aa r X i v : . [ m a t h . L O ] N ov ADDING A NON-REFLECTING WEAKLY COMPACT SET
BRENT CODY
Abstract.
For n ă ω , we say that the Π n -reflection principle holds at κ and write Refl n p κ q if and only if κ is a Π n -indescribable cardinal and everyΠ n -indescribable subset of κ has a Π n -indescribable proper initial segment.The Π n -reflection principle Refl n p κ q generalizes a certain stationary reflectionprinciple and implies that κ is Π n -indescribable of order ω . We define a forcingwhich shows that the converse of this implication can be false in the case n “ κ is p α ` q -weakly compact where α ă κ ` , thenthere is a forcing extension in which there is a weakly compact set W Ď κ having no weakly compact proper initial segment, the class of weakly compactcardinals is preserved and κ remains p α ` q -weakly compact. We also provea resurrection result for the Π -reflection principle. Contents
1. Introduction 12. Preliminaries 33. Basic consequences of the Π n -reflection principle 74. Weak compactness and forcing 85. Adding a non-reflecting weakly compact set 106. Resurrecting the weakly compact reflection principle 157. Questions 17References 181. Introduction
For a regular cardinal κ , we say that a set S Ď κ is Π n -indescribable if forevery A Ď V κ and every Π n -formula ϕ , whenever p V κ , P , A q |ù ϕ there exists α P S such that p V α , P , A X V α q |ù ϕ . Levy proved that if κ is Π n -indescribable then thecollection Π n p κ q “ t X Ď κ : X is not Π n -indescribable u is a normal proper idealon κ . The work of Hellsten as well as the recent work of Bagaria-Magidor-Sakai hasshown that various results involving the nonstationary ideal can be extended to theΠ n -indescribable ideal Π n p κ q . For example, the notion of α -Mahloness of a cardinal κ where α ď κ ` can be generalized to that of α -Π n -indescribability where α ď κ ` (see Definition 3 below). Hellsten proved [Hel06] that the Π n -indescribability idealΠ n p κ q is not κ ` -saturated if κ is κ ` -Π n -indescribable. This is analogous to aresult of Baumgartner, Taylor and Wagon [BTW77] stating that NS κ æ REG is not
Date : October 16, 2018.2010
Mathematics Subject Classification.
Primary 03E35; Secondary 03E55.The author is grateful for support from the Virginia Commonwealth University PresidentialResearch Quest Fund. κ ` -saturated if κ is κ ` -Mahlo. Extending work of Jech and Woodin [JW85] onthe nonstationary ideal, Hellsten proved [Hel10] that it is consistent relative to theexistence of a measurable cardinal κ of Mitchell order α ă κ ` that there is an α -Π -indescribable cardinal κ such that the Π -indescribable ideal Π p κ q is κ ` -saturated.Recall that for a regular cardinal κ , we say that the stationary reflection principleRefl p κ q holds if and only if every stationary subset of κ has a stationary properinitial segment. Bagaria-Magidor-Sakai [BMS15] generalized Jensen’s result [Jen72]which states that in L , the weakly compact cardinals are precisely the regularcardinals at which the stationary reflection principle holds, by providing a similarcharacterization of the Π n -indescribable cardinals in L . Additionally, extendingSolovay’s theorem on splitting stationary sets, Hellsten [Hel10, Theorem 2] hasshown (for a proof see [For10, Proposition 6.4]) that the Π n -indescribable ideal ona Π n -indescribable cardinal κ is nowhere κ -saturated. It is important to point outthat for many reasons, these generalizations require substantial effort: for example,in many situations, one must deal with the fact that a non-weakly compact set (i.e.a non-Π -indescribable set) can become weakly compact in a forcing extension, whereas a nonstationary set cannot be forced to become stationary.In this article we continue this line of research by generalizing the stationaryreflection principle Refl p κ q as follows. The stationary reflection principle Refl p κ q isformulated by referencing the nonstationary ideals NS γ for γ ď κ with cf p γ q ą ω ,and one may formulate new reflection principles by replacing these ideals withothers. Let Refl p κ q be the statement asserting that κ is inaccessible and for everystationary S Ď κ there is an inaccessible cardinal γ ă κ such that S X γ is astationary subset of γ . Since a set S Ď γ is Π -indescribable if and only if γ is inaccessible and S is stationary [Hel06], we obtain a direct generalization ofRefl p κ q as follows. For n ă ω , we say that the Π n -reflection principle holds at κ and write Refl n p κ q if and only if κ is a Π n -indescribable cardinal and for every Π n -indescribable set S Ď κ there exists a γ ă κ such that S X γ is Π n -indescribable. Forthe case in which n “
1, we say that the weakly compact reflection principle holdsat κ and write Refl wc p κ q if and only if κ is a weakly compact cardinal and everyweakly compact subset of κ has a weakly compact initial segment. Since Refl n p κ q holds whenever κ is Π n ` -indescribable (see the beginning of Section 2 below),and since Refl n p κ q implies that there are many Π n -indescribable cardinals below κ , it follows that the consistency strength of “ D κ Refl n p κ q ” is strictly greater thanthe existence of a Π n -indescribable cardinal but not greater than the existence of aΠ n ` -indescribable cardinal. The exact consistency strength of “ D κ Refl n p κ q ” is notknown, but the work of Mekler and Shelah [MS89] leads to a conjecture in Section7 below. Additionally, one may derive the consistency of “ D κ @ n ă ω Refl n p κ q ” fromthe existence of a Π n -indescribable cardinal of order ω (see Definition 2 below).In Lemma 7 below, we observe that Refl n p κ q implies that κ is Π n -indescribableof order ω . The main result of this article addresses the question: if κ is Π n -indescribable of high order γ ă κ ` , does this entail that Refl n p κ q holds? In thecase where n “ Indeed, Kunen showed [Kun78] that a non-weakly compact cardinal can become supercompactin a forcing extension by showing that the forcing to add a Cohen subset to κ is equivalent to atwo-step iteration S ˚ T in which the first step S is a forcing which adds homogeneous Suslin tree T , and then in the second step one forces with this tree. In fact, this question can also be answered by using the result of Bagaria-Magidor-Sakai[BMS15] mentioned above. Although the characterization of Π n ` -indescribable cardinals in L DDING A NON-REFLECTING WEAKLY COMPACT SET 3
Suppose κ is a weakly compact cardinal. We say that S Ď κ is a non-reflectingweakly compact subset of κ if and only if S is a weakly compact set having no weaklycompact proper initial segment; similarly, one can define the notion of non-reflecting Π n -indescribable subset of κ . The following theorem establishes the consistency ofthe failure of the weakly compact reflection principle at κ with κ being weaklycompact of any order γ ă κ ` . Theorem 1.
Suppose that
Refl wc p κ q holds, κ is weakly compact of order p α ` q where α ă κ ` and GCH holds. Then there is a cofinality-preserving forcingextension in which there is a non-reflecting weakly compact subset of κ , the class ofweakly compact cardinals is preserved and κ remains p α ` q -weakly compact. Notice that the assumption Refl wc p κ q allows us to avoid trivialities, since ifRefl wc p κ q fails then κ already has a non-reflecting weakly compact subset. Theforcing used in the proof of Theorem 1, adds a non-reflecting weakly compact sub-set of κ . Notice that many forcings throughout the literature also add non-reflectingweakly compact subsets to a weakly compact cardinal κ , such as an Easton-supportiteration to turn κ into the least weakly compact cardinal by adding non-reflectingstationary subsets to each weakly compact γ ă κ ; the new features of the forcingdeveloped here is that it preserves the class of weakly compact cardinals and alsopreserves the fact that κ is p α ` q -weakly compact. Our proof of Theorem 1, reliesheavily on the simple characterization of weak compactness in terms of elementaryembeddings, which is reviewed in Lemma 9, below. Although one can characterizeΠ n -indescribability in terms of elementary embeddings [Hau91], it is not knownwhether or not the techniques used in our proof of Theorem 1 generalize.Another important feature of the proof of Theorem 1, is that we will be concernedwith showing that various forcings do not create new instances of weak compactness,a feature which is not present when dealing with forcing to add a non-reflectingstationary set. Our main tool in this regard, will be Hamkins’ work showing thatextensions with the approximation and cover properties have no new large cardinals[Ham03]; see Lemma 18 below.In Section 6, we prove that if κ is a measurable cardinal then there is a forcingextension in which there is a weakly compact subset of E Ď κ with no weaklycompact proper initial segment and in which the Π -reflection principle can beresurrected by shooting a 1-club through κ z E .Many questions about the weakly compact and Π n -indescribable reflection prin-ciples remain open, some of which are discussed in Section 7 below.2. Preliminaries
Since there is a Π n ` -sentence ϕ such that for any γ a set S Ď γ is Π n -indescribable if and only if p V γ , P , S q |ù ϕ [Kan03, Corollary 6.9], it follows thatRefl n p κ q holds if κ is Π n ` -indescribable. Therefore “ D κ Refl n p κ q ” is consistencerelative to the existence of a Π n ` -indescribable cardinal. given by Bagaria-Magidor-Sakai does not resemble the reflection principles Refl n p κ q consideredhere, Sakai informed the author, after the current article was written, that in L , the Bagaria-Magidor-Sakai characterization is equivalent to Refl n p κ q . Bagaria-Magidor-Sakai showed that in L , a cardinal κ is Π n ` -indescribable if and only if Refl n p κ q holds. Thus, in L , Refl n p κ q fails atthe least greatly Π n -indescribable cardinal. BRENT CODY
Assuming κ is a Π n -indescribable cardinal, the Π n -idealΠ n p κ q “ t X Ď κ : X is not Π n -indescribable u is a normal ideal on κ [L´ev71]. We will denote the corresponding collection ofpositive sets by Π n p κ q ` “ t X Ď κ : X is Π n -indescribable u and the dual filter is written asΠ n p κ q ˚ “ t X Ď κ : κ z X is not Π n -indescribable u . In the case where n “
0, we obtain the nonstationary ideal NS κ “ Π p κ q . Thus,in this case, the filter Π p κ q ˚ is generated by the collection of club subsets of κ and Π p κ q ` is the collection of stationary subsets of κ . Furthermore, a cardinal κ is Π -indescribable if and only if κ is inaccessible [Kan03, Proposition 6.3]. In thecase where n “
1, we sometimes refer to Π p κ q as the weakly compact ideal . For0 ă n ă ω one can characterize a natural collection of subsets of κ which generatesthe filter Π n p κ q ˚ as follows. We say that a set C Ď κ is 0 -club if and only if C isclub. Furthermore, for n ą C is said to be n -club if C P Π n ´ p κ q ` and whenever C X α P Π n ´ p α q ` we have α P C . Note that, C is 1-club if and only if C isa stationary subset of κ which contains all of its inaccessible stationary reflectionpoints. Under the assumption that κ is a Π n -indescribable cardinal, it follows fromthe work of Sun [Sun93] and Hellsten [Hel06] that S P Π n p κ q ` if and only if forevery n -club C Ď κ we have S X C ‰ H . Thus, if κ is a Π n -indescribable cardinal,a set X Ď κ is in the filter Π n p κ q ˚ if and only if X contains an n -club.For n ă ω , Hellsten defined an operation Tr n on P p κ q analogous to Mahlo’soperation by Tr n p X q “ t α ă κ : X X α P Π n p α q ` u , as well as a transitive wellfounded partial order ă n on Π n p κ q ` analogous to Jech’sordering on stationary sets; for S, T P Π n p κ q ` we have S ă n T ðñ T z Tr n p S q P Π n p κ q . Hellsten also defined the order of a Π n -indescribable set S P Π n p κ q ` to be its rankunder ă n , o n p S q “ sup t o n p X q ` X ă n S and X P Π n p κ q ` u ;and the order of a Π n -indescribable cardinal κ to be the height of ă n , o n p κ q “ sup t o n p S q ` S P Π n p κ q ` u . Generalizing the notion of great Mahloness from [BTW77], Hellsten providedthe following (for details see [Hel06, Definition 2 and Lemma 10]).
Definition 1 ([Hel06]) . A cardinal κ is greatly Π n -indescribable if and only if o n p κ q “ κ ` .We will find it useful to consider orders of Π n -indescribability less than κ ` ,generalizing the Mahlo-hierarchy. Definition 2.
Given cardinal κ and an ordinal 0 ă γ ă κ ` , we say that κ isΠ n -indescribable of order γ if and only if o n p κ q ě γ . Hellsten denoted this operation by M n , but here we use Tr n to avoid notational confusionsince we use M to denote a κ -model. DDING A NON-REFLECTING WEAKLY COMPACT SET 5
In what follows we find it easier to work with another, slightly more concrete,characterization of the order of a Π n -indescribable cardinal, which we describe inLemma 5 below. Working towards this characterization, let us establish a few basicproperties of the operation Tr n , which are implicit in [Hel06]. Hellsten provedthat Tr n is a generalized Mahlo operation for the Π n -ideal over a regular cardinal κ . Among other things, this implies [Hel06, Theorem 2] that the Π n -indescribablefilter is closed under the operation Tr n ´ , which is analogous to the fact that theclub filter is closed under the map taking a set X P P p κ q to its set of limit points X . As an easy consequence, we obtain the following. Lemma 2.
Suppose κ is a Π n -indescribable cardinal. If Z P Π n p κ q then Tr n p Z q P Π n p κ q .Proof. Suppose Z P Π n p κ q and Tr n p Z q R Π n p κ q . Let C Ď κ be an n -club with C X Z “ H . Since the filter Π n p κ q ˚ is closed under Tr n ´ , it follows thatTr n ´ p C q “ t α ă κ : α is Π n ´ -indescribable and C X α P Π n ´ p α q ` u contains an n -club subset of κ and since C is n -club we have Tr n ´ p C q Ď C . SinceTr n p Z q P Π n p κ q ` we may choose η P Tr n p Z q X Tr n ´ p C q . Notice that Z X η P Π n p η q ` and C X η is an n -club subset of η disjoint from Z X η , a contradiction. (cid:3) Following Hellsten’s notation, elements of the boolean algebra P p κ q{ Π n p κ q arewritten as r S s n where S P Π n p κ q ` and r S s n is the equivalence class of S modulothe ideal Π n p κ q . We let ď n denote the usual ordering on P p κ q{ Π n p κ q . Hellstenproved [Hel06, Lemma 6] that Tr n is well defined as a map Tr n : P p κ q{ Π n p κ q Ñ P p κ q{ Π n p κ q where Tr n pr S s n q “ r Tr n p S qs n , and we observe that this follows directlyfrom Lemma 2. Corollary 3.
Suppose κ is a Π n -indescribable cardinal and S, T P P p κ q . If S ď n T then Tr n p S q ď n Tr n p T q . Thus, Tr n is well defined as a map P p κ q{ Π n p κ q Ñ P p κ q{ Π n p κ q . It is well known (see [BTW77] and [Hel06, Lemma 3]), that if I is a normalideal on κ then the diagonal intersection of a collection of ď κ -many subsets of κ is independent of the indexing used, modulo I , and this allows one to calculate κ ` iterates of Tr n as an operation on P p κ q{ Π n p κ q . For the readers convenience let usbriefly recall how this is done, in the context of the Π n -indescribable ideal. Given acollection tr A i s n : i ă β u Ď P p κ q{ Π n p κ q where β ă κ ` , let f : β Ñ κ be injectiveand define g : κ Ñ P p κ q by g p α q “ κ if α R f r β s and g p α q “ A i if f p i q “ α . Wedefine △ tr A i s n : i ă β u “ r △ g s n where, as usual, △ g “ t ξ ă κ : p@ i ă ξ q ξ P g p i qu . For A P Π n p κ q ` , we define asequence x Tr αn pr A s n q : α ă κ ` y in the boolean algebra P p κ q{ Π n p κ q :Tr n pr A s n q “ r A s n Tr α ` n pr A s n q “ Tr n p Tr αn pr A s n qq , Tr γn pr A s n q “ △ t Tr αn pr A s n q : α ă γ u if γ ă κ ` is a limit.Similarly, one may iterate Tr n as an operation on P p κ q , κ -many times. Notice thatfor limits γ ă κ , if we let A i be a representative of the equivalence class Tr in pr κ s n q for i ă γ , then Tr γn pr A s n q “ r Ş t A i : i ă γ us n . Observe that Tr n p κ q is the set BRENT CODY of Π n -indescribable cardinals below κ , and Tr n p κ q is the set of Π n -indescribablecardinals µ ă κ such that Tr n p κ q X µ is a Π n -indescribable subset of µ . Definition 3.
For γ ď κ ` , we say that κ is γ - Π n -indescribable if and only ifTr αn pr κ sq n ą α ă γ .Thus κ is 1-Π n -indescribable if and only if κ is Π n -indescribable, κ is 2-Π n -indescribable if and only if the collection of Π n -indescribable cardinals less than κ is a Π n -indescribable subset of κ , etc. Lemma 4.
For α ă κ we have Tr αn pr κ s n q “ rt γ ă κ : γ is α - Π n -indescribable us n . We now show that these two concepts of orders of indescribability are equivalent.
Lemma 5.
Suppose κ is a Π n -indescribable cardinal and γ ď κ ` . Then κ is Π n -indescribable with order γ if and only if κ is γ - Π n -indescribable. In other words, o n p κ q ě γ if and only if for all α ă γ we have Tr αn pr κ sq ą in the boolean algebra P p κ q{ Π n p κ q .Proof. The reverse direction follows easily from the fact that if S P Π n p κ q ` then S ă n Tr n p S q .For the forward direction, suppose κ is Π n -indescribable with order γ ă κ ` .Then for each δ ă γ there exists a Π n -indescribable set X P Π n p κ q ` such that o n p X q “ δ . This implies that there is a strictly increasing chain x X α : α ă δ y inthe poset p Π n p κ q ` , ă n q below X ; thus α ă β ă δ implies X β z Tr n p X α q P Π n p κ q .We must show that in the boolean algebra P p κ q{ Π n p κ q , we have Tr αn pr κ s n q ą α ă δ .For each α ă δ , let A α be a representative of the equivalence class Tr αn pr κ s n q .We will use induction to prove that r X α s n ď n r A α s n for all α ă δ . It will sufficeto inductively construct a sequence ~Z “ x Z α : α ă δ y of sets in Π n p κ q suchthat for each α ă δ we have X α Ď A α Y Z α . Clearly r X s n ď n r κ s n and we let Z “ X z κ “ H .If α “ β ` X β ` ď n Tr n p X β q and by the in-ductive hypothesis r X β s n ď n r A β s n . By Corollary 3, we have Tr n pr X β sq ď n Tr n pr A β s n q “ Tr β ` n pr κ sq and thus r X β ` s n ď n Tr β ` n pr κ s n q “ r A β ` s n . We let Z β ` “ X β ` z A β ` P Π n p κ q . Notice that X β ` Ď A β ` Y Z β ` .At limit stages α ă κ ` we inductively build a sequence ~z “ x z β : β ď α y of setsin Π n p κ q such that for all β ď α , X α Ď A β Y z β . Let z “ H . If β “ ξ ` ă α is a successor, by assumption we have X α ď n Tr n p X ξ q and inductively we have r X ξ s n ď n r A ξ s n . By Corollary 3, Tr n pr X ξ s n q ď n Tr n pr A ξ s n q “ r A ξ ` s n , whichimplies that r X α s n ď n r A ξ ` s n . Hence there exists z ξ ` P Π n p κ q such that X α Ď A ξ ` Y z ξ ` . If β ď α is a limit, fix an injection f : β Ñ κ and define two functions g A , g z : κ Ñ P p κ q by g A p ξ q “ κ if ξ R f r β s A i if f p i q “ ξ and g z p ξ q “ κ if ξ R f r β s z i if f p i q “ ξ DDING A NON-REFLECTING WEAKLY COMPACT SET 7
Now let z β “ ▽ t z ξ : ξ ă β u “ ▽ g z and notice that z β P Π n p κ q by normality. Bythe inductive hypothesis, X α Ď A i Y z i for all i ă β , and hence X α Ď p △ g A q Y p ▽ g z q “ n A β Y z β . Thus, we may let z β P Π n p κ q be such that X α Ď A β Y z β . This defines the sequence ~z “ x z β : β ď α y . We let Z α “ z α and note that X α Ď A α Y Z α holds byconstruction. (cid:3) Basic consequences of the Π n -reflection principle Lemma 6.
Suppose
Refl n p κ q holds and S P Π n p κ q ` . Then Tr n p S q “ t α ă κ : S X α P Π n p α q ` u is a Π n -indescribable subset of κ .Proof. Fix an n -club C Ď κ . We will show that Tr n p S q X C ‰ H . Since κ is aΠ n -indescribable cardinal, the filter Π n p κ q ˚ is normal and hence the intersectionof fewer than κ many n -club subsets of κ contains an n -club. Thus, S X C is aΠ n -indescribable subset of κ . Since Refl n p κ q holds, there is an α ă κ such that S X C X α P Π n p α q ` . This implies that S X α P Π n p α q ` . Furthermore, C X α is a Π n ´ -indescribable subset of α and thus α P C because C is n -club. Thus α P Tr n p S q X C . (cid:3) The next lemma establishes that Refl n p κ q implies that κ is ω -Π n -indescribable. Lemma 7.
Suppose
Refl n p κ q holds. Then the set Ind n X κ “ t α ă κ : α is Π n -indescribable u is a Π n -indescribable subset of κ . Thus, by Lemma 6, for each m ă ω we have Tr mn p Ind n X κ q P Π n p κ q ` ; in other words, κ is ω - Π n -indescribable.Proof. Suppose Refl n p κ q holds. Let C Ď κ be an n -club. We will prove that p Ind n X κ q X C ‰ H . Since C P Π n p κ q ˚ Ď Π n p κ q ` is Π n -indescribable, Refl n p κ q implies that there is a µ ă κ such that C X µ P Π n p µ q ` , and hence µ P Ind n X κ .Since C X µ is, in particular, a Π n ´ -indescribable subset of µ , and since C is an n -club we have µ P C . Thus µ P p
Ind n X κ q X C ‰ H . (cid:3) We now restrict our attention to weak compactness. Below we will write Tr wc todenote the operation Tr ; that is, if X P P p κ q then Tr wc p X q “ t α ă κ : X X α P Π p α q ` u . In contrast to Lemma 10 below, under Refl wc p κ q we obtain the following. Corollary 8.
Suppose
Refl wc p κ q holds. If S P Π p κ q ` then for every A Ď κ thereis a κ -model M with κ, A, S P M and an elementary embedding j : M Ñ N withcritical point κ such that N |ù S P p Π p κ q ` q N .Proof. Suppose Refl wc p κ q holds and X P Π p κ q ` . By Lemma 6, we have Tr wc p S q P Π p κ q ` . Fix A Ď κ . There exists a κ -model M with κ, A, S, Tr wc p S q P M and anelementary embedding j : M Ñ N with critical point κ such that κ P j p Tr wc p S qq ;in other words, in N , the set S “ j p S q X κ is a weakly compact subset of κ . (cid:3) BRENT CODY Weak compactness and forcing
In what follows the elementary embedding characterization of weak compactness,i.e. Π -indescribability, is an essential ingredient, and here we review the requiredproperties of this characterization. We say that M is a κ -model if and only if M isa transitive model of ZFC ´ such that | M | “ κ , κ P M and M ă κ X V Ď M .The following lemma, essentially due to Baumgartner [Bau77], is well known.For details see one may consult [Hel03, Theorem 4.13], [Kan03, Theorem 6.4] or[Cum10, Theorem 16.1]. Lemma 9.
Given a set S P P p κ q , the following are equivalent. (1) S is Π -indescribable. (2) For every κ -model M with κ, S P M there is an elementary embedding j : M Ñ N with critical point κ such that κ P j p S q and N is also a κ -model. (3) For any κ -model M with S P M , then there is a κ -complete nonprincipal M -ultrafilter U on κ such that the ultrapower j U : M Ñ N has criticalpoint κ and satisfies κ P j U p S q . (4) For all A Ď κ there is a κ -model M with κ, A, S P M and there is anelementary embedding j : M Ñ N with critical point κ such that κ P j p S q and N is also a κ -model. Remark 1. If j : M Ñ N is the ultrapower by an M -ultrafilter on κ and M ă κ X V Ď M , then N ă κ X V Ď N . Remark 2.
Below we will need to prove that if a set S Ď κ is weakly compact inthe ground model V , then it remains so in a certain forcing extension V r G s , saywhere G is p V, P q -generic for some poset P . To do this we will verify that (4) holdsin V r G s . Specifically, we will fix A P P p κ q V r G s and argue that there is a P -name A P H Vκ ` with A G “ A . To find a κ -model and an embedding in V r G s , we proceedas follows. Applying the weak compactness of S in V , let M be a κ -model with κ, S, A . . . P M and let j : M Ñ N be an elementary embedding with critical point κ such that κ P j p S q . Then we will argue that the embedding j can be extended to j : M r G s Ñ N r j p G qs . Clearly, A P M r G s and it will follow that M r G s is a κ -modelin V r G s .The next lemma will be useful for construction of master conditions later on. Lemma 10.
Suppose S P Π p κ q ` . Then for every A Ď κ there is a κ -model M with κ, A, S P M and an elementary embedding j : M Ñ N with critical point κ ,where N is also a κ -model, such that κ P j p S q and N |ù S R p Π p κ q ` q N .Proof. Suppose S Ď κ is weakly compact and fix A Ď κ . By Lemma 9, there is a κ -model M with κ, A, S P M and there is an elementary embedding j : M Ñ N with critical point κ where κ P j p S q and N is a κ -model. Choose such an embeddingwith j p κ q as small as possible. We will show that S is not weakly compact in N .Suppose S is weakly compact in N . Since N ă H κ ` , it follows that N does notsatisfy the power set axiom, however since V κ Ď M , it follows from the elementarityof j that N believes that there is a set Y such that X P Y if and only if X Ď κ . Welet P p κ q N denote this set Y . Working in N , by Lemma 9, there is a κ -model ¯ M with κ, A, S P ¯ M and there is an N - κ -complete nonprincipal ¯ M -ultrafilter F Ď P p κ q N such that the ultrapower embedding i N : ¯ M Ñ p ¯ M q κ { F “ tr h s F : h P ¯ M κ X ¯ M u has DDING A NON-REFLECTING WEAKLY COMPACT SET 9 critical point κ and satisfies κ P i N p S q . Since j p κ q is inaccessible in N , it followsthat i N p κ q ă j p κ q . Since F P N is N - κ -complete and N ă κ X V Ď N , it follows that F is κ -complete. Since | ¯ M | N “ κ we have | ¯ M | V “ κ . Thus in V , F is a κ -complete¯ M -ultrafilter and the ultrapower i : ¯ M Ñ p ¯ M q κ { F is computed the same in V asit is in N ; that is, i “ i N . Hence i p κ q ă j p κ q , which contradicts the minimality of j p κ q . (cid:3) In the proof of our main theorem we will use the following standard lemmasto argue that various instances of weak compactness are preserved in a particularforcing extension. For further discussion of these methods see [Cum10].
Lemma 11.
Suppose j : M Ñ N is an elementary embedding with critical point κ where M and N are κ -models and P P M is some forcing notion. Suppose G Ď P is a filter generic over M and H Ď j p P q is a filter generic over N . Then j extendsto j : M r G s Ñ N r H s if and only if j r G s Ď H . Using terminology from [Cum10], we say that a forcing P is κ -strategically closed if and only if Player II has a winning strategy in the game G κ p P q of length κ where Player II plays at even stages. For more information on strategic closure, see[Cum10, Section 5]. Lemma 12.
Suppose M is a κ -model in V , so in particular M ă κ X V Ď M , andsuppose P P M is a forcing notion with V |ù “ P is κ -strategically closed”. Thenthere is a filter G P V generic for P over M . Lemma 13.
Suppose that M is a κ -model in V , so that in particular M ă κ X V Ď M , P P M is some forcing notion and there is a filter G P V which is generic for P over M . Then M r G s ă κ X V Ď M r G s . Lemma 14.
Suppose that M is a κ -model in V , so in particular M ă κ X V Ď M , andsuppose P P M is κ -c.c. If G Ď P is generic over V , then M r G s ă κ X V r G s Ď M r G s . In addition to arguing that a certain forcing iteration preserves instances of weakcompactness, we will be concerned with showing that this iteration does not createnew instances of weak compactness. Our main tool in this regard will be Lemma18 below, which is due to Hamkins [Ham03], but first we review some more basicresults.
Lemma 15.
A cardinal κ is weakly compact after ď κ -distributive forcing if andonly if it was weakly compact in the ground model. Moreover, a set S Ď κ is weaklycompact after ď κ -distributive forcing if and only if it was weakly compact in theground model.Proof. The weak compactness of a set S Ď κ is witnessed by sets whose transitiveclosure has size at most κ , and since such objects are unaffected by ď κ -distributiveforcing, the result follows immediately. (cid:3) Lemma 16.
Suppose κ is weakly compact and P is a forcing of size less than κ .Then κ is weakly compact after forcing with P if and only if it was weakly compactin the ground model. Lemma 17. If κ is weakly compact after κ -strategically closed forcing, then it wasweakly compact in the ground model. Proof.
Suppose P is κ -strategically closed and G Ď P is generic over V . Any κ -tree T in V remains a κ -tree in V r G s , and thus T has a cofinal branch in V r G s . Usinga name for this branch, and the strategic closure of P , one may construct a cofinalbranch through T in V . (cid:3) Let us recall two definitions from [Ham03]. A pair of transitive classes M Ď N satisfies the δ -approximation property if whenever A Ď M is a set in N and A X a P M for any a P M of size less than δ in M , then A P M . The pair M Ď N satisfiesthe δ -cover property if for every set A in N with A Ď M and | A | N ă δ , there is a set B P M with A Ď B and | B | M ă δ . One can easily see that many Easton-supportiterations P of length greater than a Mahlo cardinal δ satisfy the δ -approximationand cover properties by factoring the iteration as P – Q ˚ R where Q is δ -c.c. and R is forced by Q to be ă δ -strategically closed. Lemma 18 (Hamkins, [Ham03]) . Suppose that κ is a weakly compact cardinal, S P P p κ q V and V Ď V satisfies the δ -approximation and cover properties for some δ ă κ . If S is a weakly compact subset of κ in V then S is a weakly compact subsetof κ in V .Proof. Suppose S P P p κ q V is a weakly compact subset of κ in V . Fix A P P p κ q V .By [Ham03, Lemma 16], there is a transitive model M P V of some large fixedfinite fragment ZFC ˚ of ZFC with | M | V “ κ such that κ, A, S P M , the model M is closed under ă κ -sequences from V and M “ M X V P V is a transitive modelof the finite fragment ZFC ˚ with | M | V “ κ . Since S is weakly compact in V , itfollows that there is an elementary embedding j : M Ñ N where N ă κ X V Ď N and κ P j p S q . Since this embedding satisfies the hypotheses of the main theoremfrom [Ham03], it follows that j æ M : M Ñ N is an elementary embedding in V with critical point κ . Since A, S P M and κ P p j æ M qp S q we see that S is a weaklycompact subset of κ in V . (cid:3) Adding a non-reflecting weakly compact set
In Lemma 7 above we proved that Refl n p κ q implies that κ is ω -Π n -indescribable.Taking n “
1, this shows that if the weakly compact reflection principle holds at κ then κ is ω -weakly compact. We now show that the converse is consistently false.Let WC denote the class of weakly compact cardinals and let WC κ “ WC X κ .Indeed we will prove that if κ is p α ` q -weakly compact, then there is an Easton-support forcing iteration P κ ` of length κ ` V P κ ` we have (1) thereis a non-reflecting weakly compact subset of κ , (2) WC Vκ “ WC V P κ ` κ and (3) κ remains p α ` q -weakly compact.For a cardinal γ and a cofinal subset W Ď γ , we define a forcing notion Q p γ, W q as follows. Let p be a condition in Q p γ, W q if and only if(1) p is a function with dom p p q ă γ and range p p q Ď η ď dom p p q is weakly compact then t α ă η : p p α q “ u is not a weaklycompact subset of η and(3) supp p p q Ď W .For p, q P Q p γ, W q let p ď q if and only if p Ě q . Lemma 19.
Assuming that the collection of weakly compact limit points of W iscofinal in γ , every condition q P Q p γ, W q can be extended to a condition p ď q suchthat dom p p q is a weakly compact cardinal and supp p p q is cofinal in dom p p q . DDING A NON-REFLECTING WEAKLY COMPACT SET 11
Proof.
Let δ be the least element of WC γ X Lim p W q greater than dom p q q . Onecan use the usual reflection arguments to show that WC δ X W is not a weaklycompact subset of δ (if it were then δ would not be the least such cardinal byelementarity). Define p : δ Ñ p æ r dom p q q , δ q be the characteristicfunction of WC δ X W X r dom p q q , δ q and p æ dom p q q “ q . (cid:3) The proof of the next lemma is very similar to that of the analogous fact aboutthe forcing to add a non-reflecting stationary set of cofinality ω ordinals in a regularcardinal [Cum10, Example 6.5]. Lemma 20.
Suppose µ ď γ is the least weakly compact cardinal such that W X µ is a weakly compact subset of µ . Then Q p γ, W q is ă µ -closed and γ -strategicallyclosed, but not ď µ -closed.Proof. It is easy to see that Q p γ, W q is ă µ -closed. Suppose δ ă µ and x p i : i ă δ y is a decreasing sequence of conditions in Q p γ, W q . Let η “ sup t dom p p i q : i ă δ u .If η is not weakly compact then Ť t p i : i ă δ u P Q p γ, W q is a lower bound of thesequence. On the other hand, if η is weakly compact, then η “ δ ă µ , whichimplies that W X η is not a weakly compact subset of η . It follows that the supportsupp p Ť t p i : i ă δ uq is a subset of W and hence is not a weakly compact subset of η . Thus Ť t p i : i ă δ u is a condition in Q p γ, W q .It is also easy to see that Q p γ, W q is not ď µ -closed. For each i ă µ let p i bethe characteristic function of W X i . Then x p i : i ă µ y is a decreasing sequence ofconditions in Q p γ, W q with no lower bound.To prove that Q p γ, W q is γ -strategically closed we must argue that Player IIhas a winning strategy in the game G γ p Q p γ, W qq . The game begins with Player IIplaying p “ H . As the game proceeds, Player II may use the following strategy.At an even stage α , Player II calculates γ α “ sup t dom p p i q : i ă α u and thendefines p α by setting dom p p α q “ γ α ` p α æ γ α “ Ť i ă α p i and p α p γ α q “
0. Tocheck that this strategy succeeds, we must verify that for all limit stages η ă γ thestrategy produces a condition p η . If η ă γ is not weakly compact then p η is clearlya condition. If η ă γ is weakly compact one of two things must occur. Either γ η “ sup t γ i : i ă η u is equal to η or γ η ą η . If γ η ą η then γ η is singular, inwhich case p η “ Ť i ă η p i Y tp γ η , qu is a condition. Otherwise, if γ η “ η , since γ η is weakly compact, in order to see that p η “ Ť i ă η p i Y tp γ η , qu is a condition, wemust check that t α ă η : p η p α q “ u is not weakly compact. The set t γ i : i ă η u is club in η “ γ η and Player II has ensured that p η p γ i q “ i ă η . Thus t α ă η : p η p α q “ u is nonstationary and hence not weakly compact. (cid:3) Lemma 21.
For every cardinal δ ă γ , there is a ă δ -directed closed open densesubset of Q p γ, W q .Proof. Suppose δ ă γ . Let D “ t p P Q p γ, W q : dom p p q ą δ u . Clearly D isopen and dense. Suppose A Ď D is a directed set of conditions and | A | ă δ . Let η “ sup t dom p p q : p P A u . Since η ą δ and cf p η q ď | A | ă δ , it follows that η is notweakly compact. Thus p “ Ť A is a condition in Q p γ, W q . (cid:3) Let W Ď WC κ be a weakly compact subset of κ and suppose Refl wc p κ q holds.Let P κ ` “ xp P γ , Q γ q : γ ď κ y be the length κ ` γ forcing is defined as follows. ‚ If γ ď κ is a Mahlo limit point of Tr wc p W q , then Q γ is a P γ -name for theforcing Q p γ, W X γ q defined above. ‚ Otherwise, Q γ is a P γ -name for trivial forcing.Next we will show that, as intended, in certain contexts, the iteration P κ ` adds a non-reflecting weakly compact set and preserves many instances of weakcompactness. In the next theorem, the assumption that Refl wc p κ q holds is made toavoid trivialities. For example: if Refl wc p κ q fails, then κ already has a non-reflectingweakly compact subset. Theorem 22.
Suppose
Refl wc p κ q holds, W Ď WC κ is a weakly compact subset of κ and GCH holds. Then there is a cofinality-preserving forcing P such that if G is p V, P q -generic then the following conditions hold. (1) In V r G s , there is a non-reflecting weakly compact set E Ď W (thus W remains weakly compact). (2) The class of weakly compact cardinals is preserved: WC V “ WC V r G s . (3) If S Ď γ ă κ is a weakly compact subset of γ in V and W X γ Ď S , then S remains weakly compact in V r G s .Proof. Let G κ ` be p V, P κ ` q -generic. Since the iteration P κ ` uses Easton-supportand at each nontrivial stage γ ď κ we have , P γ “ Q γ is γ -strategically closed”,standard arguments involving factoring the iteration show that cofinalities are pre-served.The forcing P κ ` can be factored as P κ ˚ Q p κ, W q . For γ ď κ , let H γ denotethe p V r G γ s , Q γ q -generic filter obtained from G κ ` . Thus, G κ ` “ G κ ˚ H κ . Let f “ Ť H κ : κ Ñ E “ t α ă κ : f p α q “ u .First we prove (1). Let us show that in V r G κ ` s , for each η ă κ , the set E X η is not a weakly compact subset of η . If η is not weakly compact in V r G κ ` s then neither is E X η . Suppose η is weakly compact in V r G κ ` s . Since Q p κ, W q is κ -strategically closed, it follows that E X η P V r G κ s and thus f æ η is a conditionin Q p κ, W q . By Lemma 15, η is weakly compact in V r G κ s . Hence by definitionof Q p κ, W q , the set E X η is not weakly compact in V r G κ s . Applying Lemma 15again, we conclude that E X η is not weakly compact in V r G κ ` s .We will show that in the extension V r G κ ` s , the set E Ď κ is a weakly compactsubset of κ using the method outlined in Remark 2 above. Fix A P P p κ q V r G κ ` s and let E, A P H κ ` be P κ ` -names with E “ E G κ ` and A “ A G κ ` . Working in V , we may apply Lemma 10 to find a κ -model M with κ, P κ ` , A, E, W P M and anelementary embedding j : M Ñ N with critical point κ where N is also a κ -modelsuch that κ P j p W q and W is not a weakly compact subset of κ in N . Since κ is an inaccessible limit point of Tr wc p W q N “ Tr wc p W q V in N and N ă κ X V Ď N we have j p P κ q – P κ ˚ Q p κ, W q ˚ P Nκ,j p κ q where P Nκ,j p κ q is a P κ ` -name for the tailof the iteration j p P κ q . Since P κ is κ -c.c. in V and Q p κ, W q is ă κ -distributive in V r G κ s , it follows from Lemma 14, that M r G κ ` s is closed under ă κ -sequences in V r G κ ` s . Thus, using the facts that | N | V “ κ and that in N r G κ ` s , the poset P κ,j p κ q contains a ă κ -directed closed dense subset, we can build an p N r G κ ` s , It is important that we force at some stages γ with Q p γ, W q at which W X γ is not a weaklycompact subset of γ because in a crucial part of the argument we will have that on the j -side,where j is some elementary embedding with critical point κ , the set W “ j p W qX κ is not a weaklycompact subset of κ and this is required in order for a master condition to exist. DDING A NON-REFLECTING WEAKLY COMPACT SET 13 P κ,j p κ q q -generic filter K P V r G κ ` s . Since the supports of conditions in G have sizeless than κ we have j r G s Ď G κ ˚ H κ ˚ K , and thus we can extend the embeddingto j : M r G κ s Ñ N r p G j p κ q s where p G j p κ q “ G κ ˚ H κ ˚ K . Since K P V r G κ ` s wehave N r p G j p κ q s ă κ X V r G κ ` s Ď N r p G j p κ q s . Since W is not weakly compact in N , itfollows from Lemma 18 that W is not weakly compact in N r p G j p κ q s .Let us now argue that f Y tp κ, qu is a condition in j p Q p κ, W qq . We have f, E P N r p G j p κ q s because H κ P N r p G j p κ q s . It will suffice to show that the set E is not aweakly compact subset of κ in N r p G j p κ q s . From the definition of Q p κ, W q P V r G κ s ,we have E Ď W , and since W is not a weakly compact subset of κ in N r p G j p κ q s , itfollows that E is not a weakly compact subset of κ in N r p G j p κ q s . This implies that Ť j r H κ s “ Ť H κ “ f is a condition in j p Q p κ, W qq , and thus so is p “ f Y tp κ, qu .In N r p G j p κ q s , the poset j p Q p κ, W qq contains a ď κ -closed dense subset, and hencewe can build an p N r p G s j p κ q , j p Q p κ, W qqq -generic filter p H j p κ q P V r G κ ` s with p P p H j p κ q . This guarantees that j r H κ s Ď p H j p κ q , and thus the embedding extends to j : M r G κ ˚ H κ s Ñ N r p G j p κ q ˚ p H j p κ q s . Since p κ, q P Ť p H j p κ q , it follows that κ P j p E q .Therefore E is a weakly compact subset of κ in V r G ˚ g s , and hence we haveestablished (1).For (2), let us now show that the class of weakly compact cardinals is pre-served; i.e. that WC V r G κ ` s “ WC V . By the work of Hamkins, it follows thatWC V r G κ ` s Ď WC V (see Lemma 18 above or [Ham03, Lemma 16]). We must showthat WC V r G κ ` s Ě WC V . Suppose γ is weakly compact in V . If γ ą κ , then byLemma 16, it follows that γ is weakly compact in V r G κ ` s . Suppose γ ă κ . Sincein V r G γ ` s the forcing P γ,κ ` is ď γ -distributive, it will suffice to argue that γ remains weakly compact in V r G γ ` s . There are several cases to consider. Case 1.
Suppose γ is a limit point of Tr wc p W q . Then P γ ` – P γ ˚ Q p γ, W X γ q .Working in V r G γ ˚ H γ s , fix A Ď γ and let A P H κ ` be a P γ ˚ Q p γ, W X γ q -name suchthat A G γ ˚ H γ “ A . Working in V , by Lemma 10, it follows that there is a γ -model M with γ, A, P γ ` , W X γ P M and an elementary embedding j : M Ñ N with criticalpoint γ such that W X γ is not weakly compact in N . Since γ is a Mahlo limit pointof Tr wc p W q in N , it follows that j p P γ q – P γ ˚ Q p γ, W X γ q ˚ P γ,j p γ q . It follows that G γ is p M, P γ q -generic and G γ ˚ H γ is p N, P γ ˚ Q p γ, W X γ qq -generic, and furthermorethat N r G γ ˚ H γ s ă γ X V r G γ ˚ H γ s Ď N r G γ ˚ H γ s . Thus, as before, we may build an p N r G γ ˚ H γ s , P γ,j p γ q q -generic filter K P V r G γ ˚ H γ s . Let p G j p γ q “ G γ ˚ H γ ˚ K Since j r G γ s Ď p G j p γ q the embedding extends to j : M r G γ s Ñ N r p G j p γ q s . Let m “ Ť H γ .From the definition of Q p γ, W X γ q , it follows that supp p m q Ď W X γ . Furthermore,since W X γ is not weakly compact in N , it follows by Lemma 18, that W X γ isnot weakly compact in N r p G j p γ q s . Thus, m P j p Q p γ, W X γ qq . We can build an p N r p G j p γ q s , j p Q p γ, W X γ qqq -generic filter p H j p γ q P V r G γ ˚ H γ s with m P p G j p γ q . The Of course W X γ may or may not be a weakly compact subset of γ in V , but in either casesuch an embedding exists. If W X γ is a weakly compact subset of γ , then Lemma 10 applies. If W X γ is not a weakly compact subset of γ , let C Ď γ be a 1-club with C X W ‰ H , let M bea γ -model with γ, A, P γ ` , W X γ, C P M and let j : M Ñ N be an elementary embedding withcritical point γ , where N is also a γ -model. Then since C P N and C is 1-club in N , it followsthat W X γ is not weakly compact in N . embedding extends to j : M r G γ ˚ H γ s Ñ N r p G j p γ q ˚ p H j p γ q s , and thus γ is weaklycompact in V r G γ ` s . Case 2.
Suppose γ is not a limit point of Tr wc p W q . In this case, since P γ ` – P γ ˚ Add p γ, q , one may use a standard master condition argument to show that γ is weakly compact in V r G γ ` s .Thus the class of weakly compact cardinals is preserved WC V r G κ ` s “ WC V ,establishing (2).To prove (3), suppose S Ď γ ă κ is a weakly compact subset of γ in V and W X γ Ď S . To show that S is weakly compact in V r G κ ` s it will suffice, byLemma 15, to argue that S is weakly compact in V r G γ ` s . Fix A P P p κ q V r G γ ` s and let A P H γ ` be a P γ ` -name with A G “ A . By Lemma 10, we may let M be a γ -model with γ, A, P γ ` , S, W X γ P M and let j : M Ñ N be an elementaryembedding with critical point γ where N is a γ -model such that κ P j p S q and S “ j p S q X γ is not a weakly compact subset of γ in N . Since W X γ Ď S , itfollows that W is not a weakly compact subset of γ in N , and thus we can liftthe embedding using a master condition argument as in Case 1 or Case 2 above.Thus S remains a weakly compact subset of γ in V r G γ ` s . (cid:3) We now show that Theorem 1 is a consequence of Theorem 22, which establishesthe consistency of the failure of the weakly compact reflection principle Refl wc p κ q at a weakly compact cardinal of any order γ ă κ ` . Proof of Theorem 1.
Suppose κ is p α ` q -weakly compact where ω ď α ă κ . Wemust show that there is a cofinality-preserving forcing extension in which there isa non-reflecting weakly compact subset of κ , the class of weakly compact cardinalsis preserved and κ remains p α ` q -weakly compact.Choose a sequence ~A “ x A ξ : ξ ă κ ` y of subset of κ with A “ κ such that forall ξ ă κ ` ,(1) A ξ is a representative of the equivalence class Tr ξ wc pr κ s q ,(2) A ξ ` “ Tr wc p A ξ q and(3) if ξ is a limit then A ξ “ △ t A ζ : ζ ă ξ u .Let W “ A α P Tr α wc pr κ s q . Let P κ ` be the forcing from the proof of Theorem 22,and suppose G κ ` is p V, P κ ` q -generic. Applying Theorem 22 (1), we conclude thatin V r G κ ` s , the set W is weakly compact and the set E Ď W is a non-reflectingweakly compact subset of κ . By Theorem 22 (2), the class of weakly compactcardinals is preserved. It remains to show that κ remains p α ` q -weakly compactin V r G κ ` s .Working in V r G κ ` s , let ~B “ x B ξ : ξ ă κ ` y be a sequence defined using thesame conditions that were used to define ~A , but this time we run the definition in V r G κ ` s ; that is, B “ κ and for all ξ ă κ ` ,(1) B ξ is a representative of the equivalence class Tr ξ wc pr κ s q V r G κ ` s ,(2) A ξ ` “ Tr wc p A ξ q V r G κ ` s and(3) if ξ is a limit then A ξ “ △ t A ζ : ζ ă ξ u V r G κ ` s .To show that κ is p α ` q -weakly compact in V r G κ ` s , it will suffice to show that W “ A α Ď B α . We will use induction to prove that for every ξ ď α we have A ξ Ď B ξ .By definition of the sequences we have A “ κ “ B . If ξ ď α is a limitthe result follows immediately from the inductive hypothesis that A ζ Ď B ζ for DDING A NON-REFLECTING WEAKLY COMPACT SET 15 all ζ ă ξ . If ξ “ ζ ` ď α is a successor, then A ξ “ A ζ ` “ Tr wc p A ζ q and B ξ “ B ζ ` “ Tr wc p B ζ q . Let us show that A ξ Ď B ξ . Suppose γ P A ξ , this meansthat A ζ X γ is a weakly compact subset of γ in V . By Theorem 22 (3), it followsthat A ζ X γ remains a weakly compact subset of γ in V r G κ ` s . By the inductivehypothesis, A ζ Ď B ζ and thus B ζ X γ is a weakly compact subset of γ in V r G κ ` s .Thus we have shown that Tr α wc pr κ s q “ r B α s ą V r G κ ` s and hence κ remains p α ` q -weakly compact in V r G κ ` s . (cid:3) Resurrecting the weakly compact reflection principle
Recall that by a theorem of Kunen [Kun78], mentioned in the introduction, it isconsistent relative to a supercompact cardinal that a non-weakly compact cardinal κ can become supercompact in a forcing extension. Kunen’s proof proceeds asfollows. Suppose κ is a Laver-indestructible [Lav78] supercompact cardinal, let S be Kunen’s forcing for adding a homogeneous Souslin tree and let T be an S -namefor this tree. Then in V S , κ is not weakly compact and if we force over V S with T ,thereby adding a branch through T , we obtain a further extension V S ˚ T in which κ is supercompact because S ˚ T is forcing equivalent to adding a Cohen subset to κ . A similar resurrection result can be established using a different forcing argu-ment. Suppose κ is a Laver-indestructible supercompact cardinal, let S be thenatural forcing for adding a non-reflecting stationary subset of κ and let S be an S -name for this set. Then κ is not weakly compact in V S . Working in V S , let C be the forcing to shoot a club through κ z S . Then one can argue that S ˚ C is aseparative poset of size κ which contains a ă κ -closed dense subset, and hence mustbe equivalent to the forcing to add a Cohen subset to κ . Recall that if κ is weaklycompact, then there is a forcing extension in which the weak compactness of κ isindestructible by Add p κ, q . Thus, one can resurrect the weak compactness of κ starting with the hypothesis much weaker than supercompactness.We now establish a resurrection result regarding the weakly compact reflectionprinciple Refl wc p κ q . Suppose κ is a measurable cardinal, W Ď WC κ , P κ ` – P κ ˚ Q p κ, W q is the iteration from Theorem 22 for adding a non-reflecting weaklycompact set and let E be a P κ ` -name for this set. Then Refl wc p κ q fails in V P κ ` .Working in V P κ ` , let H κ ` – H κ ˚ T p κ z E q be Hellsten’s forcing [Hel03] for shoot-ing a 1-club through the complement of E such that nontrivial forcing occurs atstage γ in H κ ` if and only if nontrivial forcing occured at stage γ in P κ ` . Wewill prove that P κ ` ˚ H κ ` is forcing equivalent to the Easton-support iteration S κ ` – S κ ˚ Add p κ, q which adds a single Cohen subset to every Mahlo limit pointof Tr wc p W q . Standard arguments show that since κ is measurable in V , then κ remain measurable in V P κ ` ˚ H κ ` “ V S κ ` , and hence Refl wc p κ q has been resur-rected.Let us recall the definition and a few basic facts regarding Hellsten’s forcing forshooting a 1-club through a weakly compact set. Theorem 23 (Hellsten, [Hel03]) . Suppose E is a weakly compact subset of κ . Thereis a forcing extension in which E contains a -club, all weakly compact subsets of E remain weakly compact and thus κ remains a weakly compact cardinal. We now describe Hellsten’s forcing. Let X Ď κ be an unbounded subset of aninaccessible cardinal κ . We define a poset T p X q as follows. Conditions in T p X q are all c Ď X such that c is bounded and 1-closed, meaning that whenever c X α isstationary in α we have α P c . The ordering on T p X q is by end extension: c ď c iff c “ c æ sup t α ` α P c u . Hellsten proved [Hel10, Lemma 3] that T p X q is κ -strategically closed. The forcing H κ ` which Hellsten used to prove Theorem 23is an Easton-support iteration x H α , C β : α ď κ ` , β ď κ y such that(1) if β ď κ is Mahlo then C β is an H β -name for T p E X β q V H β and(2) otherwise C β is an H β -name for trivial forcing. Lemma 24.
Suppose γ is an inaccessible cardinal, W Ď WC γ and γ is a limit pointof Tr wc p W q . Let E be the cannonical Q p γ, W q -name for the subset of W added byforcing with Q p γ, W q . Then Q p γ, W q ˚ T p γ z E q is forcing equivalent to Add p γ, q .Proof. Notice that since Q p γ, W q ˚ T p γ z E q is γ -strategically closed, we can viewconditions p e, c q P Q p γ, W q ˚ T p γ z E q as ordered pairs of the form p e, c q where c is some bounded 1-closed subset of γ in V . Let D be the set of conditions p e, c q P Q p γ, W q˚ T p γ z E q such that dom p e q “ sup p c q “ α ` P Tr wc p W q and supp p e qX c ‰H . First let us show that D is a dense subset of Q p γ, W q ˚ T p γ z E q . If p e, c q is anycondition, let e P Q p γ, W q be such that e ď e , α “ dom p e q ą dom p e q Y sup p c q and W X dom p e q is a weakly compact subset of dom p e q . Since supp p e q is not aweakly compact subset of dom p e q , there is a 1-club c Ď dom p e qzp sup p c q ` q suchthat supp p e q X c “ H . Then c “ c Y c ď c is a 1-club subset of dom p e q . Let e “ e Y tp α, qu and c “ c Y t α u . Then p e , c q ď p e, c q and p e , c q P D .Standard arguments show that D is a ă γ -closed subset of Q p γ, W q ˚ T p γ z E q .Since the poset Q p γ, W q˚ T p γ z E q has size γ and contains a ă γ -closed dense subset,it must be forcing equivalent to Add p γ, q . (cid:3) Theorem 25.
Suppose that κ is a measurable cardinal, and let W Ď κ be weaklycompact in V . Let V r G κ ˚ H κ s be the forcing extension of Theorem 22, in whichthere is a non-reflecting weakly compact set E “ Ť H κ Ď W . Then there is a furtherforcing extension V r G κ ˚ H κ sr C κ ˚ D κ s in which Refl wc p κ q holds and D p κ q “ Ť D κ Ď κ z E is a -club subset of κ .Proof. Let P κ ` – P κ ˚ Q p κ, W q be the forcing from Theorem 22 and suppose G κ ` – G κ ˚ H κ is p V, P κ ` q -generic, so that E “ Ť H κ is a non-reflecting weaklycompact subset of κ . Working in V r G κ ˚ H κ s , let H κ ` – H κ ˚ T p κ z E q be Hell-sten’s Easton-support forcing iteration for shooting a 1-club through κ z E , wherenontrivial forcing is used at stage γ ď κ of the preparatory forcing H κ if and onlyif nontrivial forcing was used in stage γ of P κ , that is, if and only if γ is a Mahlolimit point of Tr wc p W q . Let C κ ` – C κ ˚ D κ be p V r G κ ` s , H κ ` q -generic, where D p κ q “ Ť D κ is a 1-club subset of κ z E in V r G κ ` ˚ C κ ` s .We will use induction to argue that in V , for every γ ď κ , the forcing P γ ` ˚ H γ ` is equivalent to the Easton-support iteration S γ ` of length γ ` µ ď γ which is a Mahlo limit point of Tr wc p W q .Working in V , suppose γ ď κ is a Mahlo limit point of Tr wc p W q . Since P γ ` ,κ ˚ Q p κ, W q is ď γ -distributive in V r G γ ˚ H γ s , it follows that H γ ˚ T p γ zp E X γ qq P V r G γ ˚ H γ s . Since conditions in H γ have bounded support, and since H γ can bedefined using G γ , we have H γ P V r G γ s . Thus, we may view V r G γ ˚ C γ s as anintermediate extension. By the inductive hypothesis, P γ ˚ H γ is forcing equivalentto S γ . Since γ is a Mahlo limit of Tr wc p W q V r G γ ˚ C γ s in V r G γ ˚ C γ s , we can apply DDING A NON-REFLECTING WEAKLY COMPACT SET 17
Lemma 24 to conclude that in V r G γ ˚ C γ s , the poset Q p γ, W q ˚ T p γ z E X γ q isforcing equivalent to Add p γ, q .Thus P κ ` ˚ H κ ` is forcing equivalent to S κ ` , so we have V r G κ ` ˚ H κ ` s “ V r g κ ` s where g κ ` is the p V, S κ ` q -generic filter obtained from G κ ` ˚ H κ ` . Stan-dard arguments show that in V r g κ ` s “ V r G κ ` ˚ H κ ` s , κ is a measurable cardinaland hence Refl wc p κ q holds. (cid:3) Questions
Many questions regarding the weakly compact and Π n -reflection principles re-main open. Question 1.
What is the consistency strength of the statement “there is a car-dinal κ such that the weakly compact reflection principle Refl wc p κ q holds”? Moregenerally, what is the strength of “there is a cardinal κ such that the Π n -reflectionprinciple Refl n p κ q holds”?Mekler and Shelah [MS89] showed that the statement “there is a regular cardinal κ such that every stationary subset of κ has a stationary initial segment” is equicon-sistent with the existence of a reflection cardinal ; i.e. a cardinal κ which carries anormal ideal I coherent with the nonstationary ideal such that the I -positive setsare closed under the operation Tr where Tr p X q “ t α ă κ : X X α is stationary u .As shown in [MS89], the existence of a reflection cardinal is a hypothesis who’s con-sistency strength is strictly between that of a greatly Mahlo cardinal and a weaklycompact cardinal. Definition 4.
A cardinal κ is a weak compactness reflection cardinal if and onlyif there exists a normal ideal I on κ such that(1) I is coherent to the weakly compact ideal Π p κ q and(2) the I -positive sets I ` are closed under the operation Tr wc .We conjecture that the existence of a cardinal at which the weakly compact re-flection principle holds is equiconisistent with the existence of a weak compactnessreflection cardinal, and this hypothesis is strictly between the existence of a greatlyweakly compact cardinal and the existence of a Π -indescribable cardinal in thelarge cardinal hierarchy. Notice that the forward direction of the above conjecturedequiconsistency easily follows from Lemma 6: if Refl wc p κ q holds then κ is a weakcompactness reflection cardinal since Π p κ q is a normal ideal and Π p κ q ` is closedunder the operation Tr wc . One may be able to establish the remaining parts ofthe conjecture, by carrying out arguments similar to [MS89, Corollary 5 and The-orem 7], however many technical difficulties need to be overcome. Similarly, onecan define the notion of Π n -reflection cardinal and formulate a similar conjectureregarding the consistency strength of the Π n -reflection principle.Above we have established that Refl wc p κ q implies that κ is ω -weakly compact,and that it is consistent that Refl wc p κ q fails and κ is γ -weakly compact, for anyfixed γ ă κ ` . Question 2.
Is it consistent that there exists a greatly weakly compact cardinal κ such that Refl wc p κ q fails?Regarding Question 2, in our proof of Theorem 1, we were able to show that theweak compactness of a set W “ A α is preserved by the iteration P κ ` as follows. We showed that the set E Ď W added by the iteration P κ ` is weakly compact, bychoosing an embedding j : M Ñ N with critical point κ such that κ P j p W q and W is not weakly compact in N . This allowed us to conclude that W remains weaklycompact, since it contains E . In order to preserve the great weak compactnessof κ , one would want to argue that the weak compactness of κ ` subsets of κ ispreserved. It is not clear that the techniques used in this article could be adaptedto accomplish this.One may also be able to adapt another argument [MS89, Theorem 9] of Meklerand Shelah to answer the following question. Question 3.
Is it consistent that Refl wc p κ q holds and κ is not p ω ` q -weakly com-pact? Is it consistent that Refl wc p κ q holds at the least ω -weakly compact cardinal?Finally, we ask, can the resurrection of Refl wc p κ q as in Theorem 25 be obtainedby starting with a large cardinal hypothesis weaker than the measurability of κ ? Question 4.
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Virginia Commonwealth University, Department of Mathematics andApplied Mathematics, 1015 Floyd Avenue, PO Box 842014, Richmond, Virginia 23284
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