aa r X i v : . [ m a t h . L O ] S e p Forcing the Π -Reduction Property Stefan Hoffelner ∗ Abstract
We generically construct a model in which the Π -reduction prop-erty is true. The reduction property was introduced by K. Kuratowski in 1936 and isone of the three regularity properties of subsets of the reals which wereextensively studied by descriptive set theorists, along with the separationand the uniformization property.
Definition 1.1.
We say that a universe has the Π n -reduction property ifevery pair A , A of Π n -subsets of the reals can be reduced by a pair of Π n -sets D , D , which means that D ⊂ A , D ⊂ A , D ∩ D = ∅ and D ∪ D = A ∪ A . Classical work of M. Kondo, building on ideas of Novikov, shows thatthe Π -uniformization (and equivalently the Σ -uniformization) property istrue. As the uniformization property for some pointclass always implies thereduction property for the same pointclass, Kondo’s uniformization theoremimplies, modulo a well-known theorem, that the Π -reduction property isalso true and the Π -reduction property is wrong. This is as much as ZFC can prove about the reduction property.In Gödel’s constructible universe L , the Σ -uniformization-property (infact the Σ n -uniformization property for n ≥ ) is true, hence the Π -reductionproperty is wrong. On the other hand, due to Y. Moschovakis celebrated re-sult, determinacy assumptions for projective pointclasses outright imply the Π n +1 -reduction property. In particular, ∆ -determinacy implies the Π -reduction property, the assumption, however, carries large cardinal strength. ∗ WWU Münster. Research funded by the Deutsche Forschungsgemeinschaft (DFGGerman Research Foundation) under Germanys Excellence Strategy EXC 2044 390685587,Mathematics Münster: Dynamics-Geometry-Structure.
1t is known that ∆ -determinacy implies the existence of an inner modelwith a Woodin cardinal.For a long time, it remained open however to force models with propertiesimplied by local forms of projective determinacy. Indeed, forcing even theweakest instance of these properties, the Σ -separation property, remainedan open problem up until it was solved recently in [3]. This work continuesthis line of research. The goal is to prove Theorem 1.2.
There is a generic extension of L in which the Π -reductionproperty is true. A similar construction yields a generic extension of L inwhich the Π -property is true. There are some similarities to [3], in particular both proofs rely on thesame ground model W , which is a generic extension of L , use the samecoding method which relies on a suitably chosen ω -sequence of ω -Suslintrees and take advantage of the fact that in this particular setting, decidingto not code and leave an empty space instead, can add valuable information.However the actual construction uses several new ideas whose presentationis the goal of this paper. The forcings which we will use in the construction are all well-known. Wenevertheless briefly introduce them and their main properties.
Definition 2.1.
For a stationary S ⊂ ω the club-shooting forcing withfinite conditions for S , denoted by P S consists of conditions p which arefinite partial functions from ω to S and for which there exists a normalfunction f : ω → ω such that p ⊂ f . P S is ordered by end-extension. The club shooting forcing P S is the paradigmatic example for an S - properforcing , where we say that P is S -proper if and only if for every condition p ∈ P S , every sufficiently large θ and every countable M ≺ H ( θ ) such that M ∩ ω ∈ S and p, P S ∈ M , there is a q < p which is ( M, P S ) -generic. Lemma 2.2.
The club-shooting forcing P S generically adds a club throughthe stationary set S ⊂ ω , while being S -proper and hence ω -preserving.Moreover stationary subsets T of S remain stationary in the generic exten-sion. We will choose a family of S β ’s so that we can shoot an arbitrary patternof clubs through its elements such that this pattern can be read off fromthe stationarity of the S β ’s in the generic extension. For that it is crucialto recall that S -proper posets can be iterated with countable support and2lways yield an S -proper forcing again. This is proved exactly as in thewell-known case for plain proper forcings (see [1], 3.19. for a proof). Fact 2.3.
Let ( P α , ˙ Q α ) be a countable support iteration, assume also that atevery stage α , (cid:13) α ˙ Q α is S -proper. The the iteration is an S -proper notionof forcing again. The following coding method has been used several times already.
Lemma 2.4.
Let r ∈ ω be arbitrary, and let P be a countable supportiteration ( P α , ˙ Q α ) of length ω , inductively defined via ˙ Q α := P ω \ S · α if r ( α ) = 1 and ˙ Q α := P ω \ S (2 · α )+1 if r ( α ) = 0 . Then in the resulting genericextension V P , we have that ∀ α < ω : r ( α ) = 1 if and only if S · α isnonstationary, and r α = 0 iff S (2 · α )+1 is nonstationary.Proof. Assume first that r ( α ) = 1 in V P . Then by definition of the iter-ation we must have shot a club through the complement of S α , thus it isnonstationary in V P .On the other hand, if S · α is nonstationary in V P , then as for β = 2 · α ,every forcing of the form P S β is S · α -proper, we can iterate with countablesupport and preserve S · α -properness, thus the stationarity of S · α . So if S · α is nonstationary in V P , we must have used P S · α in the iteration, so r ( α ) = 1 .The second forcing we use is the almost disjoint coding forcing due to R.Jensen and R. Solovay. We will identify subsets of ω with their characteristicfunction and will use the word reals for elements of ω and subsets of ω respectively. Let F = { f α α < ℵ } be a family of almost disjoint subsetsof ω , i.e. a family such that if r, s ∈ F then r ∩ s is finite. Let X ⊂ κ for κ ≤ ℵ be a set of ordinals. Then there is a ccc forcing, the almost disjointcoding A F ( X ) which adds a new real x which codes X relative to the family F in the following way α ∈ X if and only if x ∩ f α is finite. Definition 2.5.
The almost disjoint coding A F ( X ) relative to an almostdisjoint family F consists of conditions ( r, R ) ∈ ω <ω × F <ω and ( s, S ) < ( r, R ) holds if and only if1. r ⊂ s and R ⊂ S .2. If α ∈ X and f α ∈ R then r ∩ f α = s ∩ f α . For the rest of this paper we let F ∈ L be the definable almost disjointfamily of reals one obtains when recursively adding the < L -least real to thefamily which is almost disjoint from all the previously picked reals. Whenever3e use almost disjoint coding forcing, we assume that we code relative tothis fixed almost disjoint family F .The last two forcings we briefly discuss are Jech’s forcing for adding aSuslin tree with countable conditions and, given a Suslin tree T , the as-sociated forcing which adds a cofinal branch through T . Recall that a settheoretic tree ( T, < ) is a Suslin tree if it is a normal tree of height ω and hasno uncountable antichain. As a result, forcing with a Suslin tree S , whereconditions are just nodes in S , and which we always denote with S again,is a ccc forcing of size ℵ . Jech’s forcing to generically add a Suslin tree isdefined as follows. Definition 2.6.
Let P J be the forcing whose conditions are countable, nor-mal trees ordered by end-extension, i.e. T < T if and only if ∃ α < height ( T ) T = { t ↾ α : t ∈ T } It is wellknown that P J is σ -closed and adds a Suslin tree. In fact moreis true, the generically added tree T has the additional property that forany Suslin tree S in the ground model S × T will be a Suslin tree in V [ G ] .This can be used to obtain a robust coding method (see also [2] for moreapplications) Lemma 2.7.
Let V be a universe and let S ∈ V be a Suslin tree. If P J isJech’s forcing for adding a Suslin tree and if T is the generic tree then V [ T ] | = T × S is Suslin.Proof. Let ˙ T be the P J -name for the generic Suslin tree. We claim that P J ∗ ˙ T has a dense subset which is σ -closed. As σ -closed forcings will alwayspreserve ground model Suslin trees, this is sufficient. To see why the claimis true consider the following set: { ( p, ˇ q ) : p ∈ P J ∧ height ( p ) = α + 1 ∧ ˇ q is a node of p of level α } . It is easy to check that this set is dense and σ -closed in P J ∗ ˙ T .A similar observation shows that a we can add an ω -sequence of suchSuslin trees with a countably supported iteration. Lemma 2.8.
Let S be a Suslin tree in V and let P be a countably supportedproduct of length ω of forcings P J . Then in the generic extension V [ G ] thereis an ω -sequence of Suslin trees ~T = ( T α : α ∈ ω ) such that for any finite e ⊂ ω the tree S × Q i ∈ e T i will be a Suslin tree in V [ ~T ] . These sequences of Suslin trees will be used for coding in our proof andget a name.
Definition 2.9.
Let ~T = ( T α : α < κ ) be a sequence of Suslin trees. We saythat the sequence is an independent family of Suslin trees if for every finiteset e = { e , e , ..., e n } ⊂ κ the product T e × T e × · · · × T e n is a Suslin treeagain. .2 The ground model W of the iteration We have to first create a suitable ground model W over which the actualiteration will take place. W will be a generic extension of L , satisfying CH and, as stated already earlier, has the property that it contains two ω -sequence ~S = ~S ∪ ~S of independent Suslin trees.To achieve this we start with Gödels constructible universe L as ourground model. Next we fix an appropriate sequence of stationary subsets of ω . Recall that ♦ holds in our ground model L , i.e. there is a Σ -definablesequence ( a α : α < ω ) of countable subsets of ω such that any set A ⊂ ω is guessed stationarily often by the a α ’s, i.e. { α < ω : a α = A ∩ α } is astationary subset of ω . The ♦ -sequence can be used to produce an easilydefinable sequence of stationary subsets: we list the reals in L in an ω sequence ( r α : α < ω ) and define for every β < ω a stationary set in thefollowing way: R β := { α < ω : a α = r β } . These stationary sets will be used to define our separating sets which willwitness the Σ -separation property. We definably split the sequence ~R = ( R β : β < ω ) into two, we let ~R := Even ( ~R ) = ( R ω · α +2 n : α < ω , n ∈ ω ) be the sequence of the even entries in ~R and ~R := Odd ( ~R ) = ( R ω · α +2 n +1 : α < ω , n ∈ ω ) be the sequence of the odd entries.Frist we add ℵ -many ℵ -sized blocks of Suslin trees with a countablysupported product of Jech’s Forcing C ( ω ) . We let R ,α := Q n ∈ ω C ( ω ) , andlet R = Q α<ω R ,α . This is a σ -closed, hence proper notion of forcing. Wedenote the generic filter of R with ~S = ( S ω · α + n : α < ω , n ∈ ω ) and notethat whenever I ⊂ ω is a set of indices then for every j / ∈ I , the Suslin tree S j will remain a Suslin tree in the universe L [ ~S ][ g ] , where g ⊂ Q i ∈ I S i denotesthe generic filter for the forcing with the finitely supported product of thetrees S i , i ∈ I (see [2] for a proof of this fact). We fix a definable bijectionbetween [ ω ] ω and ω and identify the trees in ( S ω · α + n : α < ω, n ∈ ω ) withtheir images under this bijection, so the trees will always be subsets of ω from now on. Further we partition the ω · ω -sequence of Suslin trees intothe odd and even members and let ~S := Even ( ~S ) ~S := Odd ( ~S ) In a second step we code the even and the odd trees into the accordingsequence of the even and odd definable L -stationary subsets ~R and ~R we produced earlier, using club shooting forcing. We will just describe themethod for one sequence ( S ω · α + n : α < ω , n ∈ ω ) , to not write everythingtwice for the even and the odd sequence.The forcing used in the second step will be denoted by R . Fix α < ω and n ∈ ω and consider the Suslin tree S ω · α + n . We let R ,α,n be the countablesupport product which codes S ω · α + n into the ω · α + n -th ω -block of the ω · ω -sequence of the R β ’s. So R ,α,β = Y γ ∈ S ω · α + n P ω \ R ω · ( ω · α + n )+2 · γ × Y γ / ∈ S ω · α + n P ω \ R ω · ( ω · α + n )+2 · γ +1 If we let R be some stationary subset of ω which is disjoint from all the R α ’s, e.g. R = { α < ω : a α = { ω }} , then it is obvious that for every α < ω and every n ∈ ω , R ,α,β is an R -proper forcing which additionally is ω -distributive. Then we let R be the countably supported iteration, R := ⋆ α<ω ,n ∈ ω R ,α,n which is again R -proper and ω -distributive. This way we can turn the gener-ically added sequence of Suslin trees ~S into a definable sequence of Suslintrees. Now we apply this method to code up both, the even and the oddsequence ~S and ~S into patterns of non-stationary elements of ~R and ~R .Let R now be this coding forcing, so R consists of two copies, one for ~S and one for ~S , of the forcing we denoted with R above. If H denotes thegeneric filter for R over L [ ~S ] then we obtain that in L [ ~S ][ H ] , every element S i ∈ ~S is Σ -definable over H ( ω ) with parameter ω using the definablesequence of the even L -stationary subsets ~R = ( R α : α < ω ) . ( ∗ ) S ∈ ~S if and only if there is an α < ω and a n < ω such that ∀ γ ∈ ω ( γ ∈ S iff R ω · ( ω · α + n )+2 γ is nonstationary and γ / ∈ S iff R ω · ( ω · α + n )+2 γ +1 is nonstationary).Likewise elements of ~S do have a Σ ( ω ) -definition using the elements of ~R .Note that this formula is indeed equivalent to a Σ ( ω ) formula, as alreadytransitive, ℵ -sized models of ZF − are sufficient to witness the truth of thestatement above.The Suslin trees from ~S will be used later to code information via de-stroying certain patterns of elements of ~S . Thus it is crucial that elementsfrom ~S are preserved in the process of making them definable using R . Thisis indeed the case as is shown in [2], Lemma 35.6 emma 2.10. The forcing R , defined above preserves Suslin trees. Let us set W := L [ R ∗ R ] which will serve as our ground model for asecond iteration of length ω . Note that W satisfies CH . We proceed with an informal discussion of the main ideas of the proof. Wefocus on reducing one fixed, arbitrary pair A m and A k of Π -sets. Thearguments will be uniform, so that reducing every pair of Π -sets will followimmediately.The ansatz is to use the two definable sequences of Suslin trees ~S and ~S for coding and a bookkeeping function F which lists all possible reals inour iteration. We use a finite support iteration over W of length ω . Atstages β , where F ( β ) is (the name of) a real number x , we decide whetherto code x into the ~S -sequence or the ~S -sequence. Coding here means thatwe write the characteristic function of x into an ω -block of elements of ~S ina way such that the statement “ x is coded into ~S ” is a Σ ( x ) -statement.Our goal is that for every x ∈ A m ∩ A k , either x is coded into ~S or x iscoded into ~S . The set of elements of A m which are not coded somewhereinto ~S , which will be equivalent to the set of elements which are codedsomewhere into ~S , shall form our reducing set D m,k . On the other hand,reals in A k which are not coded into ~S form the set D m,k which shalleventually reduce A k . As being coded is Σ , not being coded is Π .This set-up has the following difficulties one has to overcome: the eval-uation of Π -sets changes as we use coding forcings. In particular it couldwell be that at some stage β of the iteration we decide to code the real x into ~S , which is equivalent to put x into D m,k . Now it could happen thatthis coding forcing actually puts x out of A m , while x remains in A k . Theconsequence of this is that x witnesses that D m,k and D m,k do not reduce A m and A k as x / ∈ D m,k ∪ D m,k , yet x ∈ A m ∪ A k . This pathological situationcould also arise later in the iteration. Indeed, if we decide at some stage β ,that we put x into D m,k via coding x somewhere into the ~S -sequence andproceed with our iteration, then it could happen that at some later stagethe pathological situation described above is happening. If we attempt torepair that bad situation via coding x at stage β somewhere into ~S instead,it is not guaranteed that a similar pathological situation will happen again,maybe at some other real y .It seems that in order to overcome these difficulties, for every real x whichwe have to decide where to put it, we must be able to look ahead into thefuture to determine whether putting x into D m,k or D m,k will cause some7athologies. But even if we consider all possible future runs of our iteration,it is not guaranteed at all that there will one iteration of length ω , whichwill place every real x into either D m,k or D m,k in such a way that A m ∩ D m,k and A k ∩ D m,k will indeed reduce ( A m , A k ) .The main idea of the proof relies on the following observation which isnot completely accurate for our situation but nevertheless should give someintuition. As our iteration only consists of coding forcing which place realseither in ~S or ~S we will only talk about forcings of this particular formwhich we call allowable. If our bookkeeping function hands us a real x suchthat there is no further allowable forcing P such that (cid:13) P x / ∈ A m , thensurely it is safe to code x into the ~S sequence while not running into apathological situation for x . This can be done for all such x . Thus we canshrink the set of allowable forcings, to the allowable forcings which obey therule: whenever we hit a real x which can not be kicked out of A m with anadditional allowable forcing, then code x into ~S .But this reasoning can be iterated: as we only want to use the shrunkversion of allowable forcings, we can now consider the reals x which can notbe kicked out of A m with a forcing belonging to the shrunk class and whichcodes x into ~S . If x is such, then we code x into ~S , which again can beapplied to all reals along our iteration which will yield an even smaller classof forcings and so on. This way we can shrink our class of allowable forcingsinfinitely often, yielding better and better approximations to the right classof forcings we eventually want to use. This shrinking process will eventuallystabilize at a nonempty set of ℵ -sized, ccc posets, called ∞ -allowable, andthese forcings will be the ones which we want to use for our iteration.One crucial property of ∞ -allowable forcings is that whenever we do havean iteration which obeys the rule: if x is such that for no ∞ -allowable forcing P , (cid:13) P x / ∈ A m , then this will yield an ∞ -allowable forcing again, as otherwisethere would be a new and smaller class of ( ∞ + 1) -allowable forcings whichcontradicts the fact that ∞ -allowable forcings are stable under the shrinkingprocess. ∞ -allowable Forcings We continue with the construction of the appropriate notions of forcing whichwe want to use in our proof. The goal is to iteratively shrink the set ofnotions of forcing we want to use until we reach a fixed point. All forcingswill belong to a certain class, which we call allowable. These are just forcingswhich iteratively code reals into ω -blocks of ~S or ~S in such a way that theblocks, where the coding is happening do not overlap. We first want topresent the coding method, which we use to code a real x up, using thedefinable sequence of Suslin trees, and subsequently introduce the notionallowable.Our ground model shall be W . Let x be a real, let m, k ∈ ω and let8 < ω be an arbitrary ordinal. The forcing P ( x,m,k ) , ,γ , which codes the real w , which in turn codes the triple ( x, m, k ) into ~S at γ is defined as a twostep iteration P ( x,m,k ) , ,γ := Q ∗ ˙ Q , where Q is itself a finitely supported ω -length iteration of the factors ( S n + i : n ∈ ω, i ∈ of Suslin tree forcings,where S n + i for i = 0 if w ( n ) = 0 , and S n + i for i = 1 if w ( n ) = 1 . This way,we can read off w and hence ( x, m, k ) via looking at the block of ~S -treesstarting at γ . Let G be Q -generic, then in W [ G ] the following Σ ( ω , w ) -statement is true: ( ∗ ) n ∈ w if and only if S ω · γ +2 n +1 is not Suslin, and n / ∈ w if and only if S ω · γ +2 n is not Suslin.Indeed if n / ∈ w then we shot a branch through S ω · γ +2 n . If on the other hand S ω · γ +2 n is Suslin in W [ G ] then we must have forced with S ω · γ +2 n +1 as wealways use either S ω · γ +2 n +1 or S ω · γ +2 n and every other factor preserves thestatement “ S ω · γ +2 n is Suslin ” .We note that we can apply David’s trick in this situation. We code the ω -many clubs necessary to correctly compute for every n ∈ ω , S ωγ + n andthe ω -many branches through S ω · γ +2 n + i witnessing ( ∗ ) into just one subset X ⊂ ω . Then rewrite the information of X as a subset Y ⊂ ω using thefollowing line of reasoning. It is clear that any transitive, ℵ -sized model M of ZF − which contains X will be able to correctly decode out of X all theinformation. Consequentially, if we code the model ( M, ∈ ) which contains X as a set X M ⊂ ω , then for any uncountable β such that L β [ X M ] | = ZF − and X M ∈ L β [ X M ] : L β [ X M ] | = “The model decoded out of X M satisfies ( ∗ ) ”.In particular there will be an ℵ -sized ordinal β as above and we can fixa club C ⊂ ω and a sequence ( M α : α ∈ C ) of countable elementarysubmodels such that ∀ α ∈ C ( M α ≺ L β [ X M ] ∧ M α ∩ ω = α ) Now let the set Y ⊂ ω code the pair ( C, X M ) such that the odd entries of Y should code X M and if Y := E ( Y ) where the latter is the set of evenentries of Y and { c α : α < ω } is the enumeration of C then1. E ( Y ) ∩ ω codes a well-ordering of type c .2. E ( Y ) ∩ [ ω, c ) = ∅ .3. For all β , E ( Y ) ∩ [ c β , c β + ω ) codes a well-ordering of type c β +1 .4. For all β , E ( Y ) ∩ [ c β + ω, c β +1 ) = ∅ .We obtain 9 ∗∗ ) For any countable transitive model M of ZF − such that ω M = ( ω L ) M and Y ∩ ω M ∈ M , M can construct its version of the universe L [ Y ∩ ω ] ,and the latter will see that there is an ordinal ξ < ω and an ω -blockof Suslin trees S ω · ξ + n , computed with the help of the local ♦ -sequenceand the ω -sequence of clubs of ω M coded into Y ∩ ω M , such that forany m ∈ ω , S Mω · ξ +2 m is Suslin iff m ∈ w and S Mω · ξ +2 m +1 is Suslin iff m / ∈ w .Thus we have a local version of the property ( ∗ ) .In the next step ˙ Q , we use almost disjoint forcing A F ( Y ) relative to the < L -least almost disjoint family of reals F to code the set Y into one real r .This forcing is well-known, has the ccc and its definition only depends onthe subset of ω we code, thus the almost disjoint coding forcing A F ( Z ) willbe independent of the surrounding universe in which we define it, as long asit has the right ω and contains the set Z .We finally obtained a real r such that ( ∗∗∗ ) For any countable, transitive model M of ZF − such that ω M = ( ω L ) M and r ∈ M , M can construct its version of L [ r ] which in turn thinksthat there is an ordinal ξ < ω such that for any m ∈ ω , S Lξ +2 m isSuslin iff m ∈ w and S Lξ +2 m +1 is Suslin iff m / ∈ w .We say in this situation that the real w , which codes ( x, m, k ) is writteninto ~S , or that w is coded into ~S . Definition 3.1.
A finite support iteration P = ( P β : β < α ) is called allow-able if α ≤ ω and there exists a sequence of ordinals ( γ β ) β<α and a sequenceof P -names of reals ( ˙ x β ) β<α such that for every β < α , ˙ x β is a P β -name of areal and P ( β ) is P ˙ x β , ,γ β , where the latter is the coding forcing defined above. We add as a remark that the set of blocks used in the factors of anallowable forcing must be non-intersecting.We define next a derivative of the class of allowable forcings. Induc-tively we assume that for an ordinal α , we have already defined the notionof δ -allowable for every δ ≤ α and now aim to define the derivation of the α -allowable forcings which we call α + 1 -allowable. A δ ≤ ω -length iter-ation P is called α + 1 -allowable if it is recursively constructed following abookkeeping function F (which has three additional dimensions, i.e. F ( β ) isa quadruple (( F ( β )) , ( F ( β )) , ( F ( β )) , ( F ( β ) ) , where the first coordinate ( F ( β )) is the ususal bookkeeping part, ( F ( β )) ∈ ω , ( F ( β )) ∈ and ( F ( β )) is always a ( P β -)name for an allowable forcing) and two rules atevery stage β < ω of the iteration. We assume inductively that we alreadycreated the allowable forcing R ( β ) and our actual forcing iteration up to β , P β . We let b β be the set indices of trees used by the R ( β ) . We shall nowdefine the next forcing of our iteration P ( β ) . Using the bookkeeping F wesplit into two cases. 10a) We assume first that the first coordinate of F ( β ) , ( F ( β )) = ( ˙ x, m, k ) ,where ˙ x is the P β -name of a real and m, k are natural numbers. Fur-ther we assume that ˙ x G β = x , for G β a P β -generic filter over W and W [ G β ] | = x ∈ A m ∪ A k . We assume that in W [ G β ] , there is an allow-able forcing P such that R β × P is allowable (i.e. the set of trees usedin R β is disjoint from the set of trees used in P ) and there is an ordinal γ such that P × R β × P ( x,m,k ) , ,γ is allowable. Last, we demand thatthere is an ordinal ζ ≤ α such that W [ G β ] | = ∀ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( if R β × P × P ( x,m,k ) , ,γ (cid:13) ( Q is ζ -allowable)then ( (cid:13) P ( x,m,k ) , ,γ ∗ Q x ∈ A m )) If this is the case, the we just use the forcing P ( x,m,k ) , ,γ . Additionally,we let R β +1 := ( P × R β ) and b β +1 be the set of indices of trees used in R β +1 .Note that if G β +1 = G β ∗ G ( β ) and G ( β ) is a filter for P ( β ) , thenfor every forcing Q ∈ W [ G β +1 ] such that W [ G β +1 ] | = R β +1 (cid:13) Q is ζ -allowable, we have that W [ G β +1 ] | = Q (cid:13) x ∈ A m , by construction.Indeed if there would be such a Q and P ( x,m,k ) , ,γ ∗ Q (cid:13) x / ∈ A m then wecan follow up with the forcing R β × P and by upwards-absoluteness of Σ -statements, ( P ( x,m,k ) , ,γ ∗ Q ) ∗ ( R β × P ) (cid:13) x / ∈ A m . But ( P ( x,m,k ) , ,γ ∗ Q ) ∗ ( R β × P ) = ( P ( x,m,k ) , ,γ ∗ Q ) × ( R β × P ) = ( R β × P ) ∗ ( P ( x,m,k ) , ,γ ∗ Q ) ,which contradicts the assumption that there is no Q such that R β × P × P ( x,m,k ) , ,γ (cid:13) ( Q is ζ -allowable) and such that (cid:13) P ( x,m,k ) , ,γ ∗ Q x / ∈ A m .Likewise, if the above described situation is not true, nevertheless theabove reasoning is true for the dual situation, i.e. for the case where A m is replaced with A k and P ( x,m,k ) , ,γ is replaced with P ( x,m,k ) , ,γ thenwe proceed as described above with the obvious replacements.(b) Else code x into F ( β ) -th free (i.e. a block whose indices are not in b β ) block in ~S F ( β ) . We alter b β and R β in that situation according to F , i.e. we let R β +1 be just R β × F ( β ) .We proceed with a couple of easy observations: Lemma 3.2.
For any ordinal α , the notion α -allowable is definable over theuniverse W . Lemma 3.3. If P is α -allowable and α < β , then P is β -allowable. Thus thesequence of α -allowable forcings is decreasing with respect to the ⊂ -relation.Proof. Let α < β , let P be a β -allowable forcing and let F be the bookkeep-ing function which, together with the rules (a)+(b) using β defined above,determine P . We will show that there is a bookkeeping function F ′ ∈ W P can be seen as an α -allowable forcing determined by F ′ . Thefirst coordinate of F ′ should always coincide with the first coordinate of F ,i.e. ∀ γ (( F ( γ ) = F ′ ( γ ) ) . The second coordinate, which determines at everystage where in the ~S -sequence a coding is happening, is defined via simulat-ing the reasoning for a β -allowable forcing. This means that at every stage γ of the iteration, we pretend that we are working with β -allowable forcings,we do the reasoning described in (a) and (b) for β -allowable using F . If case(a) does apply, and P ( γ ) is some P ( x,m,k ) , ,δ while R β +1 is defined as well,then we simply let ( F ′ ( γ )) be δ and ( F ′ ( γ )) = 1 and ( F ′ ( γ )) = R β +1 / R β .That is, we let F ′ simulate the reasoning we would apply if P would be a β -allowable forcing using F , and the forget about β -allowable and just keepthe result of the reasoning. The new bookkeeping F ′ is definable from F ,and clearly P is α -allowable using F ′ . Lemma 3.4.
For any α , the set of α -allowable forcings is nonempty.Proof. By induction on α . If there are α -allowable forcings, then the rules(a) and (b) above, together with some bookkeeping F will create an α + 1 -allowable forcing. For limit ordinals α , this is clear as well.As a direct consequence of the last two observations we obtain that theremust be an ordinal α such that for every β > α , the set of α -allowableforcings must equal the set of β -allowable forcings. Indeed every allowableforcing is an ℵ -sized partial order, thus there are only set-many of them,and the decreasing sequence of α -allowable forcings must eventually stabilizeat a set which also must be non-empty. Definition 3.5.
Let α be the least ordinal such that for every β > α , theset of α -allowable forcings is equal to the set of β -allowable forcings. We saythat some forcing P is ∞ -allowable if and only if it is α -allowable. The set of ∞ -allowable forcings can also be described in the followingway. An ω -length iteration P = ( P α : α < ω ) is ∞ -allowable if it isrecursively constructed following a bookkeeping function F and two rules atevery stage β < ω of the iteration:(a) We assume that F ( β ) = ( ˙ x, m, k ) and ˙ x G β = x W [ G β ] | = x ∈ A m ∪ A k .Also we assume that the allowable forcing R β has already been defined,even though we did not use it in the iteration P β so far. We assumethat in W [ G β ] , there is an allowable forcing P and there is an ordinal γ such that P × R β × P ( x,m,k ) , ,γ is allowable. Last, we demand thatthere is an ordinal ζ such that W [ G β ] | = ∀ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( if R β × P × P ( x,m,k ) , ,γ (cid:13) ( Q is ζ -allowable)then ( (cid:13) P ( x,m,k ) , ,γ ∗ Q x ∈ A m ))
12f this is the case, the we just use the forcing P ( x,m,k ) , ,γ . Additionally,we let R β +1 := ( P × R β ) and b β +1 be the set of indices of trees used in R β +1 .If the above is wrong but there is a ζ , a P such that P × R β is allowableand contains P ( x,m,k ) , ,γ as a factor and W [ G β ] | = ∀ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( if R β × P × P ( x,m,k ) , ,γ (cid:13) ( Q is ζ -allowable)then ( (cid:13) P ( x,m,k ) , ,γ ∗ Q x ∈ A k )) then we force with P ( x,m,k ) , ,γ .(b) Otherwise, we pick the free γ and force with either P ( x,m,k ) , ,γ or P ( x,m,k ) , ,γ and let R β +1 be according to our bookkeeping F .For every ω -length iteration following some F and the rules above we cancompute the supremum of the α ’s which appear in the cases (a) of the defi-nition of the iteration. As there are only set many such iterations, there willbe an ordinal α such that we can replace item (a) in the definition with thestronger (a’) below and still end up with exactly the same set of forcings.(a’) We assume that F ( β ) = ( ˙ x, m, k ) and ˙ x G β = x W [ G β ] | = x ∈ A m ∪ A k .Also we assume that the allowable forcing R β has already been defined,even though we did not use it in the iteration P β so far. We assumethat in W [ G β ] , there is an allowable forcing P and there is an ordinal γ such that P × R β × P ( x,m,k ) , ,γ is allowable. Last, we demand thatthere is an ordinal ζ < α such that W [ G β ] | = ∀ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( if R β × P × P ( x,m,k ) , ,γ (cid:13) ( Q is ζ -allowable)then ( (cid:13) P ( x,m,k ) , ,γ ∗ Q x ∈ A m )) If the above is wrong but there is a ζ < α , a P such that P × R β isallowable and contains P ( x,m,k ) , ,γ as a factor and W [ G β ] | = ∀ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( if R β × P × P ( x,m,k ) , ,γ (cid:13) ( Q is ζ -allowable)then ( (cid:13) P ( x,m,k ) , ,γ ∗ Q x ∈ A k )) then we force with P ( x,m,k ) , ,γ .This α is exactly the ordinal where the notion of α -allowable starts tostabilize. 13 .3 Definition of the universe in which the Π reductionproperty holds The notion of ∞ -allowable will be used now to define the universe in whichthe Π -reduction property is true. We let W be our ground model and startan ω -length iteration of ∞ -allowable forcings using four rules and somebookkeeping F .1. We assume that we are at stage β < ω , the allowable forcing R β has been defined, F ( β ) = ( ˙ x, m, k ) and ˙ x G β = x and in W [ G β ] , x ∈ A m ∪ A k . If in W [ G β ] , there is an allowable P such that R β × P isallowable, R β × P contains P ( x,m,k ) , ,γ as a factor for some γ / ∈ b β andsuch that W [ G β ] | = ∀ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( if R β × P × P ( x,m,k ) , ,γ (cid:13) ( Q is ∞ -allowable)then ( (cid:13) P ( x,m,k ) , ,γ ∗ Q x ∈ A m )) then just force with P ( x,m,k ) , ,γ and set R β +1 := ( R β × P ) \ P ( x,m,k ) , ,γ .Note that this has as a direct consequence, that if we restrict ourselvesfrom now on to forcings Q ∈ W [ G β +1 ] such that R β +1 (cid:13) Q is ∞ -allowable, then x will remain an element of A m . In particular, thepathological situation that x / ∈ A m , x ∈ A k while x is coded into ~S isruled out for ( x, m, k ) .2. If there is no allowable P for A m , but there is an allowable P such that R β × P is allowable and contains P ( x,m,k ) , ,γ as a factor and such that W [ G β ] | = ∀ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( if R β × P (cid:13) ( Q is ∞ -allowable)then ( (cid:13) P ( x,m,k ) , ,γ ∗ Q x ∈ A k )) then just force with P ( x,m,k ) , ,γ and set R β +1 := ( R β × P ) \ P ( x,m,k ) , ,γ .3. If F ( β ) = ( x, m, k ) and W [ G β ] | = x ∈ A m ∩ A k and neither case 1 nor2 applies, then we obtain that for any P such that R β × P is allowableand which contains P ( x,m,k ) , ,γ as a factor, W [ G β ] | = ∃ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( R β × P (cid:13) ( Q is ∞ -allowable) and P ( x,m,k ) , ,γ ∗ Q (cid:13) x / ∈ A m ) In particular, if we let P := P ( x,m,k ) , ,γ for γ / ∈ b β , then there is a Q asabove which kicks x out of A m . We first use the forcing P ( x,m,k ) , ,γ ∗ Q and let H be the generic filter over W [ G β ] . Note that as R β × × P ( x,m,k ) , ,γ (cid:13) ( Q is ∞ -allowable ) , and by the definition of ∞ -allowable, Q will produce new R -forcings along its way. We thus let R β +1 be the R -forcing we obtain as we iterate P ( x,m,k ) , ,γ ∗ Q ∈ W [ G β ] .Likewise, as 2 fails as well, we infer that for any P such that R β × P isallowable and which contains P ( x,m,k ) , ,γ as a factor, W [ G β ] | = ∃ Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ]( R β × P (cid:13) ( Q is ∞ -allowable) and P ( x,m,k ) , ,γ ∗ Q (cid:13) x / ∈ A k ) So, in the above, if we let P := ( P ( x,m,k ) , ,γ ∗ Q ) × P ( x,m,k ) , ,γ ′ × ( R β +1 \ R β ) for some γ ′ / ∈ b β +1 , then there is a Q ∈ W [ G β ][ P ( x,m,k ) , ,γ ′ ] such thatin W [ G β ][ P ( x,m,k ) , ,γ ′ ] R β × ( P ( x,m,k ) , ,γ ∗ Q ) × P ( x,m,k ) , ,γ ′ × ( R β +1 \ R β ) (cid:13) Q is ∞ -allowableand such that P ( x,m,k ) , ,γ ′ ∗ Q (cid:13) x / ∈ A k . But R β × ( P ( x,m,k ) , ,γ ∗ Q ) × P ( x,m,k ) , ,γ ′ × ( R β +1 \ R β ) is equal to R β +1 × ( P ( x,m,k ) , ,γ ∗ Q ) × P ( x,m,k ) , ,γ ′ .We therefore apply over W [ G β ][ H ] the forcing P ( x,m,k ) , ,γ ′ ∗ Q andobtain that W [ G β ][ H ] | = R β +1 × P ( x,m,k ) , ,γ ′ (cid:13) Q is ∞ -allowable.Hence W [ G β ][ H ] | = R β +1 (cid:13) P ( x,m,k ) , ,γ ′ ∗ Q is ∞ -allowable.and W [ G β ][ H ] | = P ( x,m,k ) , ,γ ′ ∗ Q (cid:13) x / ∈ A k In summary, we obtained a forcing P ( x,m,k ) , ,γ ∗ Q ∗ P ( x,m,k ) , ,γ ′ ∗ Q ∈ W [ G β ] such that R β (cid:13) ( P ( x,m,k ) , ,γ ∗ Q ∗ P ( x,m,k ) , ,γ ′ ∗ Q ) is ∞ -allowableand such that after forcing with ( P ( x,m,k ) , ,γ ∗ Q ∗ P ( x,m,k ) , ,γ ′ ∗ Q ) , x / ∈ A m ∩ A k .So if case 3 applies, we will force with such a forcing ( P ( x,m,k ) , ,γ ∗ Q ∗ P ( x,m,k ) , ,γ ′ ∗ Q ) and kick x out of A m and A k , which of coursemeans that we do not have to worry about the triple ( x, m, k ) anymore.Finally we define the new R β +1 (we overwrite the R β +1 from above)according to what the iteration ( P ( x,m,k ) , ,γ ∗ Q ∗ P ( x,m,k ) , ,γ ′ ∗ Q ) tellsus. 15. If F ( β ) is not of the form ( x, m, k ) , then we deal with the codes whichwere added in case 3 and which created pathological situations. Notehere that in case 3 we applied a forcing P ( x,m,k ) , ,γ ∗ Q ∗ P ( x,m,k ) , ,γ ′ ∗ Q which is such that some allowable forcing thinks that P ( x,m,k ) , ,γ ∗ Q ∗ P ( x,m,k ) , ,γ ′ ∗ Q is ∞ -allowable. While running through Q and Q wewill only use the rules for ∞ -allowable forcings. In particular, therecould be stages in Q and Q where case (a) did not apply, hence weused for some ( y, m, k ) the forcing P ( y,m,k ) , ,γ , but later in the iteration y / ∈ A k will become true. In that situation we need to additionallyensure, with a ∞ -allowable forcing over W [ G β ] , that y / ∈ A m , whichcan be done as follows. Let ( y, m, k ) be as just described and let δ be the stage of the iteration, where F ( δ ) = ( y, m, k ) . As case (a) didnot apply, we know by the same reasoning as described in (3) that forany P such that R δ × P is allowable and which contains P ( y,m,k ) , ,γ asa factor, W [ G δ ] | = ∃ Q ( R δ × P (cid:13) ( Q is ∞ -allowable) and P ( y,m,k ) , ,γ ∗ Q (cid:13) y / ∈ A m ) In particular if in the above we set P := P ( y,m,k ) , ,γ × P [ δ,β ) , where P [ δ,β ) is the factor iteration we used when passing from W [ G δ ] to W [ G β ] , thenwe know that W [ G β ] | = ∃ Q ∈ W [ G β ]( R β × P ( y,m,k ) , ,γ (cid:13) Q is ∞ -allowable ∧ P ( x,m,k ) , ,γ ∗ Q (cid:13) y / ∈ A m We then use the forcing P ( y,m,k ) , ,γ ∗ Q for which W [ G β ] | = R β (cid:13) P ( y,m,k ) , ,γ ∗ Q is ∞ -allowableis true, thus W [ G β ] | = R β (cid:13) P ( y,m,k ) , ,γ ∗ Q is an ∞ -allowable which forces y / ∈ A m which is as desired. We further let R β +1 be according to the ∞ -allowable P ( y,m,k ) , ,γ ∗ Q over W [ G β ] .The just described reasoning can be iterated countably often to obtaina forcing Q such that W [ G β ] thinks that R β (cid:13) Q is ∞ -allowable andsuch that whenever G ( β ) is a Q -generic over W [ G β ] , W [ G β ][ G ( β )] := W [ G β +1 ] and the following is true W [ G β +1 ] | = ∀ ( y, m, k )(( y, m, k ) is coded into ~S ∧ y / ∈ A k then y / ∈ A m ) . Thus there are no pathological situations in W [ G β +1 ] .16his ends the definition of the iteration and we shall show that the resultinguniverse W [ G ω ] satisfies the Π -reduction property. For every pair ( m, k ) ∈ ω , we define D m,k := { x ∈ ω : ( x, m, k ) is coded into the ~S -sequence } and D m,k := { x ∈ ω : ( x, m, k ) is coded into the ~S -sequence } Our goal is to show that for every pair ( m, k ) the sets D m,k ∩ A m and D m,k ∩ A k reduce the pair of Π -sets A m and A k . Lemma 3.6. In W [ G ω ] , for every pair ( m, k ) , m, k ∈ ω and corresponding Π -sets A m and A k :1. D m,k ∩ A m and D m,k ∩ A k are disjoint.2. ( D m,k ∩ A m ) ∪ ( D m,k ∩ A k ) = A m ∪ A k .3. D m,k ∩ A m and D m,k ∩ A k are Π -definable.Proof. We prove 1 first. If x is an arbitrary real in A m ∩ A k there will bea stage β , such that F ( β ) = ( x, m, k ) . As x ∈ A m ∩ A k , we know that case1 or 2 must have applied. We argue for case 1 as case 2 is similar. In case1, P ( x,m,k ) , ,γ does code ( x, m, k ) into ~S at some free γ , while ensuring thatfor all future ∞ -allowable extensions, x will remain an element of A m . Thus x ∈ D m,k ∩ A m . The rules of the iteration ensure however that ( x, m, k ) willnever be coded into ~S , thus x / ∈ D m,k and D m,k ∩ A m and D m,k ∩ A k aredisjoint.To prove 2, let x be an arbitrary element of A m ∪ A k . Let β be the stageof the iteration where the triple ( x, m, k ) is considered first. As x ∈ A m ∪ A k ,either case 1 or case 2 were applied at stage γ . Assume first that it was case1. Then, as argued above, x ∈ D m,k ∩ A m . If at stage γ case 2 applied, then x ∈ D m,k ∩ A k and we are finished.To prove item 3, we claim that D m,k has uniformly the following Π -definition over W [ G ω ] : x ∈ D m,k ∩ A m ⇔ x ∈ A m ∧¬ ( ∃ r ∀ M ( r ∈ M ∧ ω M = ( ω L ) M ∧ M transitive → M | = L [ r ] | = ∃ α (( x, m, k ) can be read off from an ω -block of elements of ~S starting at α. )) Note that the right hand side is the conjunction of two Π -formulas, so Π as desired. To show the claim, it is sufficient to show that if x ∈ A m then x ∈ D m,k , i.e. ( x, m, k ) being coded into the ~S -sequence is equivalent to ( x, m, k ) is not coded into the ~S -sequence. But this follows again from17he way we defined our iteration. If x ∈ A m , then if β is some stage suchthat ( x, m, k ) is considered by the bookkeeping function, then we must havealways applied either case 1 or case 2 and the choice of either 1 or 2 isconstant throughout all of the iteration. Thus if x ∈ A m either x is codedinto the ~S -sequence or the ~S -sequence. Consequentially, if x ∈ A m , then ( x, m, k ) not being coded into the ~S -sequence is equivalent to ( x, m, k ) beingcoded into the ~S -sequence. References [1] M. Goldstern
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