aa r X i v : . [ m a t h . L O ] M a y NON-LINEAR ITERATIONS AND HIGHER SPLITTING
ÖMER FARUK BAĞ AND VERA FISCHER
Abstract.
We show that generalized eventually narrow sequences on a strongly inaccessiblecardinal κ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechlerforcing on κ . Moreover assuming GCH, κ <κ = κ , we show that(1) if κ is strongly unfoldable, κ + ≤ β = cf ( β ) ≤ cf ( δ ) ≤ δ ≤ µ and cf ( µ ) > κ , then there is acardinal preserving generic extension in which s ( κ ) = κ + ≤ b ( κ ) = β ≤ d ( κ ) = δ ≤ κ = µ. (2) if κ is strongly inaccessible, λ > κ + , then in the generic extension obtained as the < κ -support iteration of κ -Hechler forcing of length λ there are no κ -towers of length λ . Introduction
The topic “cardinal characteristics of the continuum” is a broad subject, which has been studiedin many research articles and surveys like [3] or [13]. Combinatorial cardinal invariants, someof which build the subject of this article, give an insight to the combinatorial and topologicalstructure of the (generalized) real line.The splitting, bounding and dominating numbers, denoted by s , b and d are due to David D.Booth, Fritz F. Rothberger and Miroslav Katětov respectively. While s , b ≤ d , the characteristics s and b are independent. Introducing the notion of an eventually narrow sequence and showing thepreservation of such sequences under the finite support iterations of Hechler poset for adjoining adominating real, Baumgartner and Dordal showed that consistently ℵ = s < b . The consistencyof b = ℵ < s = ℵ is due to S. Shelah and is in fact the first appearance of the method ofcreature forcing. Studying the existence of ultrafilters U , which have the property that for a givenunbounded family H ⊆ ω ω , the relativized Mathis poset M ( U ) preserves the family H unbounded(appearing more recently in the literature as H -Canjar filters, see [8]), the second author jointlywith J. Stepr¯ans generalized the result to an arbitrary regular uncountable κ , i.e. showed theconsistency of b = κ < s = κ + (see [7]). The more general inequality of b = κ < s = λ wasobtained only after certain developments of the method of matrix iteration, namely the appearanceof a method of preserving maximal almost disjoint families along matrix iterations introduced bythe second author and J. Brendle in [2]. Mathematics Subject Classification.
Key words and phrases. cardinal characteristics; eventually narrow sequences; generalised Baire spaces, largecardinals; forcing; non-linear iterations.
Acknowledgments. : The authors would like to thank the Austrian Science Fund (FWF) for the generous supportthrough Grant Y1012-N35.
In strong contrast to the countable case, the generalized bounding and splitting numbers are notindependent. Indeed, Raghavan and Shelah showed that s ( κ ) ≤ b ( κ ) for each regular uncountable κ . Moreover, by a result of Motoyoshi, s ( κ ) ≥ κ if and only if κ is strongly inaccessible. Later T.Suzuki showed that under the same assumption s ( κ ) ≥ κ + if and only if κ is weakly compact (seee.g. [12]). An easy diagonalization argument, shows that κ + ≤ b ( κ ) and so unless κ is weaklycompact, s ( κ ) < κ + ≤ b ( κ ) .In this article we want to further address the behaviour of s ( κ ) and b ( κ ) in the presence oflarge cardinal properties on κ . Our results are to a great extent based on the preservation of astrong splitting property, namely the preservation of generalized eventually narrow sequences inthe context of linear and non-linear iterations. Definition. (1) A sequence h a ξ : ξ < λ i , where each a ξ is in [ κ ] κ , is eventually narrow (we also say κ -eventually narrow) if ∀ a ∈ [ κ ] κ ∃ ξ < λ ∀ η > ξ a ∗ a η .(2) A sequence h a ξ : ξ < λ i , where each a ξ is in [ κ ] κ , is eventually splitting (we also say κ -eventually splitting) if ∀ a ∈ [ κ ] κ ∃ ξ < λ ∀ η > ξ a η splits a .Devising a special forcing notion D ( κ, Q ) , which can be interpreted as the non-linear iterationof the higher Hechler forcing (see Definition 9 and Definition 10), Cummings and Shelah showthat any admissible assignment to b ( κ ) , d ( κ ) and κ for κ regular uncountable, can be realized ina generic extension of a model of GCH (see Section 2). We show that if κ is strongly inaccessible,then generalized eventually narrow sequences on κ are preserved not only by linear iterations ofthe higher Hechler forcing, but also by the non-linear iterations of Cummings and Shelah, whichis our main preservation result (see Theorem 16): Theorem.
Assume GCH, κ <κ = κ , κ is strongly inaccessible, cf ( λ ) > κ . If τ = h a ξ : ξ < λ i is a κ -eventually narrow sequence in V , then τ remains eventually narrow in V D ( κ,Q ) .In analogy with [1] we introduce a notion of a derivative for a dense open subset of D ( κ, Q ) (see Definition 13 and Theorem 14), which is a key tool in obtaining the above theorem. Imposingfurther large cardinal properties on κ , we come to our main result: Theorem. (GCH, κ <κ = κ ) Assume κ is strongly unfoldable and β, δ, µ are cardinals with κ + ≤ β = cf ( β ) ≤ cf ( δ ) ≤ δ ≤ µ and cf ( µ ) > κ ; then there exists a cardinal preserving genericextension of the ground model, where s ( κ ) = κ + ∧ b ( κ ) = β ∧ d ( κ ) = δ ∧ c ( κ ) = µ holds.Our techniques modify to the countable case and so in particular establish the consistency of s = ω < b < d < c . Alternatively, one can force with B ( ω ) over a model of ω < b < d < c .Note however (at least to the best knowledge of the authors) that even though there are alreadygood higher analogues of random forcing (the poset for adjoining a single random real), we stilldo not seem to have an appropriate analogue of B ( ω ) into the uncountable.In addition, we show that in generic extensions obtained as the linear iteration of the higherHechler poset, there are no long κ -towers. ON-LINEAR ITERATIONS AND HIGHER SPLITTING 3
Corollary. (GCH, κ <κ = κ ) Assume κ is strongly inaccessible. Then there is a cardinal preservinggeneric extension where c ( κ ) = κ ++ and Spec ( t ( κ )) = { κ + } hold.Controlling s ( κ ) strictly above κ + simultaneously with b ( κ ) , d ( κ ) and κ remains an interestingopen question. For a model of ℵ < s < b = d < c (= a ) see [5].2. Preliminaries
Now we recall some preliminaries and definitions.
Definition 1.
Let κ be regular. Let a and b be elements in [ κ ] κ .(1) Then a ⊆ ∗ b holds, if | a \ b | < κ .(2) Further for a, b ∈ [ κ ] κ , we say a splits b if | b \ a | = | a ∩ b | = κ .(3) A family S ⊆ [ κ ] κ is splitting if ∀ b ∈ [ κ ] κ ∃ a ∈ S such that a splits b .(4) Finally s ( κ ) denotes the generalized splitting number: s ( κ ) = min {|S| : S ⊆ [ κ ] κ , S is splitting } . Definition 2.
Let κ be regular and let f and g be functions from κ to κ , i.e. f, g ∈ κ κ .(1) Then g eventually dominates f , denoted by f < ∗ g , if ∃ α < κ ∀ β > α f ( β ) < g ( β ) .(2) A family F ⊆ κ κ , is dominating if ∀ g ∈ κ κ ∃ f ∈ F such that g < ∗ f .(3) A family F ⊆ κ κ is unbounded if ∀ g ∈ κ κ ∃ f ∈ F such that f < ∗ g .(4) b ( κ ) and d ( κ ) denote the generalized bounding and dominating numbers respectively: b ( κ ) = min {|F | : F ⊆ κ κ, F is unbounded } d ( κ ) = min {|F | : F ⊆ κ κ, F is dominating } .(5) Finally c ( κ ) = 2 κ .In [4] it is shown that κ + ≤ b ( κ ) = cf ( b ( κ )) ≤ cf ( d ( κ )) ≤ d ( κ ) ≤ c ( κ ) holds.More generally one defines bounding and dominating for arbitrary posets as follows: Definition 3 ([4]) . Let ( P, ≤ P ) be a partial order.(1) We call U ⊆ P unbounded if ∀ p ∈ P ∃ q ∈ U q P p .(2) b ( P ) = min {| U | : U ⊆ P , U is unbounded } .(3) A subset D ⊆ P is dominating if ∀ p ∈ P ∃ q ∈ Dp ≤ P q .(4) d ( P ) = min {| D | : D ⊆ P , D is dominating } .As mentioned above it is known that the triple ( b ( κ ) , d ( κ ) , c ( κ )) can be anything not contra-dicting the known results in ZFC: Theorem 4 ([4]) . (GCH at and above κ , κ <κ = κ ) Suppose β, δ, µ are cardinals satisfying κ + ≤ β = cf ( β ) ≤ cf ( δ ) ≤ δ ≤ µ and cf ( µ ) > κ . Then there is a cardinal preserving forcing notion M ( κ, β, δ, µ ) such that V M ( κ,β,δ,µ ) (cid:15) b ( κ ) = β ∧ d ( κ ) = δ ∧ µ = c ( κ ) . Although the exact definition a strong unfoldable cardinal is not strictly necessary to under-stand the results of this article we state it for the sake of completeness. First if κ is stronglyinaccessible, then a κ -model denotes a transitive structure M of size κ , such that M (cid:15) ZF C − P , κ ∈ M and M <κ ⊆ M , i.e. M is closed under building sequences of size less than κ . ÖMER FARUK BAĞ AND VERA FISCHER
Definition 5 ([14], [10]) . (1) Let λ be an ordinal. A cardinal κ is λ -strongly unfoldable iff(a) κ is strongly inaccessible(b) for every κ -model M there is an elementary embedding j : M → N with criticalpoint κ such that λ < j ( κ ) and V κ ⊆ N .(2) A cardinal κ is called strongly unfoldable if it is θ -strongly unfoldable for every ordinal θ .A strongly unfoldable cardinal is in particular weakly compact. In the next section we will alsouse T. A. Johnstone’s theorem concerning the indestructiblity of strongly unfoldable cardinals. Theorem 6 ([10]) . Let κ be strongly unfoldable. Then there is a set forcing extension where thestrong unfoldability of κ is indestructible by forcing notions of any size which are < κ -closed andhave κ + -c.c.. Remark 7.
In Theorem 17 one could make the stronger assumption of a supercompact cardinal.Then Laver preparation can be used to make the supercompactness of κ indestructible by the κ -directed closed forcing poset D ( κ, Q ) . The straight forward proof of the later assertion is givenfor the sake of completeness. Lemma 8. D ( κ, Q ) is κ -directed closed. Proof.
Let W := { p α : α < γ } be a directed set of conditions where γ < κ . We define theircommon extension p as follows: dom ( p ) = S α<γ dom ( p α ) then | dom ( p ) | < κ by regularity. Forany α < γ and b ∈ dom ( p α ) let p α ( b ) = ( t bα , ˙ F bα ) . Then we define p ( b ) = ( t bp , ˙ F bp ) where t bp is theunion of the stems (since W is directed): t bp = S { t bα : b ∈ dom ( p α ) } and ˙ F bp is a P b -name for thepointwise supremum of the second coordinates { ˙ F bα : b ∈ dom ( p α ) } . Now it is easy to verify byan induction on the rank of b ∈ dom ( p ) (its Q -rank) that p is a common extension for W . (cid:3) Consistency of s ( κ ) = κ + < b ( κ ) < d ( κ ) < c ( κ ) In this section we want to obtain the consistency of κ + = s ( κ ) < b ( κ ) < d ( κ ) < c ( κ ) where κ is strongly unfoldable and b ( κ ) , d ( κ ) and c ( κ ) can be arbitrary uncountable cardinals β, δ, µ ≥ κ + not contradicting β = cf ( β ) ≤ cf ( δ ) ≤ δ ≤ µ and cf ( µ ) > κ . Unless otherwise specified, κ is aregular uncountable cardinal.Let κ <κ ↑ := { s ∈ κ <κ : s is strictly increasing } and κ κ ↑ := { f ∈ κ κ : f is strictly increasing } . Definition 9.
The κ -Hechler poset is defined as the set H ( κ ) = { ( s, f ) : s ∈ κ <κ ↑ , f ∈ κ κ ↑} with extension relation given by ( t, g ) ≤ H ( κ ) ( s, f ) iff s ⊆ t ∧ ∀ α ∈ κ [ g ( α ) ≥ f ( α )] ∧ ∀ α ∈ dom ( t ) \ dom ( s ) [ t ( α ) > f ( α )] . When κ is clear from the context, we write just H instead of H ( κ ) .Next we recall the definition of a non-linear forcing iteration D ( κ, Q ) from [4] with the additionalassumption that the stems and second coordinates are strictly increasing. Note that the poset isdense in the original one. ON-LINEAR ITERATIONS AND HIGHER SPLITTING 5
Definition 10.
Let ( Q, ≤ Q ) be a well-founded poset such that κ + ≤ b (( Q, ≤ Q )) . Extend Q to apartial order Q ′ = Q ∪ { m } with a maximal element m . Recursively on Q ′ , define for each a ∈ Q ′ a forcing notion P a as follows: • Fix a ∈ Q ′ and suppose for each b < Q ′ a the poset P b has been defined. Then P a consistsof functions p such that dom ( p ) ⊆ a ↓ := { c ∈ Q ′ : c < Q ′ a } and | dom ( p ) | < κ and ∀ b ∈ dom ( p ) p ( b ) = ( t, ˙ F ) where t ∈ κ <κ ↑ and ˙ F is a P b -name for an element in κ κ ↑ . • Thus for every a ∈ Q ′ each p ∈ P a is of the form p = (¯ p , ¯ p ) , where ¯ p = h s ( b ) i b ∈ dom ( p ) , ¯ p = h ˙ F b i b ∈ dom ( p ) and each pair ( s ( b ) , ˙ F b ) is a P b -name for a κ -Hechler condition. • For p ∈ P a and c ∈ Q ′ let p ↾ c ↓ = ( h s ( b ) i b ∈ dom ( p ) ∩ c ↓ , h ˙ F b i b ∈ dom ( p ) ∩ c ↓ ) .The extension relation of P a is defined as follows: p ≤ q iff dom ( q ) ⊆ dom ( p ) and ∀ b ∈ dom ( q ) [ p ↾ b (cid:13) P b (¯ p ( b ) , ¯ p ( b )) ≤ H ( κ ) (¯ q ( b ) , ¯ q ( b ))] . Finally, let D ( κ, Q ) = P m . Remark 11. (1) The fact that D ( κ, Q ) has the κ + -c.c. and κ -closed is shown in [4].(2) If λ is a regular uncountable cardinal and ( Q, ≤ Q ) = ( λ, ∈ ) , then D ( κ, Q ) = D ( κ, ( λ, ∈ )) is the < κ support iteration of H ( κ ) . Note also, that b (( λ, ∈ )) = d (( λ, ∈ )) = λ .Whenever X and Y are given sets, let f in <κ ( X, Y ) = { f : f is a partial function from X to Y , | dom ( f ) | < κ } . Definition 12.
Whenever ¯ s ∈ f in <κ ( Q, κ <κ ↑ ) , we denote by l ¯ s ∈ dom (¯ s ) κ the lengths of thesequences in ¯ s , i.e. l ¯ s = h dom (¯ s ( a )) i a ∈ dom (¯ s ) . Definition 13.
Let D be open dense in D ( κ, Q ) , i.e. ∀ p ∈ D ( κ, Q ) ∃ q ∈ D such that q ≤ p andwhenever p ∈ D and q ≤ p then q ∈ D . Define a sequence of subsets of f in <κ ( Q, κ <κ ↑ ) , referredto as a sequence of derivatives, as follows:(1) D = { ¯ s ∈ f in <κ ( Q, κ <κ ↑ ) | ∃ p ∈ D [¯ p = ¯ s ] } ,(2) D α +1 = { ¯ s ∈ f in <κ ( Q, κ <κ ↑ ) | (a) ¯ s ∈ D α , or(b) ∃ ¯ t ∈ D α ∃ ! a ∈ dom (¯ t ) such that dom (¯ s ) = dom (¯ t ) \ { a } ∧ ¯ t ↾ dom (¯ s ) = ¯ s , or(c) ∃ ¯ l ∈ f in <κ ( Q, κ ) [ dom (¯ l ) = dom (¯ s )] ∧ ∀ c ∈ dom (¯ l ) [¯ l ( c ) ≥ dom (¯ s ( c ))] ∧ ∃ c ∈ dom (¯ l ) [¯ l ( c ) > dom (¯ s ( c ))] and ∃{ ¯ t β : β < κ } ⊆ D α ∀ β < κ [¯ s ⊆ ¯ t β ∧ l ¯ t β ↾ dom (¯ l ) =¯ l ∧ ∀ b ∈ dom (¯ l ) [¯ t β ( b )( dom (¯ s ( b ))) > β ]] } , and(3) D α = S { D β | β < α } if α is a limit ordinal.Item (2b) says that the hierarchy of derivatives is closed under shortening the domain, i.e.whenever a sequence (of sequences) ¯ t appears in the hierarchy and the sequence ¯ s is obtainedfrom ¯ t only by forgetting points in the domain of ¯ t , then ¯ s also appears in the hierarchy ofderivatives (at a higher level). In item (2c) first ¯ l fixes a sequence of lengths on a domain. Thenfor every β ∈ κ a sequence of sequences ¯ t β is found such that each one’s domain contain thedomain of ¯ l and on this domain the lengths of the sequences in ¯ t β coincide with the lengths fixed ÖMER FARUK BAĞ AND VERA FISCHER by ¯ l (how each ¯ t β behaves outside this domain dom (¯ l ) doesn’t matter). Further each sequence in ¯ t β is an end-extension of the sequence in ¯ s at the same point and if the former is strictly longerthan the latter, then it goes above any value on its new domain.Due to (2a) and (3) this sequence is increasing, i.e. D α ⊆ D α +1 . Consequently this increasingsequence of derivatives has to stabilize at some index below | f in <κ ( Q, κ <κ ↑ ) | + , that is thereexists γ < | f in <κ ( Q, κ <κ ↑ ) | + such that D γ = D γ +1 . Theorem 14.
Assume GCH, κ <κ = κ and κ is strongly inaccessible. Let γ be the least such that D γ = D γ +1 . Then D γ = f in <κ ( Q, κ <κ ↑ ) .Proof. Suppose not and let ¯ s ∈ f in <κ ( Q, κ <κ ↑ ) \ D γ . For the purposes of this proof, we will usethe following notion: Definition.
A sequence ¯ t ∈ D γ is said to be a minimal extension of ¯ s if dom (¯ t ) = dom (¯ s ) , ¯ s ⊆ ¯ t and whenever ¯ l ∈ dom (¯ s ) κ is such that l ¯ s ≤ ¯ l ≤ l ¯ t pointwise and ∃ c ∈ dom (¯ s ) [¯ l ( c ) < l ¯ t ( c ))] then ¯ t ↾ ¯ l D γ , where ¯ t ↾ ¯ l = h ¯ t ( a ) ↾ ¯ l ( a ) : a ∈ dom (¯ t ) i .For the first let us claim, that for given lengths on dom (¯ s ) , there are less than κ many minimalextensions with these lengths: Claim.
For every ¯ l ∈ dom (¯ s ) κ we have that | T ¯ l | < κ where T ¯ l := { ¯ t : ¯ t is a minimal extension of ¯ s with l ¯ t = ¯ l } . Proof of the Claim.
Suppose not and let ¯ l ∈ dom (¯ s ) κ be such that | T ¯ l | ≥ κ . Then | T ¯ l | = κ , as T ¯ l ⊆ dom (¯ s ) ( κ <κ ↑ ) , | κ <κ ↑ | = κ and | dom (¯ s ) | < κ . For each a ∈ dom (¯ l ) and each ¯ t ∈ T ¯ l let ρ a (¯ t ) = sup { ¯ t ( a )( α ) : α < ¯ l ( a ) } and let ρ a = sup { ρ a (¯ t ) : ¯ t ∈ T ¯ l } . If for each a ∈ dom (¯ s ) , ρ a < κ , then | T ¯ l | < κ . Thisis due to κ <κ = κ , the regularity of κ , | T ¯ l | = κ and the inaccessibility of κ . Thus, there is a ∈ dom (¯ s ) such that ρ a = κ . Now, if for each α < ¯ l ( a ) , µ α := sup { ¯ t ( a )( α ) : ¯ t ∈ T ¯ l } < κ ,then ρ a = sup α< ¯ l ( a ) µ α < κ , which is a contradiction. Therefore, there is α < ¯ l ( a ) such that µ α = κ . Pick α least such that µ α = κ . Then in particular, |{ ¯ t ( a ) ↾ α : ¯ t ∈ T ¯ l }| < κ and sowe can find u ∈ <κ κ ↑ and T ′ ⊆ T ¯ l of cardinality κ such that for each ¯ t ∈ T ′ , ¯ t ( a ) ↾ α = u and { ¯ t ( a )( α ) : ¯ t ∈ T ′ } is unbounded in κ . Fix ¯ t ∈ T ′ . Then ¯ s ′ ∈ dom (¯ s ) ( κ <κ ↑ ) where ¯ s ′ ( a ) = u and ¯ s ′ ( b ) = ¯ t ( b ) for b = a is an element of D γ +1 = D γ , contradicting the minimality of ¯ t . (cid:3) We continue with the proof of the theorem. As there are < κ many minimal extensions for afixed sequence of lengths ¯ l and | dom (¯ s ) κ | = κ , we can define on dom (¯ s ) functions which go aboveall minimal extensions in T ¯ l . For any ¯ l ∈ dom (¯ s ) κ let ρ ¯ l := sup { ¯ l ( a ) : a ∈ dom (¯ l ) ∧ ¯ l ( a ) = dom (¯ s ( a )) } . Since | dom (¯ s ) | < κ , ρ ¯ l ∈ κ for each ¯ l ∈ dom (¯ s ) κ . First we deal with those minimal extensions ¯ t with ρ ¯ l ¯ t is a successor. Then for each a ∈ dom (¯ s ) and each dom (¯ s ( a )) ≤ α < κ the set H a,α = { ¯ t ( a )( α ) : ¯ t ∈ T ¯ l , ρ ¯ l = ¯ l ( a ) = α + 1 } is bounded by the above claim. Thus there are functions g a ∈ κ κ ↑ such that g a ( α ) > sup ( H a,α ) ON-LINEAR ITERATIONS AND HIGHER SPLITTING 7 for each dom (¯ s ( a )) ≤ α < κ . Thus, g a dominates at α the values of all minimal extensions ¯ t whosemaximal length (not equaling the lengths of ¯ s ) is the successor α + 1 which again is witnessed atpoint a ∈ dom (¯ t )(= dom (¯ s )) .Second we deal with those minimal extensions ¯ t with ρ ¯ l ¯ t is a limit. For each limit α ∈ κ with α ≥ sup { ¯ l ¯ s ( a ) : a ∈ dom (¯ s ) } and β ∈ α the set G β = { ¯ t ( a )( β ) : ¯ t ∈ T ¯ l , ρ ¯ l = α } , which is again bounded by the claim. Thus inductively for each limit α ∈ κ with α ≥ sup { ¯ l ¯ s ( a ) : a ∈ dom (¯ s ) } and β ∈ α we can define a function g ∈ κ κ ↑ such that g ( β ) > sup ( G β ) if g ( β ) isnot defined already. So, g dominates below an ordinal α suitable values of all minimal extensionswhose maximal length is the limit α .Finally for each a ∈ dom (¯ s ) , let ˙ f a be a P a -name for the pointwise maximum of g a and g .Consider the condition p ∈ D ( κ, Q ) with dom ( p ) = dom (¯ s ) and ∀ a ∈ dom ( p ) [ p ( a ) = (¯ s ( a ) , ˙ f a )] .By the density of D we can find a condition q ≤ p such that q ∈ D . So the element ¯ t ∈ dom ( q ) ( κ <κ ↑ ) with ¯ t = ¯ q is in D and ¯ s ⊆ ¯ t (by the extension relation). Then, some initial segment ¯ t ′ of ¯ t must be a minimal extension of ¯ s . If ¯ t ′ ↾ dom (¯ s ) = ¯ s , then ¯ s ∈ D γ , which is a contradiction.Otherwise ∃ ¯ l ′ ∈ dom (¯ s ) κ ∃ a ∈ dom (¯ l ′ ) [¯ l ′ ( a ) > dom (¯ s ( a ))] and l ¯ t ′ = ¯ l ′ . Let λ be sup { ¯ l ′ ( b ) | ¯ l ′ ( b ) = dom (¯ s ( b )) } . If λ = α + 1 and a ∈ dom (¯ l ′ ) [ α + 1 = ¯ l ′ ( a )] , then ¯ t ( a )( α ) = ¯ t ′ ( a )( α ) < ˙ g a ( α ) ≤ ˙ f a ( α ) which is a contradiction to q ≤ p . Suppose λ is a limit and a ∈ dom (¯ l ′ ) with ¯ l ′ ( a ) > dom (¯ s ( a )) .Take β ∈ ¯ l ′ ( a ) with β ≥ dom (¯ s ( a )) . Then ¯ t ( a )( β ) = ¯ t ′ ( a )( β ) < ˙ g ( β ) ≤ ˙ f a ( β ) which is acontradiction to q ≤ p . (cid:3) Definition 15. (1) A sequence h a ξ : ξ < λ i , where each a ξ is in [ κ ] κ , is eventually splitting if ∀ a ∈ [ κ ] κ ∃ ξ < λ ∀ η > ξ a η splits a .(2) A sequence h a ξ : ξ < λ i , where each a ξ is in [ κ ] κ , is eventually narrow if ∀ a ∈ [ κ ] κ ∃ ξ < λ ∀ η > ξ a ∗ a η .Note that τ = h a ξ : ξ < λ i is eventually splitting iff the sequence τ ′ = h b ξ : ξ < λ i , defined as b ξ = a ξ and b ξ +1 = κ \ a ξ , is eventually narrow. Theorem 16.
Assume GCH, κ <κ = κ , κ is strongly inaccessible and let cf ( λ ) > κ . Then anyeventually narrow sequence τ = h a ξ : ξ < λ i remains eventually narrow in V D ( κ,Q ) .Proof. Suppose not. Fix p ∈ D ( κ, Q ) and a name ˙ a for a subset of κ of size κ such that p (cid:13) D ( κ,Q ) ∀ ξ < λ ∃ η > ξ [ ˙ a ⊆ ∗ a η ] . Let ̺ be a regular cardinal such that D ( κ, Q ) ∈ H ( ̺ ) = { x ∈ W F : | trcl ( x ) | < ̺ } . Let N bean elementary substructure of H ( ̺ ) of size κ such that D ( κ, Q ) ∈ N , ˙ a ∈ N and ˙ f pa ∈ N , wherewe denote p ( a ) = ( s pa , ˙ f pa ) . Since τ is eventually narrow, for every a ∈ [ κ ] κ ∩ N there is a ξ < λ such that for all η > ξ we have a ∗ a η . However |N | = κ , τ is of length λ and cf ( λ ) > κ ,so this will yield κ -many ξ ’s smaller than λ ; so we can not reach λ in κ -many steps. Hence ∃ ξ ′ < λ ∀ c ∈ N ∩ [ κ ] κ ∀ η ′ ≥ ξ ′ [ c ∗ a η ′ ] . ÖMER FARUK BAĞ AND VERA FISCHER
Since p (cid:13) “ ∀ ξ < λ ∃ η > ξ ˙ a \ a η is of size less than κ ”, in particular p forces the existence of an η greater than ξ ′ (fixed two lines above) such that ˙ a \ a η is of size less than κ . By extending p (the extension is also called p ) we have that there is an α ∈ κ and η > ξ ′ such that p (cid:13) ∀ j ≥ α if j ∈ ˙ a then j ∈ a η .Let ˙ h be a D ( κ, Q ) -name such that (cid:13) “ ˙ h enumerates ˙ a ” . Then, in particular (cid:13) ∀ ζ < κ ˙ h ( ζ ) ≥ ζ .To define ˙ h we only used ˙ a which was in N and ˙ h ∈ N as well. For the purposes of this proof,we will use the following notions: Definition. (1) Let u = (¯ u , ¯ u ) ∈ D ( κ, Q ) . A sequence ¯ t ∈ f in <κ ( Q, κ <κ ↑ ) is said to be u -admissible if ¯ u ⊆ ¯ t and (¯ t ↾ dom ( u ) , ¯ u ) is a condition in D ( κ, Q ) .(2) Let ¯ t ∈ f in <κ ( Q, κ <κ ↑ ) and let ¯ τ = h ˙ g a : a ∈ dom (¯ t ) i where ∀ a ∈ dom (¯ t ) ˙ g a is a P a -namefor an element in κ κ ↑ . We say that ¯ τ is ¯ t -admissible if q (¯ t, ¯ τ ) = h (¯ t ( a ) , ˙ g a ) : a ∈ dom (¯ t ) i is a condition in D ( κ, Q ) . Claim.
Let ¯ t ∈ f in <κ ( Q, κ <κ ↑ ) be p -admissible and ζ ≥ α . Then Z ¯ t ( ζ ) = ∅ , where Z ¯ t ( ζ ) = { j : ∀ ¯ τ [¯ τ is ¯ t -admissible → ∃ r ≤ D ( κ,Q ) q (¯ t, ¯ τ ) such that r (cid:13) ˙ h ( ζ ) = j ] } . Proof.
Fix ζ ≥ α and let D = { u ∈ D ( κ, Q ) : ∃ j ∈ κ [ u (cid:13) ˙ h ( ζ ) = j ] } . Then D is dense, openand we can form the sequence of derivatives h D α i α ≤ γ where γ is the least with D γ = D γ +1 = f in <κ ( Q, κ <κ ↑ ) . We will prove the claim inductively on α ≤ γ for all p -admissible ¯ t .If ¯ t ∈ D we have ∃ u ∈ D such that ¯ u = ¯ t and ∃ j [ u (cid:13) ˙ h ( ζ ) = j ] . Let ¯ τ be ¯ t -admissible.Then q (¯ t, ¯ τ ) and u (= (¯ t, ¯ u )) are compatible with common extension r . Thus r (cid:13) ˙ h ( ζ ) = j and so Z ¯ t ( ζ ) = ∅ . For limit ordinals α the claim is true by the induction hypothesis, since D α = S { D β : β < α } . Let ¯ t ∈ D α +1 \ D α be p -admissible. By definition of D α +1 there are twopossibilities:First ∃ ¯ t ′ ∈ D α ∃ ! b ∈ Q : dom (¯ t ′ ) = dom (¯ t ) ∪ { b } ∧ ¯ t ′ ↾ dom (¯ t ′ ) \ { b } = ¯ t . Since ¯ t is p -admissible, ¯ t ′ is also p -admissible (easily seen by definition) and by induction hypothesis Z ¯ t ′ ( ζ ) = ∅ . That isfor some j ∈ κ , we have ∀ ¯ τ ′ [¯ τ ′ is ¯ t ′ -admissible → ∃ r ≤ D ( κ,Q ) q (¯ t ′ , ¯ τ ′ ) such that r (cid:13) ˙ h ( ζ ) = j ] . ( ∗ ) We claim that j ∈ Z ¯ t ( ζ ) . Indeed consider any ¯ t -admissible ¯ τ . Then τ can be extended to a ¯ t ′ -admissible ¯ τ ′ . Then q (¯ t ′ , ¯ τ ′ ) ≤ q (¯ t, ¯ τ ) and by ( ∗ ) , there is r ≤ D ( κ,Q ) q (¯ t ′ , ¯ τ ′ ) with r (cid:13) ˙ h ( ζ ) = j .Then by transitivity r ≤ D ( κ,Q ) q (¯ t, ¯ τ ) and we conclude that Z ¯ t ( ζ ) = ∅ .Second there is a sequence h ¯ t β : β ∈ κ i of elements of D α such that ∀ β < κ : dom (¯ t ) ⊆ dom (¯ t β ) and l ¯ t β ↾ dom (¯ t ) = ¯ l (for some ¯ l ∈ dom (¯ t ) κ ) and ∀ b ∈ dom (¯ t ) [¯ t β ( b )( dom (¯ t ( b ))) > β ] . Since sucha sequence exists in H ( ̺ ) and the latter was an existential statement and N H ( ̺ ) , by theTarski-Vaught-Criterion we can find a witness in N . So assume h ¯ t β : β ∈ κ i ∈ N .At this point we distinguish between two either-or cases. Case 1: There is a j ∈ κ such that j ∈ Z ¯ t β ( ζ ) for κ -many β . Let ¯ τ = h ˙ g a : a ∈ dom (¯ t ) i be ¯ t -admissible and for each β < κ let ¯ τ β be ¯ t β -admissible with ¯ τ β ↾ dom (¯ t ) = ¯ τ . We have that “ ∃ r ≤ q (¯ t β , ¯ τ β ) [ r (cid:13) ˙ h ( ζ ) = j ] ” for κ -many ¯ t β ’s, but not all of these q (¯ t β , ¯ τ β ) extend q (¯ t, ¯ τ ) . However since we have κ -many such ¯ t β ’s and ∀ b ∈ dom (¯ t ) : ¯ t β ( b )( dom (¯ t ( b ))) > β we can find one (actually infinitely many) q (¯ t β , ¯ τ β ) ≤ q (¯ t, ¯ τ ) ON-LINEAR ITERATIONS AND HIGHER SPLITTING 9 and consequently infinitely many r ≤ q (¯ t, ¯ τ ) such that j ∈ Z ¯ t β ( ζ ) ; hence j ∈ Z ¯ t ( ζ ) = ∅ . (To finda q (¯ t β , ¯ τ β ) as desired we choose β such that β > S { ˙ g a (¯ l ( a )) : a ∈ dom ( q g ) } . Then for such a β and any a ∈ dom (¯ t ) and α with dom (¯ t ( a )) ≤ α < ¯ l ( a ) : (cid:2) ¯ t β ( a )( α ) > β > ˙ g a (¯ l ( a )) > ˙ g a ( α )) (cid:3) ).Case 2: Fix by the induction hypothesis one j β ∈ Z ¯ t β ( ζ ) (e.g. choose the minimal one) andconsider the set J := { j β : β ∈ κ } . This set is of size κ , because otherwise it would have an upperbound in κ , so ∃ α < κ ∀ α, β ≥ α : j α = j β . But then we would have a j which is in all Z ¯ t β ( ζ ) ’sfor β ≥ α , so we would have a j which is in κ -many Z ¯ t β ( ζ ) ’s, which is in fact Case1. So | J | = κ ,but J consists of j β ’s which are elements of Z ¯ t β ( ζ ) and these were defined using ˙ h which was in N and the sequence h ¯ t β : β ∈ κ i which was also in N , so we may take J ∈ N . This further meansthat | J \ a η | = κ . So choose β large enough such that j β ≥ α , β ≥ S { ˙ f pa (¯ l ( a )) : a ∈ dom ( p ) } and j β a η . Then for this particular β we have u ≤ v ≤ p where ¯ v = ¯ t and ¯ v ↾ dom ( p ) = ¯ p and ¯ u = ¯ t β and ¯ u ↾ dom ( p ) = ¯ p . For the first extension relation note that for a ∈ dom ( v ) wehave ¯ t β ( a )( dom (¯ t ( a ))) > β ≥ ˙ f pa (¯ l ( a )) so this extension really holds. And since j β ∈ Z ¯ t β ( ζ ) thereis by the definition of Z ¯ t β ( ζ ) some r β ≤ u such that r β (cid:13) ˙ h ( ζ ) = j β . But then since j β ≥ α and r β (cid:13) “ ∀ j ≥ α if j ∈ ˙ a then j ∈ a η ” ( p forced this). All together we have j β ∈ a η which is acontradiction. (cid:3) By the claim Z ¯ s ( ζ ) = ∅ for ζ ≥ α where ¯ s ∈ dom ( p ) ( κ <κ ↑ ) with ¯ s = ¯ p since p ≤ p . Choose k ζ ∈ Z ¯ s ( ζ ) for each ζ ≥ α and consider the set K := { k ζ : ζ ≥ α } . Since (cid:13) ˙ h ( ζ ) ≥ ζ we have k ζ ≥ ζ for all ζ , hence K is of size κ . Since K is definable from ¯ s and other parameters of Z ¯ s ( ζ ) ,we have K ∈ N , so K \ a η has size κ . Now let k ζ ∈ K \ a η be chosen; so by definition ∃ r ≤ p such that r (cid:13) ˙ h ( ζ ) = k ζ and again r (cid:13) “ ∀ j ≥ α if j ∈ ˙ a then j ∈ a η ”, and k ζ ≥ ζ ≥ α so wehave k ζ ∈ a η which is a contradiction. (cid:3) Finally we formulate the theorem:
Theorem 17. (GCH, κ <κ = κ ) Assume κ is strongly unfoldable and β, δ, µ are cardinals with κ + ≤ β = cf ( β ) ≤ cf ( δ ) ≤ δ ≤ µ and cf ( µ ) > κ ; then there exists a forcing poset P κ,β,δ,µ suchthat the cardinal preserving generic extension V P κ,β,δ,µ satisfies s ( κ ) = κ + ∧ b ( κ ) = β ∧ d ( κ ) = δ ∧ c ( κ ) = µ. Proof.
One part follows as in [4]: In the ground model V let Q be a poset with b (( Q, ≤ q )) = β ≤ d (( Q, ≤ q )) = δ and let Q ′ be a cofinal well-founded subset of Q ( Q ′ has the same bounding anddominating numbers). Next we construct a poset which consists of a copy of ( µ, ∈ ) at the bottomand a cofinal copy of Q ′ at the top; so let R consist of pairs ( p, i ) such that either i = 0 ∧ p ∈ µ or i = 1 ∧ p ∈ Q ′ . The order relation is defined as ( p, i ) ≤ ( q, j ) iff i = 0 ∧ j = 1 or i = j = 1 ∧ p ≤ Q ′ q or i = j = 0 ∧ p ≤ q in µ . Finally let P κ,β,δ,µ = D ( κ, R ) . This forcing poset is κ -closed and has the κ + -c.c., so we start the whole construction by Lottery preparation Q and let V = V Q . Lotterypreparation preserves GCH for all cardinals ≥ κ , which suffices to apply Theorem 2 in [4]. Asshown in [4] this forcing poset satisfies V P κ,β,δ,µ (cid:15) β = b ( κ ) ≤ δ = d ( κ ) ≤ µ = c ( κ ) .Since κ is strongly unfoldable and P κ,β,δ,µ is κ -closed and has the κ + -c.c., we may assumethat κ is still strongly unfoldable in the generic extension after doing Lottery preparation, so s ( κ ) ≥ κ + . Further the last theorem shows that there is a splitting family of size κ + in theextension: Every forcing adding a dominating real, also adds a splitting real. The first κ + -manysplitting reals added by the first κ + -many steps of the Hechler iteration, build an eventuallysplitting sequence in the intermediate model V P κ + , which is preserved as a such in the finalmodel. Hence V P κ,β,δ,µ (cid:15) κ + = s ( κ ) . (cid:3) Remark 18.
Thus under the assumption that there is a strongly unfoldable κ , it is consistentthat all four characteristics are different, i.e. κ + = s ( κ ) < β = b ( κ ) < δ = d ( κ ) < µ = c ( κ ).4. Consistency of
Spec ( t ( κ )) = { κ + } ∧ c ( κ ) = κ ++ Definition 19 ([9]) . Let κ be a regular uncountable cardinal.(1) A sequence h a ξ : ξ < µ i of elements in [ λ ] λ is a descending ⊆ ∗ -sequence if for ξ < η < µ we have that a η ⊆ ∗ a ξ .(2) A family or sequence of subsets of λ has the strong intersection property (SIP) if anysubfamily of size less than λ has intersection of size λ .(3) A λ -tower is a descending ⊆ ∗ -sequence with the SIP and no pseudo- intersection of size λ , in other words ∀ a ∈ [ ω ] ω ∃ ξ < µ a ∗ a ξ .(4) Let the tower number t ( λ ) denote the minimal cardinality of a λ -tower.(5) Finally we define the spectrum of λ -towers as Spec ( t ( λ )) = {| τ | : τ is a λ -tower } .By a diagonal argument one can find a pseudo- intersection for any family F ⊆ [ λ ] λ with theSIP and |F | ≤ λ , hence λ + ≤ t ( λ ) . Remark 20.
Note that a λ -tower is eventually narrow: Let h a ξ : ξ < µ i be a λ -tower and a ∈ [ λ ] λ be arbitrary. Since h a ξ : ξ < µ i is a λ -tower ∃ ξ < µ | a \ a ξ | = λ . Let ξ ′ > ξ , since h a ξ : ξ < µ i isa descending ⊆ ∗ -sequence we have that a ξ ′ ⊆ ∗ a ξ . Now a \ a ξ ′ ⊇ [( a \ a ξ ) \ a ξ ′ ] ∪ [( a ∩ a ξ ) \ a ξ ′ ]= [( a \ a ξ ) \ ( a ξ ′ \ a ξ )] ∪ [( a ∩ a ξ ) \ a ξ ′ ] , but recall that ( a \ a ξ ) was of size λ and | ( a ξ ′ \ a ξ ) | < λ , so a \ a ξ ′ contains a subset of size λ , so itself has size λ . Since ξ ′ was arbitrary all together we have ∀ a ∈ [ λ ] λ ∃ ξ < µ ∀ ξ ′ > ξ | a \ a ξ ′ | = λ , so the λ -tower is eventually narrow.Along a non-linear iteration of Hechler forcings, as defined in Definition 10, also witnesses for t ( κ ) are preserved; so are also the witnesses for other higher analogues of t , e.g. t cl ( κ ) or t ∗ ( κ ) (see[6] for the definitions). Next we show that it is consistent that the generalized cardinal invariant t ( κ ) has no witness of size c ( κ ) . Here we use a well-ordered ( λ, ∈ ) in place of the well-foundedposet ( Q, ≤ Q ) as in Remark 11 (2). Theorem 21.
Assume GCH, κ <κ = κ and κ is strongly inaccessible. Let λ > κ + be a regular un-countable cardinal. Then in V D ( κ, ( λ, ∈ )) there are no κ -towers of length λ , but there are descending ⊆ ∗ -sequences of length λ with the SIP.Proof. Suppose τ = h a ξ : ξ < λ i is a κ -tower in V D ( κ, ( λ, ∈ )) . Then by the definition of a κ -towerwe have ∀ a ∈ [ κ ] κ ∩ V D ( κ, ( λ, ∈ )) ∃ ξ = ξ ( a ) < λ a ∗ a ξ . Since GCH holds it is easily observed thatthe following holds: ON-LINEAR ITERATIONS AND HIGHER SPLITTING 11
Claim. ∀ α < λ we have |P ( κ ) ∩ V P α | < λ .Consider the function f , where f ( α ) = sup { ξ ( a ) : a ∈ P ( κ ) ∩ V P α } . By the above claim andthe regularity of λ we have f ( α ) < λ , i.e. f ∈ λ λ ∩ V D ( κ, ( λ, ∈ )) . By the Approximation Lemma([11, Lemma IV.7.8.]) (remember that D ( κ, ( λ, ∈ )) has the κ + -c.c. and κ + < λ is regular) wecan find a function g : λ → λ in the ground model V such that ∀ α < λ f ( α ) ≤ g ( α ) .The set M = { γ < λ : h a ξ : ξ < γ i ∈ V P γ } contains a club C in λ (as the fixed points of anormal function form a club). By a well-known argument ([11, Lemma III.6.13]) we have that D := { α < λ : ∀ β < α g ( β ) < α } is a club in λ , so the intersection C ∩ D is also a club. Since E κ + λ = { α < λ : cf ( α ) = κ + } is a stationary set, there is γ such that h a ξ : ξ < γ i ∈ V P γ , ∀ α < γ g ( α ) < γ and cf ( γ ) = κ + > κ . Then h a ξ : ξ < γ i is a κ -tower in V P γ . Indeed, if a ∈ [ κ ] κ ∩ V P γ then a is already added at stage α , for some α < γ (since cf ( γ ) > κ and suchstages do not add new κ -reals). Now | a f ( α ) \ a | = κ , f ( α ) ≤ g ( α ) < γ and so h a ξ : ξ < γ i isa κ -tower in V P γ . Hence it is also a κ -tower in V D ( κ, ( λ, ∈ )) because κ -towers are preserved and V D ( κ, ( λ, ∈ )) is obtained by iterated forcing over V P γ . This yields a contradiction, since a δ for γ < δ < λ is almost contained in each a β for β < γ .For the last statement of the theorem, consider the reals f ξ ∈ κ κ where f ξ is the dominatingreal added at stage ξ of the iteration. So if ξ < η < λ we have f ξ < ∗ f η . Define each c ξ as c ξ = { ( α, β ) ∈ κ × κ : β ≥ f ξ ( α ) } and consider the sequence h c ξ : ξ < λ i . Using a bijectionbetween κ and κ × κ it is easily seen that this is a descending ⊆ ∗ -sequence with the SIP. (cid:3) Corollary 22.
Assume GCH, κ <κ = κ and κ is strongly inaccessible. Then there is a cardinalpreserving extension where c ( κ ) = κ ++ and Spec ( t ( κ )) = { κ + } hold. Proof.
It suffices to choose λ = κ ++ in Theorem 21. Then V D ( κ, ( κ ++ , ∈ )) (cid:15) Spec ( t ( κ )) = { κ + } asthere are no long κ -towers and t ( κ ) ≥ κ + . c ( κ ) ≥ κ ++ is witnessed by the κ ++ -many Hechler κ -reals added. c ( κ ) ≤ κ ++ is seen by the standard argument of counting nice names (see [11]). (cid:3) Questions
The method in Section 3 allows s ( κ ) to be κ + . However generalizing the methods in [1] seemsnot to be enough to control s ( κ ) arbitrarily above κ + . Question 23.
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Institute of Mathematics, Kurt Gödel Research Center, University of Vienna, Augasse 2-6,UZA 1 - Building 2, 1090 Wien, Austria
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