aa r X i v : . [ m a t h . L O ] A ug An Inconsistent Forcing Axiom at ω Abstract
We show that the forcing axiom for countably compact, ω -Knaster,well-met posets is inconsistent. This is supplemental to an inconsistencyresult of Shelah [5] and sets a new limit to the generalization of Martin’sAxiom to the stage of ω . As an attempt to generalize Martin’s Axiom to the stage of ω , Baumgartner [1]proved the following theorem which shows the consistency of a forcing axiomfor ω -linked posets. Theorem 1 (Baumgartner [1]) . It is consistent with CH and ω > ω that F A κ ( B ) holds for every κ < ω , in which B is the class of all σ -closed, ω -linked and well-met posets. Here a poset P is well-met if every pair of compatible conditions from P has a greatest lower bound. F A κ ( P ) is the statement that for any family, { D ξ : ξ < κ } , of dense open subsets of P , there exists a sufficiently generic filter F ⊂ P which intersects every D ξ . F A κ ( B ) states that F A κ ( P ) holds for every P ∈ B .Many applications of Baumgartner’s forcing axiom was studied by Tall [7].However, Shelah [5] showed that the full generalization of Martin’s Axiom isinconsistent with CH + 2 ω > ω . Theorem 2 (Shelah [5]) . If CH holds and ω > ω , then there is a σ -closedposet P satisfying the ω -chain conditions such that FA ω ( P ) fails. The inconsistency of such a na¨ıve generalization of Martin’s Axiom doesnot come from the chain condition directly. Actually the counterexample con-structed by Shelah is ω -linked and the inconsistency comes from the absenceof the well-met condition.The first author of this paper observed that even if we add the well-metnesscondition, the forcing axioms can still fail to be consistent due to an excessivelystrong chain condition. Namely, we show the following result.1 poset P is κ - Knaster if for every S ⊂ P with cardinality κ , there existsa subset S ⊂ S with cardinality κ such that the elements in S are pairwiselycompatible in P . A poset is countably compact if every countable decreasingchain from the poset has a greatest lower bound. Let K be the class of allcountably compact, ω -Knaster and well-met posets. Theorem 3.
F A ω ( K ) is inconsistent with ZFC + CH + ω > ω . (cid:3) ω We first introduce Shelah’s forcing axiom (S) [4].A poset P is ω - normal if for any sequence h p ξ : ξ ∈ ω i of conditions from P , there exists a club C ⊂ ω and a regressive function f : ω → ω such thatfor any α, β ∈ C with cf α, cf β > ω and f ( α ) = f ( β ), p α and p β are alwayscompatible. Theorem 4 (Shelah [4]) . It is consistent with CH + ω > ω that F A κ ( S ) holds for every κ < ω , in which S is the class of all ω -normal, countablycompact and well-met posets. Let (S) denote Shelah’s forcing axiom ∀ κ < ω , F A κ ( S ). Clearly, (S) isa direct consequence of our inconsistent axiom F A ω ( K ). We show that (S)implies (cid:3) ω . Lemma 1. (S) + CH + ω > ω implies (cid:3) ω .Proof. We study the poset P of all partial functions p ... S → P ( ω ) in which S = { α ∈ ω : cf α = ω } , satisfying:1. dom p is countable.2. For α ∈ dom p , p ( α ) is a countable subset of α and closed in α .3. If δ < γ are both in dom p and ξ is a limit point of both p ( δ ) and p ( γ ),then p ( δ ) ∩ ξ = p ( γ ) ∩ ξ .The order of the poset is defined by setting p ≤ q if and only if4. dom q ⊂ dom p .5. For α ∈ dom q , q ( α ) ⊑ p ( α ).
6. For α ∈ dom q , min p ( α ) \ q ( α ) ≥ sup [ ξ ∈ dom q ( q ( ξ ) ∩ α ) , and min p ( α ) \ q ( α ) ≥ sup(dom q ∩ α ) . q ( α ) ⊑ p ( α ) means that p ( α ) is an end extension of q ( α ). P is well-met.Given p, q ∈ P of which the compatibility is witnessed by r , define r : dom p ∪ dom q → P ( ω ) such that r ( α ) = p ( α ) ∪ q ( α ). Clearly every r ( α ) is a countableclosed subset of α and is an initial segment of r ( α ). For any δ < γ ∈ dom r and ξ , a limit point of both r ( δ ) and r ( γ ), it must be true that ξ is also a limitpoint of both r ( δ ) and r ( γ ). Since r is a condition from P , we have that ξ ∩ r ( δ ) = ξ ∩ r ( δ ) = ξ ∩ r ( γ ) = ξ ∩ r ( γ ). Therefore, r is a condition from P . Meanwhile, r extends both p and q . For any α ∈ dom r , since both p ( α )and q ( α ) are initial segments of r ( α ), r ( α ) must be an end-extension of bothof them. Condition 6 is witnessed by r and obviously r is the greatest lowerbound of p and q .To see that P is countably compact, let h p i : i ∈ ω i be a countable decreasingchain from P . Define r : S i ∈ ω dom p i → P ( ω ) such that r ( α ) = cl ( S i ∈ ω p i ( α )).It is clear that r ( α ) must be a countable subset of α since there is at most onepoint in r ( α ) − S i ∈ ω p i ( α ), which is sup( S i ∈ ω p i ( α )) if it is not already in some p i ( α ). Let’s say that this point is special for α .It suffices to verify condition 3 for r . Let ξ be a limit point of both r ( γ )and r ( δ ) and i be the minimal natural number such that both γ and δ are indom p i . Without loss of generality, we assume that γ < δ . If ξ is not specialfor either γ or δ , there exists j ∈ ω such that ξ ∈ p j ( γ ) ∩ p j ( δ ). Therefore, ξ ∩ r ( γ ) = ξ ∩ p j ( γ ) = ξ ∩ p j ( δ ) = ξ ∩ r ( δ ). If ξ is special for δ , there exists j > i such that we can find an η ∈ p j ( δ ) \ p i ( δ ) and η < ξ < γ . This contradictscondition 6. If ξ is special for γ but not special for δ , there exists j ≥ i suchthat ξ ∈ p j ( δ ). Now for any j > j , by condition 6, min p j ( γ ) \ p j ( γ ) ≥ ξ which is a contradiction.Therefore, r extends the sequence h p i : i ∈ ω i and it must be the greatestlower bound of this sequence by definition.By simple density arguments, we can see that a sufficiently generic objectmust be a sequence C = h C β : β ∈ S i such that every C β is a club subset of β with ordertype ω . We can extend the definition to the ordinals with countablecofinality. For α ∈ ω such that cf α = ω , if there is some β ∈ S such that α isa limit point of C β , then define C α = C β ∩ α . By the coherence of C given bycondition 6, it does not depend on our choice of β . If there is no such β , thenjust define C α to be any cofinal subset with ordertype ω . This extension mustbe a (cid:3) ω -sequence.Now it suffices to verify that P is ω -normal. Fix any sequence h p ξ : ξ ∈ ω i from P and a club C ⊂ ω such that dom p ξ ⊂ η for any ξ < η ∈ C .Given ξ ∈ S , define s ξ = sup((dom p ξ ∪ [ α ∈ dom p ξ p ξ ( α )) ∩ ξ ) + 1 .s ξ is always strictly less than ξ since cf ξ = ω .For every ξ ∈ S , let γ = otp(dom p ξ ∩ s ξ ) and γ = otp(dom p ξ \ s ξ ).Without loss of generality, we may assume that both of them are independent3rom the choice of ξ . Now we fix a function F : ω × ω → V such that ∀ ι ∈ ω , F ( ι, · ) : ω → V emumerates ( ι × [ ι ] ω ) γ × ([ ι ] ω ) γ . The existence of such a function F is given by our assumption CH.For ξ ∈ ω , define f ( ξ ) = h s ξ , τ ξ i in which τ ξ is the unique τ such that thefirst term of F ( s ξ , τ ) codes the restriction of the function p ξ on ξ , i.e., {h α, p ξ ( α ) i : α ∈ dom p ξ ∩ ξ } , and the second term of F ( s ξ , τ ) codes the function ξ i p ξ ( ξ i ) ∩ s ξ in which i ∈ γ and ξ i is the i -th element of dom p ξ \ s ξ . f is a desired regressive function such that whenever ξ < η are in C withcofinality ω and f ( ξ ) = f ( η ), p ξ is compatible with p η . Given ξ < η with suchproperties, we define the condition r by simply taking r ( β ) = p ξ ( β ) ∪ p η ( β ) forany β ∈ dom p ξ ∪ dom p η . Note that either p ξ ( β ) = p η ( β ) or one of them is notdefined, in which case we understand it as an empty set. Therefore, r extendsboth p ξ and p η as long as it is a condition.It suffices to verify condition 3 for r . Let s = s ξ = s η . Assume that α isa limit point of both r ( γ ) and r ( δ ). We may assume that γ ∈ dom p ξ \ s and δ ∈ dom p η \ s . By the definition of the club C , γ < η ≤ δ , so we have α < γ < δ hence α ≤ s since α is a limit point of p η ( δ ). Assuming γ = ξ i and δ = η j , wehave α ∩ p ξ ( ξ i ) = α ∩ p η ( η i ) = α ∩ p η ( η j ) , in which the first equality comes from the definition of f and the second comesfrom the coherence property of p η . ω -Aronszajn Tree We also need the following theorem from Todorˇcevi´c [8] (also see [9] Chapter 7).
Theorem 5 (Todorˇcevi´c [8]) . If θ ≥ ω and there is a (cid:3) ( θ ) -sequence, then thereis a nonspecial θ -Aronszajn tree in the form of T ( ρ C ) in which C is a nonspecial (cid:3) ( θ ) -sequence. Shelah-Stanley [6] has a similar result which states that (cid:3) ( θ ) implies theexistence of a nonspecial θ -Aronszajn tree. However, their construction is es-sentially different. The trees constructed in Shelah-Stanley [6] admit ω -ascentpaths which make the nonspeciality of those trees robust. According to thehistorical remarks in [6], Laver first observed in 1970s that an ω -Aronszajntree with an ascent path cannot be special and later Baumgartner was able toconstruct such trees from (cid:3) ω (see Devlin [2]). However, as we will see soon,trees in the form of T ( ρ C ) can be easily specialized.4 Specializing the ω -Aronszajn Trees In this section, we show that the forcing axiom
F A ω ( K ) is strong enough tospecialize all the trees in the form of T ( ρ C ) for any nontrivial (cid:3) ( ω )-sequence C . The following is a basic property of the sequence ρ . Theorem 6 (Todorˇcevi´c [9]) . The following are equivalent for any C-sequence C = h C α : α ∈ Lim( θ ) i on a regular uncountable cardinal θ and the correspond-ing ρ sequence.1. C is nontrivial.2. For every family A of θ pairwisely disjoint finite subsets of θ and everyinteger n , there is a subfamily B of A of size θ such that ρ ( α, β ) > n forall α ∈ a , β ∈ b and a = b in B . The following lemma is essentially Lemma 6.3.3 of [9]. We include the proofhere for completeness.
Lemma 2 (Todorˇcevi´c) . For any C-sequence C on ω , the tree T = T ( ρ ) hasthe property that every subset X ⊂ T of cardinality ω contains an antichain ofcardinality ω .Proof. Consider a subset X ⊂ T of cardinality ω . We may assume that X is alevel set and by replacing X = { x α : α ∈ Γ } by a set lying inside its downwardsclosure, we may assume that the set consists of successor nodes of the tree. Let K ⊂ [ ω ] be such that X = { x α : ξ ∈ Γ } = { ρ ( · , β ) ↾ ( α + 1) : { α, β } ∈ K } . Shrinking X further, we may assume that pairs in K are pairwisely disjoint and ρ is constantly n on K since ρ only takes values of natural numbers. ApplyTheorem 6 we have K ⊂ K of cardinality ω such that for all { α, β } , { γ, δ } ∈ K such that α < β, γ < δ and α < γ , we have ρ ( α, γ ) > n . Then X = { ρ ( · , β ) ↾ ( α + 1) : { α, β } ∈ K } is an antichain in T .We also need the following property of the tree T ( ρ ). Lemma 3. (CH) For any nontrivial (cid:3) ( ω ) -sequence h C α : α ∈ ω i , the tree T = T ( ρ ) has the following property:For any level sequence h p ξ : ω → ( T ) ξ | ξ ∈ ω i such that every p ξ isinjective, there is a stationary Γ ⊂ ω such that for any ξ < η ∈ Γ and any i, j ∈ ω , p ξ ( i ) < p η ( i ) ⇐⇒ p ξ ( j ) < p η ( j ) . roof. For any δ ∈ ω such that cf δ = ω and any countable set X ⊂ ω above δ , there exists a δ ′ < δ such that for any α ∈ X and ξ ∈ tr ( δ, α ), either δ ∈ Lim C ξ or C ξ ∩ δ ⊂ δ ′ .Clearly if ξ ∈ tr ( δ, α ) and δ ∈ Lim C ξ , the ξ must be the least elementof tr ( δ, α ) above δ . Therefore, δ ′ has the property that if γ ∈ [ δ ′ , δ ), then thegreatest element of tr ( γ, α ) below δ is min( C δ \ γ ) by the coherence of the squaresequence.Now if α ∈ X and γ ∈ [ δ ′ , δ ), and ξ = min( tr ( δ, α ) \ ( δ + 1)), then ρ ( γ, α ) = ( ρ ( γ, δ ) + ρ ( δ, α ) if δ ∈ Lim C ξ , ρ ( γ, δ ) + ρ ( δ, α ) − p ξ , let X ξ = { α iξ : p ξ ( i ) = ρ ( · , α iξ ) ↾ ξ } . Apply the previousargument to every ξ and get an ordinal ξ ′ < ξ . By shrinking the index set toa stationary Γ ⊂ ω , we may assume that for any ξ ∈ Γ, ξ ′ < ξ is an constantordinal s and p δ ( i ) ↾ s is constantly t i . For any ξ ∈ Γ and γ ∈ [ s, ξ ), ρ ( γ, α iξ ) = ρ ( γ, ξ ) + ρ ( ξ, α iξ ) + ǫ iξ , in which ǫ iξ is either 0 or − ρ ( ξ, α iξ ) nor ǫ iξ dependson ξ . Therefore, if ρ ( γ, α iξ ) = ρ ( γ, α iη ) for some i , it is also true for any i ,which completes the proof. Lemma 4. (CH) For any nontrivial (cid:3) ( ω ) -sequence h C α : α ∈ ω i , there is acountably compact, well-met poset with ω -Knaster poset which specializes thetree T ( ρ ) Proof.
Since there is a natural order-preserving mapping from T ( ρ ) to T ( ρ ),it suffices to specialize T = T ( ρ ).Let P be the poset of all the conditions p satisfying1. p is a partial function from ω to T ω such that dom p is countable, and2. for every α ∈ dom p , p ( α ) is a countable antichain of T ,ordered by reversed inclusion.It is clear that P is countably compact, well-met and it specializes the tree T . It suffices to verify the ω -Knaster property of P .Fix a sequence h p ξ : ξ ∈ ω i . By CH and a standard ∆-system argument,we may assume that the dom p ξ ’s are constantly D . For the sake of simplicity,we only verify the ω -Knaster property for the case that D is a singleton { d } and identify p ξ with p ξ ( d ) : ω → T . In the general case that D is a countablesubset of ω , the proof is the same since p ξ ( d ) is already a countable set.For p ξ such that cf ξ = ω , define s ξ = sup( ξ ∩ ( { ht( p ξ ( i )) : i ∈ ω } ∪ { ∆( p ξ ( i ) , p ξ ( j )) : i, j ∈ ω } )) .
6e have s ξ < ξ and without loss of generality we may assume that s ξ is con-stantly s for all ξ . By futher shrinking the set, we may assume that p ξ is afunction p ξ : ω → ( T ) ξ and there is a sequence of nodes h t i : i ∈ ω i from( T ) ht( s ) such that for any ξ , p ξ ( i ) ↾ ht( s ) = t i .Now we can directly apply Lemma 3 and Lemma 2 to this level set and thiscompletes the proof.If the forcing axiom F A ω ( K ) held, then for any (cid:3) ( ω )-sequence C , thetree T ( ρ C ) must be special. On the other hand, since this axiom implies (S),hence (cid:3) ω , there exists such a tree which is nonspecial by Theorem 5, This is acontradiction and finishes the proof of Theorem 3. As we already discussed, this inconsistency result is different from Shelah’sTheorem 2. It focuses on the chain condition rather than the well-metnesscondition. A related result can be found in L¨ucke [3].
Theorem 7 (L¨ucke [3]) . Let κ be an uncountable regular cardinal with κ = κ <κ .If κ + is not weakly compact in L , then there is a κ -closed, well-met partial order P satisfying the κ + -chain condition such that FA κ + ( P ) fails. On the other hand, Theorem 3 in together with Theorem 2 sets a pretty tightbound on the generalization of Martin’s Axiom to ω . If we compare this incon-sistent axiom, FA ω ( K ), with (S), theoretically speaking, there is an essentialgap between the ω -Knaster property and ω -normality. For any ω -sequenceof forcing conditions, the pressing-down function requires the understanding ofat least a club many of them, while Knaster property only requests an arbitarysubset with cardinality ω . The different destinies of the corresponding forcingaxioms also point to and result from that gap. However pragmatically, thisgap is narrow in the sense that many proofs involving the Knaster propertyactually factorize through constructing pressing-down functions, which suggeststhat somehow Shelah’s forcing axiom (S) is quite close to optimal. The research of the first author is partially supported by grants from NSERC(455916)and CNRS(UMR7586).The second author would like to thank Prof. Justin Moore for helpful con-versation. The research of the second author is partially supported by NSFgrant DMS-1854367.
Stevo Todorˇcevi´c,
Department of Mathematics, University of Toronto, Canada , E : [email protected] Institut de Math´ematiques de Jussieu, Paris, France , E : [email protected] Shihao Xiong,
Department of Mathematics, Cornell University, Ithaca, NY, UnitedStates , E : [email protected] eferences [1] J. Baumgartner , Iterated forcing , Surveys in set theory, (1983), pp. 1–59.[2]
K. Devlin , Reduced powers of ℵ -trees , Fundamenta Mathematicae, 118(1983), pp. 129–134.[3] P. L¨ucke , Ascending paths and forcings that specialize higher Aronszajntrees , Fundamenta Mathematicae, 239 (2017), pp. 51–84.[4]
S. Shelah , A weak generalization of MA to higher cardinals , Israel Journalof Mathematics, 30 (1978), pp. 297–306.[5]
S. Shelah and L. Stanley , Generalized Martins Axiom and SouslinsHypothesis for higher cardinals , Israel Journal of Mathematics, 43 (1982),pp. 225–236.[6] ,
Weakly compact cardinals and nonspecial Aronszajn trees , Proceed-ings of the American Mathematical Society, 104 (1988), pp. 887–897.[7]
F. D. Tall , Some applications of a generalized Martin’s Axiom , Topologyand its Applications, 57 (1994), pp. 215–248.[8]
S. Todorˇcevi´c , Special square sequences , Proceedings of the AmericanMathematical Society, (1989), pp. 199–205.[9] ,