) and d ( R ⊥ ) = b ( R ). Using duality we see that EUB is aspecial case of
COB . 21 emma 44.
COB ( R ⊥ , P , ν, ν ) and EUB ( R, P , ν ) are equivalent.Proof. Indeed, if S is < ν -directed of size ν , then S has a cofinal subset isomorphic to ν , and we may as wellassume S = ν . Now, COB ( R ⊥ , P , ν, ν ) means that there are P -names ( ˙ x α : α < ν ) for reals such that for every P -name ˙ y for a real there is α < ν such that for all β ≥ α , (cid:13) P ¬ ( ˙ x β R ˙ y )which is exactly EUB ( R, P , ν ).Using the earlier results about COB , we infer:
Corollary 45.
1. Assume ν is regular and EUB ( R, P , ν ) . Then P forces that b ( R ) ≤ ν and d ( R ) ≥ ν .2. Assume ν is regular and EUB ( R, P , ν ) . If ν = κ , then EUB ( R, Ult D ( P , B ) , ν ) . We end end this subsection with a couple of relations which we shall use in the next subsection as well as inSection 4. • Say a function ϕ : ω → [ ω ] <ω is a slalom if | ϕ ( n ) | = n for all n . The slaloms can be identified with theBaire space. For a slalom ϕ and x ∈ ω ω , let xR ϕ if for all but finitely many n, x ( n ) ∈ ϕ ( n )It is well-known [BJ, Theorem 2.3.9] that b ( R ) = add ( N ) and d ( R ) = cof ( N ). • Let x ∈ ω ω . There is a canonical way to associate a null G δ set N x with x . More explicitly, let ( U ni : i ∈ ω )list all clopen subsets of 2 ω of measure ≤ − n , and put N x = T m S n ≥ m U nx ( n ) . For x, y ∈ ω ω , let xR y if y / ∈ N x Then clearly b ( R ) = cov ( N ) and d ( R ) = non ( N ). • For x, y ∈ ω ω , let xR y if x ≤ ∗ y Then clearly b ( R ) = b and d ( R ) = d . • For x, y ∈ ω ω , let xR y if x = ∗ y if for all but finitely many n, x ( n ) = y ( n )It is well-known [BJ, Theorems 2.4.1 and 2.4.7] that b ( R ) = non ( M ) and d ( R ) = cov ( M ).These cardinals can be displayed in Cicho´n’s diagram (see [BJ, Chapter 2] or [Bl, Section 5] for details), wherecardinals grow as one moves up or right. d ( R ) = cof ( N ) non ( N ) cof ( M ) dd ( R ) = cov ( M ) add ( M ) bb ( R ) = non ( M ) cov ( N ) b ( R ) = add ( N ) ℵ c Cicho´n’s diagramThe following result forms the basis for the main results of both Subsections 3.3 and 4.2.
Theorem 46 (Goldstern, Mej´ıa, and Shelah [GMS]) . Assume GCH, and let λ ≤ λ ≤ λ ≤ λ ≤ λ beuncountable regular cardinals. There is a ccc p.o. P pre , the preparatory forcing , such that for ≤ i ≤ , EUB ( R i , P pre , ν ) for every ν with λ i ≤ ν ≤ λ , • COB ( R i , P pre , λ i , λ ) In particular, P pre forces add ( N ) = λ ≤ cov ( N ) = λ ≤ add ( M ) = b = λ ≤ non ( M ) = λ ≤ cov ( M ) = c = λ Proof Sketch.
We first note that the values for the cardinal invariants follow from Lemma 42 and part 1 ofCorollary 45.We sketch the proof for the particular case λ = λ and then make some comments on the general case. Make a finite support iteration ( P α , ˙ Q α : α < λ ) of length λ of ccc partial orders, going through1. eventually different reals forcing E cofinally often, as well as through2. all subforcings of localization forcing of size < λ ,3. all subforcings of random forcing of size < λ , and4. all subforcings of Hechler forcing of size < λ using a book-keeping argument. Then P pre = P λ . For definitions of the particular forcing notions and theirproperties, see [BJ, Chapter 3 and 7.4.B]. Standard arguments show that the family of partial generics addedby item i + 1 guarantees COB ( R i , P pre , λ i , λ ) for 1 ≤ i ≤
3. For example, the partial Hechler generics form awitness for
COB ( R , P pre , λ , λ ). Similarly, the eventually different reals witness COB ( R , P pre , λ , λ ) (since λ = λ ).Fix regular uncountable ν ≤ λ . Let ( ˙ c α : α < ν ) be the sequence of Cohen reals added in the limit stagesof the initial segment P ν of the iteration. They clearly witness EUB ( R i , P ν , ν ) for all i . If λ i ≤ ν , then standardpreservation arguments show that EUB ( R i , P α , ν ) holds for all α ≥ ν , and EUB ( R i , P pre , ν ) follows. If i = 4this is trivial (by λ = λ ). Moreover, preservation for limit ordinals α is a standard argument. For successorordinals α , if i = 1, use the fact that all ˙ Q α either are of size < λ or carry a finitely additive measure (see [Ka]for why σ -centered forcings and subforcings of random forcing carry such a measure), and that this preserves EUB ( R , P α , ν ). If i = 2, use that all ˙ Q α are either of size < λ or σ -centered and thus preserve EUB ( R , P α , ν ).If i = 3, use that all ˙ Q α either are of size < λ or are E , which preserves EUB ( R , P α , ν ) by a compactnessargument (see [Mi]). This completes the argument in the special case.In case λ < λ , one would like to go through all subforcings of eventually different reals forcing of size < λ instead. It is not clear, however, why this should preserve EUB ( R , P α , ν ) for α ≥ ν . For this reason thesubforcings of E have to be very carefully chosen in a sophisticated argument, see [GMS] for details.By this theorem, all cardinal invariants on the left-hand side of Cicho´n’s diagram can be separated simul-taneously. It is harder to separate the dual cardinals on the right-hand side. We shall present two methods fordoing this, in Subsections 3.3 and 4.2, using the fact that we already achieved separation on the left-hand side. Assume µ > κ is a regular cardinal (where κ is strongly compact as before). Let B be the completion of Fn ( µ, κ, < κ ), forcing with partial functions from µ to κ of size < κ . Note that B is κ + -cc (because 2 <κ = κ )and < κ -distributive. Let A ⊆ Fn ( µ, κ, < κ ) be a maximal antichain and w : A → κ . Let supp( A ) = S { dom( a ) : a ∈ A } , the support of A . Clearly | supp( A ) | ≤ κ . If the maximal antichain A ′ refines A we canonically extend w to A ′ by letting w ( a ′ ) = w ( a ) where a is the unique element of A above a ′ , for a ′ ∈ A ′ . If w : A → κ and w ′ : A ′ → κ are two such functions we get the Boolean value [[ w < w ′ ]] = _ { a ′′ ∈ A ′′ : w ( a ′′ ) < w ′ ( a ′′ ) } where A ′′ is a common refinement of A and A ′ . For δ < µ let A δ be the maximal antichain of singleton partialfunctions {h δ, ξ i} , ξ < κ , and define v δ : A δ → κ by v δ ( {h δ, ξ i} ) = ξ . Lemma 47.
The Boolean values [[ v δ > w ]] with δ > sup(supp(dom( w ))) form a κ -complete filter on B andtherefore can be extended to a κ -complete ultrafilter D . This special case has been known at least since the 90’s. roof. Let ν < κ and ( w ξ , δ ξ ), ξ < ν , be pairs such that δ ξ > sup(supp(dom( w ξ ))), and let A ξ = dom( w ξ ). Weneed to show that V ξ<ν [[ v δ ξ > w ξ ]] = . Enumerate { δ ξ : ξ < ν } in increasing order and without repetitions as { δ ζ : ζ < γ } for some γ ≤ ν . Let C ζ = { ξ : δ ξ = δ ζ } . Construct a decreasing chain { q ζ : ζ ≤ γ } of conditionsin Fn ( µ, κ, < κ ) as follows. q is the trivial condition and for limit ζ , q ζ is the union of the q η , η < ζ . Assume q ζ has been constructed such that dom( q ζ ) ⊆ δ ζ and let q ζ +1 be an extension such that dom( q ζ +1 ) ⊆ δ ζ + 1, q ζ +1 ↾ δ ζ extends an element a ξ ∈ A ξ for each ξ ∈ C ζ and q ζ +1 ( δ ζ ) = sup { w ξ ( a ξ ) : a ξ ∈ C ζ } + 1. Clearly q γ isan extension of V ξ<ν [[ v δ ξ > w ξ ]].By strong compactness of κ , we can now extend this filter base to a κ -complete ultrafilter on B .We assume from now on that D is constructed as in this lemma. Lemma 48.
Assume
EUB ( R, P , κ ) . Then EUB ( R, Ult D ( P , B ) , µ ) .Proof. This is like the proof of Lemma 43, but we additionally need to use the special property of the ultrafilter D given by Lemma 47. Let ( ˙ x α : α < κ ) be the eventually unbounded sequence forced by P . For δ < µ let˙ y δ = h ˙ x v δ ( a ) : a ∈ A δ i / D . We claim that Ult D ( P , B ) forces ( ˙ y δ : δ < µ ) is an eventually unbounded sequence.Indeed, let ˙ z = h ˙ z a : a ∈ A i / D be a Ult D ( P , B )-name for a real in ω ω and let δ > supp( A ). There is a function w : A → κ such that for all a ∈ A and all β ≥ w ( a ), (cid:13) P ¬ ˙ x β R ˙ z a By Lemma 47, we know that [[ v δ > w ]] ∈ D . A fortiori W { a ∈ A ′ : (cid:13) P ¬ ˙ x v δ ( a ) R ˙ z a } ∈ D where A ′ is a commonrefinement of A δ and A and therefore, (cid:13) Ult D ( P , B ) ¬ ˙ y δ R ˙ z as required. EUB and
COB are tools to compute the cardinal invariants b ( R ) and d ( R ) in Ult D ( P , B )-generic extensions. Corollary 49.
Let λ ≤ ν be regular and assume EUB ( R, P , λ ) and COB ( R, P , λ, ν ) hold.1. If λ < κ ≤ ν ≤ µ = µ κ and, additionally, EUB ( R, P , κ ) holds, then EUB ( R, Ult D ( P , B ) , λ ) , EUB ( R, Ult D ( P , B ) , µ ) ,and COB ( R, Ult D ( P , B ) , λ, µ ) , and therefore b ( R ) = λ and d ( R ) = µ in the Ult D ( P , B ) -generic extension.2. If κ < λ , and, additionally, EUB ( R, P , ν ) holds, then these properties are preserved by Ult D ( P , B ) , and b ( R ) = λ and d ( R ) = ν in the Ult D ( P , B ) -generic extension.Proof. This is immediate by Lemmas 42, 43, 48, and Corollary 45.
STRATEGY.
Assume P forces b ( R ) < d ( R ) as witnessed by EUB and
COB . For the case the strongly compactcardinal κ of the ground model lies between these two values, the first part of this corollary describes a methodfor further increasing d ( R ) while keeping the value of b ( R ) by taking the Boolean ultrapower of P . For the case κ is below the smaller cardinal, the Boolean ultrapower will preserve both cardinals by the second part of thecorollary. This provides us with a scenario for obtaining a model in which all cardinals in Cicho´n’s diagramare distinct. Namely, force the left-hand cardinals to be distinct, with strongly compact cardinals in betweenthem, and then keep stretching the right-hand cardinals while keeping the ones on the left, by repeatedly takingBoolean ultrapowers. Theorem 50 (Goldstern, Kellner, and Shelah [GKS]) . Assume the existence of four strongly compact cardinalsis consistent. Then so is the statement that all cardinals in Cicho´n’s diagram are distinct. More explicitly,assume GCH and let ℵ < κ < λ < κ < λ < κ < λ < κ < λ < λ < λ < λ < λ < λ be regularcardinals such that the κ i are strongly compact. Then there is a ccc p.o. P forcing add ( N ) = λ < cov ( N ) = λ < add ( M ) = b = λ < non ( M ) = λ << cov ( M ) = λ < d = cof ( M ) = λ < non ( N ) = λ < cof ( N ) = λ < c = λ Proof.
Assume B j is the completion of Fn ( λ j , κ j , < κ j ) for 6 ≤ j ≤
9, and let D j be the κ j -complete ultrafilteron B j obtained from Lemma 47.Let P := P pre be the ccc partial order from Theorem 46. Next let P j = Ult D j ( P j − , B j ) for 6 ≤ j ≤ P := P is as required by the theorem. Since the proof is the same for the four relations R i ,1 ≤ i ≤
4, we do it for i = 3, that is, we show that P forces b = b ( R ) = λ and d = d ( R ) = λ . First, usingpart 2 of Lemma 43, part 2 of Corollary 45 and Lemma 48, we obtain24 COB ( R , P , λ , λ ) • EUB ( R , P , λ ) and EUB ( R , P , λ )Then, using part 1 of Lemma 43 and part 2 of Corollary 45, we get • COB ( R , P j , λ , λ ) • EUB ( R , P j , λ ) and EUB ( R , P j , λ )for 7 ≤ j ≤
9. By Lemma 42 and part 1 of Corollary 45, b ( R ) = λ and d ( R ) = λ follow.With more work resulting in a somewhat different preparatory forcing, the large cardinal assumption can bereduced to three strongly compact cardinals instead of four, see [BCM]. For a simple proof of a weaker versionof Theorem 50, based on the preparatory forcing with λ = λ whose proof is sketched above (Theorem 46),using three strongly compact cardinals, and showing the consistency of ℵ < add ( N ) < cov ( N ) < b < d < non ( N ) < cof ( N ) < c we refer the reader to [KTT]. The method of the previous subsection can be used to obtain some other results where many cardinal invariantssimultaneously assume distinct values.
Theorem 51 (Kellner, Shelah, and T˘anasie [KST]) . Assume GCH and let ℵ < κ < λ < κ < λ < κ <λ < κ < λ < λ < λ < λ < λ < λ be regular cardinals such that the κ i are strongly compact. Then thereis a ccc p.o. forcing add ( N ) = λ < add ( M ) = b = λ < cov ( N ) = λ < non ( M ) = λ << cov ( M ) = λ < non ( N ) = λ < d = cof ( M ) = λ < cof ( N ) = λ < c = λ This is based on a different, more involved, preparatory forcing, see also [KST].Mixing the technique of [BCM] with small partial orders forcing specific values to m (the smallest cardinalfor which Martin’s Axiom fails) and to the pseudointersection, distributivity, and groupwise density numbers, p , h , and g (see [Bl, Section 6] for definitions), in the iteration leading to the preparatory forcing, and thentaking ultrapowers, one obtains: Theorem 52 (Goldstern, Kellner, Mej´ıa, and Shelah [GKMS3]) . Assume GCH and let ℵ ≤ λ ≤ λ ≤ λ ≤ κ < λ < κ < λ < κ < λ ≤ λ ≤ λ ≤ λ ≤ λ ≤ λ ≤ λ be cardinals such that the κ i are stronglycompact, the λ i are regular for i = 9 , , and cf( λ ) ≥ λ and cf( λ ) ≥ λ . Then there is a p.o. preservingcofinalities and forcing ℵ ≤ m = λ ≤ p = λ ≤ h = g = λ < add ( N ) = λ < cov ( N ) = λ < add ( M ) = b = λ ≤ non ( M ) = λ ≤≤ cov ( M ) = λ ≤ d = cof ( M ) = λ ≤ non ( N ) = λ ≤ cof ( N ) = λ ≤ c = λ Composing this with collapses one gets for example:
Theorem 53 (Goldstern, Kellner, Mej´ıa, and Shelah [GKMS3]) . Assume GCH and there are three stronglycompact cardinals. Then there is a p.o. forcing ℵ < m = ℵ < p = ℵ < add ( N ) = ℵ < cov ( N ) = ℵ < add ( M ) = b = ℵ < non ( M ) = ℵ << cov ( M ) = ℵ < d = cof ( M ) = ℵ < non ( N ) = ℵ < cof ( N ) = ℵ < c = ℵ In still unpublished work [RS], Raghavan and Shelah have obtained a number of consistency results abouthigher cardinal invariants, that is, cardinal invariants describing the higher Baire space λ λ for regular uncount-able λ like b ( λ ) or d ( λ ), using the Boolean ultrapower technique.25 heorem 54 (Raghavan and Shelah [RS]) . For any regular λ > ω , d ( λ ) < a ( λ ) is consistent relative to asupercompact cardinal. More specifically, suppose that ℵ < λ = λ <λ < κ and that κ is supercompact. Thenthere is a forcing extension in which κ < b ( λ ) = d ( λ ) < a ( λ ) . For a proof sketch, as in the previous subsection, let B be the completion of Fn ( µ, κ, < κ ) for some regular µ > κ + . Using the fact that κ is supercompact, one builds a κ -complete optimal ultrafilter D on B . Let P be the κ + -stage iteration of λ -Hechler forcing with supports of size less than λ . This is the canonical p.o. for forcing b ( λ ) = d ( λ ) = κ + . Next, let Q = Ult D ( P , B ). Forcing with Q preserves b ( λ ) = d ( λ ) = κ + (this is basically thesame argument as the proof of Lemma 7, see also part 2 of Corollary 49) and makes a ( λ ) = µ (this uses thecombinatorial properties of D and is the core of the argument).Further results of theirs include: Theorem 55 (Raghavan and Shelah [RS]) . Suppose that ℵ < λ = λ <λ < κ and that κ is supercompact. Thenthere is a forcing extension in which κ < b ( λ ) < d ( λ ) < a ( λ ) . Theorem 56 (Raghavan and Shelah [RS]) . Suppose that λ < κ , that κ is supercompact, and that λ is Laverindestructible supercompact. Then there is a forcing extension in which λ is still supercompact and κ < u ( λ ) < a ( λ ) . Assume P is a ccc partial order, κ is a regular uncountable cardinal, and N is a < κ -closed elementary sub-structure of H ( χ ) containing κ and P , where χ is a large enough regular cardinal. Then the restriction of P to N , P ∩ N , is again a ccc partial order, and P ∩ N completely embeds into P . In fact, if we choose N sufficientlycarefully (typically N is the union of a chain of submodels), then P ∩ N reflects the combinatorial properties of P , albeit with possibly different cardinals as witnesses. We will introduce this method and discuss its effect on COB and
EUB in Subsection 4.1.In Subsection 4.2 we present a proof of the result, due to Goldstern, Kellner, Mej´ıa, and Shelah [GKMS1],saying that on the basis of ZFC it is consistent that all cardinal invariants in Cicho´n’s diagram simultaneouslyassume distinct values (Theorem 61). Further results using the submodel method are presented in Subsection 4.3.
Assume (for the whole subsection) κ is a regular uncountable cardinal, P is a κ -cc partial order, and N (cid:22) H ( χ )is < κ -closed with κ, P ∈ N , where χ is a large enough regular cardinal. Then: Lemma 57.
1. For every antichain A ⊆ P , A ∈ N if and only if A ⊆ N .2. P ∩ N is κ -cc.3. P ∩ N < ◦ P .Proof. (1) First let A ∈ N . Since | A | < κ ≤ | N | by the assumptions on P and N , A ⊆ N follows. If, on theother hand, A ⊆ N , then by the κ -cc, | A | < κ and by < κ -closure A ∈ N .(2) Let A ⊆ P ∩ N be an antichain. By elementarity we see A is an antichain of P , and | A | < κ follows.(3) Let A ⊆ P ∩ N be a maximal antichain. Again, A is an antichain of P , and by (1) A ∈ N . Clearly, N thinks that A is a maximal antichain of P , and therefore A is maximal in P by elementarity.By (3), any P ∩ N -generic filter G over V (or, equivalently, H ( χ )) can be extended to a P -generic filter G + over V ( H ( χ ), respectively). By elementarity G + is P -generic over N as well, and N [ G + ] (cid:22) H ( χ ) V [ G + ] . Usingthis we can establish a correspondence between P ∩ N -names ˙ x ∈ V for reals and P -names ˙ y ∈ N for reals suchthat ˙ x [ G ] = ˙ y [ G + ] and for all p ∈ P ∩ N and sufficiently absolute (e.g. Borel) ϕ , p (cid:13) P ϕ ( ˙ y ) iff p (cid:13) P ∩ N ϕ ( ˙ x )In particular, N [ G + ] ∩ ω ω = V [ G ] ∩ ω ω .To see this, recall that a P ∩ N -name ˙ x for a real in ω ω is given by maximal antichains { p n,i : n ∈ λ i } ⊆ P ∩ N and numbers { k n,i : n ∈ λ i } , i ∈ ω and λ i < κ , such that p n,i (cid:13) P ∩ N ˙ x ( i ) = k n,i N is < κ -closed, {{ p n,i : n ∈ λ i } , { k n,i : n ∈ λ i } : i ∈ ω } ∈ N , and by P ∩ N < ◦ P , ˙ x can be construed as a P -name in N . On the other hand, if ˙ x ∈ N is a P -name, that is, {{ p n,i : n ∈ λ i } , { k n,i : n ∈ λ i } : i ∈ ω } ∈ N ,then by part 1 of Lemma 57, { p n,i : n ∈ λ i } ⊆ N for all i and ˙ x is a P ∩ N -name.We now investigate how COB and
EUB for P ∩ N relate to COB and
EUB for P . For a partial order h S, ≤i ,let comp ( S ), the completeness of S , be the least λ such that S is not λ -directed. Lemma 58.
1. Assume
COB ( R, P , λ, ν ) as witnessed by the partial order h S, ≤i ∈ N . Then COB ( R, P ∩ N, λ ′ , ν ′ ) whenever λ ′ ≤ comp ( S ∩ N ) and ν ′ ≥ cof ( S ∩ N ) .2. Assume EUB ( R, P , ν ) with ν ∈ N . Then EUB ( R, P ∩ N, cof ( ν ∩ N )) .Proof. (1) Assume ( ˙ z s : s ∈ S ) witnesses COB ( R, P , λ, ν ). It suffices to show that ( ˙ z s : s ∈ S ∩ N ) witnesses COB ( R, P ∩ N, λ ′ , | S ∩ N | ). For then we can replace S ∩ N by any < λ ′ -directed superset of a cofinal subset. Let˙ y be a P ∩ N -name for a real. By the previous discussion, we know that ˙ y ∈ N may be construed as a P -namefor a real. Hence, by COB ( R, P , λ, ν ) and elementarity, there is s ∈ S ∩ N such that, in N , for all t ≥ s , (cid:13) P ˙ yR ˙ z t Again by the previous discussion, this means that for all t ≥ s in N we have (cid:13) P ∩ N ˙ yR ˙ z t as required.(2) This follows from (1) and Lemma 44. Lemma 59.
Assume κ ≤ θ ≤ µ , θ regular. Let λ ≤ ν with λ regular. Next let ( N i : i < θ ) be an increasingsequence of < θ -closed elementary submodels of H ( χ ) with | N i | = µ , N i ∈ N i +1 , and µ ∪ { µ, λ, ν, P } ∈ N . Put N = S i N i .1. Assume EUB ( R, P , ν ) . Then:(a) If ν ≤ µ then EUB ( R, P ∩ N, ν ) .(b) If ν > µ then EUB ( R, P ∩ N, θ ) .2. Assume S ∈ N witnesses COB ( R, P , λ, ν ) . Then comp ( S ∩ N ) ≥ min { θ, λ } and cof ( S ∩ N ) ≤ min { µ, ν } and therefore COB ( R, P ∩ N, min { θ, λ } , min { µ, ν } ) . In particular:(a) If ν ≤ µ then COB ( R, P ∩ N, λ, ν ) .(b) If µ < λ then comp ( S ∩ N ) = cof ( S ∩ N ) = θ and thus COB ( R, P ∩ N, θ, θ ) .Proof. (1) Notice that if ν > µ is regular, then cof ( ν ∩ N ) = θ . Hence this follows from part 2 of Lemma 58.(2) comp ( S ∩ N ) ≥ min { θ, λ } holds because if A ⊆ S ∩ N with | A | < min { θ, λ } , then A ∈ N by < θ -closureof N , and N thinks that A has an upper bound by elementarity. cof ( S ∩ N ) ≤ | S ∩ N | = min { µ, ν } is obvious. COB ( R, P ∩ N, min { θ, λ } , min { µ, ν } ) then follows from part 1 of Lemma 58.(a) If ν ≤ µ , then S ⊆ N ⊆ N , so that S = S ∩ N and COB ( R, P ∩ N, λ, ν ) is immediate.(b) Assume µ < λ . We know already comp ( S ∩ N ) ≥ min { θ, λ } = θ . By elementarity and N i ∈ N i +1 , forevery i < θ there is s ∈ S ∩ N i +1 such that s ≥ t for all t ∈ S ∩ N i . Hence cof ( S ∩ N ) ≤ θ follows, and we mustactually have comp ( S ∩ N ) = cof ( S ∩ N ) = θ . Lemma 60.
Assume additionally to the assumptions of part 2 of the previous lemma that µ < λ , that θ ′ ≥ θ and µ ′ > µ are regular cardinals in N , and that ( M j : j < θ ′ ) ∈ N is a family of < µ ′ -closed elementarysubmodels of H ( χ ) with | M j | = µ ′ . Put M = S j M j . Then COB ( R, P ∩ M ∩ N, θ, θ ′ ) .Proof. comp ( S ∩ M ∩ N ) ≥ θ is straightforward. So it suffices to show cof ( S ∩ M ∩ N ) ≤ θ ′ .Clearly comp ( S ∩ M j ) ≥ min { µ ′ , λ } and cof ( S ∩ M j ) ≤ min { µ ′ , ν } , and therefore COB ( R, P ∩ M j , min { µ ′ , λ } , min { µ ′ , ν } ).By part 2 (b) of the previous lemma we see that comp ( S ∩ M j ∩ N ) = cof ( S ∩ M j ∩ N ) = θ . Let T j ⊆ S ∩ M j ∩ N be cofinal in S ∩ M j ∩ N of size θ and let T = S j<θ ′ T j . Then | T | = θ ′ , and T is cofinal in S ∩ M ∩ N . Thus cof ( S ∩ M ∩ N ) ≤ θ ′ , and COB ( R, P ∩ M ∩ N, θ, θ ′ ) follows. COB and
EUB were originally defined for ccc forcing in Subsection 3.2 but this does not really matter. TRATEGY.
Assume P forces b ( R ) < d ( R ) = ν as witnessed by EUB and
COB . Let θ < θ ′ < µ < µ ′ := b ( R )be arbitrary regular cardinals. Building first ( N ′ i : i < θ ′ ) of size µ ′ according to Lemma 59, and then analogously( N j : j < θ ) of size µ such that ( N ′ i : i < θ ′ ) ∈ N , we see by the two previous lemmata that P ∩ N ′ ∩ N forces b ( R ) = θ and d ( R ) = θ ′ . If we then further intersect P with N ′′ such that ( N ′ i : i < θ ′ ) , ( N i : i < θ ) ∈ N ′′ with µ ′′ > θ ′ , we will not change these values anymore. This provides us with a scenario for obtaining a model forCicho´n’s maximum: force the left-hand cardinals to be distinct, of large enough value, and then “collapse” dualpairs of cardinals to a priori given values, by repeatedly restricting to appropriate elementary submodels. We are ready to present a ZFC-proof of the consistency result of Theorem 50.
Theorem 61 (Goldstern, Kellner, Mej´ıa, and Shelah [GKMS1]) . Assume GCH and ( λ i : 1 ≤ i ≤ is a ≤ -increasing sequence of uncountable cardinals with λ i regular for i ≤ and λ of uncountable cofinality. Thenthere is a ccc partial order P forcing that ℵ ≤ add ( N ) = λ ≤ cov ( N ) = λ ≤ add ( M ) = b = λ ≤ non ( M ) = λ ≤≤ cov ( M ) = λ ≤ d = cof ( M ) = λ ≤ non ( N ) = λ ≤ cof ( N ) = λ ≤ c = λ Proof.
Fix an increasing sequence of cardinals ℵ ≤ θ ≤ θ ≤ θ ≤ θ ≤ θ ≤ θ ≤ θ ≤ θ ≤ θ < µ < µ < µ < µ < µ < µ < µ < µ < λ such that all cardinals with the possible exception of θ are regular and cf( θ ) ≥ ℵ . Next let λ i := µ − i for1 ≤ i ≤
4. Let P pre be the ccc partial order from Theorem 46 for λ i , 1 ≤ i ≤ P = P pre ∩ N ∗ of P pre which forces ( b ( R i ) , d ( R i )) = ( θ − i , θ − i )for 1 ≤ i ≤ c = θ .Fix N n,α for 0 ≤ n ≤ α < θ n , N n := S α<θ n N n,α , as well as N such that • all N n,α as well as N are elementary submodels of H ( χ ) containing the sequence of cardinals, P pre , aswell as the witnesses S i of COB ( R i , P pre , λ i , λ ) (from Theorem 46) for 1 ≤ i ≤ • N n,α contains ( N m,β : m < n, β < θ m ) and ( N n,β : β < α ), and N contains ( N m,β : m ≤ , β < θ m ) • the N n,α are < µ n -closed of size µ n , and N is < ℵ -closed of size θ Let N ∗ := T ≤ n ≤ N n . For 0 ≤ n ≤
8, let P n := P pre ∩ T ≤ m ≤ n N m , and let P := P = P pre ∩ N ∗ . Note that T ≤ m ≤ n N m is again an elementary submodel of H ( χ ), and therefore each P n is a complete subforcing of P pre .We first show that for all i with 1 ≤ i ≤
4, we have
EUB ( R i , P , θ − i ) and EUB ( R i , P , θ − i ). Since the proofis the same for all i , we do it for i = 3. By Theorem 46 we have • EUB ( R , P pre , µ ) and EUB ( R , P pre , µ ).By part 1 of Lemma 59, we successively get • EUB ( R , P i , µ ) and EUB ( R , P i , µ ) for i = 0 , • EUB ( R , P , µ ) and EUB ( R , P , θ ), • EUB ( R , P j , θ ) and EUB ( R , P j , θ ) for 3 ≤ j ≤ i with all 1 ≤ i ≤
4, we have
COB ( R i , P , θ − i , θ − i ). Again we consider only thecase i = 3. By Theorem 46 we have COB ( R , P pre , µ , λ ). We first show COB ( R , P , θ , θ ): for s ∈ θ × θ × θ let M s := N ,s (0) ∩ N ,s (1) ∩ N ,s (2) . Then M s is a < µ -closed elementary submodel of H ( χ ) of size µ . Also( M s : s ∈ θ × θ × θ ) ∈ N , , M := S s M s = N ∩ N ∩ N , and | θ × θ × θ | = θ . Therefore we may applyLemma 60 with θ = θ , θ ′ = θ , µ = µ , µ ′ = λ = µ , ν = λ , N α = N ,α for α < θ , and N = N to obtain COB ( R , P , θ , θ ) (note here that P = P pre ∩ M ∩ N ). An easy application of part 2 (a) of Lemma 59 yieldsthat COB ( R , P , θ , θ ) still holds.By Lemma 42 and part 1 of Corollary 45, P forces b ( R i ) = θ − i and d ( R i ) = θ − i .Finally note that | P | = | N ∗ | = | N | = θ = θ ℵ and therefore by standard arguments P forces c ≤ θ . Onthe other hand, there is a sequence ( ˙ x ξ : ξ < λ ) of P pre -names for distinct reals belonging to N ∗ . Hence( ˙ x ξ : ξ ∈ λ ∩ N ∗ ) is a sequence of P -names for distinct reals. Since | λ ∩ N ∗ | = θ , P forces c ≥ θ .28 .3 Further results Using the preparatory forcing from [KST] (cf Theorem 51) together with the submodel technique, one obtains:
Theorem 62 (Goldstern, Kellner, Mej´ıa, and Shelah [GKMS1]) . Assume GCH and ( λ i : 1 ≤ i ≤ is a ≤ -increasing sequence of uncountable cardinals with λ i regular for i ≤ and λ of uncountable cofinality. Thenthere is a ccc partial order forcing that ℵ ≤ add ( N ) = λ ≤ add ( M ) = b = λ ≤ cov ( N ) = λ ≤ non ( M ) = λ ≤≤ cov ( M ) = λ ≤ non ( N ) = λ ≤ d = cof ( M ) = λ ≤ cof ( N ) = λ ≤ c = λ Further cardinal invariants can be included in the picture (cf Theorem 52):
Theorem 63 (Goldstern, Kellner, Mej´ıa, and Shelah [GKMS2]) . Assume GCH and ( λ i : 1 ≤ i ≤ is a ≤ -increasing sequence of uncountable cardinals with λ i regular for i ≤ and λ of uncountable cofinality.Then there is a cofinality-preserving partial order forcing that ℵ ≤ m = λ ≤ p = λ ≤ h = g = λ ≤ add ( N ) = λ ≤ cov ( N ) = λ ≤ add ( M ) = b = λ ≤ non ( M ) = λ ≤≤ cov ( M ) = λ ≤ d = cof ( M ) = λ ≤ non ( N ) = λ ≤ cof ( N ) = λ ≤ c = λ Theorem 64 (Goldstern, Kellner, Mej´ıa, and Shelah [GKMS4]) . In the previous result, letting λ = ℵ and λ s and λ r two regular cardinals with λ ≤ λ s ≤ λ , λ ≤ λ r ≤ λ , and λ s ∈ [ λ i , λ i ] iff λ r ∈ [ λ − i , λ − i ] ,there is a cofinality-preserving partial order forcing the given distribution together with s = λ s and r = λ r . References [BJ] T. Bartoszy´nski and H. Judah,
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