aa r X i v : . [ m a t h . L O ] A ug DISJOINT BOREL FUNCTIONS
DAN HATHAWAY
Abstract.
For each a ∈ ω ω , we define a Baire class one function f a : ω ω → ω ω which encodes a in a certain sense. We show thatfor each Borel g : ω ω → ω ω , f a ∩ g = ∅ implies a ∈ ∆ ( c ) where c is any code for g . We generalize this theorem for g in largerpointclasses Γ. Specifically, if Γ = ∆ , then a ∈ L [ c ]. Also for all n ∈ ω , if Γ = ∆ n , then a ∈ M n ( c ). Introduction
Definition 1.1.
A challenge-response relation ( c.r.-relation ) is a triple h R − , R + , R i such that R ⊆ R − × R + . The set R − is the set of challenges ,and R + is the set of responses . When cRr , we say that r meets c . Definition 1.2.
A backwards generalized Galois-Tukey connection ( mor-phism ) from A = h A − , A + , A i to B = h B − , B + , B i is a pair h φ − , φ + i offunctions φ − : B − → A − and φ + : A + → B + such that( ∀ c ∈ B − )( ∀ r ∈ A + ) φ − ( c ) A r ⇒ c B φ + ( r ) . When there is a morphism from A to B , let us say that A is above B and B is below A . Definition 1.3.
The norm of a c.r.-relation R = h R − , R + , R i is ||R|| := min {| S | : S ⊆ R + and ( ∀ c ∈ R − )( ∃ r ∈ S ) c R r } . If there is a morphism from A to B , then ||A|| ≥ ||B|| . Challenge-response relations and morphisms between them were introduced byVojtas as a way to abstract features of the study of cardinal charcter-istics of the continuum. For more on c.r.-relations, see [2] and [6].Temporarily fix a pointclass Γ. Let F Γ be the set of functions from ω ω to ω ω in Γ. Let D be the binary relation of disjointness of functionsfrom ω ω to ω ω . That is, given two functions f, g : ω ω → ω ω , let f Dg : ⇔ f ∩ g = ∅ ⇔ ( ∀ x ∈ ω ω ) f ( x ) = g ( x ) . A portion of the results of this paper were proven during the September 2012Fields Institute Workshop on Forcing while the author was supported by the FieldsInstitute. Work was also done while under NSF grant DMS-0943832.
Let D Γ be the c.r.-relation D Γ := hF Γ , F Γ , D i . In this paper we will be interested in the c.r.-relation D Γ for variouspointclasses Γ.For example, we will be interested in computing ||D ∆ || , which is thesmallest size of a family of Borel functions from ω ω to ω ω such thateach Borel function from ω ω to ω ω is disjoint from some member of thefamily. We will show that ||D ∆ || = 2 ω by showing that D ∆ is abovea c.r.-relation whose norm is 2 ω . Specifically, we will show that D ∆ is above h ω ω, ω ω, ≤ ∆ i , where a ≤ ∆ b iff a ∈ ω ω is definable by a ∆ formula using b ∈ ω ω as a parameter. To define the φ − part of themorphism, for each a ∈ ω ω we will define a Baire class one funciton f a : ω ω → ω ω (and we will have φ − ( a ) = f a ). The φ + part of themorphism will simply map each function from ω ω to ω ω in Γ to anycode for that function. The fact that h φ − , φ + i is a morphism is thefollowing statement: for each a ∈ ω ω and Borel function g : ω ω → ω ω , f a ∩ g = ∅ ⇒ a ≤ ∆ any code for g. We will prove that there is a morphism from D ∆ to h ω ω, ω ω, ≤ ∆ i by proving a general theorem (Theorem 5.3) which provides a sufficientcondition for when there exists a morphism from an arbitrary D Γ toan arbitrary h ω ω, ω ω, ≺i , where ≺ is an ordering on ω ω . Just like thecase with D ∆ , we will use the functions f a for the φ − map, and the φ + map will be “take any code for”. Thus, if the appropriate relationshipholds between Γ and ≺ , then we will have that for each a ∈ ω ω andeach g : ω ω → ω ω in Γ, f a ∩ g = ∅ ⇒ a ≺ any code for g. We will get that there exists a morphism from D ∆ to h ω ω, ω ω, ≤ L i ,where a ≤ L b iff a ∈ L [ b ]. The analogous result for larger Γ uses largecardinals. We will have that as long as M ( b ) (the canonical innermodel containing 1 Woodin cardinal and containing b ∈ ω ω ) exists forall b ∈ ω ω , then there is a morphism from D ∆ to h ω ω, ω ω, ≤ M i , where a ≤ M b iff a ∈ M ( b ). Next, as long as M ( b ) exists for all b ∈ ω ω ,there is a morphism from D ∆ to h ω ω, ω ω, ≤ M i . The pattern continueslike this through the projective hierarchy.In this paper, we are considering functions from ω ω to ω ω in a point-class Γ. We could have instead considered functions in Γ from anarbitrary uncountable Polish space X to an arbitrary Polish space Y ,and our results would not change much. The appropriate encodingfunction f ′′ a : X → Y could be defined by first defining f ′ a : ω → ω ω ISJOINT BOREL FUNCTIONS 3 in a way similar to f a and then using an injection of ω X and asurjection of ω ω onto Y . We trust that the interested reader can workthrough the details without trouble.2. Related Results
Before considering D Γ for various Γ, we will consider related c.r.-relations. First, consider the everywhere domination ordering of func-tions from ω ω to ω . That is, given f, g : ω ω → ω , we write f ≤ g iff ( ∀ x ∈ ω ω ) f ( x ) ≤ g ( x ) . Given any pointclass Γ, let E Γ be the c.r.-relation whose challenges andresponses are Γ functions from ω ω to ω , and g meets f iff f ≤ g .Next, consider the pointwise eventual domination ordering of func-tions from ω ω to ω ω . That is, given f, g : ω ω → ω ω , we write f ≤ ∗ g iff ( ∀ x ∈ ω ω ) { n ∈ ω : f ( x )( n ) > g ( x )( n ) } is finite.Given any pointclass Γ, let R Γ be the c.r.-relation whose challengesand responses are Γ functions from ω ω to ω ω , and g meets f iff f ≤ ∗ g .It is not difficult to see that for any reasonably closed pointclass Γ,there is a morphism from E Γ to R Γ and there is a morphism from R Γ to D Γ . The relation E Γ for a fixed Γ is relatively high up in the hierarchyof c.r.-relations, as we will soon see.Given a sequence a ∈ ω ω , let [[ a ]] := { a ↾ l : l ∈ ω } . Given a tree T ⊆ <ω ω , let Exit( T ) be the (Baire class one) functionExit( T )( x ) := min { l : x ↾ l T } . The following result shows a way of constructing a morphism from E Γ to another relation in a way which does not depend on Γ: Theorem 2.1.
Fix a ∈ ω ω . If M is an ω -model ZF such that some g : ( ω ω ) M → ω in M satisfies ( ∀ x ∈ ( ω ω ) M ) Exit([[ a ]])( x ) ≤ g ( x ) , then a is ∆ definable in M using g as a predicate.Proof. Fix M and g satisfying the hypothesis of the theorem. Let B ⊆ <ω ω be the set { t ∈ <ω ω : g ( x ) ≥ | t | for all x ⊒ t in M } . Note that B is defined (in M ) by a Π formula that uses g as a pred-icate. That is, B is Π in g . We claim there is some l ∈ ω satisfying( ∀ l ′ ≥ l ) a ↾ l ′ B . If not, the poset of elements of B ordered byextension would be ill-founded, and therefore would be ill-founded in DAN HATHAWAY M , so there would exist x ∈ ( ω ω ) M satisfying ( ∃ ∞ l ′ ∈ ω ) g ( x ) ≥ l ′ ,which is impossible. Now, fix such an l .We claim that for each l ′ ≥ l , a ( l ′ ) is the unique n satisfying ( a ↾ l ′ ) ⌢ n B . Indeed, since Exit([[ a ]]) ≤ g , for each l ′ ≥ l we have( ∀ n ∈ ω ) a ( l ′ ) = n ⇒ ( a ↾ l ′ ) ⌢ n ∈ B. The other direction is given by the property we arranged l to have.Thus, we have the following definition (in M ) for a : a ( l ′ ) = ( a ( l ′ ) if l ′ < l,n if l ′ ≥ l and ( ∀ n ′ = n )( ∀ x ⊒ ( a ↾ l ′ ) ⌢ n ′ in M ) g ( x ) ≥ l ′ + 1 . Since h a ( l ′ ) : l ′ < l i can be coded by a single number, we have a Π definition (in M ) for a which uses g as a predicate. We also have a Σ variant: a ( l ′ ) = ( a ( l ′ ) if l ′ < l,n if l ′ ≥ l and ( ∃ x ⊒ ( a ↾ l ′ ) ⌢ n in M ) g ( x ) < l ′ + 1 . Thus, a is ∆ definable in M using g as a predicate. (cid:3) Let use write “
All ” to refer to the pointclass of all pointsets.
Corollary 2.2.
There is a morphism from E All to h ω ω, ( ω ω ) ω, ≤ ∆ i .Proof. Fix a ∈ ω ω . Let f a := Exit([[ a ]]). By the above theorem taking M = V , if g : ω ω → ω satisfies f a ≤ g , then a is ∆ definable using g as a predicate. (cid:3) Corollary 2.3.
There is a morphism from E ∆ to h ω ω, ω ω, ≤ ∆ i .Proof. Fix a ∈ ω ω . Let f a := Exit([[ a ]]). Let g : ω ω → ω be Borel andlet c be a code for g . If we can show that a is in every ω -model whichcontains c , we will have that a ≤ ∆ c . Let M be an arbitrary ω -modelwhich contains c . Letting ˜ g be the function in M coded by c , we havethat ˜ g = M ∩ g . Hence, in M we have f a ≤ ˜ g , so the theorem abovetells us that a ∈ M . (cid:3) Corollary 2.3 will be improved by our result that there is a mor-phism from D ∆ to h ω ω, ω ω, ≤ ∆ i . The generalizations of Corollary 2.3to larger pointclasses Γ are also improved by our main result (Theo-rem 5.3) about morphisms from D Γ to orderings h ω ω, ω ω, ≺i . On theother hand, we do not have an analogue of Corollary 2.2 with D All ;here we see a qualitative difference between E All and D All .Another difference between E All and D All is the ability to encode notjust an a ∈ ω ω but an A ⊆ ω ω : ISJOINT BOREL FUNCTIONS 5
Proposition 2.4.
Fix a set X . Fix A ⊆ X . There exists a function f A : ω X → ω such that whenever M is a transitive model of ZF with X ∈ M and M contains some g : ( ω X ) M → ω satisfying ( ∀ x ∈ ( ω X ) M ) f A ( x ) ≤ g ( x ) , then A ∈ M . Moreover, there is some t ∈ <ω X satisfying A = { z ∈ X : g ( x ) ≥ | t | + 1 for all x ⊒ t ⌢ z in M } . Proof.
It suffices to show the second claim. Let f A : ω X → ω be thefunction f A ( x ) := ( ∀ l ∈ ω ) x ( l ) A,l + 1 if x ( l ) ∈ A and ( ∀ l ′ < l ) x ( l ′ ) A. Define B := { t ∈ <ω X : g ( x ) ≥ | t | for all x ⊒ t in M } . We must find a t ∈ <ω X satisfying A = { z ∈ X : t ⌢ z ∈ B } , and we will be done. By the hypothesis on g and the definition of f A ,for each z ∈ X , z ∈ A implies h z i ∈ B . If conversely for each z ∈ X , h z i ∈ B implies z ∈ A , then we have A = { z ∈ X : h z i ∈ B } , and we are done by defining t := ∅ . If not, then fix some x ∈ X satisfying h x i ∈ B but x A .Again by the hypothesis on g and the definition of f A , for each z ∈ X , z ∈ A implies h x , z i ∈ B . Here it is important that x A . Again, ifthe converse holds that h x , z i ∈ B implies z ∈ A , then A = { z ∈ X : h x , z i ∈ B } , and we are done by defining t := h x i . If not, we may fix x ∈ X satisfying h x , x i ∈ B but x A . We may continue like this, but weclaim that the procedure terminates in a finite number of steps.Assume, towards a contradiction, that it does not terminate. Thesequence x := h x , x , ... i we have constructed has all its initial segments in B . However, x neednot be in M . We handle this situation as follows: let T be the set ofthose elements of B all of whose initial segments are also in B . The tree T is ill-founded because x is a path through it. Since being ill-foundedis absolute, T has some path x ′ in M . We now have ( ∀ l ∈ ω ) g ( x ′ ) ≥ l ,which is impossible. (cid:3) DAN HATHAWAY
We immediately have the following:
Corollary 2.5.
For each A ⊆ ω ω , there is a function f A : ω ω → ω such that whenever g : ω ω → ω is any function which satisfies f ≤ g ,then A is ∆ in a predicate for g . Thus, there is a morphism from E All to hP ( ω ω ) , P ( ω ω ) , ≤ ∆ i .Proof. Use the above theorem with X = ω ω and M = V . (cid:3) Now, a morphism from D All to hP ( ω ω ) , P ( ω ω ) , ≺i , where ≺ is anyordering such that ( ∀ B ∈ P ( ω ω )) |{ A : A ≺ B }| ≤ ω , will imply that ||D All || = 2 ω . However, it is consistent that ||D All || < ω so there canbe no such morphism. In fact, it is consistent that ||R All || < ω . Thiscontrasts with the fact that ||E All || = 2 ω .To get a model of ||R All || < ω , it suffices to get a model in which b = c (so that there is a scale in h ω ω, ≤ ∗ i of length c ) and the cofinalitycf h c c , ≤i of all functions from c to c ordered by everywhere dominationis < c . By h λ λ, ≤ ∗ i we mean the set of functioms from λ to λ orderedby domination mod < λ . By b we mean the bounding number , and c = 2 ω . To get the required model, we first force so that 1) t = c (where t is the tower number ), 2) c is regular, 3) c < c = c , and 4) c + < c . Then, we force to add c ++ Cohen subsets of c . This preserves1)-4). Finally, we force by the generalization of Hechler forcing in [3] tocofinally embed h c + , ≤i into the poset of functions from c to c orderedby eventual mod < c domination ( ≤ ∗ ). A simple observation showsthat cf h c c , ≤i = cf h c c , ≤ ∗ i , and we are done.For the last result of this section, let V ∆ be the c.r.-relation whosechallenges and responses are Borel functions from ω ω × ω ω to ω , and g meets f iff ( ∀ x ∈ ω ω )( ∃ y ∈ ω ω ) f ( x, y ) = g ( x, y ). By Theorem 5.3 wewill have that ||D ∆ || = 2 ω . It is natural to ask whether ||V ∆ || = 2 ω .The answer is no for the following reason: fix an α < ω . Using the factthat there is a universal Σ α set, we can build a function g α : ω ω × ω ω → ω whose graph is Σ α +1 such that if f : ω ω × ω ω → ω is a function whosegraph is Σ α , then g α meets f . Hence, ||V ∆ || = ω The Encoding Function
In this section we will define the function f a : ω ω → ω ω which encodes a ∈ ω ω to be used in Thorem 5.3. Definition 3.1 (The Encoding Function f a ) . Fix a ∈ ω ω . Pick some A ⊆ ω such that A = T a , A is infinite, and A ≤ T B whenever B is aninfinite subset of A . Here ≤ T means Turing reducible to and = T meansTuring equivalent to. Such a set A is easy to construct. We actually ISJOINT BOREL FUNCTIONS 7 only need A to be ∆ in every infinite subset of itself. Let η : A → ω bea function such that ( ∀ n ∈ ω ) η − ( n ) is infinite. Consider an arbitrary x = h x , x , ... i ∈ ω ω. Let i < i < ... be the sequence of indices listingwhich numbers x i are in A . That is, each x i k ∈ A , but no other x i isin A . Define f a ( x ) := h η ( x i ) , η ( x i ) , ... i If there are only finitely many x i in A , define f a ( x ) to be anything.One can check that the function f a is Baire class one (the pointwiselimit of the sequence of continuous functions). One might wonder ifwe could define f a differently to be continuous but still encode a in thesense that given any Borel g : ω ω → ω ω satisfying f a ∩ g = ∅ , a is insome countable set associated to g . The answer is no for the reasonthat the cofinality of the poset of all continuous functions from ω ω to ω ordered by everywhere domination is d , the dominating number, whichcan be consistently less than 2 ω .4. Reachability
In this section we introduce some combinatorial lemmas needed forthe main theorem. The results may be of independent interest to thereader.
Definition 4.1.
Fix h : <ω ω → ω , A ⊆ ω , and t , t ∈ <ω ω . We write t ⊒ h t and say that t is an extension of t to the right of h iff t ⊒ t and( ∀ n ∈ Dom( t ) − Dom( t )) t ( n ) ≥ h ( t ↾ n ). We write t ⊒ A t iff t ⊒ t and ( ∀ n ∈ Dom( t ) − Dom( t )) t ( n ) A . We write t ⊒ Ah t iff both t ⊒ h t and t ⊒ A t . Definition 4.2.
Given h , h : <ω ω → ω , we write h ≤ h iff( ∀ t ∈ <ω ω ) h ( t ) ≤ h ( t ) . The following notion is crucial for the ability to find ⊒ A extensionsof a node t in a set S ⊆ <ω ω . Definition 4.3.
Given t ∈ <ω ω and S ⊆ <ω ω , • t is 0- S - reachable iff t ∈ S ; • for α > t is α - S - reachable iff t is β - S -reachable for some β < α or { n ∈ ω : ( ∃ β < α ) t ⌢ n is β - S -reachable } is infinite. DAN HATHAWAY • t is S - reachable iff t is α - S -reachable for some α .A computation shows the following: • t is S -reachable iff t is α - S -reachable for some α < ω CK ( S ) . • Given α < ω CK , the set of all t that are β - S -reachable for some β < α is ∆ ( S ). Lemma 4.4 (Reachability Dichotomy) . Fix t ∈ <ω ω , S ⊆ <ω ω , and A ⊆ ω which is infinite and ∆ in every infinite subset of itself. Assume A ∆ ( S ) . • If t is not S -reachable, then ( ∃ h ∈ ∆ ( S ))( ∀ t ′ ⊒ h t ) t ′ S. • If t is S -reachable, then ( ∀ h )( ∃ t ′ ⊒ Ah t ) t ′ ∈ S. Proof.
First, consider the case that t is not S -reachable. If ˜ t is a nodewhich is not S -reachable, then there must be only finitely many ˜ t ⌢ n that are S -reachable. For each ˜ t that is not S -reachable, define h (˜ t )to be the smallest n such that ( ∀ m ≥ n ) ˜ t ⌢ m is not S -reachable. Foreach ˜ t that is S -reachable, define h (˜ t ) = 0. A computation shows that h ∈ ∆ ( S ). This function h witnesses that ( ∀ t ′ ⊒ h t ) t ′ S .Consider the second case that t is S -reachable. Fix t , S , and A asin the statement of the lemma. Assume that t is S -reachable and fix h : <ω ω → ω . We must find some t ′ ⊒ Ah t such that t ′ ∈ S .Assume that t is not 0- S -reachable, otherwise we are already done bysetting t ′ = t . Thus, fix the smallest α > t is α - S -reachable.By induction, it suffices to find some n ∈ ω such that n A , n ≥ h ( t ), and t ⌢ n is β - S -reachable for some β < α . That is, if we keep doingthis, then we will have a decreasing sequence of ordinals α > α > ... which must eventually reach 0, at which point we will be done. Let B := { n ∈ ω : ( ∃ β < α ) t ⌢ n is β - S -reachable } .B is infinite and B ∈ ∆ ( S ). If B − A is infinite, we can get the desired n . Now, B − A must be infinite because otherwise B ∩ A = T B and B ∩ A is infinite, so A ≤ ∆ B ∩ A = T B ≤ ∆ S, which implies A ≤ ∆ S , a contradiction. (cid:3) ISJOINT BOREL FUNCTIONS 9 Main Theorem
We will prove the main theorem by using a variant of Hechler forcing.In fact, we could have used a slight variant of Hechler focing where thefunctions in the conditions are required to be strictly increasing (see[1]). However, we thought the Reachability Dichotomy (Lemma 4.4)was worth presenting for its own sake, and that lemma encapsulatesthe relevant rank analysis corresponding to what was carried out in [1].
Definition 5.1. H is the poset of all pairs ( t, h ) such that t ∈ <ω ω and h : <ω ω → ω , where ( t , h ) ≤ ( t , h ) iff t ⊒ h t and h ≥ h . Given A ⊆ ω , we write ( t , h ) ≤ A ( t , h ) iff t ⊒ Ah t and h ≥ h .From the Reachability Dichotomy follows the Main Lemma. Recallthat ( ∀ x, y ∈ ω ω ) x ∈ ∆ ( y ) iff every ω -model M which contains y alsocontains x . Lemma 5.2 (Main Lemma) . Let M be an ω -model of ZF and U ∈P M ( H M ) be a set dense in H M . Let A ⊆ ω be infinite and ∆ in everyinfinite subset of itself but A M . Then ( ∀ p ∈ H M )( ∃ p ′ ≤ A p ) p ′ ∈ U. Proof.
Define S := { t ∈ <ω ω : ( ∃ h ∈ M ) ( t, h ) ∈ U } . We have S ∈ M . It must be that A ∆ ( S ), because otherwise since M is an ω -model, we would have A ∈ M .Now fix an arbitrary p = ( t, h ) ∈ H M . We must find some p ′ =( t ′ , h ′ ) ≤ A ( t, h ) such that p ′ ∈ U (and so h ′ ∈ M ). It suffices to findsome t ′ ∈ S such that t ′ ⊒ Ah t .There are two cases: t is S -reachable or not. If t is not S -reachable,then by the Reachability Dichotomy (Lemma 4.4) there is h ∈ ∆ ( S )such that ( ∀ t ′ ⊒ h t ) t ′ S . Since M is an ω -model and S ∈ M , suchan h would be in M . Unpacking the definition of S , we get that U isnot dense in H M , a contradiction.The other case is that t is S -reachable. Lemma 4.4 gives us a t ′ ∈ S such that t ′ ⊒ Ah t , which is what we wanted. (cid:3) This next theorem refers to the function f a defined in Section 3. Theorem 5.3 (Main Theorem) . Let Γ be the pointclass of all setsdefined by formulas in a certain class (so it makes sense to talk about Γ -formulas). Let ≺ be an ordering on ω ω such that whenever c, a ∈ ω ω are such that a c , then there exists an ω -model M of ZF such that • c ∈ M ; • a M ; • P M ( H M ) is countable (in V ); • for every forcing extension N (in V ) of M by H M , the truth (in V ) of Γ formulas with real parameters in N can be computed in N .Then for any a ∈ ω ω and g : ω ω → ω ω in Γ , f a ∩ g = ∅ ⇒ a ≺ ( any code for g ) . Proof.
Fix a , g , and an arbitrary code c for g . In any model N whichcontains c and which can compute the truth (in V ) of Γ formulas withreal parameters in N , let ˜ g refer to the function g ∩ N (which is in N ). Suppose a c . Fix an ω -model M as in the hypothesis of thetheorem. Let A ⊆ ω be the set from the definition of f a that is ∆ inevery infinite subset of itself and a = T A . Note that A M .We will construct an x ∈ ω ω satisfying f a ( x ) = g ( x ) and this willprove the theorem. Let h U n ∈ P M ( H M ) : n < ω i be an enumeration (in V ) of the dense subsets of H M in M . Let ˙ x be the canonical name for the generic real added by H M . We willconstruct a decreasing sequence of conditions of H M which hit each U n . The x ∈ ω ω will be the union of the stems in this sequence (and itwill be generic over M having the name ˙ x ).Starting with 1 ∈ H M , apply the Lemma 5.2 to get p ≤ A U .Then, apply Lemma 5.2 again to get p ′ ≤ A p and m ∈ ω such that( p ′ (cid:13) ˜ g ( ˙ x )(0) = ˇ m ) M . Next, extend the stem of p ′ by one to get p ′′ ≤ p ′ to ensure that f a ( x )(0) = m .Next, get p ′′ ≤ p ′ ≤ A p ≤ A p ′′ such that p ∈ U , ( p ′ (cid:13) ˜ g ( ˙ x )(1) =ˇ m ) M for some m ∈ ω , and p ′′ extends the stem of p ′ by one to ensurethat f a ( x )(1) = m . Continue forever like this.The x we have constructed is generic for H M over M . Let N = M [ x ]. For each n ∈ ω we have (˜ g ( x )( n ) = m n ) N . Since Γ-formulas areabsolute between N and V , for each n ∈ ω we have g ( x )( n ) = m n . On the other hand, for each n ∈ ω we have f a ( x )( n ) = m n . (cid:3) In the following, M n ( y ) refers to the cannonical proper class modelwith n Woodin cardinals which contains y ∈ ω ω . For each n ∈ ω and y ∈ ω ω , ω ω ∩ M n ( y ) is countable. When we write a ∈ M n ( c ), we willbe making the assumption that M n ( c ) exists, which has large cardinalstrength. ISJOINT BOREL FUNCTIONS 11
Corollary 5.4.
Fix a ∈ ω ω , Γ , g : ω ω → ω ω in Γ , and a code c for g .Assume f a ∩ g = ∅ . • Γ = ∆ ⇒ a ∈ ∆ ( c ) ; • Γ = ∆ ⇒ a ∈ L ( c ) ; • Γ = ∆ ⇒ a ∈ M ( c ) ; • Γ = ∆ ⇒ a ∈ M ( c ) ; • ...Proof. The first bullet holds because ∆ formulas are absolute between ω -models and V , and whenever a ∆ ( r ), there is some ω -model of ZFwhich contains r but not a . The second bullet holds by Shoenfield’sAbsoluteness Theorem. The last two bullets hold because a forcingextension of M n below its bottom Woodin cardinal can compute thetruth of ∆ n formulas with real parameters in N . For more informa-tion related to the last two bullets, see Lemma 4.6 of [Steel]. (cid:3) From the top bullet of this corollary, it follows that there is a mor-phism from D ∆ to h ω ω, ω ω, ≤ ∆ i . From the second bullet, it followsthat there is a morphism from D ∆ to h ω ω, ω ω, ≤ L i , etc.6. Necessity of Hypotheses
Let Γ = S n ∈ ω ∆ n be the pointclass of projective sets. By Corol-lary 5.4, if g : ω ω → ω ω is a projective function and f a ∩ g = ∅ ,then a ∈ S n<ω M n ( c ) where c is any code for g . This implies that ||D Γ || = 2 ω . It is natural to ask whether ||D Γ || = 2 ω can be provedin ZFC alone (the assumption that the M n ( c ) exist goes far beyondZFC). We can ask the following stronger question: Question 6.1.
Does ZFC prove that for each projective g : ω ω → ω ω there is a countable set G ( g ) ⊆ ω ω , and for each a ∈ ω ω there is aprojective function f a : ω ω → ω ω such that ( ∀ a ∈ ω ω )( ∀ g ) f a ∩ g = ∅ ⇒ a ∈ G ( g )?We do not know how to answer the above question. The problem isthat the functions f a for various a may have nothing to do with oneanother. We can, however, answer the following: Question 6.2.
Does ZFC prove that there exist functions f a and count-able sets G ( g ) as in the above question but with the additional require-ment that the mapping ( a, x ) f a ( x ) is projective?We will now argue that the answer to Question 6.2 is no. It sufficesto show that ZFC does not prove there is a pair of mappings a f a and g G ( g ) such that ( a, x ) f a ( x ) is projective and ( ∀ a ∈ ω ω )( ∀ g )( ∀ x ∈ ω ω ) f a ( x ) ≤ ∗ g ( x ) ⇒ a ∈ G ( g ) , because the pointwise eventual domination relation is above the dis-jointness relation.Consider a model of the following statements:1) There is a projective wellordering of the reals of ordertype 2 ω ;2) ¬ CH;3) b = 2 ω .Statement 3) is equivalent to saying that each subset of ω ω of size < ω is ≤ ∗ -dominated by a single element of ω ω . The construction ofa model in which MA + ¬ CH holds (and therefore b = 2 ω ) and thereis a projective wellordering of the reals is done in [4]. Consider a givenencoding a f a such that the map ( a, x ) f a ( x ) is projective. Themapping which takes a ∈ ω ω to a code for f a is projective. Let ≺ bethe projective wellordering given by 1). For each b ∈ ω ω , we may definethe function g b : ω ω → ω ω as follows: g b ( x ) := the ≺ -least y ∈ ω ω such that ( ∀ a ≺ b ) f a ( x ) ≤ ∗ y. Note that the prewellordering ≺ is used twice. Because b = 2 ω , thisfunction is indeed well-defined. It is also projective. Now, consider aset A ⊆ P ( ω ω ) of size ω . Since ¬ CH, we may fix a single b satisfying( ∀ a ∈ A ) a ≺ b . By definition of g b , we have( ∀ a ∈ A )( ∀ x ∈ ω ω ) f a ( x ) ≤ ∗ g b ( x ) . On the other hand, given the countable set G ( g b ) ⊆ P ( ω ω ), it cannotbe that A ⊆ G ( g B ). Hence, the encoding is not as required.7. A Forcing Free Proof
In Corollary 5.4 we showed that if g : ω ω → ω ω is Borel and c is anycode for g , then f a ∩ g = ∅ ⇒ a ∈ ∆ ( c ) , where f a is defined in Section 3. In this section we will present adifferent and forcing free proof that f a ∩ g = ∅ ⇒ a ∈ Σ ( c ) . To avoid complications, we will actually consider functions from ω ω to ω
2. The function f a can be modified into a function from ω ω to ω η : A → ω with η : A → f a . We will prove the desired result by proving the contrapositive.That is, fix a ∈ ω ω , Borel g : ω ω → ω
2, and a code c ∈ ω ω for g . Fix A ⊆ ω that is Turing equivalent to a and A is computable from every ISJOINT BOREL FUNCTIONS 13 infinite subset of itself. Assume that a Σ ( c ). We must construct an x ∈ ω ω such that f a ( x ) = g ( x ) . The following game theoretic notion is how we will get a forcing freeproof:
Definition 7.1.
Given a function j : ω ω →
2, and an m ∈ G ( j, m )is the game where Player I plays a pair ( t, h ) ∈ H that is ≤ the cur-rent pair and Player II plays a pair ( t, h ) ∈ H that is ≤ A the currentpair. After infinitely many moves, let x ∈ ω ω be the union of the firstelements of the pairs played. Player II wins iff j ( x ) = m . We say that( t, h ) ensures that j ( x ) = m iff Player II has a winning strategy for G ( j, m ) where the starting position is ( t, h ). Lemma 7.2.
If for each i ∈ ω and ( t, h ) ∈ H there exists m ∈ and ( t ′ , h ′ ) ≤ A ( t, h ) which ensures g ( x )( i ) = m , then there exists an x ∈ ω ω such that f a ( x ) = g ( x ) .Proof. Our x will be the union of the first elements of the pairs in thesequence we will construct. Start with the condition ( ∅ , h ) ∈ H where h is arbitrary. Let m ∈ t , h ) ≤ A ( t, h ) be such that ( t , h )ensures g ( x )(0) = m . Fix a winning strategy η for Player II for thecorresponding game. Have Player II play according to η for one moveto get ( t ′ , h ′ ) ≤ A ( t , h ). Extend t ′ by one to get ( t ′′ , h ′ ) ≤ ( t ′ , h ′ ) sothat f a ( x )(0) = m .Let m ∈ t , h ) ≤ A ( t ′′ , h ′ ) be such that ( t , h ) ensures g ( x )(1) = m . Fix a winning strategy η for Player II for the corre-sponding game. Have Player II play according to η for one more andaccording to η for one more (in the correct games) to get ( t ′ , h ′ ) ≤ A ( t , h ). Extend t ′ by one to get ( t ′′ , h ′ ) ≤ ( t ′ , h ′ ) so that f a ( x )(1) = m . Continue like this forever. (cid:3) Once the next lemma is proved, we will be done.
Lemma 7.3.
Assuming a Σ ( c ) , for each Borel j : ω ω → and ( t, h ) ∈ H , there exists m ∈ and ( t ′ , h ′ ) ≤ A ( t, h ) which ensures j ( x ) = m .Proof. This can be proved by induction on the rank of j within theBaire hierarchy. The base case is when j is continuous, and the proofis immediate. For the induction step, assume that h j n : n ∈ ω i is asequence of Borel functions such that( ∀ x ∈ ω ω ) j ( x ) = lim n →∞ j n ( x ) . Assume that for each n ∈ ω and (˜ t, ˜ h ) ∈ H , there exists m ′ ∈ t ′ , ˜ h ′ ) ≤ A (˜ t, ˜ h ) which ensures j n ( x ) = m ′ .Let n = 0. Let m ∈ t , h ) ≤ A ( t, h ) ensure j n ( x ) = m . Let η be a winning strategy for Player II for G ( j n , m ). Thestrategy η should be applied infinitely often for the remainder of theconstruction (assuming it does not terminate).For n ∈ ω and m ∈
2, let S ( n, m ) ⊆ <ω ω be the following set: S ( n, m ) := { t ′ ∈ <ω ω : ( ∃ n ′ ≥ n )( ∃ h ′ ) ( t ′ , h ′ ) ensures j n ′ ( x ) = m } . There are two cases: either t is S ( n + 1 , − m )-reachable or not.First, assume that it is not. We may fix ˜ h ≥ h from Lemma 4.4such that ( ∀ t ′ ⊒ ˜ h t ) t ′ S ( n + 1 , − m ). We claim that ( t , ˜ h )ensures j ( x ) = m . To see why, consider the following strategy ofPlayer II: 1) make ≤ A -extensions to either ensure the value of j n ( x )for all n ≤ m ), and 2)periodically play according to the winning strategies being producedfrom the ensuring process. When the game finishes, calling x the realconstructed, j n ( x ) = m for all n ≥ n , and so also j ( x ) = m .The other case is that t is S ( n +1 , − m )-reachable. It is importantthat t can reach S ( n + 1 , − m ) by making a ≤ A -extension, insteadof an arbitrary ≤ -extension. The set S ( n + 1 , − m ) is Σ ( c ) (becausethe definition of the set existentially quantifies over winning strategiesfor a game of real information). It cannot be that A is Σ in S ( n +1 , − m ), because if it was then by transitivity we would have that a is Σ ( c ). Since A is not Σ in S ( n + 1 , − m ), it is also not∆ in it, so by Lemma 4.4 we may fix ( t ′ , h ) ≤ A ( t , h ) such that t ′ ∈ S ( n + 1 , − m ). At this point, apply the strategy η one timeto get ( t ′′ , h ′′ ) ≤ A ( t ′ , h ). Since t ′ ∈ S ( n + 1 , − m ), get n > n , m = 1 − m , and ( t , h ) ≤ A ( t ′′ , h ′′ ) that ensures j n ( x ) = m . Let η be a winning strategy for Player II for G ( j n , m ). The strategy η ,along with η , should be applied infinitely often for the remainder ofthe construction (assuming it does not terminate).There are now two cases: either t is S ( n + 1 , − m )-reachableor not. If not, then we are done by reasoning similar to before. If t is S ( n + 1 , − m )-reachable, then we continue the construction andthe question becomes whether it ever terminates. Suppose, towards acontradiction, that the construction does not terminate. Let x ∈ ω ω be the sequence that has been constructed. For all i ∈ ω we have j n i ( x ) = m i . However, the m i ’s alternate, so the limit lim n →∞ j n ( x )cannot exist, which is a contradiction. (cid:3) ISJOINT BOREL FUNCTIONS 15 Acknowledgements
I would like to thank Andreas Blass for reading through the argu-ments here and making suggestions. I would also like to thank TreverWilson for explaining how much truth a forcing extension of M n cancompute. Finally, I would like to thank the referee for finding a signif-icant simplification in the main theorem. References [1] J. Baumgartner and P. Dordal.
Adjoining dominating functions . The Journalof Symbolic Logic 50: 94-101, 1985.[2] A. Blass. Combinatorial cardinal characteristics of the continuum. In M. Fore-man and A. Kanamori, editors,
Handbook of Set Theory Volume 1 , pages 395-489. Springer, New York, NY, 2010.[3] J. Cummings and S. Shelah.
Cardinal invariants above the continuum . Ann. ofPure and Appl. Logic 75: 251-268, 1995.[4] L. Harrington.
Long projective wellorderings . Annals of Mathematical Logic21: 1-24, 1977.[5] J. Steel.
Projectively well-ordered inner models . Ann. of Pure and Appl. Logic74: 77-104, 1995.[6] P. Vojt´aˇs.
Generalized Galois-Tukey connections between explicit relations onclassical objects of real analysis . In H. Judah, editor, Set Theory of the Reals,Volume 6 of Isreal Math. Conf. Proc. pages 619-643. Amer. Math. Soc. 1993.
Mathematics Department, University of Denver, Denver, CO 80208,U.S.A.
E-mail address ::