aa r X i v : . [ m a t h . L O ] F e b LARGE CARDINAL IDEALS
BRENT CODY
Abstract.
Building on work of Holy, L¨ucke and Njegomir [30] on small em-bedding characterizations of large cardinals, we use some classical results ofBaumgartner (see [7] and [9]), to give characterizations of several well-knownlarge cardinal ideals, including the Ramsey ideal, in terms of generic ele-mentary embeddings; we also point out some seemingly inherent differencesbetween small embedding and generic embedding characterizations of subtlecardinals. Additionally, we present a simple and uniform proof which showsthat, when κ is weakly compact, many large cardinal ideals on κ are nowhere κ -saturated. Lastly, we survey some recent consistency results concerning theweakly compact ideal as well as some recent results on the subtle, ineffableand Π -indescribable ideals on P κ λ , and we close with a list of open questions. Contents
1. Introduction 22. Preliminaries 32.1. Basic terminology and facts about ideals 32.2. Small embeddings 42.3. Embedding characterizations of stationarity 53. Large cardinal ideals and elementary embeddings 63.1. The Π mn -indescribability ideals 63.2. The subtle ideal 83.3. The almost ineffable and ineffable ideals 113.4. The Ramsey ideal 174. Splitting positive sets assuming weak compactness 225. Consistency results 235.1. n -clubs and indescribability embeddings 245.2. A theorem of Hellsten 265.3. The weakly compact reflection principle 275.4. A (cid:3) ( κ )-like principle consistent with weak compactness 296. Large cardinal ideals on P κ λ P κ λ P κ λ P κ λ P κ λ Date : February 22, 2021.2010
Mathematics Subject Classification.
Primary 03E55; Secondary 03E02, 03E05.The author would like to thank Sean Cox, Monroe Eskew, Victoria Gitman and Chris Lambie-Hanson for many helpful conversations regarding the topics of this article. The author also thanksthe anonymous referee for the detailed review which greatly improved this article.
Acknowledgment 36References 361.
Introduction
Baumgartner (see [7] and [9]) showed that many large cardinal properties canalso be viewed as properties of subsets of cardinals and not just of the cardinalsthemselves, and this leads naturally to a consideration of ideals associated to largecardinals. For example, a set X ⊆ κ is Ramsey if for every function f : [ X ] <ω → κ with f ( a ) < min( a ) for all a ∈ [ X ] <ω , there is a set H ⊆ X of size κ which is homogeneous for f , meaning that f ↾ [ H ] n is constant for all n < ω . Baumgartnershowed that if κ is a Ramsey cardinal then the collection of non-Ramsey subsetsof κ is a nontrivial normal ideal on κ . Similarly, one can define normal idealsassociated to indescribability, subtlety, ineffability and many other large cardinalnotions. In fact, Baumgartner showed that certain characterizations of almostineffability, ineffability and Ramseyness require the consideration of large cardinalideals (for example, see Remark 3.19 below).It is a well-known and often-used fact that the stationarity of a set S ⊆ κ canbe characterized in terms of certain types of elementary embeddings. For example,a set S ⊆ κ is stationary if and only if there is some forcing P such that whenever G ⊆ P is generic there is, in V [ G ], an elementary embedding j : ( V, ∈ ) → ( M, ∈ M ) ⊆ V [ G ]with critical point κ where M is well-founded up to κ + such that κ ∈ j ( S ). Recallthat such elementary embeddings are obtained from stationary sets by using genericultrapowers. Alternatively, S ⊆ κ is stationary if and only if there is a nontrivialelementary embedding j : M → H ( κ + ) such that M is transitive of size less than κ , j (crit( j )) = κ , S ∈ ran( j ) and crit( j ) ∈ S . Recall that such embeddings areobtained from stationary sets by taking the inverses of certain transitive collapsemaps. Inspired by the work of Holy-L¨ucke-Njegomir [30], in this article we addressthe question: to what extent can these elementary embedding characterizations ofthe nonstationary ideal be generalized to large cardinal ideals?In Section 2, we cover some preliminaries, including a few basic properties ofideals we will need later on, as well as some facts concerning elementary embeddingcharacterizations of stationary sets.In Section 3, we show that both the generic embedding and the transitive collapseembedding characterizations of stationarity can be generalized to many large cardi-nal ideals including the subtle, ineffable and Ramsey ideals. Let us emphasize thatfor some large cardinal notions, such as subtlety, inherent differences emerge be-tween generic embedding and transitive collapse embedding characterizations (seeProposition 3.10, Remark 3.11 and Proposition 3.11 below).In Section 4, we provide a simple and uniform proof of a folklore result, whichstates that Solovay’s splitting theorem for stationary sets can be generalized tomany large cardinal ideals on κ including the Π n -indescribable ideal, the almost in-effable ideal, the ineffable ideal, the Ramsey ideal and others, when these ideals arenontrivial. Specifically, we show that if κ is weakly compact and I is a normal ideal ARGE CARDINAL IDEALS 3 on κ which is definable over H ( κ + ), then I is nowhere κ -saturated. Let us em-phasize that this result is essentially folklore, although it may not have been widelyknown previously since Hellsten states the result only for the Π n -indescribable ideals[28, Theorem 2] and Foreman gives a different proof for the weakly compact idealwhich uses generic embeddings [23, Proposition 6.4].In Section 5, we give a survey of some consistency results concerning the weaklycompact ideal. We discuss a result of Hellsten [28] on the saturation of the weaklycompact ideal, as well as some results due to the author [17] and Cody-Sakai [20]on the weakly compact reflection principle. We also state a few theorems due toGitman, Cody and Lambie-Hanson [19] regarding forcing a (cid:3) ( κ )-like principle tohold at a weakly compact cardinal.In Section 6, we give a survey of selected results involving large cardinal idealson P κ λ . For example, we discuss two-cardinal versions of indescribability, subtletyand ineffability.Finally, in Section 7, we state several open questions.2. Preliminaries
Basic terminology and facts about ideals. An ideal I on a cardinal κ isa collection of subsets of κ which is closed under finite unions and closed undersubsets. If κ is a cardinal and I is an ideal on κ then I + = { X ⊆ κ | X / ∈ I } isthe collection of I -positive sets, I ∗ = { X ⊆ κ | κ \ X ∈ I } is the filter dual to I . If S ∈ I + then I ↾ S = { X ⊆ κ | X ∩ S ∈ I } is an ideal on κ extending I and noticethat S ∈ ( I ↾ S ) ∗ . An ideal I on κ is normal if the collection I + satisfies the Fodorproperty: for all S ∈ I + whenever f : S → κ satisfies f ( α ) < α for all α ∈ S , thenthere is an H ∈ P ( S ) ∩ I + such that f ↾ H is constant. Equivalently, I is normalif it is closed under diagonal unions, that is, whenever { X α | α < κ } ⊆ I we have ▽ α<κ X α := { β < κ | β ∈ S α<β X α } ∈ I .In Section 3 below, we will be concerned with showing that certain large cardinalideals are obtained by taking the ideal generated by a union of some other largecardinal ideals. Given a family A ⊆ P ( κ ) of subsets of κ , the ideal generated by A is defined to be A = { X ⊆ κ | ( ∃B ∈ [ A ] <ω ) X ⊆ [ B} . the collection of all subsets of κ which are contained in some union of finitelymany sets from A . In what follows we will repeatedly use both of the followingobservations. Remark 2.1.
It is easy to see that if J and K are ideals on a cardinal κ , then theideal on κ generated by J ∪ K is the collection of all X ⊆ κ which can be writtenas a disjoint union of a set from J and a set from K . That is, J ∪ K = { X ⊆ κ | ( ∃ A ∈ J )( ∃ B ∈ K )( X = A ∪ B and A ∩ B = ∅ ) } . Remark 2.2.
Suppose I , I and J are ideals on κ . If we want to prove that J = I ∪ I , part of what we must show is that J ⊇ I ∪ I , or equivalently J + ⊆ I ∪ I . Notice that we may obtain a chain of equivalences directly from the The author would like to thank Sean Cox for pointing out this result and the included proof.
BRENT CODY definitions involved: J + ⊆ I ∪ I ⇐⇒ I ∪ I ⊆ J ⇐⇒ I ∪ I ⊆ J ⇐⇒ J + ⊆ I +0 ∩ I +1 . In what follows, in order to prove that the property J + ⊆ I ∪ I (or equivalentlythe property J ⊇ I ∪ I ) holds for various ideals, we will prove J + ⊆ I +0 ∩ I +1 andinclude a reference to this remark.Given an ideal I on κ , we write P ( κ ) /I to denote the usual atomless booleanalgebra obtained from I . If G is ( V, P ( κ ) /I )-generic, then we let U G be the canonical V -ultrafilter obtained from G extending the dual filter I ∗ . The appropriate versionof Los’s Theorem can be easily verified, and thus we obtain a canonical genericultrapower embedding j : V → V κ /U G in V [ G ]. If I is a normal ideal thenthe generic ultrafilter U G is V -normal and the critical point of the corresponding,possibly illfounded, generic ultrapower j : V → V κ /U G ⊆ V [ G ] is κ . The followinglemma is a standard tool for working with generic ultrapowers (see [31, Lemma22.14] or [23, Section 2] for more details). Lemma 2.3.
Suppose I is a normal ideal on κ . Let G be ( V, P ( κ ) /I ) -generic and j : V → V κ /U G be the corresponding generic elementary embedding and let E bethe ultrapower of the ∈ relation. Then (1) E is wellfounded on the ordinals up to κ + , (2) κ = [ id ] U G , (3) for all X ∈ P ( κ ) V we have X ∈ U G if and only if κEj ( X ) and (4) for all f : κ → V with f ∈ V we have [ f ] U G = j ( f )( κ ) . Definition 2.4.
When we say there is a generic elementary embedding j : V → M ⊆ V [ G ] we mean that there is some forcing poset P such that whenever G is ( V, P )-generic then, in V [ G ], there are definable classes M , E and j such that j : ( V, ∈ ) → ( M, E ) is an elementary embedding, where (
M, E ) is possibly notwellfounded.2.2.
Small embeddings.
Suppose κ is a regular uncountable cardinal. By itera-tively taking Skolem hulls, one can build an elementary substructure X ≺ H ( κ + )such that κ ∈ X , X ∩ κ ∈ κ and | X | < κ . Let π : X → M be the transitivecollapse of X and let j = π − . Then j : M → H ( κ + ) is an elementary embed-ding with j (crit( j )) = κ . Holy, L¨ucke and Njegomir [30] used such embeddingsto give new characterizations of several large cardinal notions, including subtlety,Π n -indescribability, ineffability, measurability and λ -supercompactness, and to givenew proofs of results of Christoph Weiss [48] on the consistency strength of certaingeneralized tree properties. Definition 2.5 (Holy, L¨ucke, Njegomir [30]) . Given cardinals κ < θ , we say thata non-trivial elementary embedding j : M → H ( θ ) is a small embedding for κ if M ∈ H ( θ ) is transitive and j (crit( j )) = κ . We write P ( κ ) /I when we really mean P ( κ ) /I − { [ ∅ ] } . ARGE CARDINAL IDEALS 5
Embedding characterizations of stationarity.
In this section we give two(folklore) characterizations of stationary subsets of a cardinal κ in terms of elemen-tary embeddings, and we discuss several modest generalizations of some results of[30]. Proposition 2.6 (Folklore) . Suppose κ > ω is a regular cardinal. The followingare equivalent. (1) S ⊆ κ is stationary. (2) There is a generic elementary embedding j : V → M ⊆ V [ G ] with criticalpoint κ such that κ ∈ j ( S ) . (3) There is a small embedding j : M → H ( κ + ) for κ such that S ∈ ran( j ) and crit( j ) ∈ S .Proof. Let us show that (1) and (2) are equivalent. If S ⊆ κ is stationary, let G be ( V, P ( κ ) / (NS κ ↾ S ))-generic, let U G be the generic ultrafilter obtained from G and let j : V → M = V κ /U G be the corresponding generic ultrapower embedding.Since U G is a V -normal V -ultrafilter, the critical point of j is κ and since S ∈ U G we have κ ∈ j ( S ). Conversely, suppose there is a generic elementary embedding j : V → M ⊆ V [ G ] with critical point κ such that κ ∈ j ( S ). If C ⊆ κ is a club in V , then κ ∈ j ( S ∩ C ) and by elementarity S ∩ C = ∅ .Next we show that (1) and (3) are equivalent. Suppose S ⊆ κ is stationary.Let h X α | α < κ i be a continuous increasing elementary chain of submodels of H ( κ + ) each of cardinality less than κ such that S ∈ X , α ⊆ X α ∩ κ ∈ κ for all α < κ . Since S is stationary, there is an α < κ such that α = X α ∩ κ ∈ S . Let j : M → H ( κ + ) be the inverse of the Mostowski collapse of X α . Then j is a smallembedding for κ with S ∈ ran( j ) and crit( j ) = X α ∩ κ = α . Conversely, suppose j : M → H ( κ + ) is a small embedding for κ such that S ∈ ran( j ) and crit( j ) ∈ S .Assume S is not stationary in κ . Then S is not a stationary subset of κ in H ( κ + )and by elementarity it follows that in M there is a club C ⊆ crit( j ) such that C ∩ j − ( S ) = ∅ . Again by elementarity, j ( C ) ∩ S = ∅ , but this is impossible sincecrit( j ) ∈ j ( C ) ∩ S . (cid:3) Proposition 2.7 (Folklore) . Given an L ∈ -formula ϕ ( v , v ) , the following state-ments are equivalent for every cardinal κ and every set x . (1) κ is regular and uncountable and { α < κ | ϕ ( α, x ) } is stationary in κ . (2) There is a generic elementary embedding j : V → M ⊆ V [ G ] with criticalpoint κ such that M | = ϕ ( κ, j ( x )) . (3) There is a small embedding j : M → H ( κ + ) for κ such that ϕ (crit( j ) , x ) holds and x ∈ ran( j ) .Proof. For the equivalence of (1) and (3) see [30, Lemma 2.1]. Let us show that (1)and (2) are equivalent.Let S = { α < κ | ϕ ( α, x ) } . Since S is stationary in κ we may let G ⊆ P ( κ ) / (NS κ ↾ S ) be generic and let U G be the V -normal V -ultrafilter obtainedfrom G . Let j : V → V κ /U G ⊆ V [ G ] be the corresponding generic ultrapower.Since NS κ ↾ S is a normal ideal on κ , the critical point of j is κ , and since S ∈ (NS κ ↾ S ) ∗ ⊆ U G we have κ ∈ j ( S ). In other words, V κ /U G | = ϕ ( κ, j ( x )). Forthe converse, fix a club C ∈ P ( κ ) V . Let j : V → M ⊆ M [ G ] be a generic embeddingwith critical point κ such that M | = ϕ ( κ, j ( x )). Then κ ∈ j ( { α < κ | ϕ ( α, x ) } ∩ C ),and by elementarity { α < κ | ϕ ( α, x ) } ∩ C = ∅ . (cid:3) BRENT CODY
The following corollary of Proposition 2.7 is the analogue of [30, Corollary 2.2],which gives similar characterizations of various types of cardinals in terms of smallembeddings.
Corollary 2.8 (Folklore) . Let κ be a cardinal. (1) κ is uncountable and regular if and only if there is a generic embedding j : V → M ⊆ V [ G ] with critical point κ . (2) κ is weakly inaccessible if and only if there is a generic embedding j : V → M ⊆ V [ G ] such that M | = “ κ is a cardinal.” (3) κ is inaccessible if and only if there is a generic embedding j : V → M ⊆ V [ G ] with critical point κ such that M | = “ κ is a strong limit.” (4) κ is weakly Mahlo if and only if there is a generic embedding j : V → M ⊆ V [ G ] with critical point κ such that M | = “ κ is regular.” (5) κ is Mahlo if and only if there is a generic embedding j : V → M ⊆ V [ G ] with critical point κ such that M | = “ κ is inaccessible.” Large cardinal ideals and elementary embeddings
The Π mn -indescribability ideals. Recall that a formula is Π mn if it startswith a block of universal quantifiers of type m + 1 variables, followed by a block oftype m + 1 existential quantifiers, and so on with at most n blocks in all, followedby a formula containing unquantified variables of type at most m + 1 and quantifiedvariables of type at most m . Similarly, a formula is Σ mn if it begins with a block oftype m + 1 existential quantifiers. See [34, Chapter 0] for more details.A set S ⊆ κ is Π mn -indescribable if for every A ∈ V κ +1 and every Π mn -sentence ϕ over ( V κ , ∈ , A ), whenever ( V κ , ∈ , A ) | = ϕ there is an α ∈ S such that ( V α , ∈ , A ∩ V α ) | = ϕ . Levy [38] showed that if κ is a Π mn -indescribable cardinal then thecollection Π mn ( κ ) = { X ⊆ κ | X is not Π mn -indescribable } is a normal ideal on κ , which is referred to as the Π mn -indescribable ideal on κ (see[35, Proposition 6.11]). A set S ⊆ κ is called weakly Π mn -indescribable if for every A ⊆ κ and every Π mn -sentence ϕ over ( κ, ∈ , A ), whenever ( κ, ∈ , A ) | = ϕ there isan α ∈ S such that ( α, ∈ , A ∩ α ) | = ϕ . It is easy to check [7] that a set S ⊆ κ isΠ mn -indescribable if and only if κ is inaccessible and S is weakly Π mn -indescribable.Furthermore, when κ is weakly Π mn -indescribable, the collection e Π mn ( κ ) = { X ⊆ κ | X is not weakly Π mn -indescribable } of non-weakly Π mn -indescribable subsets of κ is a nontrivial normal ideal on κ . Remark 3.1.
Let us note here that Sharpe and Welch [43] defined a notion ofΠ ξ -indescribability of a cardinal κ where ξ < κ + by demanding that the existenceof a winning strategy for a particular player in a certain finite game played at κ implies that the same player has a winning strategy for the game played at α .Independently, Bagaria [4] defined a natural notion of Π ξ -formula for ξ ≥ ω . Forexample, a formula ϕ is Π ω if it is of the form V n<ω ϕ n where each ϕ n is Π n ,and it contains only finitely-many free second-order variables. A set S ⊆ κ is Π ξ -indescribable if and only if for all A ⊆ V κ whenever ϕ is Π ξ and ( V κ , ∈ , A ) | = ϕ ,there must be some α ∈ S such that ( V α , ∈ , A ∩ V α ) | = ϕ . Independently, Brickhill-Welch [11] and Bagaria [4] showed that if κ is Π ξ -indescribable where ξ < κ then ARGE CARDINAL IDEALS 7 the collection Π ξ ( κ ) = { X ⊆ κ | X is not Π ξ -indescribable } is a normal ideal on κ . Remark 3.2.
Recall that κ is Π -indescribable if and only if κ is inaccessible, andin this case the Π -indescribable ideal equals NS κ . Definition 3.3.
For notational convenience later on we letΠ − ( κ ) = [ κ ] <κ for all cardinals κ . Lemma 3.4. If κ is Π mn -indescribable, A ∈ V κ +1 and ϕ is a Π mn -sentence over ( V κ , ∈ , A ) such that ( V κ , ∈ , A ) | = ϕ then the set { α < κ | ( V α , ∈ , A ∩ V α ) | = ϕ } is in the Π mn -indescribable filter Π mn ( κ ) ∗ .Proof. Suppose ( V κ , ∈ , A ) | = ϕ where ϕ is a Π mn -sentence. Let Z = { α < κ | ( V α , ∈ , A ∩ V α ) | = ϕ } . The set κ \ Z = { α < κ | ( V α , ∈ , A ∩ V α ) | = ¬ ϕ } is not Π mn -indescribable because if it were then the fact that ( V κ , ∈ , A ) | = ϕ wouldimply that ( V α , ∈ , A ∩ V α ) | = ϕ for some α ∈ κ \ Z , a contradiction. Thus κ \ Z ∈ Π mn ( κ ) which implies Z ∈ Π mn ( κ ) ∗ . (cid:3) The following is implicit in the work of Holy, L¨ucke and Njegomir [30], the onlydifference is that we generalize their characterization of Π mn -indescribable cardinalsto subsets of κ and we also give a generic embedding characterization. Proposition 3.5 (Holy, L¨ucke and Njegomir [30, Lemma 4.2]) . For n, m < ω , κ aregular cardinal and S ⊆ κ the following are equivalent. (1) S is Π mn -indescribable. (2) There is a generic embedding j : V → M ⊆ V [ G ] with critical point κ such that κ ∈ j ( S ) and for every Π mn -sentence ϕ over ( V κ , ∈ , A ) where A ∈ ( V κ +1 ) V we have (( V κ , ∈ , A ) | = ϕ ) V = ⇒ (( V κ , ∈ , A ) | = ϕ ) M . (3) For all sufficiently large cardinals θ there is a small embedding j : M → H ( θ ) for κ such that (a) S ∈ ran( j ) , (b) crit( j ) ∈ S and (c) for every Π mn -sentence ϕ over ( V crit( j ) , ∈ , A ) where A ∈ M ∩ V crit( j )+1 we have (( V crit( j ) , ∈ , A ) | = ϕ ) M = ⇒ (( V crit( j ) , ∈ , A ) | = ϕ ) V . Proof.
The proof that (1) and (3) are equivalent is very similar to [30, Lemma 4.2].We leave the details to the reader.To see that (1) implies (2), suppose S ⊆ κ is Π mn -indescribable and let G begeneric for P ( κ ) / (Π mn ( κ ) ↾ S ). Let U G be the V -normal V -ultrafilter on P ( κ ) V obtained from G and note that U G extends (Π mn ( κ ) ↾ S ) ∗ , hence S ∈ U G . Let j : Note that A ∈ M since A = j ( A ) ∩ V κ and j ( A ) , V κ ∈ M . BRENT CODY V → M = Ult( V, U G ) ⊆ V [ G ] be the generic ultrapower by U G . Clearly, crit( j ) = κ and κ ∈ j ( S ). Let ϕ ( A ) be a Π mn -sentence with parameter A ∈ ( V κ +1 ) V such that(( V κ , ∈ , A ) | = ϕ ) V . Since κ is Π mn -indescribable in V , it follows from Lemma 3.4that the set C = { α < κ | ( V α , ∈ , A ∩ V α ) | = ϕ } is in the filter (Π mn ( κ ) ↾ S ) ∗ and ishence also in U G . Thus, κ ∈ j ( C ) = { α ∈ j ( κ ) | (( V α , E, j ( A ) ∩ V α ) | = ϕ ) M } , which implies (( V κ , ∈ , A ) | = ϕ ) M .For (2) implies (1), fix S ⊆ κ and suppose that j : V → M ⊆ V [ G ] is a genericembedding as in (2). We will show that S is Π mn -indescribable. Fix A ∈ ( V κ +1 ) V and let ϕ ( A ) be a Π mn -sentence over ( V κ , ∈ , A ) such that ( V κ , ∈ , A ) | = ϕ . By theassumed properties of j we obtain (( V κ , ∈ , j ( A ) ∩ V κ ) | = ϕ ( j ( A ) ∩ V κ )) M and henceby elementarity, V | = ( ∃ α ∈ S )( V α , ∈ , A ∩ V α ) | = ϕ ( A ∩ V α ). Thus, S is Π mn -indescribable. (cid:3) Remark 3.6.
Notice that in Proposition 3.5, the existence of a single genericembedding suffices to characterize the Π mn -indescribability of S ⊆ κ , whereas, it mayseem, we need to demand the existence of many small embeddings to characterizethe Π mn -indescribability of S . However, the proof [30, Lemma 4.2] shows that, infact, a single small embedding also suffices. That is, we can replace Proposition3.5(3) with the following.(3) There is a small embedding j : M → H ( i m ( κ ) + ) for κ such that(a) S ∈ ran( j ),(b) crit( j ) ∈ S and(c) for every Π mn -sentence ϕ over ( V crit( j ) , ∈ , A ) where A ∈ M ∩ V crit( j )+1 we have(( V crit( j ) , ∈ , A ) | = ϕ ) M = ⇒ (( V crit( j ) , ∈ , A ) | = ϕ ) V . Remark 3.7.
It is not too difficult to see that the characterization given in Propo-sition 3.5(2) can be extended to Π ξ -indescribable subsets of κ for all ξ < κ + . SeeRemark 3.1 for the definition of Π ξ -indescribability and [16, Proposition 8.3] for aproof of this result.3.2. The subtle ideal.
Recall that for S ⊆ κ we say that a sequence ~S = h S α | α ∈ S i is an S -list if S α ⊆ α for all α ∈ S . Jensen and Kunen defined a set S ⊆ κ to be subtle if for every S -list ~S = h S α | α ∈ S i and for every club C ⊆ κ there exist α, β ∈ S ∩ C with α < β such that S α = S β ∩ α . Although the least subtle cardinalis not weakly compact, every subtle cardinal is a stationary limit (and more) ofcardinals which are Π n -indescribable for every n < ω (this follows from Lemma3.12 below).To show that the collection of non-subtle subsets of a subtle cardinal forms anormal ideal let us fix some notation. We let Γ : ORD × ORD → ORD denotethe standard definable pairing function. If α is a closure point of Γ, meaningΓ[ α × α ] ⊆ α , and if ~A = h A ξ | ξ < α i is a sequence of subsets of α , then we canuse Γ to code the sequence into a single subset of α : A = [[ ~A ]] = { Γ( η, ξ ) | η ∈ A ξ } .The following lemma was used in [7] and its proof is straightforward. Lemma 3.8.
Suppose α, β ∈ ORD are closure points of Γ with α < β , and furthersuppose that A ⊆ α codes the sequence ~A = h A ξ | ξ < α i of subsets of α and B ⊆ β ARGE CARDINAL IDEALS 9 codes the sequence ~B = h B ξ | ξ < β i of subsets of β . In other words, A = [[ ~A ]] and B = [[ ~B ]] . Then it follows that if A = B ∩ α then A ξ = B ξ ∩ α for all ξ < α . Proposition 3.9 (Baumgartner [7]) . If κ is subtle then the collection NSub κ = { X ⊆ κ | X is not subtle } is a normal ideal.Proof. Let S ⊆ κ be subtle and suppose f : S → κ is regressive. We must showthat f is constant on a subtle subset of S , that is, we must show that for some η < κ we have f − ( η ) ∈ NSub + κ . Suppose not. Let D = { α < κ | α is closed under the pairing function Γ } . Then D is club and hence S ∩ D is subtle. By assumption, for each η < κ , we maylet ~S η = h S ηα | α ∈ f − ( η ) i be an f − ( η )-list and let C η be a club such that for all α, β ∈ f − ( η ) ∩ C η with α < β we have S ηα = S ηβ ∩ α .Define an S ∩ D -list ~S = h S α | α ∈ S ∩ D i by letting S α = [[ h S f ( α ) α , { f ( α ) }i ]]. Let C = △ η<κ C η . Since S ∩ D is subtle there exists α, β ∈ S ∩ D ∩ C with α < β and S α = S β ∩ α . Thus [[ h S f ( α ) α , { f ( α ) }i ]] = [[ h S f ( β ) β , { f ( β ) }i ]] ∩ α and since α and β areboth closed under G¨odel-pairing, it follows from Lemma 3.8 that S f ( α ) α = S f ( β ) β ∩ α and f ( α ) = f ( β ). If we let η = f ( α ) = f ( β ) then we have S ηα = S ηβ ∩ α where α, β ∈ f − ( η ) ∩ C η , a contradiction. (cid:3) In the following result we give a characterization of subtle sets in terms of ele-mentary embeddings. The fact that (1) and (3) are equivalent in the following isdue to Holy, L¨ucke and Njegomir [30, Lemma 5.2]; we provide a proof here sinceit will be referenced below in order to emphasize a difference in character betweengeneric embedding and small embedding characterizations of large cardinal ideals.
Proposition 3.10.
For all cardinals κ and all S ⊆ κ , the following are equivalent. (1) S ⊆ κ is subtle (2) There is a generic elementary embedding j : V → M ⊆ V [ G ] with criticalpoint κ and κ ∈ j ( S ) such that for every S -list ~S = h S α | α ∈ S i and everyclub C ⊆ κ in V , we have S α = j ( ~S )( κ ) ∩ α for some α ∈ S ∩ C . (3) For every S -list ~S = h S α | α ∈ S i and for every club C ⊆ κ , there is a smallembedding j : M → H ( κ + ) for κ such that S, C, ~S ∈ ran( j ) , crit( j ) ∈ S and S α = S crit( j ) ∩ α for some α ∈ C ∩ crit( j ) .Proof. For (1) implies (3), suppose S is subtle, ~S = h S α | α ∈ S i is an S -listand C ⊆ κ is a club. Let h X α | α < κ i be a continuous increasing sequence ofelementary substructures of H ( κ + ) of cardinality less than κ with ~S, C ∈ X and α ⊆ X α ∩ κ ∈ κ for all α < κ . Then D = { α ∈ C | α = X α ∩ α } is club in κ andsince S ∩ D is subtle there are α, β ∈ S ∩ D with α < β and S α = S β ∩ α . Let π : X β → M be the transitive collapse of X β . Then j = π − : M → H ( κ + ) is asmall embedding for κ with crit( j ) = β , ~S , C ∈ ran( j ) and S α = S crit( j ) ∩ α . Theconverse, namely (3) implies (1), is quite straight forward.To see that (1) implies (2), suppose S ⊆ κ is subtle. Clearly S is stationary in κ . Let G ⊆ P ( κ ) / (NSub κ ↾ S ) be generic. Let U G be the V -normal V -ultrafilterobtained from G and let j : V → V κ /U G ⊆ V [ G ] be the generic ultrapower by U G . Then the critical point of j is κ and κ ∈ j ( S ) since S ∈ U G . Fix an S -list ~S = h S α | α ∈ S i and a club C ⊆ κ . Let us show that the set X = { ξ ∈ S | ( ∃ ζ ∈ S ∩ C ∩ ξ ) S ζ = S ξ ∩ ζ } is in (NSub κ ↾ S ) ∗ ; in other words, we will show that κ \ X ∈ NSub κ ↾ S , i.e.( κ \ X ) ∩ S ∈ NSub κ . Since( κ \ X ) ∩ S = { ξ ∈ S | ( ∀ ζ ∈ S ∩ C ∩ ξ ) S ζ = S ξ ∩ ζ } we have that S α = S β ∩ α for all α, β ∈ C ∩ ( κ \ X ) ∩ S with α < β . Hence ( κ \ X ) ∩ S is not subtle. Thus X ∈ (NSub κ ↾ S ) ∗ ⊆ U G which implies that κ ∈ j ( X ) and hencethere is an α ∈ S ∩ C such that S α = j ( ~S )( κ ) ∩ α .Conversely, for (2) implies (1), fix an S -list ~S = h S α | α ∈ S i and a club C ⊆ κ and suppose j : V → M ⊆ V [ G ] is a generic elementary embedding with criticalpoint κ such that κ ∈ j ( S ) and S α = j ( ~S )( κ ) ∩ α for some α ∈ S ∩ C . Since α, κ ∈ j ( S ∩ C ), it follows by elementarity that there exist ζ, ξ ∈ S ∩ C with ζ < ξ such that S ζ = S ξ ∩ ζ . (cid:3) Remark 3.11.
Let us point out that there is a substantial difference between thegeneric embedding characterization and the small embedding characterization ofsubtlety given in the previous proposition, which is not manifested in the embeddingcharacterizations of stationarity. Namely, in the generic embedding characterizationof the subtlety of S ⊆ κ , Proposition 3.10(2), a single embedding j works for all S -lists and all clubs C . Whereas, in the small embedding characterization, Proposition3.10(3), for each S -list there is a small embedding with the desired property. Thisseems to be an inherent difference between these two characterizations because onecannot place all club subsets of κ into an elementary substructure X ≺ H ( θ ) whilestill guaranteeing that X ∩ κ ∈ κ . Indeed, it is easy to see that for regular cardinals ω < κ < θ , if X ≺ H ( θ ) and X ∩ κ ∈ κ then X does not contain all club subsets of κ . Suppose not. Let X ≺ H ( θ ) contain all club subsets of κ with X ∩ κ ∈ κ . Thencrit( j ) = X ∩ κ ∈ C for all clubs C ⊆ κ , which is a contradiction.As it will be needed below for a characterization of ineffability, let us presenta result of Baumgartner, which shows that if κ is a subtle cardinal then there aremany cardinals less than κ which are Π n -indescribable for all n < ω . Recall thatfor a cardinal κ , S ⊆ κ and an S -list ~S = h S α | α ∈ S i , a set X ⊆ κ is homogeneous for ~S if and only if whenever α, β ∈ X with α < β we have S α = S β ∩ α . Lemma 3.12 (Baumgartner [7]) . Let κ be a cardinal and S ⊆ κ . Assume S issubtle and ~S = h S α | α ∈ S i is an S -list. Let A = { α ∈ S | ∃ X ⊆ S ∩ α such that X is Π n -indescribable in α for all n < ω and X is homogeneous for ~S } Then S \ A is not subtle.Proof. Suppose S \ A is a subtle subset of κ . Fix a bijection b : V κ → κ and let C be aclub subset of κ such that for all α ∈ C we have b ↾ V α : V α → α is a bijection and α is closed under the standard definable pairing function Γ : κ × κ → κ . By normalityof the subtle ideal, it follows that E := ( S \ A ) ∩ C is a subtle subset of κ . Wedefine an E -list ~E as follows. Fix β ∈ E and let B β = { α ∈ S ∩ β | S α = S β ∩ α } .Notice that B β ∪ { β } is homogeneous for ~S . Since β ∈ E , it follows that for ARGE CARDINAL IDEALS 11 some n < ω the set B β is not Π n -indescribable in β . Let ϕ β be a Π n -sentenceand let A β ⊆ V β be such that ( V β , ∈ , Γ , A β ) | = ϕ β while for all α ∈ B β we have( V α , ∈ , Γ ↾ α, A β ∩ α ) | = ¬ ϕ β . Let E β = [[( S β , { ϕ β } , b [ A β ])]]. This defines an E -list ~E = h E β | β ∈ E i .Since E is subtle there exist α < β in E such that E α = E β ∩ α . It followsthat S α = S β ∩ α and thus α ∈ B β . Furthermore, b [ A α ] = b [ A β ] ∩ α and hence A α = A β ∩ V α . Also we have ϕ α = ϕ β = ϕ . By definition of ϕ α we see that( V α , ∈ , Γ , A α ) | = ϕ , and since α ∈ B β we have ( V α , ∈ , Γ ↾ α, A β ∩ V α ) | = ¬ ϕ , acontradiction. (cid:3) An easy corollary of the previous lemma shows that, although the definition ofsubtlety only asserts the existence of homogeneous sets of size 2, one can obtainmuch larger homogeneous sets for free; quoting Abramson-Harrington-Kleinberg-Zwicker, “a little coherence goes a long way” (see [3, Section 3.6]).
Corollary 3.13 (Baumgartner [7]) . A set S ⊆ κ is subtle if and only if for every S -list ~S = h S α | α ∈ S i and every club C ⊆ κ there is an α ∈ S ∩ C such that thereis a set H ⊆ S ∩ C ∩ α homogeneous for ~S which is Π n -indescribable for all n < ω . As another corollary of Lemma 3.12, we easily obtain an additional characteri-zation of subtlety in terms of generic elementary embeddings.
Corollary 3.14.
For all cardinals κ and S ⊆ κ the following are equivalent. (1) S is subtle. (2) There is a generic embedding j : V → M ⊆ V [ G ] with critical point κ and κ ∈ j ( S ) such that for every S -list ~S = h S α | α ∈ S i and every club C ⊆ κ in V , we have S α = j ( ~S )( κ ) ∩ α for some α ∈ S ∩ C and furthermore, M | = “there is a set H ⊆ S which is Π n -indescribable in κ for all n < ω andhomogeneous for ~S ”.Proof. For (2) implies (1) the proof is the same as that of Proposition 3.10. For(1) implies (2), assuming S is subtle, let G ⊆ P ( κ ) / (NSub κ ↾ S ) be generic, let U G be the corresponding V -normal ultrafilter and let j : V → V κ /U G ⊆ V [ G ] be thegeneric ultrapower. As in the proof of Proposition 3.10, one can check that thereis an α ∈ S ∩ C such that S α = j ( ~S )( κ ) ∩ α . For the remaining statement, let A = { α ∈ S | ∃ X ⊆ S ∩ α such that X is Π n -indescribable in α for all n < ω and X is homogeneous for ~S } and notice that by Lemma 3.12, S ∩ ( κ \ A ) ∈ NSub κ , which implies A ∈ (NSub κ ↾ S ) ∗ and hence κ ∈ j ( A ). (cid:3) The almost ineffable and ineffable ideals.
Given a regular cardinal κ andan ideal I ⊇ [ κ ] <κ we define another ideal I ( I ) by letting S / ∈ I ( I ) if and only if forevery S -list ~S = h S α | α ∈ S i there is a set H ∈ P ( S ) ∩ I + which is homogeneousfor ~S . We say that a set S ⊆ κ is almost ineffable if S ∈ I ([ κ ] <κ ) + , that is, every S -list ~S = h S α | α ∈ S i has a homogeneous set H ⊆ S of size κ . Similarly, S is ineffable if S ∈ I (NS κ ) + , that is, every S -list ~S = h S α | α ∈ S i has a homogeneousset H ⊆ S which is stationary in κ . Baumgartner [7] showed that if κ is ineffablethen the ideal I (NS κ ) is nontrivial and normal, and similarly, if κ is almost ineffablethen I ([ κ ] <κ ) is nontrivial and normal. The collection I ([ κ ] <κ ) is called the almost ineffable ideal and I (NS κ ) is called the ineffable ideal . It is easy to see that for anyideal I ⊇ [ κ ] <κ we have S ∈ I ( I ) + if and only if for every S -list ~S = h S α | α ∈ S i there is a set D ⊆ κ which is anticipated by S on a set in I + , meaning that the set { α ∈ S | S α = D ∩ α } is in I + . This easily leads to the following characterizationof the ineffability of a set S ⊆ κ ; note that below we give another characterizationof ineffability (see Proposition 3.25). Proposition 3.15.
For all cardinals κ and S ⊆ κ the following are equivalent. (1) S is ineffable. (2) For all S -lists ~S = h S α | α ∈ S i there is a generic elementary embedding j : V → M ⊆ M [ G ] with critical point κ and κ ∈ j ( S ) such that j ( ~S )( κ ) ∈ V . (3) For all S -lists ~S = h S α | α ∈ S i there is a small embedding j : M → H ( κ + ) for κ with S, ~S ∈ ran( j ) and crit( j ) ∈ S such that S crit( j ) ∈ M .Proof. The proof that (1) and (3) are equivalent is very similar to that of [30,Lemma 5.5]. For (1) implies (2), suppose S is ineffable, let ~S = h S α | α ∈ S i be an S -list and let D ⊆ κ be such that the set A = { α ∈ S | S α = D ∩ α } isstationary. Let G ⊆ P ( κ ) / (NS κ ↾ A ) and let j : V → M = V κ /U G ⊆ V [ G ] be thecorresponding generic ultrapower. Then, crit( j ) = κ and κ ∈ j ( S ). Furthermore,since A ∈ U G it follows that κ ∈ j ( A ) and hence j ( ~S )( κ ) = D ∈ V .For (2) implies (1), fix an S -list ~S = h S α | α ∈ S i . We must show that ~S has ahomogeneous set that is stationary in κ . Using (2), there is a generic elementaryembedding j : V → M ⊆ V [ G ] with critical point κ and κ ∈ j ( S ) such that D = j ( ~S )( κ ) ∈ V . Let H = { α ∈ S | S α = D ∩ α } . Clearly H ∈ V and H ishomogeneous for ~S . To see that H is stationary in κ , fix a club C ⊆ κ and noticethat κ ∈ j ( H ) ∩ j ( C ), and hence H ∩ C = ∅ by elementarity. (cid:3) By combining the arguments of Proposition 3.5 and Proposition 3.15, we obtainthe following characterization of the ideals I (Π n ( κ )) for n < ω , which generalizesthe generic embedding characterization of ineffability from Proposition 3.15; it isnot clear whether or not the small embedding characterization from Proposition3.15 can also be generalized in this way. Proposition 3.16.
For all n < ω , all cardinals κ and all S ⊆ κ , the following areequivalent. (1) S ∈ I (Π n ( κ )) + (2) For all S -lists ~S = h S α | α ∈ S i there is a generic elementary embedding j : V → M ⊆ V [ G ] with critical point κ and κ ∈ j ( S ) such that the followingproperties hold. (a) j ( ~S )( κ ) ∈ V (b) For all R ∈ V Vκ +1 and all Π n -sentences ϕ we have (( V κ , ∈ , R ) | = ϕ ) V = ⇒ (( V κ , ∈ , R ) | = ϕ ) M . Proof.
For (1) implies (2), suppose S ∈ I (Π n ( κ )) + and fix an S -list ~S = h S α | α ∈ S i . Let D ⊆ κ be such that the set A = { α ∈ S | S α = D ∩ α } is Π n -indescribable.Let G ⊆ P ( κ ) / (Π n ( κ ) ↾ A ) be generic and let j : V → M = V κ /U G ⊆ V [ G ] be thecorresponding generic ultrapower embedding. Since Π n ( κ ) ↾ S is a normal idealon κ , it follows that U G is a V -normal ultrafilter. Hence crit( j ) = κ and M is ARGE CARDINAL IDEALS 13 well-founded up to κ + . Since A ∈ (Π n ( κ ) ↾ A ) ∗ ⊆ U G , it follows that κ ∈ j ( A ) andthus j ( ~S )( κ ) = D , so (2)(a) holds. Since Π n ( κ ) ∗ ⊆ (Π n ( κ ) ↾ A ) ∗ ⊆ U G , the sameargument as that of Proposition 3.5 shows that (2)(b) holds.Conversely, suppose (2) holds. Fix an S -list ~S = h S α | α ∈ S i and let j bea generic elementary embedding as in (2). Let D = j ( ~S )( κ ). We will argue that A = { α ∈ S | S α = D ∩ α } is Π n -indescribable, which is sufficient since A is clearlyhomogeneous for ~S . By (2)(a) it follows that D ∈ V , and thus A ∈ V . Suppose R ∈ V Vκ +1 and ϕ is a Π n -sentence such that (( V κ , ∈ , R ) | = ϕ ) V . By (2)(b), it followsthat (( V κ , ∈ , R ) | = ϕ ) M . Since j ( ~S )( κ ) = j ( D ) ∩ κ , we see that κ ∈ j ( A ), and thus,it follows by elementarity that there is some α ∈ A such that (( V α , ∈ , R ∩ V α ) | = ϕ ) V .In other words, A is Π n -indescribable. (cid:3) Remark 3.17.
Notice that the case n = −
1, where I (Π − ( κ )) = I ([ κ ] <κ ) isconspicuously missing from Proposition 3.15 and Proposition 3.16. One reason forthis is that the assumption that S ⊆ κ is almost ineffable gives for each S -list ~S = h S α | α ∈ S i a set D ⊆ κ such that A = { α ∈ S | S α = D ∩ α } has size κ , andforcing with the non-normal ideal [ κ ] <κ ↾ A is problematic. Nonetheless, we givea generic embedding characterization of almost ineffable sets below in Proposition3.25.Baumgartner proved [7] that, for a cardinal κ and S ⊆ κ , demanding every S -list has a Π n -indescribable homogeneous set implies S is Π n +1 -indescribable. Weinclude a proof here for the reader’s convenience. Together with Theorem 3.23, thisresult will be use below in the proof of Proposition 3.25 to establish another genericembedding characterization of the ideal I (Π n ( κ )). Lemma 3.18 (Baumgartner [7]) . Suppose κ is a cardinal and S ⊆ κ is such thatevery S -list ~S = h S α | α ∈ S i has a homogeneous set which is Π n -indescribablewhere n ∈ {− } ∪ ω . Then S is Π n +2 -indescribable.Proof. Let us first handle the case in which n = −
1. Suppose every S -list hasa homogeneous set H in (Π − ( κ )) + = ([ κ ] <κ ) + . In other words, S is almostineffable. Let us show that S is Π -indescribable. Since the almost ineffability of S implies that κ is inaccessible, it suffices to show that S is weakly Π -indescribable(although showing that S is Π n -indescribable in κ is not much harder using abijection b : V κ → κ and a club C ⊆ κ of closure points of b ). Suppose not. Thenthere is an A ⊆ κ and a Π -sentence ∀ Xψ ( X ) such that ( κ, ∈ , A ) | = ∀ Xψ ( X ) andfor all α ∈ S there is an S α ⊆ α such that ( α, ∈ , A ∩ α, S α ) | = ¬ ψ ( S α ). Let H ⊆ S be homogeneous for ~S = h S α | α ∈ S i with | H | = κ and let X = S α ∈ S S α . Then( κ, ∈ , A, X ) | = ψ ( X ) where ψ ( X ) is a Π sentence. Since κ is inaccessible and S isstationary, it follows that S is Π -indescribable. Thus, there is an α ∈ S such that( α, ∈ , A ∩ α, X ∩ α ) | = ψ ( X ∩ α ), but since X ∩ α = S α , this is a contradiction.Now suppose n ≥ S -list ~S has a Π n -indescribable homogeneousset H . Then κ is inaccessible, so to show that S is Π n +2 -indescribable, it sufficesto show that S is weakly Π n +2 -indescribable. Suppose S is not weakly Π n +2 -indscribable. Then there is an A ⊆ κ and a Π n +2 sentence ∀ X ∃ Y ψ ( X, Y ), where ψ ( X, Y ) is a Π n formula, such that ( κ, ∈ , A ) | = ∀ X ∃ Y ψ ( X, Y ) and for all α ∈ S there is an S α ⊆ α such that ( α, ∈ , A ∩ α, S α ) | = ∀ Y ¬ ψ ( S α , Y ). Then ~S = h S α | α ∈ S i is an S -list and we may let H ⊆ S be a Π n -indescribable homogeneous set for ~S . Since ( κ, ∈ , A ) | = ∀ X ∃ Y ψ ( X, Y ), if we let X = S α ∈ H S α then there is a Y ⊆ κ such that ( κ, ∈ , A, X, Y ) | = ψ ( X, Y ). Since S is weakly Π n -indescribable,and the set C = { α < κ | ( α, ∈ , A ∩ α, X ∩ α, Y ∩ α ) | = ψ ( X, Y ) } is in the weakly Π n -indescribable filter, it follows that S ∩ C is weakly Π n -indescrib-able. Thus, there is some α ∈ S ∩ C such that( α, ∈ , A ∩ α, X ∩ α, Y ∩ α ) | = ψ ( X ∩ α, Y ∩ α ) . Since X ∩ α = S α (by homogeneity of H ), it follows that( α, ∈ , A ∩ α, S α ) | = ∃ Y ψ ( S α , Y ) , a contradiction. (cid:3) Remark 3.19.
Baumgartner showed that a cardinal κ is almost ineffable if andonly if it is subtle, Π -indescribable, and the ideal generated by the subtle idealtogether with the Π -indescribable ideal is nontrivial and equals the almost ineffa-ble ideal; moreover, the statement about the ideals cannot be removed from thischaracterization since the least cardinal which is subtle and Π -indescribable is notalmost ineffable (this follows from Theorem 3.23). Similarly, κ is ineffable if andonly if it is subtle, Π -indescribable and the ideal generated by the subtle ideal andthe Π -indescribable ideal is nontrivial and equals the ineffable ideal; as above, thestatement about ideals cannot be removed from the characterization because theleast subtle cardinal which is Π -indescribable is not ineffable (this follows fromTheorem 3.23). In fact, Baumgartner proved a more general result, which appearsas Theorem 3.23 below.Let us consider the ideals I (Π n ( κ )) where n ∈ {− } ∪ ω . Recall, using thenotation introduced above, I (Π − ( κ )) = I ([ κ ] <κ ) is the almost ineffable ideal and I (Π ( κ )) = I (NS κ ) is the ineffable ideal. By definition, a set S ⊆ κ is in I (Π n ( κ )) + if and only if every S -list has a homogeneous set H ⊆ S which is Π n -indescribablein κ . Since ineffability implies almost ineffability, which in turn implies subtlety, itis easy to see thatNSub κ ⊆ I ([ κ ] <κ ) ⊆ I (NS κ ) ⊆ I (Π ( κ )) ⊆ I (Π ( κ )) ⊆ · · · . In fact, by applying Lemma 3.18, one can show that the above containments areproper when the ideals are nontrivial. Let us show that NSub κ ( I ([ κ ] <κ ) when κ is almost ineffable (see Corollary 3.22 below). Similar arguments, which are left tothe reader, establish the remaining proper containments (for a more general resultsee [29, Theorem 1.5]).Recall that if S is stationary in κ and for each α ∈ S we have a set S α whichis stationary in α , then S α ∈ S S α is stationary in κ . The analog of this result alsoholds for large cardinal ideals. For example, the stationary limit of subtle cardinalsis subtle by the following Lemma. Lemma 3.20.
Suppose S ∈ NS + κ and that for each α ∈ S we have a set S α ∈ NSub + α . Then S α ∈ S S α ∈ NSub + κ .Proof. To show that T = S α ∈ S S α is subtle in κ , fix a T -list ~T = h T α | α ∈ T i anda club C ⊆ κ . Since S is stationary in κ , we may choose α ∈ S ∩ C ′ . Since S α issubtle in α and C ∩ α is a club subset of α , there exist ξ, ζ ∈ S α ∩ C ⊆ T ∩ C with ξ < ζ such that T ξ = T ζ ∩ ξ . Thus, T is subtle in κ . (cid:3) ARGE CARDINAL IDEALS 15
Lemma 3.21. If κ is subtle then the set T = { α < κ | α is not subtle } is subtle.Proof. For the sake of contradiction, suppose κ is the least counterexample. Thatis, κ is the least cardinal such that κ is subtle and { α < κ | α is not subtle } is notsubtle. Then S := κ \ T ∈ NSub ∗ κ and hence S ∈ NSub + κ . For each α ∈ S , by theminimality of κ , the set S α := T ∩ α is subtle in α . Thus, by Lemma 3.20, the set T = S α ∈ S S α is subtle in κ , a contradiction. (cid:3) Corollary 3.22.
Suppose κ is almost ineffable. Then NSub κ ( I ([ κ ] <κ ) . Proof.
Lemma 3.21 implies that T = { α < κ | α is not subtle } / ∈ NSub κ . Further-more, by definition of subtlety, there is a Π sentence ϕ such that for all α ≤ κ wehave α is subtle if and only if V α | = ϕ. Since κ is almost ineffable, it is also subtle, and hence V κ | = ϕ . By Lemma 3.18, κ is Π -indescribable and hence the set κ \ T = { α < κ | α is subtle } = { α < κ | V α | = ϕ } is in Π ( κ ) ∗ . By Lemma 3.18, it follows that I ([ κ ] <κ ) + ⊆ Π ( κ ) + and henceΠ ( κ ) ∗ ⊆ I ([ κ ] <κ ) ∗ . Therefore, we see that κ \ T ∈ I ([ κ ] <κ ) ∗ and now it easilyfollows that T ∈ I ([ κ ] <κ ) \ NSub κ . (cid:3) Notice that taking n = − n = 0 yields his characterization of ineffability,both of which were mentioned above in Remark 3.19. Theorem 3.23 (Baumgartner [7]) . For all n ∈ {− } ∪ ω , we have κ / ∈ I (Π n ( κ )) if and only if the following properties hold. (1) κ is subtle and Π n +2 -indescribable. (2) The ideal generated by the subtle ideal and the Π n +2 -indescribable ideal isa nontrivial normal ideal; in this case, I (Π n ( κ )) = NSub κ ∪ Π n +2 ( κ ) . Furthermore, (2) cannot be removed from this characterization because the leastcardinal κ which is subtle and Π n +2 -indescribable is in I (Π n ( κ )) .Proof. Let I = NSub κ ∪ Π n +2 ( κ ). We will show that S ∈ I (Π n ( κ )) + if and only if S ∈ I + .Suppose S ∈ I (Π n ( κ )) + . To show S ∈ I + , it suffices to show S is subtle andΠ n +2 -indescribable by Remark 2.2. Clearly, S is subtle, and by Lemma 3.18, S isΠ n +2 -indescribable. Thus S ∈ I + . Conversely, suppose S ∈ I + . For the sake ofcontradiction, suppose S ∈ I (Π n ( κ )). Then there is an S -list ~S = h S α | α ∈ S i such that every homogeneous set for ~S is in the ideal Π n ( κ ). This is expressible bya Π n +2 -sentence ϕ over ( V κ , ∈ , ~S ). Thus, it follows that the set C = { α < κ | ( V α , ∈ , ~S ∩ V α ) | = ϕ } = { α < κ | every hom. set for ~S ↾ α is in Π n ( α ) } is in the filter Π n +2 ( κ ) ∗ . Since S ∈ I + , it follows that S is not equal to the union ofa non-subtle set and a non–Π n +2 -indescribable set. Since S = ( S ∩ C ) ∪ ( S \ C ) and S \ C ∈ Π n +2 ( κ ), it follows that S ∩ C must be subtle. Thus, by Lemma 3.13, thereis some α ∈ S ∩ C for which there is an H ⊆ S ∩ C ∩ α which is Π n -indescribablein α and homogeneous for ~S . This contradicts α ∈ C .For the remaining statement, let us show that if κ / ∈ I (Π n ( κ )), then there aremany cardinals below κ which are both subtle and Π n +2 -indescribable. Suppose κ / ∈ I (Π n ( κ )). Notice that the fact that κ is subtle can be expressed by a Π -sentence ϕ over V κ and thus the set C = { α < κ | ( V α , ∈ ) | = ϕ } = { α < κ | α is subtle } is in the filter Π ( κ ) ∗ ⊆ Π n +2 ( κ ) ∗ . Furthermore, by Lemma 3.12, the set H = { α < κ | α is Π n +2 -indescribable } is in the filter NSub ∗ κ . Since I (Π n ( κ )) ⊇ NSub κ ∪ Π n +2 ( κ ), it follows that C ∩ H is in the filter I (Π n ( κ )) ∗ . (cid:3) Remark 3.24.
Let us note that the preceding result of Baumgartner can be gen-eralized to the ideals of the form I (Π ξ ( κ )) where ξ ≥ ω and Π ξ ( κ ) denotes theΠ ξ -indescribability ideal defined in Remark 3.1. For example, if κ / ∈ I (Π ω ( κ )) thenone has I (Π ω ( κ )) = NSub κ ∪ Π ω +2 ( κ ) . Furthermore, by iterating the ineffability operator one obtains ideals of the form I α (Π ξ ( κ )), and the previous Theorem of Baumgartner can also be generalized tothese ideals. Considering the large cardinal notions associated to these ideals pro-vides a strict refinement of Baumgartner’s original ineffability hierarchy [9]. Thisrefinement of the ineffability hierarchy is analogous to the refinement of the Ramseyhierarchy studied in [16].Using the previous theorem of Baumgartner, we obtain another generic embed-ding characterization of almost ineffability (taking n = −
1) and ineffability (taking n = 0). Note that in Proposition 3.15, the generic embedding was obtained byforcing with the nonstationary ideal below a particular stationary anticipation set,whereas in the following proposition, the generic embedding is obtained by forcingwith the ineffable ideal below an ineffable set. Proposition 3.25.
For all n ∈ {− } ∪ ω , for all cardinals κ and all S ⊆ κ thefollowing are equivalent. (1) S ∈ I (Π n ( κ )) + . (2) There is a generic elementary embedding j : V → M ⊆ V [ G ] with criticalpoint κ and κ ∈ j ( S ) such that the following properties hold. (a) For all A ∈ V Vκ +1 and all Π n +2 -sentences ϕ we have (( V κ , ∈ , A ) | = ϕ ) V = ⇒ (( V κ , ∈ , A ) | = ϕ ) M . (b) For every S -list ~S = h S α | α ∈ S i in V , it follows that M | = “there isa set H ⊆ S which is Π n -indescribable in κ and homogeneous for ~S ”.Proof. For (1) implies (2), suppose S ∈ I (Π n ( κ )) + . Let G ⊆ P ( κ ) / ( I (Π n ( κ )) ↾ S )be generic and let j : V → M = V κ /U G ⊆ V [ G ] be the corresponding genericultrapower. Clearly, crit( j ) = κ and κ ∈ j ( S ). By Theorem 3.23, we have ARGE CARDINAL IDEALS 17 Π n +2 ( κ ) ∗ ⊆ I (Π n ( κ )) ∗ ⊆ U G , and thus (2)(a) follows by an argument similarto that of Proposition 3.5. For (2)(b), fix an S -list ~S = h S α | α ∈ S i . Since S ∈ I (Π n ( κ )) + , there is a set H ⊆ S which is Π n -indescribable in κ and homo-geneous for ~S . Clearly, H = j ( H ) ∩ κ ∈ M , and furthermore, the fact that H is Π n -indescribable is expressible by a Π n +1 -sentence over ( V κ , ∈ , H ). Thus, by(2)(a), it follows that M | = “ H is Π n -indescribable in κ and homogeneous for ~S ”.Suppose (2) holds. Let ~S = h S α | α ∈ S i be an S -list in V . We must show that ~S has a Π n -indescribable homogeneous set. Suppose not. Recall that “ X ∈ Π n ( κ )”is expressible by a Σ n +1 -sentence over ( V κ , ∈ , X ). Thus, there is a Π n +2 -sentence ϕ over ( V κ , ∈ , ~S ) asserting that every homogeneous set for ~S is not Π n -indescribable.Now let j : V → M ⊆ V [ G ] be a generic embedding as in (2). By (2)(a), itfollows that (( V κ , ∈ , ~S ) | = ϕ ) M and hence M | = “every homogeneous set for ~S isnot Π n -indescribable”. This contradicts (2)(b). (cid:3) The Ramsey ideal.
Recall that κ > ω is a
Ramsey cardinal if for everyfunction f : [ κ ] <ω → H ⊆ κ of size κ which is homogeneous for f ,meaning that f ↾ [ H ] n is constant for all n < ω . Furthermore, for S ⊆ κ where κ isa cardinal, a function f : [ S ] <ω → κ is regressive if f ( a ) < min a for all a ∈ [ S ] <ω .Given an ideal I ⊇ [ κ ] <κ on κ we define another ideal R ( I ) on κ by letting S / ∈ R ( I )if and only if for every regressive function f : [ S ] <ω → κ there is a set H ⊆ S in I + which is homogeneous for f . Feng proved [22, Theorem 2.1], that R ( I ) is alwaysa normal ideal. A set S ⊆ κ is Ramsey if S ∈ R ([ κ ] <κ ) + , that is every regressivefunction f : [ S ] <ω → κ has a homogeneous set H ⊆ S of size κ . Baumgartnershowed that when κ is Ramsey, the collection R ([ κ ] <κ ) is a nontrivial normal idealcalled the Ramsey ideal on κ . In this section we will study the ideals R (Π n ( κ )) for n ∈ {− } ∪ ω .In order to give a characterization of sets in R (Π n ( κ )) + for n < ω which isanalogous to Proposition 3.16, let us present an alternative characterization ofRamseyness due to Feng. Indeed, Feng [22, Theorem 2.3] gave a characterizationof Ramseyness which resembles the definition of ineffability. Definition 3.26 (Feng [22]) . Suppose S ⊆ κ . For each n < ω and for all increasingsequences α < · · · < α n taken from S suppose that S α ...α n ⊆ α . Then we saythat ~S = h S α ...α n | n < ω ∧ ( α , . . . , α n ) ∈ [ S ] n i is an ( ω, S ) -list . A set H ⊆ S is said to be homogeneous for an ( ω, S )-lists ~S if forall 0 < n < ω and for all increasing sequences α < · · · < α n and β < · · · < β n taken from S with α ≤ β we have S α ··· α n = S β ··· β n ∩ α . Theorem 3.27 (Feng [22]) . Let κ be a regular cardinal and suppose I is an idealon κ such that I ⊇ NS κ . For S ⊆ κ the following are equivalent. (1) S ∈ R ( I ) + , that is, every function f : [ S ] <ω → has a homogeneous set H ∈ P ( S ) ∩ I + . (2) For all γ < κ , every function f : [ S ] <ω → γ has a homogeneous set H ∈ P ( S ) ∩ I + . (3) Every structure A in a language of size less than κ with κ ⊆ A has a set ofindiscernibles H ∈ P ( S ) ∩ I + . (4) S ∈ R ( I ) + , that is, for every regressive function f : [ S ] <ω → κ there is aset H ∈ P ( S ) ∩ I + which is homogeneous for f . (5) For every club C ⊆ κ , every regressive function f : [ S ] <ω → κ has ahomogeneous set H ∈ P ( S ∩ C ) ∩ I + . (6) For all ( ω, S ) -sequences ~S there is a set H ∈ P ( S ) ∩ I + which is homoge-neous for ~S . (7) For all ( ω, S ) -lists ~S there is a set H ⊆ S with H ∈ I + and a sequence h D n | n < ω i such that for each n < ω and for all α < · · · < α n from H we have S α ··· α n = D n ∩ α . In the previous theorem, if one weakens the assumption I ⊇ NS κ to I ⊇ [ κ ] <κ one can still prove some of the equivalences. For more on this issue see [16]. Proposition 3.28.
Suppose I ⊇ [ κ ] <κ is an ideal on a regular cardinal κ . Thenclauses (4), (5), (6) and (7) of Theorem 3.27 are equivalent. Remark 3.29.
Although the ideals I (Π n ( κ )) for n ∈ {− } ∪ κ as well as the ideals R (Π − ( κ )) = R ([ κ ] <κ ) and R (Π ( κ )) = R (NS κ ), have been well-studied (see [7],[9] and [22]), ideals of the form R (Π n ( κ )) for n > Remark 3.30.
Let κ be a cardinal and suppose I ⊇ NS κ is a normal ideal on κ .Suppose S ∈ R ( I ) + and let ~S = h S α ...α n | n < ω ∧ ( α , . . . , α n ) ∈ [ S ] n i be an( ω, S )-list. Let H ⊆ S and h D n | n < ω i be as in Theorem 3.27(7). Then for each n < ω we have H ⊆ { α ∈ S | ( ∀ α . . . α n ∈ H )( α < α < · · · < α n = ⇒ S αα ··· α n = D n ∩ α ) } . The set which contains H in the statement of Remark 3.30 can be thought ofas the set X of ordinals at which the sequence h D n | n < ω i is anticipated by ~S .We obtain the following generic embedding characterization of R (Π n ( κ )) by forcingwith P ( κ ) / Π n ( κ ) below the ‘anticipation set’ X . Theorem 3.31.
For all n < ω , all cardinals κ and all S ⊆ κ , the following areequivalent. (1) S ∈ R (Π n ( κ )) + (2) For all ( ω, S ) -lists ~S there is a set H ⊆ κ and there is a generic elementaryembedding j : V → M ⊆ V [ G ] with critical point κ and κ ∈ j ( H ) such thatthe following properties hold. (a) For any h a m | m < ω i ∈ (cid:0)Q Suppose S ∈ R (Π n ( κ )) + and let ~S be an ( ω, S )-list. Let H ⊆ S and h D m | m < ω i be the sets obtained from Theorem 3.27(7). Since H ∈ Π n ( κ ) + ,the ideal Π n ( κ ) ↾ H is normal. Let G ⊆ P ( κ ) / (Π n ( κ ) ↾ H ) be generic and let j : V → M = V κ /U G ⊆ V [ G ] be the corresponding generic ultrapower embedding.Then crit( j ) = κ and κ ∈ j ( H ) ⊆ j ( S ). To prove that (2)(a) holds, fix any h a m | m < ω i ∈ ( Q For all n ∈ {− } ∪ ω and all cardinals κ , we have κ / ∈ R (Π n ( κ )) if and only if the following properties hold. (1) κ ∈ R (Π n ( κ )) + and κ is Π n +2 -indescribable. (2) The ideal generated by R (Π n ( κ )) and the Π n +2 -indescribable ideal is anontrivial normal ideal; in this case, R (Π n ( κ )) = R (Π n ( κ )) ∪ Π n +2 ( κ ) . Furthermore, (2) cannot be removed from this characterization because the leastcardinal κ such that κ ∈ R (Π n ( κ )) + and κ is Π n +2 -indescribable is in R (Π n ( κ )) .Proof. Let I = R (Π n ( κ )) ∪ Π n +2 ( κ ). We will show that S ∈ R (Π n ( κ )) + if andonly if S ∈ I + .Suppose S ∈ R (Π n ( κ )) + . To show that S ∈ I + , it suffices to show that S ∈ Π n +2 ( κ ) + and S ∈ R (Π n ( κ )) + (see Remark 2.2). Let ~S = h S α | α ∈ S i bean S -list. By arbitrarily extending ~S to an ( ω, S )-list, we see that the fact that S ∈ R (Π n ( κ )) + implies that the S -list ~S has a homogeneous H ⊆ S which is Π n -indescribable in κ . Thus, by Lemma 3.18, it follows that S ∈ Π n +2 ( κ ) + . To seethat S ∈ R (Π n ( κ )) + , fix a regressive function f : [ S ] <ω → κ and a club C ⊆ κ .Since S ∩ C ∈ R (Π n ( κ )) + , there is set H ⊆ S ∩ C in Π n ( κ ) + which is homogeneousfor f . The fact that H ∈ Π n ( κ ) + can be expressed by a Π n +1 sentence ϕ over( V κ , ∈ , H ). Since S ∩ C is Π n +2 -indescribable, it follows that there is an α ∈ S ∩ C such that ( V α , ∈ , H ∩ V α ) | = ϕ , which implies that H ∩ α is Π n -indescribable in α .Thus S ∈ R (Π n ( κ )) + .Suppose S ∈ I + . To show that S ∈ R (Π n ( κ )) fix a regressive function f :[ S ] <ω → κ and suppose, for the sake of contradiction, that every homogeneous setfor f is not Π n -indescribable in κ . This can be expressed by a Π n +2 sentence ϕ over ( V κ , ∈ , S, f ). Thus, the set C = { α < κ | ( V α , ∈ , S ∩ α, f ∩ V α ) | = ϕ } = { α < κ | every hom. set H ⊆ S ∩ α for f is in Π n ( α ) } is in the filter Π n +2 ( κ ) ∗ . Since S ∈ I + it follows that S is not the union of a set in R (Π n ( κ )) and a set in Π n +2 ( κ ). Since S = ( S ∩ C ) ∪ ( S \ C ) and S \ C ∈ Π n +2 ,we see that S ∩ C ∈ R (Π n ( κ )) + . Hence, there is an α ∈ S ∩ C such that thereis an H ⊆ S ∩ C ∩ α which is Π n -indescribable in α and homogeneous for f . Thiscontradicts α ∈ C .For the remaining statement, let us show that if κ / ∈ R (Π n ( κ )), then there aremany cardinals α < κ such that α ∈ R (Π n ( α )) + and α ∈ Π n +2 ( α ) + . Suppose κ / ∈ R (Π n ( κ )). Notice that the fact that κ ∈ R (Π n ( κ )) + can be expressed by aΠ -sentence ϕ over V κ and thus the set C = { α < κ | ( V α , ∈ ) | = ϕ } = { α < κ | α ∈ R (Π n ( κ )) + } is in the filter Π ( κ ) ∗ ⊆ Π n +2 ( κ ) ∗ . Furthermore, by Lemma 3.12, the set H = { α < κ | α is Π n +2 -indescribable } is in the filter NSub ∗ κ ⊆ R (Π n ( κ )) ∗ . Since R (Π n ( κ )) ∗ ⊇ R (Π n ( κ )) ∗ ∪ Π n +2 ( κ ) ∗ ,it follows that C ∩ H is in the filter R (Π n ( κ )) ∗ . (cid:3) ARGE CARDINAL IDEALS 21 Remark 3.33. Comparing the statements of Theorem 3.23 and 3.32, one wouldlike to strengthen Theorem 3.32 by replacing R (Π n ( κ )) with the pre-Ramsey idealNPreRam κ . However, it seems to be unknown whether or not this is possible. Theproblem is that it is not known whether Lemma 3.12 can be generalized to thepre-Ramsey ideal. See Question 7.1 and Question 7.2 in Section 7 below. Remark 3.34. Let us note that the preceding result of Baumgartner can be gen-eralized to the ideals of the form R (Π ξ ( κ )) where ξ ≥ ω and Π ξ ( κ ) denotes theΠ ξ -indescribability ideal defined in Remark 3.1. Furthermore, by iterating theRamsey operator one obtains ideals of the form R α (Π β ( κ )), and the previous The-orem of Baumgartner can also be generalized to these ideals. For example, if κ ∈ R m (Π β ( κ )) + , one has R m (Π β ( κ )) = R ( R m − (Π β ( κ ))) ∪ Π β +2 m ( κ ) . See [16] for more details.Next we give a second generic embedding characterization of sets which arepositive for the ideal R (Π n ( κ )). Taking n = − Theorem 3.35. For all n ∈ {− } ∪ ω , for all cardinals κ and all S ⊆ κ thefollowing are equivalent. (1) S ∈ R (Π n ( κ )) + . (2) There is a generic elementary embedding j : V → M ⊆ V [ G ] with criticalpoint κ and κ ∈ j ( S ) such that the following properties hold. (a) For all A ∈ V Vκ +1 and all Π n +2 -sentences ϕ we have (( V κ , ∈ , A ) | = ϕ ) V = ⇒ (( V κ , ∈ , A ) | = ϕ ) M . (b) For every regressive function f : [ S ] <ω → κ in V , it follows that M | = “there is a set H ⊆ S which is Π n -indescribable in κ and homogeneousfor f ”.Proof. The proof is very similar to that of Proposition 3.25. For (1) implies (2),suppose S ∈ R (Π n ( κ ) + . Let G ⊆ P ( κ ) / ( R (Π n ( κ ))) ↾ S ) be generic and let j : V → M = V κ /U G ⊆ V [ G ] be the corresponding generic ultrapower. Since Π n +2 ( κ ) ∗ ⊆R (Π n ( κ )) ∗ ⊆ U G , (2)(a) holds (see the proof of Proposition 3.5 or Proposition3.25). Fix a regressive function f : [ S ] <ω → κ in V . Since S ∈ R (Π n ( κ )) + there is a set H ⊆ S in V which is Π n -indescribable in κ and homogeneous for f . Then H = j ( H ) ∩ κ ∈ M and by (2)(a) together with the fact that the Π n -indescribability of H is expressible by a Π n +1 sentence over ( V κ , ∈ , H ), we concludethat M | = “ H ⊆ S is Π n -indescribable and homogeneous for f ”.For (2) implies (1), suppose there is a generic embedding j : V → M ⊆ V [ G ] asin (2). To show that S ∈ R (Π n ( κ )) + , fix a regressive function f : [ S ] <ω → κ . Forthe sake of contradiction assume every homogeneous set for f is in Π n ( κ ). Recallthat “ X ∈ Π n ( κ )” is expressible by a Σ n +1 sentence over ( V κ , ∈ , X ). Thus thereis a Π n +2 sentence ϕ over ( V κ , ∈ , f ) asserting that every homogeneous set for f isnot Π n -indescribable. By (2)(a), it follows that (( V κ , ∈ , f ) | = ϕ ) M and hence M | =“every homogeneous set for f is not Π n -indescribable”. This contradicts (2)(b). (cid:3) Splitting positive sets assuming weak compactness In this section we present a folklore result which states that assuming κ is weaklycompact, many large cardinal ideals on κ are nowhere κ -saturated. We put togethertechniques used by Hellsten [28, Theorem 2] in the context of Π n -indescribabilityand Brickhill-Welch [11] in the context of Π γ -indescribability, and note that previ-ously known methods allow for more general conclusions than what may have beenknown. Hellsten [28] attributes the following result to Tarski [45]. Theorem 4.1 (Tarski) . If κ is a weakly compact cardinal and I is a κ -completeideal on κ such that for every S ∈ I + there are S , S ∈ I + such that S = S ⊔ S ,then I is nowhere κ -saturated.Proof. Suppose S ∈ I + . We will show that I ↾ S is not κ -saturated. We define atree ( T, ⊇ ) where T ⊆ I + ∩ P ( S ) as follows. We will inductively define Σ α ⊆ α α ( T ) ⊆ I + such that for each j ∈ Σ α there is some S j ∈ Lev α ( T ) andLev α ( T ) = { s j | j ∈ Σ α } . Let Σ = { ∅ } and S ∅ = S . Suppose Σ α and Lev α ( T )have been defined. For each j ∈ Σ α we have S j ∈ Lev α ( T ) ⊆ I + . Let S j a , S j a ∈ I + be such that S j = S j a ⊔ S j a and add j a , j a α +1 . In other words,Σ α +1 = { j a i | j ∈ Σ α ∧ i = 0 , } . Now, suppose Σ α and Lev α ( T ) have beendefined for all α < η where η is a limit ordinal. For each j ∈ η 2, if j ↾ α ∈ Σ α forall α < η and T α<η S j ↾ α ∈ I + , then let S j = T α<η S j ↾ α and add j to Σ η . Thiscompletes the definition of the tree ( T, ⊇ ). It is relatively straight forward to showthat ( S, ⊇ ) is a κ -tree, and thus, applying the weak compactness of κ , must have acofinal branch b ⊆ T , which provides a partition of S into κ disjoint I + -sets. (cid:3) The following result is attributed to L´evy and Silver in [36]. Corollary 4.2 (L´evy-Silver) . If κ is weakly compact and not measurable then everynormal ideal on κ is nowhere κ -saturated.Proof. Suppose κ is weakly compact and not measurable. Fix a normal ideal I on κ . By Theorem 4.1, it suffices to show that for every S ∈ I + there are S , S ∈ I + such that S = S ⊔ S . Suppose S ∈ I + does not split. Then U := ( I ↾ S ) ∗ = { X ⊆ κ | ( κ \ X ) ∩ S ∈ I } is a normal measure on κ , a contradiction. (cid:3) Corollary 4.3 (Folklore) . If κ is weakly compact and I is a normal ideal which isdefinable over H + κ , then I is nowhere κ -saturated. Proof. Suppose κ is weakly compact and I is a normal ideal on κ which is definableover H κ + . By Theorem 4.1, it suffices to show that every I -positive set splits.Choose S ∈ I + and suppose S does not split. Then ( I ↾ S ) ∗ is a normal measureon κ . Let j : V → M be the ultrapower by ( I ↾ S ) ∗ . Then H κ + = H Mκ + whichimplies ( I ↾ S ) ∗ ∈ M , which is a contradiction since a normal measure cannot bean element of its own ultrapower. (cid:3) Since the assumption that a cardinal κ is Π n -indescribable, almost ineffable orRamsey implies that κ is weakly compact, the next corollary follows directly from The author would like to thank Sean Cox for pointing out the statement and proof of thisresult. ARGE CARDINAL IDEALS 23 Corollary 4.3. Let us note that Hellsten attributes Corollary 4.4(1) to Solovay andTarski (see the end of Section 1 in [28]). Corollary 4.4. The following hold. (1) (Hellsten [28] ) For n < ω , if κ is Π n -indescribable then the Π n -indescribableideal on κ is nowhere κ -saturated. (2) If κ is (almost) ineffable then the (almost) ineffable ideal on κ is nowhere κ -saturated. (3) If κ is Ramsey then the Ramsey ideal on κ is nowhere κ -saturated. (4) If κ is weakly compact and subtle then the subtle ideal on κ is nowhere κ -saturated. (5) For n < ω , if κ is Π n -indescribable then the Π n -indescribable ideal on κ isnowhere κ -saturated. See Section 7 for several questions related to Corollary 4.4 which appear toremain open.The next result can be used to show that if a cardinal κ satisfies a large enoughlarge cardinal property, then many ideals on κ will not be κ + -saturated. This resultbears some similarity to work of Leary [37] on ideal families and is implicit in thework of Hellsten [27]. Proposition 4.5 (Hellsten [27]) . Suppose that for each α ≤ κ , I α is a normal idealon α and let I = h I α | α ≤ κ i . We define the trace operation Tr I : P ( κ ) → P ( κ ) by Tr I ( X ) = { α < κ | X ∩ α ∈ I + α } and suppose that the following conditions hold. (1) For every S ∈ I + κ we have S \ Tr I ( S ) ∈ I + κ . (2) Suppose there is a normal ideal J ⊇ I κ on κ and there is an A ∈ J + suchthat the filter ( J ↾ A ) ∗ is closed under Tr I .Then I κ ↾ A is not κ + -saturated.Proof. Suppose J ⊇ I κ is a normal ideal on κ and A ∈ J + is such that ( J ↾ A ) ∗ is closed under Tr I . Suppose I κ ↾ A is κ + -saturated. Recall that by [10, Theorem3.1], an ideal I on κ is κ + -saturated if and only if the ideals I ↾ S for S ∈ I + are the only normal ideals on κ which extend I . Therefore, since I κ ↾ A ⊆ J ↾ A and J ↾ A is a normal ideal, it follows that J ↾ A = ( I κ ↾ A ) ↾ B for some B ∈ ( I κ ↾ A ) + . Thus J ↾ A = I κ ↾ ( A ∩ B ). Now A ∩ B ∈ I + κ which implies( A ∩ B ) \ Tr I ( A ∩ B ) ∈ I + κ by assumption. Furthermore, since A ∩ B ∈ ( J ↾ A ) ∗ ,it follows that Tr I ( A ∩ B ) ∈ ( J ↾ A ) ∗ , which implies κ \ Tr I ( A ∩ B ) ∈ J ↾ A = I κ ↾ ( A ∩ B ) and hence ( A ∩ B ) \ Tr I ( A ∩ B ) ∈ I κ , a contradiction. (cid:3) From Proposition 4.5 we can easily derive the following result of Hellsten [27]. Corollary 4.6 (Hellsten [27]) . If κ is κ + - Π n -indescribable (i.e. Tr α Π n ( κ ) ( κ ) ∈ Π n ( κ ) + for all α < κ + ) then the Π n -indescribable ideal on κ is not κ + -saturated. Consistency results Recently progress has been made in generalizing forcing constructions from thenonstationary ideal to the weakly compact ideal. Such generalizations often re-quire a substantial amount of effort because there are some significant differencesbetween stationarity and weak compactness: for example, one cannot force ground Here we allow for ideals to be trivial. For example, when α is singular we have I α = P ( α ). model nonstationary sets to become stationary in an extension, whereas, in somesettings, ground model non–weakly compact sets can become weakly compact inafter certain kinds of forcing [36]. The main reasons for the success in this direc-tion are twofold. First, weak compactness has a simple characterization in termsof elementary embeddings which allows one to use standard techniques to arguethat weak compactness is preserved after certain Easton-support forcing iterations(see [21]). Second, weakly compact sets have a characterization in terms of 1-clubswhich resembles the definition of stationarity, so constructions which involve clubshooting forcings can sometimes be generalized to forcings which shoot 1-clubs. Inthe current section we briefly discuss these characterizations of weakly compactsets as well as some forcing constructions concerning the weakly compact ideal. Wealso discuss related characterizations of Π n -indescribability and state a few openquestions.5.1. n -clubs and indescribability embeddings. A transitive set M | = ZFC − ofsize κ with κ ∈ M and M <κ = κ is called a κ -model . The following folklore resultfollows easily from [9, Section 2]. Lemma 5.1. For all cardinals κ and S ⊆ κ the following are equivalent. (1) S is Π -indescribable. (2) For all A ⊆ κ there are κ -models M and N with A, S ∈ M such that thereis an elementary embedding j : M → N with critical point κ and κ ∈ j ( S ) . Definition 5.2 (Hauser [25]) . Suppose κ is inaccessible. For n ≥ 0, a κ -model N is Π n -correct at κ if and only if V κ | = ϕ ⇐⇒ ( V κ | = ϕ ) N for all Π n -formulas ϕ whose parameters are contained in N ∩ V κ +1 . Remark 5.3. Notice that every κ -model is Π -correct at κ .An easy adaptation of the arguments in [25] establishes the following. Theorem 5.4 (Hauser [25]) . The following statements are equivalent for everyinaccessible cardinal κ , every subset S ⊆ κ , and all < n < ω . (1) S is Π n -indescribable.(2) For every A ⊆ κ there is a κ -model M with A, S ∈ M for which there is aΠ n − -correct κ -model N and an elementary embedding j : M → N withcrit( j ) = κ such that κ ∈ j ( S ).(3) For every A ⊆ κ there is a κ -model M with A, S ∈ M for which there is aΠ n − -correct κ -model N and an elementary embedding j : M → N withcrit( j ) = κ such that κ ∈ j ( S ) and j, M ∈ N .Recall that κ is inaccessible if and only if it is Π -indescribable, and in this casethe Π -indescribable ideal equals NS κ . For κ a regular cardinal, we say that a set C ⊆ κ is 1 -closed below κ if for all inaccessible α < κ , C ∩ α stationary in α implies α ∈ C . Furthermore, we say that C ⊆ κ is 1 -club if it is stationary in κ and 1-closedbelow κ . More generally, for n < ω we say that C ⊆ κ is ( n + 1) -closed below κ iffor all Π n -indescribable α < κ , C ∩ α being Π n -indescribable in α implies α ∈ C .We say that C ⊆ κ is ( n + 1) -club if C is Π n -indescribable in κ and n -closed below κ . ARGE CARDINAL IDEALS 25 The following result, due to Sun [44] in the case n = 1 and due to Hellsten [26]for n > 1, provides a characterization of Π n -indescribability which resembles thedefinition of stationarity. Theorem 5.5 (Sun, Hellsten) . Suppose κ is Π n -indescribable. The following areequivalent for all S ⊆ κ . (1) S is Π n -indescribable. (2) For all n -clubs C ⊆ κ we have S ∩ C = ∅ . An easy consequence of Theorem 5.5 is that when κ is Π n -indescribable, the filterdual to the Π n -indescribable ideal on κ equals the filter generated by the collectionof n -club subsets of κ . In other words, we haveΠ n ( κ ) ∗ = { X ⊆ κ | there is an n -club C ⊆ κ such that C ⊆ X } Let us point out here that it is consistent that the 1-club filter on κ is nontrivialwhen κ is not weakly compact (see [44, Theorem 1.18]). Remark 5.6. Recall that when κ is inaccessible and S ⊆ κ is stationary there isa natural forcing to add a club subset of S , namelyCU( S ) = { p | p ⊆ S , | p | < κ and p is a closed set of ordinals } where conditions in CU( S ) are ordered by letting q ≤ p if and only if q is an endextension of p , meaning that p = q ∩ sup { α + 1 | α ∈ p } . A generic filter for CU( S )yields a club subset of κ which is contained in S . In general, there is no reason toexpect that this forcing preserves cofinalities. However, if S contains the singularcardinals below κ , then the forcing CU( S ) is well-behaved: if SING ∩ κ ⊆ S thenfor every regular γ < κ the poset CU( S ) contains a dense set that is γ -closed. See[33] for more details.Suppose γ is an inaccessible cardinal and A ⊆ γ is cofinal. For n ≥ 1, we definea poset T n ( A ) consisting of all bounded n -closed c ⊆ A ordered by end extension: c ≤ d if and only if d = c ∩ sup α ∈ d ( α + 1). A generic filter for T n ( A ) will in manycases provide an n -club contained in A . Notice that these results show that n -clubshooting forcings are, in general, more well-behaved than club shooting forcings.Indeed, let us show that T n ( A ) often has a closure property which club shootingforcings lack.Given a forcing P and an ordinal α , we define G α ( P ), a two-player game of perfectinformation as follows. Player I and Player II take turns to play conditions from P for α many moves, where Player I plays at odd stages and Player II plays at evenstages (including all limit stages). Let p β be the condition played at move β . Theplayer who played p β loses immediately unless p β ≤ p γ for all γ < β . If neitherplayer loses at any stage β < α , then Player II wins. As in [21, Definition 5.15], wesay that P is κ -strategically closed if and only if Player II has a winning strategyfor G κ ( P ). Lemma 5.7 (Hellsten [28]) . For n ≥ , if γ is inaccessible and A ⊆ γ is cofinal,then T n ( A ) is γ -strategically closed.Proof. We describe a winning strategy for player II in the game G κ ( T n ( A )). PlayerII begins the game by playing c = ∅ . At an even successor stage α + 2, player IIchooses a condition c α +2 ∈ T n ( A ) such that c α +2 (cid:12) c α +1 . At limit stages α < γ ,player II records an ordinal γ α = S β<α c β , chooses an element η α ∈ A \ ( γ α + 1) and plays c α = (cid:16)S β<α c β (cid:17) ∪ { η α } . In order to argue that c α is a condition in T n ( A ),we need to verify, letting c = S β<α c β , that c is not a Π n − -indescribable subsetof γ α . We can assume that γ α is Π n − -indescribable, as otherwise c ∩ γ α is clearlynot Π n − -indescribable. But then, by construction, { γ ξ | ξ < α is a limit ordinal } is a club (and hence an ( n − γ α disjoint from c , which implies that c isnot a Π n − -indescribable subset of γ α . Thus, c α is a valid play by Player II, andwe have described a winning strategy in G κ ( T n ( A )). (cid:3) The following result is due to Hellsten (see [26] and [28]) in the case in which n = 1 and to [19] for n > 1. Notice the forcing is the same for all n < ω in theproof of the following. The difficult part for n > n -indescribablesets are preserved. Theorem 5.8 (Hellsten (for n = 1), Cody, Gitman, Lambie-Hanson (for n > . Suppose that n ≥ and S ⊆ κ is Π n -indescribable. Then there is a cofinality-preserving forcing extension in which S contains a -club and all Π n -indescribablesubsets of S from V remain Π n -indescribable. Proof. We only provide a definition of the forcing. The reader should consult [19]for details.Let P κ +1 = h ( P α , ˙ Q β ) | α ≤ κ + 1 , β ≤ κ i be an Easton-support iteration suchthat • if γ ≤ κ is inaccessible and S ∩ γ is cofinal in γ , then ˙ Q γ = ( T ( S ∩ γ )) V P γ ; • otherwise, ˙ Q γ is a P γ -name for trivial forcing. (cid:3) A theorem of Hellsten. Recall that the Mahlo operation M : P ( κ ) → P ( κ )defined by M ( X ) = { α < κ | X ∩ α is stationary in α } , can be iterated as followsto define the hierarchy of α -Mahlo cardinals for α < κ + . Baumgartner, Taylor andWagon [8] defined M α : P ( κ ) / NS κ → P ( κ ) / NS κ for α < κ + by letting M ([ X ]) =[ X ], M ([ X ]) = [ M ( X )], M α +1 ([ X ]) = M ( M α ([ X ]) and M β ([ X ]) = △{ M α ([ X ]) | α < β } where β < κ + is a limit ordinal and the diagonal intersection is takenrelative to some function f : κ → { M α ([ X ]) | α < β } . It is well known that suchdiagonal intersections are independent of the indexing used when the ideal one isworking with is normal. A cardinal κ is called α -Mahlo , where α ≤ κ + , if for all β < α we have M β ([ κ ]) > 0. A cardinal κ is called greatly Mahlo if it is κ + -Mahlo.If κ is κ + -Mahlo then there is a natural decomposition of κ into κ + almostdisjoint stationary sets [8], and henceNS κ ↾ REG = { X ⊆ κ | X ∩ REG ∈ NS κ } is not κ + -saturated. Jech and Woodin [33] showed that this result is sharp in thesense that NS κ ↾ REG can be saturated when κ is α -Mahlo where α < κ + , relativeto the existence of a measurable cardinal of Mitchell order α . In fact, Jech andWoodin gave an equiconsistency. If κ is α -Mahlo where α < κ + and NS κ ↾ REG is κ + -saturated, then there is an inner model with a measurable cardinal of Mitchellorder α . Furthermore, if κ is measurable with Mitchell order α < κ + then thereis a forcing extension in which κ is α -Mahlo and NS κ ↾ REG is κ + -saturated. ByGitik-Shelah [24], restricting to the regulars is necessary. Notice that the conclusion “ S contains a 1-club” directly implies that “ S contains an n -club”because for n < ω , every n -club is an ( n + 1)-club. ARGE CARDINAL IDEALS 27 Hellsten showed that the forcing construction of Jech and Woodin mentionedabove can be generalized from the nonstationary ideal NS κ to the the weaklycompact ideal Π ( κ ). First, Hellsten iterated the generalized Mahlo operationTr : P ( κ ) → P ( κ ) defined byTr ( X ) = { α < κ | X ∩ α is weakly compact in α } modulo the weakly compact ideal in order to define the notion of κ + –weak com-pactness. For A ⊆ κ we let [ A ] denote the equivalence class of A modulo theweakly compact ideal and, as in the definition of α -Mahlo cardinals and the non-stationary ideal, working modulo the weakly compact ideal we can iterate Tr todefine a sequence h Tr α ([ A ] ) | α < κ + i in the boolean algebra P ( κ ) / Π ( κ ). Definition 5.9. A cardinal κ is α -weakly compact , where α ≤ κ + , if and only ifTr β ([ κ ] ) > β < α .As in the case of κ + -Mahloness and the nonstationary ideal, it is easy to seethat if κ is κ + –weakly compact then the weakly compact ideal Π ( κ ) is not κ + -saturated. Hellsten used an Easton support iteration of iterated 1-club shootings toprove that, relative to the existence of a measurable cardinal, the weakly compactideal Π ( κ ) can be κ + -saturated when κ is weakly compact (take α = 1 in thefollowing). Theorem 5.10 (Hellsten [28]) . If κ is measurable with Mitchell order α ∈ κ + \ { } then there is a forcing extension in which κ is α - Π -indescribable and the weaklycompact ideal Π ( κ ) is κ + -saturated. The weakly compact reflection principle. The weakly compact reflectionprinciple holds at κ , which we write as Refl ( κ ) or Refl wc ( κ ), if and only if κ isa weakly compact cardinal and every weakly compact subset of κ has a weaklycompact proper initial segment; in other words, κ is weakly compact and S ∈ Π ( κ ) + implies there is an α < κ such that S ∩ α ∈ Π ( α ) + . It is easy to see thatthe weakly compact reflection principle Refl ( κ ) implies that κ is ω -weakly compact(see [17]). In this section we address the question: what is the relationship betweenthe weakly compact reflection principle Refl ( κ ) and the α –weak compactness of κ ? Several related questions remain open.It follows from a result of Bagaria, Magidor and Sakai [5] that, in L , a cardinal κ is Π -indescribable if and only if Refl ( κ ). Since there are many ω -weakly compactcardinals below any Π -indescribable cardinal, the Bagaria-Magidor-Sakai resultshows that κ being ω -weakly compact does not imply Refl ( κ ). The author [17]gave a forcing construction for adding a non-reflecting weakly compact set whichalso establishes that the α -weak compactness of κ , where α < κ + , does not implyRefl ( κ ). Theorem 5.11 (Cody [17]) . Suppose κ is ( α + 1) -weakly compact where α < κ + .Then there is a cofinality-preserving forcing extension in which κ remains ( α + 1) -weakly compact and there is a weakly compact subset of κ with no weakly compactproper initial segment, thus Refl ( κ ) fails in the extension. One may also wonder, does the weakly compact reflection principle Refl ( κ )imply that κ must be ( ω + 1)-weakly compact? The author and Sakai [20] provedthat the answer is no. Theorem 5.12 (Cody-Sakai [20]) . Suppose the weakly compact reflection principle Refl ( κ ) holds. Then there is a forcing extension in which Refl ( κ ) holds and κ isthe least ω -weakly compact cardinal. In the course of proving Theorem 5.12, the authors generalized the well-knownfact that after κ -c.c. forcing the nonstationary ideal of the extension equals theideal generated by the ground model nonstationary ideal, to the weakly compactideal. Theorem 5.13 (Cody-Sakai [20]) . Suppose κ is a weakly compact cardinal, assumethat P = h ( P α , ˙ Q α ) | α < κ i ⊆ V κ is a good Easton-support forcing iteration suchthat for each α < κ , (cid:13) P α “ ˙ Q α is α -strategically closed”. Let ˙Π ( κ ) be a P κ -namefor the weakly compact ideal of the extension V P κ and let ˇΠ ( κ ) be a P κ -check namefor the weakly compact ideal of the ground model. Then (cid:13) P κ ˙Π ( κ ) = ˇΠ ( κ ) .Proof. Since ground-model 1-club subsets of κ remain 1-club after κ -c.c. forcing, itis easy to see that (cid:13) P κ ˙Π ( κ ) ⊇ ˇΠ ( κ ).To show that (cid:13) P κ ˙Π ( κ ) ⊆ ˇΠ ( κ ), we will show that if p (cid:13) P κ ˙ X ∈ ˙Π ( κ ), thenthere is B ∈ Π ( κ ) V with p (cid:13) P κ ˙ X ⊆ B . Suppose p (cid:13) P κ ˙ X ∈ ˙Π ( κ ). Since P κ ⊆ V κ is κ -c.c. we may assume that ˙ X ∈ H κ + . By the fullness principle, take a P κ -name˙ A ∈ H κ + for a subset of κ such that p (cid:13) P κ “for every κ -model M with κ, ˙ A, ˙ X ∈ M and for every ( ∗ )elementary embedding j : M → N where N is a κ -modelwe have κ / ∈ j ( ˙ X )”Let B = { α < κ | ∃ q ∈ P κ ( q ≤ p ) ∧ ( q (cid:13) P κ α ∈ ˙ X ) } and notice that B ∈ V and p (cid:13) P κ ˙ X ⊆ B . Thus, to complete the proof it will suffice to show that B ∈ Π ( κ ) V .Suppose B / ∈ Π ( κ ) V . Using the weak compactness of B in V , let M be a κ -modelwith κ, B, ˙ A, P κ , ˙ X, p, . . . ∈ M and let j : M → N be an elementary embeddingwith critical point κ such that κ ∈ j ( B ) where N is a κ -model. Since κ ∈ j ( B ),it follows by elementarity that there is a condition r ∈ j ( P κ ) with r ≤ j ( p ) = p such that r (cid:13) j ( P κ ) κ ∈ j ( ˙ X ). Let G ⊆ P κ be generic over V with r ↾ κ ∈ G . Since P κ is κ -c.c. the model N [ G ] is closed under <κ -sequences in V [ G ]. Furthermore,the poset j ( P κ ) /G is κ -strategically closed in N [ G ]. Thus, working in V [ G ] wecan build a filter H ⊆ j ( P κ ) /G which is generic over N [ G ] with r/G ∈ H . Letˆ G denote the filter for j ( P κ ) obtained from G ∗ H and notice that r ∈ ˆ G . Sinceconditions in P κ have support bounded below the critical point of j , it follows that j [ G ] ⊆ ˆ G . Thus the embedding extends to j : M [ G ] → N [ ˆ G ]. Since r ∈ ˆ G and r (cid:13) j ( P κ ) κ ∈ j ( ˙ X ), we have κ ∈ j ( ˙ X G ). Notice that p ∈ G ; this contradicts ( ∗ )since M [ G ] and N [ ˆ G ] are κ -models. (cid:3) See Section 7 for some relevant open questions. Such an iteration is good if if for all α < κ , if α is inaccessible, then ˙ Q α is a P α -name fora poset such that (cid:13) P α ˙ Q α ∈ ˙ V κ , where ˙ V κ is a P α -name for ( V κ ) V P α and, otherwise, ˙ Q α is a P α -name for trivial forcing. ARGE CARDINAL IDEALS 29 A (cid:3) ( κ ) -like principle consistent with weak compactness. Recall thatTodorˇcevi´c’s principle (cid:3) ( κ ) [46] implies that κ is not weakly compact. In thissection we survey some results from [19], which concern forcing a (cid:3) ( κ )-like principleto hold at a weakly compact cardinal κ .Recall that Todorˇcevi´c’s principle (cid:3) ( κ ) asserts that there is a κ -length coher-ent sequence of clubs ~C = h C α | α ∈ lim( κ ) i that cannot be threaded. For anuncountable cardinal κ , a sequence ~C = h C α | α ∈ lim( κ ) i of clubs C α ⊆ α iscalled coherent if whenever β is a limit point of C α we have C β = C α ∩ β . Givena coherent sequence ~C , we say that C is a thread through ~C if C is a club subsetof κ and C ∩ α = C α for every limit point α of C . A coherent sequence ~C iscalled a (cid:3) ( κ ) -sequence if it cannot be threaded. Gitman, Lambie-Hanson and theauthor [19], and independently Welch and Brickhill [11], introduced generalizationsof (cid:3) ( κ ), by replacing the use of clubs with n -clubs. The Brickhill-Welch principle (cid:3) n ( κ ) is defined differently from the principle (cid:3) n ( κ ) from [19] which we presenthere. Although it is easy to see that when κ is Π n -indescribable the Brickhill-Welchprinciple (cid:3) n ( κ ) implies (cid:3) n ( κ ), but it is not known whether the converse holds. See[19] for more information on the relationship between (cid:3) n ( κ ) and (cid:3) n ( κ ).For n < ω and X ⊆ κ , we define the n -trace of X to beTr n ( X ) = { α < κ | X ∩ α ∈ Π n ( α ) + } . Notice that when X = κ , Tr n ( κ ) is the set of Π n -indescribable cardinals below κ , and in particular Tr ( κ ) is the set of inaccessible cardinals less than κ . Foruniformity of notation, let us say that an ordinal α is Π − -indescribable if it isa limit ordinal, and if α is a limit ordinal, S ⊆ α is Π − -indescribable if it isunbounded in α . Thus, if X ⊆ κ , thenTr − ( κ ) = { α < κ | α is a limit ordinal and sup( X ∩ α ) = α } . Definition 5.14. Suppose n < ω and Tr n − ( κ ) is cofinal in κ . A sequence ~C = h C α | α ∈ Tr n − ( κ ) i is called a coherent sequence of n -clubs if(1) for all α ∈ Tr n − ( κ ), C α is an n -club subset of α and(2) for all α < β in Tr n − ( κ ), C β ∩ α ∈ Π n − ( α ) + implies C α = C β ∩ α .We say that a set C ⊆ κ is a thread through a coherent sequence of n -clubs ~C = h C α | α ∈ Tr n − ( κ ) i if C is n -club and for all α ∈ Tr n − ( κ ), C ∩ α ∈ Π n − ( α ) + implies C α = C ∩ α . Acoherent sequence of n -clubs ~C = h C α | α ∈ Tr n − ( κ ) i is called a (cid:3) n ( κ ) -sequence ifthere is no thread through ~C . We say that (cid:3) n ( κ ) holds if there is a (cid:3) n ( κ )-sequence ~C = h C α | α ∈ Tr n − ( κ ) i . Remark 5.15. Note that (cid:3) ( κ ) is simply (cid:3) ( κ ). For n = 1, the principle (cid:3) ( κ )states that there is a coherent sequence of 1-clubs h C α | α < κ is inaccessible i that cannot be threaded.Generalizing the fact that (cid:3) ( κ ) implies κ is not weakly compact, we prove thefollowing fact from [19]. Proposition 5.16. For every n < ω , (cid:3) n ( κ ) implies that κ is not Π n +1 -indescriba-ble. Proof. Suppose ~C = h C α | α ∈ Tr n ( κ ) i is a (cid:3) n ( κ )-sequence and κ is Π n +1 -indescr-ibable. Let M be a κ -model with ~C ∈ M . Since κ is Π n +1 -indescribable, we may let j : M → N be an elementary embedding with critical point κ and a Π n -correct N asin Theorem 5.4(2). By elementarity, it follows that j ( ~C ) = h ¯ C α | α ∈ Tr Nn − ( j ( κ )) i is a (cid:3) n ( j ( κ ))-sequence in N . Since N is Π n -correct, we know that κ ∈ Tr Nn − ( j ( κ ))and ¯ C κ must also be n -club in V . Since j ( ~C ) is a (cid:3) n ( j ( κ ))-sequence in N , it followsthat for every Π n − -indescribable α < κ if ¯ C κ ∩ α ∈ Π n − ( α ) + , then ¯ C κ ∩ α = C α ,and hence ¯ C κ is a thread through ~C , a contradiction. (cid:3) Remark 5.17. It follows easily that (cid:3) ( κ ) holds trivially at weakly compact car-dinals κ below which stationary reflection fails (see [19]). Thus, the task at handis not just to force (cid:3) ( κ ) to hold while κ is weakly compact, but to force (cid:3) ( κ ) tohold while κ is weakly compact and stationary reflection holds often below κ , sothat the coherence requirement in (cid:3) ( κ ) is nontrivial.By adapting techniques used to force the existence of (cid:3) ( κ )-sequences, Gitman,Lambie-Hanson and the author [19] proved the following two results. Theorem 5.18 (Cody, Gitman and Lambie-Hanson [19]) . If κ is κ + -weakly com-pact and the GCH holds, then there is a cofinality-preserving forcing extension inwhich (1) κ remains κ + -weakly compact and (2) (cid:3) ( κ ) holds. Theorem 5.19 (Cody, Gitman and Lambie-Hanson [19]) . Suppose that κ is Π -indescribable and the GCH holds. Then there is a cofinality-preserving forcingextension in which (1) (cid:3) ( κ ) holds, (2) Refl ( κ ) holds and (3) κ is κ + -weakly compact. Additionally, one can prove [19], that (cid:3) n ( κ ) is incompatible with simultaneousreflection of Π n -indescribable sets; a similar result was proven by Brickhill andWelch [11] using their principle (cid:3) n ( κ ). Theorem 5.20. Suppose that ≤ n < ω , κ is Π n -indescribable and (cid:3) n ( κ ) holds.Then there are two Π n -indescribable subsets S , S ⊆ κ that do not reflect simulta-neously, i.e., there is no β < κ such that S ∩ β and S ∩ β are both Π n -indescribablesubsets of β . Large cardinal ideals on P κ λ In this section we survey some results on two-cardinal versions of subtlety, inef-fability and indescribability. Recall that, for cardinals κ ≤ λ , Jech defined notionsof closed unbounded and stationary subsets of P κ λ . Carr [12] proved that, givena regular cardinal κ , Jech’s nonstationary ideal NS κ,λ is the minimal normal fine κ -complete ideal on P κ λ . Using a two-cardinal version of the cumulative hierarchyup to κ , Baumgartner introduced a notion of Π n -indescribability for sets S ⊆ P κ λ (see Section 6.2 below), which led to the consideration of associated idealsΠ n ( κ, λ ) = { X ⊆ P κ λ | X is not Π n -indescribable } . ARGE CARDINAL IDEALS 31 Recall that a cardinal κ is inaccessible if and only if it is Π -indescribable, andfurthermore, when κ is inaccessible we have Π ( κ ) = NS κ . Thus, one might expectthat when P κ λ is Π -indescribable the Π -indescribable ideal Π ( κ, λ ) equals Jech’snonstationary ideal on P κ λ . However, this is not the case. The author proved [18]that, when it is nontrivial, the ideal Π ( κ, λ ) equals the minimal strongly normal ideal on P κ λ , which does not equal NS κ , and which consists of all non– stronglystationary subsets of P κ λ (see Section 6.1 for a discussion of strong stationarityand Section 6.2 for more details regarding this result). This leads to a definition of1 -club subset of P κ λ (see Definition 6.7 below), and the following characterizationof Π -indescribable subsets of P κ λ : when P κ λ is Π -indescribable, a set S ⊆ P κ λ is Π -indescribable if and only if S ∩ C = ∅ for all 1-clubs C ⊆ P κ λ (see Section6.2 below).In Section 6.3 we survey some results on two-cardinal versions of subtlety andineffability.Finally, in Section 6.4, motivated by [30], we give generic embedding character-izations of two-cardinal indescribability, subtlety and ineffability.6.1. Stationary vs. strongly stationary subsets of P κ λ . Throughout thissection we assume κ ≤ λ are cardinals and κ is a regular cardinal. Recall that anideal I on P κ λ is normal if for every X ∈ I + and every function f : P κ λ → λ with { x ∈ X | f ( x ) ∈ x } ∈ I + there is a Y ∈ P ( X ) ∩ I + such that f ↾ Y is constant.Equivalently, an ideal I on P κ λ is normal if and only if for every { X α | α < κ } ⊆ I the set ▽ α<κ X α = def { x | x ∈ X α for some α ∈ x } is in I . An ideal I on P κ λ is fine if and only if { x ∈ P κ λ | α ∈ x } ∈ I ∗ for every α < λ . Jech [32] generalizedthe notion of closed unbounded and stationary subsets of cardinals to subsets P κ λ and proved that the nonstationary ideal NS κλ is a normal fine κ -complete ideal on P κ λ . Carr [12] proved that, when κ is a regular cardinal, the nonstationary idealNS κ,λ is the minimal normal fine κ -complete ideal on P κ λ .When considering ideals on P κ λ for κ inaccessible, it is quite fruitful to workwith a different notion of closed unboundedness obtained by replacing the structure( P κ λ, ⊆ ) with a different one. For x ∈ P κ λ we define κ x = | x ∩ κ | and we define anordering ( P κ λ, < ) by letting x < y if and only if x ∈ P κ y y. Given a function f : P κ λ → P κ λ we let C f = def { x ∈ P κ λ | x ∩ κ = ∅ ∧ f [ P κ x x ] ⊆ P κ x x } . We say that a set C ⊆ P κ λ is weakly closed unbounded if there is an f such that C = C f . Moreover, X ⊆ P κ λ is called strongly stationary if for every f we have C f ∩ X = ∅ . An ideal I on P κ λ is strongly normal if for any X ∈ I + and function f : P κ λ → P κ λ such that f ( x ) < x for all x ∈ X there is Y ∈ P ( X ) ∩ I + such that f ↾ Y is constant. It follows easily that an ideal I on P κ λ is strongly normal if andonly if for any { X a | a ∈ P κ λ } ⊆ I the set ▽ < X a = def { x | x ∈ X a for some a < x } is in I . Note that an easy argument shows that if κ is λ -supercompact then theprime ideal dual to a normal fine ultrafilter on P κ λ is strongly normal. Matet [40]showed that if κ is Mahlo then the collection of non–strongly stationary setsNSS κ,λ = def { X ⊆ P κ λ | ∃ f : P κ λ → P κ λ such that X ∩ C f = ∅ } is the minimal strongly normal ideal on P κ λ . Improving this, Carr, Levinski andPelletier obtained the following. Theorem 6.1 (Carr-Levinski-Pelletier [15]) . P κ λ carries a strongly normal idealif and only if κ is Mahlo or κ = µ + where µ <µ = µ ; moreover, in this case NSS κ,λ is the minimal such ideal. In these cases, since every strongly normal ideal on P κ λ is normal, we haveNS κ,λ ⊆ NSS κ,λ . The following lemma, due to Zwicker (see the discussion on page 61 of [15]), showsthat if κ is weakly inaccessible the previous containment is strict. Lemma 6.2. If κ is weakly inaccessible then NS κ,λ is not strongly normal. Corollary 6.3. If κ is Mahlo then NSS κ,λ is nontrivial and NS κ,λ ( NSS κ,λ . Indescribable subsets of P κ λ . According to [1] and [13], in a set of hand-written notes, Baumgartner [6] defined a notion of indescribability for subsets of P κ λ as follows. Give a regular cardinal κ and a set of ordinals A ⊆ ORD, considerthe hierarchy: V ( κ, A ) = AV α +1 ( κ, A ) = P κ ( V α ( κ, A )) ∪ V α ( κ, A ) V α ( κ, A ) = [ β<α V β ( κ, A ) for α a limitClearly V κ ⊆ V κ ( κ, A ) and if A is transitive then so is V α ( κ, A ) for all α ≤ κ . See[13, Section 4] for a discussion of the restricted axioms of ZFC satisfied by V κ ( κ, λ )when κ is inaccessible. Definition 6.4 (Baumgartner [6]) . Let S ⊆ P κ λ . We say that S is Π n -indescriba-ble if for every R ⊆ V κ ( κ, λ ) and every Π n -formula ϕ such that ( V κ ( κ, λ ) , ∈ , R ) | = ϕ ,there is an x ∈ S such that x ∩ κ = κ x and ( V κ x ( κ x , x ) , ∈ , R ∩ V κ x ( κ x , x )) | = ϕ .It is not too difficult to see that κ is supercompact if and only if P κ λ is Π -indescribable for all λ ≥ κ . See [18, Section 1] for a more detailed discussion ofthe way in which the Π -indescribability of P κ λ fits in with other large cardinalnotions. Remark 6.5. As noted in [13], standard arguments using the coding apparatusavailable in V κ ( κ, λ ) show that if λ <κ = λ , then we can replace R ⊆ V κ ( κ, λ ) inDefinition 6.4 by any finite sequence R , . . . , R k of subsets of V κ ( κ, λ ).Abe [1, Lemma 4.1] showed that if P κ λ is Π n -indescribable thenΠ n ( κ, λ ) = { X ⊆ P κ λ | X is not Π n -indescribable } is a strongly normal proper ideal on P κ λ .As mentioned above, the next theorem suggests that one should use strong sta-tionarity instead of stationarity when generalizing the notion of 1-club subset of κ to that of P κ λ . Theorem 6.6 (Cody [18]) . If κ is Mahlo then S ⊆ P κ λ is in NSS + κ,λ if and onlyif S is Π -indescribable (i.e. first-order indescribable); in other words, Π ( κ, λ ) = NSS κ,λ . ARGE CARDINAL IDEALS 33 Following [44], the author showed [18] that the Π -indescribable subsets of P κ λ can be characterized using the following notion of 1-club subsets of P κ λ . Definition 6.7 (Cody [18]) . We say that C ⊆ P κ λ is 1 -club if and only if(1) C ∈ NSS + κ,λ and(2) C is 1 -closed , that is, for every x ∈ P κ λ , if κ x is an inaccessible cardinaland C ∩ P κ x x ∈ NSS + κ x ,x then x ∈ C . Proposition 6.8 (Cody [18]) . Suppose P κ λ is Π -indescribable and S ⊆ P κ λ .Then S is Π -indescribable if and only if S ∩ C = ∅ for every -club C ⊆ P κ λ . The author also proved [18] that two-cardinal indescribability can be character-ized using elementary embeddings which resemble those considered by Schanker[42] in his work on nearly λ -supercompact cardinals. We say that a set W ⊆ P κ λ is weakly compact if for every A ⊆ λ there is a transitive M | = ZFC − with λ, A, W ∈ M and M <κ ∩ V ⊆ M , a transitive N and an elementary embedding j : M → N with critical point κ such that j ( κ ) > λ and j ” λ ∈ j ( W ). Theorem 6.9 (Cody [18]) . Suppose κ ≤ λ are cardinals with λ <κ = λ . A set W ⊆ P κ λ is Π -indescribable if and only if it is weakly compact. See [18, Section 1] for a detailed discussion of how the weak compactness of P κ λ ,or in Schanker’s terminology, the near λ -supercompactness of κ fits in with otherlarge cardinal notions.6.3. Subtle, strongly subtle and ineffable subsets of P κ λ . Menas [41] defineda set S ⊆ P κ λ to be subtle if for every club C ⊆ P κ λ and every function ~S = h S x | x ∈ S i where S x ⊆ x for all x ∈ S , there are x, y ∈ C ∩ S such that x ( y and S x = S y ∩ x . Menas proved that no matter how large λ is, the subtlety of P κ λ isnot a stronger assumption than the subtlety of κ . Theorem 6.10 (Menas [41]) . If κ is subtle then P κ λ is subtle for all λ ≥ κ . Menas also showed that the subtlety of P κ λ implies the usual two-cardinal di-amond principle ♦ κ,λ , and thus the subtlety of κ implies ♦ κ,λ for all λ ≥ κ . Asurprising result of Usuba demonstrates that Menas’s version of two-cardinal sub-tlety does not behave as expected: Usuba proved that, in general, the subtlety of P κ λ , where λ ≥ κ , does not imply the subtlety of κ . Theorem 6.11 (Usuba [47]) . Suppose λ is a measurable cardinal. Then there is aregular uncountable κ < λ such that P κ λ is subtle but κ is not subtle. As noted above, when studying P κ λ combinatorics where κ is inaccessible, it canbe fruitful to shift attention from the structure ( P κ λ, ⊆ ) to the structure ( P κ λ, < )where x < y if and only if x ∈ P κ y y (see Section 6.1 above). Abe [2] defined aset S ⊆ P κ λ to be strongly subtle if for every function ~S = h S x | x ∈ S i where S x ⊆ P κ x x for x ∈ S , and for every weak club C ∈ NSS ∗ κ,λ there exist x, y ∈ S ∩ C with x < y (meaning x ∈ P κ y y ) such that S x = S y ∩ P κ x x . Abe proved [2] that thecollection NSSub κ,λ = { X ⊆ P κ λ | X is not strongly subtle } of non–strongly subtle subsets of P κ λ is a strongly normal ideal on P κ λ when it isnontrivial. Although Abe’s definition of strong subtlety of P κ λ for λ > κ still doesnot provide a hypothesis stronger than the subtlety of κ , it does not possess thesame unexpected behavior as Menas’s notion of subtlety. Theorem 6.12 (Abe [2]) . If κ is subtle then P κ λ is subtle for all λ ≥ κ . Further-more, if P κ λ is strongly subtle for some λ ≥ κ then κ is subtle. Jech [32] defined a natural two-cardinal version of ineffability, which was usedby Magidor [39] to characterize supercompactness. Carr [14] showed that thereis an associated normal ideal. A set S ⊆ P κ λ is said to be ineffable if for everyfunction ~S = h S x | x ∈ S i where S x ⊆ x for x ∈ S , there is a D ⊆ λ such that { x ∈ S | S x = D ∩ x } ∈ NS + κ,λ . Carr proved P κ λ is ineffable if and only if thecollection NIn κ,λ = { X ⊆ P κ λ | X is not ineffable } is a normal ideal on P κ λ .6.4. Generic embedding characterizations of large cardinal ideals on P κ λ . In this section we provide generic embedding characterizations of various largecardinal ideals on P κ λ . First, let us consider the following well-known result (see[23]). Lemma 6.13 (Folklore) . Suppose κ is regular, κ ≤ λ with λ <κ = λ , and I is a κ -complete normal fine ideal on P κ λ . If G ⊆ P ( P κ λ ) /I is generic and j : V → M = V P κ λ /G ⊆ V [ G ] is the corresponding generic ultrapower then the followingconditions hold. (1) G extends the filter I ∗ dual to I . (2) crit( j ) = κ and j ( κ ) > λ . (3) [id] G = j ” λ ∈ M and thus for all X ∈ P ( P κ λ ) V we have X ∈ G if and onlyif j ” λ ∈ M j ( X ) . (4) For every function f : P κ λ → V in V we have j ( f )( j ” λ ) = [ f ] G . (5) M is wellfounded up to ( λ + ) V . This previous lemma easily leads to a generic embedding characterization ofstationary subsets of P κ λ . Proposition 6.14 (Folklore) . A set S ⊆ P κ λ is stationary if and only if there isa generic elementary embedding j : V → M ⊆ V [ G ] with critical point κ such that j ( κ ) > λ and j ” λ ∈ j ( S ) ∩ M .Proof. Suppose S is stationary and let G ⊆ P ( κ ) / (NS κ,λ ↾ S ) be generic. SinceNS κ,λ ↾ S is a κ -complete normal ideal on P κ λ we have crit( j ) = κ and [id] G = j ” λ .Hence j ” λ ∈ M . Since λ = j ( f )( j ” λ ) = [ f ] G where f ( x ) = ot( x ) and j ( κ ) = [ c κ ] G we have λ < j ( κ ). Clearly S ∈ (NS κ,λ ↾ S ) ∗ ⊆ G and hence κ ∈ j ( S ).Conversely, suppose we have such a j added by forcing with P . Fix a club C ⊆ P κ λ in V . Then I = { X ∈ P ( P κ λ ) | (cid:13) P j ” λ / ∈ j ( X ) } is a normal ideal in V and since NS κ,λ is the minimal κ -complete normal ideal on P κ λ we have NS ∗ κ,λ ⊆ I ∗ . This implies that C ∈ I ∗ , in other words, (cid:13) P j ” λ ∈ j ( C ).Thus M | = j ( S ) ∩ j ( C ) = ∅ and by elementarity S ∩ C = ∅ . (cid:3) We close by stating generic embedding characterizations of certain two-cardinalversions of indescribability, subtlety and ineffability. The proofs are similar to thosein Section 3 above. Proposition 6.17 below should be compared with [30, Lemma5.5]. ARGE CARDINAL IDEALS 35 Proposition 6.15. For n, m < ω , κ regular, λ ≥ κ and S ⊆ P κ λ , the followingare equivalent. (1) S is Π mn -indescribable. (2) There is a generic embedding j : V → M ⊆ V [ G ] with critical point κ suchthat crit( j ) = κ , j ( κ ) > λ , j ” λ ∈ j ( S ) ∩ M and for every Π mn -sentence ϕ over ( V κ ( κ, λ ) , ∈ , A ) where A ∈ P ( V κ ( κ, λ )) V we have (( V κ ( κ, λ ) , ∈ , A ) | = ϕ ) V = ⇒ (( V κ ( κ, j ” λ ) , ∈ , j ( A ) ∩ V κ ( κ, j ” λ )) | = ϕ ) M . Proposition 6.16. A set S ⊆ P κ λ is subtle if and only if there is a genericelementary embedding j : V → M ⊆ V [ G ] with critical point κ such that j ( κ ) > λ , j ” λ ∈ j ( S ) ∩ M and for every function ~S = h S x | x ∈ S i where S x ⊆ x for x ∈ S ,and every club C ⊆ P κ λ , we have j ( ~S )( x ) = j ( ~d )( j ” λ ) ∩ x for some x ∈ j ( S ∩ C ) with x ( j ” λ . Proposition 6.17. A set S ⊆ P κ λ is ineffable if and only if for every function ~S = h S x | x ∈ S i where S x ⊆ x for x ∈ S , there is a generic elementary embedding j : V → M ⊆ V [ G ] with critical point κ such that j ( κ ) > λ , j ” λ ∈ j ( S ) ∩ M and j ( ~S )( j ” λ ) = j ” D for some set D ∈ P ( λ ) V . Questions In this section we state several open questions relating to the topics of this article.Before stating each question we refer the reader to the relevant section above formore background information and motivation.See Section 3.4 and Remark 3.33 for background concerning the first two ques-tions. Question 7.1. For n > 1, if κ ∈ R (Π n ( κ )) + , does it follow that R (Π n ( κ )) = NPreRam κ ∪ Π n +2 ( κ )?It seems that in order to answer Question 7.1 one would need an answer to thefollowing. Question 7.2. Can Baumgartner’s Lemma 3.12 be generalized to the pre-Ramseyideal? Specifically, if S ⊆ κ is pre-Ramsey and f : [ S ] <ω → κ is a regressivefunction, does it follow that S \ A is not pre-Ramsey where A = { α ∈ S | ∃ X ⊆ S ∩ α such that X is Π n -indescribable in α for all n < ω and X is homogeneous for f } ?Baumgartner actually proved a version of Lemma 3.12 for the n -subtle ideal (for n < ω ), however the proof does not seem to generalize to the pre-Ramsey ideal.As far as the author is aware, the following two questions are open. See Section4 above for more information. Question 7.3. If κ is subtle (and not weakly compact), is the subtle ideal on κ nowhere κ -saturated?Note that Abe [2] proved that when λ > κ the subtle ideal on P κ λ is not λ -saturated.However, the proof does not seem to give information about the subtle ideal on κ . Question 7.4 (Hellsten) . For n < ω , if κ is weakly Π n -indescribable (and notinaccessible), is the weakly Π n -indescribable ideal nowhere κ -saturated? As pointed out by Hellsten, the techniques used in [28] do not seem to providean answer to the following. Question 7.5 (Hellsten) . Can the Π -indescribable ideal on κ be κ + -saturated?The following question remains open, see Section 5.3 or [17] for more details. Question 7.6. If κ is κ + -weakly compact, is there a forcing extension in which κ remains κ + -weakly compact and there is a weakly compact subset of κ with noweakly compact proper initial segment?It is known [19] that κ -strategically closed forcing cannot make ground modelweakly compact sets become weakly compact in the extension, it is also knownthat this can fail for Π -indescribable sets, in other words, a set which is notΠ -indescribable in the ground model can become so after κ -strategically closedforcing (see [19]). The following related question concerning the Π -indescribableideal remains open. See Theorem 5.13 above or [20] for more details. Question 7.7. Suppose κ is Π -indescribable and P = h ( P α , ˙ Q α ) | α < κ i ⊆ V κ is a good Easton-support iteration of length κ as in Theorem 5.12. Is the Π -indescribable ideal of the extension by P equal to the ideal generated by the groundmodel Π -indescribable ideal?As is the case for Question 7.5, using current techniques, namely those used in[19], we seem to be unable to answer the following. Question 7.8 ([19]) . From some large cardinal hypothesis on κ , can one force (cid:3) ( κ ) to hold nontrivially while preserving the Π -indescribability of κ ? Acknowledgment The author would like to thank Sean Cox, Monroe Eskew, Victoria Gitman andChris Lambie-Hanson for many helpful conversations regarding the topics of thisarticle. The author also thanks the anonymous referee for the detailed review whichgreatly improved this article. References [1] Yoshihiro Abe. Combinatorial characterization of Π -indescribability in P κ λ . Arch. Math.Logic , 37(4):261–272, 1998.[2] Yoshihiro Abe. Notes on subtlety and ineffability in P κ λ . Arch. Math. Logic , 44(5):619–631,2005.[3] F. G. Abramson, L. A. Harrington, E. M. Kleinberg, and W. 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