The full basis theorem does not imply analytic wellordering
aa r X i v : . [ m a t h . L O ] F e b The full basis theorem does not imply analyticwellordering
Vladimir Kanovei ∗† Vassily Lyubetsky ‡§ March 18, 2019
Abstract
We make use of a finite support product of ω clones of the Jensenminimal Π singleton forcing to define a model in which every non-emptylightface analytically definable set of reals contains a lightface analyticallydefinable real (the full basis theorem), but there is no lightface analyticallydefinable wellordering of the continuum. ∗ IITP RAS and MIIT, Moscow, Russia, [email protected] — contact author. † Thankful the Department of Philosophy, Linguistics and Theory of Science at the Universityof Gothenburg and the Erwin Schrodinger International Institute for Mathematics and Physics(ESI) at Vienna for their hospitality and support in resp. May 2015 and December 2016. ‡ IITP RAS, Moscow, Russia, [email protected] . § Supported in part by RNF Grant Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 The structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I Basic constructions 6 II Refinements 13
III
Structure of real names 21
13 Real names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114 Direct forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215 Locking real names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316 Non-principal names and avoiding refinements . . . . . . . . . . . . . . . . . . . . . . . . . 2317 Generic refinements avoid non-principal names . . . . . . . . . . . . . . . . . . . . . . . . 2418 Consequences for reals in generic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 2619 Combining refinement types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 IV The forcing notion 29
20 Increasing sequences of small multiforcings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2921 Layer restrictions of multiforcings and deciding sets . . . . . . . . . . . . . . . . . . . . . . 3022 Auxiliary diamond sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3223 Key sequence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3324 Key product forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 V Auxiliary forcing relation 38
25 Auxiliary forcing: preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3826 Auxiliary forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3927 Forcing simple formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4028 Tail invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4229 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4330 Forcing with subsequences of the key multisequence . . . . . . . . . . . . . . . . . . . . . 45 VI The model 48
31 Key generic extension and subextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4832 Definability of generic reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4933 Elementary equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5134 Non-wellorderability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5335 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References 56Index 58 Introduction
The uniformization problem, introduced by Luzin [22, 23], as well as the relatedbasis problem, are well known in modern set theory. (See Moschovakis [24],Kechris [21], Hauser and Schindler [7] for both older and more recent studies.) Inparticular, it is known that every non-empty Σ set of reals contains a ∆ real,but on the other hand, it is consistent that there exists a non-empty Π set ofreals containing even no ordinal-definable real.The negative part of this result was strengthened in [19] to the effect that thecounter-example set X ⊆ ω ω is a Π E -equivalence class (hence, a countableset), see related discussions at the Mathoverflow exchange desk and at FOM .Recall that E is an equivalence relation on ω ω defined so that x E y iff x ( n ) = y ( n ) for all but finite n .As for the positive direction, the most transparent way to get a basis resultis to make use of an analytically definable wellordering < of the reals, whichenables one to pick the < -least real in each non-empty set of reals. This leads tothe question: is the existence of an analytically definable wellordering < of thereals independent of the basis theorem. We answer it in the positive: Theorem 1.1.
In a suitable generic extension of L , it is true that in which everynon-empty lightface analytically definable set of reals contains a lightface analyt-ically definable real (the full basis theorem), but there is no lightface analyticallydefinable wellordering of the continuum.More precisely, there is a cardinal-preserving generic extension L [ X ] of L ,such that X = h x ξk i ξ<ω L ∧ k<ω , where each x ξk is a real in ω , and in addition (I) if m < ω then the submodel L [ X m ] admits a ∆ m +3 wellordering of thereals of length ω , where X m = h x ξk i ξ<ω L ∧ k To prove the theorem, we define, in L , a system of forcing notions P ξk , ξ < ω and k < ω , whose finite-support product P = Q ξ,k P ξk adds an array X = h x ξk i ξ<ω ,k<ω of reals x ξk to L , such that (I), (II), (III), (IV) hold in L [ X ].Regarding the history of this research, in goes down to Jensen [10], wherea forcing J = S α<ω J α is defined in L , the constructible universe, such thateach J α is a countable set of perfect trees in 2 <ω , the canonical J -generic realis a single J -generic real in the extension, and ‘being a J -generic real’ is a Π property, so as a result we get a Π nonconstructible singleton in any J -genericextension of L . See 28A in [8] for a more modern exposition of Jensen’s forcing.A nonconstructible Π singleton also was defined in [9] by means of thealmost-disjoint forcing, yet the construction in [10] has the advantage of minimal-ity of J -generic reals and some other advantages (as well as some disadvantages).Jensen’s forcing construction (including its iterations) was exploited by Abra-ham [1, 2], including a definable minimal collapsing real. Another modificationof Jensen’s forcing construction in [11] yields such a forcing notion in L that anyextension of L , containing two generic reals x = y , necessarily satisfies ω L < ω .See [3, 15] on some other modifications in coding purposes.A different modification of Jensen’s forcing construction was engineered in[16] in order to define an extension of L in which, for a given n ≥ 2, there is anonconstructible Π n singleton while all Σ reals are constructible. (An abstractappeared in [14].) The idea is to complicate the inductive construction of Jensen’ssequence ~ J = h J α i α<ω in L by the requirement that it intersects any set, of acertain definability level, dense in the collection of all possible countable initialsteps of the construction. The same inner genericity idea, with respect to theJensen – Johnsbraten forcing notion in [11], was developed in [18].Such an inner genericity modification of the Jensen – Solovay almost-disjointforcing [9] was developed in [6] towards some great results which unfortunatelyhave never been published in a mathematical journal. Except for a one result, amodel in which the set of all analytically definable reals is equal to the set of allconstructible reals, independently obtained in [17]. We employ the inner definablegenericity idea here in such a way that if m < ω then the m -tail h P ξk i ξ<ω ∧ k ≥ m of the forcing construction, bears an amount of inner definable genericity whichstrictly depends on m . (See Definition 21.1, where a key concept is introduced.)Ali Enayat (Footnote 2) conjectured that some definability questions can besolved by finite-support products of Jensen’s [10] forcing J . Enayat demonstratedin [4] that a symmetric part of the J ω -generic extension of L definitely yields amodel of ZF (not a model of ZFC !) in which there is a Dedekind-finite infinite Π set of reals with no OD elements. Following the conjecture, we proved in [12]that indeed it is true in a J ω -generic extension of L that the set of J -generic4eals is a countable non-empty Π set with no OD elements. We also provedin [13] that the existence of a Π E -class with no OD elements is consistentwith ZFC , using a E -invariant version of Jensen’s forcing. We further employedanother finite-support product of Jensen’s forcing to define a generic extensionof L where there is a Π set P ⊆ ω ω × ω ω which has countable cross-sections P x = { y : h x, y i ∈ P } and is non-uniformizable by any projective set [20]. Acknowledgement. The idea of making use of a suitable finite-support productof Jensen-like forcing notions in order to obtain a model, in which the full basistheorem holds but there is no lightface analytically definable wellordering of thecontinuum, was communicated to an author of this paper (VK) by Ali Enayat in2015, and we thank Ali Enayat for fruitful discussions and helpful ideas. The general organization of the paper is as follows. Chapter I contains a generalformalism related to forcing by perfect trees and finite-support products, con-venient for our goals. Following Jensen [10], we consider forcing notions of theform P = S α<λ P α , where λ < ω and each P α is a countable set of perfect treesin 2 <ω . Each term P α has to satisfy some routine conditions of refinement withrespect to the previous terms, in particular, to make sure that each P α remainspre-dense at further steps. Also, each P α has to lock some dense sets in S ξ<α P ξ so that they remain pre-dense at further steps as well. And this procedure has tobe extended from single forcing notions to their finite-support products. Theseissues are dealt with in Chapter II.Then we consider real names with respect to finite-support products of perfect-tree forcing notions in Chapter III. Here the key issue is to make sure that if P is a factor in a product forcing considered then there is no other P -generic realin the whole product extension except for the obvious one.In Chapter IV we define the forcing notion PPP = Q ξ<ω ,k<ω P ξk to prove themain theorem, in the form of a limit of a certain increasing sequence of countableproducts of countable perfect-tree forcing notions. Quite a complicated construc-tion of this sequence in L involves ideas related to diamond-style constructions,as well as to some sort of definable genericity, as explained above.The forcing P is not analytically definable; basically, each k -th layer h P ξk i ξ<ω belongs to ∆ k +4 . But it is a key property that the PPP -forcing relation restrictedto Σ n formulas is essentially Σ n . We prove this in Chapter V, with the help ofan auxiliary forcing notion forc . We also establish the invariance of forc withrespect to countable-support permutations of ω × ω .We finally prove Theorem 1.1 in Chapter VI, on the base of the results ob-tained in two previous chapters. 5 Basic constructions We begin with some basic things: perfect trees in the Cantor space 2 ω , perfecttree forcing notions (those which consist of perfect trees), their finite-supportproducts, and a splitting construction of perfect trees. Let 2 <ω be the set of all strings (finite sequences) of numbers 0 , 1. If t ∈ <ω and i = 0 , t a k is the extension of t by k . If s, t ∈ <ω then s ⊆ t meansthat t extends s , while s ⊂ t means proper extension. If s ∈ <ω then lh ( s ) isthe length of s , and 2 n = { s ∈ <ω : lh ( s ) = n } (strings of length n ).A set T ⊆ <ω is a tree iff for any strings s ⊂ t in 2 <ω , if t ∈ T then s ∈ T .Every non-empty tree T ⊆ <ω contains the empty string Λ. If T ⊆ <ω is atree and s ∈ T then put T ↾ s = { t ∈ T : s ⊆ t ∨ t ⊆ s } .Let PT be the set of all perfect trees ∅ = T ⊆ <ω . Thus a non-empty tree T ⊆ <ω belongs to PT iff it has no endpoints and no isolated branches. Thenthere is a largest string s ∈ T such that T = T ↾ s ; it is denoted by s = stem ( T )(the stem of T ); we have s a ∈ T and s a ∈ T in this case. Definition 4.1 (perfect sets) . If T ∈ PT then [ T ] = { a ∈ ω : ∀ n ( a ↾ n ∈ T ) } is the set of all paths through T , a perfect set in 2 ω . Conversely if X ⊆ ω is aperfect set then tree ( X ) = { a ↾ n : a ∈ X ∧ n < ω } ∈ PT and [ tree ( X )] = X .Trees T, S ∈ PT are almost disjoint , a.d. for brevity, iff the intersection S ∩ T is finite; this is equivalent to just [ S ] ∩ [ T ] = ∅ .The simple splitting of a tree T ∈ PT consists of smaller trees T ( → 0) = T ↾ stem ( T ) a and T ( → 1) = T ↾ stem ( T ) a in PT , so that [ T ( → i )] = { x ∈ [ T ] : x ( h ) = i } , where h = lh ( stem ( T )). We let T ( → u ) = T ( → u (0))( → u (1))( → u (2)) . . . ( → u ( n − u ∈ <ω , lh ( u ) = n ; and separately T ( → Λ) = T . Lemma 4.2. Suppose that T ∈ PT . Then :(i) if u ∈ <ω then there is a string s ∈ <ω such that T ( → u ) = T ↾ s ;(ii) if s ∈ <ω then there is a string u ∈ <ω such that T ↾ s = T ( → u ) ;(iii) if ∅ = U ⊆ [ T ] is a (relatively) open subset of [ T ] , or at least U has a non-empty interior in [ T ] , then there is a string s ∈ T such that T ↾ s ⊆ U . T ∈ PT and a ∈ ω then the intersection T ( → a ) = T n<ω T ( → a ↾ n ) = { h T ( a ) } is a singleton, and the map h T is a canonical homeomorphism from 2 ω onto [ T ]. Accordingly if S, T ∈ PT then the map h ST ( x ) = h T ( h S − ( x )) is a canonical homeomorphism from [ S ] onto [ T ]. A perfect-tree forcing notion is any non-empty set P ⊆ PT such that if s ∈ T ∈ P then T ↾ s ∈ P , or equivalently, by Lemma 4.2, if u ∈ <ω then T ( → u ) ∈ P . Let PTF be the set of all such forcing notions P ⊆ PT . Example 5.1. If s ∈ <ω then the tree [ s ] = { t ∈ <ω : s ⊆ t ∨ t ⊆ s } belongsto PT . The set P coh = { [ s ] : s ∈ <ω } of all such trees (the Cohen forcing) is aregular perfect-tree forcing notion. Lemma 5.2. Let P ∈ PTF . If T ∈ P and a set X ⊆ [ T ] is (relatively) open(resp., clopen) in [ T ] , then there is a countable (resp., finite) set S of pairwisea.d. trees S ∈ P , satisfying S S ∈ S [ S ] = X . Lemma 5.3. (i) If s ∈ T ∈ P ∈ PTF then T ↾ s ∈ P . (ii) If P , P ′ ∈ PTF , T ∈ P , T ′ ∈ P ′ , then there are trees S ∈ P , S ′ ∈ P ′ suchthat S ⊆ T , S ′ ⊆ T ′ , and [ S ] ∩ [ S ′ ] = ∅ . Proof. (i) use Lemma 4.2. (ii) If T = T ′ then let S = T ( → S ′ = T ( → 1) .If say T T ′ then let s ∈ T r T ′ , S = T ↾ s , and simply S ′ = T ′ . Definition 5.4. A set A ⊆ PT is an antichain iff any trees T = T ′ in A area.d., that is, [ T ] ∩ [ T ′ ] = ∅ . A forcing notion P ∈ PTF is: small , if it is countable; special , if there is an antichain A ⊆ P such that P = { T ↾ s : s ∈ T ∈ A } — notethat A is unique if exists; we write A = base ( P ) (the base of P ); regular , if for any S, T ∈ P , the intersection [ S ] ∩ [ T ] is clopen in [ S ] or clopenin [ T ] (or clopen in both [ S ] and [ T ]). Lemma 5.5. Let P ∈ PTF . If P is special and S, T ∈ P are not a.d., thenthey are comparable : S ⊆ T or T ⊆ S .If P is special then P is regular. If P is regular, then (i) if S, T ∈ P are not a.d., then they are compatible in P , that is, there is atree R ∈ P such that R ⊆ S ∩ T . (ii) if S , . . . , S k ∈ P then there is a finite set of pairwise a.d. trees R , . . . , R n ∈ P such that [ S ] ∩ . . . ∩ [ S k ] = [ R ] ∪ . . . ∪ [ R n ] . if S , . . . , S k are finite collections of trees in P then there is a finite set oftrees R , . . . , R n ∈ P such that S S ∈ S [ S ] ∩ . . . ∩ S S ∈ S k [ S ] = [ R ] ∪ . . . ∪ [ R n ] ,and for any S i and R j , there is S ∈ S i such that R j ⊆ S . Proof. (iii) Apply (ii) to every set of the form [ S ] ∩ . . . ∩ [ S k ], where S i ∈ S i , ∀ i , then gather all trees R i obtained in one finite set. Remark 5.6. Any set P ∈ PTF can be considered as a forcing notion (if T ⊆ T ′ then T is a stronger condition); then P adds a real x ∈ ω . Lemma 5.7. If a set G ⊆ P is generic over a ground set universe V ( resp.,over a transitive model, e. g. L ) then (i) the intersection T T ∈ G [ T ] contains a single real x = x [ G ] ∈ ω , and (ii) this real x is P - generic , in the sense that if D ⊆ P is dense in P andbelongs to V ( resp., to the ground model ) then x ∈ S T ∈ D [ T ] . As usual, a set D ⊆ P is: − open in P , if for any trees T ⊆ S in P , T ∈ D = ⇒ S ∈ D ; − dense in P , if for any T ∈ P there is S ∈ D , S ⊆ T ; − pre-dense in P , if the set D ′ = { T ∈ P : ∃ S ∈ D ( T ⊆ S ) } is dense in P . We proceed with an important splitting/fusion construction of perfect trees bymeans of infinite splitting systems of such trees. Definition 6.1. Let FSS be the set of all finite splitting systems , that is, systemsof the form ϕ = h T s i s ∈ ≤ n , where n = hgt ( ϕ ) < ω (the height of ϕ ), each value T s = T ϕs = ϕ ( s ) is a tree in PT , and( ∗ ) if s ∈ Let P ∈ PTF . Then there is an ≺ -increasing sequence h ϕ n i n<ω of systems in FSS ( P ) . And if h ϕ n i n<ω is such then :(i) the limit system ϕ = S n ϕ n = h T s i s ∈ <ω satisfies ( ∗ ) of Definition 6.1 onthe whole domain of strings s ∈ <ω ;(ii) T = T n S s ∈ n T s is a perfect tree in PT and [ T ] = T n S s ∈ n [ T s ] ;(iii) if u ∈ <ω then T ( → u ) = T ∩ T u = T n ≥ lh ( u ) S s ∈ n ,u ⊆ s T s . We’ll systematically make use of finite support products of perfect tree forcingsin this paper. The following definitions introduce suitable notation.Call a multiforcing any map π : | π | → PTF , where | π | = dom π ⊆ ω × ω .Thus each set π ( ξ, k ), h ξ, k i ∈ | π | , is a perfect tree forcing notion. Such a π is: − small , if both | π | and each forcing π ( ξ, k ), h ξ, k i ∈ | π | , are countable; − special , if each π ( ξ, k ) is special in the sense of Definition 5.4; − regular , if each π ( ξ, k ) is regular, in the sense of Definition 5.4.Let MF be the set of all multiforcings.Let a multitree be any map p : | p | → PT , such that | p | = dom p ⊆ ω × ω is finite and each value T p ξk = p ( ξ, k ) is a tree in PT . In this case we define acofinite-dimensional perfect cube in 2 ω × ω [ p ] = { x ∈ ω × ω : ∀ h ξ, k i ∈ | p | ( x ( ξ, k ) ∈ [ T p ξk ]) } == { x ∈ ω × ω : ∀ h ξ, k i ∈ | p | ∀ m ( x ( ξ, k ) ↾ m ∈ T p ξk ) } . Let MT be the set of all multitrees. We order MT componentwise: q p ( q is stronger) iff | p | ⊆ | q | and T q ξk ⊆ T p ξk for all h ξ, k i ∈ | p | ; this is equivalent to[ q ] ⊆ [ p ], so that stronger multitrees correspond to smaller cubes. The weakestmultitree Λ ∈ MT is just the empty map; | Λ | = ∅ and [ Λ ] = 2 ω × ω .Multitrees p , q are somewhere almost disjoint , or s.a.d. , if, for at least onepair of indices h ξ, k i ∈ | p |∩ | q | , the trees T p ξk , T q ξk are a.d., that is, [ T p ξk ] ∩ [ T q ξk ] = ∅ , or equivalently, T p ξk ∩ T q ξk is finite. 9 orollary 7.1 (of Lemma 5.5(i)) . If π is a regular multiforcing and multitrees p , q ∈ MT ( π ) are not s.a.d., then p , q are compatible in MT ( π ) , so that thereis a multitree r ∈ MT ( π ) with r p , r q . If π is a multiforcing then a π -multitree is any multitree p with | p | ⊆ | π | and T p ξk ∈ π ( ξ, k ) for all h ξ, k i ∈ | p | . Let MT ( π ) be the set of all π -multitrees;it is equal to the finite support product Q h ξ,k i∈| π | π ( ξ, k ).The following is similar to Lemma 5.5(iii). Lemma 7.2. If a multiforcing π is regular, ξ ⊆ | π | is finite, and U , . . . , U k are finite collections of multitrees in MT ( π ) with | p | = ξ for all p ∈ S i U i , thenthere is a finite set of multitrees u , . . . , u n ∈ MT ( π ) such that | u j | = ξ , ∀ j , S p ∈ U [ p ] ∩ . . . ∩ S p ∈ U k [ p ] = [ u ] ∪ . . . ∪ [ u n ] , and for any U i and u j , there is p ∈ U i such that [ u j ] ⊆ [ p ] . We consider sets of the form MT ( π ) in the role of product forcing notions .A set D ⊆ MT ( π ) is: − open in MT ( π ), if for any p q in MT ( π ), q ∈ D = ⇒ p ∈ D ; − dense in MT ( π ), if for any p ∈ MT ( π ), there is q ∈ D , q p ; − pre-dense in MT ( π ), if the set D ′ = { p ∈ MT ( π ) : ∃ q ∈ D ( p q ) } isdense in MT ( π ). Remark 7.3. As a forcing notion, each MT ( π ) adds an array h x ξk i h ξ,k i∈| π | of reals, where each real x ξk ∈ ω is a π ( ξ, k )-generic real. Namely if a set G ⊆ MT ( π ) is generic over the ground set universe V then each factor G ( ξ, k ) = { T p ξk : p ∈ G ∧ h ξ, k i ∈ | p |} ⊆ π ( ξ, k )(where h ξ, k i ∈ | π | ) is accordingly a set π ( ξ, k )-generic over V , the real x ξk = x ξk [ G ] = x [ G ( ξ, k )] ∈ ω is the only real satisfying x ξk ∈ T T ∈ G ( ξ,k ) [ T ], and x ξk is π ( ξ, k )-generic over V as in Lemma 5.7.The reals of the form x ξk [ G ] will be called principal generic reals in V [ G ]. Definition 7.4. A componentwise union of multiforcings π , ϙ is a multiforcing π ∪ cw ϙ satisfying | ( π ∪ cw ϙ ) | = | π | ∪ | ϙ | and( π ∪ cw ϙ )( ξ, k ) = π ( ξ, k ) , whenever h ξ, k i ∈ | π | r | ϙ | ϙ ( ξ, k ) , whenever h ξ, k i ∈ | ϙ | r | π | π ( ξ, k ) ∪ ϙ ( ξ, k ) , whenever h ξ, k i ∈ | π | ∩ | ϙ | Similarly, if π = h π α i α<λ is a sequence of multiforcings then define a multiforcing π = S cw π = S cw α<λ π α so that | π | = S α<λ | π α | and if h ξ, k i ∈ | π | then π ( ξ, k ) = S α<λ, h ξ,k i∈| π α | π α ( ξ, k ). 10 Multisystems The next definition introduces multisystems , a multi version of the splitting/fusiontechnique of Section 6, whose intention is to define suitable multiforcings, as willbe shown in Section 11 below. Definition 8.1. A multisystem is any map ϕ : | ϕ | → FSS , such that | ϕ | ⊆ ω × ω × ω is finite. This amounts to(1) the map h ϕ ( ξ, k, m ) = hgt ( ϕ ( ξ, k, m )) : | ϕ | → ω , and(2) the finite collection of trees T ϕ ξk,m ( s ) = ϕ ( ξ, k, m )( s ), where h ξ, k, m i ∈| ϕ | and s ∈ ≤ h ϕ ( ξ,k,m ) , such that if h ξ, k, m i ∈ | ϕ | then ϕ ( ξ, k, m ) = h T ϕ ξk,m ( s ) i s ∈ ≤ h ϕ ( ξ,k,m ) is a finite splitting system in FSS .If π is a multiforcing, | ϕ | ⊆ ( | π | ) × ω , and ϕ ( ξ, k, m ) ∈ FSS ( π ( ξ, k )) for all h ξ, k, m i ∈ | ϕ | (or equivalently, T ϕ ξk,m ( s ) ∈ π ( ξ, k ) whenever h ξ, k, m i ∈ ϕ and s ∈ ≤ h ϕ ( ξ,k,m ) ), then say that ϕ is a π - multisystem , ϕ ∈ MS ( π ).Let ϕ , ψ be multisystems. Say that ϕ extends ψ , symbolically ψ ϕ , if | ψ | ⊆ | ϕ | , and, for every h ξ, k, m i ∈ | ψ | , ϕ ( ξ, k, m ) extends ψ ( ξ, k, m ), that is, h ϕ ( ξ, k, m ) ≥ h ψ ( ξ, k, m ) and T ϕ ξk,m ( s ) = T ψ ξk,m ( s ) for all s ∈ ≤ h ψ ( ξ,k,m ) .It will be demonstrated in Section 11 that a suitably increasing infinite se-quence ϕ ϕ ϕ . . . of multisystems in some MS ( π ) leads to a “limit”multiforcing ϙ with | ϙ | = S n | ϕ n | , such that each factor ϙ ( ξ, k ), h ξ, k i ∈ | π | , isfilled in by trees Q ξk,m , m < ω , in such a way, that the ( ξ, k, m )-components ofthe systems ϕ n are responsible for the construction of the tree Q ξk,m .The next lemma introduces different ways to extend a given multisystem.Say that a multisystem ϕ is if [ T ϕξk,m ( s )] ∩ [ T ϕηℓ,n ( t )] = ∅ forall triples h ξ, k, m i 6 = h η, ℓ, n i in | ϕ | and all s ∈ h ϕ ( ξ,k,m ) and t ∈ h ϕ ( η,ℓ,n ) . Lemma 8.2. Let π be a multiforcing and ϕ ∈ MS ( π ) . (i) If h ξ, k, m i ∈ | ϕ | and h = h ϕ ( ξ, k, m ) then the extension ψ of ϕ by h ψ ( ξ, k, m ) = h + 1 and T ψ ξk,m ( s a i ) = T ϕ ξk,m ( s )( → i ) for all s ∈ h and i = 0 , , belongs to MS ( π ) and ϕ ψ . (ii) If h ξ, k, m i / ∈ | ϕ | then the extension ψ of ϕ by | ψ | = | ϕ | ∪ {h ξ, k, m i} , h ψ ( ξ, k, m ) = 0 and T ψ ξk,m (Λ) = T , where T ∈ π ( ξ, k ) and Λ is the emptystring, belongs to MS ( π ) and ϕ ψ . (iii) If h ξ, k, m i ∈ | ϕ | and a set D ⊆ π ( ξ, k ) is open dense in π ( ξ, k ) thenthere is a multisystem ψ ∈ MT ( π ) such that | ψ | = | ϕ | , ϕ ψ , and T ψ ξk,m ( s ) ∈ D whenever s ∈ h ψ ( ξ,k,m ) . There is a 2wise disjoint ψ ∈ MT ( π ) such that | ψ | = | ϕ | and ϕ ψ . Proof. To prove (iii) first use (i) to get a multisystem ψ ∈ MS ( π ) with ϕ ψ and h ψ ( ξ, k, m ) = h + 1, where h = h ϕ ( ξ, k, m ). Then replace each tree T ψ ξk,m ( s ) = ψ ( ξ, k, m )( s ), s ∈ h +1 , with a suitable tree T ′ ∈ D , T ′ ⊆ T ψ ξk,m ( s ).To prove (iv) first apply (i) to get a multisystem ψ ∈ MS ( π ) with ϕ ψ , | ψ | = | ϕ | , and h ψ ( ξ, k, m ) = h ϕ ( ξ, k, m ) + 1 for all h ξ, k, m i ∈ | ϕ | . Now if h ξ, k, m i 6 = h η, ℓ, n i are triples in | ϕ | and s ∈ h ϕ ( ξ,k,m )+1 , t ∈ h ϕ ( η,ℓ,n )+1 ,then, by Lemma 5.3(ii), there are trees S ∈ π ( ξ, k ) and T ∈ π ( η, ℓ ) satisfying[ S ] ∩ [ T ] = ∅ and S ⊆ T ψ ξk,m ( s ), T ⊆ T ψ ηℓ,n ( t ). Replace the trees T ψ ξk,m ( s ), T ⊆ T ψ ηℓ,n ( t ) with resp. S , T . Iterate this shrinking construction for all triples h ξ, k, m i 6 = h η, ℓ, n i and strings s, t as above.12 I Refinements Here we consider refinements of perfect tree forcings and multiforcings, the keytechnical tool of definition of various forcing notions in this paper. If T ∈ PT (a perfect tree) and D ⊆ PT then X ⊆ fin S D will mean that thereis a finite set D ′ ⊆ D such that T ⊆ S D ′ , or equivalently [ T ] ⊆ S S ∈ D ′ [ S ]. Definition 9.1. Let P , Q ∈ PTF be perfect tree forcing notions. Say that Q isa refinement of P (symbolically P ❁ Q ) if(1) the set Q is dense in P ∪ Q : if T ∈ P then ∃ Q ∈ Q ( Q ⊆ T );(2) if Q ∈ Q then Q ⊆ fin S P ;(3) if Q ∈ Q and T ∈ P then [ Q ] ∩ [ T ] is clopen in [ Q ] and T Q . Lemma 9.2. (i) If P ❁ Q and S ∈ P , T ∈ Q , then [ S ] ∩ [ T ] is meager in [ S ] , therefore P ∩ Q = ∅ and Q is open dense in P ∪ Q ; . (ii) if P ❁ Q ❁ R then P ❁ R , thus ❁ is a strict partial order ;(iii) if h P α i α<λ is a ❁ -increasing sequence in PTF and < µ < λ then P = S α<µ P α ❁ Q = S µ ≤ α<λ P α ;(iv) if h P α i α<λ is a ❁ -increasing sequence in PTF and each P α is special then P = S α<λ P α is a regular forcing in PTF ;(v) in (iv) , each P γ is pre-dense in P = S α<λ P α . Proof. (i) Otherwise there is a string u ∈ S such that S ↾ u ⊆ [ T ] ∩ [ S ]. But S ↾ u ∈ P , which contradicts to 9.1(3).(ii), (iii) Make use of (i) to establish 9.1(3).(iv) To check the regularity let S ∈ P α , T ∈ P β , α ≤ β . If α = β then, as P α is special, the trees S, T are either a.d. or ⊆ -comparable by Lemma 5.5. If α < β then [ S ] ∩ [ T ] is clopen in [ T ] by 9.1(3).(v) Let S ∈ P α , α = γ . If α < γ then by 9.1(1) there is a tree T ∈ P γ , T ⊆ S . Now let γ < α . Then S ⊆ fin S P γ by 9.1(2), in particular, there is atree T ∈ P γ such that [ S ] ∩ [ T ] = ∅ . However [ S ] ∩ [ T ] is clopen in [ S ] by 9.1(3).Therefore S ↾ u ⊆ T for a string u ∈ S . Finally S ↾ u ∈ P α since P α ∈ PTF .Note that if P , Q ∈ PTF and P ❁ Q then a dense set D ⊆ P is not necessarilydense or even pre-dense in P ∪ Q . Yet there is a special type of refinement whichpreserves at least pre-density. We modify the relation ❁ as follows.13 efinition 9.3. Let P , Q ∈ PTF and D ⊆ P . Say that Q locks D over P ,symbolically P ❁ D Q , if P ❁ Q holds and every tree S ∈ Q satisfies S ⊆ fin S D .Then simply P ❁ Q is equivalent to P ❁ P Q .As we’ll see now, a locked set has to be pre-dense both before and after therefinement. The additional importance of locking refinements lies in fact that,once established, it preserves under further simple refinements, that is, ❁ D istransitive in a combination with ❁ in the sense of (ii) of the following lemma: Lemma 9.4. (i) If P ❁ D Q then D is pre-dense in P ∪ Q , and if in addition P is regular then D is pre-dense in P as well ;(ii) if P ❁ D Q ❁ R (note: the second ❁ is not ❁ D !) then P ❁ D R ;(iii) if h P α i α<λ is a ❁ -increasing sequence in PTF , < µ < λ , and P = S α<µ P α ❁ D P µ , then P ❁ D Q = S µ ≤ α<λ P α . Proof. (i) To see that D is pre-dense in P ∪ Q , let T ∈ P ∪ Q . By 9.1(1), thereis a tree T ∈ Q , T ⊆ T . Then T ⊆ fin S D , in particular, there is a tree S ∈ D with X = [ S ] ∩ [ T ] = ∅ . However X is clopen in [ T ] by 9.1(3). Therefore,by Lemma 5.2, there is a tree T ′ ∈ Q with [ T ′ ] ⊆ X , thus T ′ ⊆ S ∈ D and T ′ ⊆ T ⊆ T . We conclude that T is compatible with S ∈ D in P ∪ Q .To see that D is pre-dense in P (assuming P is regular), let S ∈ P . Itfollows from the above that S is compatible with some S ∈ D , hence, S and S are not absolutely incompatible. It remains to use Lemma 5.5(i).To prove (ii) on the top of Lemma 9.2(ii), let R ∈ R . Then R ⊆ fin S Q , buteach T ∈ Q satisfies T ⊆ fin S D . The same for (iii).The existence of ❁ D -refinements will be established below. 10 Refining multiforcings Let π , ϙ be multiforcings. Say that ϙ is an refinement of π , symbolically π ❁ ϙ , if | π | ⊆ | ϙ | and π ( ξ, k ) ❁ ϙ ( ξ, k ) whenever h ξ, k i ∈ | π | . Corollary 10.1 (of Lemma 9.2) . If π ❁ ϙ ❁ ρ then π ❁ ρ .If π ❁ ϙ then the multiforcing MT ( ϙ ) is open dense in MT ( π ∪ cw ϙ ) . Our next goal is to introduce a version of Definition 9.3 suitable for multi-forcings; we expect an appropriate version of Lemma 9.4 to hold.First of all, we accomodate the definition of the relation ⊆ fin in Section 9for multitrees. Namely if u is a multitree and D a collection of multitrees, then u ⊆ fin W D will mean that there is a finite set D ′ ⊆ D satisfying 1) | v | = | u | for all v ∈ D ′ , and 2) [ u ] ⊆ S v ∈ D ′ [ v ].14 efinition 10.2. Let π , ϙ be multiforcings, and π ❁ ϙ . Say that ϙ locks a set D ⊆ MT ( π ) over π , symbolically π ❁ D ϙ if the following condition holds:( ∗ ) if p ∈ MT ( π ), u ∈ MT ( ϙ ), | u | ⊆ | π | , | u | ∩ | p | = ∅ , then there is q ∈ MT ( π ) such that q p , still | q | ∩ | u | = ∅ , and u ⊆ fin W D | u | q , where D | u | q = { u ′ ∈ MT ( π ) : | u ′ | = | u | and u ′ ∪ q ∈ D } . Note that if p , u , D , q are as indicated then still u ∪ q ⊆ fin W D holds viathe finite set D ′ = { u ′ ∪ q : u ′ ∈ D | u | q } ⊆ D . Anyway the definition of ❁ D in10.2 looks somewhat different and more complex then the definition of ❁ D in9.3, which reflects the fact that finite-support products of forcing notions in PTF behave differently (and in more complex way) than single perfect-tree forcings.Accordingly, the next lemma, similar to Lemma 9.4, is way harder to prove. Lemma 10.3. Let π , ϙ , σ be multiforcings and D ⊆ MT ( π ) . Then :(i) if π ❁ D ϙ then D is dense in MT ( π ) and pre-dense in MT ( π ∪ cw ϙ ) ;(ii) if π ❁ D ϙ and D ⊆ D ′ ⊆ MT ( π ) then π ❁ D ′ ϙ ;(iii) if π is regular, π ❁ D i ϙ for i = 1 , . . . , n , all sets D i ⊆ MT ( π ) are opendense in MT ( π ) , and D = T i D i , then π ❁ D ϙ ;(iv) if D is open dense in MT ( π ) and π ❁ D ϙ ❁ σ then π ❁ D σ ;(v) if h π α i α<λ is a ❁ -increasing sequence in MF , < µ < λ , π = S cw α<µ π α , D is open dense in MT ( π ) , and π ❁ D π µ , then π ❁ D ϙ = S cw µ ≤ α<λ π α . Proof. (i) To check that D is pre-dense in MT ( π ∪ cw ϙ ), let r ∈ MT ( π ∪ cw ϙ ).Due to the product character of MT ( π ∪ cw ϙ ), we can assume that | r | ⊆ | π | . Let X = {h ξ, k i ∈ | r | : T r ξk ∈ MT ( ϙ ) } , Y = {h ξ, k i ∈ | r | : T r ξk ∈ MT ( π ) } . Then r = u ∪ p , where u = r ↾ X ∈ MT ( ϙ ), p = r ↾ Y ∈ MT ( π ). As ϙ locks D , there is a multitree q ∈ MT ( π ) such that q p , | q | ∩ | u | = ∅ , and u ⊆ fin S D | u | q . By an easy argument, there is a multitree u ′ ∈ D | u | q compatiblewith u in MT ( ϙ ); let w ∈ MT ( ϙ ), w u , w u ′ , | w | = | u ′ | = | u | . Then themultitree r ′ = w ∪ q ∈ MT ( π ∨ ϙ ) satisfies r ′ r and r ′ u ′ ∪ q ∈ D .To check that D is dense in MT ( π ), suppose that p ∈ MT ( π ). Let u = Λ (the empty multitree) in ( ∗ ) of Definition 10.2, so that | u | = ∅ and D | u | q = D .(ii) is obvious. To prove (iii), let p ∈ MT ( π ), u ∈ MT ( ϙ ), | u | ⊆ | π | , | u | ∩ | p | = ∅ . Iterating ( ∗ ) for D i , i = 1 , . . . , n , we find a multitree q ∈ MT ( π )such that q p , | q | ∩ | u | = ∅ , and u ⊆ fin W ( D i ) | u | q for all i , where( D i ) | u | q = { u ′ ∈ MT ( π ) : | u ′ | = | u | and u ′ ∪ q ∈ D i } . U i ⊆ ( D i ) | u | q such that [ u ] ⊆ S v ∈ U i [ v ] for all i .Using the regularity assumption and Lemma 7.2, we refine multitrees in S i U i ,getting a finite set W ⊆ MT ( π ) such that still | w | = | u | for all w ∈ W , T i S v ∈ U i [ v ] = S w ∈ W [ w ], and if i = 1 , . . . , n and w ∈ W then [ w ] ⊆ [ v ] forsome v ∈ U i — therefore w ∪ q ∈ D i . We conclude that if w ∈ W then w ∪ q ∈ D , hence w ∈ D | u | q . Thus W ⊆ D | u | q . However [ u ] ⊆ S w ∈ W [ w ] by thechoice of W . We conclude that u ⊆ fin W D | u | q , as required.(iv) It follows from Corollary 10.1 that π ❁ σ , hence it remains to checkthat σ locks D over π . Assume that u ∈ MT ( σ ), | u | ⊆ | π | , p ∈ MT ( π ), | u | ∩ | p | = ∅ . As ϙ ❁ σ , there is a finite set U ⊆ MT ( ϙ ) such that | v | = | u | for all v ∈ U , and [ u ] ⊆ S v ∈ U [ v ]. As π ❁ D ϙ , by iterated application of( ∗ ) of Definition 10.2, we get a multitree q ∈ MT ( π ) such that q p , still | q | ∩ | u | = ∅ , and if v ∈ U then v ⊆ fin W D | u | q , where D | u | q = { v ′ ∈ MT ( π ) : | v ′ | = | v | = | u | ∧ v ′ ∪ q ∈ D } . Note finally that u ⊆ fin W U by construction, hence u ⊆ fin W D | u | q as well.(v) We have to check that ϙ locks D over π . Let u ∈ MT ( ϙ ), | u | ⊆ | π | , p ∈ MT ( π ), | u | ∩ | p | = ∅ . As above, there is a finite set U ⊆ MT ( π µ ) such that | v | = | u | for all v ∈ U and [ u ] ⊆ S v ∈ U [ v ]. And so on as in the proof of (iv). 11 Generic refinement of a multiforcing Here we introduce a construction, due to Jensen in its original form, which impliesthe existence of refinements of forcings and multiforcings, of types ❁ D and ❁ D . Definition 11.1. 1. Suppose that π is a small multiforcing, and M ∈ HC isany set. (Recall that HC = all hereditarily countable sets.) This is the input.2. The set M + of all sets X ∈ HC , ∈ -definable in HC by formulas withsets in M as parameters, is still countable. Therefore there exists a -increasingsequence h ϕ ( j ) i j<ω of multisystems ϕ ( j ) ∈ MS ( π ), M + - generic in the sensethat it intersects any set ∆ ⊆ MS ( π ), ∆ ∈ M + , dense in MS ( π ). (The densitymeans that for any ψ ∈ MS ( π ) there is a multisystem ϕ ∈ ∆ with ψ ϕ .)Let us fix any such a M + -generic sequence Φ = h ϕ ( j ) i j<ω .3. Suppose that h ξ, k i ∈ | π | and m < ω . In particular, the sequence Φ intersects every (dense by Lemma 8.2(i),(ii)) set of the form∆ ξkmh = { ϕ ∈ MS ( π ) : h ϕ ( ξ, k, m ) ≥ h } ∈ M + , where h < ω . Hence a tree T Φ ξk,m ( s ) ∈ π ( ξ, k ) can be associated to any s ∈ <ω , such that, forall j , if h ξ, k, m i ∈ | ϕ ( j ) | and lh ( s ) ≤ h ϕ ( j ) ( ξ, k, m ) then T ϕ ( j ) ξk,m ( s ) = T Φ ξk,m ( s ).16. Then it follows from Lemma 6.3 that each set Q Φ ξk,m = T h S s ∈ h T Φ ξk,m ( s )is a tree in PT (not necessarily in π ( ξ, k )), as well as the trees Q Φ ξk,m ( s ) = T n ≥ lh ( s ) S t ∈ n , s ⊆ t T Φ ξk,m ( t ) , and obviously Q Φ ξk,m = Q Φ ξk,m (Λ). Let Q Φ ξk = { Q Φ ξk,m ( s ) : m < ω ∧ s ∈ <ω } .5. If h ξ, k i ∈ | π | then let ϙ ( ξ, k ) = Q Φ ξk = { Q Φ ξk,m ( s ) : m < ω ∧ s ∈ <ω } .6. Finally if ϙ = ϙ [ Φ ] is obtained this way from an M + -generic sequence Φ of multisystems in MS ( π ), then ϙ is called an M - generic refinement of π . Proposition 11.2 (by the countability of M + ) . If π is a small multiforcingand M ∈ HC then there is an M -generic refinement ϙ of π . Theorem 11.3. If π is a small multiforcing, a set M ∈ HC contains π , | π | ⊆ M , and ϙ is an M -generic refinement of π , then :(i) ϙ is a small special multiforcing, | ϙ | = | π | , and π ❁ ϙ ;(ii) if h ξ, k i ∈ | π | and a set D ∈ M , D ⊆ π ( ξ, k ) is pre-dense in π ( ξ, k ) then π ( ξ, k ) ❁ D ϙ ( ξ, k ) ;(a) if h ξ, k i ∈ | π | , m < ω , and s ∈ <ω then Q Φ ξk,m ( s ) = Q Φ ξk,m ( → s ) ;(b) if h ξ, k i ∈ | π | , m < ω , and s ∈ <ω then Q Φ ξk,m ( s ) ⊆ T Φ ξk,m ( s ) ;(c) if h ξ, k i ∈ | π | , m < ω , and strings t ′ = t in <ω are ⊆ -incomparable then [ Q Φ ξk,m ( t ′ )] ∩ [ Q Φ ξk,m ( t )] = [ T Φ ξk,m ( t ′ )] ∩ [ T Φ ξk,m ( t )] = ∅ ;(d) if h ξ, k, m i 6 = h η, ℓ, n i then [ Q Φ ξk,m ] ∩ [ Q Φ ηℓ,n ] = ∅ ;(e) if h ξ, k i ∈ | π | , S ∈ ϙ ( ξ, k ) and T ∈ π ( ξ, k ) then [ S ] ∩ [ T ] is clopen in [ S ] and T S , in particular, π ( ξ, k ) ∩ ϙ ( ξ, k ) = ∅ ;(f) if h ξ, k i ∈ | π | then the set ϙ ( ξ, k ) is open dense in ϙ ( ξ, k ) ∪ π ( ξ, k ) .If in addition π = S cw α<λ π α , where λ < ω , h π α i α<λ is a ❁ -increasing sequenceof small special multiforcings, and M contains h π α i α<λ and all α < λ , then (iii) if α < λ then π α ❁ ϙ . Proof. Let ϙ = ϙ [ Φ ] be obtained from an M + -generic sequence Φ of multisys-tems in MS ( π ), as above. We argue in the notation of Definition 11.1.If h ξ, k i ∈ | π | and m < ω then by construction the system of trees T Φ ξk,m ( s ) ∈ π ( ξ, k ), s ∈ <ω , satisfies 6.1( ∗ ) on the whole domain s ∈ <ω . This leads to (a),(b) (essentially corollaries of Lemma 6.3) and (c).To prove (d) note that the set ∆ of all 2wise disjoint multisystems ϕ suchthat | ϕ | contains both h ξ, k, m i and h η, ℓ, n i , is dense in MS ( π ) by Lemma 8.2,17nd obviously ∆ ∈ M + . Therefore there is j < ω such that ϕ ( j ) ∈ ∆. Let h = h ϕ ( j ) ( ξ, k, m ) and h ′ = h ϕ ( j ) ( η, ℓ, n ). Then the sets A = S s ∈ h [ T ϕ ( j ) ξk,m ( s )] = S s ∈ h [ T Φ ξk,m ( s )] , B = S t ∈ h ′ [ T ϕ ( j ) ξℓ,n ( t )] = S t ∈ h ′ [ T Φ ξℓ,n ( t )]are disjoint as ϕ ( j ) ∈ ∆. However [ Q Φ ξk,m ] ⊆ A and [ Q Φ ηℓ,n ] ⊆ B .(i) It follows that the sets ϙ ( ξ, k ) = Q Φ ξk are special PTF s (Definition 5.4),and hence ϙ is a small special multiforcing, as in (i), and | ϙ | = | π | .(e) To prove the clopenness claim, note that the set ∆ of all multisystems ϕ ∈ MS ( π ) such that h ξ, k, m i ∈ | ϕ | and if s ∈ h , where h = h ϕ ( ξ, k, m ), theneither T ϕ ξk,m ( s ) ⊆ T or [ T ϕ ξk,m ( s )] ∩ [ T ] = ∅ , is dense. To prove T S , the set ∆ ′ of all multisystems ϕ ∈ MS ( π ) such that h ξ, k, m i ∈ | ϕ | and T S s ∈ h T ϕ ξk,m ( s ),where h = h ϕ ( ξ, k, m ), is dense. Note that ∆ , ∆ ′ ∈ M + and argue as above.(f) Density . If T ∈ π ( ξ, k ) then the set ∆( T ) of all multisystems ϕ ∈ MS ( π ),such that T ϕ ξk,m (Λ) = T for some m , is dense in MS ( π ) by Lemma 8.2(ii),therefore ϕ ( j ) ∈ ∆( T ) for some j . Then T Φ ξk,m (Λ) = T for some m < ω .However Q Φ ξm,k (Λ) ⊆ T Φ ξk,m (Λ). Openness . Suppose that S ∈ ϙ ( ξ, k ), T ∈ ϙ ( ξ, k ) ∪ π ( ξ, k ), T ⊆ S . Then T / ∈ π ( ξ, k ) by (e). Therefore T ∈ ϙ ( ξ, k ).(i), continuation. To establish π ❁ ϙ , let h ξ, k i ∈ | π | . We have to prove that π ( ξ, k ) ❁ ϙ ( ξ, k ). This comes down to conditions (1), (2), (3) of Definition 9.1, ofwhich (1) follows from (f) and (3) from (e), and (2) is obvious since Q Φ ξk,m ( s ) ⊆ T Φ ξk,m ( s ) ∈ π ( ξ, k ) for all m .(ii) As π ❁ ϙ has been checked, it remains to prove Q Φ ξk,m ⊆ fin S D forall m . It follows from the pre-density of D that the set D ′ = { T ∈ π ( ξ, k ) : ∃ S ∈ D ( T ⊆ S ) } is open dense in π ( ξ, k ), and still D ′ ∈ M + . Then theset ∆ ∈ M + of all multisystems ϕ ∈ MS ( π ) such that h ξ, k, m i ∈ | ϕ | and T ϕ ξk,m ( s ) ∈ D for all s ∈ h ϕ ( ξ, k, m ), is dense in MS ( π ) by Lemma 8.2(iii). Thus ϕ ( j ) ∈ ∆ for some j , which witnesses Q Φ ξk,m ⊆ fin S D .(iii) We have to prove that π α ( ξ, k ) ❁ ϙ ( ξ, k ) whenever h ξ, k i ∈ | π α | . And as π ( ξ, k ) ❁ ϙ ( ξ, k ) has been checked, it suffices to prove that Q Φ ξk,m ⊆ fin S π α ( ξ, k )for all m . However D = π α ( ξ, k ) is pre-dense in π ( ξ, k ) by Lemma 9.2(v), andstill D ∈ M + , hence we can refer to (ii). Corollary 11.4. In the assumptions of Proposition 11.2, if | π | ⊆ Z ⊆ ω × ω and Z is at most countable then there is a small special multiforcing ϙ such that | ϙ | = Z and π ❁ ϙ . Proof. If | π | = Z then let M be any countable set containing π , pick ϙ byProposition 11.2, and apply Theorem 11.3. If | π | $ Z then we trivially extendthe construction by ϙ ( ξ, k ) = P coh (see Example 5.1) for all h ξ, k i ∈ Z r | π | .18 orollary 11.5. Suppose that λ < ω , and h P α i α<λ is an ❁ -increasing sequenceof countable special forcings in PTF . Then there is a countable special forcing Q ∈ PTF such that P α ❁ Q for each α < λ . Proof. If α < λ then let a multiforcing π α be defined by | π α | = {h , i} andby π α (0 , 0) = P α . By Proposition 11.2 and Theorem 11.3 there is a multiforcing ϙ satisfying | ϙ | = {h , i} and π α ❁ ϙ , ∀ α . Let Q = ϙ (0 , 12 Preservation of density This Section proves a special consequence of M + -genericity of multiforcing re-finements, the relation ❁ of Definition 10.2 between a multiforcing and its re-finement. Theorem 12.1. In the assumptions of Theorem 11.3, if D ∈ M + , D ⊆ MT ( π ) ,and D is open dense in MT ( π ) , then π ❁ D ϙ . Proof. We suppose that ϙ = ϙ [ Φ ] is obtained from an increasing M + -genericsequence Φ of multisystems in MS ( π ), as in Definition 11.1, and argue in thenotation of 11.1.Suppose that p ∈ MT ( π ), u ∈ MT ( ϙ ), | u | ∩ | p | = ∅ , as in ( ∗ ) of Defi-nition 10.2; the extra condition | u | ⊆ | π | holds automatically as we still have | ϙ | = | π | . Let X = | u | , Y = | π | r X. If h ξ, k i ∈ X then T u ξk = Q Φ ξk,m ξk ( s ξk ),where m ξk < ω and s ξk ∈ <ω . By obvious reasons we can assume that s ξk = Λ,hence T u ξk = Q Φ ξk,m ξk , for all h ξ, k i ∈ X .Consider the set ∆ of all multisystems ϕ ∈ MS ( π ) such that there is anumber H > q ∈ MS ( π ) satisfying (1), (2), (3), (4) below.(1) | q | ∩ X = ∅ and q p ;(2) if h ξ, k i ∈ X then h ξ, k, m ξk i ∈ | ϕ | ;(3) if h ξ, k, m i ∈ | ϕ | then h ϕ ( ξ, k, m ) = H .To formulate the last requirement, we need one more definition. Suppose that τ = h t ξk i h ξ,k i∈ X is a system of strings τ ( ξ, k ) = t ξk ∈ H , symbolically τ ∈ (2 H ) X . Define a multitree s ( ϕ , τ ) ∈ MT ( π ) so that | s ( ϕ , τ ) | = X and T s ( ϕ ,τ ) ξk = T ϕ ξk,m ξk ( t ξk ) for all h ξ, k i ∈ X . Note that | s ( ϕ , τ ) | = | u | , and hence the multitree s ( ϕ , τ ) ∪ q belongs to MT ( π ) as well. Now goes the last condition.(4) If τ ∈ (2 H ) X then s ( ϕ , τ ) ∪ q ∈ D . Here, if p , q are multitrees satisfying | p |∩| q | = ∅ (disjoint domains), then p ∪ q , a disjointunion , is a multitree such that | p ∪ q | = | p | ∪ | q | and T p ∪ q ξk = T p ξk whenever h ξ, k i ∈ | p | but T p ∪ q ξk = T q ξk whenever h ξ, k i ∈ | q | . emma 12.2. The set ∆ is dense in MS ( π ) . Proof (Lemma) . Suppose that ψ ∈ MS ( π ); we have to find a multisystem ϕ ∈ MS ( π ) with ψ ϕ . First of all, by Lemma 8.2(i)(ii) we can assume that(a) if h ξ, k i ∈ X then h ξ, k, m ξk i ∈ | ψ | ;(b) there is a number g > h ψ ( ξ, k, m ) = g for all h ξ, k, m i ∈ | ψ | .Let H = g + 1. Define χ ∈ MS ( π ) so that | χ | = | ψ | , and h χ ( ξ, k, m ) = H , T χ ξk,m ( s a i ) = T ψ ξk,m ( s )( → i ) for all h ξ, k, m i ∈ | ψ | and s a i ∈ H ; then ψ χ .It follows from the open density of D that there is a multitree q ∈ MT ( π )satisfying (1), and a multisystem ϕ ∈ MS ( π ) satisfying (4) and such that still | ϕ | = | ψ | and h ϕ ( ξ, k, m ) = H for all h ξ, k, m i ∈ | ψ | , and in addition(c) if h ξ, k i ∈ X and s ∈ H then T ϕ ξk,m ξk ( s ) ⊆ T χ ξk,m ξk ( s );(d) T ϕ ξk,m ( s ) = T χ ξk,m ( s ) for all applicable ξ, k, m, s not covered by (c).Namely to achieve (4) for one particular τ ∈ (2 H ) X , consider the multitree r = s ( χ , τ ) ∪ p . There is a multitree r ′ ∈ D , r ′ r . Let a new multisystem χ ′ be obtained from χ by the reassignment T χ ′ ξk,m ξk ( τ ( ξ, k )) = T r ′ ξk for all h ξ, k i ∈ X .To get the input for the next step, let p ′ = r ′ ↾ Y, so that r ′ = s ( χ ′ , τ ) ∪ p ′ ∈ D .Now consider another τ ′ ∈ (2 H ) X and the multitree r ′ = s ( χ ′ , τ ′ ) ∪ p ′ . Thereis r ′′ ∈ D , r ′′ r ′ . Define χ ′′ from χ ′ by the reassignment T χ ′′ ξk,m ξk ( τ ′ ( ξ, k )) = T r ′′ ξk for all h ξ, k i ∈ X . Let p ′′ = r ′′ ↾ Y, so that r ′′ = s ( χ ′′ , τ ′ ) ∪ p ′′ ∈ D .And so on. The final multisystem and multitree of this construction will be ϕ and q satisfying (1), (2), (3), (4). Note that ψ ϕ , as we only amend the H -th level of χ absent in ψ . (cid:3) ( Lemma )Note that ∆ is defined in HC using sets D , π , p , X , and the map h ξ, k i → m ξk : X → ω as parameters. Now, D , π belong to M + straightforwardly, X belongs to M + since it is a finite subset of a set | π | ⊆ M , and p belongs to M + by similar reasons. It follows that ∆ belongs to M + as well.Therefore, by the lemma and the choice of Φ , there is an index j such thatthe multisystem ϕ ( j ) belongs to ∆, which is witnessed by a number H > q ∈ MT ( π ) satisfying (1), (2), (3), (4) for ϕ ( j ) instead of ϕ . Toprove that u ⊆ fin W D | u | q , note that the multitrees s ( ϕ ( j ) , τ ) ∪ q , τ ∈ (2 H ) X ,belong to D by (4), and easily [ u ] ⊆ S τ ∈ (2 H ) X [ s ( ϕ ( j ) , τ )]. Here r ′ ↾ Y is the plain restriction of the function r ′ : | r ′ | → PT to the set | r ′ | ∩ Y . II Structure of real names Here we turn to some details of the structure of reals in models of MT ( π )-generictype, π being a multiforcing. We are going focus on non-principal reals, i. e. ,those different from the principal generic reals x ξk [ G ] (Remark 7.3). We’ll worktowards the goal of making every non-principal real to be non-generic with respectto each of the factor forcing notions π ( ξ, k ). 13 Real names Our next goal is to introduce a suitable notation related to names of reals in thecontext of forcing notions of the form MT ( π ). Definition 13.1. A real name is any set c ⊆ MT × ( ω × ω ) such that the sets K c ni = { p ∈ MT : h p , n, i i ∈ c } satisfy the following:( ∗ ) if n, k, ℓ < ω , k = ℓ , and p ∈ K c nk , q ∈ K c nℓ , then p , q are a.d..A real name c is small if each set K c ni is at most countable — then the sets dom c = S n,i K c ni ⊆ MT and | c | = S n,i S p ∈ K c ni | p | ⊆ ω × ω , and c itself, arecountable, too. Definition 13.2. Let c be a real name and G ⊆ MT a pairwise compatible set.Define the evaluation c [ G ] ∈ ω ω so that c [ G ]( n ) = i iff: − either ∃ p ∈ G ∃ q ∈ K c ni ( p q ) (recall that p q means p is stronger), − or just i = 0 and ¬ ∃ p ∈ G ∃ q ∈ S i K c ni ( p q ) (default case). Definition 13.3. Let π be a multiforcing. A real name c is said to be a π -realname if, in addition to ( ∗ ) above, the following condition holds:( † ) each set K c n = S i K c ni is pre-dense for MT ( π ), in the sense that the set K c n ↑ π = { p ∈ MT ( π ) : ∃ q ∈ K c n ( p q ) } is dense (then obviously opendense) in MT ( π ).Generally speaking, we do not assume that K c n ⊆ MT ( π ). However if, inaddition to ( ∗ ), ( † ) above, K c n ⊆ MT ( π ) holds for all n , then say that c is a true π -real name . Then each set K c n = S i K c ni is a pre-dense subset of MT ( π ). Remark 13.4. Let π be a multiforcing, c be a π -real name, and a set G ⊆ MT ( π ) be MT ( π )-generic over the collection of all sets K c n ↑ π as in ( † ) (All of K c n ↑ π are dense by the choice of c .) Then the “or” case in Definition 13.2 neverhappens as we have G ∩ ( K c n ↑ π ) = ∅ by the choice of G .21 emark 13.5. If π is a regular multiforcing then the notions of being a.d. andbeing incompatible in MT ( π ) are equivalent by Lemma 5.5(i), so that a true π -real name is the same as a MT ( π )-name for an element of ω ω in the generaltheory of (unramified) forcing. Example 13.6. If ξ < ω , k < ω , then . x ξk is a real name such that if i = 0 , C ξkni = K . x ξk ni consists of a lone multitree r = r ξkni with | r | = {h ξ, k i} and T r ξk = { t ∈ <ω : lh ( t ) ≤ n ∨ t ( n ) = i } , and if i ≥ C ξkni = ∅ . Remark 13.7. If π ∈ MT and h ξ, k i ∈ | π | then . x ξk is a π -real name ofthe real x ξk = x ξk [ G ] ∈ ω , the ( ξ, k )th term of a MT ( π )-generic sequence h x ξk [ G ] i h ξ,k i∈| π | . That is, if G ⊆ MT ( π ) is generic then the real x ξk [ G ] definedby 7.3 coincides with the real . x ξk [ G ] defined by 13.2. 14 Direct forcing The following definition of the direct forcing relation is not explicitly associatedwith any concrete forcing notion, but in fact the direct forcing relation (in allthree instances) is compatible with any forcing notion of the form MT ( π ).Let c be a real name. Let us say that a multitree p : • directly forces c ( n ) = i , where n, i < ω , iff there is a multitree q ∈ K c ni such that p q (meaning: p is stronger); • directly forces s ⊂ c , where s ∈ ω <ω , iff for all n < lh ( s ), p directly forces c ( n ) = i , where i = s ( n ); • directly forces c / ∈ [ T ], where T ∈ PT , iff there is a string s ∈ ω <ω r T such that p directly forces s ⊂ c . Lemma 14.1. If π is a multiforcing and c is a π -real name, p ∈ MT ( π ) , h ξ, k i ∈ | π | , T ∈ PT , n < ω , then (i) there is a number i < ω and a multitree q ∈ MT ( π ) , q p , which directlyforces c ( n ) = i ;(ii) there is a multitree q ∈ MT ( π ) , q p , which directly forces c / ∈ [ T ( → or directly forces c / ∈ [ T ( → . Note that if T ∈ π ( ξ, k ) then the trees T ( → i ), i = 0 , π ( ξ, k ). Proof. (i) Use the density of sets K c n ↑ π by 13.3( † ) above.(ii) Let r = stem ( T ), n = lh ( r ). By (i), there is a multitree q ∈ MT ( π ) , p ′ p , and, for any m ≤ n , — a number i m = 0 , 1, such that q directly forces c ( m ) = i m , ∀ m < n . Let s ∈ n +1 be defined by s ( m ) = i m for every m ≤ n .Then q directly forces s ⊂ c . On the other hand, s cannot belong to both T ( → 0) and T ( → 1) . 22 The next definition extends Definition 10.2 to real names. Definition 15.1. Assume that π , ϙ are multiforcings, c is a real name, and π ❁ ϙ . Say that ϙ locks c over π , symbolically π ❁ c ϙ , if ϙ locks, over π ,each set K c n ↑ π defined in 13.3( † ), in the sense of Definition 10.2. Corollary 15.2 (of Theorem 12.1) . In the assumptions of Theorem 11.3, if c ∈ M + and c is a π -real name then π ❁ c ϙ . Proof. Note that each set K c n ↑ π belongs to M + (as so do c and π ) and isdense in MT ( π ), so it remains to apply Theorem 12.1. Lemma 15.3. Let π , ϙ , σ be multiforcings and c be a real name. Then (i) if π ❁ c ϙ then c is a π -real name and a ( π ∪ cw ϙ ) -real name ;(ii) if π ❁ c ϙ ❁ σ then π ❁ c σ ;(iii) if h π α i α<λ is a ❁ -increasing sequence in MF , < µ < λ , π = S cw α<µ π α ,and π ❁ c π µ , then π ❁ c ϙ = S cw µ ≤ α<λ π α . Proof. (i) By definition, we have π ❁ K c n ↑ π ϙ for each n , therefore K c n ↑ π isdense in MT ( π ) (then obviously open dense) and pre-dense in MT ( π ∪ cw ϙ ) byLemma 10.3(i). It follows that K c n ↑ ( π ∪ cw ϙ ) is dense in MT ( π ∪ cw ϙ ).To check (ii), (ii) apply Lemma 10.3(iv),(v). 16 Non-principal names and avoiding refinements Let π be a multiforcing. Then MT ( π ) adds a collection of reals x ξk , h ξ, k i ∈ | π | ,where each principal real x ξk = x ξk [ G ] is π ( ξ, k )-generic over the ground setuniverse. Obviously many more reals are added, and given a π -real name c , onecan elaborate different requirements for a condition p ∈ MT ( π ) to force that c is a name of a real of the form x ξk or to force the opposite. But we are mostlyinterested in simple conditions related to the “opposite” part. The next definitionprovides such a condition. Definition 16.1. Let π be a multiforcing, h ξ, k i ∈ | π | . A real name c is non-principal over π at ξ, k if the following set is open dense in MT ( π ): D π ξk ( c ) = { p ∈ MT ( π ) : p directly forces c / ∈ [ T p ξk ] } . We’ll show below (Theorem 18.2(i)) that the non-principality implies c being not a name of the real x ξk [ G ]. And further, the avoidance condition in the nextdefinition will be shown to imply c being a name of a non-generic real.23 efinition 16.2. Let π , ϙ be multiforcings, π ❁ ϙ , h ξ, k i ∈ | π | . Say that ϙ avoids a real name c over π at ξ, k , in symbol π ❁ c ξk ϙ , if for each Q ∈ ϙ ( ξ, k ), ϙ locks, over π , the set D ( c , Q, π ) = { r ∈ MT ( π ) : r directly forces c / ∈ [ Q ] } , in the sense of Definition 10.2 — formally π ❁ D ( c ,Q, π ) ϙ . Lemma 16.3. Assume that π , ϙ , σ are multiforcings, h ξ, k i ∈ | π | , and c is a π -real name. Then :(i) if π ❁ c ξk ϙ and Q ∈ ϙ ( ξ, k ) then the set D ( c , Q, π ) is open dense in MT ( π ) and pre-dense in MT ( π ∪ cw ϙ ) ;(ii) if π ❁ c ξk ϙ ❁ σ then π ❁ c ξk σ ;(iii) if h π α i α<λ is a ❁ -increasing sequence in MF , < µ < λ , π = S cw α<µ π α ,and π ❁ c ξk π µ , then π ❁ c ξk ϙ = S cw µ ≤ α<λ π α . Proof. (i) Apply Lemma 10.3(i).(ii) Let h ξ, k i ∈ | π | and S ∈ σ ( ξ, k ). Then, as ϙ ❁ σ , there is a finite set { Q , . . . , Q m } ⊆ ϙ ( ξ, k ) such that S ⊆ Q ∪· · ·∪ Q m . We have π ❁ D ( c ,Q i , π ) ϙ forall i since π ❁ c ξk ϙ , therefore π ❁ D ( c ,Q i , π ) σ , ∀ i , by Lemma 10.3(iv). Note that T i D ( c , Q i , π ) ⊆ D ( c , S, π ) since S ⊆ S i Q i . We conclude that π ❁ D ( c ,S, π ) σ by Lemma 10.3(ii),(iii). Therefore π ❁ c ξk σ . 17 Generic refinements avoid non-principal names The following theorem says that generic refinements as in Section 11 avoid non-principal names. It resembles Theorem 12.1 to some extent, yet the latter is notdirectly applicable here as both the multitree Q and the set D ( c , Q, π ) dependon ϙ , and hence the sets D ( c , Q, π ) do not necessarily belong to M + . Howeverthe proof will be based on rather similar arguments. Theorem 17.1. In the assumptions of Theorem 11.3, if h η, K i ∈ | π | ⊆ M and c ∈ M is a π -real name non-principal over π at η, K then π ❁ c ηK ϙ . Proof. Assume that ϙ = ϙ [ Φ ] is obtained from an M + -generic sequence Φ ofmultisystems in MS ( π ), as in Definition 11.1. We stick to the notation of 11.1.Let Q ∈ ϙ ( η, K ); we have to prove that ϙ locks the set D ( c , Q, π ) over π . Byconstruction Q = Q Φ ηK, e m ( s ) ⊆ Q Φ ηK, e m for some e m < ω ; it can be assumed that Q = Q Φ ηK, e m . Following the proof of Theorem 12.1, we suppose that p ∈ MT ( π ), u ∈ MT ( ϙ ), | u | ∩ | p | = ∅ , define X = | u | , Y = | π | r X, and assume that T u ξk = Q Φ ξk,m ξk , where m ξk < ω , for each h ξ, k i ∈ X .Consider the set ∆ of all multisystems ϕ ∈ MS ( π ) such that there is anumber H > q ∈ MS ( π ) satisfying conditions241) | q | ∩ X = ∅ and q p ;(2) if h ξ, k i ∈ X then h ξ, k, m ξk i ∈ | ϕ | ;(3) if h ξ, k, m i ∈ | ϕ | then h ϕ ( ξ, k, m ) = H ;(but not (4) though) as in the proof of Theorem 12.1, along with two morerequirements(5) h η, K, e m i ∈ | ϕ | — hence still h ϕ ( η, K, e m ) = H by (3);(6) if s ∈ H and τ ∈ (2 H ) X then s ( ϕ , τ ) ∪ q directly forces c / ∈ [ T ϕ ηK, e m ( s )]. Lemma 17.2. ∆ is dense in MS ( π ) . Proof. Suppose that ψ ∈ MS ( π ); we can assume that ψ already satisfies(a) if h ξ, k i ∈ X then h ξ, k, m ξk i ∈ | ψ | ;(b) there is a number g < ω such that h ψ ( ξ, k, m ) = g for all h ξ, k, m i ∈ | ψ | ;as in Lemma 12.2, and in addition h η, K, e m i ∈ | ψ | .Let H = g + 1. Define a multisystem χ ∈ MS ( π ) so that | χ | = | ψ | ,and h χ ( ξ, k, m ) = H , T χ ξk,m ( s a i ) = T ψ ξk,m ( s )( → i ) for all h ξ, k, m i ∈ | ψ | and s a i ∈ H ; then ψ χ . We claim that there is a multitree q ∈ MT ( π ) satisfying(1), and a multisystem ϕ ∈ MS ( π ) satisfying (6) and such that still | ϕ | = | ψ | and h ϕ ( ξ, k, m ) = H for all h ξ, k, m i ∈ | ψ | , and in addition(c) if h ξ, k i ∈ X and s ∈ H then T ϕ ξk,m ξk ( s ) ⊆ T χ ξk,m ξk ( s ), and we also have T ϕ ηK, e m ( s ) ⊆ T χ ηK, e m ( s );(d) T ϕ ξk,m ( s ) = T χ ξk,m ( s ) for all applicable ξ, k, m, s not covered by (c).To achieve (6) in one step for one particular τ ∈ (2 H ) X , consider the multitree r = s ( χ , τ ) ∪ p . By Lemma 14.1 and the density assumption of the theorem, thereis a multitree r ′ ∈ MT ( ϕ ), r ′ r , which directly forces c / ∈ [ T r ′ ηK ], and thereare multitrees U s ∈ MT ( ϕ ), s ∈ H , such that U s ⊆ T χ ηK, e m ( s ) and r ′ directlyforces c / ∈ [ U s ], ∀ s . Let χ ′ be obtained from χ by the following reassignment.(I) We set T χ ′ ξk,m ξk ( τ ( ξ, k )) = T r ′ ξk for all h ξ, k i ∈ X .(II) If s ∈ H , and either h η, K i / ∈ X , or e m = m ηK , or s = τ ( η, K ) then weset T χ ′ ηK, e m ( s ) = U s . (Note that if h η, K i ∈ X and e m = m ηK then the tree T χ ′ ηK, e m ( τ ( η, K )) = T r ′ ηK is already defined by (I).)25et p ′ = r ′ ↾ Y, so that r ′ = s ( χ ′ , τ ) ∪ p ′ . By construction the tree p ′ satisfies(6), for the system τ chosen, in the case h η, K i ∈ X , e m = m ηK , s = τ ( η, K ) by(I) and in all other cases by (II).Now consider another τ ′ ∈ (2 H ) X and the multitree r ′ = s ( χ ′ , τ ′ ) ∪ p ′ . Thereis a multitree r ′′ ∈ MT ( π ), r ′′ r ′ , which which directly forces c / ∈ [ T r ′ ηK ] and c / ∈ [ U ′ s ] for each s ∈ H , where U ′ s ∈ MT ( ϕ ) and U ′ s ⊆ T χ ′ ηK, e m ( s ) . Let χ ′′ beobtained from χ ′ by the the same reassignment (for τ ′ instead of τ ).And so on. The final multisystem and multitree of this construction will be ϕ and q satisfying (1), (2), (3), (5), (6). (cid:3) ( Lemma )Come back to the theorem. Note that ∆ ∈ M + , similarly to the proof ofTheorem 12.1. Therefore, by the lemma, there is an index j such that thesystem ϕ ( j ) belongs to ∆. Let this be witnessed by a number H > q ∈ MT ( π ), such that conditions (1), (2), (3), (5), (6) are satisfied for ϕ = ϕ ( j ).It remains to prove that u ⊆ fin W D ( c , Q, π ) | u | q . Let V consist of all multi-trees v = s ( ϕ ( j ) , τ ), where τ ∈ (2 H ) X ; [ u ] ⊆ S v ∈ V [ v ] by construction.Further, if s ∈ H and v ∈ V then v ∪ q directly forces c / ∈ [ T ϕ ( j ) ηK, e m ( s )] by(6), that is, directly forces c / ∈ [ T Φ ηK, e m ( s )] in the notation of Definition 11.1.Therefore v ∪ q directly forces c / ∈ [ Q Φ ηK, e m ( s )] since Q Φ ηK, e m ( s ) ⊆ T Φ ηK, e m ( s ) byLemma 11.3(b). However Q = Q Φ ηK, e m = S s ∈ H Q Φ ηK, e m ( s ) by Lemma 11.3(a). Itfollows that v ∪ q directly forces c / ∈ [ Q ], that is, v ∈ D ( c , Q, π ) | u | q .We conclude that V is a (finite) subset of D ( c , Q, π ) | u | q . And this accom-plishes the proof of u ⊆ fin W D ( c , Q, π ) | u | q . 18 Consequences for reals in generic extensions We first prove a result saying that all reals in MT ( π )-generic extensions areadequately represented by real names. Then Theorem 18.2 will show effects ofthe property of being a non-principal name. Proposition 18.1. Suppose that π is a regular multiforcing, G ⊆ MT ( π ) isgeneric over the ground set universe V , and x ∈ V [ G ] ∩ ω ω . Then (i) there is a true π -real name c ∈ V such that x = c [ G ] ;(ii) if MT ( π ) is a CCC forcing in V then there is a small true π -real name d ∈ V with x = d [ G ] . Proof. (i) is an instances of a general forcing theorem (see Remark 13.5 on theeffect of regularity). To prove (ii), pick a real name c by (i), extend each set26 c n = S i K c ni to an open dense set O n by closing strongwards, choose maximalantichains A n ⊆ O n in those sets — which have to be countable by CCC, andthen let A ni = A n ∩ K c ni and d = {h p , n, i i : p ∈ A ni } . Theorem 18.2. Let π be a regular multiforcing. Then (i) if a set G ⊆ MT ( π ) is generic over the ground set universe V , h ξ, k i ∈ | π | ,and x ∈ V [ G ] ∩ ω ω , then x = x ξk [ G ] if and only if there is a true π -realname c , non-principal over π at ξ, k and such that x = c [ G ] . (ii) if c is a π -real name, h ξ, k i ∈ | π | , ϙ is a multiforcing, π ❁ c ξk ϙ , and aset G ⊆ MT ( π ∪ cw ϙ ) is generic over V then c [ G ] / ∈ S Q ∈ ϙ ( ξ,k ) [ Q ] . Proof. (i) Suppose that x = x ξk [ G ]. By a known forcing theorem, there is atrue π -real name c such that x = c [ G ] and MT ( π ) forces that c = x ξk [ G ]. Itremains to show that c is a non-principal name over π at ξ, k . We have to provethat the set D π ξk ( c ) = { p ∈ MT ( π ) : p directly forces c / ∈ [ T p ξk ] } . is open dense in MT ( π ). The openness is clear, let us prove the density. Consideran arbitrary q ∈ MT ( π ). Then q MT ( π )-forces c = x ξk [ G ] by the choiceof c , hence we can assume that, for some n , it is MT ( π )-forced by q that c ( n ) = x ξk [ G ]( n ). Then by Lemma 14.1(i) there is a multitree p ∈ MT ( π ), p q , and a string s ∈ ω n +1 , such that p MT ( π )-forces s ⊆ c . Now itsuffices to show that s / ∈ T p ξk . Suppose otherwise: s ∈ T p ξk . Then the tree T = T p ξk ↾ s still belongs to MT ( π ). Therefore the multitree r defined by T r ξk = T and T r ξ ′ k ′ = T p ξ ′ k ′ for each pair h ξ ′ , k ′ i 6 = h ξ, k i , belongs to MT ( π ) and satisfies r p q . However r directly forces both c ( n ) and x ξk [ G ]( n ) to be equal toone and the same value ℓ = s ( n ), which contradicts to the choice of n .To prove the converse let c ∈ V be a real name non-principal over π at ξ, k ,and x = c [ G ]. Assume to the contrary that h ξ, k i ∈ | π | and x = x ξk [ G ]. There isa multitree q ∈ G which MT ( π )-forces c = x ξk [ G ]. As c is non-principal, thereis a stronger multitree p ∈ G ∩ D π ξk ( c ), p q . Thus p directly forces c / ∈ [ T p ξk ],and hence MT ( π )-forces the same statement. Yet p MT ( π )-forces . x ξk ∈ [ T p ξk ],of course, and this is a contradiction.(ii) Suppose towards the contrary that Q ∈ ϙ ( ξ, k ) and c [ G ] ∈ [ Q ]. Bydefinition, ϙ locks, over π , the set D ( c , Q, π ) = { r ∈ MT ( π ) : r directly forces c / ∈ [ Q ] } . Therefore in particular D ( c , Q, π ) is pre-dense in MT ( π ∪ cw ϙ ) by Lemma 10.3.We conclude that G ∩ D ( c , Q, π ) = ∅ . In other words, there is a multitree r ∈ MT ( π ) which directly forces c / ∈ [ Q ]. It easily follows that c [ G ] / ∈ [ Q ],which is a contradiction. 27 Here we summarize the properties of generic refinements considered above. Thenext definition combines the refinement types ❁ D , ❁ D , ❁ c ξk . Definition 19.1. Suppose that π ❁ ϙ are multiforcings and M ∈ HC is anyset. Let π ❁❁ M ϙ mean that the four following requirements hold:(1) if h ξ, k i ∈ | π | , D ∈ M , D ⊆ π ( ξ, k ), D is pre-dense in π ( ξ, k ), then π ( ξ, k ) ❁ D ϙ ( ξ, k );(2) if D ∈ M , D ⊆ MT ( π ), D is open dense in MT ( π ), then π ❁ D ϙ ;(3) if c ∈ M is a π -real name then π ❁ c ϙ ;(4) if h ξ, k i ∈ | π | and c ∈ M is a π -real name, non-principal over π at ξ, k ,then π ❁ c ξk ϙ . Corollary 19.2 (of lemmas 9.4, 10.3, 15.3, 16.3) . Let π , ϙ , σ be multiforcingsand M be a countable set. Then :(i) if π ❁❁ M ϙ ❁ σ then π ❁❁ M σ ;(ii) if h π α i α<λ is a ❁ -increasing sequence in MF , < µ < λ , π = S cw α<µ π α ,and π ❁❁ M π µ , then π ❁❁ M ϙ = S cw µ ≤ α<λ π α . Corollary 19.3. If π is a small multiforcing, M ∈ HC , and ϙ is an M -genericrefinement of π (exists by Proposition 11.2!) , then π ❁❁ M ϙ . Proof. We have π ❁❁ M ϙ by a combination of 11.3(ii), 12.1, 15.2, and 17.1.28 V The forcing notion In this chapter we define the forcing notion to prove the main theorem. It willhave the form MT ( Π ), for a certain multiforcing Π with | Π | = ω × ω . Themultiforcing Π will be equal to the componentwise union of terms of a certainincreasing sequence Π of small multiforcings. And quite a complicated construc-tion of this sequence in L will make use of some ideas related to diamond-styleconstructions, as well as to some sort of definable genericity. 20 Increasing sequences of small multiforcings Recall that MF is the set of all multiforcings (Section 7). Let sMF ⊆ MF bethe set of all small special multiforcings; s accounts for both small and special .Thus a multiforcing π ∈ MF belongs to sMF if | π | is (at most) countable andif h ξ, k i ∈ | π | then π ( ξ, k ) is a small special (Definition 5.4) forcing in PTF . Definition 20.1. Let sMF , resp., sMF ω be the set of all ❁ -increasing sequences π = h π α i α<κ of multiforcings π α ∈ sMF , of length κ = dom ( π ) < ω , resp., κ = ω , which are domain-continuous , in the sense that if λ < κ is a limit ordinalthen | π λ | = S α<λ | π α | . Sequences in sMF ∪ sMF ω are called multisequences .We order sMF ∪ sMF ω by the usual relations ⊆ and ⊂ of extension of sequences. • Thus π ⊂ ϙ iff κ = dom ( π ) < λ = dom ( ϙ ) and π α = ϙ α for all α < κ . • In this case, if M is any set, and ϙ κ (the first term of ϙ absent in π )satisfies π ❁❁ M ϙ κ , where π = S cw α<κ π α , then we write π ⊂ M ϙ .If π is a multisequence in sMF ∪ sMF ω then let MT ( π ) = MT ( π ), where π = S cw π = S cw α<κ π α (componentwise union), and κ = dom π . Accordingly a(true) π -real name will mean a (true) π -real name. Corollary 20.2. Suppose that κ < λ < ω , M is a countable set, and π = h π α i α<κ is a multisequence in sMF . Then :(i) the componentwise union π = S cw π = S cw α<κ π α is a regular multiforcing ;(ii) there is a multisequence ϙ ∈ sMF satisfying dom ( ϙ ) = λ and π ⊂ M ϙ ;(iii) if moreover h s α i α<λ is an ⊂ -increasing sequence of countable sets s α ⊆ ω × ω , s α = | π α | for all α < κ , and s γ = S α<γ s α for all limit γ < λ ,then there is a multisequence ϙ ∈ sMF satisfying dom ( ϙ ) = λ , | ϙ α | = s α for all α < λ , and π ⊂ M ϙ ;(iv) if π , ρ , ϙ ∈ sMF and π ⊂ M ρ ⊆ ϙ then π ⊂ M ϙ ;29v) if ϙ = h ϙ α i α<λ ∈ sMF and π ⊂ M ϙ then π = S cw α<κ π α ❁❁ M ϙ β when-ever λ ≤ β < µ , and also π ❁❁ M ϙ ′ = S cw λ ≤ β<µ ϙ β , therefore (a) MT ( ϙ ′ ) is open dense in MT ( ϙ ) , (b) if h ξ, k i ∈ | π | , D ∈ M , D ⊆ π ( ξ, k ) , D is pre-dense in π ( ξ, k ) , then D remains pre-dense in π ( ξ, k ) ∪ ϙ ( ξ, k ) , (c) if D ∈ M , D ⊆ MT ( π ) , D is open dense in MT ( π ) , then D ispre-dense in MT ( π ∪ cw ϙ ′ ) = MT ( ϙ ) . Proof. (i) Make use of Lemma 9.2(iv).(ii) We define terms ϙ α of the multisequence ϙ required by induction.Naturally put ϙ α = π α for each α < κ .Now suppose that κ ≤ γ < λ , multiforcings ϙ α , α < γ , are defined, and ρ = h ϙ α i α<γ is a multisequence in sMF . To define ϙ γ , assume first that γ is limit. Let ρ = S cw ρ = S cw α<γ ϙ α (componentwise union). We can assumethat M contains ρ and satisfies γ ⊆ M (otherwise take a bigger set). ByProposition 11.2, there is an M -generic refinement ϙ of ρ . By Theorem 11.3, ϙ is small special multiforcing, ρ ❁ ϙ , and ρ α ❁ ϙ for all α < γ . In addition ρ ❁❁ M ϙ by Corollary 19.3. We let ρ γ = ϙ . The extended multisequence ρ + = h ρ α i α<γ +1 belongs to sMF and satisfies ρ ⊂ M ρ + .(iii) The proof is similar, with the extra care of | ϙ α | = s α .To prove the main claim of (v) make use of Corollary 19.2.(iv) The relation π ⊂ M ϙ involves only the first term of ϙ absent in π .To prove (v)(a) apply Corollary 10.1.(v)(b) As π ❁❁ M ϙ ′ and D ∈ M , we have π ( ξ, k ) ❁ D ϙ ( ξ, k ). Therefore D is pre-dense in ϙ ( ξ, k ) by Lemma 9.4(ii).(v)(c) Similarly π ❁ D ϙ ′ , D is pre-dense in MT ( ϙ ) by Lemma 10.3(i).Our plan regarding the forcing notion for Theorem 1.1 will be to define acertain multisequence Π in sMF ω and the ensuing multiforcing Π = S cw Π withremarkable properties related to definability and its own genericity of some sort.But we need first to introduce an important notion involved in the construction. 21 Layer restrictions of multiforcings and deciding sets The construction of the mentioned multiforcing Π will be maintained in such away that different layers h Π ( k, ξ ) i ξ<ω , k < ω , appear rather independent of eachother, albeit the principal inductive parameter will be ξ rather than k . To reflectthis feature, we introduce here a suitable notation related to layer restrictions. If30 < ω then, using a special “layer restriction” symbol ↾↾ to provide a transparentdistinction from the ordinary restriction ↾ , we define sets of multitrees: MT ↾↾ Assume that m < ω . A multisequence π ∈ sMF m - decides aset W if either π ↾↾ ≥ m belongs to W ( positive decision) or there is no multise-quence ϙ ∈ W ∩ sMF ↾↾ ≥ m extending π ↾↾ ≥ m ( negative decision). Lemma 21.2. If π ∈ sMF , M is countable, W is any set, and m < ω, thenthere is a multisequence ϙ ∈ sMF such that π ⊂ M ϙ and ϙ m -decides W. Proof. By Corollary 20.2, there is a multisequence ρ ∈ sMF such that π ⊂ M ρ . Then either ρ outright m -decides W negatively, or there is a sequence σ ∈ W ∩ sMF ↾↾ ≥ m satisfying ρ ↾↾ ≥ m ⊆ σ .31n the other hand, using Corollary 20.2(iii), we get a multisequence σ ′ ∈ sMF ↾↾ 22 Auxiliary diamond sequences Recall that HC is the set of all hereditarily countable sets (those with finite orcountable transitive closures).The next theorem employs the technique of diamond sequences in L . Theorem 22.1 (in L ) . There exist ∆ HC1 sequences h π ⌈ µ ⌉i µ<ω , h D ⌈ µ ⌉i µ<ω , h z ⌈ µ ⌉i µ<ω , such that, for every µ , D ⌈ µ ⌉ and z ⌈ µ ⌉ are sets in HC , π ⌈ µ ⌉ ∈ sMF , dom ( π ⌈ µ ⌉ ) = µ , and in addition if Π = h Π ν i ν<ω ∈ sMF ω , z ∈ HC , and D ⊆ MT ( Π ) , then the set M of all ordinals µ < ω such that (a) z ⌈ µ ⌉ = z ;(b) π ⌈ µ ⌉ is equal to the restricted sub-multisequence Π ↾ µ = h Π ν i ν<µ ;(c) D ⌈ µ ⌉ = D ∩ MT ( Π ↾ µ ) ; is stationary in ω . Proof. Arguing in L , the constructible universe , we let L be the canonicalwellordering of L . It is known that L orders HC similarly to ω , and that L is ∆ HC1 and has the goodness property: the set of all L -initial segments I x ( L ) = { y : y L x } , x ∈ HC , is still ∆ HC1 .We begin with a ∆ HC1 sequence of sets S α ⊆ α , α < ω , such that(A) if X ⊆ HC then the set { α < ω : S α = X ∩ α } is stationary in ω .This is a well-known instance of the diamond principle true in L . The additionaldefinability property can be achieved by taking the L -least possible S α at eachstep α . We get the following two results as easy corollaries.First, let A µ = { c α : α ∈ S µ } , where c α is the α -th element of HC in thesense of the ordering L . Then h A µ i µ<ω is still a ∆ HC1 sequence, and(B) if X α ∈ HC for all α < ω then the set { µ : A µ = { X α : α < µ }} isstationary in ω .Second, for any α , if A α = h a γ i γ<α , where each a γ itself is equal to an ω -sequence h a nγ i n<ω , then let B nα = h a nγ i γ<α for all n . Otherwise let B nα = ∅ , ∀ n .Then (cid:10) B α (cid:11) n<ωα<ω is still a ∆ HC1 system of sets in HC, such that32C) if X nα ∈ HC for all α < ω , n < ω , then, for every µ < ω , the set { µ : ∀ n ( B nµ = { X nα : α < µ }} is stationary in ω .Now things become more routinely complex.Let µ < ω . We define z ⌈ µ ⌉ = S B µ . If B µ ∈ sMF and dom ( B µ ) = µ then let π ⌈ µ ⌉ = B µ ; otherwise let π ⌈ µ ⌉ be equal to the L -least multisequence in sMF of length µ . (Those exist by Corollary 20.2(ii).) Finally we let D ⌈ µ ⌉ = S B µ +1 .Let’s show that the sequences of sets π ⌈ µ ⌉ , D ⌈ µ ⌉ , z ⌈ µ ⌉ prove the theorem.Suppose that Π = h Π ν i ν<ω ∈ sMF ω , z ∈ HC , and D ⊆ MT ( Π ). Let X α = z , X α = h α, Π α i , X α = D ∩ MT ( Π ↾ α ) for all α . The set M = { µ < ω : B nµ = { X nα : α < µ } for n = 0 , , } is stationary by (C). Assume that µ ∈ M . Then B µ = { X α : α < µ } = { z } ,therefore z ⌈ µ ⌉ = z . Further B µ = { X α : α < µ } = {h α, Π α i : α < µ } = Π ↾ µ ∈ sMF , therefore π ⌈ µ ⌉ = Π ↾ µ . Finally we have D ⌈ µ ⌉ = S B µ +1 = S α ≤ µ X α = D ∩ MT ( Π ↾ µ ), as required. 23 Key sequence theorem Now we prove a theorem which introduces the key multisequence Π . Theorem 23.1 ( V = L ) . There exists a multisequence Π = h Π α i α<ω ∈ sMF ω satisfying the following requirements :(i) if m < ω then the multisequence Π ↾↾ m belongs to the class ∆ HC m +2 ;(ii) if m ′ < ω and W ⊆ sMF is a Σ HC m ′ +1 set then there is an ordinal γ < ω such that the multisequence Π ↾ γ m ′ -decides W ;(iii) if a set D ⊆ MT ( Π ) is dense in MT ( Π ) , then the set Z of all ordinals γ < ω such that Π ↾ γ ⊂ { D ∩ MT ( Π ↾ γ ) } Π , is stationary in ω . Proof. If m < ω then let un m ( p, x ) be a canonical universal Σ m +1 formula,so that the family of all Σ HC m +1 sets X ⊆ HC (those definable in HC by Σ m +1 formulas with parameters in HC) is equal to the family of all sets of the formΥ m ( p ) = { x ∈ HC : HC | = un m ( p, x ) } , p ∈ HC .(I) Fix ∆ HC1 sequences h π ⌈ µ ⌉i µ<ω , h D ⌈ µ ⌉i µ<ω , and h z ⌈ µ ⌉i µ<ω satisfyingTheorem 22.1; the terms D ⌈ µ ⌉ , z ⌈ µ ⌉ , π ⌈ µ ⌉ of the sequences belong to HC ,and in addition π ⌈ µ ⌉ ∈ sMF , dom ( π ⌈ µ ⌉ ) = µ .(II) Let µ < ω . If z ⌈ µ ⌉ is a pair of the form z ⌈ µ ⌉ = h m, p i then let m ⌈ µ ⌉ = m and p ⌈ µ ⌉ = p , otherwise let m ⌈ µ ⌉ = p ⌈ µ ⌉ = 0.33III) If m < ω then let, by Lemma 21.2, π ⌈ µ, m ⌉ ∈ sMF be the L -least multi-sequence in sMF which satisfies π ⌈ µ ⌉ ⊂ { D ⌈ µ ⌉} π ⌈ µ, m ⌉ and m -decides theset Υ m ( p ⌈ µ ⌉ ). Let ⌈ µ, m ⌉ + = dom ( π ⌈ µ, m ⌉ ); then µ < ⌈ µ, m ⌉ + < ω . Proposition 23.2 (in L ) . The sequences h m ⌈ µ ⌉i µ<ω and h p ⌈ µ ⌉i µ<ω belongto the definability class ∆ HC1 . If m < ω then the sequences h π ⌈ µ, m ⌉i µ<ω and h⌈ µ, m ⌉ + i µ<ω belong to the class ∆ HC m +2 . Proof. Routine. Note that π ⌈ µ, m ⌉ and ⌈ µ, m ⌉ + depend on m through theformulas un m ( · , · ), whose complexity strictly increases with m → ∞ .Now define a multisequence Π = h Π α i α<ω ∈ sMF ω and a family of strictlyincreasing, continuous maps µ m : ω → ω , m < ω , as follows:1 ◦ . Let µ m (0) = 0 and µ m ( λ ) = sup γ<λ µ m ( γ ) for all m and all limit λ < ω .2 ◦ . Suppose that m < ω , γ < ω , µ = µ m ( γ ), and the twofold-restrictedsequence ( Π ↾ µ ) ↾↾ m = ( Π ↾↾ m ) ↾ µ is already defined. If the following holds:( ∗ ) m ≥ m ′ = m ⌈ µ ⌉ and ( Π ↾ µ ) ↾↾ m coincides with π ⌈ µ ⌉ ↾↾ m ,then let µ m ( γ + 1) = ⌈ µ, m ′ ⌉ + and ( Π ↾ ⌈ µ, m ′ ⌉ + ) ↾↾ m = π ⌈ µ, m ′ ⌉ ↾↾ m .3 ◦ . In the assumptions of 2 ◦ , if 2 ◦ ( ∗ ) fails, then let ρ is the L -least multi-sequence in sMF with ( Π ↾ µ ) ↾↾ m ⊂ ρ (we refer to Corollary 20.2), anddefine µ m ( γ + 1) = dom ( ρ ) and ( Π ↾ µ m ( γ + 1)) ↾↾ m = ρ ↾↾ m . To conclude, given γ < ω and m , if an ordinal µ = µ m ( γ ), and a multisequence( Π ↾ µ ) ↾↾ m = ( Π ↾↾ m ) ↾ µ are defined, then items 2 ◦ , 3 ◦ define a bigger ordinal µ m ( γ +1) > µ = µ m ( γ ) and a longer multisequence ( Π ↾ µ m ( γ + 1)) ↾↾ m satisfying( Π ↾ µ ) ↾↾ m ⊂ ( Π ↾ µ m ( γ + 1)) ↾↾ m . Thus overall items 1 ◦ , 2 ◦ , 3 ◦ of the definitioncontain straightforward instructions as how to uniquely define the layers Π ↾↾ m and maps µ m for different m < ω , independently from each other.From now on, fix a multisequence Π = h Π α i α<ω ∈ sMF ω of multiforcings Π α ∈ sMF and increasing continuous maps µ m : ω → ω defined by 1 ◦ , 2 ◦ , 3 ◦ .As the maps µ m are continuous, the following holds: Proposition 23.3 (in L ) . C = { γ < ω : ∀ m ( γ = µ m ( γ )) } is a club in ω . To show that Π proves Theorem 23.1, we check items (i), (ii), (iii).(i) Let m < ω . Then the multisequence Π ↾↾ m and the map µ m belong tothe class ∆ HC m +2 by Proposition 23.2; a routine proof is omitted.(ii) Suppose that m ′ < ω and W ⊆ sMF is a Σ HC m ′ +2 set. Pick p ∈ HC suchthat W = Υ m ′ ( p ). Let z = h m ′ , p i . As C is a club, it follows from the choice of34erms π ⌈ µ ⌉ , D ⌈ µ ⌉ , and z ⌈ µ ⌉ , by (I) and Theorem 22.1, that there is an ordinal γ ∈ C such that π ⌈ γ ⌉ = Π ↾ γ and z ⌈ γ ⌉ = z — hence, m ⌈ γ ⌉ = m ′ and p ⌈ γ ⌉ = p .Let µ = γ ; then also µ = µ m ( γ ), ∀ m — since γ ∈ C , and Π ↾ µ = π ⌈ µ ⌉ .Then it follows from the choice of Π that item 2 ◦ of the construction appliesfor the ordinal γ chosen and all m ≥ m ′ . It follows that the multisequence ρ = π ⌈ µ, m ′ ⌉ and the ordinal ν = µ m ( γ + 1) = ⌈ µ, m ′ ⌉ + satisfy ν = dom ( ρ )and ( Π ↾ ν ) ↾↾ m = ρ ↾↾ m for all m ≥ m ′ . In other words, ( Π ↾ ν ) ↾↾ ≥ m ′ = ρ ↾↾ ≥ m ′ .However by definition ρ m ′ -decides the set W = Υ m ′ ( p ), and the definitionof this property depends only on ρ ↾↾ ≥ m ′ .(iii) Assume that a set D ⊆ MT ( Π ) is dense in MT ( Π ), and C ⊆ C is aclub in ω . Following the proof of (ii), we find an ordinal γ ∈ C such that π ⌈ γ ⌉ = Π ↾ γ , m ⌈ γ ⌉ = 0, and D ⌈ γ ⌉ = D ∩ MT ( Π ↾ γ ), Note that γ = µ m ( γ ), ∀ m . We have π ⌈ γ ⌉ ⊂ { D ⌈ γ ⌉} π ⌈ γ, ⌉ by (III) (with µ = γ ), that is, π ⌈ γ ⌉ ⊂ { D ∩ MT ( Π ↾ γ ) } π ⌈ γ, ⌉ . ( † )Yet it follows from the choice of γ that condition 2 ◦ ( ∗ ) holds (for µ = γ ) forall m ≥ 0. Then, by definition 2 ◦ , the ordinal µ + = ⌈ γ, m ⌉ + satisfies µ + = µ m ( γ + 1) and ( Π ↾ µ + ) ↾↾ m = ( π ⌈ γ, ⌉ ) ↾↾ m for all m , that is, just Π ↾ µ + = π ⌈ γ, ⌉ . We conclude that Π ↾ γ ⊂ { D ∩ MT ( Π ↾ γ ) } Π ↾ µ + by ( † ), therefore we have Π ↾ γ ⊂ { D ∩ MT ( Π ↾ γ ) } Π , as required. (cid:3) ( Theorem 23.1 ) Definition 23.4 (in L ) . From now on we fix a multisequence Π = h Π α i α<ω ∈ sMF ω satisfying requirements of Theorem 23.1, that is,(i) if m < ω then the multisequence Π ↾↾ m belongs to the class ∆ HC m +2 ;(ii) if m ′ < ω and W ⊆ sMF is a Σ HC m ′ +1 set then there is an ordinal γ < ω such that the multisequence Π ↾ γ m ′ -decides W ;(iii) if a set D ⊆ MT ( Π ) is dense in MT ( Π ), then the set Z of all ordinals γ < ω such that Π ↾ γ ⊂ { D ∩ MT ( Π ↾ γ ) } Π , is stationary in ω .We call Π the key multisequence .A set U ⊆ sMF ↾↾ ≥ m is dense in sMF ↾↾ ≥ m if for each π ∈ sMF ↾↾ ≥ m there isa multisequence ϙ ∈ U satisfying π ⊆ ϙ . Lemma 23.5. If m < ω and W ⊆ sMF ↾↾ ≥ m is a Σ HC m +1 set dense in sMF ↾↾ ≥ m then there is an ordinal γ < ω such that ( Π ↾ γ ) ↾↾ ≥ m ∈ W . In particular, if W ⊆ sMF is a Σ HC1 set dense in sMF then there is γ < ω such that Π ↾ γ ∈ W . Proof. Apply 23.4(ii). The negative decision is impossible by the density.35 We continue to argue in L , and we’ll make use of the key multisequence Π = h Π α i α<ω introduced by Definition 23.4. Definition 24.1 (in L ) . Define the multiforcings Π = S cw Π = S cw α<ω Π α ∈ MF , Π <γ = S cw ( Π ↾ γ ) = S cw α<γ Π α ∈ sMF , for each γ < ω Π ≥ γ = S cw ( Π ↾ ( ω r γ )) = S cw γ ≤ α<ω Π α ∈ MF , for each γ < ω . We further define PPP = MT ( Π ) = MT ( Π ), and, for all γ < ω , PPP <γ = MT ( Π <γ ) = MT ( Π ↾ γ ) , PPP ≥ γ = MT ( Π ≥ γ ) = MT ( Π ↾ ( ω r γ )) . The multiforcing PPP will be our principal forcing notion, the key forcing . Lemma 24.2 (in L ) . Π is a regular multiforcing. In addition, | Π | = ω × ω , thusif ξ < ω and k < ω then there is an ordinal α < ω such that h ξ, k i ∈ | Π α | .Therefore PPP = Q ξ<ω , k<ω Π ( ξ, k ) (with finite support). Proof. To prove the additional claim, note that the set W of all multisequences π ∈ sMF satisfying h ξ, k i ∈ | S cw π | is Σ HC1 (with ξ as a parameter of definition).In addition W is dense in sMF . (First extend π by Corollary 20.2 so that is hasa non-limit length and the last term, then make use of Corollary 11.4.) Thereforeby Lemma 23.5 there is an ordinal γ < ω such that Π ↾ γ ∈ W , as required.If ξ < ω and k < ω then, following the lemma, let α ( ξ, k ) < ω be theleast ordinal α satisfying h ξ, k i ∈ | Π α | . Thus a forcing Π α ( ξ, k ) ∈ PTF is de-fined whenever α satisfies α ( ξ, k ) ≤ α < ω , and h Π α ( ξ, k ) i α ( ξ,k ) ≤ α<ω is a ❁ -increasing sequence of countable special forcings in PTF .Note that Π ( ξ, k ) = S α ( ξ,k ) ≤ α<ω Π α ( ξ, k ) by construction. Corollary 24.3 (in L ) . If k < ω then the sequence of ordinals h α ( ξ, k ) i ξ<ω and the sequence of multiforcings h Π α ( ξ, k ) i ξ<ω , α ( ξ,k ) ≤ α<ω are ∆ HC k +2 . Proof. By construction the following double equivalence holds: α < α ( ξ, k ) ⇐⇒ ∃ π ( π = Π α ↾↾ k ∧ h ξ, k i ∈ dom π ) ⇐⇒⇐⇒ ∀ π ( π = Π α ↾↾ k = ⇒ h ξ, k i ∈ dom π ) . However π = Π α ↾↾ k is a ∆ HC k +2 relation by Theorem 23.1(i). It follows thatso is the sequence h α ( ξ, k ) i ξ<ω . The second claim easily follows by the sameDefinition 23.4(i). 36 orollary 24.4 (in L , of Lemma 9.2(v)) . If ξ < ω , k < ω , and α ( ξ, k ) ≤ α <ω then the set Π α ( ξ, k ) is pre-dense in Π ( ξ, k ) and in Π . In spite of Lemma 24.2, the sets | Π <γ | can be quite arbitrary (countable)subsets of ω × ω . However we easily get the next corollary: Corollary 24.5 (in L , of Lemma 24.2) . The set C ′ = { γ < ω : | Π <γ | = γ × ω } is a club in ω . Lemma 24.6 (in L ) . PPP is CCC. Proof. Let A ⊆ PPP be a maximal antichain in PPP . The set C = { γ < ω : A ∩ PPP <γ is a maximal antichain in PPP <γ } is a club in ω . Let D = { p ∈ PPP : ∃ q ∈ A ( p q ) } ; this is an open dense set. ByDefinition 23.4(iii), there is an ordinal γ ∈ C such that Π ↾ γ ⊂ { D ∩ PPP <γ } Π . Recallthat γ ∈ C , hence A ∩ PPP <γ is a maximal antichain in PPP <γ , thus D ∩ PPP <γ is opendense in PPP <γ . Therefore the set D ∩ PPP <γ is pre-dense in the forcing MT ( Π ) = PPP by Corollary 20.2(v)(c). We claim that A = A ∩ PPP <γ , so A is countable.Indeed suppose that r ∈ A r PPP <γ . Then r is compatible with some q ∈ D ∩ PPP <γ ; let p ∈ D ∩ PPP <γ , p q , p r . As q ∈ D , there is some r ′ ∈ A with q r ′ . Then r = r ′ as A is an antichain; thus q r ∈ A r PPP <γ . However q ∈ PPP <γ and A ∩ PPP <γ is a maximal antichain in PPP <γ , thus q , and hence r , iscompatible with some r ′′ ∈ A ∩ PPP <γ . Which is a contradiction. Corollary 24.7 (in L ) . If a set D ⊆ PPP is pre-dense in PPP then there is anordinal γ < ω such that D ∩ PPP <γ is already pre-dense in PPP . Proof. We can assume that in fact D is dense. Let A ⊆ D be a maximalantichain in D ; then A is a maximal antichain in PPP because of the density of D .Then A ⊆ PPP <γ for some γ < ω by Lemma 24.6. But A is pre-dense in PPP .37 Auxiliary forcing relation Recall that PPP = MT ( Π ), the key forcing, is a product forcing notion defined(in L ) in Section 24. Its components Π ( ξ, k ) have different complexity in HC ,depending on k by Corollary 24.3, hence there is no way the forcing notion PPP (or Π ) as a whole is definable in HC . Somewhat surprisingly, the PPP -forcing relationturns out to be definable in HC when restricted to analytic formulas of a certainlevel of complexity within the usual hierarchy. This will be established on thebase of an auxiliary forcing relation. 25 Auxiliary forcing: preliminaries We argue in L . Consider the 2nd order arithmetic language, with variables k, l, m, n, . . . of type 0 over ω and variables a, b, x, y, . . . of type 1 over ω ω ,whose atomic formulas are those of the form x ( k ) = n . Let L be the extensionof this language, which allows to substitute free variables of type 0 with naturalnumbers (as usual) and free variables of type 1 with small real names c ∈ L . By L -formulas we understand formulas of this extended language.We define natural classes L Σ n , L Π n ( n ≥ 1) of L -formulas. Let L ( Σ + Π ) be the closure of L Σ ∪ L Π under ¬ , ∧ , ∨ and quantifiers over ω . If ϕ is aformula in L Σ n (resp., L Π n ), then let ϕ − be the result of canonical transfor-mation of ¬ ϕ to the L Π n (resp., L Σ n ) form.If ϕ is a L -formula and G ⊆ MT is a pairwise compatible set of multitreesthen let ϕ [ G ] be the result of substitution of c [ G ] for any name c in ϕ . (RecallDefinition 13.2.) Thus ϕ [ G ] is an ordinary 2nd order arithmetic formula, whichmay include natural numbers and elements of ω ω as parameters.We are going to define a relation p forc π ϕ between multitrees p , multise-quences π , and L -formulas ϕ , which suitably approximates the true PPP -forcingrelation. But it depends on a two more definitions. Definition 25.1. If m < ω then sMF [ Π ↾↾ ZFC .38 efinition 25.3. Let ZFL – be the theory containing all axioms of ZFC − (minusthe Power Set axiom) plus the axiom of constructibility V = L . Any transitivemodel (TM) of ZFL – has the form L α , where α ∈ Ord . Therefore it is truein L that for any set x there is a least TM M = M ( x ) of ZFL – containing x . If x ∈ HC (HC = all hereditarily countable sets) then M ( x ) is a countable transitive model (CTM), equal to the least CTM of ZFL – containing x . 26 Auxiliary forcing The definition of p forc π ϕ goes on by induction on the complexity of ϕ .1 ◦ . Let ϕ is a closed L ( Σ + Π ) formula, π ∈ sMF , p ∈ MT , but p ∈ MT ( π )is not necessarily assumed. We define:(a) p forc π ϕ iff there is a CTM M | = ZFL – , an ordinal ϑ < dom π ,and a multitree p ∈ MT ( π ↾ ϑ ), such that p p (meaning: p isstronger), the model M contains π ↾ ϑ (then contains MT ( π ↾ ϑ ) aswell) and contains ϕ (that is, all names in ϕ ), π ↾ ϑ ⊂ M π , and p MT ( π ↾ ϑ )-forces ϕ [ G ] over M (in the usual sense) ;(b) p wforc π ϕ (weak forcing) iff there is no multisequence π ′ ∈ sMF and p ′ ∈ MT ( π ′ ) such that π ⊆ π ′ , p ′ p , and p ′ forc π ′ ¬ ϕ .2 ◦ . If ϕ ( x ) is a L Π n formula, n ≥ 1, then we define p forc π ∃ x ϕ ( x ) iff thereis a small real name c such that p forc π ϕ ( c ).3 ◦ . If ϕ is a closed L Π n formula, n ≥ 2, then we define p forc π ϕ iff π ∈ sMF [ Π ↾↾ With p and ϑ given, the premise “ p MT ( π ↾ ϑ )-forces ϕ [ G ]over M ” of 1 ◦ a does not depend on the choice of a CTM M containing π ↾ ϑ and ϕ , since if ϕ is L ( Σ + Π ) then the formula ϕ [ G ] (in which all names areevaluated by some MT ( π ↾ ϑ )-generic set G as in 13.2) in simultaneously true orfalse in all transitive models by the Shoenfield absoluteness theorem. Remark 26.2. It easily holds by induction that if p forc π ϕ then π ∈ sMF , ϕ is a closed formula in one of the classes L ( Σ + Π ) , L Σ n , L Π n , n ≥ 2, and if n ≥ ϕ ∈ L Π n ∪ L Σ n +1 then π ∈ sMF [ Π ↾↾ Relations like “being an MSP”, “being a formula in L ( Σ + Π ) , L Σ n , L Π n ”, p ∈ MT ( ρ ), forcing over a CTM, etc. are definable in HC by boundedformulas, hence ∆ HC1 . On the top of this, the model M can be tied by both ∃ and ∀ in 1 ◦ a, see Remark 26.1. This wraps up the ∆ HC1 estimation for L ( Σ + Π ) .The inductive step by 2 ◦ is quite simple.Now the step by 3 ◦ . Assume that n ≥ 2, and it is already established that FORC [ L Σ n ] ∈ Σ HC n − . Then h π , p , ϕ i ∈ FORC [ L Π n ] iff π ∈ sMF [ Π ↾↾ 27 Forcing simple formulas We still argue in L . The following results are mainly related to the relation forc with respect to formulas in the class L ( Σ + Π ) . Lemma 27.1 (in L ) . Assume that π ∈ sMF , ϙ ∈ sMF ∪ sMF ω , π ⊆ ϙ , p ∈ MT ( π ) , ϕ is a formula in L ( Σ + Π ) , N | = ZFL – is a TM containing ϙ and ϕ , and p forc π ϕ . Then p MT ( ϙ ) -forces ϕ [ G ] over N . roof. By definition there is an ordinal ϑ < dom π , a multitree p ∈ MT ( π ↾ ϑ ),and a CTM M | = ZFL – containing ϕ and π ↾ ϑ , such that π ↾ ϑ ⊂ M π , p p ,and p MT ( π ↾ ϑ )-forces ϕ [ G ] over M . We can w. l. o. g. assume that M ⊆ N (by the same reference to Shoenfield as in Remark 26.1).Now suppose that G ⊆ MT ( ϙ ) is a set MT ( ϙ )-generic over N and p ∈ G — then p ∈ G , too. We have to prove that ϕ [ G ] is true in N [ G ].We claim that the set G ′ = G ∩ MT ( π ↾ ϑ ) is MT ( π ↾ ϑ )-generic over M .Indeed, let a set D ∈ M , D ⊆ MT ( π ↾ ϑ ), be open dense in MT ( π ↾ ϑ ). Then,as π ↾ ϑ ⊂ M ϙ by Corollary 20.2(iv), D is pre-dense in MT ( ϙ ) by 20.2(v)(c),and hence G ∩ D = ∅ by the choice of G . It follows that G ′ ∩ D = ∅ .We claim that c [ G ] = c [ G ′ ] for any name c ∈ M , in particular, for any namein ϕ . Indeed, as G ′ ⊆ G , the otherwise occurs by Definition 13.2 only if forsome n, i and q ′ ∈ K c ni there is q ∈ G satisfying q q ′ , but there is no such q in G ′ . Let D consist of all multitrees r ∈ MT ( π ↾ ϑ ) either satisfying r q ′ or somewhere a.d. with q ′ . Then D ∈ M and D is open dense in MT ( π ↾ ϑ ).Therefore D ∩ G ′ = ∅ by the above, so let r ∈ D ∩ G ′ . If r q ′ then we get acontradiction with the choice of q ′ . If r is somewhere a.d. with q ′ then we geta contradiction with the choice of q as both q , r belong to the generic filter G .It follows that ϕ [ G ] coincides with ϕ [ G ′ ].Note also that p ∈ G ′ . We conclude that ϕ [ G ′ ] is true in M [ G ′ ] as p forces ϕ [ G ] over M . The same formula ϕ [ G ] is true in N [ G ] by Shoenfield. Lemma 27.2. Let π ∈ sMF , p ∈ MT ( π ) , ϕ be a formula in L ( Σ + Π ) . Then (i) p forc π ϕ and p forc π ¬ ϕ cannot hold together ;(ii) if p forc π ϕ then p wforc π ϕ ;(iii) if p wforc π ϕ then there is a multisequence ϙ ∈ sMF such that π ⊂ M ( π ) ϙ and p forc ϙ ϕ ; (iv) p wforc π ϕ and p wforc π ¬ ϕ cannot hold together. Proof. (i) Otherwise p MT ( π )-forces both ϕ [ G ] and ¬ ϕ [ G ] over a large enoughCTM M , by Lemma 27.1, which cannot happen.(ii) Assume that p wforc π ϕ fails, hence there is a multisequence ϙ ∈ sMF and a multitree q ∈ MT ( ϙ ) such that q p and q forc π ¬ ϕ . But Lemma 26.3implies q forc π ϕ , which contradicts to (i).(iii) Let M | = ZFL – be a CTM containing ϕ and π . By Corollary 20.2(ii),there is a multisequence multisequence ϙ ∈ sMF with π ⊂ M ϙ . We claim that p MT ( π )-forces ϕ [ G ] over M in the usual sense — then by definition p forc ϙ ϕ (via ϑ = dom π ), and we are done. To prove the claim suppose otherwise. Thenthere is a multitree q ∈ MT ( π ) such that q p and q MT ( π )-forces ¬ ϕ [ G ]over M , thus q forc ϙ ¬ ϕ . But this contradicts to p wforc π ϕ .41iv) There is a multisequence ϙ ∈ sMF by (iii), such that π ⊂ ϙ and p forc ϙ ϕ . Note that still p wforc ϙ ¬ ϕ by Lemma 26.3. Extend ϙ onceagain, getting a contradiction with (i). Corollary 27.3. Let n ≥ , π ∈ sMF , p ∈ MT ( π ) , ϕ be a formula in L Σ n .Then p forc π ϕ and p forc π ϕ − cannot hold together. Proof. If n = 1 then apply Lemma 27.2(i). If n ≥ ◦ in Section 26). Corollary 27.4 (in L ) . Assume that π ∈ sMF , p ∈ MT ( π ) , ϕ is a formula in L ( Σ + Π ) , N | = ZFL – is a TM containing π and ϕ , and p wforc π ϕ . Then p MT ( π ) -forces ϕ [ G ] over N in the usual sense. This looks like the case ρ = π in Lemma 27.1, but forc is weakened to wforc , and ϕ ∈ M (automatic in Lemma 27.1) is added, in the premise. Proof. Otherwise there is a multitree q ∈ MT ( π ), q p , that MT ( π )-forces ¬ ϕ [ G ] over N . On the other hand, by Lemma 27.2(iii), there is a multise-quence ϙ ∈ sMF such that π ⊂ M ( π ) ϙ and p forc ϙ ϕ , hence, q forc ϙ ϕ byLemma 26.3. However we have q forc ϙ ¬ ϕ by definition (1 ◦ a in Section 26with ϑ = dom π ), which contradicts to Lemma 27.2(i). 28 Tail invariance If π = h π α i α<λ ∈ sMF and γ < λ = dom π then let the γ - tail π ↾ ≥ γ be therestriction π ↾ [ γ, λ ) to the ordinal semiinterval [ γ, λ ) = { α : γ ≤ α < λ } . Thenthe multiforcing MT ( π ↾ ≥ γ ) = S cw γ ≤ α<λ π ( α ) is open dense in MT ( π ) by Corol-lary 20.2(v)(a). Therefore it can be expected that if ϙ is another multisequenceof the same length λ = dom ϙ , and ϙ ↾ ≥ γ = π ↾ ≥ γ , then the relation forc π coincides with forc ϙ . And indeed this turns out to be the case (almost). Theorem 28.1. Assume that π , ϙ are multisequences in sMF , γ < λ = dom π = dom ϙ , ϙ ↾ ≥ γ = π ↾ ≥ γ , p ∈ MT , and ϕ is an L -formula. Then (i) if ϕ ∈ L ( Σ + Π ) then p wforc π ϕ iff p wforc ϙ ϕ ;(ii) if n ≥ , π , ϙ ∈ sMF [ Π ↾↾ 2, let ϕ ( x ) be a formula in L Π n .Assume that p forc π ∃ x ϕ ( x ). By definition (see 2 ◦ in Section 26), there is asmall real name c such that p forc π ϕ ( c ). Then we have p forc ϙ ϕ ( c ) by theinductive assumption, thus p forc ϙ ∃ x ψ ( x ).To carry out the step L Σ n → L Π n , n ≥ 3, assume that ϕ is a L Π n formula, p forc π ϕ , but to the contrary p forc ϙ ϕ fails. Then by 3 ◦ of Section 26,as ϙ ∈ sMF [ Π ↾↾ 29 Permutations Still arguing in L , we let PERM be the set of all bijections h : ω × ω onto −→ ω × ω , such that the kernel | h | = {h ξ, k i : h ( ξ, k ) = h ξ, k i} is at most countable.43lements of PERM will be called permutations . If m < ω then let PERM m consist of those permutations h ∈ PERM satisfying | h | ⊆ ω × ( ω r m ). Inother words, if h ∈ PERM m and ξ < ω , k < m , then h ( ξ, k ) = h ξ, k i .Let h ∈ PERM. We extend the action of h as follows. • if p is a multitree then hp is a multitree, | hp | = h ” p = { h ( ξ, k ) : h ξ, k i ∈| p |} , and ( hp )( h ( ξ, k )) = p ( ξ, k ) whenever h ξ, k i ∈ | p | , in other words, hp coincides with the superposition p ◦ ( h − ); • if π ∈ MT is a multiforcing then h · π = π ◦ ( h − ) is a multiforcing, | h · π | = h ” π and ( h · π )( h ( ξ, k )) = π ( ξ, k ) whenever h ξ, k i ∈ | π | ; • if c ⊆ MT × ( ω × ω ) is a real name, then put h c = {h hp , n, i i : h p , n, i i ∈ c } ,thus easily h c is a real name as well; • if π = h π α i α<κ is a multisequence, then h π = h h · π α i α<κ , still a multise-quence. • if ϕ := ϕ ( c , . . . , c n ) is a L -formula (with all names explicitly indicated),then h ϕ is ϕ ( h c , . . . , h c n ).Many notions and relations defined above are clearly PERM-invariant, e. g. , p ∈ MT ( π ) iff hp ∈ MT ( h · π ), π ❁ ϙ iff h · π ❁ h · ϙ , et cetera . The invariancepersists even with respect to the relation forc , at least to some extent. Theorem 29.1. Assume that π ∈ sMF , p ∈ MT ( π ) , ϕ is an L -formula, and h ∈ PERM . Then (i) if ϕ belongs to L ( Σ + Π ) and p forc π ϕ , then ( hp ) wforc h π ( h ϕ ) ;(ii) if n ≥ , h ∈ PERM n − , and ϕ belongs to L Π n ∪ L Σ n +1 , then p forc π ϕ iff ( hp ) forc h π ( h ϕ ) . Proof. Let ϙ = h π , q = hp , ψ := h ϕ .(i) Suppose to the contrary that p wforc π ϕ , but q wforc ϙ ψ fails, so thatthere is a multisequence ϙ ′ ∈ sMF and q ′ ∈ MT ( ϙ ′ ) such that ϙ ⊂ ϙ ′ , q ′ q ,and q ′ forc ϙ ′ ¬ ψ . The multisequence π ′ = h − ϙ ′ then satisfies π ⊂ π ′ ,and the multitree p ′ = h − q ′ belongs to MT ( π ′ ) and p ′ p , hence we have p ′ wforc π ′ ϕ by Lemma 26.3.Now let M | = ZFL – be an arbitrary CTM containing π ′ , ϙ ′ , ϕ, ψ, h ↾ | h | .Then, by Corollary 27.4, p ′ MT ( π ′ )-forces ϕ [ G ], but q ′ MT ( ϙ ′ )-forces ψ [ G ],over M . However the sets MT ( π ′ ), MT ( ϙ ′ ) belong to the same model M , wherethey are order-isomorphic via the isomorphism induced by h ↾ | h | . Therefore, andsince q = hp and ψ = h ϕ , it cannot happen that both p MT ( π ′ )-forces ϕ [ G ]and q MT ( ϙ ′ )-forces ¬ ψ [ G ]. But this contradicts to the above.44ii) Consider first the L Π case. Assume that ϕ ( x ) is a L Σ formula, ψ ( x ) := h ϕ ( x ), p forc π ∀ x ϕ ( x ), but to the contrary q forc ϙ ∀ x ψ ( x ) fails.Thus there is a multisequence ϙ ′ ∈ sMF and a multitree q ′ ∈ MT ( ϙ ′ ) suchthat ϙ ⊂ ϙ ′ , q ′ q , and q ′ forc ϙ ′ ∃ x ψ − ( x ). By definition there is a smallreal name d such that q ′ forc ϙ ′ ψ − ( d ). The multisequence π ′ = h − ϙ ′ thensatisfies π ⊂ ϙ , the multitree p ′ = h − q ′ belongs to MT ( π ′ ) and p ′ p , c = h − d is a small real name, and we have p ′ wforc π ′ ϕ − ( c ) by (i). Then byLemma 27.2 there is a longer multisequence σ ∈ sMF satisfying π ′ ⊂ σ and p ′ forc σ ϕ − ( c ), that is, we have p ′ forc σ ∃ x ϕ − ( x ). But by definition (3 ◦ inSection 26) this contradicts to the assumption p forc π ∀ x ϕ ( x ).To carry out the step L Π n → L Σ n +1 , n ≥ 2, let ϕ ( x ) be a formula in L Π n , ψ ( x ) := h ϕ ( x ), and h ∈ PERM n − . Assume that p forc π ∃ x ϕ ( x ).By definition (see 2 ◦ in Section 26), there is a small real name c such that p forc π ϕ ( c ). Then we have q forc ϙ ψ ( d ) by inductive assumption, where d = h c is a small real name itself. Thus q forc ϙ ∃ x ψ ( x ).To carry out the step L Σ n → L Π n , n ≥ 3, let ϕ be a formula in L Π n ,and h ∈ PERM n − . Let p forc π ϕ , in particular π ∈ sMF [ Π ↾↾ 30 Forcing with subsequences of the key multisequence We argue in L . The key multisequence Π = h Π α i α<ω ∈ sMF ω , satisfying (i),(ii), (iii) of Theorem 23.1, was fixed by 23.4, and PPP = MT ( Π ) is our forcingnotion. If γ < ω then the subsequence Π ↾ γ belongs to sMF [ Π ↾↾ We write p forc α ϕ instead of p forc Π ↾ α ϕ , for the sake ofbrevity. Let p forc ϕ mean: p forc α ϕ for some α < ω . Lemma 30.2 (in L ) . Assume that p ∈ PPP , α < ω , and p forc α ϕ . Then :(i) if α ≤ β < ω , q ∈ PPP <β = MT ( Π ↾ β ) , and q p , then q forc β ϕ ;(ii) if q ∈ PPP , q p , then q forc β ϕ for some β ; α ≤ β < ω ;(iii) if q ∈ PPP and q forc ϕ − then p , q are somewhere almost disjoint ;45iv) therefore, 1st, if p , q ∈ PPP , q p , and p forc ϕ then q forc ϕ , and 2nd, p forc ϕ , p forc ϕ − cannot hold together. Proof. To prove (i) apply Lemma 26.3. To prove (ii) let β satisfy α < β < ω and q ∈ MT ( Π ↾ β ), and apply (i). Finally to prove (iii) note that p , q have to beincompatible in PPP , as otherwise (i) leads to contradiction, but the incompatibilityin PPP implies being somewhere a.d. by Corollary 7.1.Now we are going to prove that the auxiliary relation forc essentially coin-cides with the usual PPP -forcing relation over L . Lemma 30.3. If n < ω , ϕ is a closed formula as in 26.2, and p ∈ PPP , then p PPP -forces ϕ [ G ] over L in the usual sense if and only if p forc ϕ . Proof. Let k− denote the usual PPP -forcing relation over L . Part 1 : ϕ is a formula in L ( Σ + Π ) . If p forc ϕ then p forc Π ↾ γ ϕ forsome γ < ω , and then p k− ϕ [ G ] by Lemma 27.1 with ϙ = Π and N = L .Suppose now that p k− ϕ [ G ]. There is an ordinal γ < ω such that p ∈ PPP γ = MT ( Π ↾ γ ) and ϕ belongs to M ( Π ↾ γ ). The set U of all multitrees π ∈ sMF such that γ < dom π and there is an ordinal ϑ , γ < ϑ < dom π ,such that π ↾ ϑ ⊂ M ( π ↾ ϑ ) π , is dense in π by Corollary 20.2(ii), and is ∆ HC1 .Therefore by Corollary 23.5 there is an ordinal γ < ω such that π = Π ↾ γ ∈ U .Let this be witnessed by an ordinal ϑ , so that γ < ϑ < γ = dom π and π ↾ ϑ ⊂ M ( π ↾ ϑ ) π . We claim that p MT ( π ↾ ϑ )-forces ϕ [ G ] over M ( π ↾ ϑ ) in theusual sense — then by definition p forc π ϕ , and we are done.To prove the claim, suppose otherwise. Then there is a multitree q ∈ MT ( Π ↾ ϑ ), q p , which MT ( π ↾ ϑ )-forces ¬ ϕ [ G ] over M ( π ↾ ϑ ). Then bydefinition we have q forc π ¬ ϕ , hence q forc ¬ ϕ , which implies q k− ¬ ϕ [ G ](see above), with a contradiction to p k− ϕ [ G ]. Part 2 : the step L Π n → L Σ n +1 ( n ≥ L Π n formula ϕ ( x ).Assume p forc ∃ x ϕ ( x ). By definition there is a small real name c such that p forc ϕ ( c ). By inductive hypothesis, p k− ϕ ( c )[ G ], that is, p k− ∃ x ϕ ( x )[ G ].Conversely, assume that p k− ∃ x ϕ ( x )[ G ]. As PPP is CCC, there is a small realname c (in L ) such that p k− ϕ ( c )[ G ]. We have p forc ϕ ( c ) by the inductivehypothesis, hence p forc ∃ x ϕ ( x ). Part 3 : the step L Σ n → L Π n ( n ≥ L Σ n formula ϕ . Assume that p forc ϕ − . By Lemma 30.2(iv), there is no multitree q ∈ PPP , q p , with q forc ϕ . This implies p k− ϕ − by the inductive hypothesis.Conversely, suppose that p k− ϕ − . There is an ordinal γ < ω such that p ∈ PPP γ = MT ( Π ↾ γ ) and ϕ belongs to M ( Π ↾ γ ). Consider the set U of allmultisequences of the form π ↾↾ ≥ n − , where π ∈ sMF [ Π ↾↾ Let U consist of all multisequences of the form π ↾↾ ≥ m , where π ∈ sMF [ Π ↾↾ In this conclusive section we gather the results obtained above towards the proofof Theorem 1.1. We begin with the analysis of definability of key generic reals in PPP -generic extensions of L , which will lead to (I) and (II) of Theorem 1.1. Thenwe proceed to (III) (elementary equivalence) and (IV) (the non-wellorderability). 31 Key generic extension and subextensions Recall that the key multisequence Π = h Π α i α<ω of small multiforcings Π α isdefined in L by 23.4, Π = S cw α<ω is a multiforcing, | Π | = ω × ω in L , and PPP = MT ( Π ) = MT ( Π ) ∈ L is our key forcing notion, equal to the finite-supportproduct Q ξ<ω ,k<ω Π ( ξ, k ) of perfect-tree forcings Π ( ξ, k ) in L . See Section 24,where some properties of PPP are established, including CCC and definability ofthe factors Π ( ξ, k ).From now on, we’ll typically argue in L and in PPP -generic extensions of L , soby Lemma 24.6 it will always be true that ω L = ω . This allows us to still thinkthat | Π | = ω × ω (rather than ω L × ω ).Recall that Π ∈ L and PPP = MT ( Π ) is a forcing notion in L . Definition 31.1. Let a set G ⊆ PPP be generic over the constructible set universe L . If h ξ, k i ∈ ω × ω then following Remark 7.3, we − define G ( ξ, k ) = { T p ξk : p ∈ G ∧ h ξ, k i ∈ | p |} ⊆ Π ( ξ, k ); − let x ξk = x ξk [ G ] ∈ ω be the only real in T T ∈ G ( ξ,k ) [ T ].Thus PPP adds an array X = h x ξk i h ξ,k i∈ ω × ω of reals, where each real x ξk = x ξk [ G ] ∈∈ ω ∩ L [ G ] is a Π ( ξ, k )-generic real over L , and L [ G ] = L [ X ].Let G ⊆ PPP be a set (filter) PPP -generic over L . If m < ω then following thenotation in Section 21 we define G ↾↾ Assume that a set G ⊆ PPP is PPP -generic over L , ξ < ω , k < ω ,and x ∈ L [ G ] ∩ ω ω . The following are equivalent :(1) x = x ξk [ G ] ;(2) x is Π ( ξ, k ) -generic over L ;(3) x ∈ T α ( ξ,k ) ≤ α<ω S T ∈ Π α ( ξ,k ) [ T ] . Proof. (1) = ⇒ (2) is a routine (see Remark 7.3). To check (2) = ⇒ (3) recallthat each set Π α ( ξ, k ) is pre-dense in Π ( ξ, k ) by Lemma 9.2(v). It remains toestablish (3) = ⇒ (1) . Suppose towards the contrary that a real x ∈ L [ G ] ∩ ω satisfies (3) but x = x ξk [ G ]. By Theorem 18.2(i) there is a true Π -real name c = h C ni i n,i<ω , non-principal over Π at ξ, k and such that x = c [ G ]. Beingnon-principal means that the next set is open dense in PPP = MT ( Π ): D Π ξk ( c ) = { p ∈ PPP = MT ( Π ) : p directly forces c / ∈ [ T p ξk ] } . And as PPP = MT ( Π ) is a CCC forcing by Lemma 24.6, we can assume that thename c is small, that is, each set C ni ⊆ PPP is countable. Then there is an ordinal γ < ω such that C ni ⊆ PPP <γ for all n, i . Then c is a true Π <γ -real name.Moreover we can assume by Corollary 24.7 that D Π ξk ( c ) ∩ PPP <γ is pre-dense in PPP .Now consider the set W of all multisequences π = h π α i α< dom ( π ) ∈ sMF suchthat dom ( π ) > γ and − either (I) Π ↾ γ π ; − or (II) Π ↾ γ ⊂ π and c is not non-principal over π = S cw π at ξ, k ; − or (III) Π ↾ γ ⊂ π , dom ( π ) = δ + 1 is a successor, and S cw α<δ π α ❁ c ξk π δ .We assert that W is dense in sMF : any multisequence π ∈ sMF can be extendedto some ϙ ∈ W . Indeed first extend π by Corollary 20.2 so that is has a length dom ( π ) = δ > γ . If now Π ↾ γ π then immediately π ∈ W via (I), sowe assume that Π ↾ γ ⊂ π . We can also assume that c is non-principal over π = S cw π at ξ, k by similar reasons related to (II). The multisequence π canbe extended, by Corollary 20.2, by an extra term π δ , so that the extendedmultisequence π + satisfies π ⊂ { c } π + , that is, π ❁❁ { c } π δ . By definition(Definition 19.1) and the nonprincipality of c , we get π ❁ c ξk π δ . It follows that π + ∈ W via (III). 49ince W is Σ HC1 , by Definition 23.4(ii) there is an ordinal γ < ω suchthat the multisequence Π ↾ γ W . However the negative decision isimpossible by the density (see the proof of Lemma 24.2). We conclude that Π ↾ γ ∈ W ; hence, γ > γ . Option (I) for π = Π ↾ γ clearly fails, and (II) failseither because the set D Π ξk ( c ) ∩ PPP <γ is pre-dense in PPP and γ > γ . Therefore Π ↾ γ belongs to W via (III), that is, γ = δ + 1 and Π <δ = S cw α<δ Π α ❁ c ξk Π δ .Then Π <δ ❁ c ξk Π ≥ δ = S cw δ ≤ α<ω Π δ by Lemma 16.3(iii).Now we make use of Theorem 18.2(ii) with π = Π <δ and ϙ = Π ≥ δ ; note that π ∪ cw ϙ = Π . It follows that x = c [ G ] / ∈ S Q ∈ Π ≥ δ ( ξ,k ) [ Q ], which clearly contradictsto the assumption (3). Corollary 32.2. Assume that k < ω and G ⊆ PPP is PPP -generic over L . Then W k = {h ξ, x ξk [ G ] i : ξ < ω } ⊆ ω × ω is a set of definability class Π HC k +2 in L [ G ] and in any transitive model M | = ZFC satisfying L ⊆ M ⊆ L [ G ] and { x ξk [ G ] : ξ < ω } ⊆ M . Proof. By the theorem, it is true in L [ G ] that h ξ, x i ∈ W k iff ∀ α < ω ∃ T ∈ Π α ( ξ, k ) (cid:0) α ( ξ, k ) ≤ α = ⇒ x ∈ [ T ] (cid:1) , which can be re-written as ∀ α < ω ∀ µ < ω ∀ X ∃ T ∈ X (cid:0) µ = α ( ξ, k ) ∧ X = Π α ( ξ, k ) ∧ µ ≤ α = ⇒ x ∈ [ T ] (cid:1) . Here the equality µ = α ( ξ, k ) (with a fixed k ) is ∆ HC k +2 by Corollary 24.3, and sois the equality X = Π α ( ξ, k ) by Corollary 24.3. It follows that the whole relationis Π HC k +2 , since the quantifier ∃ T ∈ X is bounded.The next corollary is the first cornerstone in the proof of Theorem 1.1. Corollary 32.3 (= (I), (II) of Theorem 1.1) . Assume that m < ω and a set G ⊆ PPP is PPP -generic over L . Then ω ω ∩ L [ G ↾↾ Theorem 33.1. Assume that m < ω and a set G ⊆ PPP is PPP -generic over L .Then L [ G ↾↾ Suppose that this is not the case. Then there is a Π m +1 formula ϕ ( r, x )with r ∈ ω ω ∩ L [ G ↾↾ 1) = Π α ( ξ, k ).We claim that π = h π α i α<λ is a multisequence, that is, if α < β < λ then π α ❁ π β . This amounts to the folowing: if h η, k i ∈ | π α | then π α ( η, k ) ❁ π β ( η, k ).Note that π α ( η, k ) = Π α ( η, k ) in case h η, k i / ∈ R .52hus it remains to check that π α ( η, m − ❁ π β ( η, m − 1) whenever α <β < λ , h η, m − i = h ( ξ, k ) ∈ R ∩ | π α | , and h ξ, k i ∈ D . If now α < γ then R ∩ | π α | = ∅ by the choice of ν , so it remains to consider the casewhen γ ≤ α . Then the pairs h ξ, k i , h η, m − i belong to | π α | by construction,and we have π α ( η, m − 1) = Π α ( ξ, k ) and π β ( η, m − 1) = Π β ( ξ, k ). Therefore π α ( ξ, m ) ❁ π β ( ξ, m ) since Π is a multisequence, and we are done.Now we claim that the multisequence π = h π α i α<λ satisfies (A), (B), (C).Indeed as the difference between each π α and the corresponding Π α is fullylocated in the domain R = {h ξ, m − i : ν ≤ ξ < ν } , we have π ↾↾ 34 Non-wellorderability We finally prove that the reals are not wellorderable by a (lightface) analyticallydefinable relation in PPP -generic extensions, that is, (IV) of Theorem 1.1. Theorem 34.1. Assume that m < ω and a set G ⊆ PPP is PPP -generic over L .Then it is true in L [ G ] that the reals are not wellorderable by an analyticallydefinable relation. Proof. Suppose to the contrary that, in L [ G ], a Σ m +2 relation ≪ strictlywellorders ω ω , m ≥ 1. Let ψ ( x, y ) be a parameter-free Σ m +2 formula, whichdefines ≪ , so that x ≪ y iff ψ ( x, y ) in L [ G ]. Note that ≪ is essentially a ∆ m +2 relation, since x ≪ y ⇐⇒ y x ∧ x = y .Of all nonconstructible reals x ξm [ G ], ξ < ω , there is a ≪ -least one. Wesuppose that x m [ G ] is such. (If it is some x ξ m [ G ], ξ = 0, then the argumentssuitably change in obvious way.) That is, x m [ G ] ≪ x ξm [ G ] whenever ξ > p ∈ G that PPP -forces, over L , that(1) ≪ (that is, the relation defined by ψ ) is a wellordering of ω ω , and(2) ∀ ξ > x m [ G ] ≪ x ξm [ G ]).Therefore, if ξ > p PPP -forces . x m [ G ] ≪ . x ξm [ G ]) over L . (We make useof the real names . x ξk introduced by 13.6, 13.7.)By Lemma 30.3, we can assume that p forc (1) ∧ (2), so that in fact we have p forc Π ↾ γ (1) ∧ (2), for some γ < ω . Then p ∈ MT ( Π ↾ γ ) = MT ( Π <γ ),53here Π <γ = S cw ξ<γ Π ξ is a small multiforcing. Let δ < ω be the least ordinalsatisfying | Π <γ | ⊆ δ × ω . It follows then that | p | ⊆ δ × ω .By Lemma 30.4, there is an ordinal γ , γ < γ < ω , such that if ξ < ω then p forc Π ↾ γ ( . x ξm ≪ . x m ) − . We can enlarge γ , if necessary, using Lemma 24.2,to make sure that h , m i ∈ | Π ↾ γ | , that is, h , m i ∈ | Π α ′ | for some α ′ < γ .If ξ < ω then let h ξ ∈ PERM m be the permutation of h , m i and h ξ, m i ,such that | h ξ | = {h , m i , h ξ, m i} , h ξ (0 , m ) = h ξ, m i , h ξ ( ξ, m ) = h , m i , h ξ ( η, n ) = h η, n i for any pair h η, n i different from both h , m i and h ξ, m i .The remainder of the proof is very similar to the proof of Theorem 33.1. Let U consist of all multisequences of the form π ↾↾ ≥ m , where(A) π ∈ sMF [ Π ↾↾ 35 Proof of the main theorem Proof (Theorem 1.1) . We consider a PPP -generic extension L [ G ] of L and presentit in the form L [ G ] = L [ X ] as in Section 31, where X = h x ξk i h ξ,k i∈ ω × ω , andeach x ξk = x ξk [ G ] is a real in 2 ω ∩ L [ G ]. 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Jacques Hadamard d’uniformisation des en-sembles. Mathematica, Cluj , 4:54–66, 1930.[24] Yiannis N. Moschovakis. Descriptive set theory. Studies in Logic and the Foun-dations of Mathematics, Vol. 100. Amsterdam, New York, Oxford: North-HollandPublishing Company, 637 p., 1980. Luzin grants the uniformization problem to Hadamard with a reference to Hadamard’sobservations related to the axiom of choice in the famous Cinq Lettres [5]. ndex a.d. (trees), 6antichain, 7avoids, 24base, base ( P ), 7canonical homeomorphism h ST , 7 h T , 7Cohen forcing, P coh , 7componentwise union π ∪ cw ϙ , 10 S cw π = S cw α<λ π α , 10CTM, countable transitive model, 39decision m -decides, 31negative, 31positive, 31dense, 8, 10diamond sequences π ⌈ µ, m ⌉ , ⌈ µ, m ⌉ + , 34 m ⌈ µ ⌉ , p ⌈ µ ⌉ , δ ⌈ µ ⌉ , 33 π ⌈ µ ⌉ , D ⌈ µ ⌉ , z ⌈ µ ⌉ , 32directly forces, 22domain-continuous, 29finite splitting system, FSS , 8empty system, Λ , 8extension proper, ψ ≺ ϕ , 8extension, ψ ϕ , 8height, hgt ( ϕ ), 8over P , FSS ( P ), 8tree occurs in, 8forcing forc , 39 wforc , 39formula L -formula, 38 L Σ n , L Π n , L ( Σ + Π ) n , 38 ϕ − , 38generic P -generic real, 8key elements C , 34 C ′ , 37key maps µ m , 34 µ m ( γ ), 34 PPP , 36 PPP <γ , 36 PPP ≥ γ , 36 Π , 36 Π , 34 Π α , 34 Π <γ , 36 Π ≥ γ , 36layer restriction sMF ↾↾ PPP <γ , 36 PPP ≥ γ , 36 Π , 36 Π , 34 Π α , 34 Π <γ , 36 Π ≥ γ , 36refinementgeneric, 17refinement, π ❁ ϙ , 14refinement, π ❁ c ϙ , 23refinement, π ❁ D ϙ , 15refinement, π ❁ c ξk ϙ , 24refinement, π ❁❁ M ϙ , 28regular, 9small, 9special, 9multisequence, 29 key multisequence, 35multisystem, 112wise disjoint, 11extension, ψ ϕ , 11 h ϕ ( ξ, k, m ), 11 MS ( π ), 11 π -multisystem, 11 T ϕ ξk,m ( s ), 11multitree, 9 MT , 9 MT ( π ), 10 MT ( π ), 29disjoint union, p ∪ q , 19empty multitree, Λ , 9 π -multitree, 10restriction, p ↾ X , 20somewhere almost disjoint, s.a.d.,9[ p ], 9 T p ξk , 9occurs, 8open, 8, 10perfect-tree forcing, PTF , 7base, base ( P ), 7refinement, P ❁ Q , 13refinement, P ❁ D Q , 14regular, 7small, 7special, 7permutationPERM, 44action, 44 P -generic, 8pre-dense, 8, 10principal generic reals, x ξk [ G ], 10real P -generic, 8real name, 2159valuation, c [ G ], 21 π -real name, 29true, 29 π -real name, 21true, 21small, 21true π -real name, 21 . x ξk , 22realsprincipal generic reals, x ξk [ G ], 10refinementgeneric, 17locks, 14refinement, P ❁ D Q , 14refinement, π ❁ c ϙ , 23refinement, π ❁ D ϙ , 15refinement, π ❁ c ξk ϙ , 24refinement, π ❁❁ M ϙ , 28regular, 7, 9s.a.d., somewhere a.d., 9set C , 34 C ′ , 37set of multitreesdense, 10open, 10pre-dense, 10set of treesdense, 8open, 8pre-dense, 8sets Υ m ( p ), 33small, 7, 9s.a.d., somewhere a.d., 9somewhere almost disjoint, s.a.d., 9special, 7, 9splitting, 6stem, stem ( T ), 6strings, 6empty string, Λ, 6 γ -tail, 42tree, 6 tree ( X ), 6almost disjoint, 6a.d. trees, 6 Q Φ ξk,m , 17 Q Φ ξk,m ( s ), 17 T p ξk , 9universal formula, un m ( p, x ), 332 n , 6 L , 32 base ( P ), 7 C , 34 C ′ , 37 c [ G ], 21 | c | , 21 D ⌈ µ ⌉ , 32 D | u | q , 15 forc , 39 FSS , 8 FSS ( P ), 8HC , 32 h ϕ ( ξ, k, m ), 11 h ST , 7 h T , 7 K c n , 21 K c ni , 21Λ, 6 dom ( π ), 29 L -formula, 38 lh ( s ), 6 L Σ n , L Π n , L ( Σ + Π ) n , 38 sMF , 29 sMF , 29 sMF ↾↾ PPP <γ , 36 PPP ≥ γ , 36PERM, 44 ϕ − , 38 Π , 36 Π , 34 Π α , 34 π ↾↾