Club Stationary Reflection and the Special Aronszajn Tree Property
aa r X i v : . [ m a t h . L O ] F e b CLUB STATIONARY REFLECTION AND THE SPECIALARONSZAJN TREE PROPERTY
OMER BEN-NERIA AND THOMAS GILTON
Abstract.
We prove that it is consistent that
Club Stationary Reflection andthe
Special Aronszajn Tree Property simultaneously hold on ω , thereby con-tributing to the study of the tension between compactness and incompactnessin set theory. The poset which produces the final model follows the collapseof a weakly compact cardinal first with an iteration of club adding (with an-ticipation) and second with an iteration specializing Aronszajn trees.In the first part of the paper, we prove a general theorem about specializingAronszajn trees after forcing with what we call F WC -Strongly Proper posets .This type of poset, of which the Levy collapse is a degenerate example, usessystems of exact residue functions to create many strongly generic conditions.We prove a new result about stationary set preservation by quotients of thiskind of poset; as a corollary, we show that the original Laver-Shelah modelsatisfies a strong stationary reflection principle, though it fails to satisfy the fullClub Stationary Reflection. In the second part, we show that the compositionof collapsing and club adding (with anticipation) is an F WC -Strongly Properposet. After proving a new result about Aronszajn tree preservation, we showhow to obtain the final model. Introduction
This work is a contribution to the study of the tension between compactnessand incompactness principles in Set Theory. We focus on the second uncountablecardinal, ω , and consider the strong compactness principle of Club StationaryReflection and the strong incompactness principle known as the Special AronszajnTree Property (these are defined below).The two properties have been shown to be consistent separately by Magidor [31]and Laver and Shelah [30], respectively. Since the properties represent strong formsof opposing phenomena (compactness and incompactness) it is natural to suspectthat they are jointly inconsistent. The main result of this paper shows, on thecontrary, that the conjunction of the two principles is consistent. More precisely,we prove: Theorem 1.1.
It is consistent relative to the existence of a weakly compact cardinalthat Club Stationary Reflection and the Special Aronszajn Tree Property simulta-neously hold at ω . Our work also shows that a weaker version of stationary reflection holds in theoriginal Laver-Shelah model:
Date : February 15, 2021.2010
Mathematics Subject Classification.
Primary 03E05, 03E35.
Key words and phrases. forcing, Aronszajn Trees, stationary reflection, compactness,specialization.The first author was partially supported by the Israel Science Foundation (Grant 1832/19).
Theorem 1.2.
In the original Laver-Shelah model, the following stationary reflec-tion principle holds: For every sequence h S α | α < ω i of stationary subsets of ω ∩ cof( ω ) there is β < ω so that S α ∩ β is stationary in β , for every α < β .However, CSR ( ω ) fails. We proceed to define the relevant terms and contextualize our result. If ν isa regular cardinal, we use cof( ν ) denote the class of ordinals with cofinality ν .We recall that if cf( α ) > ω , then S ⊆ α is stationary if S ∩ C = ∅ for each club C ⊆ α . We say that S reflects if there is some β < α with cf( β ) > ω so that S ∩ β isstationary in β . If κ is regular, we say that stationary reflection holds at κ ++ if everystationary S ⊆ κ ++ ∩ cof( ≤ κ ) reflects. Baumgartner originally showed ([5]) thatstationary reflection at ω is consistent from a weakly compact cardinal. Harringtonand Shelah ([20]) later improved this, showing that the optimal assumption of aMahlo cardinal suffices. One obtains stronger principles by requiring that multiplestationary sets reflect simultaneously. Recall that a collection { S i | i < τ } of τ < α stationary subsets of α is said to reflect simultaneously if there is some β < α withcf( β ) > ω so that S i ∩ β is stationary in β for every i < τ . Magidor ([31]) hasshown that the consistency strength of “any two stationary subsets of ω ∩ cof( ω )simultaneous reflect” implies the consistency of a weakly compact cardinal. Onemay also consider stronger diagonal versions of the above, defined in the naturalway.We are interested in the following very strong form of stationary reflection whichimplies all of the above: Definition 1.3.
Suppose that κ is regular. We say that Club Stationary Re-flection holds at κ ++ if for any stationary S ⊆ κ ++ ∩ cof( ≤ κ ), there exists a club C ⊆ κ ++ so that for all β ∈ C ∩ cof( κ + ), S reflects at β . We write CSR ( κ ++ ).We will concern ourselves with the case κ = ω . Most relevant for us, Magidor([31]) showed that CSR ( ω ) is consistent from a weakly compact cardinal; by theabove remarks, this is the optimal hypotheses.Extensions of CSR to other cardinals have been shown to have limitations. Forexample, Jech and Shelah [23] proved that for every n < ω , if every stationary subsetof ω n +3 ∩ cof( ω n +1 ) reflects then CSR ( ω n +2 ) fails. However, Jech and Shelah [23],and later Cummings and Wylie [13], also show that best possible club reflectionproperties below ℵ ω , under the known limitations, are known to be consistent.Limitations on stationary reflection emerge from incompactness principles. Oneof the most prominent of these is Jensen’s (cid:3) κ . In [24], Jensen showed that (cid:3) κ holdsin L and implies the existence of many nonreflecting stationary sets.Further studies showed that variations of (cid:3) κ place limitations on the cofinality ofreflection points, as well as the amount of simultaneous reflection. For instance, in[37], Schimmerling introduced the hierarchy of square principles, (cid:3) κ,λ , 1 ≤ λ ≤ κ + .As λ increases, this hierarchy is strictly decreasing in strength; see Jensen for κ regular and [10] for κ singular. For a regular cardinal κ and λ ≤ κ , Schimmerlingand independently Foreman and Magidor have observed that if κ <λ = κ and (cid:3) κ,<λ holds then every stationary subset of κ + has a stationary subset which does notreflect at any point of cofinality ≥ λ ; see [12]. In particular, κ <κ = κ and (cid:3) κ,<κ imply that every stationary subset of κ + has a stationary subset which does notreflect at any point in κ + ∩ cof( κ ). In [10], the Cummings, Foreman, and Magidorextended these results and developed the theory for κ singular. LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 3
Other notable weakenings of (cid:3) κ were introduced and developed by Todorcevic([39]). These principles, denoted (cid:3) ( κ + ) and (cid:3) ( κ + , λ ), place refined limitations onthe extent of stationary reflection. See [14], [22], and [35].The weakest nontrivial form of square studied by Jensen is the so-called WeakSquare, denoted (cid:3) ∗ κ , Remarkably, (cid:3) ∗ κ is equivalent to a key incompactness phe-nomenon, the existence of a special Aronszajn tree. Let us recall the relevantdefinitions. A tree is a partially ordered set ( T, ≤ T ) so that for each x ∈ T , theset of ≤ T -predecessors of x is well-ordered; we refer to the level of x in T as theordertype of this set. If α is an ordinal, we use Lev α ( T ) to denote all x ∈ T ofheight α . The height of T is the least ordinal α so that T has no elements of height α . A branch through T is a linearly ordered subset of T , and a cofinal branch is abranch which intersects every level below the height of T .Let κ be regular. A κ -tree is a tree T of height κ so that each level has size < κ ;we will always assume that for each such tree, each node in the tree has incompatibleextensions to all higher levels. κ is said to have the tree property if every κ -tree hasa cofinal branch. K¨onig showed ([25]) that ω has the tree property, while Aronszjanhas shown that the tree property fails at ω (the result was communicated in [29]).The extent of the tree property on cardinals greater than ω , a famous questionof Magidor’s, is independent of ZFC . A watershed in our understanding is due toMitchell and Silver ([34]) who showed that the tree property at ω is consistentfrom a weakly compact cardinal.A tree which witnesses the failure of the tree property is said to be Aronszajn (i.e., a κ -tree which has no cofinal branches); the existence of such a tree is aninstance of incompactness. A particularly strong witness that a tree is Aronszajnis given by a specializing function : in the case that κ = λ + , a specializing functionis an f : T −→ λ so that if x < T y , then f ( x ) = f ( y ). (For an exploration of theseconcepts at an arbitrarily regular cardinal, see [27].) Having a specializing functionis a particularly strong witness to being Aronszajn since the function witnesses that T remains Aronszajn in any cardinal preserving extension of that model. T is saidto be a special Aronszajn tree if there is a specializing function for T . The propertyof interest to us is the following: Definition 1.4.
Let κ be regular. We say that κ + has the special Aronszajntree property if there are Aronszajn trees on κ + , and if every Aronszajn tree on κ + is special. We denote this property by SATP ( κ + ).By a result of Specker ([38]), if κ <κ holds, then there is a special Aronszajn treeon κ + ; in particular, if the CH holds, then there is a special Aronszajn tree on ω . Jensen ([24]) later showed that the principle (cid:3) ∗ κ holds if and only if there isa special Aronszajn tree on κ + ; since κ <κ implies (cid:3) ∗ κ , this strengthens Specker’sresult.With regards to constructing specializing functions by forcing, Baumgartner,Malitz, and Reinhardt showed ([7]) that MA + ¬ CH implies SATP ( ω ). Later,Laver and Shelah showed ([30]) that SATP ( ω ) is consistent from a weakly com-pact cardinal. Generalizing this further, Golshani and Hayut have recently shown([19]), using posets which specialize with anticipation, that it is consistent that,simultaneously, for every regular cardinal κ , SATP ( κ + ) holds. Krueger has gen-eralized the result of Laver-Shelah (and also Abraham-Shelah, [2]) in a differentdirection ([28]), showing that it is consistent with the CH that any two countablyclosed Aronszjan trees on ω are club isomorphic. And finally, Asper´o and Golshani OMER BEN-NERIA AND THOMAS GILTON ([3]) have announced a positive solution to the question of whether
SATP ( ω ) isconsistent with the GCH . This work continues the study of the tension between dif-ferent manifestations of compactness and incompactness phenomena in set theory,which together with the study of tension with other fundamental principles such asapproximation principles (e.g., [8], [18]) and cardinal arithmetic (e.g., [15], [36]) iscentral to our understanding of their extent and limitations.We proceed to describe our result in general terms and highlight the challengesthat appear in the process. Let κ be a weakly compact cardinal in a ground model V of GCH . We obtain the model which witnesses Theorem 1.1 by first defining, inthe extension by P = Col( ω , < κ ), a κ + -length iteration C κ + = h C τ , C ( τ ) | τ < κ + i of adding clubs which will eventually witnesses CSR ( ω ).After forcing with C κ + , we then force with a κ + -iteration S κ + = h S τ , S ( τ ) | τ <κ + i specializing the desired Aronszajn trees ˙ T τ , i.e., S ( τ ) = S ( ˙ T τ ) (see Section 1.1for precise definitions of the posets).To make this strategy work, we need, among other things, that all stationarysubsets of ω ∩ cof( ω ) which appear in the final generic extension by P ∗ ˙ C κ + ∗ ˙ S κ + reflect as in the definition of CSR ( ω ). Consequently, the club adding posets mustanticipate names for stationary sets added by the later specializing iteration. Inorder to carry this through, we define the names ˙ C τ and ˙ S τ , for τ < κ + , simul-taneously. More precisely, for each τ < κ + , given that the P -name ˙ C τ and the( P ∗ ˙ C τ )-name ˙ S τ have been defined, we use a bookkeeping function to pick the( P ∗ ˙ C τ ∗ ˙ S τ )-name ˙ S τ of a stationary subset of κ ∩ cof( ω ), and we set ˙ C ( τ ) to be the( P ∗ ˙ C τ )-name for the poset to add, with ˙ S τ -anticipation, the desired club. Thenwe select the ( P ∗ ˙ C τ +1 ∗ ˙ S τ )-name ˙ T τ for an Aronszajn tree on κ .As expected, the tension between compactness and incompactness gives rise totension between the different parts of the forcing construction. We list three notablemanifestations: (1) Working with Intermediate Generic Extensions. A central propertyof the Laver-Shelah forcing ([30]) is the existence of intermediate forcing extensionsin which regular cardinals α < κ becomes ω and the relevant portion ˙ T τ ∩ ( α × ω ) ofthe trees are Aronszajn trees on α . Accompanying this is a machinery of projectingconditions of P ∗ ˙ S τ to those intermediate extensions. In [30] the existence of suchintermediate extensions is secured by the weak compactness of κ , and the fact that P ∗ ˙ S τ is κ -c.c. However, in our case, the presence of the poset ˙ C τ prevents the initialsegment P ∗ ˙ C τ from being κ -c.c. To overcome this difficulty, we use the fact thatthe full collapse poset P absorbs many restricted subforcings of P ∗ ˙ C τ , which allowsus to place upper bounds on various generic filters of the restricted poset. We thencouple this in Section 6 with a generalization of a result of Abraham’s ([1]) that(stated in current language) if ˙ Q is an Add( ω, ω )-name for an ω -closed poset, thenAdd( ω, ω ) ∗ ˙ Q is strongly proper. This secures the existence of sufficiently manystrongly generic conditions (and in turn, the existence of intermediate extensions). (2) Preservation of Stationary Sets by Quotients. The ability to adda closed unbounded set through the reflection points α < κ of a stationary set˙ S τ ⊆ κ ∩ cof( ω ) hinges upon the fact that many such points exist. The weakcompactness of κ guarantees that for many α < κ , ˙ S τ ∩ α is a stationary subset I.e., it is positive with respect to the weakly compact filter on κ in the ground model V . LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 5 of α in the restricted generic extension where α = ω . The forcing constructionof [31] uses the fact that the related quotient of P ∗ ˙ C τ by its initial segment is σ -closed, and an argument of Baumgartner’s shows that σ -closed posets preservethe stationarity of stationary sets of countable cofinality ordinals. By contrast, forus the stationary sets ˙ S τ further rely on the specializing poset ˙ S τ , and althoughthe poset P ∗ ˙ C τ ∗ ˙ S τ is, σ -closed, it does not in general admit σ -closed quotientsby its natural restrictions to heights α < κ . Nevertheless, in Section 4 we analyzethe Laver-Shelah iteration ˙ S τ to prove that the relevant quotients preserve thestationary of ˙ S τ ∩ α for many suitable α < κ . (3) Preservation of Aronszajn Trees. The organization of the posets ˙ C τ and˙ S τ , described above, guarantees that for each τ < κ + , ˙ T τ , is a ( P ∗ ˙ C τ +1 ∗ ˙ S τ )-nameof an Aronszajn tree on κ , which is specialized by P ∗ ˙ C τ +1 ∗ ˙ S τ +1 .However, in the final forcing construction, S τ +1 follows the extended iteration P ∗ ˙ C κ + , and on its face, P ∗ ˙ C κ + ∗ ˙ S τ might introduce a cofinal branch to T τ , causingthe specializing poset S ( τ ) to collapse κ . To guarantee that this cannot occur, anAronszajn preservation theorem is required for the quotient of P ∗ ˙ C κ + ∗ ˙ S τ by P ∗ ˙ C τ +1 ∗ ˙ S τ . The fact that no new reals are added during the iteration, andthat ˙ C κ + is not κ -closed, presents us from using known preservation arguments(for instance those of [40]). Therefore, in Section 6 we develop an alternativepreservation argument which fits the properties of the poset P ∗ ˙ C κ + , and we applythem in Section 7 to show that the tree T τ remains Aronszajn. Structure of this work:
In the rest of this section, we review relevant prelim-inaries regarding forcing and weakly compact cardinals. The first part of the workconsists of Sections 2 through 4. In Section 2 we develop the notion and funda-mental properties of posets which are strongly proper with respect to the weaklycompact filter F WC on κ . We will later verify that initial segments of the form P ∗ ˙ C τ , for τ < κ + , are members this class. Section 3 studies an iteration of spe-cializing posets S τ , following an F WC -strongly proper poset P ∗ . We prove that themain results of the Laver-Shelah analysis applies in this context as well. Section 4is devoted to showing that suitable quotients of specializing iterations of the form P ∗ ∗ ˙ S τ , where P ∗ is strongly proper in the sense specified in Section 2, preservestationary subsets of countable cofinality ordinals. In Part 2 of the paper, we con-struct specific posets playing the role of P ∗ above, and we prove our theorem. InSection 5, we introduce the complementary notion of posets which are completelyproper with respect to F WC , and later we apply this analysis to ˙ C τ , τ < κ + . Weshow that the composition of the collapse and a poset which is completely properwith respect to F WC is strongly proper with respect to F WC . Section 6 developsthe main properties of the club adding iteration C τ . And finally, we combine theresults of the previous sections in Section 7 to prove Theorem 1.1.1.1. Forcing.
In this subsection, we review our conventions about forcing andprovide explicit definitions of posets which we will use throughout the paper.To begin, in order to anticipate working with iterations later, we will work with pre-orderings (i.e., relations which are transitive and reflexive) rather than partialorders. Moreover, we will use the
Jerusalem convention for forcing . Thus weview a forcing poset as a triple ( Q , ≤ Q , Q ), where ≤ Q is a pre-ordering and where0 Q is a smallest element; for conditions p, q ∈ Q , we will write p ≥ Q q to indicatethat p is an extension of q . When context is clear, we will drop explicit mention of OMER BEN-NERIA AND THOMAS GILTON Q in 0 Q and ≤ Q . Given that we are only working with pre-orderings rather thanpartial orders, we will often have conditions p, q ∈ Q so that p ≤ Q q and q ≤ Q p but q and p are not literally equal as sets. In this case, we will write p = ∗ Q q , or simply p = ∗ q if Q is clear from context.If Q is a poset, we say that Q is ω -closed with sups if for any increasingsequence h q n : n ∈ ω i of conditions in Q , there exists a ≤ Q -least upper bound q ofthe sequence. Any such q is referred to as a sup of the sequence. Note that thisdoes not say that any two compatible conditions in Q have a sup. Moreover, italso does not require that a sup of an increasing ω -sequence is unique. However, if q and q are two sups of such a sequence, then q = ∗ Q q . These observations willbe important later when we deal with iterations of posets with this property.If Q is a poset and q ∈ Q , then we use Q /q to denote all conditions in Q whichextend q . If π : Q −→ Q is an isomorphism and if ˙ τ is a Q -name, then we use π [ ˙ τ ] to denote the Q -name defined recursively as π [ ˙ τ ] := {h π [ ˙ σ ] , π ( q ) i : h ˙ σ, q i ∈ ˙ τ } .We call π [ ˙ τ ] the π -shift of ˙ τ ; it names the same object as ˙ τ .Let M ≺ H ( θ ) be an elementary substructure and U ∈ M a poset. A condition u in a poset U is ( K, U )- completely generic , if the set { ¯ u ∈ U ∩ K : u ≥ ¯ u } ofweaker conditions in K meets all dense subsets D ⊆ U which belong to M , andthus, forms a ( K, U )-generic filter.For the remainder of the paper, we fix a weakly compact cardinal κ ; we willreview facts about weak compactness in the next subsection.Throughout the paper, we will use P to denote the Levy collapse Col( ω , < κ ).If α < κ is inaccessible, we use P ↾ α to denote the collapse Col( ω , < α ). Weview conditions in Col( ω , < κ ) as countable functions p so that dom( p ) ⊆ κ andso that for each ν ∈ dom( p ), p ( ν ) is a countable, partial function from ω to ν .If G is a V -generic filter over P , then we use G ↾ α to denote the V -generic filter { p ↾ α : p ∈ G } over P ↾ α .For adding clubs, we generalize the club-adding poset of Magidor ([31]) by in-corporating anticipation; we only state the definition in the generality needed forour paper. Recall that if S is stationary in α , then the trace of S , denoted tr( S ),consists of all β < α so that S reflects at β . Definition 1.5.
Let S be a cardinal-preserving poset in some model W , and let ˙ S be an S -name for a stationary subset of ω ∩ cof( ω ). We let CU ( ˙ S, S ) denote theposet, defined in W , where conditions are closed, bounded subsets c of ω so that (cid:13) S ˇ c ⊆ tr( ˙ S ) ∪ ( ω ∩ cof( ω )) . The ordering is end-extension.We emphasize that in order to be a condition in CU ( ˙ S, S ), a given closed, boundedsubset of ω must be outright forced by S to be contained, mod cofinality ω points,in tr( ˙ S ). Since any condition c in CU ( ˙ S, S ) can be extended by placing an ordinalof cofinality ω above max( c ), we see that CU ( ˙ S, S ) does add a club subset of ω of the model. Moreover, the poset is trivially ω -closed, so preserves ω . Howeverpreservation of ω is a non-trivial matter.We now review the definition of the poset which we will use to specialize Aron-szajn trees on ω . The poset itself will decompose such a tree into a union of ω -many antichains, which in this case is equivalent to having a specializing func-tion. LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 7
Definition 1.6.
Suppose that T is an Aronszajn tree on ω . Let S ( T ) denote theposet where conditions are functions f with countable domain dom( f ) ⊆ ω , andwhere for each α ∈ dom( f ), f ( α ) ⊆ T is a countable antichain in < T . Recallingthat we are using the Jerusalem convention for ordering, we say that g extends f ,written f ≤ g , if dom( f ) ⊆ dom( g ) and if for all α ∈ dom( f ), f ( α ) ⊆ g ( α ).It is clear that S ( T ) is ω -closed. Moreover, if a tree T ′ is not Aronszajn, thenthe analogously defined poset S ( T ′ ) will collapse ω V .1.2. Weak Compactness.
In this final subsection, we review facts about the weakcompactness of κ , beginning with the definition of the weakly compact filter F WC on κ : Definition 1.7. F WC is the filter generated by subsets A of κ for which there issome U ⊆ V κ and a Π -statement Φ, satisfied by ( V κ , ∈ , U ), so that A = { α < κ | α is regular, and ( V α , ∈ , U ∩ V α ) | = Φ } . The filter F WC is κ -closed as well as normal. We will need the following corollaryof normality, the proof of which is routine. Lemma 1.8.
Suppose that A ∈ F +WC and f : A −→ κ is a regressive function.Then there exists some H ∈ F WC so that f is constant on A ∩ H . It will be helpful at later parts in our argument to phrase membership in theweakly compact filter in terms of embeddings. The idea will be that B ∈ F WC ifffor “most” κ -models M with j : M −→ N elementary and crit( j ) = κ , κ ∈ j ( B ).We make this precise in the following few items. Definition 1.9.
Suppose that α is an inaccessible cardinal. We say that a transitiveset M is an α -model if M | = ZFC − , | M | = α , α ∈ M , and <α M ⊆ M .We recall the following result from [21] about the weak compactness of κ : Proposition 1.10.
For any κ -model M , there exists an elementary embedding j : M −→ N for some transitive N so that j, M ∈ N and crit( j ) = κ . Elementary embeddings of κ -models are naturally associated with M -normalultrafilters U on κ . A filter U ⊆ P ( κ ) ∩ M is an M -normal ultrafilter if U is an M -ultrafilter, and for every A ∈ U and regressive function f : A → κ in M thereexists some A ′ ⊆ A in U so that f ↾ A ′ is constant. We note that being M -normalimplies that U is closed under intersections of < κ -sequences in M consisting ofsets in U .It is routine to verify that each elementary embedding j : M → N as in theproposition above, gives rise to an M -normal ultrafilter U j = { A ∈ P ( κ ) ∩ M | κ ∈ j ( A ) } . Conversely, we can associate to each M -normal ultrafilter U its ultrapowerembedding j U : M → N ∼ = Ult( M, U ). Proposition 1.11.
Let M be a κ -model and B ∈ M a subset of κ . If B ∈ U forevery M -normal ultrafilter U on κ then B ∈ F WC .Proof. Let
M, B be as in the statement of the proposition. To prove that B ∈ F WC ,it suffices to show that there is a Π statement Ψ satisfied by ( V κ , ∈ , E M , B ) so thatthe set { α < κ : α is regular, and ( V α , ∈ , E M ∩ V α , B ∩ α ) | = Ψ } is contained in B .Fix a subset E M ⊆ κ × κ so that ( κ, E M ) is isomorphic to M . Then the assertionsthat ” E M is well-founded”, that ”( κ, E M ) is isomorphic to a transitive κ -model”, OMER BEN-NERIA AND THOMAS GILTON and that ” B is represented in ( κ, E M ) by b ∈ κ ”, are all within the class of Π formulas over ( V κ , ∈ , E M , B ) (see [26], Section 2 for details). Let Φ denote theirconjunction.We also note that for a subset U M ⊆ κ , the assertion that U M codes a subsetof M ∩ P ( κ ) which is an M -normal ultrafilter, is Σ ω . Therefore the assertion Φ stating that“ ∀ U M ⊆ κ , if U M codes an M -normal ultrafilter, then B ∈ U M ”is Π . Let Φ be the conjunction Φ ∧ Φ , a Π formula satisfied in ( V κ , ∈ , E M , B ).Define X := { α < κ : α is regular, and ( V α , ∈ , E M ∩ V α , B ∩ α ) | = Φ } , and weshow that X ⊆ B . Fix some α ∈ X . Then the relation E M ↾ α = E M ∩ V α ⊆ α × α is well-founded, and it codes an α -model M α with B ∩ α represented in ( α, E M ↾ α )by the same element b ∈ κ which represents B in ( κ, E M ). Let i α : M α → M κ bethe elementary embedding resulting from the identifications M α ∼ = ( α, E M ↾ α ) ≺ ( κ, E M ) ∼ = M . It is straightforward to verify that cp( i α ) = α , i α ( α ) = κ , and i α ( B ∩ α ) = B .It follows that U α = { A ⊆ α : α ∈ i α ( A ) } is an M α -normal ultrafilter. Let j α : M α → N α be the induced ultrapower embedding, and let k α : N α → M bethe factor map given by k α ([ f ] U α ) = i α ( f )( α ). We note that cp( k α ) > α . Since( V α , ∈ , E M ↾ α, B ∩ α ) satisfies Φ, B ∩ α ∈ U α . The last implies that α ∈ j α ( B ∩ α ),which in turn implies that α = k α ( α ) ∈ k α ◦ j α ( B ∩ α ) = i α ( B ∩ α ) = B . (cid:3) F WC -strongly proper posets In this section we transition into the main body of the paper. After brieflyreviewing some important facts about strong genericity in Subsection 2.1, we thendefine, in Subsection 2.2, the class of F WC -strongly proper posets. This class, whichincludes the collapse P , consists of posets for which we may build various residuesystems and thereby obtain strongly generic conditions for models of interest.2.1. Review of Strongly Generic Conditions.
Here we review the definitionand basic properties of strongly generic conditions. Much of this material wasoriginally developed by Mitchell [34]. Parts of our exposition here summarize theexposition in [28], Section 1, to which we refer the reader for proofs.
Definition 2.1.
Let N ≺ H ( θ ), where θ is regular. Let Q ∈ N be a poset. Acondition q ∈ Q is said to be a strongly ( N, Q ) -generic condition if for any set D which is dense in Q ∩ N , D is predense above q in Q . Remark 2.2.
Note that if Q ∈ N , with N as above, then any strongly ( N, Q ) -generic condition is also an ( N, Q ) -generic condition. Moreover, q is a strongly ( N, Q ) -generic condition iff q (cid:13) Q ˙ G Q ∩ N is a V -generic filter over Q ∩ N . We now review a combinatorial characterization of strongly generic conditions,implicit in [34], Proposition 2.15, in terms of the existence of residue functions.
Definition 2.3.
Suppose that Q ∈ N ≺ H ( θ ), q ∈ Q , and s ∈ Q ∩ N . s is said tobe a residue of q to N if for all t ≥ Q ∩ N s , t and q are compatible in Q .A residue function for N above q is a function f N defined on Q /q so thatfor each r ∈ Q /q , f N ( r ) is a residue of r to N .Finally, if q, r ∈ Q and s ∈ Q ∩ N , we say that s is a dual residue of q and r to N if s is a residue for both q and r to N . LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 9
Lemma 2.4. q ∈ Q is ( N, Q ) -strongly generic iff there is a residue function for N above q . In the next subsection, we will isolate further properties of residue functions ofinterest. For now, we review the process by which strongly generic conditions allowus to break apart the forcing Q into a two-step iteration. Notation 2.5.
Let Q be a poset and q ∈ Q . Suppose Q ∈ N ≺ H ( θ ) and q is a strongly ( N, Q )-generic condition. Fix a V -generic filter ¯ G over Q ∩ N . In V [ ¯ G ], let ( Q /q ) / ¯ G denote the poset where conditions are all r ∈ ( Q /q ) which are Q -compatible with every condition in ¯ G . The ordering is the same as in Q .The following two results originate in [33]; our formulation of them follows [28]. Lemma 2.6.
Suppose Q ∈ N ≺ H ( θ ) and q is a strongly ( N, Q ) -generic condition.Then for all r ≥ q and s ∈ Q ∩ N , s is a residue of r to N iff s (cid:13) Q ∩ N r ∈ ( Q /q ) / ˙ G Q ∩ N . Lemma 2.7.
Suppose Q ∈ N ≺ H ( θ ) and q is a strongly ( N, Q ) -generic condition.Then(1) If r ≥ q , s ∈ Q ∩ N , and r and s are Q -compatible, then there exists t ≥ Q ∩ N s so that t is a residue of r to N ;(2) if D ⊆ Q is dense above q , then Q ∩ N forces that D ∩ ( Q /q ) / ˙ G Q ∩ N is densein ( Q /q ) / ˙ G Q ∩ N . Exact Residue Functions and F WC -Strong Properness. Following Nee-man ([17]), we next isolate the properties of residue functions (see Definition 2.3) ofinterest. Neeman also connected this with countable closure of the quotient forcing([17], subsection 2.2), which will apply in this work to the iteration P ∗ ˙ C , con-sisting of the collapse poset P followed by a Magidor-style, club-adding iteration˙ C . However, we were not able to apply this analysis to the final poset, which alsoincludes the specializing iteration, as we do not know if the quotients involving thespecializing iteration are even strategically closed. This will lead us later, in Section4, to an ad-hoc proof that the quotients of the final poset preserve stationary setsconsisting of countable cofinality ordinals, without having σ -closed quotients.Recalling that we are working with pre-orders (in anticipation of working withiterations later), we begin our discussion with the following definition. Definition 2.8.
Let Q be a poset which is ω -closed with sups. D ⊆ Q is said tobe countably = ∗ -closed if(a) for each q ∈ D and r ∈ Q , if r = ∗ q , then r ∈ D ;(b) if h q n : n ∈ ω i is an increasing sequence of conditions all of which are in D and if q ∗ is a sup of the sequence, then q ∗ ∈ D .Such sets D will arise later as the domains of exact (see below) residue functions,whose domains need not in general be all of the poset under consideration, but onlya dense, = ∗ -closed subset. We will construct such functions in Proposition 5.6.The following is Neeman’s notion of an exact strong residue function for N withdense domain below q ([17], Definitions 1.6, 2.10), but with the requirement ofstrategic continuity strengthened to continuity. Definition 2.9.
Let Q be a poset which is ω -closed with sups, and fix N with Q ∈ N ≺ H ( θ ). Let q ∈ Q . A partial function f : Q /q ⇀ Q ∩ N is said to be an exact, strong residuefunction for N above q if it satisfies the following properties:(1) (dense domain) the domain of f is a dense, countably = ∗ -closed subset D of Q /q ;(2) (projection) r ≥ f ( r ) for all r ∈ D ;(3) (order preservation) for all r ∗ , r ∈ D , if r ∗ ≥ r , then f ( r ∗ ) ≥ f ( r );(4) (strong residue) for any r ∈ D and any u ∈ Q ∩ N so that u ≥ f ( r ), thereexists r ∗ ≥ r with r ∗ ∈ D so that f ( r ∗ ) ≥ u ;(5) (countable continuity) if h r n : n ∈ ω i is an increasing sequence of conditionsin D with a sup r ∗ , then f ( r ∗ ) is a sup of h f ( r n ) : n ∈ ω i . We call such a pair h q, f i a residue pair for ( N, Q ), or just a residue pair for N if Q is clear from context.The following appears in [17] (Lemma 2.11). Lemma 2.10.
Suppose that Q is separative, q ∈ Q , and that f : Q /q −→ Q ∩ N is a function satisfying properties (2) and (4) of Definition 2.9. Then f is order-preserving on its domain. Example 2.11.
Let α < κ be inaccessible. Then the function f : P −→ P ↾ α given by f ( p ) = p ↾ α is an exact, strong residue function for any M ≺ H ( θ ) with M ∩ κ = α above the condition ∅ and has all of P as its domain.Our next task is to isolate the models which for us will play the role of “ N ” inDefinition 2.9. First some notation which we will fix for the remainder of the paper. Notation 2.12.
Let ⊳ be a fixed well-order of H ( κ + ).In the following definitions and claims make a standard use of continuous se-quences of elementary substructures M α , where | M α | = α < κ and M α ∩ κ = α , toform natural restrictions P ∗ ∩ M α of posets P ∗ which are members of the modelson the chain. For ease of notation in describing such chains, we use terminologysimilar to [28] and introduce the notion of a P -suitable sequence, for a parameter P . Definition 2.13.
Let P ∈ H ( κ + ) be a parameter. We say that a sequence h M α : α ∈ A i is P -suitable if(1) A ∈ F WC ;(2) for each α ∈ A , α is inaccessible, M α ∩ κ = α , <α M α ⊆ M α , and | M α | = α ;(3) for each α ∈ A , M α ≺ ( H ( κ + ) , ∈ , ⊳ ) and P ∈ M α ;(4) if α < β are in A , then M α ∈ M β , and also if γ ∈ A ∩ lim( A ), then M γ = S { M δ : δ ∈ A ∩ γ } .We refer to a single model M satisfying (2) and (3) as a P -suitable model .It is clear from the definition that if ~M is P -suitable and B ⊆ dom( ~M ) is in F WC , then h M α : α ∈ B i is also P -suitable. It is also clear that for any P ∈ H ( κ + ),there exists a P -suitable sequence.The next definition is the main item of this section; it specifies a class of posetswhich contains the Levy collapse P and each of which can play the role of a prepara-tory forcing for a ℵ -c.c. iteration specializing Aronszajn trees. (The work in Note that r ∗ is in D by (1) and also that the sequence h f ( r n ) : n ∈ ω i is increasing by (3). LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 11
Section 3 is devoted to showing this.) This type of poset uses systems of residuefunctions which are similar to the exact residue systems of [17].
Definition 2.14.
Let P ∗ be a poset of size κ which is ω -closed with sups and whichcollapses all cardinals in the interval ( ω , κ ). We say that P ∗ is F WC -stronglyproper if for any P ∗ -suitable sequence ~M there exist an A ∈ F WC with A ⊆ dom( ~M ), a sequence h p ∗ ( M α ) : α ∈ A i of conditions in P ∗ , and a sequence h ϕ M α : α ∈ A i of functions satisfying the following properties, for each α ∈ A :(1) p ∗ ( M α ) is compatible with every condition in P ∗ ∩ M α ;(2) ϕ M α is an exact, strong residue function for ( M α , P ∗ ) above p ∗ ( M α ), anddom( ϕ M α ) ⊆ P ∗ /p ∗ ( M α );(3) if β ∈ A is greater than α , then p ∗ ( M α ) and ϕ M α are members of M β .We will refer to the sequence of pairs hh p ∗ ( M α ) , ϕ M α i : α ∈ A i as a residue system for ~M ↾ A and P ∗ .A condition p ∗ ( M α ) satisfying item (1) of Definition 2.14 was called universal in[9]. Example 2.15.
The Levy Collapse poset P is an example of an F WC -stronglyproper poset. Indeed, letting ~M be any suitable sequence, we may take the set A in Definition 2.14 to just be dom( ~M ), which is in F WC by Definition 2.13(1).Then we define p ( M α ) := ∅ for all α ∈ A and define ϕ M α on the entire poset P by ϕ M α ( p ) = p ↾ α . As stated in Example 2.11, each ϕ M α is an exact, strong residuefunction for M α above ∅ . The remaining properties of Definition 2.14 are trivial.In our intended applications, the posets playing the role of P ∗ in Definition 2.14will be of the form P ∗ ˙ C , where ˙ C is a P -name for an iteration of club-adding withanticipation.We now check, by a standard argument, that forcing with an F WC -stronglyproper poset preserves κ . Lemma 2.16.
Suppose that P ∗ is F WC -strongly proper. Then forcing with P ∗ preserves κ .Proof. By Definition 2.14, we know that P ∗ preserves ω and collapses all cardinalsin the interval ( ω , κ ). Thus if P ∗ does not preserve κ , then we may find a condition p ∈ P ∗ and a P ∗ -name ˙ f for a function from ω which p forces is cofinal in κ . Since P ∗ has size κ , we may assume that the name ˙ f is a member of H ( κ + ). Let ~M be a n ˙ f , p, P ∗ o -suitable sequence. Also let A ⊆ dom( ~M ) witness Definition 2.14, and let N denote the least model on the sequence ~M ↾ A . By Definition 2.14, we may finda condition p ∗ ( N ) and an exact, strong residue function for ( N, P ∗ ) above p ∗ ( N ).Since p ∈ N , Definition 2.14(1) implies that p ∗ ( N ) and p are compatible. So let q be an extension of them both. Then since q is a ( N, P ∗ )-strongly generic conditionand ˙ f ∈ N , q forces that ran( ˙ f ) ⊆ N ∩ κ < κ . But this contradicts the fact that p forces that ˙ f is unbounded in κ . (cid:3) The remainder of the subsection is dedicated to proving lemmas about howsuitable sequences interact with the weak compactness of κ . Lemma 2.17.
Let P ∈ H ( κ + ) and ~M be P -suitable. Then P ⊆ S α ∈ dom( ~M ) M α . Proof.
By definition of a suitable sequence, each model on the sequence is elemen-tary with respect to the fixed well-order ⊳ on H ( κ + ), and therefore each modelcontains the ⊳ -least surjection ψ from κ onto P . Then P = ψ [ κ ] = [ α ∈ dom( ~M ) ψ [ α ] ⊆ [ α ∈ dom( ~M ) M α . (cid:3) Lemma 2.18.
Let P ∈ H ( κ + ) , and let ~M be P -suitable. Suppose that M ∗ is a κ -model containing ~M and j : M ∗ −→ N a weakly compact embedding. Then(1) κ ∈ dom( j ( ~M )) , and j ( ~M )( κ ) = S α ∈ dom( ~M ) j [ M α ] ;(2) j ( P ) ∩ j ( ~M )( κ ) = j [ P ]; (3) S α ∈ dom( ~M ) M α is transitive, and j − ↾ M κ is the transitive collapse of M κ ;(4) let Q ∈ H ( κ + ) be a poset and ˙ τ a nice Q -name for a subset of X where X ∈ H ( κ + ) . Suppose also that ~M is { Q , ˙ τ } -suitable. Then the ( j ↾ Q ) -shiftof ˙ τ (defined in Subsection 1.1) equals j ( ˙ τ ) ∩ j ( ~M )( κ ) = j [ ˙ τ ] . In particular, j [ ˙ τ ] is a j [ Q ] -name.Proof. Let B := dom( ~M ). For item (1), we first observe that since B ∈ M ∗ ∩ F WC , B ∈ U j := { X ∈ P ( κ ) ∩ M ∗ : κ ∈ j ( X ) } , by the remarks preceding Proposition 1.11. Thus κ ∈ dom( j ( ~M )), and so we maylet M κ := j ( ~M )( κ ). Additionally, j ( B ) ∩ κ = B , and so by Definition 2.13(4), M κ = S α ∈ B j ( M α ). But | M α | = α < κ for each α ∈ B , and hence j ( M α ) = j [ M α ].Thus M κ = [ α ∈ B j [ M α ] , completing the proof of (1). (2) follows immediately.For (3), observe that if x ∈ M := S α ∈ dom( ~M ) M α , then a tail of the sequence ~M is x -suitable; then x ⊆ M by Lemma 2.17. Since j − ↾ M κ is an ∈ -isomorphismwhose range (namely M ) is transitive, j − is the transitive collapse.For (4), let π := j ↾ Q . By (1) and (2), j ( ˙ τ ) ∩ j ( ~M )( κ ) = j [ ˙ τ ], so we showthat π [ ˙ τ ], the π -shift ˙ τ , equals j [ ˙ τ ], the pointwise image of ˙ τ under j . Since ˙ τ is anice Q -name for a subset of X , for each x ∈ X we may find an antichain A x ⊆ Q so that τ = S {{ ˇ x } × A x : x ∈ X } ; here ˇ x is the canonical Q -name for x givenrecursively by ˇ x := {h ˇ y, Q i : y ∈ x } . Since ˇ x is the canonical Q -name for x , π (ˇ x )is the canonical π [ Q ]-name for x , which equals the canonical j ( Q )-name for x since0 π [ Q ] = 0 j ( Q ) . From this it follows that the π -shift of ˙ τ equals j [ ˙ τ ]. (cid:3) Lemma 2.19.
Suppose that ~M is P ∗ -suitable and that hh p ∗ ( M α ) , ϕ M α i : α ∈ dom( ~M ) i is a residue system for ~M and P ∗ . Then we may find some A ⊆ dom( ~M ) with A ∈ F WC so that for all α ∈ A , P ∗ ∩ M α (cid:13) ˇ α = ˙ ℵ .Proof. Suppose that ~M and hh p ∗ ( M α ) , ϕ M α i : α ∈ dom( ~M ) i are as in the statementof the lemma. Let A be the set of β ∈ dom( ~M ) so that P ∗ ∩ M β forces that ˇ β = ˙ ℵ .We apply Proposition 1.11 to show that A ∈ F WC . Let M ∗ be a κ -modelcontaining ~M , A , P ∗ , and the sequence hh p ∗ ( M α ) , ϕ M α i : α ∈ dom( ~M ) i . Let j : M ∗ −→ N be a weakly compact embedding, and we verify that κ ∈ j ( A ). Let LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 13 M κ = j ( ~M )( κ ). By Lemma 2.16, P ∗ forces that ˇ κ = ˙ ℵ . Thus j [ P ∗ ] forces thesame, which implies that j ( P ∗ ) ∩ M κ forces ˇ κ = ˙ ℵ by Lemma 2.17(2). (cid:3) We recall that in Definition 2.14(1), the condition p ∗ ( M α ) is required to becompatible with every condition in P ∗ ∩ M α . A practical corollary of this is thatany generic for P ∗ contains plenty of conditions of the form p ∗ ( M α ). Lemma 2.20.
Suppose that ~M is P ∗ -suitable, that hh p ∗ ( M α ) , ϕ M α i : α ∈ dom( ~M ) i is a residue system for ~M and P ∗ , and that B ⊆ dom( ~M ) is in F +WC . Supposethat there is a condition ¯ p ∈ P ∗ and that for each α ∈ B , there is a condition p α ∈ dom( ϕ M α ) so that ¯ p = ∗ ϕ M α ( p α ) for all α ∈ B ∗ . Then ¯ p (cid:13) ˙ X := n α < κ : p α ∈ ˙ G P ∗ o is unbounded in κ. In particular, taking p α := p ∗ ( M α ) with ¯ p the trivial condition, P ∗ (cid:13) n α < κ : p ∗ ( M α ) ∈ ˙ G P ∗ o is unbounded in κ. Moreover, letting B denote dom( ~M ) , if α ∈ B ∩ lim( B ) , then (cid:13) P ∗ ∩ M α n ξ < α : p ∗ ( M ξ ) ∈ ˙ G P ∗ ∩ M α o is unbounded in α. Proof.
Let p ∈ P ∗ be a condition extending ¯ p and γ < κ . We find an extension of p which forces that ˙ X \ γ = ∅ . Since B is unbounded in κ and ~M is P ∗ -suitable,Lemma 2.17 implies that p ∈ M β for some β ∈ B \ ( γ + 1). By definition of anexact, strong residue function, p β is compatible with p ≥ ϕ M β ( p β ) (in the casewhere p ∗ ( M β ) = p β , we use Definition 2.14(1)). Therefore, let p ∗ be a commonextension; then p ∗ (cid:13) β ∈ ˙ X \ γ .The proof in the case α ∈ B ∩ lim( B ) is identical, using the fact that, by Definition2.13, M α = S ξ ∈ B ∩ α M ξ in this case. (cid:3) F WC -Strongly Proper Posets and Specializing Aronszajn Trees on ω In this section we will prove that if P ∗ is an F WC -strongly proper poset, thenwe can iterate to specialize Aronszajn trees on κ in the extension by P ∗ . Recallfrom Lemma 2.16 that P ∗ forces that κ becomes ℵ and also preserves the CH ; thusthere are in fact Aronszajn trees on κ in any P ∗ -extension. By Example 2.15, thecollapse poset P is F WC -strongly proper, and therefore our results here generalizethose of [30].We consider a countable support iteration ˙ S = h ˙ S ξ , ˙ S ( ξ ) : ξ < κ + i of length κ + , specializing Aronszajn trees on κ in the P ∗ -extension. More precisely, ˙ S is a P ∗ -name for an iteration with countable support so that for any ξ < κ + , ˙ S ( ξ ) isan ˙ S ξ -name for the poset ˙ S ( ˙ T ξ ), where ˙ T ξ is a nice ˙ S ξ -name for an Aronszajn treeon κ ; see Definition 1.6. The iteration is constructed using the fixed well-order ⊳ of H ( κ + ) from Notation 2.12 as a bookkeeping function. In particular, for each ξ < κ + , the name ˙ S ξ is definable in ( H ( κ + ) , ∈ , ⊳ ) from P ∗ and ξ , and consequentlyit is a member of any model which is suitable with respect to P ∗ and ξ . We willuse R ξ to abbreviate P ∗ ∗ ˙ S ξ for each ξ < κ + .Since the poset R ξ is ω -closed, ˙ f has countable support, and ˙ f ( α ) is forced to bea countable sequence of countable sets of pairs of ordinals for each α ∈ dom( f ), the poset R ξ has a dense set of determined conditions, i.e., conditions ( p, ˙ f ) for whichthere is some function f in V so that p (cid:13) P ∗ ˙ f = ˇ f . The dense set of determinedconditions is also closed under sups of countable increasing sequences. Thus we willassume that all future conditions are determined. Notation 3.1.
Strictly speaking, the domain of a (determined) condition f in R ξ is a countable subset of ξ , and for each ζ ∈ dom( f ), f ( ζ ) is itself a functionwhose domain is a countable subset of ω . However, we will often make an abuseof notation and write f ( ζ, ν ) to mean the countable set of tree nodes f ( ζ )( ν ).The main goal of this section is to prove that ˙ S is forced to be κ -c.c. For this itsuffices to prove the following: Theorem 3.2.
For every ρ < κ + it is forced by the trivial condition of P ∗ that ˙ S ρ is κ -c.c. We will prove Theorem 3.2 by induction on ρ . Doing so will require two inductionhypotheses, the first of which is the following: Inductive Hypothesis I:
For each ξ < ρ , P ∗ (cid:13) ˙ S ξ is κ -c.c.We will assume Inductive Hypothesis I throughout the entire section. Later inthe section, after developing more of the theory, we will introduce a second, moretechnical inductive hypothesis; we state this after Remark 3.15. Though we assumethe first inductive hypothesis throughout, we will only use the second inductivehypothesis once it is introduced, and the results prior to the statement thereof donot require it.The rest of the section will proceed as follows. We will first establish, in Propo-sition 3.4, that for all ξ < ρ , there are plenty of intermediate extensions between V and the full R ξ -extension in which various restrictions of Aronszajn trees areAronszajn in the intermediate model. In light of this, we will define analogues ofthe “hashtag” and “star” principles from [30]; the former will say that two con-ditions have the same restriction to a given model, whereas the latter says thattwo conditions have a dual residue to a given model. Afterwards, we define thenotion of a splitting pair of conditions, a notion which will play a key role in lateramalgamation arguments. Next, we will state our second induction hypothesis,which describes the interplay between the star and hashtag principles mentionedabove. Using the second induction hypothesis, we will prove that splitting pairsexist and isolate sufficient conditions under which they can be amalgamated (seeLemma 3.17). Finally, we show that ˙ S ρ is forced to be κ -c.c., and we verify thatthe second induction hypothesis holds at ρ . Definition 3.3.
Let G ∗ be V -generic over P ∗ . If ξ ≤ ρ , f ∈ S ξ , and ¯ f is a function(not necessarily a condition), we write f ≥ ¯ f to mean that dom( ¯ f ) ⊆ dom( f ) andfor all h ζ, ν i ∈ dom( ¯ f ), ¯ f ( ζ, ν ) ⊆ f ( ζ, ν ).The next item establish the existence of the desired intermediate extensionsbetween V and V [ R ξ ] for ξ < ρ , and in turn the existence of plenty of residues. Werecommend recalling Lemma 2.18(3) before reading the proof. Proposition 3.4.
Suppose that ~M is R ρ -suitable. Then there exists B ∗ ⊆ dom( ~M ) with B ∗ ∈ F WC so that for any α ∈ B ∗ , for any residue pair h p ∗ ( M α ) , ϕ M α i for ( M α , P ∗ ) , and for any ζ ∈ M α ∩ ρ the following are true: LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 15 (1) ( P ∗ ∩ M α ) (cid:13) ( ˙ S ζ ∩ M α ) is α -c.c. Furthermore, ( R ζ ∩ M α ) (cid:13) ˙ T ζ ∩ M α is anAronszajn tree on α = ˙ ℵ ;(2) ( p ∗ ( M α ) , ˙ S ζ ) forces that ˙ G R ζ ∩ M α is a V -generic filter for R ζ ∩ M α ;(3) ( p ∗ ( M α ) , ˙ S ζ ) forces that ˙ T ζ ∩ (ˇ ω × ˇ α ) = ( ˙ T ζ ∩ M α )[ ˙ G R ζ ∩ M α ] .Proof. Fix an R ρ -suitable sequence ~M , and let B := dom( ~M ) so that B ∈ F WC byDefinition 2.13. By Proposition 1.11, it suffices to show that if M ∗ is any κ -modelwhich contains R ρ and ~M as elements, and if j : M ∗ −→ N is a weakly compactembedding, then κ satisfies the j -image of items (1)-(3) in N . So fix such an M ∗ and j : M ∗ −→ N . By the remarks before Proposition 1.11, F WC ∩ M ∗ ⊆ U j := { E ∈ M ∗ : κ ∈ j ( E ) } , and therefore κ ∈ j ( B ).Let M κ := j ( ~M )( κ ), and fix ζ ∗ ∈ M κ ∩ j ( ρ ) for the remainder of the proof.Since M κ = j (cid:2)S α ∈ B M α (cid:3) , by Lemma 2.18, we have that ζ ∗ ∈ ran( j ), and so ζ ∗ = j ( ζ ) for some ζ < ρ . Moreover, M κ ∩ j ( P ∗ ) = j [ P ∗ ]. Finally, by Lemma2.18(3), M κ ∩ j ( ˙ S ζ ) = j [ ˙ S ζ ] equals the ( j ↾ P ∗ )-shift of ˙ S ζ , which implies that j [ P ∗ ] (cid:13) j [ ˙ S ζ ] is κ -c.c. Consequently, j ( P ∗ ) ∩ M κ (cid:13) j ( ˙ S ζ ) ∩ M κ is κ -c.c.Similarly, applying Lemma 2.18(3) to R ζ and the nice name ˙ T ζ , j ( R ζ ) ∩ M κ (cid:13) j ( ˙ T ζ ) ∩ M κ is an Aronszajn tree on κ = ˙ ℵ , since R ζ forces that ˙ T ζ is an Aronszajn tree on κ = ˙ ℵ . This completes the proofthat κ satisfies the j -image of (1) in N .We now turn to verifying that κ satisfies the j -image of (2) in N . Towards thisend, fix a residue pair h p ∗ ( M κ ) , ϕ M κ i for ( M κ , j ( P ∗ )). To see that ( p ∗ ( M κ ) , j (˙ S ζ ) )forces the desired statement, fix an extension ( q ∗ , ˙ g ) of ( p ∗ ( M κ ) , j (˙ S ζ ) ) in j ( R ζ ).Let A ∗ ∈ N be a maximal antichain of j ( R ζ ) ∩ M κ = j [ R ζ ], and we will find someextension of ( q ∗ , ˙ g ) which forces that A ∗ ∩ ˙ G j ( R ζ ) = ∅ . Since q ∗ extends p ∗ ( M κ ) in j ( P ∗ ) and ϕ M κ is an exact, strong residue function, we may extend and relabel, ifnecessary, to assume that q ∗ ∈ dom( ϕ M κ ). Then ϕ M κ ( q ∗ ) ∈ j ( P ∗ ) ∩ M κ = j [ P ∗ ].Therefore ϕ M κ ( q ∗ ) = j ( q ) for some q ∈ P ∗ .Now let A := j − [ A ∗ ]; since A ∗ is a maximal antichain in j [ R ζ ], A is a maximalantichain in R ζ . However, note that since R ζ is not necessarily κ -c.c., A couldvery well have size κ and therefore we cannot assume that it is an element of M ∗ .Until after the proof of the next claim, we will work in V , not M ∗ . Let ˙ A (1) bethe P ∗ -name for n f ∈ ˙ S ζ : ( ∃ p ∈ ˙ G P ∗ ) ( p, f ) ∈ A o . Then q (cid:13) ˙ A (1) is a maximalantichain in ˙ S ζ .Since P ∗ (cid:13) ˙ S ζ is κ -c.c., by Inductive Hypothesis I, we may extend q in P ∗ to somecondition q ′ and find an ordinal β < κ and a sequence h ˙ f γ : γ < β i of P ∗ -names sothat q ′ (cid:13) V P ∗ ˙ A (1) = n ˙ f γ : γ < β o . Since a given ˙ f γ needn’t be a member of M ∗ , we show how to replace these nameswith ones that are in M ∗ , up to extending q ′ . Claim 3.5.
There exist a condition u ≥ P ∗ q ′ and a sequence of P ∗ -names h ˙ h γ : γ <β i in M ∗ so that u (cid:13) V ( ∀ γ < β ) ˙ h γ = ˙ f γ . Proof.
To find u , let G ∗ be a V -generic filter over P ∗ containing q ′ . Then S ζ :=˙ S ζ [ G ∗ ] ⊆ S η ∈ B M η [ G ∗ ], since R ζ ⊆ S η ∈ B M η . Since κ = ℵ V [ G ∗ ]2 and β < κ , thereexists some η ∈ B so that for all γ < β , ˙ f γ [ G ∗ ] ∈ M η [ G ∗ ]. By Lemma 2.20, thereexists a δ ≥ η so that p ∗ ( M δ ) ∈ G ∗ . Now let p ∗ ∈ G ∗ be an extension of p ∗ ( M δ )and q ′ so that p ∗ (cid:13) ( ∀ γ < β ) ˙ f γ ∈ M δ [ ˙ G P ∗ ].Back in V we define new names ˙ h γ for each γ < β ; recalling Notation 3.1, weview conditions in ˙ S ζ as having a domain which is a countable subset of ζ × ω sothat each element in the range is a countable subset of κ × ω .For each γ < β , ¯ ζ ∈ ζ ∩ M δ (an iteration stage), ν < ω (corresponding to the ν th tree antichain), and θ ∈ δ × ω (a node with height below δ ), let A ( γ, ¯ ζ, ν, θ )be a maximal antichain in P ∗ ∩ M δ of conditions p which decide whether or not θ is a member of ˙ f γ (¯ ζ, ν ). Let ˙ h γ be the P ∗ -name which is interpreted in anarbitrary generic extension via some G ∗ as follows: θ ∈ h γ (¯ ζ, ν ) iff there is some p ∈ A ( γ, ¯ ζ, ν, θ ) ∩ G ∗ which forces that θ ∈ ˙ f γ (¯ ζ, ν ). Otherwise h γ is undefined.We claim that u (cid:13) V ( ∀ γ < β ) ˙ h γ = ˙ f γ . To see this, let G ∗ be a V -generic filtercontaining u . Fix ¯ ζ < ζ and ν < ω , and we verify that f γ (¯ ζ, ν ) = h γ (¯ ζ, ν ). On theone hand, if θ ∈ h γ (¯ ζ, ν ), then by definition f γ (¯ ζ, ν ) is defined and also containsthe node θ .On the other hand, if τ ∈ f γ (¯ ζ, ν ) is a node, then since f γ ∈ M δ [ G ∗ ] has acountable domain, ¯ ζ ∈ M δ [ G ∗ ], and since f γ (¯ ζ, ν ) ∈ M δ [ G ∗ ] is countable, τ ∈ M δ [ G ∗ ] too. But M δ [ G ∗ ] ∩ V = M δ , since u ∈ G ∗ is a (strongly) ( M δ , P ∗ )-genericcondition (as it extends p ∗ ( M δ )). Hence ¯ ζ, τ ∈ M δ . Thus ¯ ζ ∈ M δ ∩ ζ and τ hasheight below M δ ∩ κ = δ . Next, since u is a strongly ( M δ , P ∗ )-generic condition and u ∈ G ∗ , we know that A ( γ, ¯ ζ, ν, τ ) ∩ G ∗ = ∅ , say with ¯ u in the intersection. But as τ ∈ f γ (¯ ζ, ν ), we must have that ¯ u forces that τ ∈ ˙ f γ (¯ ζ, ν ), and hence τ ∈ h γ (¯ ζ, ν ).This completes the proof that u (cid:13) ˙ f γ = ˙ h γ for each γ < β .Finally, since M ∗ is < κ -closed, the sequence of antichains h A ( γ, ¯ ζ, ν, θ ) : γ < β, ¯ ζ ∈ M δ ∩ ζ, ν < ω , θ ∈ ( δ × ω ) i is a member of M ∗ . Therefore, the sequence h ˙ h γ : γ < β i is a member of M ∗ too. (cid:3) (Claim 3.5)Continuing with the main body of the argument, let u ≥ P ∗ q ′ and h ˙ h γ : γ < β i witness the above claim. Since q ′ (cid:13) P ∗ ˙ A (1) = n ˙ f γ : γ < β o and u ≥ q ′ , we have( ∗ ) u (cid:13) P ∗ n ˙ h γ : γ < β o is a maximal antichain in ˙ S ζ . Since h ˙ h γ : γ < β i and u are in M ∗ , ( ∗ ) is satisfied in M ∗ . Applying j , j ( u ) (cid:13) Nj ( P ∗ ) n j ( ˙ h γ ) : γ < β o is a maximal antichain in j ( ˙ S ζ ) . Next, u ≥ P ∗ q ′ ≥ P ∗ q so j ( u ) ≥ j ( P ∗ ) j ( q ) = ϕ M κ ( q ∗ ), and j ( u ) ∈ j [ P ∗ ] ⊆ M κ so j ( u ) and q ∗ are compatible in j ( P ∗ ). Let q ∗∗ be a condition extending both of themwith ϕ M κ ( q ∗∗ ) ≥ j ( u ). Since q ∗∗ extends q ∗ , which forces that ˙ g is a conditionin j ( ˙ S ζ ), q ∗∗ forces this too. As q ∗∗ ≥ j ( u ) also forces that n j ( ˙ h γ ) : γ < β o is a LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 17 maximal antichain in j ( ˙ S ζ ), we may find an extension r ∗ of q ∗∗ , a j ( P ∗ )-name ˙ g ∗ ,and an ordinal γ < β so that r ∗ (cid:13) ˙ g ∗ ≥ ˙ g, j ( ˙ h γ ) . We may also extend, if necessary, to assume that r ∗ ∈ dom( ϕ M κ ), since r ∗ ≥ q ∗ ≥ p ∗ ( M κ ). Let r ∈ P ∗ so that ϕ M κ ( r ∗ ) = j ( r ).Now r ∗ ≥ j ( P ∗ ) q ∗∗ are both in dom( ϕ M κ ). Since ϕ M κ is order-preserving, j ( r ) = ϕ M κ ( r ∗ ) ≥ ϕ M κ ( q ∗∗ ) ≥ j ( u ). Then r ≥ u . As a result, r (cid:13) V ˙ h γ = ˙ f γ ∈ ˙ A (1). Bydefinition of ˙ A (1), we may find some P ∗ -extension r ′ of r so that ( r ′ , ˙ h γ ) extendssome element ( r ′ , ˙ f ) of A . Then j ( r ′ , ˙ f ) ∈ A ∗ , as A = j − [ A ∗ ]. Since r ′ extends r in P ∗ , we get that j ( r ′ ) ≥ j ( P ∗ ) j ( r ) = ϕ M κ ( r ∗ ). So j ( r ′ ) and r ∗ are compatiblein j ( P ∗ ). Let r ∗∗ be a condition extending both of them. Then ( r ∗∗ , ˙ g ∗ ) extends j ( r ′ , ˙ f ). Indeed, r ∗∗ extends j ( r ′ ) which extends j ( r ′ ). Furthermore, r ∗∗ extends r ∗ which forces that ˙ g ∗ ≥ j ( ˙ h γ ), and r ∗ extends j ( r ′ ) which forces that j ( ˙ h γ ) ≥ j ( ˙ f ).Thus ( r ∗∗ , ˙ g ∗ ) extends j ( r ′ , ˙ f ), and therefore( r ∗∗ , ˙ g ∗ ) (cid:13) j ( R ζ ) j ( r ′ , ˙ f ) ∈ A ∗ ∩ ˙ G j ( R ζ ) = ∅ . However, ( r ∗∗ , ˙ g ∗ ) also extends the starting condition ( q ∗ , ˙ g ). This finishes the proofthat κ satisfies the j -image of item (2).We now show that κ satisfies the j -image of item (3). Let G ∗ be V -generic for j ( R ζ ) containing the condition ( p ∗ ( M κ ) , j (˙ S ζ ) ), and let G = G ∗ ∩ M κ which, byitem (2), is a V -generic filter for j ( R ζ ) ∩ M κ = j [ R ζ ]. Let j ( T ζ ) denote j ( ˙ T ζ )[ G ∗ ],and let T ζ denote ( j ( ˙ T ζ ) ∩ M κ )[ G ]. We need to show that j ( T ζ ) ∩ ( κ × ω ) = T ζ .Since ˙ T ζ is a nice R ζ -name for a tree order on κ , for each τ, θ ∈ κ × ω , thereexists an antichain B θ,τ of R ζ so that, letting op( θ, τ ) denote the canonical namefor h θ, τ i , < ˙ T ζ = [ {{ op( θ, τ ) } × B θ,τ : θ, τ < κ } . Now if θ < T ζ τ , then that implies that j [ B θ,τ ] ∩ G = ∅ . Since j [ B θ,τ ] ⊆ j ( B θ,τ ) and G ⊆ G ∗ , we conclude that j ( B θ,τ ) ∩ G ∗ = ∅ , and therefore θ < j ( T ζ ) τ . On the otherhand, suppose that θ is not < T ζ -below τ . Then j [ B θ,τ ] ∩ G = ∅ . By considering amaximal antichain extending B θ,τ , we may find some condition u ∈ G so that u isincompatible with all conditions in j [ B θ,τ ]. Since u ∈ M κ ∩ j ( R ζ ) = j [ R ζ ], fix some u ∈ R ζ so that j ( u ) = u . By elementarity, u is incompatible with all conditionsin B θ,τ . But B θ,τ is a member of M ∗ . Consequently, by the elementarity of j ,we conclude that u = j ( u ) is incompatible with every condition in j ( B θ,τ ). Since u ∈ G ⊆ G ∗ , we conclude that j ( B θ,τ ) ∩ G ∗ = ∅ , and therefore θ is not < j ( T ζ ) -below τ . (cid:3) In both the previous result and Definitions 2.14 and 5.8, there was the apparentnecessity of refining the domain of a suitable sequence so that various desired be-havior obtains on each level of the refined sequence. The next item amalgamatesthis into one definition which we use frequently throughout.
Definition 3.6.
Let ~M be an R ρ -suitable sequence. We say that ~M is in pre-splitting configuration up to ρ if dom( ~M ) satisfies Definition 2.14 as well asthe conclusion of Proposition 3.4. Definition 3.7.
Suppose that ~M is in pre-splitting configuration up to ρ , ξ ≤ ρ ,and α ∈ dom( ~M ) so that ξ ∈ M α . Fix a residue pair h p ∗ ( M α ) , ϕ M α i for ( M α , P ∗ ). (1) For a (determined) condition ( p, f ), we define f ↾ M α to be the function ¯ f with domain dom( f ) ∩ M α so that for each h ζ, ν i ∈ dom( f ) ∩ M α ,¯ f ( ζ, ν ) = f ( ζ, ν ) ∩ M α . (2) We define D ( ϕ M α , ξ ) to be the set of conditions ( p, f ) ∈ R ξ so that p ∈ dom( ϕ M α ) and ( ϕ M α ( p ) , f ↾ M α ) is a condition in R ξ (and hence in R ξ ∩ M α ).(3) If ( p, f ) ∈ D ( ϕ M α , ξ ), we make a slight abuse of notation and define ( p, f ) ↾ M α to be the pair ( ϕ M α ( p ) , f ↾ M α ), when ϕ M α is clear from context.We observe that in general, for ( p, f ) ∈ D ( ϕ M α , ξ ), although ( p, f ) ↾ M α ∈ R ξ ∩ M α is a condition, it need not be residue of ( p, f ) to M α in the sense that it ispossible for some ( p ′ , f ′ ) ∈ R ξ ∩ M α which extends ( p, f ) ↾ M α to not be compatiblewith ( p, f ). However, ( p, f ) ↾ M α must have some extension (¯ p, ¯ f ) ∈ R ξ ∩ M α whichis a residue of ( p, f ). This is because if G ⊆ R ξ is V -generic and contains ( p, f ), thenby Proposition 3.4, ¯ G := G ∩ M α is V -generic over R ξ ∩ M α and ( p, f ) ↾ M α ∈ ¯ G .Since ( p, f ) ∈ G is compatible with every condition in ¯ G , there must be some(¯ p, ¯ f ) ∈ ¯ G which extends ( p, f ) ↾ M α and is a residue of ( p, f ), by Lemma 2.6. Lemma 3.8.
Suppose that ~M is in pre-splitting configuration up to ρ . Then foreach α ∈ dom( ~M ) , each residue pair h p ∗ ( M α ) , ϕ M α i for ( M α , P ∗ ) , and each ξ ≤ ρ with ξ ∈ M α , D ( ϕ M α , ξ ) is dense and = ∗ -countably closed in ( P ∗ /p ∗ ( M α )) ∗ ˙ S ξ .Proof. We will first prove the result for ξ < ρ , and then use this to prove the resultat ρ . Note that the = ∗ -countable closure of D ( ϕ M α , ξ ), in either case for ξ , followsfrom the continuity of ϕ M α and the closure of the posets. We therefore concentrateon showing density.Let ( p , f ) ∈ R ξ be given with p ≥ p ∗ ( M α ). By the observations followingDefinition 3.7, we may build increasing sequences h ( p n , f n ) : n ∈ ω i and h (¯ p n , ¯ f n ) : n ∈ ω i so that (i) (¯ p n , ¯ f n ) is a residue of ( p n , f n ) to M α ; (ii) ( p n +1 , f n +1 ) ≥ ( p n , f n ) , (¯ p n , ¯ f n ); (iii) p n +1 ∈ dom( ϕ M α ) and ϕ M α ( p n +1 ) ≥ ¯ p n .Now let ( p ∗ , f ∗ ) be a sup of h ( p n , f n ) : n ∈ ω i . Note that p ∗ ∈ dom( ϕ M α ) since p n ∈ dom( ϕ M α ) for each n and also that ϕ M α ( p ∗ ) is a sup of h ϕ M α ( p n ) : n ∈ ω i . Weclaim that ( ϕ M α ( p ∗ ) , f ∗ ↾ M α ) is a condition. Now ϕ M α ( p ∗ ) ≥ ϕ M α ( p n +1 ) ≥ ¯ p n foreach n , so ϕ M α ( p ∗ ) forces that h ¯ f n : n ∈ ω i is an increasing sequence of conditionsin ˙ S ξ , and therefore forces that S n ¯ f n is a condition too. The claim follows since S n ¯ f n = S n ( f n ↾ M α ) = ( S n f n ) ↾ M α = f ∗ ↾ M α . Now we show that the lemma holds for ξ = ρ . We deal with ρ limit first. Ifcf( ρ ) > ω , then the result holds since any ( p, f ) ∈ R ρ is in R ξ for some ξ < ρ . On theother hand, if cf( ρ ) = ω , then let h ξ n : n ∈ ω i be an increasing sequence of ordinalsin M α which is cofinal in ρ . By applying the lemma below ρ , we define an increasingsequence of extensions h ( p n , f n ) : n ∈ ω i of ( p, f ) so that f n ↾ [ ξ n , ρ ) = f ↾ [ ξ n , ρ )and so that ( p n , f n ↾ ξ n ) ∈ D ( ϕ M α , ξ n ). Now let p ∗ be a sup of h p n : n ∈ ω i and f ∗ := S n f n . Then ( p ∗ , f ∗ ) extends ( p, f ) and is a member of D ( ϕ M α , ρ ).Finally, assume that ρ = ρ + 1 is a successor, and let ( p, h ) ∈ R ρ be given.By the remarks after Definition 3.7, we may find a residue (¯ p, ¯ h ) of ( p, h ↾ ρ )to M α . Recalling Notation 3.1, we use h ( ρ ) ∩ M α in what follows as an abuseof notation for h h ( ρ )( ν ) ∩ M α : ν ∈ dom( h ( ρ )) i . By the elementarity of M α and the fact that h ( ρ ) ∩ M α is a member of M α , we may find an extension of(¯ p, ¯ h ), say (¯ p ′ , ¯ h ′ ), which either forces that h ( ρ ) ∩ M α ∈ ˙ S ( ρ ) or forces that LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 19 h ( ρ ) ∩ M α / ∈ ˙ S ( ρ ). Since (¯ p ′ , ¯ h ′ ) extends (¯ p, ¯ h ), it is compatible with ( p, h ↾ ρ ),and hence it must force that h ( ρ ) ∩ M α ∈ ˙ S ( ρ ). Finally, since D ( ϕ M α , ρ ) is denseand ρ ∈ M α , we may find an extension ( q, g ) of (¯ p ′ , ¯ h ′ ) and ( p, h ↾ ρ ) which isin D ( ϕ M α , ρ ). Then ( q, g ) ↾ M α is a condition which forces that h ( ρ ) ∩ M α isa condition in ˙ S ( ρ ). Thus ( ϕ M α ( q ) , ( g ↾ M α ) ⌢ h h ( ρ ) ∩ M α i ) is a condition andequals ( q, g ⌢ h h ( ρ ) i ) ↾ M α . (cid:3) Notation 3.9.
We will often find it useful to denote conditions in R ξ by the letters u, v and w . If u ∈ R ξ , we write p u and f u to denote the objects so that u = ( p u , f u ).Furthermore, if ζ ≤ ξ , then we write u ↾ ζ to denote the pair ( p u , f u ↾ ζ ), whichrestricts the length. This should not be confused with u ↾ M α = ( ϕ M α ( p ) , f ↾ M α )from Definition 3.7, which restricts the height.The following definitions of ∗ are taken for from [30] and modified to thecurrent presentation. The dual residue property defined below is equivalent to thestatement that “ ∗ ” at α from [30]. Definition 3.10.
Suppose that ~M is in pre-splitting configuration up to ρ , that α ∈ dom( ~M ), and that ζ ≤ ρ is in M α .(1) Fix conditions u, v ∈ R ζ and w ∈ R ζ ∩ M α . Fix a residue pair h p ∗ ( M α ) , ϕ M α i for ( M α , P ∗ ).(a) We say that ζϕ Mα ( u, v, w )holds if u, v ∈ D ( ϕ M α , ζ ) and u ↾ M α = ∗ w = ∗ v ↾ M α . (b) We say that ∗ ζϕ Mα ( u, v, w )holds if u, v ∈ D ( ϕ M α , ζ ) and if w ≥ u ↾ M α , v ↾ M α is a dual residuefor u and v (see Definition 2.3). (2) We say that R ζ satisfies the dual residue property at M α if for anyresidue pair h p ∗ ( M α ) , ϕ M α i for ( M α , P ∗ ) and any conditions u, v, w so that ζϕ Mα ( u, v, w ) holds, there exists w ∗ ≥ R ζ ∩ M α w so that ∗ ζϕ Mα ( u, v, w ∗ )holds. Lemma 3.11.
Suppose that ∗ ζϕ Mα ( u, v, w ) holds and that D is dense and countably = ∗ -closed in R ζ / ( p ∗ ( M α ) , ˙ S ζ ) . Then:(1) there exist u ′ ≥ u , v ′ ≥ v with u ′ , v ′ ∈ D and there exists w ′ ≥ w so that u ′ ↾ M α ≥ w , v ′ ↾ M α ≥ w , and ∗ ζϕ Mα ( u ′ , v ′ , w ′ ) hold;(2) there exist u ∗ ≥ u and v ∗ ≥ v with u ∗ , v ∗ ∈ D , and there exists w ∗ ≥ w sothat ζϕ Mα ( u ∗ , v ∗ , w ∗ ) holds.Proof. First define E to be the set of conditions s in R ζ / ( p ∗ ( M α ) , ˙ S ζ ) so that s ∈ D ( ϕ M α , ζ ) ∩ D and so that either s ↾ M α ≥ w or s ↾ M α is incompatiblewith w ; then E is dense in R ζ / ( p ∗ ( M α ) , ˙ S ζ ). Now fix a V -generic filter ¯ G over R ζ ∩ M α containing w , and note that u and v are in ( R ζ / ( p ∗ ( M α ) , ∅ )) / ¯ G . ByLemma 2.7(3), we can find u ′ ≥ u and v ′ ≥ v so that u ′ , v ′ are in E as well Note that if ( p, f ) and ( q, g ) are (determined) conditions with ( p, f ) = ∗ ( q, g ), then f = g . One can in fact argue that if w is a dual residue, then it follows that w ≥ u ↾ M α and w ≥ v ↾ M α ; however, we don’t need this fact. as in ( R ζ / ( p ∗ ( M α ) , ˙ S ζ )) / ¯ G . We next observe that u ′ ↾ M α ∈ ¯ G . Indeed, since u ′ ∈ D ( ϕ M α , ζ ), u ′ ↾ M α is a condition in R ζ ∩ M α . Additionally, since u ′ ∈ ( R ζ / ( p ∗ ( M α ) , ˙ S ζ )) / ¯ G and u ′ ≥ u ′ ↾ M α , we have that u ′ ↾ M α (a condition) iscompatible with every condition in ¯ G . Thus u ′ ↾ M α ∈ ¯ G . However, by definitionof E , and since w ∈ ¯ G , u ′ ↾ M α must extend w . A symmetric argument shows that v ′ ↾ M α ≥ w .Now let w ′ ∈ ¯ G be a condition extending w which forces that u ′ , v ′ are in( R ζ / ( p ∗ ( M α ) , ˙ S ζ )) / ˙¯ G . By Lemma 2.6, we have that ∗ ζϕ Mα ( u ′ , v ′ , w ′ ) holds. Since u ′ ↾ M α and v ′ ↾ M α both extend w , this completes the proof of (1).For (2), suppose that we are given conditions so that ∗ ζϕ Mα ( u , v , w ) holds. Byrepeatedly applying (1), we may define an increasing sequence hh u n , v n , w n i : n ∈ ω i so that for all n ∈ ω , u n +1 ≥ u n and v n +1 ≥ v n ; u n +1 ↾ M α ≥ w n and v n +1 ↾ M α ≥ w n ; and ∗ ζϕ Mα ( u n , v n , w n ) holds. Let u ∗ be a sup of h u n : n ∈ ω i , and let v ∗ and w ∗ be defined similarly. Since ∗ ζϕ Mα ( u n , v n , w n ) holds for each n , by definition wehave that w n ≥ u n ↾ M α , v n ↾ M α . Therefore the sequences h u n ↾ M α : n ∈ ω i and h v n ↾ M α : n ∈ ω i are each intertwined with h w n : n ∈ ω i , and consequently,they have the same sups. It follows by the continuity of ϕ M α that ζϕ Mα ( u ∗ , v ∗ , w ∗ )holds. (cid:3) Suppose that ~M is in pre splitting configuration up to ρ , that α ∈ dom( ~M ) andthat ζ ∈ M α ∩ ρ . Fix θ ∈ ( κ \ α ) × ω , a node in the tree ˙ T ζ of level greater thanor equal to α .Let ˙ b ζ ( θ, α ) denote the R ζ -name for n ¯ θ ∈ α × ω : ¯ θ < ˙ T ζ θ o . By Proposition3.4(3), the condition ( p ∗ ( M α ) , ˙ S ζ ) forces that ˙ b ζ ( θ, α ) is a cofinal branch through( ˙ T ζ ∩ M α )[ ˙ G R ζ ∩ M α ]. Note that by Proposition 3.4 and the definition of pre-splittingconfiguration, ( ˙ T ζ ∩ M α )[ ˙ G R ζ ∩ M α ] is an Aronsjzan tree on α in the V -generic ex-tension V [ ˙ G R ζ ∩ M α ] over R ζ ∩ M α . In light of this, we make the following definition: Definition 3.12.
Let ζ < ρ , α < κ , and h θ, τ i be a pair of tree nodes (possiblyequal) at or above level α , which we view as nodes in the tree ˙ T ζ . We say that twoconditions u and v in R ζ split h θ, τ i below α in ˙ T ζ if there exist a level ¯ α < α and distinct nodes ¯ θ, ¯ τ on level ¯ α so that u (cid:13) ¯ θ < ˙ T ζ θ and v (cid:13) ¯ τ < ˙ T ζ τ . More generally,if ζ ≤ ξ < ρ and u ′ , v ′ ∈ R ξ , then we say that u ′ and v ′ split h θ, τ i below α in ˙ T ζ if u = u ′ ↾ ζ and v = v ′ ↾ ζ do. Lemma 3.13.
Suppose that ζ ≤ ξ < ρ and R ξ satisfies the dual residue propertyat some M α , where ζ ∈ M α (see Definition 3.10). Fix u, v ∈ R ξ so that for some w ∈ R ξ ∩ M α , ξϕ Mα ( u, v, w ) . Let h θ, τ i be a pair of tree nodes (possibly equal) eachof which is at or above level α . Then there exist extensions u ∗ ≥ u , v ∗ ≥ v , and w ∗ ≥ w so that ξϕ Mα ( u ∗ , v ∗ , w ∗ ) and so that u ∗ and v ∗ split h θ, τ i below α in ˙ T ζ .Proof. Since R ξ satisfies the dual residue property at M α , R ζ does too, and sowe may find some w ′ ∈ R ζ ∩ M α so that w ′ ≥ R ζ w ↾ ζ and ∗ ζϕ Mα ( u ↾ ζ, v ↾ ζ, w ′ ). Fix a V -generic filter ¯ G over R ζ ∩ M α containing w ′ . As a result, u ↾ ζ and v ↾ ζ are in ( R ζ / ( p ∗ ( M α ) , ˙ S ζ )) / ¯ G . By the discussion preceding Definition3.12, we know that u ↾ ζ forces in the quotient that ˙ b ζ ( θ, α ) is a cofinal branch LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 21 through ¯ T := ( ˙ T ζ ∩ M α )[ ¯ G ], which by Proposition 3.4 is an Aronszajn tree on α in V [ ¯ G ]. Consequently, we may find two conditions u , u which extend u ↾ ζ in( R ζ / ( p ∗ ( M α ) , ˙ S ζ )) / ¯ G , some level ¯ α < α , and two distinct nodes θ , θ on level ¯ α of¯ T so that u i forces in ( R ζ / ( p ∗ ( M α ) , ˙ S ζ )) / ¯ G that θ i < ˙ T ζ θ . Since v ↾ ζ also forcesthat ˙ b ζ ( τ, α ) is a cofinal branch through ¯ T , we may find some extension v of v ↾ ζ in the quotient so that v decides the < ˙ T ζ -predecessor, say ¯ τ , of τ on level ¯ α of ¯ T .As θ = θ , there exists some i ∈ θ i = τ . Set ¯ θ = θ i .Now fix an extension w ′′ of w ′ in ¯ G so that w ′′ forces the following statements:(i) u i , v are in the quotient; (ii) u i forces in the quotient that ¯ θ < ˙ T ζ θ ; (iii) v forces in the quotient that ¯ τ < ˙ T ζ τ .By two applications of Lemma 2.7(3), we may find conditions ¯ u, ¯ v in the quotientso that ¯ u extends u i and w ′′ , so that ¯ v extends v and w ′′ , and so that ¯ u, ¯ v ∈ D ( ϕ M α , ζ ).We now see that ¯ u (cid:13) R ζ ¯ θ < ˙ T ζ θ , since w ′′ (cid:13) R ζ ∩ M α (cid:18) u i (cid:13) ( R ζ / ( p ∗ ( M α ) , ˙ S ζ )) / ˙¯ G ¯ θ < ˙ T ζ θ (cid:19) and since ¯ u ≥ w ′′ , u i . Similarly, ¯ v (cid:13) R ζ ¯ τ < ˙ T ζ τ .Finally, let ¯ w ≥ w ′′ be a condition in ¯ G which forces that ¯ u and ¯ v are in thequotient, so that by Lemma 2.6, ∗ ζϕ Mα (¯ u, ¯ v, ¯ w ) holds. By Lemma 3.11, we can findsome ¯ u ∗ ≥ ¯ u , ¯ v ∗ ≥ ¯ v , and ¯ w ∗ ≥ w so that ζϕ Mα (¯ u ∗ , ¯ v ∗ , ¯ w ∗ ) holds. Now let u ∗ bethe condition where p u ∗ = p ¯ u ∗ , and where f u ∗ = f ¯ u ∗ ⌢ f u ↾ [ ζ, ξ ). Let v ∗ and w ∗ bedefined similarly. Then ξϕ Mα ( u ∗ , v ∗ , w ∗ ), and u ∗ and v ∗ split h θ, τ i below α . (cid:3) One of the most important uses of the dual residue property is to obtain splittingpairs of conditions. Obtaining such conditions will also crucially use the “exactness”conditions of Definition 2.14.
Definition 3.14.
Suppose that ~M is in pre-splitting configuration up to ρ .(1) Let α ∈ dom( ~M ) and ξ ∈ M α ∩ ρ . Fix a residue pair h p ∗ ( M α ) , ϕ M α i for( M α , P ∗ ) and conditions u, v ∈ R ξ . We say that u and v are a splittingpair for ( ϕ M α , ξ ) if • for some w ∈ R ξ ∩ M α , ξϕ Mα ( u, v, w ); • for any h ζ, ν i ∈ dom( f u ) ∩ dom( f v ) ∩ M α and any h θ, τ i ∈ ( f u ( ζ, ν )) × ( f v ( ζ, ν )) , both at or above level α , u and v split h θ, τ i below α in ˙ T ζ .(2) Given fixed enumerations f u ( ζ, ν ) \ ( α × ω ) = { θ n | n < ω } and f v ( ζ, ν ) \ ( α × ω ) = { τ m | m < ω } (possibly with repetitions in the case the sets are fi-nite, nonempty), we define a splitting function to be a function Σ withdomain dom(Σ) = (dom( f u ) ∩ dom( f v ) ∩ M α ) × ω × ω, so that for any h ζ, ν, m, n i ∈ dom(Σ), Σ( ζ, ν, m, n ) is a pair h ¯ θ, ¯ τ i of treenodes satisfying Definition 3.12 with respect to h θ m , τ n i . We will denote ¯ θ by Σ( ζ, ν, m, n )( L ) and ¯ τ by Σ( ζ, ν, m, n )( R ). Remark 3.15.
Let Σ be as in Definition 3.14. Recall our notation from 3.1 regarding conditions f u and their domains. (1) We emphasize the fact that if h ζ, ν, m, n i ∈ dom(Σ) , then Σ( ζ, ν, m, n )( L ) = Σ( ζ, ν, m, n )( R ) are two distinct tree nodes on the same level . We will usually suppressexplicit mention of the level.(2) Any splitting function Σ is a member of M α since M α is countably closedand since Σ maps from a countable subset of M α into M α .(3) We only require the splitting pair in Definition 3.14 to split nodes comingfrom coordinates which are members of M α . As we will see from Lemma3.17 and later thinning out arguments, this is sufficient for our purposes. Now we are ready to state our second inductive hypothesis, the point of whichis to provide plenty of instances of the dual residue property. We will assume thesecond inductive hypothesis for the rest of the section.
Inductive Hypothesis II : Let ξ < ρ and suppose that ~M is R ξ -suitable. Then n α ∈ dom( ~M ) : R ξ satisfies the dual residue property at M α o ∈ F WC . Now we move to the main part of the proof that P ∗ satisfies inductive hypothesesI and II with respect to ρ . In broad outline, we will first verify inductive hypothesisI for ρ and then use this to verify inductive hypothesis II at ρ .At important parts of the following proofs we will need to understand the cir-cumstances under which we can amalgamate conditions in R ρ , and in particular, in˙ S ρ . We will often be interested in the following strong sense of amalgamation:
Definition 3.16.
Let u, v ∈ R ρ so that p u and p v are compatible in P ∗ . We saythat f u and f v are strongly compatible over p u and p v if for any condition q ∈ P ∗ which extends p u and p v , q forces that ˇ f u ∪ ˇ f v ∈ ˙ S ρ .The next lemma gives sufficient conditions under which we may amalgamateconditions in R ρ whose specializing parts are strongly compatible as above. Lemma 3.17.
Suppose that ~M is in pre-splitting configuration up to ρ , that α <β are in dom( ~M ) , and that h p ∗ ( M α ) , ϕ M α i and h p ∗ ( M β ) , ϕ M β i are residue pairsfor ( M α , P ∗ ) and ( M β , P ∗ ) , respectively. Let h u α , v α i and h u β , v β i be two pairs ofconditions in R ρ which satisfy the following:(1) h u α , v α i is a splitting pair for ( ϕ M α , ρ ) with splitting function Σ α ;(2) h u β , v β i is a splitting pair for ( ϕ M β , ρ ) with splitting function Σ β ;(3) Σ α = Σ β ;(4) there exists w ∈ M α so that ρϕ Mα ( u α , v α , w ) and ρϕ Mβ ( u β , v β , w ) bothhold; and(5) u α , v α ∈ M β .Then u α and v β are compatible in R ρ ; in fact, f u α is strongly compatible with f v β over p u α and p v β .Proof. We first observe that p u α and p v β are compatible in P ∗ . Indeed, by (4), ϕ M α ( p u α ) = ∗ p w = ∗ ϕ M β ( p v β ), and by (5), p u α ∈ M β . Thus as p u α ≥ ϕ M α ( p u α ) ≥ ϕ M β ( p v β ), and as ϕ M β is a residue function, p u α is compatible with p v β .Now let q ∈ P ∗ be any common extension of p u α and p v β . We will argue byinduction on ζ ≤ ρ that q (cid:13) ( ˇ f u α ∪ ˇ f v β ) ↾ ζ ∈ ˙ S ζ . Limit stages are immediate. LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 23
For the successor stage, suppose that h ζ, ν i ∈ dom( f u α ) ∩ dom( f v β ) and that wehave proven that q (cid:13) ( ˇ f u α ∪ ˇ f v β ) ↾ ζ ∈ ˙ S ζ . Since f u α ∈ M β by (5), h ζ, ν i ∈ M β .Thus h ζ, ν i ∈ dom( f v β ) ∩ M β = dom( f w ), since w = ∗ v β ↾ M β . Since we also havethat w = ∗ u β ↾ M β , it follows that h ζ, ν i ∈ dom( f u β ). Thus h ζ, ν i ∈ dom( f u β ) ∩ dom( f v β ) ∩ M β = dom( f u α ) ∩ dom( f v α ) ∩ M α , with equality holding by (3) and thedefinition of a splitting function. Moreover, ζ ∈ M α since h ζ, ν i ∈ dom( f w ) ⊆ M α .Now pick a pair of distinct nodes h θ, τ i ∈ f u α ( ζ, ν ) × f v β ( ζ, ν ), and we will showthat ( q, ( f u α ∪ f v β ) ↾ ζ ) forces in R ζ that θ and τ are ˙ T ζ -incompatible. If θ is belowlevel α , then θ ∈ ( f u α ↾ M α )( ζ, ν ) = f w ( ζ, ν ) ⊆ f v β ( ζ, ν ). Thus ( q, f v β ↾ ζ ) (cid:13) θ, τ are ˙ T ζ -incompatible, and so ( q, ( f u α ∪ f v β ) ↾ ζ ) forces this too. A similar argumentapplies if τ is below level β .We therefore assume that θ is at or above level α and τ is at or above level β .Let m and n be chosen so that θ is the m th node in f u α ( ζ, ν ) \ ( α × ω ) and τ isthe n th node in f v β ( ζ, ν ) \ ( β × ω ). By assumption (3), letting Σ := Σ α = Σ β , weknow that Σ( ζ, ν, m, n )( L ) and Σ( ζ, ν, m, n )( R ) are two distinct nodes on the samelevel and also that ( p u α , f u α ↾ ζ ) (cid:13) Σ( ζ, ν, m, n )( L ) < ˙ T ζ θ and ( p v β , f v β ↾ ζ ) (cid:13) Σ( ζ, ν, m, n )( R ) < ˙ T ζ τ. Therefore ( q, ( f u α ∪ f v β ) ↾ ζ ) forces that τ and θ are incompatible in ˙ T ζ , as weintended to show. (cid:3) The following item shows how we can obtain the desired splitting pairs of con-ditions.
Lemma 3.18.
Suppose that ~M is in pre-splitting configuration up to ρ . Fix α ∈ dom( ~M ) , and suppose that h p ∗ ( M α ) , ϕ M α i is a residue pair for ( M α , P ∗ ) . Finally,fix u, v, w so that ρϕ Mα ( u, v, w ) . Then there exist extensions u ∗ ≥ u and v ∗ ≥ v sothat u ∗ , v ∗ are a splitting pair for ( ϕ M α , ρ ) .Proof. Fix u, v, w as in the statement of the lemma. We define an increasing se-quence of triples hh u n , v n , w n i : n ∈ ω i of conditions and a sequence hh ζ n , ν n , θ n , τ n i : n ∈ ω i of tuples of ordinals and tree nodes so that h u , v , w i = h u, v, w i and sothat for each n , • ρϕ Mα ( u n , v n , w n ) holds; • h ζ n , ν n i ∈ dom( f u n ) ∩ dom( f v n ) ∩ M α and h θ n , τ n i ∈ ( f u n ( ζ n , ν n ) \ ( α × ω )) × ( f v n ( ζ n , ν n ) \ ( α × ω )); and • u n +1 and v n +1 split h θ n , τ n i below α .This is done with respect to some bookkeeping device in such a way that if u ∗ is a supof h u n : n ∈ ω i (and similarly for v ∗ ), then for each h ζ, ν i ∈ dom( f u ∗ ) ∩ dom( f v ∗ ) ∩ M α and each h θ, τ i ∈ ( f u ∗ ( ζ, ν ) \ ( α × ω )) × ( f v ∗ ( ζ, ν ) \ ( α × ω )), h ζ, ν, θ, τ i appearsas the n th tuple for some n .To show the successor step, suppose that u n , v n and w n are given, and con-sider h ζ n , ν n , θ n , τ n i . Note that ζ n ϕ Mα ( u n ↾ ζ n , v n ↾ ζ n , w n ↾ ζ n ) also holds. Thenby applying Lemma 3.13, we may find conditions u ′ n ≥ u n ↾ ζ n , v ′ n ≥ v n ↾ ζ n , and w ′ n ≥ w n ↾ ζ n so that ζ n ϕ Mα ( u ′ n , v ′ n , w ′ n ) and so that u ′ n and v ′ n split h θ n , τ n i below α . Now define f u n +1 to be the function f ⌢u ′ n ( f u n ↾ [ ζ n , ρ )), and let u n +1 be the pair ( p u ′ n , f u n +1 ). Let v n +1 and w n +1 be defined similarly. Then ρϕ Mα ( u n +1 , v n +1 , w n +1 ) holds and u n +1 and v n +1 split h θ n , τ n i below α .This completes the construction of the sequence. Fix sups u ∗ , v ∗ , w ∗ . Since ρϕ Mα ( u n , v n , w n ) holds for all n , ρϕ Mα ( u ∗ , v ∗ , w ∗ ) also holds. By the choice ofbookkeeping, u ∗ , v ∗ is a splitting pair for ( ϕ M α , ρ ), completing the proof. (cid:3) Lemma 3.19.
Suppose that ~M is in pre-splitting configuration up to ρ . Let B ⊆ dom( ~M ) be in F +WC and suppose that for each α ∈ B , there exist u α , v α which area splitting pair for ( ϕ M α , ρ ) , where h p ∗ ( M α ) , ϕ M α i is a residue pair for ( M α , P ∗ ) .Then there exists B ∗ ⊆ B in F WC so that for any α < β in B ∗ , u α , v α ∈ M β , u α ↾ M α = ∗ v β ↾ M β , and u α is compatible with v β .Proof. Suppose that for each α ∈ B , we have a splitting pair u α , v α for ( ϕ M α , ρ );we also let w α ∈ R ρ ∩ M α be a condition witnessing ρϕ Mα ( u α , v α , w α ). Let Σ α be asplitting function for ( u α , v α ) with respect to M α , as in Definition 3.14. By Remark3.15, Σ α ∈ M α . Now the function on B defined by α
7→ h w α , Σ α i is regressive (sincethe pair can be coded by an ordinal below α ). By Lemma 1.8 there exists some B ∗ ⊆ B with B ∗ ∈ F WC for which that function takes a constant value, say h ¯ w, Σ i .Moreover, by intersecting with a club and relabelling if necessary, we may assumethat if α < β are in B ∗ , then u α , v α ∈ M β . But then for any α < β in B ∗ , we havethat u α ↾ M α = ∗ ¯ w = ∗ v β ↾ M β . Therefore, for all α < β in B ∗ , the assumptionsof Lemma 3.17 are satisfied, and consequently u α and v β are compatible. (cid:3) Proposition 3.20. P ∗ (cid:13) ˙ S ρ is κ -c.c.Proof. Let p ∈ P ∗ be a condition, and suppose that p (cid:13) h ˙ f γ : γ < κ i is a sequenceof conditions in ˙ S ρ . We will find some extension p ∗ of p which forces that thissequence does not enumerate an antichain.Let ~M be a sequence which is suitable with respect to the three parameters R ρ , p and h ˙ f γ : γ < κ i , and which is in pre-splitting configuration up to ρ . Let B := dom( ~M ), and let hh p ∗ ( M α ) , ϕ M α i : α ∈ B i be a residue system witnessingthat P ∗ is F WC -strongly proper. For each α ∈ B , p ∈ P ∗ ∩ M α , and therefore wemay find some extension p α of p so that p α ∈ dom( ϕ M α ). We may also assume thatfor some function f α in V , p α (cid:13) P ∗ ˙ f α = ˇ f α . Now extend h p α , f α i to a condition u α in D ( ϕ M α , ρ ). By Lemma 3.18, we may further extend u α to a splitting pair h u ∗ α , v ∗ α i for ( ϕ M α , ρ ). By Lemma 3.19, we may find some B ∗ ⊆ B with B ∗ ∈ F WC so that for all α < β in B ∗ , u ∗ α and v ∗ β are compatible. Let w be a conditionextending them both. Then p w forces that ˇ f w extends both ˇ f u ∗ α and ˇ f v ∗ β and henceextends ˙ f α and ˙ f β . Therefore p w forces that ˙ f α and ˙ f β are compatible in ˙ S ρ . (cid:3) We are now ready to verify that the second induction hypothesis holds at ρ . Proposition 3.21.
Suppose that ~M is in pre-splitting configuration up to ρ . Then n α ∈ dom( ~M ) : R ρ satisfies the dual residue property at M α o ∈ F WC . Proof.
Suppose otherwise, for a contradiction. Then B := n α ∈ dom( ~M ) : R ρ does not satisfy the dual residue property at M α o is in F +WC . For each α ∈ B , we fix a residue pair h p ∗ ( M α ) , ϕ M α i for ( M α , P ∗ )and a triple h u α , v α , w α i which witnesses that R ρ does not satisfy the dual residue LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 25 property at M α . Thus ρϕ Mα ( u α , v α , w α ) holds, but for any w ∗ ≥ R ρ ∩ M α w α , either w ∗ is not a residue for u α to M α or w ∗ is not a residue for v α to M α . In particular,for each such w ∗ , we may find a further extension which is either incompatible with u α or incompatible with v α in R ρ .By Lemma 3.18, we may extend h u α , v α , w α i to another triple h u ∗ α , v ∗ α , w ∗ α i sothat u ∗ α and v ∗ α are a splitting pair for M α . By Lemma 3.19, we may find some B ∗ ⊆ B with B ∗ ∈ F +WC so that for all α, β ∈ B ∗ with α < β , w ∗ α = ∗ w ∗ β and u ∗ α is compatible with v ∗ β . We let ¯ w ∗ denote a condition which is = ∗ equal to w ∗ α for α ∈ B ∗ .Next, fix α ∈ B ∗ , and we will define a function with domain B ∗ \ ( α + 1). Fixan arbitrary β ∈ B ∗ above α . Since u ∗ α ∈ M β is compatible with v ∗ β , there is anextension w ∗ α,β of u ∗ α in R ρ ∩ M β which is a residue for v ∗ β to R ρ ∩ M β . Since β ∈ B and since w ∗ α,β is a residue for v ∗ β , we may further extend (and relabel if necessary)to assume that w ∗ α,β is incompatible with u ∗ β .Since for each β ∈ B \ ( α + 1), w ∗ α,β ∈ M β , the function β w ∗ α,β is regressiveand hence by Lemma 1.8 there exists some H α ∈ F WC so that for all β ∈ H α ∩ B ∗ , β > α and the function β w ∗ α,β takes a constant value, say w ∗∗ α , on H α ∩ B .Now let H := ∆ α ∈ B ∗ H α so that H ∈ F WC . Consequently, H ∩ B ∗ ∈ F +WC . Nextobserve that the function on H ∩ B ∗ defined by α ϕ M α ( p w ∗∗ α ) is regressive, andtherefore there exists some F WC -positive B ∗∗ ⊆ H ∩ B ∗ so that the function takesa constant value, say p ∗∗ , on B ∗∗ .Next, let α < β both be in B ∗∗ . We claim that p w ∗∗ α and p w ∗∗ β are compatibleconditions in P ∗ . Indeed, p w ∗∗ α ≥ P ∗ ∩ M β p ∗∗ = ϕ M β ( p w ∗∗ β ), and since p w ∗∗ α ∈ P ∗ ∩ M β ,Definition 2.14 implies that p w ∗∗ α and p w ∗∗ β are compatible in P ∗ . With the same α, β , fix any condition q ∈ P ∗ which extends both p w ∗∗ α and p w ∗∗ β . Recall that w ∗∗ α = w ∗ α,β and so w ∗∗ α extends u ∗ α and is incompatible with u ∗ β ; we also have that w ∗∗ β extends u ∗ β . Therefore w ∗∗ α and w ∗∗ β are incompatible conditions in R ρ . Since q extends p w ∗∗ α and p w ∗∗ β , we must have that q (cid:13) P ∗ ˇ f w ∗∗ α ⊥ ˙ S ρ ˇ f w ∗∗ β . From the previous two paragraphs, we may conclude that if α < β are bothin B ∗∗ , then p w ∗∗ α and p w ∗∗ β are compatible conditions in P ∗ , and any condition in P ∗ above them both forces that ˇ f w ∗∗ α ⊥ ˙ S ρ ˇ f w ∗∗ β . We use this to create a κ -sizedantichain in ˙ S ρ . Namely, let G P ∗ be a V -generic filter over P ∗ which contains thecondition p ∗∗ . Since p ∗∗ is a residue for p w ∗∗ α to M α for all α ∈ B ∗∗ , Lemma 2.20implies that X := (cid:8) α ∈ B ∗∗ : p w ∗∗ α ∈ G P ∗ (cid:9) is unbounded in κ . Therefore, if α < β are in X , then f w ∗∗ α and f w ∗∗ β are incompat-ible conditions in S := ˙ S ρ [ G P ∗ ] which contradicts Proposition 3.20. This completesthe proof of the given proposition. (cid:3) We have now completed the proof that Theorem 3.2. We conclude with a corol-lary which adds to that theorem an additional clause about the dual residue prop-erty; this will be useful later.
Corollary 3.22.
Suppose that P ∗ is F WC -strongly proper and that ˙ S κ + is a κ + -length, countable support iteration specializing Aronszajn trees on κ . Then (1) for all ρ < κ + , P ∗ forces that ˙ S ρ is κ -c.c.; and(2) if ~M is in pre-splitting configuration up to ρ , then there is some B ⊆ dom( ~M ) with B ∈ F WC so that for all α ∈ B , R ρ satisfies the dual residueproperty at M α . Hence, for all ζ ∈ M α ∩ ( ρ + 1) , R ζ satisfies the dualresidue property at M α .Proof. If the corollary is false, let ρ be the least such that it fails at ρ . ThenInduction Hypotheses I and II hold up to ρ , so Propositions 3.20 and 3.21 showthat (1) and (2) hold at ρ , a contradiction. (cid:3) F WC -Strongly Proper Posets and Preserving Stationary Sets In this section, we will prove that the appropriate quotients preserve stationarysets of cofinality ω ordinals. We will apply this result in Section 6 when we showthat our intended club-adding iteration is F WC -completely proper (see Definition5.8). In the first part of this section, we will prove some helpful lemmas whichwe use in the second part to complete proof of the preservation of the relevantstationary sets.For the remainder of this section, we fix an F WC -strongly proper poset P ∗ and aniteration ˙ S ρ of length ρ < κ + specializing Aronszajn trees in the extension by P ∗ ;see the beginning of Section 3 for a more precise definition and relevant notation.Note that the conclusions of Corollary 3.22 hold.We first prove two lemmas which describe how the residue functions with respectto two models on a suitable sequence interact. More precisely, suppose we have asuitable sequence ~M , where α < β are both in dom( ~M ) and M α and M β haverespective residue pairs h p ∗ ( M α ) , ϕ M α i and h p ∗ ( M β ) , ϕ M β i with respect to P ∗ . Anatural question is whether, on a dense set, ϕ M α ( ϕ M β ( q )) = ∗ ϕ M α ( q ), i.e., whetherthe M α -residue of the M β -residue is equivalent to the M α -residue. Proposition 4.2below shows that this is the case.By applying a standard dovetailing construction, which combines the countablecontinuity of the residue functions with the strong genericity property, we obtainthe following result. Lemma 4.1.
Suppose that ~M is P ∗ -suitable with residue system hh p ∗ ( M γ ) , ϕ M γ i : γ ∈ dom( ~M ) i and that α < β are in dom( ~M ) .(1) For every p ∈ P ∗ that extends both p ∗ ( M α ) and p ∗ ( M β ) , there is an exten-sion p ∗ ≥ P ∗ p with p ∗ ∈ dom( ϕ M α ) ∩ dom( ϕ M β ) .(2) D ( ϕ M α , ϕ M β ) := (cid:8) q ∈ dom( ϕ M α ) ∩ dom( ϕ M β ) : ϕ M β ( q ) ∈ dom( ϕ M α ) (cid:9) is = ∗ -countably closed and dense in P ∗ / { p ∗ ( M α ) , p ∗ ( M β ) } . Proposition 4.2.
Suppose that ~M is P ∗ -suitable with residue system hh p ∗ ( M γ ) , ϕ M γ i : γ ∈ dom( ~M ) i , and let α < β be in dom( ~M ) . Then E ( ϕ M α , ϕ M β ) := (cid:8) p ∈ P ∗ : ϕ M β ( p ) ∈ dom( ϕ M α ) ∧ ϕ M α ( ϕ M β ( p )) = ∗ ϕ M α ( p ) (cid:9) is = ∗ -countably closed and dense in P ∗ / { p ∗ ( M α ) , p ∗ ( M β ) } . This denotes the set of r ∈ P ∗ which extend both p ∗ ( M α ) and p ∗ ( M β ). LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 27
Proof.
We begin by observing that if q ∈ D ( ϕ M α , ϕ M β ), then ϕ M α ( q ) extends ϕ M α ( ϕ M β ( q )). Indeed, since q ∈ dom( ϕ M β ), q ≥ ϕ M β ( q ), and since ϕ M α is order-preserving and both q and ϕ M β ( q ) are in dom( ϕ M α ), we conclude that ϕ M α ( q ) ≥ ϕ M α ( ϕ M β ( q )).With this observation in mind, let p ∈ P ∗ extend both p ∗ ( M α ) and p ∗ ( M β ),and by extending further if necessary, we may assume that p is in D ( ϕ M α , ϕ M β ).We will define by recursion an increasing sequence of conditions h p n : n ∈ ω i in D ( ϕ M α , ϕ M β ) with p = p so that for all n , ϕ M α ( ϕ M β ( p n +1 )) ≥ ϕ M α ( p n ) ≥ ϕ M α ( ϕ M β ( p n ));note that all of the above items are defined, by definition of D ( ϕ M α , ϕ M β ).Suppose we are given p n . As observed earlier, ϕ M α ( p n ) ≥ ϕ M α ( ϕ M β ( p n )). Bythe strong residue condition in Definition 2.9 and item (3) of Definition 2.14, wemay find a condition q ∈ M β extending ϕ M β ( p n ) so that ϕ M α ( q ) ≥ ϕ M α ( p n ). Since q ∈ M β extends ϕ M β ( p n ), there is an r ≥ p n so that ϕ M β ( r ) ≥ q . Finally, let p n +1 ≥ r be a condition in D ( ϕ M α , ϕ M β ). Then ϕ M β ( p n +1 ) ≥ ϕ M β ( r ) ≥ q , andhence ϕ M α ( ϕ M β ( p n +1 )) ≥ ϕ M α ( q ) ≥ ϕ M α ( p n ). This completes the construction ofthe desired sequence.Let p ∗ be a sup of h p n : n ∈ ω i . It is straightforward to verify that it witnessesthe lemma. (cid:3) The last lemma that we will need before turning to the main result of this sectionis a technical refinement of Lemma 3.18 which isolates circumstances in which for α < β < κ as above, we can find splitting pairs u, v for ( M α , ρ ) with the additionalproperty that u ↾ M β and v ↾ M β also form a splitting pair for ( M α , ρ ). Moreover, u ↾ M β and v ↾ M β will split the nodes on levels between α and β in the same waythat u and v do. For the statement of the next result, recall the way we denoterestriction of (iteration) length p ↾ ξ , and restriction in the poset height, p ↾ M α ,from Notation 3.9. Lemma 4.3.
Suppose that ~M is in pre-splitting configuration up to ρ and that dom( ~M ) satisfies the conclusion of Corollary 3.22(2). Suppose α < β are both in dom( ~M ) and that h p ∗ ( M α ) , ϕ M α i and h p ∗ ( M β ) , ϕ M β i are residue pairs for ( M α , P ∗ ) and ( M β , P ∗ ) respectively. Finally, fix a condition u ∈ R ρ with p u ∈ dom( ϕ M α ) ∩ dom( ϕ M β ) . Then there exist a splitting pair ( u ∗ , v ∗ ) for ( ϕ M α , ρ ) extending u anda splitting function Σ satisfying the following:(1) p u ∗ and p v ∗ are both in E ( ϕ M α , ϕ M β ) (see Proposition 4.2);(2) ( u ∗ ↾ M β , v ∗ ↾ M β ) is also an ( M α , ρ ) -splitting pair, and for any tuple ( ζ, ν, m, n ) ∈ dom(Σ) so that the m -th node of f u ∗ ( ζ, ν ) \ ( α × ω ) and the n -th node of f v ∗ ( ζ, ν ) \ ( α × ω ) , are both in M β , ( ϕ M β ( p u ∗ ) , ( f u ∗ ↾ M β ) ↾ ζ ) (cid:13) R ζ Σ( ζ, ν, m, n )( L ) < ˙ T ζ θ and ( ϕ M β ( p v ∗ ) , ( f v ∗ ↾ M β ) ↾ ζ ) (cid:13) R ζ Σ( ζ, ν, m, n )( R ) < ˙ T ζ τ. Proof.
By Lemma 3.8, we know that D ( ϕ M α , ρ ) ∩ D ( ϕ M β , ρ ) is dense and = ∗ -countably closed in R ρ / { p ∗ ( M α ) , p ∗ ( M β ) } . Moreover, by Proposition 4.2 (withthe notation from the statement thereof), E ( ϕ M α , ϕ M β ) is dense and countably = ∗ -closed in P ∗ / { p ∗ ( M α ) , p ∗ ( M β ) } . Consequently, E ∗ ( ϕ M α , ϕ M β , ρ ) := (cid:8) v ∈ R ρ : v ∈ D ( ϕ M α , ρ ) ∩ D ( ϕ M β , ρ ) ∧ p v ∈ E ( ϕ M α , ϕ M β ) (cid:9) is dense and countably = ∗ -closed in R ρ / { p ∗ ( M α ) , p ∗ ( M β ) } . For use later, we alsolet E ∗ ( ϕ M α , ϕ M β , ζ ) be defined similarly, with ζ replacing ρ in the above definition.Let u be as in the assumption of the current lemma. We may extend and relabelif necessary to assume that u ∈ E ∗ ( ϕ M α , ϕ M β , ρ ). We set u := v := u and w := u ↾ M α . We will now define an increasing sequence of triples hh u n , v n , w n i : n ∈ ω i and a sequence of tuples hh ζ n , ν n , θ n , τ n i : n ∈ ω i (with respect to some bookkeepingdevice) of tree nodes and ordinals so that the following conditions are satisfied:(1) ρϕ Mα ( u n , v n , w n );(2) u n , v n ∈ D ( ϕ M β , ρ );(3) h ζ n , ν n i ∈ dom( f u n ) ∩ dom( f v n ) ∩ M α , and h θ n , τ n i ∈ ( f u n ( ζ n , ν n ) \ ( α × ω )) × ( f v n ( ζ n , v n ) \ ( α × ω ));(4) u n +1 and v n +1 split h θ n , τ n i below α in ˙ T ζ n , and if θ n and τ n are both belowlevel β , then in fact u n +1 ↾ M β and v n +1 ↾ M β split h θ n , τ n i below α in ˙ T ζ n .Moreover, in this case, there is a pair of nodes h ¯ θ n , ¯ τ n i below level α whichwitnesses the splitting for both u n +1 and v n +1 as well as their restrictionsto M β ;(5) p u n and p v n are in E ( ϕ M α , ϕ M β ) (see Proposition 4.2).For n = 0, we have that (1), (2), and (5) hold because u ∈ E ∗ ( ϕ M α , ϕ M β , ρ ). (3)holds by definition and (4) is vacuous.Suppose, then, that we have defined u n , v n , and w n . By Lemma 3.13, we mayfind extensions u ′ n ≥ u n , v ′ n ≥ v n , and w ′ n ≥ w n so that ρϕ Mα ( u ′ n , v ′ n , w ′ n ) holdsand so that u ′ n and v ′ n split h θ n , τ n i below α in ˙ T ζ n . Let ¯ θ n and ¯ τ n be nodes belowlevel α which witness the splitting.We now define conditions u ∗∗∗ n , v ∗∗∗ n , and w ∗∗∗ n (the superscript for later nota-tional purposes) which extend, respectively, u ′ n ↾ ζ n , v ′ n ↾ ζ n , and w ′ n ↾ ζ n . Ifeither θ n or τ n are at or above level β (namely, outside of M β ), then we simply set u ∗∗∗ n := u ′ ↾ ζ n , v ∗∗∗ n := u ′ ↾ ζ n . Since dom( ~M ) satisfies the conclusion of Corollary3.22 (2), and since ζ n ∈ M α , we may find a dual residue w ∗∗∗ n of u ∗∗∗ n and v ∗∗∗ n to M α . This completes the definition of the triple ( u ∗∗∗ n , v ∗∗∗ n , w ∗∗∗ n ) in the case thateither θ n or τ n are at or above level β .Suppose on the other hand that θ n and τ n are both below level β and thereforeare in M β . Since dom( ~M ) satisfies the conclusion of Corollary 3.22 (2), and since ζ n ∈ M α ∩ ρ and ζ n ϕ Mα ( u ′ n ↾ ζ n , v ′ n ↾ ζ n , w ′ n ↾ ζ n ), we may find a condition w ∗ n ∈ M α ∩ R ζ n which is a dual residue of u ′ n ↾ ζ n and v ′ n ↾ ζ n to M α . We next extend u ′ n ↾ ζ n to a condition which extends not only w ∗ n but also some residue to M β .Let u ∗∗ n be an extension in D ( ϕ M β , ζ n ) of u ′ n ↾ ζ n and w ∗ n . By the remarks beforeLemma 3.8, we may let ¯ u ∗∗ n ∈ M β be a residue of u ∗∗ n to M β in R ζ n . Finally, let u ∗∗∗ n be a condition in D ( ϕ M β , ζ n ) ∩ D ( ϕ M α , ζ n ) which extends u ∗∗ n and ¯ u ∗∗ n andwhich satisfies that u ∗∗∗ n ↾ M β ≥ R ζn ¯ u ∗∗ n .By definition of ¯ θ n above, we know that u ′ n ↾ ζ n (cid:13) R ζn ¯ θ n < ˙ T ζn θ n , and hence theextension u ∗∗ n of u ′ n forces this too. Since ¯ u ∗∗ n is a residue of u ∗∗ n to M β in R ζ n and¯ θ n , θ n are nodes in M β , we conclude that ¯ u ∗∗ n also forces that ¯ θ n < ˙ T ζn θ n . Finally,since u ∗∗∗ n ↾ M β is a condition (because u ∗∗∗ n ∈ D ( ϕ M β , ζ n )) which extends ¯ u ∗∗ n , we LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 29 conclude that u ∗∗∗ n ↾ M β forces that ¯ θ n < ˙ T ζn θ n . This completes the first round ofextensions of u ′ n ↾ ζ n .We now turn to extending v ′ n ↾ ζ n . Since u ∗∗∗ n ∈ D ( ϕ M α , ζ n ), we may let w ∗∗ n bea residue of u ∗∗∗ n to M α which extends w ∗ n . Since w ∗∗ n ≥ w ∗ n and w ∗ n is a residue of v ′ n ↾ ζ n to M α , w ∗∗ n is also a residue of v ′ n ↾ ζ n to M α . Applying the same argumentas in the previous two paragraphs to v ′ n ↾ ζ n with w ∗∗ n playing the role of w ∗ n andwith ¯ τ n and τ n playing the respective roles of ¯ θ n and θ n , we may find an extension v ∗∗∗ n of v ′ n ↾ ζ n in D ( ϕ M β , ζ n ) ∩ D ( ϕ M α , ζ n ) and a condition w ∗∗∗ n so that v ∗∗∗ n ↾ M β forces ¯ τ n < ˙ T ζn τ n and so that w ∗∗∗ n is a residue of v ∗∗∗ n to M α in R ζ n which extends w ∗∗ n . Note that since w ∗∗∗ n ≥ w ∗∗ n , w ∗∗∗ n is also a residue of u ∗∗∗ n to M α in R ζ n .To summarize, we now have extensions u ∗∗∗ n and v ∗∗∗ n of u ′ n ↾ ζ n and v ′ n ↾ ζ n respectively which are both in D ( ϕ M β , ρ ) and which satisfy that u ∗∗∗ n ↾ M β and v ∗∗∗ n ↾ M β split h θ n , τ n i as witnessed by h ¯ θ n , ¯ τ n i . Moreover, w ∗∗∗ n is a commonresidue of u ∗∗∗ n and v ∗∗∗ n to M α in R ζ n , i.e., ∗ ζ n ϕ Mα ( u ∗∗∗ n , v ∗∗∗ n , w ∗∗∗ n ). This completesthe definition of the triple ( u ∗∗∗ n , v ∗∗∗ n , w ∗∗∗ n ) in the case that θ n and τ n are belowlevel β .We now apply Lemma 3.11, to find u n +1 ↾ ζ n ≥ u ∗∗∗ n , v n +1 ↾ ζ n ≥ v ∗∗∗ n and w n +1 ↾ ζ n ≥ w ∗∗∗ n so that ζ n ϕ Mα ( u n +1 ↾ ζ n , v n +1 ↾ ζ n , w n +1 ↾ ζ n ) and so that u n +1 ↾ ζ n , v n +1 ↾ ζ n ∈ E ∗ ( ϕ M α , ϕ M β , ζ n ). Define u n +1 := ( u n +1 ↾ ζ n ) ⌢ ( u ′ n ↾ [ ζ n , ρ )),with v n +1 and w n +1 defined similarly. u n +1 , v n +1 , and w n +1 then satisfy (1)-(5),completing the successor step of the construction.If we now let u ∗ , v ∗ , w ∗ be sups of their respective sequences, it is straightforwardto see that they satisfy the lemma, using (4) to secure the desired splitting function. (cid:3) Having laid the groundwork of the previous results, we next turn to analyzingwhen quotients of R ρ preserve stationary sets of cofinality ω ordinals. We will provethe following proposition: Proposition 4.4.
Suppose that ~M is in pre-splitting configuration up to ρ and that dom( ~M ) satisfies Corollary 3.22. Then there exists some B ∗ ⊆ dom( ~M ) in F WC sothat for any α ∈ B ∗ , any ( R ρ ∩ M α ) -name ˙ S for a stationary subset of α ∩ cof( ω ) ,and any residue pair h p ∗ ( M α ) , ϕ M α i for ( M α , P ∗ ) , the poset R ρ / ( p ∗ ( M α ) , ˙ S ρ ) forcesthat ˙ S remains stationary. Thus the quotient forcing of R ρ above the condition ( p ∗ ( M α ) , ˙ S ρ ) preserves thestationarity of ˙ S . The remainder of the section is devoted to the proof. Proof.
To begin, we define the set B ∗ := tr( B ) ∩ B. Since B ∈ F WC , tr( B ) is alsoin F WC , and therefore B ∗ is too.Now fix, for the rest of the proof, an ordinal α ∈ B ∗ and a residue pair h p ∗ ( M α ) , ϕ M α i for ( M α , P ∗ ); since ~M is in pre-splitting configuration up to ρ , wemay also fix, for each γ ∈ B ∩ α , a residue pair h p ∗ ( M γ ) , ϕ M γ i for ( M γ , P ∗ ).Next, fix a condition ( p, f ) in R ρ / ( p ∗ ( M α ) , ˙ S ρ ) and an R ρ -name ˙ C for a closedunbounded subset of α . We will find some extension ( p ∗ , f ∗ ) of ( p, f ) which forcesin R ρ that ˙ C ∩ ˙ S = ∅ . By Lemma 3.8, we may assume that ( p, f ) ∈ D ( ϕ M α , ρ ).In V , let θ > κ + be a large enough regular cardinal, and let K ≺ H ( θ ) bechosen so that | K | = κ , <κ K ⊆ K , and so that K has the following parameters aselements: (i) the sequences ~M and hh p ∗ ( M γ ) , ϕ M γ i : γ ∈ B ∩ ( α + 1) i , the set B ∗ , theposet R ρ , the R ρ -condition ( p, f ), the R ρ -name ˙ C , and the ( R ρ ∩ M α )-name˙ S ;(ii) the fixed well-order ⊳ of H ( κ + ) from Notation 2.12.Finally, let K denote the tuple ( K, ∈ , α, ~M , B ∗ , R ρ , ( p, f ) , ˙ C, ˙ S, ⊳ ) . Define E to be the club of β < α so that Sk K ( β ) ∩ α = β . Since α ∈ B ∗ , weknow that B ∩ α is stationary in α , and therefore E := lim( E ∩ B ) is a club in α .Recalling that ( p, f ) ∈ D ( ϕ M α , ρ ), we can find a residue (¯ p, ¯ f ) of ( p, f ) to M α which extends the condition ( ϕ M α ( p ) , f ↾ M α ). Let ˙ X be the ( R ρ ∩ M α )-namefor n β ∈ E ∩ B ∩ α : ( p ∗ ( M β ) , ˙ S ρ ) ∈ ˙ G R ρ ∩ M α o ; by Lemma 2.20 we know that ˙ X is forced by R ρ ∩ M α to be unbounded in α . Since ˙ S is an ( R ρ ∩ M α )-name of astationary subset of α ∩ cof( ω ), then it is forced to contain a limit point of ˙ X . This,combined with the fact that R ρ ∩ M α does not add new ω -sequences, implies that wecan find an extension ( q, g ) ≥ R ρ ∩ M α (¯ p, ¯ f ) and an increasing sequence h β n : n ∈ ω i in E ∩ B with sup n β n = ν ∈ E ∩ cof( ω ), so that (i) ( q, g ) (cid:13) R ρ ∩ M α ν ∈ ˙ S , and (ii)for all n ∈ ω , q ≥ P ∗ p ∗ ( M β n ).For the rest of the proof, we will fix ( q, g ) ∈ R ρ ∩ M α , h β n | n < ω i , and ν withthe above properties. Define K β n := Sk K ( β n ) , noting that K β n ∩ α = β n because β n ∈ E .We proceed to find an extension ( p ∗ , f ∗ ) of ( p, f ) which is compatible with ( q, g )and forces that ν ∈ ˙ C . We will secure this by building two increasing ω -sequencesof conditions, one above ( p, f ) and another above ( q, g ), in such a way that thelimits of each sequence can be amalgamated; the resulting condition will then force ν into ˙ S ∩ ˙ C . Let ( p , f ) := ( p, f ) and ( q , g ) := ( q, g ). Claim 4.5.
There exist increasing sequences h ( p n , f n ) : n ∈ ω i of conditions in R ρ and h ( q n , g n ) : n ∈ ω i of conditions in R ρ ∩ M α so that for each n ∈ ω ,(1) ( p n , f n ) ∈ K β n ;(2) ( p n +1 , f n +1 ) (cid:13) R ρ ˙ C ∩ ( β n , ν ) = ∅ ;(3) ( p n , f n ) ∈ D ( ϕ M α , ρ );(4) q n ≥ R ρ ∩ M α ϕ M α ( p n );(5) f n +1 and g n +1 are strongly compatible (Definition 3.16) over p n +1 and q n +1 .Before we prove this claim, we show that proving it suffices to obtain the desiredcondition ( p ∗ , f ∗ ). So suppose that Claim 4.5 is true. Let ( p ∗ , f ∗ ) be a sup of h ( p n , f n ) : n ∈ ω i , and let ( q ∗ , g ∗ ) be a sup of h ( q n , g n ) : n ∈ ω i .Observe that by item (2) of Claim 4.5 and the fact that the sequence h β n : n ∈ ω i is cofinal in ν , we have that ( p ∗ , f ∗ ) (cid:13) R ρ ν ∈ lim( ˙ C ) and hence forces that ν ∈ ˙ C as ˙ C names a club. Also, since ( q ∗ , g ∗ ) ≥ ( q , g ) and since ( q , g ) = ( q, g ) forcesthat ν ∈ ˙ S , ( q ∗ , g ∗ ) forces that ν ∈ ˙ S too. We claim that p ∗ and q ∗ are compatiblein P ∗ , from which it follows by item (5) of Claim 4.5 that f ∗ and g ∗ are stronglycompatible over p ∗ and q ∗ . Indeed, ( p ∗ , f ∗ ) ∈ D ( ϕ M α , ρ ) since this set is closedunder sups of increasing ω -sequences by Lemma 3.8. Furthermore, by the countablecontinuity of ϕ M α , ϕ M α ( p ∗ ) is a sup of the increasing sequence h ϕ M α ( p n ) : n ∈ ω i .Thus to show that p ∗ and q ∗ are compatible, since q ∗ ∈ P ∗ ∩ M α , it suffices to showthat q ∗ ≥ P ∗ ∩ M α ϕ M α ( p ∗ ). However, we know that q ∗ ≥ q n for all n and so by (4) LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 31 of Claim 4.5, q ∗ ≥ ϕ M α ( p n ) for all n . Therefore q ∗ extends ϕ M α ( p ∗ ), by definitionof a supremum.Now let ( p ∗∗ , f ∗∗ ) be a condition in R ρ above both ( p ∗ , f ∗ ) and ( q ∗ , g ∗ ). Thenbecause ( p ∗∗ , f ∗∗ ) extends ( p ∗ ( M α ) , ˙ S ρ ) as well as ( q ∗ , g ∗ ), which in turn forces in R ρ ∩ M α that ν ∈ ˙ S , we have that ( p ∗∗ , f ∗∗ ) (cid:13) R ρ ν ∈ ˙ S . And finally, as ( p ∗∗ , f ∗∗ )extends ( p ∗ , f ∗ ) which forces in R ρ that ν ∈ ˙ C , we conclude that ( p ∗∗ , f ∗∗ ) (cid:13) R ρ ν ∈ ˙ S ∩ ˙ C . Thus it suffices to prove Claim 4.5 in order to finish the proof of Proposition4.4. Proof. (of Claim 4.5) We will construct the sequences satisfying (1)-(5) of Claim 4.5recursively. For the base case n = 0, items (2) and (5) hold vacuously. For item (1),we have that ( p , f ) = ( p, f ) ∈ K β as ( p, f ) = ( p , f ) was chosen to be definableby a constant in the language of K . We also ensured that ( p , f ) ∈ D ( ϕ M α , ρ ),which establishes (3). Finally, q ≥ P ∗ ∩ M α ¯ p ≥ P ∗ ∩ M α ϕ M α ( p ) , which establishes(4).Suppose, then, that we have defined ( p n , f n ) and ( q n , g n ) satisfying (1)-(5). Wefirst observe that ( p n , f n ) and ( q n , g n ) are compatible. If n = 0, this holds since( q , g ) is in R ρ ∩ M α and extends (¯ p, ¯ f ), which is a residue of ( p , f ) to R ρ ∩ M α .If n >
0, then we have that q n ≥ P ∗ ∩ M α ϕ M α ( p n ) and therefore p n and q n are P ∗ -compatible. Moreover, f n and g n are strongly compatible over the compatibleconditions p n and q n , and therefore ( p n , f n ) and ( q n , g n ) are compatible in R ρ .Next choose some condition ( r, h ) in R ρ which extends ( p n , f n ) and ( q n , g n ), andby extending if necessary, we may assume that there is some ordinal µ > β n sothat ( r, h ) (cid:13) R ρ µ ∈ ˙ C \ ( β n + 1). Since r ≥ p n ≥ p ∗ ( M α ) and since r ≥ q n ≥ q ≥ p ∗ ( M β n +1 ), we may also extend if necessary to assume, by Lemma 4.1(1), that r ∈ dom( ϕ M βn +1 ) ∩ dom( ϕ M α ).We now apply Lemma 4.3, with α and β n +1 playing the respective roles of “ β ”and “ α ” in the statement thereof, to find extensions ( r L , h L ) and ( r R , h R ) of ( r, h )which satisfy the conclusion of that lemma. We let (¯ r, ¯ h ) be a condition so that ρϕ Mβn +1 (( r L , h L ) , ( r R , h R ) , (¯ r, ¯ h )).Let Σ be a splitting function for ( r L , h L ) and ( r R , h R ) with respect to the model M β n +1 which satisfies Lemma 4.3. For Z ∈ { L, R } , set x Z := dom( h Z ) ∩ M β n +1 ;this is a countable subset of M β n +1 and therefore is a member of M β n +1 . Moreover, x Z is also an element of K β n +1 .We are now in a position to reflect into the model K β n +1 . We observe that in H ( θ )the following statement is true in the parameters β n , Σ , ¯ r, ¯ h, R ρ , ( p n , f n ) , α, B, ˙ C, x L ,and x R , all of which are in K β n +1 : there exists a condition ( r ∗ , h ∗ ) in R ρ and apair ( r ∗ Z , h ∗ Z ) Z ∈{ L,R } of conditions above ( r ∗ , h ∗ ) in R ρ as well as ordinals µ ∗ , η , sothat(i) ( r ∗ , h ∗ ) ≥ R ρ ( p n , f n );(ii) η ∈ B ;(iii) ρϕ Mη (( r ∗ L , h ∗ L ) , ( r ∗ R , h ∗ R ) , (¯ r, ¯ h ));(iv) for each Z ∈ { L, R } , ( r ∗ Z , h ∗ Z ) and ( r ∗ , h ∗ ) are in E ∗ ( ϕ M η , ϕ M α ) (see Propo-sition 4.2);(v) ( r ∗ , h ∗ ) (cid:13) R ρ µ ∗ ∈ ˙ C \ ( β n + 1); (vi) dom( h ∗ Z ) ∩ M η = x Z ;(vii) ( r ∗ L , h ∗ L ) and ( r ∗ R , h ∗ R ) are an ( M η , ρ )-splitting pair, and Σ is a splittingfunction for ( r ∗ L , h ∗ L ) and ( r ∗ R , h ∗ R ) with respect to the model M η .This statement is true in H ( θ ) as witnessed by the conditions ( r Z , h Z ) Z ∈{ L,R } and( r, h ), the ordinal µ playing the role of µ ∗ , and the ordinal β n +1 playing the role of η . Since the parameters of this statement are in K β n +1 , we may therefore find, in K β n +1 , conditions ( r ∗ Z , h ∗ Z ) Z ∈{ L,R } extending some ( r ∗ , h ∗ ) ≥ ( p n , f n ), an ordinal µ ∗ , and an ordinal η ∈ B so that (i)-(vii) above are satisfied of these objects.We now define ( p n +1 , f n +1 ) := ( r ∗ L , h ∗ L ). We need to extend the condition( ϕ M α ( r R ) , h R ↾ M α ) a bit more before defining ( q n +1 , g n +1 ). The following claimwill help us do this: Subclaim 4.6. ϕ M α ( p n +1 ) and ϕ M α ( r R ) are compatible in P ∗ ∩ M α . Proof.
Both the condition p n +1 and the function ϕ M α are members of K β n +1 .Therefore ϕ M α ( p n +1 ) ∈ K β n +1 ∩ M α ∩ P ∗ . Recall that K contained the fixed well-order ⊳ of H ( κ + ) and that all suitable models are elementary in H ( κ + ) with respectto ⊳ . Thus if we let e P ∗ denote the ⊳ -least bijection from κ onto P ∗ , then we havethat e P ∗ is in M α and in K β n +1 . Since M α is elementary and contains e P ∗ , we seethat ϕ M α ( p n +1 ) = e P ∗ ( ζ ) for some ζ < α . But then by the elementarity of K β n +1 ,we see that ζ ∈ K β n +1 ∩ α = β n +1 . Therefore ϕ M α ( p n +1 ) = e P ∗ ( ζ ) ∈ M β n +1 .Furthermore, we know that¯ r = ∗ ϕ M βn +1 ( r R ) = ∗ ϕ M βn +1 ( ϕ M α ( r R ))where the first equality holds by definition of ¯ r and the second because r R satisfiesLemma 4.3. Applying (iii) and (iv) above we also have that,¯ r = ∗ ϕ M η ( p n +1 ) = ∗ ϕ M η ( ϕ M α ( p n +1 )) . Additionally, since ϕ M η is an exact, strong residue function and ϕ M α ( p n +1 ) ∈ dom( ϕ M η ), we know that ϕ M α ( p n +1 ) ≥ ϕ M η ( ϕ M α ( p n +1 )) = ∗ ¯ r. Therefore, as ϕ M α ( p n +1 ) ∈ M β n +1 extends ¯ r , which is a residue of ϕ M α ( r R ) to M β n +1 , we conclude that ϕ M α ( p n +1 ) is compatible with ϕ M α ( r R ) in P ∗ ∩ M α . (cid:3) (Subclaim 4.6)Using Subclaim 4.6, we may fix some condition q n +1 in P ∗ ∩ M α which is aboveboth ϕ M α ( p n +1 ) and ϕ M α ( r R ). We finally set g n +1 := h R ↾ M α , noting that( q n +1 , g n +1 ) ∈ M α .We next verify that items (1)-(5) of Claim 4.5 hold for n + 1. We have that( p n +1 , f n +1 ) ≥ ( p n , f n ) by (i) of the reflection, more precisely, since( p n +1 , f n +1 ) = ( r ∗ L , h ∗ L ) ≥ ( r ∗ , h ∗ ) ≥ ( p n , f n ) . Additionally, because q n +1 ≥ ϕ M α ( r R ) ≥ ϕ M α ( r ) ≥ q n , we have that q n +1 ≥ q n ; moreover, g n +1 extends g n as a function. Thus ( q n +1 , g n +1 )extends ( q n , g n ). (1) of Claim 4.5 holds because we found the witnesses in themodel K β n +1 . For (2), µ ∗ ∈ K β n +1 ∩ α = β n +1 ⊆ ν , and since µ ∗ > β n , we havethat ( p n +1 , f n +1 ) (cid:13) µ ∗ ∈ ˙ C ∩ ( β n , ν ). For (3), we have ( p n +1 , f n +1 ) ∈ D ( ϕ M α , ρ ) LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 33 by (iii) and the definition of ρϕ Mη (see Definition 3.10). For (4), we have that q n +1 ≥ ϕ M α ( p n +1 ) by choice of q n +1 .It remains therefore to check that item (5) of Claim 4.5 holds. Since q n +1 ≥ P ∗ ∩ M α ϕ M α ( p n +1 ), we know that p n +1 and q n +1 are compatible in P ∗ ; let p ∗ ∈ P ∗ be anycondition extending both. We claim that p ∗ (cid:13) P ∗ ˇ f n +1 ∪ ˇ g n +1 ∈ ˙ S ρ . Suppose by induction on δ < ρ that h δ, ν i ∈ dom( f n +1 ) ∩ dom( g n +1 ) and that u forces that the union of ˇ f n +1 ↾ δ and ˇ g n +1 ↾ δ is a condition in ˙ S δ . Again using thefixed well-order ⊳ of H ( κ + ), we may let ψ be the ⊳ -least bijection from κ onto ρ ,so that ψ is a member of K β n +1 as well as every model on the R ρ -suitable sequence ~M . Since h δ, ν i ∈ dom( g n +1 ) and g n +1 ∈ M α , δ ∈ M α ∩ ρ = ψ [ α ]. Furthermore,since h δ, ν i ∈ dom( f n +1 ) and f n +1 = h ∗ L ∈ K β n +1 , we have that δ ∈ K β n +1 . Thus δ ∈ ψ [ α ] ∩ K β n +1 = ψ [ K β n +1 ∩ α ] = ψ [ β n +1 ] ⊆ M β n +1 . Therefore (recalling that g n +1 = h R ↾ M α ), h δ, ν i ∈ dom( h R ) ∩ M β n +1 = x R = dom( h ∗ R ) ∩ M η . Continuing, fix a pair h θ, τ i ∈ f n +1 ( δ, ν ) × g n +1 ( δ, ν )with θ = τ . We need to show that θ and τ are forced to be incompatible nodesin the tree ˙ T δ by the condition (cid:0) p ∗ , ( f n +1 ∪ g n +1 ) ↾ δ (cid:1) . Recall, going forward,that (¯ r, ¯ h ) equals both ( r R , h R ) ↾ M β n +1 and ( p n +1 , f n +1 ) ↾ M η ; in particular, f n +1 and h R ↾ M α = g n +1 both extend ¯ h . Continuing, if τ is below level β n +1 ,then τ ∈ ¯ h ( δ, ν ) ⊆ f n +1 ( δ, ν ) and we are done. Furthermore, if θ is below level η ,then θ ∈ ¯ h ( δ, ν ) ⊆ g n +1 ( δ, ν ) and we are done in this case too. Thus we assumethat θ is at or above level η and that τ is at or above level β n +1 . With respectto the fixed enumerations, let k and m be chosen so that θ is the k th element of f n +1 ( δ, ν ) \ ( η × ω ) and τ is the m th element of h R ( δ, ν ) \ ( β n +1 × ω ). Then becausethe function Σ is the same for both pairs of splitting conditions, we know that( p n +1 , f n +1 ↾ δ ) (cid:13) R δ Σ( δ, ν, k, m )( L ) < ˙ T δ θ and that ( r R , h R ↾ δ ) (cid:13) R δ Σ( δ, ν, k, m )( R ) < ˙ T δ τ. However, τ ∈ g n +1 ( δ, ν ) = ( h R ↾ M α )( δ, ν ), and therefore τ is below level α of thetree ˙ T δ . Therefore by Lemma 4.3, we have that (cid:16) ϕ M α ( r R ) , ( h R ↾ M α ) ↾ δ (cid:17) (cid:13) R δ Σ( δ, ν, k, m )( R ) < ˙ T δ τ. Since q n +1 ≥ ϕ M α ( r R ) and g n +1 = h R ↾ M α , we conclude that( q n +1 , g n +1 ↾ δ ) (cid:13) R δ Σ( δ, ν, k, m )( R ) < ˙ T δ τ. Finally, since (cid:0) p ∗ , ( f n +1 ∪ g n +1 ) ↾ δ (cid:1) is above both ( p n +1 , f n +1 ↾ δ ) and ( q n +1 , g n +1 ↾ δ ), it follows that (cid:0) p ∗ , ( f n +1 ∪ g n +1 ) ↾ δ (cid:1) (cid:13) Σ( δ, ν, k, m )( R ) < ˙ T δ τ ∧ Σ( δ, ν, k, m )( L ) < ˙ T δ θ. Since the distinct nodes Σ( δ, ν, k, m )( L ) and Σ( δ, ν, k, m )( R ) are on the same level, (cid:0) p ∗ , ( f n +1 ∪ g n +1 ) ↾ δ (cid:1) therefore forces that θ and τ are incompatible nodes in thetree ˙ T δ . This completes the proof that f n +1 and g n +1 are strongly compatible over p n +1 and q n +1 . Therefore the proof of Claim 4.5 is now complete. (cid:3) (Claim 4.5)As remarked earlier, this completes the proof of Proposition 4.4. (cid:3) As a corollary of Proposition 4.4, we can now prove Theorem 1.2.
Proof.
Recall that the Laver Shelah model V [ G ∗ F ] is obtained by starting froma ground model V with a weakly compact cardinal κ , and forcing with the Levycollase P followed by a countable support iteration S = h S τ , S ( τ ) | τ < κ + i ofspecializing posets S ( τ ) = S ( ˙ T τ ) of Aronszajn trees on κ , chosen by a bookkeepingfunction. Since S satisfies κ .c.c, every sequence of stationary sets h S α | α < κ i asin the statement of Theorem 1.2, belongs to an intermediate extension V [ G ∗ F τ ],where F τ := F ∩ S τ , for some τ < κ + . Now work in V , and take a P ∗ ˙ S τ -name h ˙ S α | α < κ i for the sequence of stationary sets. Let ~M be suitable with respectto these parameters. By the weak compactness of κ , let β ∈ dom( ~M ) be suchthat h ˙ S α ∩ V β | α < β i are names for stationary subsets of β in the restricted poset( P β ∗ ˙ S τ ) ∩ M β . By Proposition 4.4, which applies to the F WC -strongly proper poset P (see Example 2.15) with restrictions to P ↾ β as residue functions, we concludethat each S α ∩ β remains stationary in the full P ∗ ˙ S τ generic extension V [ G ∗ F τ ]and hence in V [ G ∗ F ].To see that CSR ( ω ) fails, observe that in the ground model, there exist station-ary sets S ⊆ κ ∩ cof( ω ) and T ⊆ κ ∩ cof( ω ) so that S does not reflect at any pointin T (see Proposition 1.1 of [23]). The stationarity of S and T are preserved by the κ -c.c. forcing of Laver and Shelah, and since ω is also preserved, we have that S and T witness the failure of CSR ( ω ) in the final model. (cid:3) F WC -completely proper posets In this section, we will specify what P -names ˙ C for posets are such that P ∗ ˙ C is F WC -strongly proper, and we will draw some conclusions from this. Recall that P denotes the Levy collapse poset Col( ω , < κ ), where κ is weakly compact. Twomain ideas come into play in this section. The first is an axiomatization of variousproperties of the iterated club adding ˙ C Magidor from [31], which will allow us toplace upper bounds on various “local” filters added by the Levy collapse. Wethen couple this axiomatization with a generalization of a result of Abraham’s ([1])that, in current language, if ˙ Q is an Add( ω, ω )-name for an ω -closed poset, thenAdd( ω, ω ) ∗ ˙ Q is strongly proper; see [17] for a proof of this fact as stated here.The strong properness results from using so-called “guiding reals.”We recall that in [31], to show that an iteration ˙ C Magidor of length < κ + addingthe desired clubs is κ -distributive, Magidor argued, in part, as follows: let j : M −→ N be a weakly compact embedding, where M has the relevant parameters. Let G ∗ be N -generic over j ( P ) and G := G ∗ ∩ P , so that in N [ G ∗ ], we may construct an M [ G ]-generic filter H for C Magidor . Moreover, j [ H ] has a least upper bound in j ( C Magidor ), namely, the function obtained by placing κ on top of each coordinatein the domain of j [ H ]; by the closure of the quotient (which implies the preservationof the stationary sets appearing along the way in the definition of C Magidor ), this isindeed a condition.The property of ˙ C Magidor which we will axiomatize is a reflection of the aboveto an F WC -measure one set of α < κ . Roughly, we want to say that for many α , LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 35 if you “cut off” P ∗ ˙ C at α , then many generics added by the tail of the collapsefor “ ˙ C cut off at α ” have upper bounds in the full poset ˙ C . More precisely, given a P -name ˙ C in H ( κ + ) for a poset which is ω -closed with sups and given a ˙ C -suitablemodel (see Definition 2.13) M , say with M ∩ κ = α < κ , we consider the poset π M ( ˙ C ), where π M denotes the transitive collapse of M to ¯ M . An easy absolutenessargument shows that π M ( ˙ C ) is a name in π M ( P ) = P ↾ α . Appealing to the closureof ¯ M under < α -sequences, and hence ω -sequences, we see that π M ( ˙ C ) is forced by P ↾ α to be ω -closed with sups. The desired condition on P -names ˙ C can now bestated a bit more precisely: we will demand that after forcing with P , say to add thegeneric G , for many α as above and many V [ G P ↾ α ]-generics H for π M ( ˙ C )[ G P ↾ α ] in V [ G ], π − M [ G ] [ H ] has an upper bound in ˙ C [ G ]. Note that we are implicitly appealingto the properness of P with respect to M to see that π M : M −→ ¯ M lifts to π M [ G ] : M [ G ] −→ ¯ M [ G P ↾ α ]; we discuss this more later.The first step to making this work is to isolate exactly which filters we will use;for reasons related to building strong, exact residue functions later, we will notconsider all filters added by the tail of the collapse for π M ( ˙ C ). The definition ismeant to capture the behavior of filters generated by using the generic surjectionsto guide choices of conditions, similar to how Abraham used guiding reals in [1].5.1. Residue Functions from Local Filters.Definition 5.1.
Let α < κ be inaccessible, and let ˙ Q be a ( P ↾ α )-name for aposet of size α which is ω -closed with sups. Since P ↾ α is α -c.c there exists a list h ˙ γ i : i < α i of ( P ↾ α )-names, which is forced enumerate all conditions in ˙ Q . We saythat a sequence ˙ s = h ˙ d ν : ν < ω i of P -names (not ( P ↾ α )-names) for conditions in˙ Q is guided by the collapse at α for ˙ Q if the following conditions are satisfied:(1) (cid:13) P h ˙ d ν : ν < ω i is ≤ ˙ Q -increasing, ˙ d is the weakest condition in ˙ Q , and if ν is limit, then ˙ d ν is a sup of h ˙ d µ : µ < ν i ;(2) if p ∈ P and dom( p ( α )) is an ordinal ν < ω , then there exists p ′ ≥ p with p ′ ↾ [ α, κ ) = p ↾ [ α, κ ) and a sequence h β ( µ ) : µ ≤ ν i of ordinals in V so that p ′ (cid:13) ˙ d µ = ∗ ˙ γ β ( µ ) for all µ ≤ ν . In this case we will say that p ′ determinesan initial segment of ˙ s ;(3) if p ′ as in (2) determines an initial segment of ˙ s and if ˙ γ is a ( P ↾ α )-namefor a ˙ Q -extension of ˙ γ β ( ν ) , then there exists p ∗ ≥ p ′ so that p ∗ (cid:13) ˙ d ν +1 ≥ ˙ γ . Lemma 5.2.
Suppose that ˙ s := h ˙ d ν : ν < ω i is guided by the collapse at α . Let H ( ˙ s ) be the P -name for the filter on ˙ Q generated by ˙ s . Then P forces that H ( ˙ s ) is V [ ˙ G ↾ α ] -generic over ˙ Q .Proof. Fix a condition p ∈ P and a P -name ˙ D for a dense subset of ˙ Q which is amember of V [ ˙ G ↾ α ]. We find an extension of p which forces that ˙ D ∩ H ( ˙ s ) = ∅ .By extending p and applying (2) of Definition 5.1 if necessary, we may assume thefollowing:(1) there is a ( P ↾ α )-name ˙ D for a dense subset of ˙ Q so that p (cid:13) ˙ D = ˙ D ;(2) dom( p ( α )) is an ordinal ν , and there is a sequence h β ( µ ) : µ ≤ ν i of ordinalsin V so that p (cid:13) ˙ d µ = ∗ ˙ γ β ( µ ) for all µ ≤ ν . Let ˙ γ be a ( P ↾ α )-name for a condition in ˙ D forced to extend ˙ γ β ( ν ) . By item (3)of Definition 5.1, we may find an extension p ∗ of p so that p ∗ (cid:13) ˙ d ν +1 ≥ ˙ γ . Then p ∗ (cid:13) ˙ γ ∈ ˙ D ∩ H ( ˙ s ), finishing the proof. (cid:3) Definition 5.3.
Let ˙ H be a P -name for a filter on ˙ Q . We say that ˙ H is guidedby the collapse at α if there is a sequence ˙ s = h ˙ d ν : ν < ω i of conditions guidedby the collapse at α so that ˙ H = H ( ˙ s ).Suppose that M is a suitable model. Let α := M ∩ κ , and let π M : M → ¯ M be the transitive collapse map of M . Let G ⊆ P be generic over V , and set G α = G ∩ ( P ↾ α ). We have that P ↾ α = π M ( P ) ∈ ¯ M , and G α ⊂ π M ( P ) is genericfor ¯ M . Moreover, setting M [ G ] = { ˙ x [ G ] | ˙ x ∈ M is a name } , we have that ¯ M [ G α ]is the transitive collapse of M [ G ], with the transitive collapse map π M [ G ] being thenatural extension of π M , given by π M [ G ] ( ˙ x [ G ]) = π M ( ˙ x )[ G α ]. Lemma 5.4.
Suppose that M is a ( P ∗ ˙ C ) -suitable model, where ˙ C is a P -namefor a poset on κ which is ω -closed with sups. Let α := M ∩ κ and π M be thetransitive collapse map of M . Suppose that there is a P -name ˙ H for a subset of π M ( ˙ C ) which is generic over V [ ˙ G α ] , and that there is a P -name ˙ c for a conditionin ˙ C which is forced to be an upper bound for π − M [ ˙ G ] [ ˙ H ] . Then P forces that ˙ c is an ( M [ ˙ G ] , ˙ C ) -completely-generic condition.Proof. Fix G . To see that c = ˙ c G is ( M [ G ] , C )-completely generic, fix a dense, open E ⊆ C with E ∈ M [ G ]. We show that c extends some condition in E .By the elementarity of π M [ G ] : M [ G ] → ¯ M [ G α ], we know that π M [ G ] ( E ) is densein π M [ G ] ( C ) = π M ( ˙ C )[ G α ]. Since H := ˙ H [ G ] is a V [ G ↾ α ]-generic filter, by Lemma5.2, H ∩ π M [ G ] ( E ) = ∅ , and thus π − M [ G ] [ H ] ∩ E = ∅ . (cid:3) There are two particularly useful properties of this class of names for genericfilters. On the one hand, filters in this class will allow us to generate strong, exactresidue functions by isolating the information which a given conditions determinesabout the filters. On the other hand, the class of such filters for one poset, suchas a two-step iteration, often projects to the class of such filters for another poset,such as the first step in a two-step iteration. This property will be particularlyuseful in Section 6 when we want to show, by induction, that our club adding posetis well-behaved.It is straightforward to verify that the notion P -names of filters, which are guidedby the collapse at a given cardinal, factor well in iterations. Lemma 5.5.
Suppose that α < κ is inaccessible and that ˙ Q ∗ ˙ Q is a ( P ↾ α ) -namefor a two-step poset of size α which is ω -closed with sups. Let h ˙ d ν : ν < ω i bea sequence of P -names which is guided by the collapse at α for ˙ Q ∗ ˙ Q . Then thesequence h ˙ d ν (0) : ν < ω i of P -names of conditions in ˙ Q is guided by the collapseat α for ˙ Q . The following proposition shows how to generate exact, strong residue functionsfrom the filters discussed above.
Proposition 5.6.
Suppose that ˙ C is a P -name in H ( κ + ) for a separative poset ofsize κ which is ω -closed with sups, and set P ∗ := P ∗ ˙ C . Let M be a ( P ∗ ˙ C ) -suitablemodel, say with α := M ∩ κ < κ , and let π M denote the transitive collapse of M . LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 37
Also let ˙ s = h ˙ d ν : ν < ω i be a sequence of P -names guided by the collapse at α for π M ( ˙ C ) . Additionally, suppose that there is a P -name ˙ d ∗ for a condition in ˙ C whichis forced to be an upper bound for the sequence π − M [ ˙ G ] [ ˙ s ] .Define p ∗ ( M ) to be the condition p ∗ ( M ) := (0 P , ˙ d ∗ ) , and let D ( M ) := n ( p, ˙ d ) ≥ p ∗ ( M ) : p determines an initial segment of ˙ s o . Finally, define ϕ M on D ( M ) by ϕ M ( p, ˙ d ) = ( p ↾ α, π − M ( ˙ γ β )) , where β < α is the least so that p (cid:13) ˙ d dom( p ( α )) = ∗ ˙ γ β ( β exists by definition of “ p determines an initial segment of ˙ s ”). Then(a) D ( M ) is a dense, countably = ∗ -closed subset of P ∗ /p ∗ ( M ) ;(b) p ∗ ( M ) is compatible with every condition in P ∗ ∩ M ; and(c) ϕ M is an exact, strong residue function from D ( M ) to M ∩ P ∗ .Proof. Let h ˙ γ i : i < α i be a sequence of ( P ↾ α )-names which is forced to enumerateall conditions in π M ( ˙ C ), and with ˙ d ∗ , satisfies Definition 5.1, witnessing that ˙ s = h ˙ d ν : ν < ω i is guided by the collapse at α for π M ( ˙ C ).We first prove item (a). Given a condition ( p, ˙ d ) in P ∗ /p ∗ ( M ), by item (2) ofDefinition 5.1, we may find an extension p ′ of p so that p determines an initialsegment of ˙ s . Then ( p ′ , ˙ d ) ≥ ( p, ˙ d ) is in D ( M ), proving density. Similarly, D ( M ) is= ∗ -closed: if ( p , ˙ d ) ∈ D ( M ) and ( p , ˙ d ) = ∗ ( p , ˙ d ), then p determines an initialsegment of ˙ s , because p = p (recall these are collapse conditions) and because( p , ˙ d ) ≥ ( p , ˙ d ) ≥ (0 P , ˙ d ∗ ).To see that D ( M ) is closed under sups of increasing ω -sequences, suppose that h ( p n , ˙ c n ) : n ∈ ω i is an increasing sequence of conditions in D ( M ), and let ( p ∗ , ˙ c ∗ )be a sup. Set ν n := dom( p n ( α )) and ν ∗ := dom( p ∗ ( α )). If ν ∗ = ν m for some m ∈ ω , then because p m determines ˙ s up to ν m , we have that p ∗ determines ˙ s upto ν ∗ = ν m . Thus p ∗ determines an initial segment of ˙ s in this case. So considerthe case that ν ∗ > ν m for all m ; in particular, ν ∗ is a limit ordinal. Since for all n ∈ ω , p n determines an initial segment of ˙ s and p ∗ ≥ p n , we may find a sequence h β ( µ ) : µ < ν ∗ i in V so that p ∗ (cid:13) ˙ d µ = ∗ ˙ γ β ( µ ) for all µ < ν ∗ . Now let β ( ν ∗ ) bechosen so that ˙ γ β ( ν ∗ ) is forced to be a sup of h ˙ γ β ( µ ) : µ < ν i , if this sequence isincreasing, and equals the trivial condition otherwise. Since ν ∗ is a limit, (cid:13) P ˙ d ν ∗ isa sup of h ˙ d µ : µ < ν ∗ i . But p ∗ forces that ˙ d µ = ∗ ˙ γ β ( µ ) for all µ < ν ∗ , and therefore p ∗ forces that ˙ d ν ∗ = ∗ ˙ γ β ( ν ∗ ) . Thus, in either case, p ∗ determines an initial segmentof ˙ s , which finishes the proof of (a).Now we verify item (b). Fix a condition ( u, ˙ c ) in P ∗ ∩ M , and we will showthat it is compatible with p ∗ ( M ). We observe that, trivially, u determines an initialsegment of ˙ s since dom( u ( α )) = 0 and (cid:13) P ˙ d is the trivial condition in π M ( ˙ C ), by(1) of Definition 5.1. By (3) of the same definition, we may find an extension p ≥ u s.t. p (cid:13) ˙ d ≥ π M ( ˙ c ). Then ( p, ˙ d ∗ ) extends ( u, ˙ c ) since p forces that ˙ d ∗ is an upperbound for the sequence π − M [ ˙ s ] and that π − M ( ˙ d ) ≥ ˙ c .It therefore remains to verify that ϕ M is an exact, strong residue function. Condi-tion (1) of Definition 2.9 holds since, by (a), D ( M ) is dense and countably = ∗ -closed in P ∗ /p ∗ ( M ). For the projection condition of Definition 2.9, fix ( p, ˙ c ) ∈ D ( M ), andlet ˙ γ be the ( P ↾ α )-name so that ϕ M ( p, ˙ c ) = ( p ↾ α, π − M ( ˙ γ )). Since ( p, ˙ c ) ∈ D ( M ), p determines an initial segment of ˙ s , and therefore p (cid:13) ˙ d dom( p ( α )) = ∗ ˙ γ . Since p alsoforces that ˙ c ≥ ˙ d ∗ , p forces that ˙ c is an upper bound for π − M [ ˙ s ] and therefore that ˙ c extends π − M ( ˙ d dom( p ( α )) ) = ∗ π − M ( ˙ γ ). Therefore ( p, ˙ c ) ≥ ( p ↾ α, π − M ( ˙ γ )) = ϕ M ( p, ˙ c ).By Lemma 2.10, condition (3) of Definition 2.9 (that ϕ M is order-preserving)will follow once we prove that ϕ M has the strong residue property (condition (4)of Definition 2.9). Thus fix ( p, ˙ c ) ∈ D ( M ), where we let ν := dom( p ( α )) and˙ γ so that ϕ M ( p, ˙ c ) = ( p ↾ α, π − M ( ˙ γ )). Fix a condition ( u, ˙ δ ) in P ∗ ∩ M with( u, ˙ δ ) ≥ ( p ↾ α, π − M ( ˙ γ )), and we will verify that ( u, ˙ δ ) is compatible with ( p, ˙ c ). Let p ′ := u ∪ p , a condition in P , and observe that p ′ still determines an initial segmentof ˙ s and ν = dom( p ′ ( α )). By item (3) of Definition 5.1, we may find some p ∗ ≥ p ′ so that p ∗ (cid:13) ˙ d ν +1 ≥ π M ( ˙ δ ). Then ( p ∗ , ˙ c ) extends both ( p, ˙ c ) and ( u, ˙ δ ).We now check that ϕ M is ω -continuous, which will finish the proof of (c) andthereby the proof of the proposition. Fix an increasing sequence of conditions h ( p n , ˙ c n ) : n ∈ ω i in D ( M ), and let ( p ∗ , ˙ c ∗ ) be a supremum of this sequence.Then p ∗ := S n p n , and ˙ c ∗ is forced by p ∗ to be a sup of h ˙ c n : n ∈ ω i . By item(a) of the proposition, ( p ∗ , ˙ c ∗ ) ∈ D ( M ). We need to show that ϕ M ( p ∗ , ˙ c ∗ ) is asup of h ϕ M ( p n , ˙ c n ) : n ∈ ω i . For each n < ω , set ν n := dom( p n ( α )), and alsoset ν ∗ := dom( p ∗ ( α )). Additionally, for each n , fix the least ordinal β ( ν n ) with p n (cid:13) ˙ d ν n = ∗ ˙ γ β ( ν n ) , so that ϕ M ( p n , ˙ c n ) = ( p n ↾ α, π − M ( ˙ γ β ( ν n ) )). Finally let β ( ν ∗ )be least so that ϕ M ( p ∗ , ˙ c ∗ ) = ( p ∗ ↾ α, π − M ( ˙ γ β ( ν ∗ ) )).We claim that p ∗ (cid:13) ˙ γ β ( ν ∗ ) is a sup of h ˙ γ β ( ν n ) : n ∈ ω i . Note that proving thisclaim suffices: indeed, then p ∗ ↾ α forces that ˙ γ β ( ν ∗ ) is a sup of h ˙ γ β ( ν n ) : n ∈ ω i ,and as a result ϕ M ( p ∗ , ˙ c ∗ ) = ( p ∗ ↾ α, π − M ( ˙ γ β ( ν ∗ ) )) is a sup of h ϕ M ( p n , ˙ c n ) : n ∈ ω i .To prove the claim, we have two cases on ν ∗ . Since ν ∗ = sup m ν m , either ν ∗ > ν m for all m , or ν ∗ = ν m for almost all m . In the first case, ν ∗ is a limit, and so P forces that ˙ d ν ∗ is a sup of h ˙ d ν : ν < ν ∗ i . Since p n (cid:13) ˙ d ν n = ∗ ˙ γ β ( ν n ) for each n , p ∗ forces that h ˙ γ β ( ν n ) : n ∈ ω i is cofinal in h ˙ d ν : ν < ν ∗ i . Therefore p ∗ forces that thesetwo sequences have the same sups. Consequently, p ∗ forces that ˙ γ β ( ν ∗ ) = ∗ ˙ d ν ∗ is asup of h ˙ γ β ( ν n ) : n ∈ ω i . For the second case, ν ∗ = ν m for all m above some k . Then P forces that h ˙ d ν n : n ∈ ω i is eventually equal to ˙ d ν ∗ . Because p n (cid:13) ˙ d ν n = ∗ ˙ γ β ( ν n ) for all n and p ∗ ≥ p n , we have that p ∗ (cid:13) h ˙ γ β ( ν n ) : n ∈ ω i is eventually equal to˙ d ν ∗ . Finally, p ∗ (cid:13) ˙ γ β ( ν ∗ ) = ∗ ˙ d ν ∗ , and therefore p ∗ forces that h ˙ γ β ( ν n ) : n ∈ ω i iseventually = ∗ -equal to ˙ γ β ( ν ∗ ) , and therefore that ˙ γ β ( ν ∗ ) is a sup. This finishes theproof of the claim and thereby the proof that ϕ M is ω -continuous. (cid:3) The final result in this section shows that we can create filters which are guided bythe collapse at α by using the generic surjection from ω onto α to guide extensionsin the second coordinate. This combines ideas of collapse absorption with, aspreviously mentioned, Abraham’s use of guiding reals. Lemma 5.7.
Suppose that ˙ Q is a ( P ↾ α ) -name for a poset of size α , which is ω -closed with sups, and let h ˙ γ i : i < α i be forced to enumerate all conditions in ˙ Q .Let ˙ f α be the P -name for the standard surjection added from ω onto α .Suppose that ˙ s = h ˙ d ν : ν < ω i is a sequence of P -names for conditions in ˙ Q forced by P to satisfy the following properties: LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 39 (1) ˙ s is ≤ ˙ Q -increasing, ˙ d names the trivial condition, and if ν is a limit, then ˙ d ν is a ≤ ˙ Q -sup of h ˙ d µ : µ < ν i ;(2) if ν < ω and ˙ γ ˙ f α ( ν ) extends ˙ d ν in ˙ Q , then ˙ d ν +1 extends ˙ γ ˙ f α ( ν ) ;(3) for each ν < ω , the sequence h ˙ d µ : µ ≤ ν i is definable in V [ ˙ G ↾ α ] from ˙ f α ↾ ( ν + 1) .Then ˙ s is guided by the collapse at α for ˙ Q .In particular, there exists a P -name for a sequence which is guided by the collapseat α for ˙ Q .Proof. We will verify that items (1)-(3) of Definition 5.1 hold. Item (1) of thedefinition is immediate from assumption (1) of the lemma.For item (2) of Definition 5.1, suppose that p ∈ P is a condition where dom( p ( α ))is an ordinal ν < ω . Let G be V -generic over P containing p , and let ¯ G := G ∩ ( P ↾ α ). For each µ ≤ ν , let d µ := ˙ d µ [ G ], an ordinal < α , and let β ( µ ) sothat d µ = ˙ γ β ( µ ) [ ¯ G ]. By assumption (3) of the lemma, the sequence h d µ : µ ≤ ν i isdefinable in V [ ¯ G ] from f α ↾ ν = p ( α ). Therefore, there exists a condition ¯ p ≥ p ↾ α with ¯ p ∈ ¯ G so that ¯ p forces that h ˙ γ β ( µ ) : µ ≤ ν i satisfies the definition with respectto p ( α ). Then p ′ := p ∪ ¯ p witnesses item (2) of Definition 5.1, since p ′ forces that h ˙ γ β ( µ ) : µ ≤ ν i and h ˙ d µ : µ ≤ ν i both satisfy the same definition in V [ ˙¯ G ] with theparameter ˙ f α ↾ ν = p ( α ).Turning to item (3) of Definition 5.1, let p ′ be a condition as in the previousparagraph. Fix a ( P ↾ α )-name ˙ γ for a ˙ Q -extension of ˙ γ β ( ν ) , and let δ < α so that (cid:13) P ↾ α ˙ γ = ˙ γ δ . Define p ∗ to be the minimal extension of p so that p ∗ (cid:13) ˙ f α ( ν ) = δ .Then p ∗ (cid:13) ˙ γ ˙ f α ( ν ) ≥ ˙ γ β ( ν ) = ∗ ˙ d ν , so there exists an extension p ∗∗ of p ∗ so that p ∗∗ forces ˙ d ν +1 ≥ ˙ γ ˙ f α ( µ ) = ˙ γ .For the final claim of the lemma, define a sequence ˙ s by recursion so that itsatisfies (1) and so that if ν < ω , then ˙ d ν +1 is forced to be equal to γ ˙ f α ( ν ) if thisextends ˙ d ν , and otherwise equals ˙ d ν . Then (2) and (3) are also satisfied, so ˙ s isguided by the collapse at α for ˙ Q . (cid:3) F WC -Complete Properness. We are now ready to isolate a sufficient con-dition on names ˙ C so that P ∗ ˙ C is F WC -strongly proper. Definition 5.8.
Let ˙ C be a P -name in H ( κ + ) for a separative poset forced tobe ω -closed with sups. We say that ˙ C is F WC - Completely Proper if for any( P ∗ ˙ C )-suitable sequence ~M there is some A ∈ F WC with A ⊆ dom( ~M ) so that foreach α ∈ A and each P -name ˙ H for a V [ ˙ G P ↾ α ]-generic filter over π M α ( ˙ C ) which isguided by the collapse at α , there exists a P -name ˙ c ˙ H for a condition in ˙ C which isforced to be a least upper bound for π M α [ ˙ G P ] [ ˙ H ].We recall that a poset U is λ -distributive if forcing with U adds no sequences ofordinals of length less than λ . If U is separative, this is equivalent to showing thatthe intersection of fewer than λ -many dense, open subsets of U is dense, open. Lemma 5.9.
Suppose that ˙ C is F WC -completely proper. Then P forces that ˙ C is κ -distributive.Proof. Let ~D = h ˙ D i : i < ω i be a sequence of P -names for dense, open subsetsof ˙ C . We show that the intersection is forced to be non-empty. Fix a sequence ~M which is suitable with respect to P ∗ ˙ C and ~D so that dom( ~M ) satisfies theconclusion of Definition 5.8. Let α ∈ dom( ~M ). By Lemma 5.7, there exists a P -name ˙ H for a filter which is guided by the collapse at α for π M α ( ˙ C ). Since dom( ~M )satisfies Definition 5.8, we may find a P -name ˙ d for a condition which is forced tobe an upper bound in ˙ C for π M [ ˙ G P ] [ ˙ H ]. Then, by Lemma 5.4, ˙ d is forced to bean ( M α [ ˙ G ] , ˙ C )- completely generic condition. But ˙ D i ∈ M α for each i < ω and isdense, open. Therefore it is forced that ˙ d ∈ T i ∈ ω ˙ D i . (cid:3) By combining Lemma 5.7, Proposition 5.6, and Lemma 2.19, we conclude thefollowing key result:
Proposition 5.10.
Suppose that ˙ C is F WC -completely proper. Then P ∗ ˙ C is F WC -strongly proper. Properties of the Club-Adding Poset
In this section, we have two main tasks. In the first subsection, we will provethat our intended club adding iteration, as well as useful variants thereof, are F WC -completely proper, and in the second subsection, we will prove that our intendedclub adding iteration does not add branches through various Aronszajn trees. Eachof these results will be used as part of a larger inductive argument in the finalsection in which we prove Theorem 1.1.6.1. Adding Clubs is F WC -Completely Proper. In order to anticipate argu-ments in the next subsection, where we show that appropriate F WC -completelyproper posets do not add branches through certain Aronszajn trees, we will needto not only show that our club adding poset is F WC -completely proper, but alsoshow that variants of it have this property. These variants are created by iteratingthe process of taking an initial segment of the iteration followed by products offinitely-many copies of the tail.The following iteration follows Magidor’s work [31] on adding clubs throughreflection points of stationary subsets of a weakly compact cardinal κ , which hasbeen collapsed to become ω .Let ρ < κ + , and suppose that we have defined a P -name for an iteration h ˙ C σ , ˙ C ( η ) : σ ≤ ρ, η < ρ i and a ( P ∗ ˙ C ρ )-name h ˙ S σ , ˙ S ( η ) : σ ≤ ρ, η < ρ i for aniteration so that for all σ < ρ the following assumptions are satisfied:(1) ˙ S σ is a ( P ∗ ˙ C σ )-name;(2) ˙ C ( σ ) is a ( P ∗ ˙ C σ )-name for CU ( ˙ S σ , ˙ S σ ) (see Definition 1.5), where ˙ S σ is a( P ∗ ˙ C σ ∗ ˙ S σ )-name for a stationary subset of κ ∩ cof( ω ) and ˙ C σ +1 = ˙ C σ ∗ ˙ C ( σ );(3) ˙ S ( σ ) is a ( P ∗ ˙ C σ +1 ∗ ˙ S σ )-name for S ( ˙ T σ ) (see Definition 1.6), where ˙ T σ is a( P ∗ ˙ C σ +1 ∗ ˙ S σ )-name for an Aronszajn tree on κ ;(4) ˙ C σ is F WC -completely proper (and hence P ∗ ˙ C σ is F WC -strongly proper, byProposition 5.10), and P ∗ ˙ C σ forces that ˙ S σ is a countable support iterationspecializing Aronszajn trees, as defined in Section 3.Working in an arbitrary generic extension by P , we now define the variationsof ˙ C ρ mentioned above; we call these Doubling Tail Products . These will be theposets C ρ ( ~δ ), where ~δ = h δ , δ , . . . , δ n − i ∈ [ ρ ] n is a strictly decreasing sequenceof ordinals. We use [ ρ ] <ω dec to denote the set of all finite strictly decreasing, finitetuples from ρ ; [ ρ ] n dec is defined similarly. LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 41
We first introduce an auxiliary name for a poset ˙ C δ ∗ ,ρ ( ~δ ), where ~δ is as aboveand δ ∗ ≤ δ n − is an additional ordinal. This is done by recursion on n = | ~δ | asfollows: • For n = 0 (i.e., ~δ = ∅ ) we define ˙ C δ ∗ ,ρ ( ∅ ) = ˙ C δ ∗ ,ρ to be the tail-segment ofthe iteration C ρ , starting from stage δ ∗ . • For n ≥ ~δ ∈ [ ρ ] n , and δ ∗ ≤ δ n − , the poset C δ ∗ ,ρ ( ~δ ) is given by-˙ C δ ∗ ,ρ ( ~δ ) = ˙ C δ ∗ ,δ n − ∗ ( ˙ C δ n − ,ρ ( ~δ ↾ n − , where ˙ C δ ∗ ,δ n − is the segment of the iteration C ρ starting from (and includ-ing) stage δ ∗ to stage δ n − , and ~δ ↾ ( n −
1) = h δ , . . . , δ n − i .We can now define C ρ ( ~δ ). Definition 6.1.
For ~δ ∈ [ ρ ] <ω dec , define C ρ ( ~δ ) = C ,ρ ( ~δ ) (i.e., as C δ ∗ ,ρ ( ~δ ) with δ ∗ = 0).For example, if ~δ = h δ i is a singleton, then C ρ ( h δ i ) = C δ ∗ ˙ C δ ,ρ . Similarly, if ~δ = h δ , δ i has two elements δ > δ then C ρ ( h δ , δ i ) = C δ ∗ ˙ C δ ,ρ ( h δ i ) = C δ ∗ (cid:16) ˙ C δ ,δ ∗ ˙ C δ ,ρ (cid:17) We refer to posets C ρ ( ~δ ) as the doubling tail products of C ρ . We now returnto working in V , in particular with the statement of the next item. Proposition 6.2. (Given assumptions (1)-(4) stated at the beginning of the sub-section) For each ρ < κ + and ~δ = h δ , . . . , δ n − i ∈ [ ρ ] <ω dec , the doubling tail product ˙ C ρ ( ~δ ) is F WC -completely proper. In particular, ˙ C ρ is F WC -completely proper. We will explain the necessity of proving Proposition 6.2 for the doubling tailproducts of ˙ C ρ in the final section of the paper. Proof.
We will first work with ˙ C ρ , rather than with the doubling tail products, inorder to establish that a certain statement ( ∗ ) (see below) holds. We will then showthat this statement ( ∗ ) can be used to prove the desired result for the iterated tailproducts. We use R σ , for σ ≤ ρ , to denote P ∗ ˙ C σ ∗ ˙ S σ .To begin, we fix an R ρ -suitable sequence ~M ; by removing an F WC -measure 0 setof points, we may assume that ~M is in pre-splitting configuration up to ρ .Let M be a κ -model so that M contains the relevant parameters, including ~M ,and let j : M −→ N be a weakly compact embedding with j, M ∈ N by Proposition1.10. Since dom( ~M ) ∈ F WC ∩ M , we know that κ ∈ j (dom( ~M )); let M κ be the κ -th model on this sequence.Fix a V -generic filter G ∗ over j ( P ), and let G := G ∗ ∩ P . For notational simplicity,we continue using j to denote the lifted map j : M [ G ] −→ N [ G ∗ ]. Recall by Lemma2.18 that j − ↾ M κ [ G ∗ ] is the transitive collapse map π M κ [ G ∗ ] of M κ [ G ∗ ].Suppose that σ ≤ ρ and that ˙ H is a j ( P )-name in N for a generic filter over π M κ [ G ∗ ] ( j ( ˙ C σ )) = ˙ C σ . We define the κ -flat function for (the pull-back of ) ˙ H to be the j ( P )-name for the function with domain j [ σ ], so that for each η < σ ,˙ r ( j ( η )) is forced to be equal to (cid:16)S π − M κ [ ˙ G ∗ ] h ˙ H ( η ) i(cid:17) ∪ { κ } .We will prove the following proposition ( ∗ ) by induction: Note that j [ σ ] ∈ N since M, j ∈ N . ( ∗ ) for any σ ≤ ρ , if ˙ H is a j ( P )-name in N for a generic filter over ˙ C σ = π M κ [ G ∗ ] ( j ( ˙ C σ )) which is guided by the collapse at κ (see Definition 5.3),then it is forced by j ( P ) that the κ -flat function for ˙ H is a condition in j ( ˙ C σ ).We first consider the case that σ ≤ ρ is limit. Suppose that we know the resultfor all η < σ . We use throughout the fact that j − equals the transitive collapsemap of M κ [ G ∗ ].Let H ∈ N [ G ∗ ] be a filter over C σ which is guided by the collapse at κ , and let r be the κ -flat function for H . Since | dom( r ) | N < j ( κ ) and j ( C σ ) is taken with < j ( κ )-supports, in order to see that r ∈ j ( C σ ), it suffices to show that for all η < σ , r ↾ j ( η ) ∈ j ( C η ) . So let η < σ be fixed. Since C σ ∼ = C η ∗ ˙ C η +1 ,σ and since H is guided by the collapse at κ over C σ , we have by Lemma 5.5 that H ↾ C η isalso guided by the collapse at κ over C η . By induction, this implies that the κ -flatcondition for H ↾ C η , namely r ↾ j ( η ), is a condition in j ( C η ). This completes theproof of ( ∗ ) in the limit case.Now suppose that σ +1 ≤ ρ and that we know that ( ∗ ) holds at σ . Let H ∈ N [ G ∗ ]be a filter over C σ +1 which is guided by the collapse at κ , and let H σ denote therestriction of H to C σ . Again appealing to Lemma 5.5, we know that H σ is guidedby the collapse at κ .Let r be the κ -flat function for H , and let ¯ r denote r ↾ j ( σ ), the κ -flat functionfor H σ . Since H σ is guided by the collapse at κ , we may apply the inductionhypothesis to conclude that ¯ r is a condition in j ( C σ ). By Proposition 5.6, since H σ is guided by the collapse at κ , we know that in N we may find a residue pair h (0 j ( P ) , ˙¯ r ) , ϕ M κ i for the pair ( M κ , j ( P ∗ )), where ˙¯ r is a j ( P )-name in N for ¯ r . Weuse p ∗ ( M κ ) to denote (0 j ( P ) , ˙¯ r ).Since ¯ r is an upper bound for π − M κ [ G ∗ ] [ H σ ] = j [ H σ ], we conclude that ¯ r forcesin j ( C σ ) over N [ G ∗ ] that S j [ H ( σ )] is unbounded in κ . Therefore, to see that r ∈ j ( C σ +1 ) (which finishes the proof of ( ∗ ) in the successor case), it suffices toshow that ¯ r (cid:13) N [ G ∗ ] j ( C σ ) (cid:16)[ j [ H ( σ )] ∪ { κ } (cid:17) ∈ j ( ˙ C ( σ )) . Since j ( ˙ C ( σ )) is a j ( P ∗ ˙ C σ )-name for CU ( j ( ˙ S σ ) , j ( ˙ S σ )), the above holds if and onlyif (¯ r, j (˙ S σ ) ) (cid:13) N [ G ∗ ] j ( C σ ∗ ˙ S σ ) (cid:16)[ j [ H ( σ )] ∪ { κ } (cid:17) ⊆ (cid:16) tr (cid:16) j ( ˙ S σ ) (cid:17) ∪ ( j ( κ ) ∩ cof( ω )) (cid:17) . By the elementarity of j , we see that(¯ r, j (˙ S σ ) ) (cid:13) N [ G ∗ ] j ( C σ ∗ ˙ S σ ) [ j [ H ( σ )] ⊆ (cid:16) tr (cid:16) j ( ˙ S σ ) (cid:17) ∪ ( j ( κ ) ∩ cof( ω )) (cid:17) . Since κ has cofinality ω after forcing with j ( C σ ∗ ˙ S σ ), it therefore suffices to showthat (¯ r, j (˙ S σ ) ) (cid:13) N [ G ∗ ] j ( C σ ∗ ˙ S σ ) ( j ( ˙ S σ ) ∩ κ ) is stationary in κ. Before continuing, we recall that R σ denotes P ∗ ˙ C σ ∗ ˙ S σ . By Proposition 3.4, weknow that( p ∗ ( M κ ) , j (˙ S σ ) ) (cid:13) Nj ( R σ ) j ( ˙ S σ ) ∩ κ = (˙ j ( ˙ S σ ) ∩ M κ )[ ˙ G j ( R σ ) ∩ M κ ] . LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 43
However, j ( R σ ) ∩ M κ = j [ R σ ] is isomorphic to R σ , and j ( ˙ S σ ) ∩ M κ = j [ ˙ S σ ]. Since˙ S σ is a nice R σ -name for a stationary subset of κ ∩ cof( ω ), j [ ˙ S σ ] is therefore a j ( R σ ) ∩ M κ -name for a stationary subset of κ ∩ cof( ω ).We now verify that the assumptions in the statement Proposition 4.4 hold in N to see that the stationarity of j [ ˙ S σ ] is preserved by the desired condition. Byassumption (4) of this section, P ∗ ˙ C σ is F WC -strongly proper and that ˙ S σ is a P ∗ ˙ C σ -name for an iteration specializing Aronszajn trees. Hence we may find some B ⊆ dom( ~M ), in M , so that B witnesses the conclusion of Corollary 3.22. ByProposition 4.4 applied in M to ~M ↾ B , we can find some B ∗ ⊆ B in M witnessingthe proposition in M . Since B ∗ ∈ F WC ∩ M , we know that κ ∈ j ( B ∗ ) and that j ( B ∗ ) witnesses that the conclusion of Proposition 4.4 holds with respect to j ( ~M )in N . Finally, applying the observations in the previous paragraph, we may nowconclude that ( p ∗ ( M κ ) , j (˙ S σ ) ) forces over N that j [ ˙ S σ ] is stationary in κ . Recallingthat p ∗ ( M κ ) = (0 j ( P ) , ˙¯ r ), we now conclude that(¯ r, j (˙ S σ ) ) (cid:13) N [ G ∗ ] j ( C σ ∗ ˙ S σ ) j ( ˙ S σ ) ∩ κ = j [ ˙ S σ ][ ˙ G j ( R σ ) ∩ M κ ] is stationary . This completes the proof that r , the κ -flat function for H , is a condition in j ( C σ )and also finishes the proof that ( ∗ ) holds.To finish the proof of Proposition 6.2, we prove by induction on k < ω that forany ~δ = h δ , . . . , δ k − i ∈ [ ρ ] k dec , the poset C ρ ( ~δ ) is F WC -completely proper. Wewill verify that Definition 5.8 holds by applying Proposition 1.11. Note that bythe previous discussion, we know that the result holds for k = 0, i.e., that ˙ C ρ is F WC -completely proper.Suppose that we know the result for k < ω . Let ~δ = h δ , . . . , δ k − , δ k i ∈ [ ρ ] k +1dec ,and ~M be suitable with respect to ~δ , and hence P ∗ ˙ C ρ ( ~δ ↾ k )-suitable. where ~δ ↾ k = h δ , . . . , δ k − i ∈ [ ρ ] k dec .As usual, let M be a κ -model containing ~M , let j : M −→ N be a weaklycompact embedding with j, M ∈ N , and let M κ := j ( ~M )( κ ). Fix an N -genericfilter G ∗ over j ( P ∗ ), and let G := G ∗ ∩ P . Finally, lift j : M [ G ] −→ N [ G ∗ ]. Weagain recall that j − ↾ M κ [ G ∗ ] = π M κ [ G ∗ ] .In N [ G ∗ ], let H ∗ be a filter over C ρ ( ~δ ) which is guided by the collapse at κ .For simplicity of notation, we write C ρ ( ~δ ↾ k ) = C δ k − ∗ ˙ D so that C ρ ( ~δ ) = C δ k ∗ (cid:16) ˙ C [ δ k ,δ k − ) ∗ ˙ D (cid:17) . The filter H ∗ adds generics J , J over C ρ ( ~δ ↾ k ) so that J and J agree on C δ k (recall that δ k < δ k − ) but are mutually generic afterwards. Since H ∗ is guided by the collapse at κ , each J i , i <
2, is also guided by the collapseat κ . Hence, our inductive assumption implies that π − M κ [ G ∗ ] [ J i ] has a sup r i in j ( C ρ ( ~δ ↾ k )). By the agreement among the J i up to stage δ k < δ k − , we know that r ↾ j ( C δ k ) = r ↾ j ( C δ k ). We let ¯ r denote the common value. Finally, we define r ∗ to be the function ¯ r ⌢ h r i ↾ j ( C ρ ( ~δ ↾ k )) : i < i . Then r ∗ is a condition in j ( C ρ ( ~δ ))which is a sup of π M κ [ G ∗ ] [ H ∗ ].This completes the inductive step and the proof of the proposition. (cid:3) No New Branches.
In this subsection, we we will show that various F WC -completely proper posets do not add branches through Aronszajn trees of interest.We will use this general result to show, in particular, that the club adding poset C ρ does not add any branches to trees ˙ T which are Aronszajn trees in an interme-diate extension obtained by forcing with C σ , for σ < ρ . This will ensure that thetree specializing iteration of length ρ above is in fact an iteration of specializing Aronszajn trees in the extension by C ρ , a conclusion which is essential in order tosee that the specializing iteration does not collapse κ .Arguments for securing that certain posets do not add new cofinal branches totrees play a crucial role in consistency results concerning the tree property, goingback to the work of Mitchell and Silver ([34]), and Magidor and Shelah ([32]).Lemma 6 of Unger [40] provides such an argument with respect to closed posetsand trees named by posets with reasonable chain condition, given constraints on thecontinuum function. Here, we prove a version of these results, in which the relevantposets (which in practice are variants of the club-adding poset) are F WC -completelyproper (and thus κ -distributive) but not κ -closed.The statement of the following Proposition involves (names) of posets, P , ˙ Q , ˙ Q ,˙ S . To relate the statement to our scenario, we suggest keeping in mind the followingassignments of the posets: Fixing ρ < ρ ∗ < κ + , consider P = Col( ω , < κ ), ˙ Q = ˙ C ρ is the ( P -name) of the first ρ steps of the club adding iteration, ˙ S = ˙ S ρ is the P ∗ ˙ C ρ -name of the first ρ steps of the iteration specializing trees, and ˙ Q = ˙ C [ ρ,ρ ∗ ) is the P ∗ ˙ C ρ -name of the segment of the final iteration from (and including stage ρ tostage ρ ∗ (i.e. ˙ Q ∗ ˙ Q = ˙ C ρ ∗ ). Proposition 6.3.
Suppose that ˙ Q is a P -name and that ˙ Q and ˙ S are ( P ∗ ˙ Q ) -names so that P ∗ ˙ Q ∗ ˙ Q is F WC -completely proper and so that P ∗ ˙ Q ∗ ˙ Q forcesthat ˙ S is κ -c.c. Let ˙ T be a ( P ∗ ˙ Q ∗ ˙ S ) -name for an Aronszajn tree on κ . Then P ∗ ˙ Q ∗ ( ˙ Q × ˙ S ) forces that ˙ T is an Aronszajn tree. That is to say, forcing with ˙ Q after P ∗ ˙ Q ∗ ˙ S does not add branches to ˙ T . Toshow this, we will follow the standard approach and show that if ˙ Q were to addsuch a branch, then we can find some model in which a level of the tree has toomany nodes.For the rest of this subsection, we suppose for a contradiction that ˙ b is ( P ∗ ˙ Q ∗ ( ˙ Q × ˙ S ))-name for a branch through ˙ T , where ˙ T is a ( P ∗ ˙ Q ∗ ˙ S )-namefor an Aronszajn tree on κ . In the context of working with the forcing R ∗ := P ∗ ˙ Q ∗ ( ˙ Q × ˙ S ), for which a typical generic looks like G ∗ Q ∗ ( Q L × Q R × F ), wewill use ˙ b L to denote the ( P ∗ ˙ Q ∗ ( ˙ Q × ˙ S ))-name for ˙ b [ ˙ G ∗ ˙ Q ∗ ( ˙ Q L × ˙ F )], i.e., theinterpretation of ˙ b using the left generic filter added by ˙ Q . ˙ b R is defined similarly.The next lemma will be used as a successor step in obtaining a tree of conditionsforcing incompatible information about a branch. Lemma 6.4. (Under the assumptions of Proposition 6.3) P forces that for each ˙ Q -name ˙ d for a condition in ˙ Q , there is a dense, open set of c in ˙ Q satisfyingthe following property: there exist names ˙ d L , ˙ d R for conditions in ˙ Q and an ordinal ξ < κ so that(1) c (cid:13) ˙ d Z ≥ ˙ d for each Z ∈ { L, R } ;(2) h c, ˙ d L , ˙ d R , ˙ S i (cid:13) V [ ˙ G ]˙ Q ∗ ( ˙ Q × ˙ Q × ˙ S ) ˙ b L ( ξ ) = ˙ b R ( ξ ) .Proof. We work in V [ G ]. Fix c ∈ Q and a Q -name ˙ d for a condition in ˙ Q . Let Q be V [ G ]-generic over Q containing c , and let Q L × Q R be V [ G ∗ Q ]-genericover Q containing ( d, d ). LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 45
We first claim that 0 S forces over V [ G ∗ Q ∗ ( Q L × Q R )] that ˙ b L = ˙ b R . Thus let F be an arbitrary V [ G ∗ Q ∗ ( Q L × Q R )]-generic filter for S . Since S and Q bothlive in V [ G ∗ Q ], the product lemma implies that Q L × Q R is V [ G ∗ Q ∗ F ]-genericover Q . Since Q L and Q R are mutually V [ G ∗ Q ∗ F ]-generic filters over Q , weconclude that V [ G ∗ Q ∗ F ] = V [ G ∗ Q ∗ ( F × Q L )] ∩ V [ G ∗ Q ∗ ( F × Q R )] . Therefore, if b := b L = b R , then b is in V [ G ∗ Q ∗ F ], and therefore T is not anAronszajn tree in that model, a contradiction.We now claim that there is an ordinal ξ so that 0 S forces over V [ G ∗ Q ∗ ( Q L × Q R )]that ˙ b L ( ξ ) = ˙ b R ( ξ ). Let A ⊆ S be a maximal antichain in V [ G ∗ Q ∗ ( Q L × Q R )]consisting of conditions g ∈ S so that for some ζ g < κ , g (cid:13) V [ G ∗ Q ∗ ( Q L × Q R )] S ˙ b L ( ζ g ) = ˙ b R ( ζ g ) . Because ˙ b L and ˙ b R name branches in ˙ T , we see that for any g ∈ A and ζ ≥ ζ g , g forces that ˙ b L ( ζ ) = ˙ b R ( ζ ). Since S is still κ -c.c. after forcing to add Q L × Q R , weknow that A has size < κ in V [ G ∗ Q ∗ ( Q L × Q R )]. Therefore, letting ξ := sup g ∈ A ζ g , ξ < κ . Then ξ witnesses the claim: indeed, if f ∈ S is any condition, we may extendit to f ∗ so that f ∗ is above some g ∈ A . By the remarks above and since ξ ≥ ζ g ,we know that f ∗ (cid:13) ˙ b L ( ξ ) = ˙ b R ( ξ ), completing the proof of the second claim.Since ( c, ˙ d, ˙ d ) ∈ Q ∗ ( Q L × Q R ), we may find an extension ( c ∗ , ˙ d L , ˙ d R ) of ( c, ˙ d, ˙ d ) aswell as an ordinal ξ < κ so that ( c ∗ , ˙ d L , ˙ d R ) forces that 0 ˙ S forces that ˙ b L ( ξ ) = ˙ b R ( ξ ).Then c ∗ ≥ c is in the desired dense set. (cid:3) Next, let ~M = h M α : α ∈ B i be a sequence which is suitable with respect toall parameters of interest, and fix M ∗ ≺ H ( κ ++ ), where M ∗ has size κ , is closedunder < κ -sequences, and contains ~M as an element; note that H ( κ + ) ∈ M ∗ , a factwhich we will use below. Let M denote the transitive collapse of M ∗ , and since M is a κ -model, fix a weakly compact embedding j : M −→ N . As usual, we use M κ to denote j ( ~M )( κ ). The following claim shows that we can build the desired treeof conditions forcing incompatible information about the branch. Claim 6.5. j ( P ) forces over N that there exist sequences h ˙ c ν : ν < ω i and h ˙ d s : s ∈ <ω i so that the following properties hold:(1) for each f ∈ (2 ω ) N [ ˙ G j ( P ) ] , the sequence h ( ˙ c ν , ˙ d f ↾ ν ) : ν < ω i is increasingin j ( ˙ Q ∗ ˙ Q ) ∩ M κ [ G ∗ ] whose pointwise π M κ [ ˙ G j ( P ) ] -image is guided by thecollapse at κ (see Definition 5.3);(2) for each ν < ω , the sequences h ˙ c µ : µ < ν i and h ˙ d s : s ∈ <ν i are membersof M κ [ ˙ G j ( P ) ];(3) if s = t are in 2 ν for some ν < ω , then h ˙ c ν , ˙ d s , ˙ d t , j (˙ S ) i (cid:13) N [ ˙ G j ( P ) ] j ( ˙ Q ∗ ( ˙ Q × ˙ Q × ˙ S ) ) j (˙ b ) L ↾ κ = j (˙ b ) R ↾ κ. Proof.
The definition is by recursion. Let G ∗ be an arbitrary V -generic over j ( P ),and let G = G ∗ ∩ P . Let ( c , ˙ d ) be the trivial condition in j ( Q ∗ ˙ Q ). Supposethat ν is a limit and that for all µ < ν and all s ∈ µ , we have defined c µ and ˙ d s .Since j ( P ) is an ω -closed poset in N and M κ is closed under countable sequencesfrom N , M κ [ G ∗ ] is still closed under countable sequences in N [ G ∗ ]. By (2) forordinals below ν , we know that for each µ < ν , the sequences h c ¯ µ : ¯ µ < µ i and h ˙ d s : s ∈ <µ i are members of M κ [ G ∗ ], and so by countable closure, h c µ : µ < ν i and h ˙ d s : s ∈ <ν i are also members of M κ [ G ∗ ]. Since M κ [ G ∗ ] ≺ j ( H ( κ + ) M )[ G ∗ ],we may apply the elementarity of M κ [ G ∗ ] to find a condition c ν ∈ j ( Q ) ∩ M κ [ G ∗ ]so that c ν is a sup of the previous c µ . Similarly, for each t ∈ ν , we may find a j ( Q )-name in M κ [ G ∗ ] which is forced by c ν to be a sup of h ˙ d t ↾ µ : µ < ν i . Note thatitem (3) in the claim still holds since if s = t are in 2 ν , then there exists some µ < ν so that s ↾ µ = t ↾ µ . So h c µ , ˙ d s ↾ µ , ˙ d t ↾ µ , j (˙ S ) i forces that j (˙ b ) L ↾ κ = j (˙ b ) R ↾ κ .Hence the extension h c ν , ˙ d s , ˙ d t , j (˙ S ) i also forces this.Now for the successor step. Suppose that we have defined c ν and ˙ d s for all s ∈ ν . In order to ensure that the assumptions of Lemma 5.7 are satisfied, andthereby ensure that the sequences are guided by the collapse at κ (which in turnwill guarantee they have an upper bound), we will first define an auxiliary extension c ∗ ν ≥ c ν and for each s ∈ ν , a j ( Q )-name ˙ d ∗ s forced by c ∗ ν to extend ˙ d s . Towards thisend, let γ := f κ ( ν ), where f κ is the standard surjection added by G ∗ from ω onto κ .Let u γ be so that π M κ [ G ∗ ] ( u γ ) is the γ -th condition in π M κ [ G ∗ ] [ j ( Q ∗ Q )] = Q ∗ Q ,and write u γ as h c γ , ˙ d γ i . If c γ does not extend c ν in j ( Q ), set c ∗ ν = c ν and ˙ d ∗ s = ˙ d s .On the other hand, if c γ ≥ c ν , we set c ∗ ν = c γ . Then, given s ∈ ν , if c ∗ ν (cid:13) ˙ d γ ≥ ˙ d s ,we set ˙ d ∗ s = ˙ d γ , and otherwise we set ˙ d ∗ s = ˙ d s . Note that there is at most one s that falls into the first of these, since c ∗ ν (cid:13) n ˙ d s : s ∈ ν o is an antichain in j ( ˙ Q ).Now we move to defining c ν +1 and ˙ d t for all t ∈ ν +1 . By Lemma 6.4, for each s ∈ ν , the set D s of all c ∈ j ( Q ) for which there exist names ˙ d L and ˙ d R and anordinal ξ < j ( κ ) satisfying(i) c (cid:13) ˙ d Z ≥ ˙ d s for each Z ∈ { L, R } ; and(ii) h c, ˙ d L , ˙ d R , j (˙ S ) i (cid:13) j (˙ b ) L ( ξ ) = j (˙ b ) R ( ξ )is dense, open in j ( Q ). Moreover, D s ∈ M κ [ G ∗ ] by elementarity and since theparameters in the definition of D s are members of M κ [ G ∗ ]. By Lemma 5.9, j ( Q )is j ( κ )-distributive, and therefore there exists an extension of c ν inside T s ∈ ν D s .Since the sequence h D s : s ∈ ν i is a member of M κ [ G ∗ ], we may find some c ν +1 ≥ c ν with c ν +1 ∈ M κ [ G ∗ ] ∩ \ s ∈ ν D s . By the elementarity of M κ [ G ∗ ], for each s ∈ ν , we may find in M κ [ G ∗ ] an ordinal ξ s < j ( κ ) and j ( Q )-names ˙ d s ⌢ h i and ˙ d s ⌢ h i so that c ν +1 (cid:13) ˙ d s ⌢ h i i ≥ ˙ d s and sothat h c ν +1 , ˙ d s ⌢ h i , ˙ d s ⌢ h i , j (˙ S ) i forces that j (˙ b ) L ( ξ s ) = j (˙ b ) R ( ξ s ). However, ξ s ∈ j ( κ ) ∩ M κ = κ , and so h c ν +1 , ˙ d s ⌢ h i , ˙ d s ⌢ h i , j (˙ S ) i forces that j (˙ b ) L ↾ κ = j (˙ b ) R ↾ κ .This completes the proof. (cid:3) Now that we have proven the above claim, we can finish the proof of Proposition6.3.
Proof. (of Proposition 6.3) Let G ∗ be N -generic over j ( P ), let G := G ∗ ∩ P , and fixsequences h c ν : ν < ω i and h ˙ d s : s ∈ <ω i as in Claim 6.5. For each f ∈ (2 ω ) N [ G ∗ ] , h ( c ν , ˙ d f ↾ ν ) : ν < ω i is guided by the collapse at κ , and since j ( Q ∗ ˙ Q ) is forcedto be j ( F WC )-completely proper, we may find a condition ( c ∗ , ˙ d f ) which is a sup in j ( Q ∗ ˙ Q ) (note that c ∗ is independent of f since any two sups are = ∗ -equal). By LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 47 item (3), we know that if f = g are in (2 ω ) ∩ N [ G ∗ ], then h c ∗ , ˙ d f , ˙ d g , j (˙ S ) i forcesthat j (˙ b ) L ↾ κ = j (˙ b ) R ↾ κ .Now let Q ∗ ∗ F ∗ be N [ G ∗ ]-generic over j ( Q ∗ ˙ S ) with Q ∗ containing c ∗ . Applyingitem (3) again, we know that if f = g are in (2 ω ) ∩ N [ G ∗ ], then h d f , d g i forces in j ( Q ) that j (˙ b ) L ↾ κ = j (˙ b ) R ↾ κ . We note here that the tree of interest, namely T ∗ := j ( ˙ T )[ G ∗ ∗ Q ∗ ∗ F ∗ ], is a member of N [ G ∗ ∗ Q ∗ ∗ F ∗ ], i.e., exists prior to forcingwith j ( Q ).For each f ∈ (2 ω ) ∩ N [ G ∗ ], let d ∗ f be an extension of d f in j ( Q ) which decidesthe value of j (˙ b )( κ ), say as α f . We claim that if f = g are in (2 ω ) ∩ N [ G ∗ ], then α f = α g . Indeed, suppose for a contradiction that there were f = g with α f = α g .Then force in j ( Q ) above the condition h d ∗ f , d ∗ g i to obtain a pair ¯ Q L × ¯ Q R ofmutually generic filters for j ( Q ). Since α f = α g , the branch of T ∗ below α f isthe same as the branch of T ∗ below α g . But the branch of T ∗ below α f equals( j (˙ b )[ ¯ Q L ]) ↾ κ and the branch of T ∗ below α g equals ( j (˙ b )[ ¯ Q R ]) ↾ κ , contradictingthe fact that h d ∗ f , d ∗ g i forces that the interpretations diverge below κ .Since (2 ω ) ∩ N [ G ∗ ] has size j ( κ ) in N [ G ∗ ] and j ( Q ∗ ˙ S ) preserves j ( κ ), thisset still has size j ( κ ) in N [ G ∗ ∗ Q ∗ ∗ F ∗ ]. Thus in the model N [ G ∗ ∗ Q ∗ ∗ F ∗ ], thefunction taking f ∈ (2 ω ) ∩ N [ G ∗ ] to α f is an injection. Therefore level κ of T ∗ hassize j ( κ ) which contradicts the fact that j ( κ ) is ℵ in N [ G ∗ ∗ Q ∗ ∗ F ∗ ] and that T ∗ is an Aronszajn tree on j ( κ ). (cid:3) Putting it all Together
Up to this point in the paper, we have worked to establish a number of isolatedresults. In this section, we will now define the poset which will witness Theorem1.1. Each of the previous sections will function as a component in the inductiveverification that this poset has the desired properties.We recall that P denotes Col( ω , < κ ), the Levy collapse of the weakly compactcardinal κ . We define a P -name ˙ C κ + for a κ + -length iteration adding clubs, andwe also define a ( P ∗ ˙ C κ + )-name ˙ S κ + for an iteration which specializes Aronszajntrees. This is done in such a way that for all ρ < κ + , the ( P ∗ ˙ C κ + )-name ˙ S ρ forthe first ρ -stages of ˙ S κ + is actually a ( P ∗ ˙ C ρ )-name.More precisely, we define by recursion on ρ ≤ κ + the names ˙ C ρ and ˙ S ρ . Supposethat ρ = ρ + 1 is a successor and that ˙ C ρ and ˙ S ρ are both defined. Usingthe fixed well-order ⊳ from Notation 2.12 as a bookkeeping device, we select a( P ∗ ˙ C ρ ∗ ˙ S ρ )-name ˙ S ρ for a stationary subset of κ ∩ cof( ω ), and we define ˙ C ρ :=˙ C ρ ∗ CU ( ˙ S ρ , ˙ S ρ ); see Definition 1.5. Next, we use ⊳ to select a ( P ∗ ˙ C ρ ∗ ˙ S ρ )-name ˙ T ρ for an Aronszajn tree on κ , and we set ˙ S ρ to be the ( P ∗ ˙ C ρ )-name for˙ S ρ ∗ S ( ˙ T ρ ); see Definition 1.6.Now suppose that ρ is a limit and that we have defined the sequences h ˙ C ξ , ˙ C ( ξ ) : ξ < ρ i and h ˙ S ξ , ˙ S ( ξ ) : ξ < ρ i . We first let ˙ C ρ be the < κ -support limit of h ˙ C ξ , ˙ C ( ξ ) : ξ < ρ i . Second, we see that P ∗ ˙ C ρ forces that h ˙ S ξ , ˙ S ( ξ ) : ξ < ρ i names an iterationwith countable support: by induction, if ξ < ρ is a limit, then ˙ S ξ is the ( P ∗ ˙ C ξ )-name for the countable support limit of h ˙ S ζ , ˙ S ( ζ ) : ζ < ξ i . But ˙ C ρ is ω -closed,and consequently, the countable support limit of h ˙ S ζ , ˙ S ( ζ ) : ζ < ξ i is the same inboth the extension by P ∗ ˙ C ξ and the extension by P ∗ ˙ C ρ . In light of this, we let˙ S ρ denote the countable support limit of h ˙ S ξ , ˙ S ( ξ ) : ξ < ρ i , noting that this is an ω -closed poset in the extension by P ∗ ˙ C ρ . This completes the definitions of thenames. We may now define R ∗ := P ∗ ˙ C κ + ∗ ˙ S κ + .We begin our analysis of R ∗ with some simple remarks. First, R ∗ is ω -closed,since all the posets under consideration are (and since our iterations were takenwith supports which are at least countable). Additionally, R ∗ is κ + -c.c. Indeed, P trivially is. Furthermore, κ <κ = κ after forcing with P , and so if β < κ + , ˙ C β isforced to be a poset of size κ . Since direct limits in the iteration ˙ C κ + are taken atall stages in κ + ∩ cof( κ ), standard arguments (e.g., see [6]) show that ˙ C κ + is κ + -c.c.Finally, since for every β < κ + , ˙ S β is forced to have size κ by P ∗ ˙ C κ + , and since˙ S κ + is taken with countable supports, a standard ∆-System argument shows that˙ S κ + is κ + -c.c.We next claim that if R ∗ preserves κ , then it forces all Aronszajn trees on κ are special, that such trees exist, and that every stationary subset S ⊆ κ ∩ cof( ω )reflects on every ordinal of cofinality ω in some closed unbounded subset on κ .First, suppose that ˙ T is an R ∗ -name for an Aronszajn tree on κ . Because R ∗ is κ + -c.c., ˙ T is an ( P ∗ ˙ C γ ∗ ˙ S γ )-name for some γ < κ + , and hence names an Aronszajntree in any extension between that given by P ∗ ˙ C γ ∗ ˙ S γ and the full R ∗ -extension. Byour bookkeeping device, there is some δ ≥ γ so that ˙ S ( δ ) is forced by P ∗ ˙ C δ +1 ∗ ˙ S δ to equal ˙ S ( ˙ T ). Hence R ∗ forces that ˙ T is special. Similarly, if ˙ S is an R ∗ -namefor a stationary subset of κ ∩ cof( ω ), then there is some α < κ + so that ˙ S is a( P ∗ ˙ C α ∗ ˙ S α )-name, and hence there is some β ≥ α so that ˙ C ( β ) is forced by P ∗ ˙ C β to equal CU ( ˙ S, ˙ S β ). Thus in the extension by P ∗ ˙ C β +1 ∗ ˙ S β , ˙ S reflects almosteverywhere, and since the forcing to complete P ∗ ˙ C β +1 ∗ ˙ S β to R ∗ is ω -closed, ˙ S still reflects almost everywhere in the full R ∗ -extension.As a result of the previous discussion, we see that in order to show that R ∗ witnesses Theorem 1.1, we need to prove that R ∗ preserves κ . To achieve this weverify that (i) P forces ˙ C κ + is κ -distributive, and that (ii) P ∗ ˙ C κ + forces that ˙ S κ + is κ -c.c.To this end, we consider the following simplifications. First, concerning (i), wenote that since ˙ C κ + is forced to be κ + -c.c, it is sufficient to verify that ˙ C ρ is forcedto be κ -distributive for all ρ < κ + to show that (i) holds. We will use Lemma 5.9to verify this, by proving that for every ρ < κ + , ˙ C ρ is F WC -completely proper.Second, concerning (ii), since ˙ S κ + names a countable support iteration, any κ -sized antichain would witness that some proper initial segment ˙ S γ is not κ -c.c.Therefore, it is sufficient to verify that P ∗ ˙ C κ + forces ˙ S γ is κ -c.c for every γ < κ + .Howover, as ˙ C κ + names a κ + -poset, every ( P ∗ ˙ C κ + )-name for a subset of ˙ S γ of size κ is equivalent to a ( P ∗ ˙ C ρ )-name, for some ρ ≥ γ . Clearly, if P ∗ ˙ C ρ forces ˙ S γ failsto satisfy the κ -c.c for some γ ≤ ρ , then it forces ˙ S ρ is not κ -c.c. We conclude that(ii) follows from the assertion that for every ρ < κ + , P ∗ ˙ C ρ forces that ˙ S ρ is κ -c.c.Combining the two simplifications, it remains to prove the next claim. Claim 7.1.
The following holds for every ρ < κ + :(1) ˙ C ρ is F WC -completely proper, and(2) P ∗ ˙ C ρ forces ˙ S ρ is κ -c.c.We prove the claim by induction on ρ < κ + . Let ρ < κ + , and suppose that theclaim holds for every σ < ρ . In particular, if σ < ρ , then since P ∗ ˙ C σ forces that˙ S σ is a countable support iteration specializing trees and since (2) holds at σ , we LUB STATIONARY REFLECTION AND THE SPECIAL ARONSZAJN TREE PROPERTY 49 must have that P ∗ ˙ C σ forces that ˙ S σ is a countable support iteration specializing Aronszajn trees. Hence we see that assumptions (1)-(4) from the beginning ofsection 6 hold. Applying Proposition 6.2, it follows that ˙ C ρ is F WC -completelyproper, and moreover, so is C ρ ( ~δ ) for every finite, decreasing sequence ~δ of ordinalsbelow ρ . We use this to prove that (2) of the claim holds at ρ .We aim to apply Corollary 3.22 to the F WC -strongly proper poset P ∗ := P ∗ ˙ C ρ ,and to do so, we need to verify that for each σ < ρ , P ∗ ∗ ˙ S σ (cid:13) ˙ T σ is an Aronszjantree. This is where the doubling tail products come into play. Indeed, we considera slightly more general statement, which would allow us to use Proposition 6.3.For each decreasing sequence ~δ = h δ , . . . , δ n − i in ρ , we let ⋆ ρ ( ~δ ) be the statementthat P ∗ ˙ C ρ ( ~δ ) forces ˙ S δ n − is κ -c.c. (note that if ~δ = ∅ then the statement holdsvacuously). We prove by induction on the reverse lexicographic order < rLex on[ ρ ] <ω dec that ⋆ ρ ( ~δ ) holds. The base case of the induction, where ~δ = ∅ , trivially holds,as mentioned above. For the induction step, fix ~δ = h δ n − , . . . , δ i ∈ [ ρ ] <ω dec andsuppose that ⋆ ρ ( ~γ ) holds for every ~γ < rLex ~δ . To show that ⋆ ρ ( ~δ ) holds, we need toverify that P ∗ ˙ C ρ ( ~δ ) forces that ˙ S δ n − is κ -c.c. For this in turn, by Corollary 3.22,it is sufficient to verify that for every γ < δ n − , P ∗ ˙ C ρ ( ~δ ) ∗ ˙ S γ forces that ˙ T γ is anAronszajn tree on κ . If δ n − = 0 there is nothing to prove. Otherwise, let γ < δ n − ,and set ~γ := ~δ ⌢ h γ i . By Proposition 6.3 and the definition of ˙ C ρ ( ~γ ), it suffices toverify that P ∗ ˙ C ρ ( ~γ ) forces that ˙ S γ is κ -c.c, to conclude that P ∗ ˙ C ρ ( ~δ ) ∗ ˙ S γ forcesthat ˙ T γ is Aronszajn. However, the last is just ⋆ ρ ( ~γ ), which holds by our inductiveassumption and the fact that ~γ < rLex ~δ . This concludes the proof of Claim 7.1,which in turn finishes the proof of Theorem 1.1. References [1] Uri Abraham. On forcing without the continuum hypothesis.
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