aa r X i v : . [ m a t h . L O ] M a r FAKE REFLECTION
GABRIEL FERNANDES, MIGUEL MORENO, AND ASSAF RINOT
Abstract.
We introduce a generalization of stationary set reflection which wecall filter reflection , and show it is compatible with the axiom of constructibilityas well as with strong forcing axioms. We prove the independence of filterreflection from ZFC, and present applications of filter reflection to the studyof canonical equivalence relations of the higher Cantor and Baire spaces. Introduction
Throughout the paper, κ denotes an uncountable cardinal satisfying κ <κ = κ .Motivated by questions arising from the study of the higher Cantor and Bairespaces (also known as the generalized Cantor and Baire spaces), 2 κ and κ κ , respec-tively, in this paper, we introduce and study a new notion of reflection which wecall filter reflection . Definition 1.1.
Suppose X and S are stationary subsets of κ , and ~ F = hF α | α ∈ S i is a sequence such that, for each α ∈ S , F α is a filter over α .(1) We say that ~ F captures clubs iff, for every club C ⊆ κ , the set { α ∈ S | C ∩ α / ∈ F α } is non-stationary;(2) We say that X ~ F -reflects to S iff ~ F captures clubs and, for every stationary Y ⊆ X , the set { α ∈ S | Y ∩ α ∈ F + α } is stationary;(3) We say that X f -reflects to S iff there exists a sequence of filters ~ F over astationary subset S ′ of S such that X ~ F -reflects to S ′ .It is not hard to verify (see Lemma 2.8 below) that if X f -reflects to S , thenthe equivalence relation = X on the space κ κ (see Definition 2.5 below) admits aLipschitz reduction to the equivalence relation = S . A detailed study of the effectof filter reflection (and stronger forms of which) on the higher Baire and Cantorspaces is given in Section 2 below, but as we feel that the notion of filter reflectionis of independent interest, let us take a moment to inspect Definition 1.1. • A sequence ~ F over a stationary S ⊆ cof( >ω ) that captures clubs can beobtained in ZFC ; for every α ∈ S , simply take F α to be the club filter over α , CUB( α ).An alternative way to consistently get a sequence ~ F that captures clubsis by consulting a ♦ ∗ S -sequence, in which case, each filter F α will have asmall base. • If we omit the requirement that ~ F captures clubs from Definition 1.1(2),then again such a sequence may be constructed in ZFC , with each F α beinga principal ultrafilter. However, if S is an ineffable set (see Definition 3.16below), then capturing implies that F α ⊇ CUB( α ) for most α ∈ S .An alternative way to consistently get “reflection minus capturing” goesthrough forcing axioms; for instance, it follows from the main result of[Moo06] that MRP (a principle weaker than the Proper Forcing Axiom)
Mathematics Subject Classification.
Primary 03E35. Secondary 03E05, 54H05.
Key words and phrases.
Filter reflection, Lipschitz reduction, higher Baire space. entails that for X := ω ∩ cof( ω ) and S := ω ∩ cof( ω ), there exists asequence ~ F with F α ⊆ CUB( α ) for all α ∈ S , and satisfying that for everystationary Y ⊆ X , the set { α ∈ S | Y ∩ α ∈ F + α } is stationary.Thus, filter reflection is the conjunction of two requirements, each being a conse-quence of ZFC , and the challenge is in simultaneously satisfying both. The specialcase in which F α = CUB( α ) for all α ∈ S ′ is better known as the assertion that“every stationary subset of X reflects in S ” that goes back to some of the milestonepapers in the history of forcing with large cardinals [Bau76, Mag82, HS85, MS89].In this paper, we pay special attention to the case in which F α + CUB( α ) forall α ∈ S ′ ; we regard this special case by the name fake reflection . The obviousadvantage of fake reflection over genuine reflection is that the former makes senseeven for stationary sets S consisting of points of countable cofinality. The other —somewhat unexpected — advantage is that its consistency does not require largecardinals: Theorem A.
For all stationary subsets X and S of κ , there exists a <κ -closed κ + -cc forcing extension, in which X f -reflects to S . We shall also show that (a strong form of) fake reflection is a consequence of twoaxioms of contradictory nature, that is, the axiom of constructibility ( V = L ) andMartin’s Maximum ( MM ).After realizing that fake reflection is so prevalent, we tried for a while to provethat it is a consequence of ZFC , or, at least, a consequence of ♦ + S , but in vain.Eventually, we discovered that this is not the case: Theorem B.
There exists a cofinality-preserving forcing extension in which for alltwo disjoint stationary subsets
X, S of κ , X does not f -reflect to S . Theorems A and B give two extreme configurations of filter reflection. The nexttheorem allows considerably more subtle configurations, as it gives rise to modelsin which filter reflection is no more general than the classic notion of reflection: Theorem C. If κ is strongly inaccessible (e.g., a Laver-indestructible large cardi-nal), then, in the forcing extension by Add( κ, κ + ) , for all two disjoint stationarysubsets X, S of κ , the following are equivalent: • X f -reflects to S ; • every stationary subset of X reflects in S . As hinted earlier, filter reflection has a strong connection with the Borel-reducibilityhierarchy of the higher Cantor and Baire spaces. To exemplify: Theorem D.
For all two disjoint stationary subsets
X, S of κ . If X f -reflects to S and S f -reflects to X , then there is a map f : κ κ → κ κ simultaneously witnessing = X ֒ → = S & = S ֒ → = X . Filter reflection also provides us with tools to answer a few questions from theliterature. For instance, the following answers Question 2.12 of [AHKM19] for thecase κ = ω : Thereom E.
The consistency of MM implies the consistency of = ω ∩ cof( ω ) ֒ → = ω ∩ cof( ω ) & = ω ∩ cof( ω ) ֒ → BM = ω ∩ cof( ω ) . See also Corollary 5.15 below. See Section 2 for missing definitions.
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Organization of this paper.
In Section 2, we introduce the notion of filterreflection and its variations and study their effect on canonical equivalence relationsover the higher Baire and Cantor spaces. The proof of Theorem D will be foundthere.In Section 3, we study strong and simultaneous forms of filter reflection, provingvarious sufficient and equivalent conditions for these principles to hold.In Section 4, we present a poset that forces filter and fake reflection to hold,thus, proving Theorem A.In Section 5, we present a poset that forces filter and fake reflection to fail, alongthe way, proving Theorems B and C.In Section 6, we extend the analysis of Section 5 to get strong failures of Bairemeasurable reductions. Along the way, we answer a few questions from the litera-ture, and prove Theorem E.1.2.
Notation and conventions.
For sets
X, Y , their difference { x ∈ X | x / ∈ Y } is denoted by X \ Y , whereas the notation X − Y is reserved for the relative ofShelah’s approachability ideal that we introduce in Section 5.By a filter we always mean a proper filter, so that is does not contain the emptyset. For a filter F over a set X , we let F + := { Y ∈ P ( X ) | X \ Y / ∈ F} . For anideal I over a set X and a subset Y ⊆ X , we write I ↾ Y for I ∩ P ( Y ).The class of ordinals is denoted by OR. The class of ordinals of cofinality µ isdenoted by cof( µ ), and the class of ordinals of cofinality greater than µ is denotedby cof( >µ ). The class of infinite singular ordinals is denoted by Sing. The classof infinite regular cardinals is denoted by Reg. We write Sing( κ ) for Sing ∩ κ , andReg( κ ) for Reg ∩ κ . For m < n < ω , we denote S nm := ℵ n ∩ cof( ℵ m ).For a set of ordinals a , we write cl( a ) := { sup( a ∩ α ) | α ∈ OR , a ∩ α = ∅} ,acc( a ) := { α ∈ a | sup( a ∩ α ) = α > } , acc + ( a ) := { α < sup( a ) | sup( a ∩ α ) = α > } and nacc( a ) := a \ acc( a ). For a stationary Y ⊆ κ , we write Tr( Y ) := { α ∈ κ ∩ cof( >ω ) | Y ∩ α is stationary in α } .We let Col( λ, <κ ) denote L´evy’s notion of forcing for collapsing κ to λ + , and letAdd( κ, θ ) denote Cohen’s notion of forcing for adding θ many functions from κ to2. For all α < κ , p : α →
2, and i <
2, we denote by p y i the unique function p ′ extending p satisfying dom( p ′ ) = α + 1 and p ′ ( α ) = i .2. Filter reflection and Lipschitz reductions
Throughout this section, we let
S, X denote stationary subsets of κ , and considersequences ~ F = hF α | α ∈ S i , where, for all α ∈ S , F α is a filter over α . Recall that ~ F is said to captures clubs iff, for every club C ⊆ κ , the set { α ∈ S | C ∩ α / ∈ F α } is non-stationary. Definition 2.1 (Filter reflection) . (1) We say that X ~ F -reflects to S iff ~ F captures clubs and, for every stationary Y ⊆ X , the set { α ∈ S | Y ∩ α ∈ F + α } is stationary;(2) We say that X strongly ~ F -reflects to S iff ~ F captures clubs and, for everystationary Y ⊆ X , the set { α ∈ S | Y ∩ α ∈ F α } is stationary;(3) We say that X ~ F -reflects with ♦ to S iff ~ F captures clubs and there existsa sequence h Y α | α ∈ S i such that, for every stationary Y ⊆ X , the set { α ∈ S | Y α = Y ∩ α & Y ∩ α ∈ F + α } is stationary. Definition 2.2.
We say that X f -reflects to S whenever there exists a stationary S ′ ⊆ S and sequence of filters ~ F = hF α | α ∈ S ′ i such that X ~ F -reflects to S ′ .The same convention applies to strong f -reflection and to f -reflection with ♦ . GABRIEL FERNANDES, MIGUEL MORENO, AND ASSAF RINOT
Notice that a priori there is no reason to believe that “ X f -reflects to S ” impliesthe existence of a sequence ~ F for which “ X ~ F -reflects to S ”. To put our finger atthe challenge, it is in the requirement of capturing clubs. Proposition 2.3.
For stationary subsets X and S of κ , (1) = ⇒ (2) = ⇒ (3) : (1) X f -reflects with ♦ to S ; (2) X strongly f -reflects to S ; (3) X f -reflects to S .Proof. (1) = ⇒ (2) Suppose ~ F = hF α | α ∈ S ′ i and h Y α | α ∈ S ′ i witness togetherthat X f -reflects with ♦ to S . Let S ′′ := { α ∈ S ′ | Y α ∈ F + α } . For each α ∈ S ′′ , let¯ F α be the filter over α generated by F α ∪ { Y α } . Evidently, h ¯ F α | α ∈ S ′′ i witnessesthat X strongly f -reflects to S . (cid:3) The following is obvious:
Lemma 2.4 (Monotonicity) . For stationary sets Y ⊆ X ⊆ κ and S ⊆ T ⊆ κ : (1) If X f -reflects to S , then Y f -reflects to T ; (2) If X strongly f -reflects to S , then Y strongly f -reflects to T ; (3) If X f -reflects with ♦ to S , then Y f -reflects with ♦ to T . (cid:3) Definition 2.5. (1) For all η, ξ ∈ κ κ , denote∆( η, ξ ) := min( { α < κ | η ( α ) = ξ ( α ) } ∪ { κ } ) . (2) For every θ ∈ [2 , κ ], the equivalence relation = θS over θ κ is defined via η = θS ξ iff { α ∈ S | η ( α ) = ξ ( α ) } is non-stationary . (3) The special case = κS is denoted by = S .The above equivalence relation is an important building block in the studyof the connection between model theory and generalized descriptive set theory(cf. [FHK14, Chapter 6] and [HKM17, Theorem 7]). Definition 2.6.
For i <
2, let X i be some space from the collection { θ κ | θ ∈ [2 , κ ] } .Let R and R be binary relations over X and X , respectively. A function f : X → X is said to be:(a) a reduction of R to R iff, for all η, ξ ∈ X , η R ξ iff f ( η ) R f ( ξ ) . (b) 1- Lipschitz iff for all η, ξ ∈ X ,∆( η, ξ ) ≤ ∆( f ( η ) , f ( ξ )) . The existence of a function f satisfying (a) and (b) is denoted by R ֒ → R . Welikewise define R ֒ → c R , R ֒ → B R and R ֒ → BM R once we replace clause (b)by a continuous, Borel, or Baire measurable map, respectively. The following isobvious.
Lemma 2.7 (Monotonicity) . For • p ∈ { , c, B, BM } , • stationary sets X ⊆ X ′ ⊆ κ and Y ⊆ Y ′ ⊆ κ , and • ordinals ≤ θ ≤ θ ′ ≤ κ and ≤ λ ≤ λ ′ ≤ κ , = θ ′ X ′ ֒ → p = λY entails = θX ֒ → p = λ ′ Y ′ . (cid:3) Lemma 2.8. If X f -reflects to S , then = X ֒ → = S . See Section 6 for definitions and topological properties of the higher Cantor and Baire spaces.
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Proof.
Suppose that ~ F = hF α | α ∈ S ′ i witnesses that X f -reflects to S . For every α ∈ S ′ , define an equivalence relation ∼ α over κ α by letting η ∼ α ξ iff there is W ∈ F α such that W ∩ X ⊆ { β < α | η ( β ) = ξ ( β ) } . As there are at most | κ α | many equivalence classes and as κ <κ = κ , we may attach to each equivalence class[ η ] ∼ α a unique ordinal (a code ) in κ , which we shall denote by p [ η ] ∼ α q . Next, definea map f : κ κ → κ κ by letting for all η ∈ κ κ and α < κ : f ( η )( α ) := ( p [ η ↾ α ] ∼ α q , if α ∈ S ′ ;0 , otherwise . As, for all η ∈ κ κ and α < κ , f ( η )( α ) depends only on η ↾ α , f is 1-Lipschitz. Tosee that it forms a reduction from = X to = S , let η, ξ be arbitrary elements of κ κ .There are two main cases to consider: ◮ If η = X ξ , then let us fix a club C such that C ∩ X ⊆ { β < κ | η ( β ) = ξ ( β ) } .Since ~ F captures clubs, let us fix a club D ⊆ κ such that, for all α ∈ D ∩ S ′ , C ∩ α ∈ F α . We claim that D is disjoint from { α ∈ S | f ( η )( α ) = f ( ξ )( α ) } , so thatindeed f ( η ) = S f ( ξ ). To see this, let α ∈ D be arbitrary. ◮◮ If α S ′ , then f ( η )( α ) = 0 = f ( ξ )( α ). ◮◮ If α ∈ S ′ , then for W := C ∩ α , we have that W ∈ F α and W ∩ X ⊆ { β < α | η ( β ) = ξ ( β ) } , so that [ η ↾ α ] ∼ α = [ ξ ↾ α ] ∼ α and f ( η )( α ) = f ( ξ )( α ). ◮ If η = X ξ , then Y := { β ∈ X | η ( β ) = ξ ( β ) } is stationary. In effect, T := { α ∈ S ′ | Y ∩ α ∈ F + α } is stationary. Now, for every α ∈ T , it must bethe case that any W ∈ F α meets Y , so that W ∩ X * { β < α | η ( β ) = ξ ( β ) } ,and [ η ↾ α ] ∼ α = [ ξ ↾ α ] ∼ α . It follows that { α ∈ S ′ | f ( η )( α ) = f ( ξ )( α ) } covers thestationary set T , so that f ( η ) = S f ( ξ ). (cid:3) Lemma 2.9. If X strongly f -reflects to S , then, for every θ ∈ [2 , κ ] , = θX ֒ → = θS .Proof. By the preceding lemma and by the implication (2) = ⇒ (3) of Propo-sition 2.3, we may assume that θ ∈ [2 , κ ). Suppose ~ F = hF α | α ∈ S ′ i is asequence witnessing that X strongly f -reflects to S . Define a map f : θ κ → θ κ asfollows. For every α ∈ S ′ and η ∈ θ κ , if there exists W ∈ F α and i < θ such that W ∩ X ⊆ { β < α | η ( β ) = i } , then it is unique (since F α is a filter), and so we let f ( η )( α ) := i . If there is no such i or if α / ∈ S ′ , then we simply let f ( η )( α ) := 0.As, for all η ∈ κ κ and α < κ , f ( η )( α ) depends only on η ↾ α , f is 1-Lipschitz. Tosee that it forms a reduction, let η, ξ be arbitrary elements of κ κ . There are twocases to consider: ◮ If η = θX ξ , then since ~ F captures clubs, we infer that f ( η ) = θS f ( ξ ), very muchlike the proof of this case in Lemma 2.8. ◮ If η = θX ξ , then, as θ < κ , we may find i = j for which Y := { β ∈ X | η ( β ) = i & ξ ( β ) = j } is stationary. In effect, T := { α ∈ S ′ | Y ∩ α ∈ F α } isstationary. Now, for every α ∈ T , we have that f ( η )( α ) = i while f ( ξ )( α ) = j . Itfollows that f ( η ) = S f ( ξ ). (cid:3) Given Lemmas 2.8 and 2.9, it is natural to ask whether = X ֒ → = S implies= X ֒ → = S . The following provides a sufficient condition. Lemma 2.10. If κ is not a strong limit, then = X ֒ → = S implies = X ֒ → = S .Proof. Suppose that κ is not a strong limit. As we are working under the hypothesisthat κ <κ = κ , this altogether means that we may fix a bijection h : κ ↔ λ forsome cardinal λ < κ . In effect, there is a bijection π : κ κ ↔ (2 κ ) λ satisfying that,for every η ∈ κ κ , π ( η ) = h η i | i < λ i iff, for all i < λ and α < κ : η i ( α ) = h ( η ( α ))( i ) . GABRIEL FERNANDES, MIGUEL MORENO, AND ASSAF RINOT
Suppose that f : 2 κ → κ is a continuous reduction from = X to = S . Define amap g : κ κ → κ κ via g ( η ) := ζ iff π ( ζ ) = h f ( η i ) | i < λ i . Claim 2.10.1. g is -Lipschitz.Proof. Notice that π is 1-Lipschitz and g is equivalent to the function π − ◦ f ◦ π .The claim follows from the fact that f is 1-Lipschitz. (cid:3) To see that g is a reduction from = X to = S , let η, ξ ∈ κ κ be arbitrary. Denote h η i | i < λ i := π ( η ) and h ξ i | i < λ i := π ( ξ ). ◮ If η = X ξ , then let us fix a club C ⊆ κ such that C ∩ X ⊆ { α < κ | η ( α ) = ξ ( α ) } . It easy to see that for every i < λ , we have C ∩ X ⊆ { α < κ | η i ( α ) = ξ i ( α ) } , so that η i = X ξ i and we may pick a club D i ⊆ κ such that D i ∩ S ⊆ { α < κ | f ( η i )( α ) = f ( ξ i )( α ) } . Then D := T i<λ D i is a club, and wehave D ∩ S ⊆ { α < κ | g ( η )( α ) = g ( ξ )( α ) } , so that g ( η ) = S g ( ξ ). ◮ If ¬ ( η = X ξ ), then Y := { α ∈ X | η ( α ) = ξ ( α ) } is stationary. Since π isa bijection, for every α ∈ Y , there is some i < λ such that η i ( α ) = ξ i ( α ). Itfollows that there exists some i < λ such that Y i := { α ∈ X | η i ( α ) = ξ i ( α ) } isstationary. Consequently, the set T := { α ∈ S | f ( η i )( α ) = f ( ξ i )( α ) } is stationary.In particular, the set { α ∈ S | g ( η )( α ) = g ( ξ )( α ) } covers T , so that ¬ ( g ( η ) = S g ( ξ )). (cid:3) Remark . The same holds true once replacing ֒ → in the preceding by ֒ → p forany choice of p ∈ { c, B, BM } . Definition 2.12. (1) The quasi-order ≤ S over κ κ is defined via η ≤ S ξ iff { α ∈ S | η ( α ) > ξ ( α ) } is non-stationary . (2) The quasi-order ⊆ S over 2 κ is defined via: η ⊆ S ξ iff { α ∈ S | η ( α ) > ξ ( α ) } is non-stationary . Remark . Note that ⊆ S is nothing but ≤ S ∩ (2 κ × κ ).In [FMR19], we addressed the problem of universality of ⊆ S . The next theoremaddresses a specific instance of this problem. Theorem 2.14. If X f -reflects with ♦ to S , then ≤ X ֒ → ⊆ S .Proof. Let ~ F = hF α | α ∈ S ′ i and h Y α | α ∈ S ′ i witness together that X f -reflectswith ♦ to S . Let S ′′ := { α ∈ S ′ | Y α ∈ F + α } . For each α ∈ S ′′ , let ¯ F α be the filterover α generated by F α ∪ { Y α } . Claim 2.14.1.
There exists a sequence h η α | α ∈ S ′′ i such that, for every stationary Y ⊆ X and every η ∈ κ κ , the set { α ∈ S ′′ | η α = η ↾ α & Y ∩ α ∈ ¯ F α } is stationary.Proof. Let C := acc + ( X ) and B := X \ C , so that C is a club in κ and B is anon-stationary subset of κ of cardinality κ . As | B | = κ <κ , let us fix an injectiveenumeration { a β | β ∈ B } of the elements of κ <κ . Then, for each α ∈ S ′′ , let η α := ( S { a β | β ∈ Y α ∩ B } ) ∩ ( α × α ).To see that h η α | α ∈ S ′′ i is as sought, fix an arbitrary η ∈ κ κ and an arbitrarystationary Y ⊆ X . Let f : κ → B be the unique function to satisfy that, for all ǫ < κ , a f ( ǫ ) = η ↾ ǫ . Evidently, Y ∩ C is a stationary subset of X disjoint fromIm( f ). In particular, Y ′ := ( Y ∩ C ) ⊎ Im( f ) is a stationary subset of X , and hence G := { α ∈ S ′ | Y α = Y ′ ∩ α & Y ′ ∩ α ∈ F + α } is a stationary subset of S ′′ . Now, as ~ F captures clubs, let us fix a club D ⊆ κ such that, for all α ∈ D ∩ S ′ , C ∩ α ∈ F α . By convention, we always assume that κ <κ = κ , but this time this also follows from thehypothesis that X f -reflects with ♦ to S . AKE REFLECTION 7
Consider T := { α ∈ G ∩ D | f [ α ] ⊆ α & η [ α ] ⊆ α } which is a stationary subset of S ′′ . Let α ∈ T be arbitrary. • As α ∈ D , C ∩ α ∈ F α ⊆ ¯ F α , and as α ∈ G , Y ′ ∩ α = Y α ∈ ¯ F α . Therefore,the intersection Y ′ ∩ C ∩ α is in ¯ F α . But Y ′ ∩ C ∩ α = Y ∩ C ∩ α , and hence thesuperset Y ∩ α is in ¯ F α , as well. • As α ∈ G , Y α = Y ′ ∩ α , and we infer that Y α ∩ B = Im( f ) ∩ α . As f [ α ] ⊆ α ,we get that f [ α ] ⊆ Y α ∩ B ⊆ Im( f ). As η [ α ] ⊆ α , we get that η ↾ α = η ∩ ( α × α ).Recalling the definition of f and the definition of η α , it follows that η ↾ α ⊆ η α ⊆ η ,so that η α = η ↾ α . (cid:3) For every α ∈ S ′′ , define a quasi-order (cid:22) α over κ α by letting η (cid:22) α ξ iff there is W ∈ ¯ F α such that W ∩ X ⊆ { β < α | η ( β ) ≤ ξ ( β ) } . Let h η α | α ∈ S ′′ i be given bythe preceding claim. Define a map f : κ κ → κ by letting for all η ∈ κ κ and α < κ : f ( η )( α ) := ( , if α ∈ S ′′ & η α (cid:22) α η ↾ α ;0 , otherwise . As, for all η ∈ κ κ and α < κ , f ( η )( α ) depends only on η ↾ α , f is 1-Lipschitz. Tosee that it forms a reduction from ≤ X to ⊆ S , let η, ξ be arbitrary elements of κ κ .There are two cases to consider: ◮ If η ≤ X ξ , then let us fix a club C such that C ∩ X ⊆ { β < κ | η ( β ) ≤ ξ ( β ) } .Since ~ F captures clubs, let us a fix a club D ⊆ κ such that, for all α ∈ D ∩ S ′ , C ∩ α ∈ F α . We claim that D is disjoint from { α ∈ S | f ( η )( α ) > f ( ξ )( α ) } , sothat indeed f ( η ) ⊆ S f ( ξ ). Let α ∈ D with f ( η )( α ) = 1. Then for W := C ∩ α ,we have that W ∈ F α ⊆ ¯ F α and W ∩ X ⊆ { β < α | η ( β ) ≤ ξ ( β ) } , so that η α (cid:22) α η ↾ α (cid:22) α ξ ↾ α , and f ( ξ )( α ) = 1. ◮ If ¬ ( η ≤ X ξ ), then Y := { β ∈ X | η ( β ) > ξ ( β ) } is stationary. In effect, T := { α ∈ S ′′ | η α = η ↾ α & Y ∩ α ∈ ¯ F α } is stationary. Now, for every α ∈ T ,the set W := Y ∩ α is in ¯ F α so that f ( η )( α ) = 1 and f ( ξ )( α ) = 0. Consequently, ¬ ( f ( η ) ⊆ S f ( ξ )). (cid:3) Remark . The notion of “ X ♦ -reflects to S ” from [FHK14, Definition 57] is thespecial case of “ X ~ F -reflects with ♦ to S ” in which, for every α ∈ S , cf( α ) > ω and F α = CUB( α ). By [FHK14, Theorem 58], if X ♦ -reflects to S , then = X ֒ → c = S .So the preceding theorem is an improvement not only because CUB is replaced byabstract filters (allowing the case S ⊆ cof( ω )), but also because ≤ X ֒ → ⊆ S entails= X ֒ → = S , so that = X ֒ → = X ֒ → = S and hence also = X ֒ → c = S . Remark . Claim 2.14.1 establishes another interesting fact: If there exists astationary X ⊆ κ such that X f -reflects with ♦ to S , then ♦ S holds.2.1. Nilpotent reductions.
Now we turn to prove Theorem D. It is clear fromTheorem 2.14 that if X f -reflects with ♦ to X, then there exists a 1-Lipschitz mapwhich is different from the identity, yet, it witnesses = X ֒ → = X . In certain cases,we can obtain a similar result with only f -reflection (without the need for ♦ ). Lemma 2.17.
Suppose
X, Y, Z are stationary subsets of κ , with X ∩ Y non-stationary. (1) If X f -reflects to Y and Y f -reflects to X , then there is a function simulta-neously witnessing = X ֒ → = Y & = Y ֒ → = X . (2) If Z f -reflects to Y and Z f -reflects to X , then there is a function simulta-neously witnessing = Z ֒ → = Y & = Z ֒ → = X . GABRIEL FERNANDES, MIGUEL MORENO, AND ASSAF RINOT
Proof.
We only prove Clause (1). The proof of Clause (2) is very similar.Let X ′ ⊆ X and Y ′ ⊆ Y be stationary subsets such that there are ~ H = hH α | α ∈ Y ′ i and ~ G = hG α | α ∈ X ′ i witnessing that X f -reflects to Y , and Y f -reflectsto X , respectively. Let us define ~ F = hF α | α ∈ X ′ △ Y ′ i via F α := ( H α , if α ∈ X ′ ; G α , if α ∈ Y ′ . For every α ∈ X ′ △ Y ′ , define the equivalence relation ∼ α over κ α and the codes p [ η ] ∼ α q as in Lemma 2.8. Next, define a map f : κ κ → κ κ by letting for all η ∈ κ κ and α < κ : f ( η )( α ) := ( p [ η ↾ α ] ∼ α q , if α ∈ X ′ △ Y ′ ;0 , otherwise . At this point, the analysis is the same as in the proof of Lemma 2.8. (cid:3)
Corollary 2.18.
Suppose that
X, Y are stationary subsets of κ , with X ∩ Y non-stationary. If X f -reflects to Y and Y f -reflects to X , then there is a function f : κ κ → κ κ different from the identity witnessing = X ֒ → = X .Proof. Take the square of a function witnessing Lemma 2.17(1). (cid:3)
Corollary 2.19.
Suppose V = L and µ ∈ Reg( κ ) . Then there is a function simul-taneously witnessing = κ ∩ cof( µ ) ֒ → = κ ∩ cof( ν ) for all ν ∈ Reg( κ ) .Proof. By Lemma 2.17 and Corollary 4.6 below. (cid:3) Strong and simultaneous forms of filter reflection
Definition 3.1.
For stationary sets
X, S ⊆ κ and a cardinal θ ≤ κ , the principle f -Refl( θ, X, S ) asserts the existence of a sequence ~ F = hF α | α ∈ S ′ i with S ′ ⊆ S ,such that each F α is a filter over α , ~ F captures clubs, and, for every sequence h Y i | i < θ i of stationary subsets of X , the set { α ∈ S ′ | ∀ i < max { θ, α } ( Y i ∩ α ∈ F + α ) } is stationary. Remark . (1) In the special case θ = 1, we shall omit θ , writing f -Refl( X, S ).Note that the latter coincides with the notion of “ X f -reflects to S ”.(2) In the special case in which F α = CUB( α ) for all α ∈ S , we shall omit f ,writing, e.g., “Refl( θ, X, S )” and “ X reflects with ♦ to S ”.Another standard notion of reflection is Friedman’s problem , FP( κ ), assertingthat every stationary subset of κ ∩ cof( ω ) contains a closed copy of ω . Lemma 3.3. If FP( κ ) holds, then κ ∩ cof( ω ) strongly f -reflects to κ ∩ cof( ω ) .Proof. For each α ∈ κ ∩ cof( ω ), let F α := CUB( α ). (cid:3) The following forms a partial converse to the implication (1) = ⇒ (2) of Propo-sition 2.3. Proposition 3.4.
Suppose X strongly f -reflects to S .If ♦ X holds, then so does ♦ S .Proof. Fix ~ F = hF α | α ∈ S ′ i with S ′ ⊆ S such that X strongly ~ F -reflects to S ′ . Claim 3.4.1.
The set A := { α ∈ S ′ | ∃ B ∈ F α (sup( B ) < α ) } is non-stationary.Proof. Suppose not. Fix ǫ < κ for which A ǫ := { α ∈ S ′ | ∃ B ∈ F α (sup( B ) = ǫ ) } is stationary. Now, consider the club C := κ \ ( ǫ + 1). As ~ F captures clubs and A ǫ is stationary, there must exist α ∈ A ǫ such that C ∩ α ∈ F α . As α ∈ A ǫ , let uspick B ∈ F α with sup( B ) = ǫ . Then ( C ∩ α ) ∩ B ∈ F α , contradicting the fact theformer is empty. (cid:3) AKE REFLECTION 9
Suppose that ♦ X holds. Fix a ♦ X -sequence h Z β | β ∈ X i . For every α ∈ S ′ \ A ,let Z α := { Z ⊆ α | { β ∈ X ∩ α | Z ∩ β = Z β } ∈ F α } . Claim 3.4.2. Z α contains at most a single set.Proof. Towards a contradiction, suppose that Z = Z ′ are elements of Z α . Let B := { β ∈ X ∩ α | Z ∩ β = Z β } and B ′ := { β ∈ X ∩ α | Z ′ ∩ β = Z β } . Fix ζ ∈ Z △ Z ′ .As Z, Z ′ ∈ Z α , B ∩ B ′ ∈ F α . As α / ∈ A , we may find some β ∈ B ∩ B ′ above ζ .But then Z ∩ β = Z ′ ∩ β , contradicting the fact that ζ ∈ ( Z ∩ β ) △ ( Z ′ ∩ β ). (cid:3) For each α ∈ S , fix a subset Z α ⊆ α such that, if α ∈ S ′ \ A , then Z α ⊆ { Z α } . Tosee that h Z α | α ∈ S i is a ♦ S -sequence, let Z be an arbitrary subset of κ . Considerthe stationary set Y := { β ∈ X | Z ∩ β = Z β } . Pick α ∈ S ′ ∩ acc + ( Y ) \ A suchthat Y ∩ α ∈ F α . Then Z α = Z ∩ α . (cid:3) The preceding provides a way of separating f -reflection from strong f -reflection.Indeed, by the main result of [Hau92], it is consistent that for some weakly compactcardinal κ , ♦ κ holds, but ♦ Reg( κ ) fails. It follows that in any such model, κ doesnot strongly f -reflects to Reg( κ ), while Refl( κ, Reg( κ )) does hold (see Lemma 3.7(1)below).A similar configuration can also be obtained at the level of accessible cardinals assmall as ℵ . In [Zha20, § GCH in which Refl( S , S ) holds, but ♦ S fails. By[Gre76, Lemma 2.1], GCH implies that ♦ X holds for any stationary X ⊆ S . Thus: Corollary 3.5.
Assuming the consistency of a weakly compact cardinal, it is con-sistent that the two hold together: • Every stationary subset of S reflects in S ; • There exists no stationary subset of S that strongly f -reflects to S . (cid:3) The rest of this section is motivated by Theorem 2.14.Evidently, if X f -reflects with ♦ to S , then f -Refl( X, S ) and ♦ S both hold. Thenext lemma deals with the converse implication. Lemma 3.6.
Let X ⊆ κ and S ⊆ κ ∩ cof( >ω ) . For P := Add( κ, , the followingare equivalent: (1) V P | = Refl( X, S ) ; (2) V P | = X reflects with ♦ to S .Proof. It is clear from the definition that (2) = ⇒ (1).(1) = ⇒ (2): For all α < κ , we define a P -name for a subset of α , as follows:˙ D α := { ( ˇ β, p ) | β < α, p ∈ <κ \ ≤ α + β , p ( α + β ) = 1 } . Now, suppose that ˙ Y is a P -name, p ∈ P and p forces that ˙ Y is a stationary subsetof X ; we shall find an extension r of p and an ordinal α ∈ S such that r forces that˙ D α is a stationary subset of α which is also an initial segment of ˙ Y . Claim 3.6.1.
There are a condition q extending p and an ordinal α ∈ S such that: • dom( q ) = α ; • q decides ˙ Y up to α , and decides it to be a stationary subset of α .Proof. Let G be a P -generic over V , with p ∈ G . Work in V [ G ]. Let g := S G . As P does not add bounded subsets of κ , we may define a function f : κ → κ such that, forall ǫ < κ , g ↾ ( f ( ǫ )) decides ˙ Y up to ǫ . Consider the club C := { α < κ | f [ α ] ⊆ α } ,and note that, for all α ∈ C , g ↾ α decides ˙ Y up to α . As V P | = Refl( X, S ), p ∈ G , and p forces that ˙ Y is a stationary subset of X , R := { α ∈ C ∩ S | ( ˙ Y G ) ∩ α is stationary in α } is stationary in V [ G ]. Fix α ∈ R with α > dom( p ).Then α and q := g ↾ α are as sought. (cid:3) Let α and q be given by the claim. Fix a stationary d ⊆ α such that q forcesthat ˙ Y up to α is equal to ˇ d . Define a function r : α + α → ǫ < α + α , r ( ǫ ) := q ( ǫ ) , if ǫ < α ;1 , if ǫ = α + β & β ∈ d ;0 , otherwise . Then r extends p and forces that ˙ D α is a stationary subset of α which is also aninitial segment of ˙ Y . (cid:3) The following is well-known, the second item is pointed out in [Mag82]. As wewill need the proof later on (in proving Theorem 4.5), we do include it.
Lemma 3.7.
Suppose that κ is weakly compact. Then: (1) Refl( κ, κ, Reg( κ )) holds; (2) For every λ ∈ Reg( κ ) , in the forcing extension by Col( λ, <κ ) , κ = λ + and Refl( κ, κ ∩ cof( <λ ) , κ ) holds.Proof. (1) We shall consider a few first-order sentences in the language with unarypredicate symbols O and D , and binary predicate symbols ǫ, A and F . Specifically,let ϕ and ϕ denote first-order sentences such that: • h V α , ∈ , O, F i | = ϕ iff F is a function from some β ∈ O to O ; • h V α , ∈ , O, F i | = ϕ iff F is a function with Im( F ) ⊆ γ for some γ ∈ O .Evidently, Φ := ∀ F ( ϕ → ϕ ) is a Π -sentence such that h V α , ∈ , OR ∩ α i | = φ iff α is a regular cardinal.Also, let ψ be a first-order sentence such that, for every α ∈ cof( >ω ), h V α , ∈ , OR ∩ α, D i | = ψ iff D is a club in α . Then, let Ψ be the following Π -sentence: ∀ D ∀ ι (( ψ ( D ) ∧ O ( ι )) → ∃ β ( D ( β ) ∧ ( A ( ι, β )))) . Now, to verify Refl( κ, κ,
Reg( κ )), fix an arbitrary sequence h Y ι | ι < κ i of sta-tionary subsets of κ , and we shall find an α ∈ Reg( κ ) such that Y ι ∩ α is stationaryfor all ι < α .Set A := { ( ι, β ) | ι < κ & β ∈ Y ι } . A moment reflection makes it clear that h V κ , ∈ , OR ∩ κ, A i | = Φ ∧ Ψ . As weak compactness is equivalent to Π -indescribability (cf. [Kan08, Theorem 6.4]),there exists an uncountable α < κ such that: h V α , ∈ , OR ∩ α, A ∩ ( α × α ) i | = Φ ∧ Ψ . Clearly, α is as sought.(2) Let λ ∈ Reg( κ ). As in [AHKM19, Claim 2.11.1], we work with a partial order P which is isomorphic to Col( λ, <κ ), but, in addition, P ⊆ V κ . Namely, P = ( P, ≤ ),where P := { r ↾ (sup(supp( r )) + 1) | r ∈ Col( λ, <κ ) } , and q ≤ p iff q ⊇ p . Note that, for every α ∈ Reg( κ ) above λ , P is also isomorphicto P α × P ≥ α , where • P α := ( P α , ≤ ), with P α := { r ↾ (sup(supp( r )) + 1) | r ∈ Col( λ, <α ) } , and • P ≥ α := ( P ≥ α , ≤ ), with P ≥ α := { r ↾ (sup(supp( r )) + 1) | r ∈ Col( λ, [ α, κ )) } .Now, suppose p ∈ P and ˙ f is a P -name such that p forces ˙ f is a function withdomain κ , and, for each ι < κ , ˙ f ( ι ) is a stationary subset of κ ∩ cof( <λ ).Define a set H to consist of all quadruples ( ι, β, p, q ) such that: • ι < κ , • β ∈ κ ∩ cof( <λ ), • p, q ∈ V κ ; AKE REFLECTION 11 • if p ∈ P and p ≤ p , then q ∈ P , q ≤ p and q (cid:13) P ˇ β ∈ ˙ f (ˇ ι ).Let op( ˙ x, ˙ y ) denote the canonical name for the ordered pair whose left element is˙ x and right element is ˙ y , and set ˙ A := { (op(ˇ ι, ˇ β ) , q ) | ∃ p ≤ p ( α, β, p, q ) ∈ H } ,so that the interpretation of ˙ A plays the role of the set A from the proof of theprevious clause. Note that p (cid:13) ˙ A = ˙ f .Let ψ be a first-order sentence as in the previous clause. Then, let Ψ be thefollowing Π -sentence: ∀ p ∀ D ∀ ι (( ψ ( D ) ∧ O ( ι )) → ∃ q ∃ β ( D ( β ) ∧ ( H ( ι, β, p, q ))) . Recalling that for any condition p ≤ p , any club D in κ , and any ι < κ , thereis a condition q ≤ p deciding the existence of an ordinal in ˙ f ( ι ) and in D , we have: h V κ , ∈ , OR ∩ κ, H i | = Ψ . Since κ is Π -indescribable, we may fix a strongly inaccessible cardinal α < κ such that ( V α , ∈ , OR ∩ α, H ∩ V α ) | = Ψ . As P α has the α -cc, every club in α in V P α covers a club in α from V . Therefore: p (cid:13) P α h ˙ f ( ι ) ∩ α | ι < α i is an α -sequence of stationary subsets of α ∩ cof( <λ ) . By [She79, Theorem 20] and [She91, Lemma 4.4(1)] (see also [Eis10, § λ regular, every stationary subset of λ + ∩ cof( <λ ) is preserved by a <λ -closednotion of forcing. In V P α , α = λ + for the regular cardinal λ , and P ≥ α is <λ -closed.Therefore: p (cid:13) P α × P ≥ α h ˙ f ( ι ) ∩ α | ι < α i is an α -sequence of stationary subsets of α. (cid:3) The upcoming corollary was announced first by Shelah and V¨a¨an¨anen in [SV05]without a proof. It amounts to the special case of “ κ reflects with ♦ to Reg( κ )”,when κ is weakly compact (see [SV05, Definition 9]). Corollary 3.8 (Hellsten, [Hel03, Lemma 5.2.3]) . If κ is weakly compact, then, insome cofinality-preserving forcing extension, κ reflects with ♦ to Reg( κ ) .Proof. Recall that Silver proved that for every weakly compact cardinal κ , there isa forcing extension V [ G ] such that V [ G ][ H ] | = κ is weakly compact, whenever H is an Add( κ, V [ G ]. (cf. [Cum10, Example 16.2]). Now, appeal toLemma 3.7(1) and Lemma 3.6. (cid:3) The preceding took care of weakly compact cardinals. Building on the work ofHayut and Lambie-Hanson from [HLH17], it is also possible to obtain reflectionat the level of successor of singulars and small inaccessibles. It is of course alsopossible to obtain reflection with diamond at the level of successors of regulars, butthis is easier, and will be done in Corollary 3.13 below.
Corollary 3.9. (1)
If the existence of infinitely many supercompact cardinalsis consistent, then it is consistent that ℵ ω +1 reflects with ♦ to ℵ ω +1 ; (2) If the existence of an inaccessible limit of supercompact cardinals is consis-tent, then it is consistent that, letting κ be the least inaccessible cardinal, κ reflects with ♦ to κ .Proof. Following [HLH17], for a stationary X ⊆ κ , we let Refl ∗ ( X ) assert thatwhenever P is a <κ -closed-directed forcing notion of size ≤ κ , (cid:13) P Refl(
X, κ ). Inparticular, if Refl ∗ ( X ) holds and κ <κ = κ , then for P := Add( κ, V P | = Refl( X, κ ), and then by Lemma 3.6, furthermore, V P | = X reflects with ♦ to κ .Now, by [HLH17, Theorem 3.23], the hypothesis of Clause (1) yields the consis-tency of Refl ∗ ( ℵ ω +1 ) holds. Likewise, by [HLH17, Theorem 3.24], the hypothesis of Clause (2) yields the consistency of the statement that, there is an inaccessiblecardinal and the least inaccessible cardinal κ satisfies Refl ∗ ( κ ). (cid:3) Another approach for adjoining diamond to reflection is as follows.
Lemma 3.10.
Suppose f - Refl( κ, X, S ) holds for stationary subsets X, S of κ . If ♦ ∗ S holds, then X f -reflects with ♦ to S .Proof. Let ~ F = hF α | α ∈ S ′ i be a sequence witnessing f -Refl( κ, X, S ′ ), so that S ′ ⊆ S . Suppose that ♦ ∗ S (or just ♦ ∗ S ′ ) holds. It follows that we may fix a matrix h Z iα | i < α < κ i such that, for every Z ⊆ κ , for club many α ∈ S ′ , there is i < α with Z iα = Z ∩ α . Fix a bijection π : κ × κ ↔ κ . For any pair i < α < κ , let Y iα := { β < α | π ( β, i ) ∈ Z iα } . Claim 3.10.1.
There exists i < κ such that for every stationary Y ⊆ X , the set { α ∈ S ′ | Y iα = Y ∩ α & Y ∩ α ∈ F + α } is stationary.Proof. Suppose not. Then, for every i < κ , we may find a stationary Y i ⊆ X forwhich { α ∈ S | Y iα = Y i ∩ α & Y i ∩ α ∈ F + α } is non-stationary. Let Z := { π ( β, i ) | i < κ, β ∈ Y i } . Fix a club C ⊆ κ such that, for all α ∈ C ∩ S : • π [ α × α ] = α , and • there exits i < α with Z iα = Z ∩ α .By the hypothesis, T := { α ∈ S ′ ∩ C | ∀ i < α ( Y i ∩ α ∈ F + α ) } must be stationary.By Fodor’s lemma, we may fix i ∗ < κ and a stationary T ′ ⊆ T such that, for all α ∈ T ′ , Z i ∗ α = Z ∩ α . For all α ∈ T ′ , we have: • α ∈ T , so that, in particular, Y i ∗ ∩ α ∈ F + α ; • α ∈ C , so that Y i ∗ α = { β < α | π ( β, i ∗ ) ∈ Z i ∗ α } = { β < α | π ( β, i ∗ ) ∈ Z ∩ α } = Y i ∗ ∩ α. It thus follows that { α ∈ S ′ | Y i ∗ α = Y i ∗ ∩ α & Y i ∗ ∩ α ∈ F + α } covers the stationaryset T ′ , contradicting the choice of Y i ∗ . (cid:3) This clearly completes the proof. (cid:3)
Remark . (1) Note that by the definition of ♦ ∗ S , if κ = λ + is a successorcardinal, then the above argument establishes that for stationary subsets X, S of κ : If f -Refl( λ, X, S ) and ♦ ∗ S both hold, then X f -reflects with ♦ to S .(2) Note that the same proof establishes the corresponding fact for the f -free(that is, genuine) versions of reflection. Corollary 3.12.
Suppose that λ is a regular uncountable cardinal, and ♦ ∗ λ + ∩ cof( λ ) holds. For every stationary X ⊆ λ + ∩ cof( <λ ) , if Refl( λ, X, λ + ) holds, then X reflects with ♦ to λ + ∩ cof( λ ) .Proof. Suppose we are given X ⊆ λ + ∩ cof( <λ ) for which f -Refl( λ, X, λ + ) holds.By Remark 3.11, it suffices to prove that Refl( λ, X, λ + ∩ cof( λ )) holds. For this, let h Y i | i < λ i be some sequence of stationary subsets of X . Fix an arbitrary partition ~X = h X i | i < λ i of X into stationary sets. As Refl( λ, X, λ + ) holds, the set A := { α ∈ λ + | ∀ i < λ ( Y i ∩ α and X i ∩ α are stationary in α ) } is stationary. As the elements of ~X are pairwise disjoint, it follows that A ⊆ λ + ∩ cof( λ ). (cid:3) Corollary 3.13.
Suppose that κ is a weakly compact cardinal.For every λ ∈ Reg( κ ) , in the forcing extension by Col( λ, <κ ) , λ + ∩ cof( <λ ) reflects with ♦ to λ + ∩ cof( λ ) . AKE REFLECTION 13
Proof.
Let λ ∈ Reg( κ ), and work in the forcing extension by Col( λ, <κ ). As theproof of [BR17, Example 1.26] shows, ♦ ∗ λ + holds. In addition, by Lemma 3.7(2),Refl( λ + , λ + ∩ cof( <λ ) , λ + ) holds. So, by the preceding corollary, λ + ∩ cof( <λ )reflects with ♦ to λ + ∩ cof( λ ). (cid:3) Remark . It thus follows from Theorem 2.14 that in the model of the preceding,for every µ ∈ Reg( λ ), ≤ λ + ∩ cof( µ ) ֒ → ⊆ λ + ∩ cof( λ ) , and hence also = λ + ∩ cof( µ ) ֒ → = λ + ∩ cof( λ ) . This improves [FHK14, Theorem 55], as the result is not limited todouble successors, and as we do not need to assume that our ground model is L .A variation of the proof of Lemma 3.10 yields the following: Lemma 3.15.
Suppose Martin’s Maximum ( MM ) holds, κ ≥ ℵ , X ⊆ κ ∩ cof( ω ) is stationary, and S = κ ∩ cof( ω ) . If ♦ X holds, then X reflects with ♦ to S .Proof. Suppose that ♦ X holds, as witnessed by h Z γ | γ ∈ X i . For every Z ⊆ κ , let G Z := { γ ∈ sup( Z ) ∩ X | Z ∩ γ = Z γ } . Let α ∈ S be arbitrary. Fix a strictly increasing function π α : ω → α whose imageis a club in α , and then let Z α := { Z ⊆ α | π − α [ G Z ] is stationary in ω } . Evidently, for all two distinct
Z, Z ′ ∈ Z α , π − α [ G Z ] and π − α [ G Z ′ ] are almost disjointstationary subsets of ω . As MM implies that NS ω is saturated, we infer that |Z α | ≤ ℵ . Thus, let { Z iα | i < ω } be some enumeration (possibly, with repetitions)of Z α . Fix a bijection π : κ × ω ↔ κ . For each α ∈ S and i < ω , let Y iα := { β < α | π ( β, i ) ∈ Z iα } . Claim 3.15.1.
There exists i < ω such that, for every stationary Y ⊆ κ , { α ∈ S ∩ Tr( Y ) | Y ∩ α = Y iα } is stationary.Proof. Suppose not. For every i < ω , fix a stationary Y i ⊆ X for which { α ∈ S ∩ Tr( Y i ) | Y i ∩ α = Y iα } is non-stationary. Let Z := { π ( β, i ) | i < ω , β ∈ Y i } . Considerthe stationary set G := { γ ∈ X | Z ∩ γ = Z γ } and the club C := { α ∈ acc + ( Z ) | π [ α × ω ] = α } . As MM implies Refl( ω , X, S ), the following set is stationary S ′ := { α ∈ C ∩ S | G ∩ α is stationary, and, for all i < ω Y i ∩ α is statioanry } . For every α ∈ S ′ , since α ∈ Tr( G ) ∩ acc + ( Z ), G Z ∩ α covers the stationary set G ∩ α , so there exists i < ω such that Z iα = Z ∩ α . Now, fix i ∗ < ω and astationary T ⊆ S ′ such that, for all α ∈ T , Z i ∗ α = Z ∩ α . For all α ∈ T , we have: • α ∈ S ′ , and hence α ∈ S ∩ Tr( Y i ∗ ); • α ∈ C , and hence Y i ∗ α = { β < α | π ( β, i ∗ ) ∈ Z i ∗ α } = { β < α | π ( β, i ∗ ) ∈ Z ∩ α } = Y i ∗ ∩ α. So { α ∈ S ∩ Tr( Y i ∗ ) | Y i ∗ ∩ α = Y i ∗ α } covers the stationary set T , contradictingthe choice of Y i ∗ . (cid:3) Let i be given by the preceding claim. Then h Y iα | α ∈ S i witnesses that X h CUB( α ) | α ∈ S i -reflects with ♦ to S . (cid:3) Definition 3.16.
A stationary subset S of κ is said to be:(1) ineffable iff for every sequence h A α | α ∈ S i , there exists A ⊆ κ for which { α ∈ S | A ∩ α = A α ∩ α } is stationary.(2) weakly ineffable iff for every sequence h A α | α ∈ S i , there exists A ⊆ κ forwhich { α ∈ S | A ∩ α = A α ∩ α } is cofinal in κ . Recall that by a theorem of Shelah [She10], if κ = κ <κ is the successor of a cardinal uncount-able cofinality, then ♦ X holds for every stationary X ⊆ κ ∩ cof( ω ). (3) weakly compact iff for every Π -sentence φ and every A ⊆ V κ such that h V κ , ∈ , A i | = φ , there exists α ∈ S such that h V α , ∈ , A ∩ V α i | = φ . Definition 3.17 (Sun, [Sun93]) . For a weakly compact subset S ⊆ κ , ♦ S assertsthe existence of a sequence h Z α | α ∈ S i such that, for every Z ⊆ κ , the set { α ∈ S | Z ∩ α = Z α } is weakly compact.The proof of [Sun93, Theorem 2.11] makes clear that, for every weakly ineffable S ⊆ κ , ♦ S holds. It is also easy to see that ♦ S implies that κ reflects with ♦ to S .Therefore: Corollary 3.18.
For every weakly ineffable S ⊆ κ , κ reflects with ♦ to S . (cid:3) We conclude this section by proving that ♦ -reflection is equivalent to variousseemingly stronger statements. For instance, the concept of Clause (2) of the nextlemma is implicit in [AHKM19], as the principle WC ∗ κ from [AHKM19, Lemma 3.4]is equivalent to the instance X := κ and S := Reg( κ ). Likewise, the question ofwhether Clause (1) implies Clause (3) is implicit in the statement of [AHKM19,Claim 2.11.1]. The fact that the two clauses are equivalent allows to reduce thehypothesis of “there is a Π λ + -indescribable cardinal” of [AHKM19, Theorem 2.11]down to “there is a weakly compact cardinal” via Corollary 3.13 above. Lemma 3.19.
Let X ⊆ κ and S ⊆ κ ∩ cof( >ω ) . Then the following are equivalent: (1) X reflects with ♦ to S ; (2) there exists a sequence h f α | α ∈ S i such that, for every g ∈ κ κ and everystationary Y ⊆ X , the set { α ∈ S ∩ Tr( Y ) | g ↾ α = f α } is stationary; (3) there exists a partition h S i | i < κ i of S such that, for all i < κ , X reflectsto S i with ♦ .Proof. (1) = ⇒ (2): This is a special case of the proof of Claim 2.14.1.(2) = ⇒ (3) Let h f α | α ∈ S i be as in Clause (3). Without loss of generality, forall α ∈ S , f α is a function from α to κ . For all i < κ , let S i := { α ∈ S | f α (0) = i } ,so that h S i | i < κ i is a partition of S . For each α ∈ S , let Y α := { β < κ | ( β = 0 ∧ f α (1) is odd) ∨ ( β = 1 ∧ f α (1) ≥ ∨ ( β > ∧ f α ( β ) = 4) } . Let i < κ be arbitrary. We claim that h Y α | α ∈ S i i witnesses that X reflectswith ♦ to S i . To see this, fix an arbitrary stationary subset Y of X . Define afunction g : κ → κ as follows: g ( β ) := i, if β = 0;1 , if β = 1 and Y ∩ { , } = { } ;2 , if β = 1 and Y ∩ { , } = { } ;3 , if β = 1 and Y ∩ { , } = { , } ;4 , if β > β ∈ Y ;0 , otherwise;Consider the stationary set G := { α ∈ S ∩ Tr( Y ) | g ↾ α = f α } . Let α ∈ G ∩ acc( κ )be arbitrary. We have f α (0) = g (0) = i , so that α ∈ S i . Finally, g ↾ ( κ \ { , } )forms the characteristic function of Y \ { , } and g (1) encodes Y ∩ { , } , so that Y α = Y ∩ α .(3) = ⇒ (1): This is trivial. (cid:3) Question 3.20.
Suppose X (resp. strongly) f -reflects to S . Must there exist a par-tition h S i | i < κ i of S into stationary sets such that, for all i < κ , X (resp. strongly) f -reflects to S i ? AKE REFLECTION 15 Forcing fake reflection
In this section, we focus on the consistency of fake reflection , i.e., X f -reflects to S and yet there exists a stationary subset of X that does not reflect (in the classicalsense) to S . By the work of Jensen [Jen72], in G¨odel’s constructible universe, L ,stationary reflection fails at any non weakly compact cardinal, but, as we will see,filter reflection holds everywhere in L . We will also show that fake reflection isforceable.4.1. A diamond reflecting second-order formulas.
A Π n -sentence φ is a for-mula of the form ∀ X ∃ X · · · X n ϕ where ϕ is a first-order sentence over a relationallanguage L as follows: • L has a predicate symbol ǫ of arity 2; • L has a predicate symbols X i , i ≤ n , of arity m ( X i ); • L has infinitely many predicate symbols ( A n ) n ∈ ω , each A m is of arity m ( A m ). Definition 4.1.
For sets N and x , we say that N sees x iff N is transitive, p.r.-closed, and x ∪ { x } ⊆ N .Suppose that a set N sees an ordinal α , and that φ = ∀ X ∃ X · · · ϕ is a Π n -sentence, where ϕ is a first-order sentence in the above-mentioned language L . Forevery sequence ( A m ) m ∈ ω such that, for all m ∈ ω , A m ⊆ α m ( A m ) , we write h α, ∈ , ( A m ) m ∈ ω i | = N φ to express that the two hold:(1) ( A m ) m ∈ ω ∈ N ;(2) N | = ( ∀ X ⊆ α m ( X ) )( ∃ X ⊆ α m ( X ) ) · · · [ h α, ∈ , ( A m ) m ∈ ω , X , X , . . . i | = ϕ ], where: • ∈ is the interpretation of ǫ ; • X i is the interpretation of X i ; • for all m ∈ ω , A m is the interpretation of A m . Convention 4.2.
We write α + for | α | + , and write h α, ∈ , ( A n ) n ∈ ω i | = φ for h α, ∈ , ( A n ) n ∈ ω i | = H α + φ. Definition 4.3 (Fernandes-Moreno-Rinot, [FMR19]) . For a stationary S ⊆ κ anda positive integer n , Dl ∗ S (Π n ) asserts the existence of a sequence ~N = h N α | α ∈ S i satisfying the following:(1) for every α ∈ S , N α is a set of cardinality < κ that sees α ;(2) for every X ⊆ κ , there exists a club C ⊆ κ such that, for all α ∈ C ∩ S , X ∩ α ∈ N α ;(3) whenever h κ, ∈ , ( A m ) m ∈ ω i | = φ , with φ a Π n -sentence, there are stationarilymany α ∈ S such that | N α | = | α | and h α, ∈ , ( A m ∩ ( α m ( A m ) )) m ∈ ω i | = N α φ . Remark . The principle Dl + S (Π n ) is defined by strengthening Clause (2) in thedefinition of Dl ∗ S (Π n ) to require that C ∩ α be in N α , as well.The Todorcevic-V¨a¨an¨anen principle ♦ + S (Π n ) from [TV99] is obtained by strength-ening Clause (1) in the definition of Dl + S (Π n ) to require that | N α | = max {ℵ , | α |} . Lemma 4.5.
Suppose S ⊆ κ is stationary for which Dl ∗ S (Π ) holds. Then: (1) f - Refl( κ, κ, S ) ; (2) κ f -reflects with ♦ to S . Proof. (1) Let h N α | α ∈ S i witness the validity of Dl ∗ S (Π ). As in the proof ofLemma 3.7(1), let Φ be a Π -sentence, such that, for every ordinal α , ( h α, ∈i | = Φ)iff ( α is a regular cardinal). Likewise, let Ψ be a Π -sentence such that for everyordinal α and every A ⊆ α × α , h α, ∈ , A i | = Φ ifffor every ι < α, { β < α | ( ι, β ) ∈ A } is stationary in α. Now, let S ′ denote the set of all α ∈ S such that:(i) α > ω ;(ii) | N α | = | α | ;(iii) h α, ∈i | = N α Φ.For each α ∈ S ′ , by Clause (iii), F α := CUB( α ) ∩ N α is a filter over α . Claim 4.5.1.
Suppose h Y ι | ι < κ i is a sequence of stationary subsets of κ . Thenthere exist stationarily many α ∈ S ′ such that, for all ι < α , Y ι ∩ α ∈ F + α .Proof. Set A := { ( ι, β ) | ι < κ & β ∈ Y ι } . Clearly, h κ, ∈ , A i | = Φ ∧ Ψ, Thus,recalling Clause (3) of Definition 4.3, the set T of all α ∈ S such that | N α | = | α | and h α, ∈ , A ∩ ( α × α ) i | = N α Φ ∧ Ψ is stationary. Evidently, every α ∈ T \ ( ω + 1)is an element of S ′ satisfying that, for all ι < α , Y ι ∩ α ∈ F + α . (cid:3) It follows in particular that S ′ is stationary. Finally, recalling Clause (2) ofDefinition 4.3, ~ F := hF α | α ∈ S ′ i captures clubs.(2) Continuing the proof of Clause (1), we see that h N α ∩ P ( α ) | α ∈ S ′ i isa ♦ ∗ S ′ -sequence, and ~ F witnesses that f -Refl( κ, κ, S ′ ) holds. The conclusion nowfollows from Lemma 3.10. (cid:3) In [FMR19], we proved that Dl ∗ S (Π ) holds in L for any stationary subset S ofany regular uncountable cardinal κ . Therefore: Corollary 4.6.
Suppose V = L . Then, for every stationary S ⊆ κ , κ f -reflects with ♦ to S . In particular, fake reflection holds at any non weakly compact cardinal. (cid:3) Furthermore, in [FMR19], we proved that Dl ∗ S (Π ) follows from a forceable con-densation principle called “Local Club Condensation” ( LCC ). In effect, f -reflectionis forceable (without assuming any large cardinals). In this short section, we shallpresent an alternative and simpler poset for forcing Dl ∗ S (Π ) to hold. The idea isto connect the latter with the following strong form of diamond due to Sakai. Definition 4.7 (Sakai, [Sak11a]) . ♦ ++ asserts the existence of a sequence h K α | α < ω i satisfying the following:(1) for every α < ω , K α is a countable set;(2) for every X ⊆ ω , there exists a club C ⊆ ω such that, for all α ∈ C , C ∩ α, X ∩ α ∈ K α ;(3) the following set is stationary in [ H ω ] ω : { M ∈ [ H ω ] ω | M ∩ ω ∈ ω ∧ clps( M, ∈ ) = ( K M ∩ ω , ∈ ) } . First, we generalize Sakai’s principle in the obvious way.
Definition 4.8.
For a stationary S ⊆ κ , ♦ ++ S asserts the existence of a sequence h K α | α ∈ S i satisfying the following:(1) for every infinite α ∈ S , K α is a set of size | α | ;(2) for every X ⊆ κ , there exists a club C ⊆ κ such that, for all α ∈ C ∩ S , C ∩ α, X ∩ α ∈ K α ;(3) the following set is stationary in [ H κ + ] <κ : { M ∈ [ H κ + ] <κ | M ∩ κ ∈ S & clps( M, ∈ ) = ( K M ∩ κ , ∈ ) } . Recall Convention 4.2.
AKE REFLECTION 17
Remark . For a structure M , clps( M ) denotes its Mostowski collapse. Hereafter, ZF − denotes ZF without the powerset axiom. Lemma 4.10.
For every stationary S ⊆ κ , ♦ ++ S implies ♦ + S (Π ) .Proof. Suppose h K α | α ∈ S i is a ♦ ++ S -sequence. Define a sequence ~N = h N α | α ∈ S i by letting N α be the p.r.-closure of K α ∪ ( α + 1). By the way the sequence ~N was constructed, N α sees α for all α ∈ S , and by Clause (1) of Definition 4.8,for every infinite α ∈ S , | N α | = | α | . In addition, for every X ⊆ κ , there exists aclub C ⊆ κ such that C ∩ α, X ∩ α ∈ K α ⊆ N α for all α ∈ C ∩ S .Let us show that ~N satisfies Clause (3) of Definition 4.3 with n = 2. To this end,let φ = ∀ X ∃ Y ϕ be a Π -sentence and ( A m ) m ∈ ω be such that h κ, ∈ , ( A m ) m ∈ ω i | = φ .Given an arbitrary club C ⊆ κ , we consider the following set C := { M ≺ H κ + | M ∩ κ ∈ C & ( A m ) m ∈ ω ∈ M } . Claim 4.10.1. C is a club in [ H κ + ] <κ .Proof. By the L¨owenheim–Skolem theorem, for every B ∈ [ H κ + ] <κ , we know that M B = { M ∈ [ H κ + ] <κ | B ⊆ M ≺ H κ + & ( A m ) m ∈ ω ∈ M } is a club in [ H κ + ] <κ , so that κ ∩ { M ∩ κ | M ∈ M B } is a club in κ . Consequently, C is cofinal in [ H κ + ] <κ .To see that C is closed, assume we are given a chain M ⊆ M ⊆ · · · of length α < κ of elements of C . As this is a chain of elementary submodels of H κ + of sizesmaller than κ , M ∗ := S i<α M i is an elementary submodel of H κ + with M ∗ ∩ κ ∈ C ,so that M ∗ ∈ C . (cid:3) Since ~K a ♦ ++ S -sequence, we may now pick M in the following intersection C ∩ { M ∈ [ H κ + ] <κ | M ∩ κ ∈ S & clps( M, ∈ ) = ( K M ∩ κ , ∈ ) } . So, M ≺ H κ + , ( A m ) m ∈ ω ∈ M , α := M ∩ κ is in S ∩ C , and clps( M, ∈ ) = ( K α , ∈ ).As M ∩ ( κ + 1) = α ∪ { κ } , α ∪ { α } is a subset of the collapse of M , so that K α sees α and N α = K α . Let π : M → N α denote the transitive collapsing map. Notethat(i) π ↾ α = id α ,(ii) π ( κ ) = α , and(iii) π ( A m ) = A m ∩ α , for all m ∈ ω .Since h κ, ∈ , ( A m ) m ∈ ω i | = φ , by definition, h κ, ∈ , ( A m ) m ∈ ω i | = H κ + ∀ X ∃ Y ϕ . Thatis, H κ + | = “ ∀ X ⊆ κ m ( X ) ∃ Y ⊆ κ m ( Y ) h κ, ∈ , ( A m ) m ∈ ω i | = ϕ ” . By elementarity and the fact that “ ∀ X ⊆ κ m ( X ) ∃ Y ⊆ κ m ( Y ) ( h κ, ∈ , ( A m ) m ∈ ω i | = ϕ )”is equivalent to ∀ X (( ∀ x ( x ∈ X → x ∈ κ m ( X ) )) → ( ∃ Y (( ∀ y ( y ∈ Y → y ∈ κ m ( Y ) )) ∧ h κ, ∈ , ( A m ) m ∈ ω i | = ϕ ))) , which is a first-order formula in the parameters m ( X ), m ( Y ), κ , h κ, ∈ , ~A i and ϕ ,we have M | = “ ∀ X ⊆ κ m ( X ) ∃ Y ⊆ κ m ( Y ) ( h κ, ∈ , ( A m ) m ∈ ω i | = ϕ )” . Since π is an elementary embedding, π [ M ] | = “ ∀ X ⊆ π ( κ m ( X ) ) ∃ Y ⊆ π ( κ m ( Y ) )( h π ( κ ) , ∈ , ( π ( A m )) m ∈ ω i | = ϕ )” . By the properties (i),(ii) and (iii) of π it follows that N α | = “ ∀ X ⊆ α m ( X ) ∃ Y ⊆ α m ( Y ) ( h α, ∈ , ( A m ∩ ( α m ( A m ) )) m ∈ ω i | = ϕ )” . We conclude h α, ∈ , ( A n ∩ ( α m ( A n ) )) n ∈ ω i | = N α φ , as sought. (cid:3) Remark . An obvious tweaking of the above proof shows that h N α | α ∈ S i infact witnesses ♦ + S (Π n ) for every positive integer n .The following answers a question of Thilo Weinert: Corollary 4.12.
It is consistent that ♦ + S holds, but ♦ ++ S fails.Proof. In [FMR19, § ♦ + S holds for S := ω ∩ cof( ω )but Dl ∗ S (Π ) fails. By Lemma 4.10, ♦ ++ S fails in this model, as well. (cid:3) In [Sak11a, Definition 3.1], Sakai presented a poset for forcing ♦ ++ to hold. Thefollowing is an obvious generalization (and a minor simplification) of Sakai’s poset. Definition 4.13.
Let S be the poset of all pairs ( k, B ) with the following properties:(1) k is a function such that dom( k ) < κ ;(2) for each α ∈ dom( k ) , k ( α ) is a transitive model of ZF − of size ≤ max {ℵ , | α |} ,with k ↾ α ∈ k ( α );(3) B is a subset of P ( κ ) of size ≤ dom( k );( k ′ , B ′ ) ≤ ( k, B ) in P if the following holds:(i) k ′ ⊇ k , and B ′ ⊇ B ;(ii) for any B ∈ B and any α ∈ dom( k ′ ) \ dom( k ), B ∩ α ∈ k ′ ( α ).It is clear that S is <κ -closed. Also, since we assume κ <κ = κ , S has the κ + -cc.Finally, Sakai’s proof of [Sak11a, Lemma 3.4] makes clear that the following holds. Proposition 4.14.
For every stationary S ⊆ κ , V S | = ♦ ++ S . (cid:3) Note that while Sakai’s forcing is considerably simpler than the poset to force
LCC to hold, it only yields “ κ f -reflects to S ” for stationary subsets S ⊆ κ from theground model, whereas, LCC imply that κ f -reflects to S for any stationary S ⊆ κ . Remark . For stationary subsets
X, S of κ , if X f -reflects to S , then for everynotion of forcing P of size <κ , V P | = X f -reflects to S . It takes a little more effort,but it can be shown that if X f -reflects with ♦ to S , then for every notion of forcing P of size <κ , V P | = X f -reflects with ♦ to S .In this section and in the previous one, we have collected a long list of sufficientconditions for filter reflection to hold. We have seen it is compatible with largecardinals, strong forcing axioms, but also with inner models like L , in which anti-reflection principles like (cid:3) λ hold. This suggests it is not trivial to destroy filterreflection. The next section is dedicated to demonstrating it is nevertheless possible.5. Killing fake reflection
Definition 5.1.
Let X ⊆ κ . We define a collection I [ κ − X ], as follows.A set Y is in I [ κ − X ] iff Y ⊆ κ and there exists a sequence h a β | β < κ i ofelements of [ κ ] <κ along with a club C ⊆ κ such that, for every δ ∈ Y ∩ C , there isa cofinal subset A ⊆ δ of order-type cf( δ ) such that(1) { A ∩ γ | γ < δ } ⊆ { a β | β < δ } , and(2) acc + ( A ) ∩ X = ∅ . Remark . Note that I [ κ − X ] is an ideal, and that X ⊆ X ′ entails I [ κ − X ] ⊇ I [ κ − X ′ ]. Shelah’s approachability ideal I [ κ ] is equal to I [ κ − ∅ ] ↾ Sing (cf. [Eis10]).In particular, for every µ ∈ Reg( κ ), I [ κ ] ↾ cof( µ ) coincides with I [ κ − ∅ ] ↾ cof( µ ). Fact 5.3 (folklore) . Every separative <κ -closed notion of forcing of size κ is forcingequivalent to Add( κ, . Communicated in person to the third author in 2017.
AKE REFLECTION 19
Theorem 5.4.
Suppose
X, S are disjoint stationary subsets of κ , with S ∈ I [ κ − X ] .For every ~ F = hF α | α ∈ S i , V Add( κ, | = X does not ~ F -reflect to S .Proof. Towards a contradiction, suppose that ~ F is a counterexample. As Add( κ, X, S, ~ F live in the ground model, it follows that, infact, V Add( κ, | = X ~ F -reflects to S .Let R denote the set of all pairs ( p, q ) ∈ <κ × <κ such that: • dom( p ) = dom( q ) is in nacc( κ ); • { α ∈ dom( p ) | p ( α ) = q ( α ) = 1 } is disjoint from X ; • { α ∈ dom( q ) | q ( α ) = 1 } is a closed set of ordinals.We let R := ( R, ≤ ) where ( p ′ , q ′ ) ≤ ( p, q ) iff p ′ ⊇ p and q ′ ⊇ q . Claim 5.4.1. R is <κ -closed.Proof. Given θ ∈ acc( κ ) and a strictly decreasing sequence h ( p i , q i ) | i < θ i ofconditions in R , let p := ( S i<θ p i ) y q := ( S i<θ q i ) y
1. Clearly, ( p, q ) is alegitimate condition extending ( p i , q i ) for all i < θ . (cid:3) It thus follows from Fact 5.3 that R is forcing equivalent to Add( κ, P := { p | ∃ q ( p, q ) ∈ R } . It is easy to see that P := ( P, ⊇ ) is <κ -closed, so that P is, as well, forcing equivalent to Add( κ, G be R -generic over V . Let G denote the projection of G to the first coordinate, so that G is P -generic over V . In V [ G ], let Q := { q ∈ <κ | ∃ p ∈ G ( p, q ) ∈ R } . Clearly, Q := ( Q, ⊇ ) isisomorphic to the quotient forcing R /G . It follows that, in V [ G ], we may read a Q -generic set G over V [ G ] such that, in particular, V [ G ] = V [ G ][ G ].Denote η := S G and let Y := { α ∈ X | η ( α ) = 1 } . Claim 5.4.2. In V [ G ] , Y is stationary.Proof. We run a density argument for P in V . Let ˙ Y be the P -name for Y , that is,˙ Y := { (ˇ α, p ) | p ∈ P, α ∈ X ∩ dom( p ) , p ( α ) = 1 } . Let p be an arbitrary condition that P -forces that some ˙ D is a P -name for a clubin κ ; we shall find p • ⊇ p such that p • (cid:13) P ˙ D ∩ ˙ Y = ∅ .Recursively define a sequence h ( p i , α i ) | i < κ i as follows: ◮ Let ( p , α ) be such that p ⊇ p and p (cid:13) P ˇ α ∈ ˙ D . ◮ Suppose that i < κ for which h ( p j , α j ) | j ≤ i i has already been defined.Set ε i := max { α i , dom( p i ) } + 1. Then pick p i +1 ⊇ p i and α i +1 < κ suchthat ε i ∈ dom( p i +1 ) and p i +1 (cid:13) P ˇ α i +1 ∈ ˙ D \ ˇ ε i . ◮ Suppose that i ∈ acc( κ ) and that h ( p j , α j ) | j < i i has already been defined.Evidently, sup j
1, so that p i is a legitimate condition satis-fying dom( p i ) = α i + 1 and p i ( α i ) = 1.This completes the recursive construction. Evidently, E := { α i | i < κ } is a club, soas X is stationary, we may pick β ∈ X such that α β = β . Then p β (cid:13) P ˇ β ∈ ˙ D ∩ ˇ X ,so that, from p β ( β ) = 1, we infer that p β (cid:13) P ˙ D ∩ ˙ Y = ∅ . (cid:3) Work in V [ G ]. Since X ~ F -reflects to S , T := { α ∈ S | Y ∩ α ∈ F + α } is stationary. Claim 5.4.3. In V [ G ][ G ] , T is stationary. Proof.
Fix ~a , C in V that witness together that S is in I [ κ − X ]. As P is cofinality-preserving, in V [ G ], the above two still witness together that S is in I [ κ − X ].Work in V [ G ]. As T is a subset of S , ~a , C also witness together that T is in I [ κ − X ].We now run a density argument for Q in V [ G ]. Let q be an arbitrary conditionthat Q -forces that some ˙ D is a Q -name for a club in κ ; we shall find q • ⊇ q suchthat q • (cid:13) Q ˙ D ∩ ˇ T = ∅ .Fix a large enough regular cardinal Θ and some well-ordering < Θ of H Θ . ByClaim 5.4.2, T is stationary, so we may find an elementary submodel N ≺ ( H Θ , < Θ )such that ~a, C, Q , q, ˙ D ∈ N and δ := N ∩ κ is in T .As C ∈ N , we altogether have δ ∈ C ∩ T . Thus, we may pick a cofinal subset A ⊆ δ with otp( A ) = cf( δ ) and acc + ( A ) ∩ X = ∅ such that: { A ∩ γ | γ < δ } ⊆ { a β | β < δ } . In particular, any proper initial segment of A is in N .Let h δ i | i < cf( δ ) i be the increasing enumeration of A . For every initial segment a of A , we recursively define the following sequence h ( q i , α i ) | i ≤ σ ( a ) i , where σ ( a )will the length of the recursion (see the second case below). ◮ Let q be the < Θ -least condition in Q extending q for which there is α < κ such that q (cid:13) Q ˇ α ∈ ˙ D . Now, let α be the < Θ -least ordinal α such that q (cid:13) Q ˇ α ∈ ˙ D . ◮ Suppose that h ( q j , α j ) | j ≤ i i has already been defined. If a \ max { α i , dom( q i ) , δ i } is empty, then we terminate the recursion, and set σ ( a ) := i .Otherwise, let ε i be the < Θ -least element of a \ max { α i , dom( q i ) , δ i } , andthen let q i +1 be the < Θ -least condition in Q extending q i satisfying ε i ∈ dom( q i +1 ) and satisfying that there is α < κ such that q i +1 (cid:13) Q ˇ α ∈ ˙ D \ ε i .Now, let α i +1 be the < Θ -least ordinal α such that q i +1 (cid:13) Q ˇ α ∈ ˙ D \ ε i . ◮ Suppose that i is a limit ordinal and that h ( q j , α j ) | j < i i has already beendefined. Evidently,sup j
Work in V [ G ][ G ]. Fix a club C disjoint from Y . Since X ~ F -reflects to S ,in particular, ~ F capture clubs, so that { α ∈ S | C ∩ α / ∈ F α } is non-stationary.Recalling that T is stationary, we now fix α ∈ T such that C ∩ α ∈ F α . By definitionof T , we also have Y ∩ α ∈ F + α , so that ( C ∩ α ) ∩ ( Y ∩ α ) is nonempty, contradictingthe fact that C is disjoint from Y . (cid:3) Corollary 5.5.
Suppose
X, S are disjoint stationary subsets of κ , with S ∈ I [ κ − X ] .After forcing with Add( κ, κ + ) , X does not f -reflect to S .Proof. Let G be Add( κ, κ + )-generic over V . For every ι ≤ κ + , let G ι denote theprojection of G into the ι th stage.Work in V [ G κ + ]. Towards a contradiction, suppose that S ′ is a stationary sub-set of S , and ~ F = hF α | α ∈ S ′ i is a sequence such that X ~ F -reflects to S ′ .As Add( κ, κ + ) does not add bounded subsets of κ , ~ F ⊆ ( H κ ) V . In addition,Add( κ, κ + ) has the κ + -cc, so that, altogether, ~ F admits a nice name of size κ .It follows that we may find a large enough ι < κ + such that ~ F is in V [ G ι ].Now, by Theorem 5.4, V [ G ι +1 ] | = “ X does not ~ F -reflect to S ′ ”. Recalling that V [ G κ + ] | = “ X ~ F -reflect to S ′ ”, it must be the case that there exists a stationarysubset of κ in V [ G ι +1 ] that ceases to be stationary in V [ G κ + ]. However, the quo-tient forcing Add( κ, κ + ) /G ι +1 is isomorphic to Add( κ, κ + ) and the latter preservesstationary subsets of κ . This is a contradiction. (cid:3) Lemma 5.6.
Suppose that κ is strongly inaccessible or κ = λ + with λ <λ = λ . Forevery stationary X, Y ⊆ κ such that Tr( X ) ∩ Y is non-stationary, Y ∈ I [ κ − X ] .Proof. Let θ := sup(Reg( κ )), so that κ ∈ { θ, θ + } , and, for every γ < κ , | [ γ ] <θ | < κ .Let h a β | β < κ i be some enumeration of [ κ ] <θ , and then define a function f : κ → κ via: f ( γ ) := min { τ < κ | [ γ ] <θ ⊆ { a β | β < τ }} . Now, fix arbitrary
X, Y ⊆ κ for which Tr( X ) ∩ Y is non-stationary. To see that Y ∈ I [ κ − X ], fix a subclub C of { δ < κ | f [ δ ] ⊆ δ } disjoint from Tr( X ) ∩ Y . Let δ ∈ Y ∩ C be arbitrary. Fix a club A in δ of order-type cf( δ ) such that A ∩ X = ∅ . Bydefinition of θ , we have cf( δ ) ≤ θ , so that, for every γ < δ , we have otp( A ∩ γ ) < θ and f ( γ ) < δ , and hence there must exist some β < δ with A ∩ γ = a β . (cid:3) We are now ready to derive Theorem C:
Corollary 5.7. If κ is strongly inaccessible, then in the forcing extension by Add( κ, κ + ) , for all two disjoint stationary subsets X, S of κ , the following areequivalent: (1) X f -reflects to S ; (2) every stationary subset of X reflects in S .Proof. The implication (2) = ⇒ (1) holds true in any model, since if X reflects in S , then S ′ := S \ cof( ω ) must be stationary, and X h CUB( α ) | α ∈ S ′ i -reflects to S ′ . In particular, X f -reflects to S . Thus, we shall focus on the other implication.Let G be Add( κ, κ + )-generic over V . For every ι ≤ κ + , we let G ι denote theprojection of G into the ι th stage. Work in V [ G ]. We verify that ¬ (2) = ⇒ ¬ (1).Suppose that X and S are disjoint stationary subsets of κ such that X admits astationary subset Z ⊆ X that does not reflect in S . As Z and S are elements of H κ + and as Add( κ, κ + ) has the κ + -cc, we may find a large enough ι < κ + such that Z and S are in V [ G ι ]. As the quotient forcing Add( κ, κ + ) /G ι is isomorphic to Add( κ, κ + )and the latter does not add bounded subsets of κ and does preserve stationarysubsets of κ , also, in V [ G ι ], Z does not reflect in S . Now, by Lemma 5.6, S ∈ I [ κ − Z ]. So, since Add( κ, κ + ) /G ι is isomorphic to Add( κ, κ + ), Corollary 5.5 implies that Z does not f -reflect to S in V [ G ]. But Z ⊆ X , contradicting MonotonicityLemma 2.4. (cid:3) Definition 5.8.
Let ~C = h C α | α ∈ Γ i be some sequence, with Γ ⊆ OR. • ~C is said to be a C -sequence over Γ iff, for every α ∈ Γ, C α is a closedsubset of α with sup( C α ) = sup( α ); • ~C is said to be coherent iff, for all α ∈ Γ and ¯ α ∈ acc( C α ), ¯ α ∈ Γ and C ¯ α = C α ∩ ¯ α ; • ~C is said to be regressive iff otp( C α ) < α for all α ∈ Γ. Remark . (1) Jensen proved [Jen72] that if V = L , then there exists a coher-ent regressive C -sequence over Sing (the class of infinite singular ordinals).(2) Jensen’s principle (cid:3) λ is equivalent to the assertion that there exists a co-herent regressive C -sequence over a club in λ + .(3) By [She91], for every regular uncountable cardinal λ , there exists a sequence h Γ i | i < λ i such that S i<λ Γ i = acc( λ + \ λ ) ∩ cof( <λ ) and, for all i < λ ,there exists a coherent regressive C -sequence over Γ i .(4) By [Sak11b], MM implies the existence of a coherent regressive C -sequenceover some Γ ⊆ ω for which Γ ∩ cof( ω ) is stationary. Lemma 5.10.
Let Γ ⊆ Sing( κ ) be stationary. If there exists a coherent regressive C -sequence over Γ , then, for every stationary X ⊆ Γ , there exists a stationary Z ⊆ X with Γ ∈ I [ κ − Z ] .Proof. Suppose that h C α | α ∈ Γ i is a coherent regressive C -sequence. For every ǫ < κ , let Γ ǫ := { α ∈ Γ | otp( C α ) = ǫ } . By Fodor’s lemma, for every stationary X ⊆ Γ, there must exist some ǫ < κ such that X ∩ Γ ǫ is stationary. Thus, we shallfocus on proving that Γ ∈ I [ κ − Γ ǫ ] for all ǫ < κ .For all η ∈ κ ∩ cof( >ω ) and ǫ < κ , fix a subclub d η,ǫ of acc( η ) of order-type cf( η )such that ǫ / ∈ d η,ǫ ; then, for every α ∈ Γ, let C η,ǫα := { ζ ∈ C α | otp( C α ∩ ζ ) ∈ d η,ǫ } . To help the reader digest the above definition, we mention that each such a set C η,ǫα is a closed (possibly empty) subset of acc( C α ) of order-type ≤ otp( d η,ǫ ).Next, let ~a = h a β | β < κ i be some enumeration of[ κ ] <ω ∪ { C η,ǫα | α ∈ Γ , η ∈ κ ∩ cof( >ω ) , ǫ < κ } . Fix a club D in κ such that, for all δ < κ : • [ δ ] <ω ⊆ { a β | β < δ } , and • { C η,ǫα | α ∈ Γ ∩ δ, η ∈ δ ∩ cof( >ω ) , ǫ < δ } ⊆ { a β | β < δ } .Let ǫ < κ . We claim that ~a and D \ ( ǫ + 1) witness together that Γ ∈ I [ κ − Γ ǫ ].To this end, let δ ∈ Γ ∩ D \ ( ǫ + 1) be arbitrary. There are two cases to consider: ◮ If cf( δ ) = ω , then let A be an arbitrary cofinal subset of δ of order-type ω .Clearly, acc + ( A ) = ∅ . In addition, by δ ∈ D , any proper initial segment of A isindeed listed in { a β | β < δ } . ◮ If cf( δ ) > ω , then let η := otp( C δ ), so that η < δ . As C δ is a club in δ ,cf( η ) = cf( δ ). As d η,ǫ is a subclub of acc( η ) of order-type cf( η ) = cf( δ ), A := C η,ǫδ is a subclub of acc( C δ ) of order-type cf( δ ). Now, if α ∈ A ∩ Γ ǫ , then α ∈ acc( C δ ),so that α ∈ Γ and C δ ∩ α = C α , and also α ∈ Γ ǫ so that otp( C δ ∩ α ) = otp( C α ) = ǫ .Recalling that α ∈ A = C η,ǫδ = { ζ ∈ C δ | otp( C δ ∩ ζ ) ∈ d η,ǫ } , this means that ǫ ∈ d η,ǫ , contradicting the choice of d η,ǫ . It follows in particular that acc + ( A ) ∩ Γ ǫ = ∅ .Finally, let γ < δ , and we shall show that A ∩ γ ∈ { a β | β < δ } . Put α :=min( A \ γ ), so that A ∩ γ = A ∩ α . As observed earlier, the fact that α ∈ A entails C α = C δ ∩ α . But then C η,ǫα = C η,ǫδ ∩ α = A ∩ γ . Recalling that δ ∈ D and AKE REFLECTION 23 max { α, η, ǫ } < δ , we infer that the proper initial segment A ∩ γ is indeed listed in { a β | β < δ } . (cid:3) Corollary 5.11.
Suppose that there exists a coherent regressive C -sequence overa stationary subset Γ of κ . After forcing with Add( κ, κ + ) , for all two disjointstationary subsets X, S of Γ , X does not f -reflect to S .Proof. Let G be Add( κ, κ + )-generic over V . As in the proof of Corollary 5.5, forevery ι ≤ κ + , we let G ι denote the projection of G into the ι th stage.Work in V [ G ]. Suppose that X and S are disjoint stationary subsets of Γ. As X and S are elements of H κ + and as Add( κ, κ + ) has the κ + -cc, we may find a largeenough ι < κ + such that X and S are in V [ G ι ]. Now, in V , and hence also in V [ G ι ], there exists a coherent regressive C -sequence over Γ. So, by Lemma 5.10,there is a stationary Z ⊆ X such that Y ∈ I [ κ − Z ]. Finally, since Add( κ, κ + ) /G ι isisomorphic to Add( κ, κ + ), Corollary 5.5 implies that, in V [ G ], Z does not f -reflectto S . As Z ⊆ X , it follows from Monotonicity Lemma 2.4 that, in V [ G ], X doesnot f -reflect to S . (cid:3) We can now derive Theorem B:
Corollary 5.12 (Dense non-reflection) . There exists a cofinality-preserving forcingextension in which: (1)
For all stationary subsets
X, S of κ , there exist stationary subsets X ′ ⊆ X and S ′ ⊆ S such that X ′ does not f -reflect to S ′ ; (2) There exists an injection h : P ( κ ) → NS + κ such that, for all X, S ∈ P ( κ ) , X ⊆ S iff h ( X ) f -reflect to h ( S ) ; (3) For all two disjoint stationary subsets
X, S of κ , X does not f -reflect to S .Proof. A moment reflection makes it clear that Clauses (1) and (2) both followfrom Clause (3), so we focus on proving the latter.Suppose first that κ is a successor cardinal, say, κ = λ + . Let S denote the stan-dard cofinality-preserving notion of forcing for adding a (cid:3) λ -sequence (see [CFM01, § V [ H ][ G ], where H ∗ G is S ∗ Add( κ, κ + )-generic over V . As, in V [ H ], there exists a coherent regressive C -sequence over a club in κ , Corollary 5.11entails that, in V [ H ][ G ], for all two disjoint stationary subsets X, S of κ , X doesnot f -reflect to S .Next, suppose that κ is inaccessible. Let S := ( S , ≤ ), where c ∈ S iff c isclosed bounded subset of κ disjoint from Reg( κ ), and d ≤ c iff d end-extends c .It is easy to see that S is <κ -distributive, of size κ <κ = κ , and shoots a clubthrough Sing( κ ), so that, S is cofinality-preserving, and, in V S , κ is not Mahlo.Now, let ˙ S be the S -name for the poset from [CS02, §
6] for adding a coherentregressive C -sequence over Sing( κ ). Then S := S ∗ ˙ S is a cofinality-preservingnotion of forcing that adds a coherent regressive C -sequence over a club in κ . So,as in the previous case, any forcing extension by S ∗ Add( κ, κ + ) gives the desiredmodel. (cid:3) Lemma 5.13.
Let X ⊆ κ be stationary, and µ ∈ Reg( κ ) . (1) κ ∩ cof( ω ) ∈ I [ κ − X ] ; (2) If X ⊆ cof( ≥ µ ) , then I [ κ − ∅ ] ↾ cof( ≤ µ ) = I [ κ − X ] ↾ cof( ≤ µ ) ; (3) If µ <µ < κ , then S ∈ I [ κ − X ] ↾ cof( ≤ µ ) iff S ∈ I [ κ − ∅ ] ↾ cof( ≤ µ ) and Tr( X ) ∩ S is non-stationary. (4) If <λ = λ and X is non-reflecting, then I [ λ + − ∅ ] = I [ λ + − X ] . For every condition c , and every θ ∈ Reg( κ ), d := c ∪ { sup( c ) + θ } is a condition extending c such that S ↓ d is θ + -closed. Proof. (1) Let h a β | β < κ i be any enumeration of all finite subsets of κ , andconsider the club C := { δ < κ | [ δ ] <ω = { a β | β < δ }} .(2) The witness is the same.(3) The forward implication is clear, so we focus on the converse. Suppose S ∈ I [ κ − ∅ ] ↾ cof( ≤ µ ) and Tr( X ) ∩ S is non-stationary. Fix a list h a β | β < κ i anda club C witnessing together that S ∈ I [ κ − ∅ ]. By shrinking C , we may assumethat Tr( X ) ∩ S ∩ C = ∅ . As S ⊆ cof( ≤ µ ), we may also assume that | a β | < µ forall β < κ . Thus, assuming µ <µ < κ , we may let h a • β | β < κ i be some enumerationof S {P (cl( a β )) | β < κ } . Now, define a function f : κ → κ by letting for all β < κ : f ( β ) := min { α < κ | P (cl( a β )) ⊆ { a • γ | γ < α }} . Put D := { δ ∈ C | f [ δ ] ⊆ δ } . To see that h a • β | β < κ i and D witness togetherthat S ∈ I [ κ − X ], let δ ∈ S ∩ D be arbitrary. In particular, δ ∈ S ∩ C , and we mayfix a cofinal subset A ⊆ δ of order-type cf( δ ) such that any proper initial segment of A is listed in { a β | β < δ } . As δ ∈ S ∩ C , δ / ∈ Tr( X ), so we may fix a subclub A • ofacc + ( A ) which is disjoint from X . Now, let γ < δ be arbitrary. Put α := min( A \ γ )and then find β < δ such that A ∩ α = a β . It follows that A • ∩ γ ⊆ cl( a β ). So, as f [ δ ] ⊆ δ , we infer that A • ∩ γ is indeed listed in { a β | β < δ } .(4) This follows from (3). (cid:3) Corollary 5.14.
Suppose X ⊆ κ and S ⊆ κ ∩ cof( ω ) are stationary sets, for which X \ S is stationary. After forcing with Add( κ, κ + ) , X does not f -reflect to S .Proof. Consider the stationary set Z := X \ S . By Lemma 5.13(1), κ ∩ cof( ω ) ∈ I [ κ − Z ], in particular, S ∈ I [ κ − Z ]. By Corollary 5.5, we conclude that afterforcing with Add( κ, κ + ), Z does not f -reflect to S . Now, Monotonicity Lemma 2.4finishes the proof. (cid:3) Thus, we get a slight improvement of Corollary 4.12:
Corollary 5.15.
It is consistent that for S := ℵ ∩ cof( ω ) and X := ℵ ∩ cof( ω ) , ♦ + S holds, but X does not f -reflect to S .Proof. Suppose κ = ℵ = 2 ℵ , and work in the forcing extension by Add( κ, κ + ).By [Gre76, Lemma 2.1], ♦ + S holds. By Corollary 5.14, X does not f -reflect to S . (cid:3) Dense non-reduction
In the previous section, we showed how adding κ + many Cohen subsets of κ couldensure the failure of instances of f -refl at the level of κ . In this section, motivatedby Lemma 2.8, we shall derive a stronger conclusion.This section builds heavily on the ideas of Chapter 4, Section 4 of [FHK14],where, given X, Y stationary subsets of κ , it is forced under certain hypothesisthat = X ֒ → B = Y . Here we adapt their arguments to get = X ֒ → BM = Y , let alone= X ֒ → B = Y or = X ֒ → B = Y . In addition, our proof takes advantage of the ideal I [ κ − X ] from the previous section, hence, the findings here are applicable also for κ successor of singular in which Y concentrates on points of cofinality above thecofinality of the singular. Convention 6.1.
We denote elements of κ <κ by English letters (e.g., p and q ),and elements of κ κ by Greek letters (e.g., η and ξ ). Subsets of κ will be denotedby X, Y, Z and S . For all η ∈ κ κ and A ⊆ κ , we denote A η := { α ∈ A | η ( α ) = 0 } .We also let ~ κ -sequence with value 0. Definition 6.2.
A function F : 2 <κ × κ <κ → encode a map from κ to κ κ iff for all p ∈ <κ and q, q ′ ∈ κ <κ , F ( p, q ) = F ( p, q ′ ) = 1 entails that q ∪ q ′ ∈ κ <κ . AKE REFLECTION 25
The interpretation of F is the function F ∗ : 2 κ → κ κ defined as follows. Given η ∈ κ , if there exists ξ ∈ κ κ satisfying that, for all ε < κ , there is a tail of δ < κ ,such that F ( η ↾ δ, ξ ↾ ε ) = 1, then ξ is unique and we let F ∗ ( η ) := ξ . Otherwise, welet F ∗ ( η ) := ~ Definition 6.3.
Let θ ∈ [2 , κ ]. A basic open set in the space θ κ is a set of the form N p := { η ∈ θ κ | p ⊆ η } for some p ∈ θ <κ . A binary relation A ⊆ θ κ × θ κ is saidto be analytic iff there is a closed subset F of the product space θ κ × θ κ × θ κ suchthat A is equal to the projection pr( F ) := { ( η, ξ ) ∈ θ κ × θ κ | ∃ ζ ∈ θ κ ( η, ξ, ζ ) ∈ F } . Definition 6.4.
A subset D ⊆ κ is said to be comeager if D ⊇ T D for somenonempty family D of at most κ -many dense open subsets of 2 κ .A subset of 2 κ is said to be meager iff its complement is comeager. Fact 6.5 (Generalized Baire Category Theorem, [V¨a¨a11, Theorem 9.87]) . For anynonempty family D of at most κ -many dense open subsets of κ , T D is dense.In particular, the collection M κ of all meager subsets of κ forms a κ + -additiveproper ideal, and the intersection of a comeager set with a set of the form N p (asin Definition 6.3) is nonempty. Definition 6.6.
A function H : κ × κ → <κ is said to encode a comeager set iff,for every i < κ , the fiber H [ { i } × κ ] is cofinal in (2 <κ , ⊆ ).The interpretation of H is the set: H ∗ := \ i<κ [ j<κ N H ( i,j ) . The following is obvious.
Lemma 6.7.
For any comeager set D ⊆ κ , there is a function H : κ × κ → <κ encoding a comeager subset H ∗ of D . (cid:3) Definition 6.8.
A subset B ⊆ κ is said to have the Baire property iff there is anopen set U ⊆ κ for which the symmetric difference U △ B is meager.We denote by B κ the collection of all subsets of 2 κ having the Baire property. Proposition 6.9. B κ is an algebra which is closed under unions of length ≤ κ .Proof. Given B ∈ B κ , to see that B c := 2 κ \ B is in B κ , fix an open set U such that U △ B is meager, and let V denote the interior of U c := 2 κ \ U ; then B c △ V ⊆ ( B c △ U c ) ∪ ( U c \ V ) = ( B △ U ) ∪ ( U c \ V ) which is the union of two comeager sets.To see that B κ is closed under union of length ≤ κ , note that since M κ forms a κ + -additive ideal, this follows from the fact that, for any sequence of pairs h ( B i , U i ) | i < κ i , S { B i | i < κ } △ S { U i | i < κ } ⊆ S { B i △ U i | i < κ } . (cid:3) Definition 6.10.
A function f : 2 κ → κ κ is said to be Baire measurable iff forany open set U ⊆ κ , f − [ U ] has the Baire property. The existence of a Bairemeasurable reduction of R to R is denoted by R ֒ → BM R . Lemma 6.11 (folklore) . Baire measurable functions are continuous on a comeagerset.Proof.
Let f : 2 κ → κ κ be a Baire measurable function. For any p ∈ κ <κ , choosean open set U p ⊆ κ such that U p △ f − [ N p ] is meager. As κ <κ = κ , the followingset is comeager: D := κ κ \ [ p ∈ κ <κ ( U p △ f − [ N p ]) . Now, to see that f ↾ D is continuous, it suffices to show that, for every p ∈ κ <κ , D ∩ f − [ N p ] is an open set in D . But this is clear, since D ∩ f − [ N p ] = D ∩ U p . (cid:3) Definition 6.12.
For stationary subsets
X, Y of κ , we say that ( F, H ) is an (
X, Y ) -pair iff all of the following hold:(1) F encodes a map from 2 κ to κ κ ;(2) H encodes a comeager set;(3) For every η ∈ H ∗ , X η is stationary ⇐⇒ Y F ∗ ( η ) is stationary . Our next task is to show that if = X ֒ → BM = Y , then there exists an ( X, Y )-pair.For this, we first introduce the notion of positivity of a reduction and prove a lemmaabout it.
Definition 6.13.
For every function f : 2 κ → κ κ , we define the positivity of f , f + : 2 κ → κ κ , via: f + ( η )( α ) := , if f ( η )( α ) = f ( ~ α ); f ( η )( α ) , if f ( η )( α ) = f ( ~ α ) and f ( η )( α ) = 0; f ( ~ α ) , if f ( η )( α ) = f ( ~ α ) and f ( η )( α ) = 0 . Lemma 6.14.
Suppose f : 2 κ → κ κ is a Baire measurable function. Then so is f + .Proof. Since every open set is the union of at most κ <κ = κ many sets of the form N p , and since B κ is closed under unions of length ≤ κ , it suffices to verify that forevery p ∈ κ <κ , ( f + ) − [ N p ] has the Baire property. Given p ∈ κ <κ , if there are no η ∈ κ and β < κ such that p = f + ( η ) ↾ β , then ( f + ) − [ N p ] is empty, and we aredone. Thus, it suffices to consider p ’s of the form f + ( η ) ↾ β for η ∈ κ and β < κ .Let η ∈ κ and β < κ be arbitrary. Denote ξ := f + ( η ). In order to compute( f + ) − [ N ξ ↾ β ], we consider the following sets defined to handle each case in thedefinition of f + :a) B := { α < β | f ( η )( α ) = f ( ~ α ) } ;b) B := { α < β | f ( η )( α ) = f ( ~ α ) and f ( η )( α ) = 0 } ;c) B := { α < β | f ( η )( α ) = f ( ~ α ) and f ( η )( α ) = 0 } .Next, consider the following sets:A) Q := { q ∈ κ β | q ↾ B = f ( ~ ↾ B } ;B) Q := { q ∈ κ β | q ↾ B = f ( η ) ↾ B } ;C) Q := { q ∈ κ β | q ↾ B = f ( η ) ↾ B } .As f is Baire measurable, for each i < W i := [ { f − [ N q ] | q ∈ Q i } is the union of ≤ κ <κ = κ many sets, each having the Baire property. So, byProposition 6.9, W ∩ W ∩ W has the Baire property. Thus, the next claimfinishes the proof. Claim 6.14.1. ( f + ) − [ N ξ ↾ β ] = W ∩ W ∩ W .Proof. ( = ⇒ ) Suppose ζ ∈ W ∩ W ∩ W . Then f ( ζ ) ↾ B = f ( ~ ↾ B = f ( η ) ↾ B , f ( ζ ) ↾ B = f ( η ) ↾ B and f ( ζ ) ↾ B = f ( η ) ↾ B , which implies that f + ( ζ ) ∈ N ξ ↾ β .( ⇐ = ) Suppose ζ ∈ κ with f + ( ζ ) ∈ N ξ ↾ β . Then f + ( ζ ) ↾ β = ξ ↾ β = f + ( η ) ↾ β = ( ~ ↾ B ) ∪ ( f ( η ) ↾ B ) ∪ ( f ( ~ ↾ B ) . This implies that • f ( ζ ) ↾ B = f ( ~ ↾ B , • f ( ζ ) ↾ B = f ( η ) ↾ B , and • f ( ζ ) ↾ B = f ( η ) ↾ B = ~ ↾ B . AKE REFLECTION 27
Thus f ( ζ ) ∈ T i< S q ∈ Q i N q , so that ζ ∈ W ∩ W ∩ W . (cid:3) This completes the proof. (cid:3)
Lemma 6.15. If = X ֒ → BM = Y , then there exists an ( X, Y ) -pair.Proof. Suppose f : 2 κ → κ κ is a Baire measurable reduction from = X into = Y .Let f + be the positivity of f , so that, by Lemma 6.14, f + is Baire measurable.By Lemma 6.11, Baire measurable functions are continuous on a comeager set,so we may choose a function H : κ × κ → <κ encoding a comeager set H ∗ andsatisfying that f + ↾ H ∗ is continuous. Now, define a function F : 2 <κ × κ <κ → F ( p, q ) := 1 iff f + [ N p ∩ H ∗ ] ⊆ N q . Claim 6.15.1. ( F, H ) is an ( X, Y ) -pair.Proof. We go over the clauses of Definition 6.12:(1) Suppose p ∈ <κ , q, q ′ ∈ κ <κ and F ( p, q ) = 1 = F ( p, q ′ ). Then f + [ N p ∩ H ∗ ] ⊆ N q and f + [ N p ∩ H ∗ ] ⊆ N q ′ . Hence f + [ N p ∩ H ∗ ] ⊆ N q ∩ N q ′ .By Remark 6.5, the former is nonempty, so the latter is nonempty and q ∪ q ′ ∈ κ <κ . Altogether, F encodes a map from 2 κ to κ κ .Let us show that F ∗ ↾ H ∗ = f + ↾ H ∗ . For this, let η ∈ H ∗ be arbitrary.Set ξ := f + ( η ). By the continuity of f + ↾ H ∗ , for every ε < κ , there is atail of δ < κ such that N η ↾ δ ∩ H ∗ ⊆ ( f + ) − [ N ξ ↾ ε ] , so f + [ N η ↾ δ ∩ H ∗ ] ⊆ N ξ ↾ ε . Thus F ( η ↾ δ, ξ ↾ ε ) = 1 for a tail of δ < κ , and itfollows from Definition 6.2 that F ∗ ( η ) = ξ .(2) By its very choice, H encodes the comeager set H ∗ .(3) Since F ∗ ↾ H ∗ = f + ↾ H ∗ , we fix an arbitrary η ∈ H ∗ and prove that X η isstationary iff Y f + ( η ) is stationary. The following are equivalent:(a) X η is stationary;(b) { α ∈ X | η ( α ) = ~ α ) } is stationary;(c) ¬ ( η = X ~ ¬ ( f ( η ) = Y f ( ~ { α ∈ Y | f ( η )( α ) = f ( ~ α ) } is stationary;(f) { α ∈ Y | f + ( η )( α ) = 0 } is stationary.(g) Y f + ( η ) is stationary.The equivalence of (c) and (d) follows from the fact that f reduces = X to = Y . The equivalence of (e) and (f) follows from Definition 6.13 (cid:3) This completes the proof. (cid:3)
Theorem 6.16.
Suppose
X, Y are disjoint stationary subsets of κ , with Y ∈ I [ κ − X ] . For every pair ( F, H ) , V Add( κ, | = ( F, H ) is not an ( X, Y ) -pair.Proof. Towards a contradiction, suppose that (
F, H ) is a counterexample. AsAdd( κ,
1) is almost homogeneous and
X, Y, F, H live in the ground model, it followsthat, in fact, V Add( κ, | = ( F, H ) is an (
X, Y )-pair.Let R , P , G, G , Q , and G be all defined as in the proof of Theorem 5.4. Denote η := S G . Claim 6.16.1. In V [ G ] , η ∈ H ∗ .Proof. Let i < κ . The set H [ { i } × κ ] is cofinal in (2 <κ , ⊆ ), so, in particular, D i := { p y | p ∈ H [ { i } × κ ] } is dense in P . Pick p ∈ G ∩ D i . Then η ∈ N p ⊆ N p ↾ max(dom( p )) and the latter is equal to N H ( i,j ) for some j < κ . It thus followsthat η ∈ T i<κ S j<κ N H ( i,j ) = H ∗ . (cid:3) Claim 6.16.2. In V [ G ] , Y F ∗ ( η ) is stationary.Proof. In V [ G ], ( F, H ) is an (
X, Y )-pair, thus, to prove that Y F ∗ ( η ) is stationary,it suffices to prove that X η is stationary. But the latter is precisely what is provedin Claim 5.4.2. (cid:3) Claim 6.16.3. In V [ G ][ G ] , Y F ∗ ( η ) is stationary.Proof. This is the content of Claim 5.4.3. (cid:3)
Claim 6.16.4. In V [ G ][ G ] , X η is non-stationary.Proof. This is the content of Claim 5.4.4. (cid:3)
By Claims 6.16.1, 6.16.3 and 6.16.4 we infer that, in V [ G ], ( F, H ) is not an(
X, Y )-pair, contradicting the choice of (
F, H ). (cid:3) Corollary 6.17.
Suppose
X, Y are disjoint stationary subsets of κ , with Y ∈ I [ κ − X ] . After forcing with Add( κ, κ + ) , = X ֒ → BM = Y .Proof. Let G be Add( κ, κ + )-generic over V . For every ι ≤ κ + , let G ι denote theprojection of G into the ι th stage.Work in V [ G κ + ]. Towards a contradiction, suppose that f : 2 κ → κ κ is a Bairemeasurable reduction from = X to = Y . By Lemma 6.15, we may fix an ( X, Y )-pair, say (
F, H ). As F and H are elements of H κ + and as Add( κ, κ + ) has the κ + -cc, we may find a large enough ι < κ + such that ( F, H ) is in V [ G ι ]. Now, byTheorem 6.16, V [ G ι +1 ] | = “( F, H ) is not an (
X, Y )-pair”. As V and V [ G κ + ] havethe same bounded subsets of κ , the fact that ( F, H ) is an (
X, Y )-pair in V [ G κ + ],but not in V [ G ι +1 ] must mean that Clause (3) of Definition 6.12 fails in V [ G ι +1 ].It thus follows that there exists a stationary subset of κ in V [ G ι +1 ] that ceasesto be stationary in V [ G κ + ]. However, the quotient forcing Add( κ, κ + ) /G ι +1 isisomorphic to Add( κ, κ + ) and the latter preserves stationary subsets of κ . This isa contradiction. (cid:3) Corollary 6.18.
Suppose that there exists a coherent regressive C -sequence overa stationary subset Γ of κ . After forcing with Add( κ, κ + ) , for all two disjointstationary subsets X, Y of Γ , = X ֒ → BM = Y .Proof. Let G be Add( κ, κ + )-generic over V . As in the proof of Corollary 6.17, forevery ι ≤ κ + , we let G ι denote the projection of G into the ι th stage.Work in V [ G ]. Suppose that X and Y are disjoint stationary subsets of Γ. As X and Y are elements of H κ + and as Add( κ, κ + ) has the κ + -cc, we may find alarge enough ι < κ + such that X and Y are in V [ G ι ]. Now, in V , and hence alsoin V [ G ι ], there exists a coherent regressive C -sequence over Γ. So, by Lemma 5.10,there is a stationary Z ⊆ X such that Y ∈ I [ κ − Z ]. Finally, since Add( κ, κ + ) /G ι isisomorphic to Add( κ, κ + ), Corollary 6.17 implies that, in V [ G ], = Z ֒ → BM = Y . As Z ⊆ X , it follows from Monotonicity Lemma 2.7 that, in V [ G ], = X ֒ → BM = Y . (cid:3) Corollary 6.19 (Dense non-reduction) . There exists a cofinality-preserving forcingextension in which: (1)
For all stationary subsets
X, Y of κ , there exist stationary subsets X ′ ⊆ X and Y ′ ⊆ Y such that = X ′ ֒ → BM = Y ′ ; (2) There exists an injection h : P ( κ ) → NS + κ such that, for all X, Y ∈ P ( κ ) , X ⊆ Y iff = h ( X ) ֒ → BM = h ( Y ) ; (3) For all two disjoint stationary subsets
X, Y of κ , = X ֒ → BM = Y . AKE REFLECTION 29
Proof.
This is the same model of Corollary 5.12. That is, we force to add a coherentregressive C -sequence over a club in κ , and then add κ + many Cohen subsets of κ . The only difference is that this time we appeal to Corollary 6.18 instead of toCorollary 5.11. (cid:3) Corollary 6.20.
Suppose X ⊆ κ and S ⊆ κ ∩ cof( ω ) are stationary sets, for which X \ S is stationary. After forcing with Add( κ, κ + ) , = X ֒ → BM = S .Proof. Consider the stationary set Z := X \ S . By Lemma 5.13(1), κ ∩ cof( ω ) ∈ I [ κ − Z ], in particular, S ∈ I [ κ − Z ]. By Corollary 6.17, we conclude that afterforcing with Add( κ, κ + ), = Z ֒ → BM = S . Now, Monotonicity Lemma 2.7 finishes theproof. (cid:3) Based on the work done thus far, we are now able to answer a few questionsfrom the literature.
Definition 6.21.
An equivalence relation R is said to be Σ -complete iff it isanalytic and, for every analytic equivalence relation E , E ֒ → B R . Question 6.22 (Aspero-Hyttinen-Kulikov-Moreno, [AHKM19, Question 4.3]) . Isit consistent that κ is inaccessible and = S is not Σ -complete for some stationary S ⊆ κ ?We answer the preceding in the affirmative: Theorem 6.23. If κ is an inaccessible cardinal, then there exists a cofinality-preserving forcing extension in which ( κ is inaccessible, and) for every stationaryco-stationary S ⊆ κ , = S is not a Σ -complete equivalence relation.Proof. This is the forcing extension of Corollary 6.19. In this model, for all twodisjoint stationary subsets
X, S of κ , = X ֒ → BM = S . In particular = X ֒ → BM = S ,so that = S is not Σ -complete. (cid:3) Question 6.24 (Moreno, [Mor17, Question 4.16]) . Is it consistent that= κ ∩ cof( µ ) ֒ → B = κ ∩ cof( ν ) holds for all infinite regular cardinals µ = ν below κ ?We answer the preceding in the affirmative: Theorem 6.25.
There is a cofinality-preserving forcing extension, in which, forall infinite regular cardinals µ = ν below κ , = κ ∩ cof( µ ) ֒ → BM = κ ∩ cof( ν ) .Proof. This is the forcing extension of Corollary 6.19. For all infinite regular car-dinals µ = ν below κ , κ ∩ cof( µ ) and κ ∩ cof( ν ) are disjoint stationary subsets of κ ,and hence = κ ∩ cof( µ ) ֒ → BM = κ ∩ cof( ν ) . In particular, = κ ∩ cof( µ ) ֒ → BM = κ ∩ cof( ν ) . (cid:3) Question 6.26 (Aspero-Hyttinen-Kulikov-Moreno, [AHKM19, Question 2.12]) . Isit consistent that, for all infinite regular µ < ν < κ , the following hold?= κ ∩ cof( µ ) ֒ → B = κ ∩ cof( ν ) & = κ ∩ cof( ν ) ֒ → B = κ ∩ cof( µ ) . We answer the preceding in the affirmative, along the way, proving Theorem E:
Theorem 6.27.
Suppose MM holds. After forcing with Add( ω , ω ) , MM stillholds, and so are all of the following: (1) = S ֒ → = S ; (2) For every stationary X ⊆ S , = X ֒ → BM = S ; (3) There are stationary subsets X ⊆ S and Y ⊆ S such that = X ֒ → BM = Y ; (4) There is a stationary Y ⊆ S such that = S ֒ → BM = Y ; (5) = S ֒ → = S and = S ֒ → BM = S .Proof. We start with a model of V | = MM , and pass to V [ G ], where G is Add( ω , ω )-generic over V . By [Lar00, Theorem 4.3], MM is preserved by <ω -directed-closedforcing, so that V [ G ] | = MM .(1) MM implies that 2 ℵ = ℵ , which, by a Theorem of Shelah [She10], impliesthat ♦ S holds. Now appeal to Lemma 3.15.(2) This is Corollary 6.20.(3) Work in V . By [Sak11b], MM implies the existence of a coherent regressive C -sequence over some Γ ⊆ ω for which Y := Γ ∩ cof( ω ) is stationary. A momentreflection makes it clear that X := Γ ∩ cof( ω ) must be stationary, as well. Now, byCorollary 6.18, in V [ G ], = X ֒ → BM = Y .(4) By Clauses (1) and (3).(5) By Clauses (1) and (2). (cid:3) Acknowledgements
This research was partially supported by the European Research Council (grantagreement ERC-2018-StG 802756). The third author was also partially supportedby the Israel Science Foundation (grant agreement 2066/18). At the end of thepreparation of this article, the second author was visiting the Kurt G¨odel ResearchCenter supported by the Vilho, Yrj¨o and Kalle V¨ais¨al¨a Foundation of the FinnishAcademy of Science and Letters.We thank Andreas Lietz, Benjamin Miller, Ralf Schindler and Liuzhen Wu fortheir feedback on a preliminary version of this manuscript.
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Department of Mathematics, Bar-Ilan University, ramat-gan 5290002, Israel.
URL : http://u.math.biu.ac.il/~zanettg Department of Mathematics, Bar-Ilan University, ramat-gan 5290002, Israel.
URL : http://u.math.biu.ac.il/~morenom3 Department of Mathematics, Bar-Ilan University, ramat-gan 5290002, Israel.
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