aa r X i v : . [ m a t h . L O ] J a n DENSE IDEALS AND CARDINAL ARITHMETIC
MONROE ESKEW
Abstract.
From large cardinals we show the consistency of normal, fine, κ -complete λ -dense ideals on P κ ( λ ) for successor κ . We explore the interplaybetween dense ideals, cardinal arithmetic, and squares, answering some openquestions of Foreman. Most large cardinals are characterizable in terms of elementary embeddings be-tween models of set theory that have a certain amount of agreement with the fulluniverse V . A typical large cardinal is the least ordinal moved by a nontrivial map j : V → M , where M is a transitive class, and the strength of the large cardinalassumption tends to increase as M gets closer to V . Such cardinals are inaccessi-ble and much more. This phenomenon can however be realized at small cardinalswhen the embedding j : V → M is defined in a forcing extension V [ G ]. The na-ture of the forcing adds another dimension to these “generic large cardinals,” andtheir strength tends to increase as the three models V , M , and V [ G ] more closelyresemble one another.Here, we consider generic versions of supercompactness at successor cardinalsthat are optimal in the sense that the forcing poset needed to produce the ele-mentary embedding is of the smallest possible size. We show that relative to asuper-almost-huge cardinal, there can exist a successor cardinal κ such that forevery regular λ ≥ κ , there is a normal, fine, κ -complete, λ -dense ideal on P κ ( λ ).As far as the author knows, this is the first result exhibiting the consistency of even saturated normal and fine ideals on P κ ( λ ) for a fixed successor κ and several valuesof λ simultaneously. The method used also has immediate application to show thenon-absoluteness of some cardinal characteristics of the powerset of a fixed regularcardinal µ , even between models with the same cardinals and same µ -sequences.Generic large cardinals can have strong influence over the combinatorial struc-ture of the universe in their vicinity. We explore the interplay between dense ideals,cardinal arithmetic, nonregular ultrafilters, and stationary reflection. We answertwo open questions posed by Foreman in [7] and provide a “global” counterexam-ple to an old conjecture in model theory. We also show some limitations of denseideals near singular cardinals, establishing the optimality some aspects of our con-sistency results. Finally we show that in contrast to traditional supercompactness,the strong forms of generic supercompactness considered here are compatible withJensen’s principle (cid:3) . 1. Preliminaries
Forcing.
First we review some essential facts about forcing. We refer thereader to [10] and [13] for background and details.A partial order P is said to be separative when p (cid:2) q ⇒ ( ∃ r ≤ p ) r ⊥ q . Everypartial order P has a canonically associated equivalence relation ∼ s and a separative quotient P s , which is isomorphic to P if P is already separative. In most cases wewill assume our partial orders are separative. For every separative partial order P ,there is a canonical complete boolean algebra B ( P ) with a dense set isomorphic to P . A map e : P → Q is an embedding when it preserves order and incompatibility.An embedding is said to be regular when it preserves the maximality of antichains.A order-preserving map π : Q → P is called a projection when π (1 Q ) = 1 P , and p ≤ π ( q ) ⇒ ( ∃ q ′ ≤ q ) π ( q ′ ) ≤ p . Lemma 1.1.
Suppose P and Q are partial orders.(1) G is a generic filter for P iff { [ p ] s : p ∈ G } is a generic filter for P s .(2) e : P → Q is a regular embedding iff for all q ∈ Q , there is p ∈ P such that forall r ≤ p , e ( r ) is compatible with q .(3) The following are equivalent:(a) There is a regular embedding e : P s → B ( Q s ) .(b) There is a projection π : Q s → B ( P s ) .(c) There is a Q -name ˙ g for a P -generic filter such that for all p ∈ P , there is q ∈ Q such that q (cid:13) p ∈ ˙ g .(4) Suppose e : P → Q is a regular embedding. If G is a filter on P let Q /G = { q : ¬∃ p ∈ G ( e ( p ) ⊥ q ) } . The following are equivalent:(a) H is Q -generic over V .(b) G = e − [ H ] is P -generic over V , and H is Q /G -generic over V [ G ] . Lemma 1.2.
Suppose P and Q are partial orders. B ( P s ) ∼ = B ( Q s ) iff the followingholds. Letting ˙ G, ˙ H be the canonical names for the generic filters for P , Q respec-tively, there is a P -name for a function ˙ f and a Q -name for a function ˙ f suchthat:(1) (cid:13) P ˙ f ( ˙ G ) is a Q -generic filter,(2) (cid:13) Q ˙ f ( ˙ H ) is a P -generic filter,(3) (cid:13) P ˙ G = ˙ f ˙ f ( ˙ G )1 ( ˙ f ( ˙ G )) , and (cid:13) Q ˙ H = ˙ f ˙ f ( ˙ H )0 ( ˙ f ( ˙ H )) .An isomorphism is given by p
7→ || ˇ p ∈ ˙ f ( ˙ H ) || B ( Q s ) . For a broader notion of “forcing equivalence,” the best that can be said in generalis the following:
Lemma 1.3.
Suppose P and Q are partial orders.(1) If e : P → Q is a regular embedding, and any Q -generic H yields V [ H ] = V [ e − [ H ]] , then there is a predense set A ⊆ B ( Q s ) such that B ( P s ) ∼ = B ( Q s ) ↾ a for all a ∈ A .(2) P and Q yield the same generic extensions iff for a dense set of p ∈ P , there is q ∈ B ( Q s ) such that B ( P s ) ↾ p ∼ = B ( Q s ) ↾ q . A partial order P is said to be κ -distributive if for any collection of maximalantichains in P , { A α : α < β < κ } , there is a maximal antichain A such that A refines A α for all α < β . P is called ( κ, λ ) -distributive if the same holds restrictedto antichains of size ≤ λ . Forcing with P adds adds no new functions from any α < κ to λ iff B ( P ) is ( κ, λ )-distributive.A strictly stronger property than distributivity is strategic closure. For a partialorder P and an ordinal α , we define a game G α ( P ) with two players Even and
Odd . Even starts by playing some element p ∈ P . At successor stages β + 1, the next ENSE IDEALS AND CARDINAL ARITHMETIC 3 player must play some element p β +1 ≤ p β . Even plays at limit stages β if possible,by playing a p β that is ≤ p γ for all γ < β . If Even cannot play at some stagebelow α , the game is over and Odd wins; otherwise
Even wins. We say that P is α -strategically closed if for every p ∈ P , Even has a winning strategy with first move p . Note that under this definition, every partial order is trivially ω -strategicallyclosed.A stronger property that κ -strategic closure is κ -closure. P is κ -closed when anydescending chain of length less than κ has a lower bound. P is κ -directed closed when any directed set of size <κ has a lower bound.For any partial order P , the saturation of P , sat( P ), is the least cardinal κ suchthat every antichain in P has size less than κ . Erd˝os and Tarski [6] proved thatsat( P ) is always regular. The density of P , d( P ), is the least cardinality of a densesubset of P . Clearly sat( P ) ≤ d( P ) + for any P . We say P is κ -saturated if sat( P ) ≤ κ ,and P is κ -dense if d( P ) ≤ κ . A synonym for κ -saturation is the κ -chain condition( κ -c.c.). The properties of distributivity, strategic closure, saturation, and density arerobust in the sense that they are absolute between P and B ( P ) for any separativepartial order P , and often inherited by intermediate forcings: Lemma 1.4.
Suppose e : P → Q is a regular embedding and κ is a cardinal.(1) If Q is κ -strategically closed, then so is P .(2) Q is κ -distributive iff P is κ -distributive and (cid:13) P Q / ˙ G is κ -distributive.(3) Q is κ -saturated iff P is κ -saturated and (cid:13) P Q / ˙ G is κ -saturated.(4) Q is κ -dense iff P is κ -dense and (cid:13) P Q / ˙ G is κ -dense. For any forcing P and any P -name ˙ X for a set of ordinals, there is a canonicallyassociated complete subalgebra A ˙ X ⊆ B ( P ) that captures ˙ X . It is the smallestcomplete subalgebra containing all elements of the form || ˇ α ∈ ˙ X || for α an ordinal. A ˙ X has the property that whenever G ⊆ P is generic, ˙ X G and G ∩ A ˙ X are definablefrom each other using the parameters B ( P ) and its powerset, as computed in theground model. In this case, we have V [ ˙ X G ] = V [ G ∩ A ˙ X ]. See [10, p. 247] fordetails.1.2. Ideals.
Let Z be any set. An ideal I on Z is a collection of subsets of Z closedunder taking subsets and pairwise unions. If κ is a cardinal, I is called κ -complete if it is also closed under unions of size less than κ . “Countably complete” is taken assynonymous with “ ω -complete.” I is called nonprincipal if { z } ∈ I for all z ∈ Z ,and proper if Z / ∈ I . Hereafter we will assume all our ideals are nonprincipal andproper.Let X = S Z . I is called fine if for all x ∈ X , { z : x / ∈ z } ∈ I . I is called normal if for any sequence h A x : x ∈ X i ⊆ I , the “diagonal union” { z : ∃ x ( x ∈ z ∈ A x ) } isin I . It is well-known that I is normal iff for any A ∈ P ( Z ) \ I and any function f on A such that f ( z ) ∈ z for all z ∈ A , there is an x such that f − ( x ) / ∈ I .To fix notation, let I ∗ = { Z \ A : A ∈ I } (the I -measure one sets ), I + = P ( Z ) \ I (the I -positive sets ), ˆ x = { z : x ∈ z } , and denote diagonal unions by ∇ x ∈ X A x . Notethat ∇ x ∈ X A x = S x ∈ X ˆ x ∩ A x .The following basic fact seems to have been previously overlooked–see, for ex-ample, the hypotheses of several theorems in [7] and [8]. Proposition 1.5.
All normal and fine ideals are countably complete.
MONROE ESKEW
Proof.
Let I be a normal and fine ideal on Z ⊆ P ( X ). If { x α : α < κ } is anenumeration of distinct elements of X , and T α<κ ˆ x α ∈ I ∗ , then I is κ + -complete.For suppose that { A α : α < κ } ⊆ I , but A = S α<κ A α ∈ I + . Then by hypothesis, A ∩ ( T α<κ ˆ x α ) ∈ I + . Let f : A → X be defined by f ( z ) = x α , where α is the leastordinal such that z ∈ A α . By normality, there is some A α ∈ I + , a contradiction.So it suffices to find an infinite set { x n : n < ω } ⊆ X such that T n<ω ˆ x n ∈ I ∗ .Since we assume I is proper and nonprincipal, X is infinite. We show that anyinfinite set of distinct elements of X suffices.Let { x n : n < ω } be distinct elements of X , and suppose the contrary, that B = { z : { x n : n < ω } * z } ∈ I + . By fineness, B ∩ ˆ x ∈ I + . For each z ∈ B ∩ ˆ x ,let n z be the largest integer such that { x , ..., x n z } ⊆ z . Let f : B ∩ ˆ x → X bedefined by f ( z ) = x n z . By normality, there is an n such that f − ( x n ) ∈ I + . Thenfor all z ∈ f − ( x n ), x n +1 / ∈ z . This contradicts fineness. (cid:3) Proposition 1.6. If I is a normal, fine, κ -complete ideal on Z ⊆ P ( κ ) , then κ ∈ I ∗ .Proof. Suppose A = { z ∈ Z : z is not an ordinal } ∈ I + . Let f : A → κ be such that f ( z ) is the least α ∈ z such that α * z . Then for some α , f − ( α ) ∈ I + . However, { z : α ⊆ z } ∈ I ∗ by fineness and κ -completeness. (cid:3) Proofs of the following facts can be found in [7]. If I is an ideal on Z , say A ∼ I B if the symmetric difference A ∆ B is in I . Let [ A ] I denote the equivalence class of A mod ∼ I . The equivalence classes form a boolean algebra under the obviousoperations, which we denote by P ( Z ) /I . Normality ensures a certain amount ofcompleteness of the algebra: Proposition 1.7.
Suppose I is a normal and fine ideal on Z ⊆ P ( X ) . If { A x : x ∈ X } ⊆ P ( Z ) , then ∇ A x is the least upper bound of { [ A x ] I : x ∈ X } in P ( Z ) /I . If we force with this algebra, we get a generic ultrafilter G on Z extending I ∗ . Wecan form the ultrapower V Z /G . If this ultrapower is well-founded for every generic G , then I is called precipitous. A combinatorial characterization of precipitousnessis given by the following: Theorem 1.8 (Jech-Prikry) . I is a precipitous ideal on Z iff the following holds:For any sequence h A n : n < ω i ⊆ P ( I + ) , such that for each n ,(1) B n = { [ a ] I : a ∈ A n } is a maximal antichain in P ( Z ) /I ,(2) B n +1 refines B n ,there is a function f with domain ω such that for all n , f ( n ) ∈ A n , and T n<ω f ( n ) = ∅ . For an ideal I , the saturation, density, distributivity, and strategic closure of I refers to that of the corresponding boolean algebra. The next proposition isimmediate from Theorem 1.8: Proposition 1.9. If I is an ω -complete, ω -distributive ideal, then I is precipi-tous. Proposition 1.10.
Suppose I is a κ -complete precipitous ideal on Z , and thereis no A ∈ I + such that I ↾ A is κ + -complete. Let G be P ( Z ) /I -generic, and let j : V → M be the associated elementary embedding, where M is the transitivecollapse of V Z /G . Then the critical point of j is κ . ENSE IDEALS AND CARDINAL ARITHMETIC 5
Proposition 1.11.
Let I be an ideal Z ⊆ P ( X ) . Then I is normal and fine iff (cid:13) P ( Z ) /I [id] ˙ G = j [ X ] . Proposition 1.12.
Suppose I is an ideal on Z ⊆ P ( X ) . If I is κ -complete and κ + -saturated, or if I is normal, fine, and | X | + -saturated, then every antichain in P ( Z ) /I has a system of pairwise disjoint representatives.Proof. If I is κ -complete, and { A α : α < κ } is an antichain, replace each A α with A α \ S β<α A β . If I is normal and fine, and { A x : x ∈ X } is an antichain, replace A x by A x ∩ ˆ x \ S y = x A y ∩ ˆ y . (cid:3) Theorem 1.13.
Suppose I is a countably complete ideal on Z , and every antichainin P ( Z ) /I has a system of pairwise disjoint representatives. Then:(1) I is precipitous.(2) P ( Z ) /I is a complete boolean algebra.(3) If G is generic over P ( Z ) /I , j : V → M is the associated embedding, and j [ λ ] ∈ M , then M is closed under λ -sequences from V [ G ] . If κ = µ + and I is a normal, fine, κ -complete ideal on P κ ( λ ), then I is not λ -saturated. For otherwise, let j : V → M ⊆ V [ G ] be a generic embedding arisingfrom I . Then M | = | [id] | = µ , and [id] = j [ λ ], so λ has cardinality µ in V [ G ]. Sothe smallest possible density of such an ideal is λ .1.3. Elementary embeddings.
Proofs of the following can be found in [11].
Lemma 1.14.
Suppose M and N are models of ZF − , j : M → N is an elementaryembedding, P ∈ M is a partial order, G is P -generic over M , and H is j ( P ) -genericover N . Then j has a unique extension ˆ j : M [ G ] → N [ H ] with ˆ j ( G ) = H iff j [ G ] ⊆ H . Lemma 1.15.
Suppose M , N are transitive models of ZFC with the same ordinals,and j : M → N is an elementary embedding. Then either j has a critical point, or j is the identity and M = N . Dense ideals from large cardinals
Here we show that it is consistent relative to an almost-huge cardinal that there isa normal, κ -complete, λ -dense ideal on P κ ( λ ), where κ is the successor of a regularcardinal µ , and λ ≥ κ is regular, for many particular choices for µ, λ . We also showthat relative to a super-almost-huge cardinal, there can exist a successor cardinal κ such that for every regular λ ≥ κ , there is a normal, κ -complete, λ -dense ideal on P κ ( λ ). This generalizes a theorem of Woodin about the relative consistency of an ℵ -dense ideal on ℵ , and has the following additional advantages: (1) An explicitforcing extension is taken, rather than an inner model of an extension. (2) Carefulconstructions within a model where the axiom of choice fails, as presented in [7],are avoided.Let us first recall the essential facts about almost-huge cardinals (see [11], 24.11).A cardinal κ is almost-huge if there is an elementary embedding j : V → M withcritical point κ , such that M The following are equivalent:(1) κ carries an almost-huge embedding j such that j ( κ ) = δ .(2) δ is inaccessible, and there is a sequence h U α : κ ≤ α < δ i such that: MONROE ESKEW (a) each U α is a normal, fine, κ -complete ultrafilter on P κ ( α ) ,(b) for α < β , U α = { A ⊆ P κ ( α ) : { z ∈ P κ ( β ) : z ∩ α ∈ A } ∈ U β } , and(c) for all α < δ and all f : P κ ( α ) → κ such that { z : f ( z ) ≥ ot( z ) } ∈ U α ,there is β such that α ≤ β < δ and { z : f ( z ∩ α ) = ot( z ) } ∈ U β .Furthermore, if a system as in (2) is given, the direct limit model and embeddingwitness the almost-hugeness of κ with target δ . A system as in (2) will be called an almost-huge tower . Almost-huge towerscapture almost-hugeness in a minimal way: Corollary 2.2. If κ has an almost-huge tower of height δ , and j : V → M is theembedding derived from the tower, then we have δ < j ( δ ) < δ + , and j [ δ ] is cofinalin j ( δ ) .Proof. For each α < δ , let M α be the transitive collapse of V P κ ( α ) /U α , and let j α : V → M α and k α : M α → M be the canonical embeddings, with j = k α ◦ j α .Since δ is inaccessible, j α ( κ ) < δ and j α ( δ ) = δ for each α < δ .If γ < j ( δ ), then there are some α, β < δ such that k α ( β ) = γ . Thus there areonly δ ordinals below j ( δ ). Also, there is η < δ such that j α ( η ) > β , so j ( η ) > γ ,and thus j [ δ ] is cofinal in j ( δ ). (cid:3) A super-almost-huge cardinal is a cardinal κ such that for all λ ≥ κ , there is analmost huge tower of height ≥ λ . The next result follows from considering the setof closure points under witnesses to property (c) in the tower characterization. Corollary 2.3. If κ has an almost-huge tower of Mahlo height δ , then for stationarymany α < δ , V α | = ZF C + κ is super-almost-huge. There is a vast gap in strength between almost-huge and huge: Theorem 2.4. If κ is a huge cardinal, then there is a stationary set S ⊆ κ suchthat for all α < β in S , α has an almost-huge tower of height β .Proof. Suppose j : V → M is an elementary embedding with critical point κ , j ( κ ) = δ , and M δ ⊆ M . Then κ carries an almost-huge tower ~U of length δ , and ~U ∈ M . Let F be the ultrafilter on κ defined by F = { X ⊆ κ : κ ∈ j ( X ) } . Let A = { α < κ : α carries an almost-huge tower of height κ } . Since κ ∈ j ( A ), A ∈ F .Now let c : κ → c ( α, β ) = 1 if α carries an almost-huge towerof height β , and c ( α, β ) = 0 otherwise. By Rowbottom’s theorem, let H ∈ F behomogeneous for c . We claim c takes constant value 1 on H . For if α ∈ A ∩ H ,then { α, κ } ∈ [ j ( A ∩ H )] , and j ( c )( α, κ ) = 1. (cid:3) Layering and absorption.Definition. We will call a partial order P ( µ, κ )-nicely layered when there is acollection L of atomless regular suborders of P such that:(1) for all Q ∈ L , Q is µ -closed and has size < κ ,(2) for all Q , Q ∈ L , if Q ⊆ Q , then (cid:13) Q Q / ˙ G is µ -closed, and(3) for all P -names ˙ f for a function from µ to the ordinals, and all Q ∈ L , thereis an Q ∈ L and an Q -name ˙ g such that Q ⊆ Q , and (cid:13) P ˙ f = ˙ g .We will say P is ( µ, κ )-nicely layered with collapses, ( µ, κ ) -NLC, when addi-tionally for all α < κ and all Q ∈ L , there is Q ∈ L such that Q ⊆ Q and (cid:13) Q | Q | = µ . ENSE IDEALS AND CARDINAL ARITHMETIC 7 Proposition 2.5. If L witnesses that P is ( µ, κ ) -nicely layered, then P is κ -c.c.and S L is dense in P .Proof. Suppose that { p α : α < η } ⊆ P , η ≥ κ , is a maximal antichain. Let ˙ f be aname of a function with domain { } such that f (0) = α iff p α ∈ G . There cannotbe a regular suborder Q of size < κ and a Q -name ˙ g that is forced to be equal to˙ f , since such a ˙ g would have < κ possible values for its range.Similarly, let p ∈ P be arbitrary, and let { p α : α < δ } be a maximal antichainwith p = p . Let ˙ f be a name of a function with domain { } such that f (0) = α iff p α ∈ G . If Q is a regular suborder and ˙ g is a Q -name such that (cid:13) P ˙ f = ˙ g , thenthere is some q ∈ Q forcing ˙ g (0) = 0, so q ≤ p . (cid:3) Proposition 2.6. If there exists a ( µ, κ ) -NLC poset, then α <µ < κ for all µ < κ .Proof. Let P be ( µ, κ )-NLC with layering L , and let α < κ . If Q ∈ L collapses α to µ , then we can build a µ -closed tree T ⊆ Q of height µ such that each level isan antichain of size ≥ α . α <µ ≤ | T | < κ . (cid:3) Lemma 2.7 (Folklore) . If P is a µ -closed partial order such that (cid:13) P | P | = µ , then B ( P ) ∼ = B (Col( µ, | P | )) .Proof. Pick a P -name ˙ f for a bijection from µ to ˙ G . We build a tree T ⊆ P thatis isomorphic to a dense subset of Col( µ, | P | ), and show that it is dense in P . Eachlevel will be a maximal antichain in P . Let the first level T = { P } . If levels { T β : β < α + 1 } are defined, below each p ∈ T α , pick a | P | -sized maximal antichainof conditions deciding ˙ f ( α ), and let T α +1 be the union of these antichains. If { T β : β < λ } is defined up to a limit λ , pick for each descending chain b throughthe previous levels, a | P | -sized maximal antichain of lower bounds to b , and set T λ equal to the union of these anithchains. It is easy to check that T λ is a maximalantichain. Let T = S α<µ T α . To show T is dense, let p ∈ P . Let q ≤ p be suchthat for some α < µ , q (cid:13) ˙ f ( α ) = p . q is compatible with some r ∈ T α +1 . Since r decides ˙ f ( α ) and forces it in ˙ G , r ≤ p . (cid:3) Lemma 2.8. Suppose µ < κ are regular, and P is ( µ, κ ) -NLC. If G is P -genericover V , then there is a forcing R ∈ V [ G ] such that R adds a filter H ⊆ Col( µ, <κ ) which is generic over V and such that (Ord µ ) V [ G ] = (Ord µ ) V [ H ] .Proof. Let L witness the ( µ, κ )-NLC property. In V [ G ], let R be the collectionof filters h ⊆ Col( µ, <α ) for α < κ which are generic over V , such that for some Q ∈ L , V [ h ] = V [ G ∩ Q ]. The ordering is end-extension.Let h ∈ R with Q ∈ L a witness, and let and α < κ be arbitrary. Let α < β < κ and Q ⊇ Q in L be such that in V [ h ], | Q / ( G ∩ Q ) | = | β | , and Q collapses β to µ . By the definition and Lemma 2.7, Q / ( G ∩ Q ) is equivalent in V [ h ] to Col( µ, β ),which is equivalent to the <µ -support product of Col( µ, γ ) for α ≤ γ ≤ β . Thefilter G ∩ Q therefore gives a filter h ′ ⊇ h on Col( µ, < β + 1) that is generic over V , with V [ h ′ ] = V [ Q ∩ G ].Let h ∈ R with Q ∈ L a witness, and let f : µ → Ord in V [ G ] be arbitrary.By the definition of ( µ, κ )-NLC, we can find some Q ⊇ Q in L such that f ∈ V [ G ∩ Q ]. By the previous paragraph, we may find Q ⊇ Q in L equivalent tosome Col( µ, < α ), and some filter h ′ ⊆ Col( µ, <α ) generic over V , extending h , andsuch that V [ G ∩ Q ] = V [ h ′ ]. MONROE ESKEW If F is generic over R , let H = S h ∈ F h . Since Col( µ, <κ ) is κ -c.c., H is generic,since any maximal antichain from V intersects some h ∈ F . By the above argu-ments, any f : µ → Ord in V [ G ] is in V [ H ]. Conversely, any f : µ → Ord in V [ H ]lives in some V [ h ] with h ∈ R , so is in V [ G ]. (cid:3) The anonymous collapse. Let κ be a regular cardinal whose regularity ispreserved by a forcing P . Let A ( P ) be the complete subalgebra of B ( P ∗ Add( κ ))generated by the canonical name for the Add( κ )-generic set. More precisely, if e : P ∗ Add( κ ) → B ( P ∗ Add( κ )) is the canonical dense embedding, A ( P ) is com-pletely generated by the elements of the form e ( h , ˙ {h α, i}i ). By [10, p. 247], wehave a canonical correspondence between such Add( κ )-generic sets X which comeafter forcing with κ -preserving posets P , and A ( P )-generic filters H . We will movebetween the two by writing, for example, X H and H X .In the case that α <µ < κ for all α < κ and P = Col( µ, <κ ), denote A ( P ) by A ( µ, κ ), and write B ( µ, κ ) for B (Col( µ, <κ ) ∗ Add( κ )). Lemma 2.9. If P is ( µ, κ ) -NLC, and H ⊆ A ( P ) is generic over V , then B ( P ∗ Add( κ )) /H is κ -distributive in V [ H ] .Proof. V [ H ] = V [ X H ] for the canonically associated X H ⊆ κ , and by forcing with B ( P ∗ Add( κ )) /H over V [ H ], we recover a filter G ∗ X H for P ∗ Add( κ ), generic over V . If G ∗ X is P ∗ Add( κ )-generic over V , then X codes all subsets of µ that livein V [ G ]. By the definition of ( µ, κ )-NLC, every z ∈ (Ord µ ) V [ G ] occurs in somesubmodel of the form V [ G ∩ Q ], where Q is isomorphic to Col( µ, α ) for some α < κ .Thus z ∈ V [ y ] for some y ⊆ µ in V [ G ], so (Ord µ ) V [ X ] ⊇ (Ord µ ) V [ G ] . Since Add( κ )adds no µ -sized sets of ordinals, (Ord µ ) V [ G ] = (Ord µ ) V [ G ∗ X ] ⊇ (Ord µ ) V [ X ] . Thus B ( P ∗ Add( κ )) /H is κ -distributive. (cid:3) Lemma 2.10. Let V be a countable transitive model of ZFC (or just assume genericextensions are always available), and assume (cid:13) V P κ is regular. If X ⊆ κ , thefollowing are equivalent:(1) X is A ( P ) -generic over V .(2) There is G ⊆ P such that G is generic over V , and X is Add( κ ) -generic over V ( P ) , where P = P κ ( κ ) V [ G ] .Proof. If X is A ( P )-generic then force with B ( P ∗ Add( κ )) /H X over V [ X ], obtaining G such that G ∗ X is P ∗ Add( κ )-generic over V . Then X is Add( κ )-generic over V [ G ], and since Add( κ ) V [ G ] = Add( κ ) V ( P ) , X is Add( κ )-generic over V ( P ).Suppose G ⊆ P is generic over V , and X is Add( κ )-generic over V ( P ), but not A ( P )-generic over V . Then some p ∈ Add( κ ) V ( P ) forces this with dom( p ) = α < κ ,and X ↾ α = p . Take Y ⊆ κ such that Y ↾ α = p that is Add( κ )-generic over thelarger model V [ G ]. Then Y is A ( P )-generic over V , and V ( P )[ Y ] can see this, butthis contradicts the property of p . So X was A ( P )-generic over V . (cid:3) Theorem 2.11. For any P that is is ( µ, κ ) -NLC, there is an isomorphism ι : A ( P ) → A ( µ, κ ) such that ι ( || α ∈ ˙ X || A ( P ) ) = || α ∈ ˙ X || A ( µ,κ ) for all α < κ .Proof. Let X be A ( P )-generic over V . There is a κ -distributive forcing over V [ X ]to get G such that G ∗ X is P ∗ Add( κ )-generic over V . By Lemma 2.8, we can dofurther forcing to obtain H ⊆ Col( µ, <κ ) generic over V such that (Ord µ ) V [ H ] =(Ord µ ) V [ G ] . By Lemma 2.10, X is also A ( µ, κ )-generic over V . ENSE IDEALS AND CARDINAL ARITHMETIC 9 Conversely, every A ( µ, κ )-generic X is A ( P )-generic. For suppose X is a coun-terexample. Then there is some ( p, ˙ q ) ∈ Col( µ, <κ ) ∗ Add( κ ) such that ( p, ˙ q ) (cid:13) ˙ X is not A ( P )-generic over V . Let Y be any A ( P )-generic set, and let P = P ( µ ) V [ Y ] .By the above, Y is A ( µ, κ )-generic over V . Thus we can force over V [ Y ] to get H ⊆ Col( µ, <κ ) such that H ∗ Y is B ( µ, κ )-generic over V . By the homogeneity ofthe Levy collapse, there is some automorphism π ∈ V such that p ∈ π [ H ] = H ′ .By the homogeneity of Cohen forcing, there is some automorphism σ in V ( P ) suchthat σ [ Y ] is a generic Y ′ such that Y ′ ↾ dom( ˙ q H ′ ) = ˙ q H ′ . Y ′ is also A ( P )-genericover V . However, ( p, ˙ q ) ∈ H ′ ∗ Y ′ , so we have a contradiction.This implies that we have a canonical correspondence between A ( P )- and A ( µ, κ )-generic filters, i.e. definable functions f, g such that for any generic H for A ( P ), f ( H ) is the generic for A ( µ, κ ) computed from X H , and vice versa, and g ( f ( H )) = H . For p ∈ A ( P ), put ι ( p ) = || p ∈ g ( ˙ H ) || A ( µ,κ ) . It is easy to see that ι is a completeembedding. For any q ∈ A ( µ, κ ), there is p ∈ A ( P ) forces that q ∈ f ( ˙ H ). Thus if H is generic for A ( µ, κ ) and ι ( p ) ∈ H , then p ∈ g ( H ), so q ∈ f ( g ( H )) = H , hence ι ( p ) ≤ q . The range of ι is dense, so it is an isomorphism. By the way we construct f and g , ι ( || α ∈ ˙ X || A ( P ) ) = || α ∈ ˙ X || A ( µ,κ ) . (cid:3) This machinery has some interesting applications to the absoluteness of someproperties of a given powerset. First, it is easy to see for regular µ < κ such that α <µ < κ for all α < κ , Col( µ, <κ ) × Add( µ, λ ) is ( µ, κ )-NLC for every λ . Thus if X is A ( µ, κ )-generic, then for any λ , we may further force to obtain a model which isa (Col( µ, <κ ) × Add( µ, λ )) ∗ Add( κ )-generic extension with the same Ord µ . Takinginner models given by such Col( µ, <κ ) × Add( µ, λ )-generic sets, we produce manymodels with the same cardinals and same P ( µ ), each assigning a different cardinalvalue for 2 µ . For example, if we add ω Cohen reals to any model of M of ZFC,this is the same as forcing with Col( ω, <ω ). There is for each uncountable ordinal α ∈ M , a generic extension with the same reals and same cardinals, in which itappears we have added α many Cohen reals.By using weakly compact cardinals, we can get even more dramatic examples. If κ is weakly compact, every κ -c.c. partial order captures small sets in small factors.To show this, first consider a partial order P of size κ . We can code P as A ⊆ κ , andby weak compactness, there is some transitive elementary extension ( V κ , ∈ , A ) ≺ ( M, ∈ , B ). If µ < κ , then any P -name for function f : µ → Ord has an equivalentname τ ∈ V κ by the κ -c.c. Since A ∈ M and M sees A as a regular suborder of B , M thinks that τ is a Q -name for some regular suborder Q of B . By elementarity, V κ thinks that τ is a Q -name for some regular Q of A . For P of arbitrary size, let τ be a P -name of size < κ , take some regular θ such that P , τ ∈ H θ , and take anelementary M ≺ H θ with P , τ ∈ M such that | M | = κ and M <κ ⊆ M . It is easyto see that M ∩ P is a regular suborder of P , and so the above considerations applyto show that there is some regular Q ⊆ P ∩ M ⊆ P of size < κ such that τ is a Q -name.Therefore, if κ is weakly compact and P is κ -c.c., the collection L of all regularsuborders of P of size < κ witnesses that P is ( ω, κ )-nicely layered. If P also forces κ = ℵ , then this collection also witnesses that P is ( ω, κ )-NLC. To check this, takeany Q ∈ L , any P -name τ of size < κ , and α < κ . Let H ⊆ Q be generic. Since κ is still weakly compact in V [ H ], there is some regular Q ⊆ P /H of size < κ in V [ H ] such that the ( P /H )-name associated to τ is a Q -name. Let β ≥ max { α, | Q ∗ ˙ Q |} .Since P / ( Q ∗ ˙ Q ) adds a generic for Col( ω, β ), we have Q ∈ L extending Q ∗ ˙ Q such that Q ∼ Col( ω, β ).In particular, if κ is weakly compact, then Col( ω, < κ ) ∗ ˙ Q , where ˙ Q is forcedto be c.c.c., is ( ω, κ )-NLC. Thus an extremely wide variety of forcing extensionswith very different theories can be obtained, each sharing the same reals and samecardinals.2.1.2. An unfortunate reality. Despite the universality of A ( µ, κ ), it is difficult tocharacterize its combinatorial structure. While it absorbs all of the small sets addedby a ( µ, κ )-NLC forcing, no such forcing completely embeds into it. The reader mayopt to skip this section, as later results will not depend it.To show this, we first isolate two properties of a forcing extension that dependon two regular cardinals µ < κ . The author is grateful to Mohammad Golshani forbringing these properties to his attention.(1) Levy( µ, κ ): ( ∃ A ∈ [ κ ] κ )( ∀ y ∈ [ κ ] µ ∩ V ) y * A .(2) Silver( µ, κ ): ( ∃ A ∈ [ κ ] κ )( ∀ X ∈ [ κ ] κ ∩ V )( ∃ y ∈ [ X ] µ ∩ V ) y ∩ A = ∅ .Note that these are both Σ properties of the parameters ([ κ ] µ ) V and ([ κ ] κ ) V .For any partial order P , and collection of dense subsets D ⊆ P ( P ) the statement,“There is a filter G ⊆ P that is D -generic,” is also a Σ property of P and D .Now the following proposition either holds or fails for a given partial order P andcardinals µ < κ :( ∗ ) µ,κ : ( ∀ X ∈ [ P ] κ )( ∃ y ∈ [ X ] µ ) y has a lower bound in P . Lemma 2.12. If P is a separative partial order that satisfies ( ∗ ) µ,κ , preserves theregularity of κ , and such that d( P ↾ p ) = κ for all p ∈ P , then P forces Silver ( µ, κ ) .Proof. Let { p α : α < κ } be a dense subset of P . Inductively build a dense D ⊆{ p α : α < κ } , putting p α ∈ D just in case there is no β < α such that p β ∈ D and p β ≤ p α . D has the property that for all p ∈ D , |{ q ∈ D : p ≤ q }| < κ .Fixing a bijection f : D → κ , we claim that if G ⊆ P is generic, A = f [ G ] witnesses Silver ( µ, κ ). Note that since P is nowhere < κ -dense, A is an unbounded subsetof κ . Now let p ∈ D and X = { q α : α < κ } ∈ [ D ] κ be arbitrary. There is some B ∈ [ κ ] κ such that for all α ∈ B , p (cid:2) q α . For each α ∈ B , choose r α ≤ p such that r α ⊥ q α . By ( ∗ ), there is some y ∈ [ B ] µ such that { r α : α ∈ y } has a lower bound r . We have r (cid:13) { q α : α ∈ ˇ y } ∩ ˙ G = ∅ . As p and X were arbitrary, Silver ( µ, κ ) isforced. (cid:3) Lemma 2.13. If P is a κ -c.c. separative partial order of size κ satisfying ¬ ( ∗ ) µ,κ ,then some p ∈ P forces Levy ( µ, κ ) .Proof. Suppose X ∈ [ P ] κ witnesses ¬ ( ∗ ) µ,κ . By the κ -c.c., there is some p suchthat p (cid:13) | ˇ X ∩ ˙ G | = κ . If y ∈ [ X ] µ , then 1 (cid:13) ˇ y * ˙ G , since otherwise some q is alower bound to y . Hence p forces that X ∩ G witnesses Levy ( µ, κ ). (cid:3) Lemma 2.14. Suppose µ < κ , µ is regular for all α < κ , α µ < κ . There are two ( µ, κ ) -NLC partial orders P and P such that P forces Levy ∧ ¬ Silver , and P forces ¬ Levy ∧ Silver .Proof. Let P be the Levy collapse Col( µ, <κ ), and let P be the Silver collapse, { p : ( ∃ α < µ )( ∃ x ∈ [ κ ] µ ) p : x × α → κ, and p ( β, γ ) < β for all ( β, γ ) ∈ dom p } ENSE IDEALS AND CARDINAL ARITHMETIC 11 It is easy to see that P satisfies ( ∗ ) µ,κ , while P fails this property, as witnessedby X = P . Hence by the previous lemmas, P forces Levy ( µ, κ ), and P forces Silver ( µ, κ ). We must show that the respective negations are also forced.Let ˙ A be a P -name such that 1 (cid:13) ˙ A ∈ [ κ ] κ . Let p ∈ P be arbitrary, and let γ < κ be such that supp( p ) ⊆ γ . Let X = { α < κ : p α / ∈ ˙ A } . For each α ∈ X , picksome q α ≤ p such that q α (cid:13) α ∈ ˙ A . By a delta-system argument, let X ∈ [ X ] κ besuch that there is r ≤ p such that for all α ∈ X , q α ↾ γ = r , and for α = β in X ,(supp( q α ) \ γ ) ∩ (supp( q β ) \ γ ) = ∅ . For any q ≤ r and y ∈ [ X ] µ , q ˇ y ∩ ˙ A = ∅ . Thisis because for such q , there is some α ∈ y such that (supp( q α ) \ γ ) ∩ supp( q ) = ∅ ,so q is compatible with q α . Hence r (cid:13) ( ∃ X ∈ [ κ ] κ ∩ V )( ∀ y ∈ [ X ] µ ∩ V ) y ∩ ˙ A = ∅ .As ˙ A and p were arbitrary, ¬ Silver ( µ, κ ) is forced.Now let ˙ A be a P -name such that 1 (cid:13) ˙ A ∈ [ κ ] κ , and let p ∈ P be arbitrary.Form X , { q α : α ∈ X } , and X like above. We can take a y ∈ [ X ] µ such that S α ∈ y q α = q ∈ P . Then q (cid:13) ˇ y ⊆ ˙ A , so q forces ¬ Levy ( µ, κ ). (cid:3) Corollary 2.15. Suppose µ , κ , P , and P are as above. Let G be P -generic and H be P -generic over V . Let Q ∈ V be a partial order. If Q forces Levy ( µ, κ ) , then V [ H ] has no Q -generic, and if Q forces Silver ( µ, κ ) , then V [ G ] has no Q -generic.If Q is κ -c.c. and of size κ , then no κ -closed forcing extension of V [ G ] or V [ H ] canintroduce a generic for Q .Proof. Since V [ H ] satisfies ¬ Levy , and Levy is a Σ property with parametersin V , no inner model of V [ H ] containing V can satisfy Levy . Likewise, no innermodel of V [ G ] containing V can satisfy Silver . To see that the non-existence of Q -generics is preserved by κ -closed forcing, suppose that for some such forcing R ∈ V [ G ], r (cid:13) V [ G ] R ˙ K is Q -generic over V . Since Q has size κ , we can build adescending sequence { r α : α < κ } below r such that for all q ∈ Q , there is r α deciding whether q ∈ K . Let K ′ = { q : ( ∃ α < κ ) r α (cid:13) q ∈ ˙ K } . Any maximalantichain A ∈ V contained in Q has size < κ , thus some r α completely decides A ∩ K . Since r α (cid:13) ˇ A ∩ ˙ K = ∅ , we must have K ′ ∩ A = ∅ , so K ′ is Q -generic over V . The argument for κ -closed forcing over V [ H ] is the same. (cid:3) Theorem 2.16. Suppose µ < κ are regular and α µ < κ for all α < κ . No ( µ, κ ) -NLC forcing regularly embeds into A ( µ, κ ) . Further, a generic extension by A ( µ, κ ) has no generic filters for any κ -c.c. forcing Q such that d( Q ↾ q ) ≥ κ for all q ∈ Q .Proof. First note that we only need to consider Q such that d( Q ↾ q ) = κ for all q ∈ Q . For if p ∈ A ( µ, κ ) is such that p (cid:13) ˙ K is Q -generic, then there would be some q ∈ B ( Q ) and some p ′ ≤ p such that B ( Q ) ↾ q completely embeds into A ( µ, κ ) ↾ p ′ .Since d ( A ( µ, κ ) = κ , this implies B ( Q ) ↾ q ≤ κ .Let Q be any κ -c.c. forcing such that d( Q ↾ q ) = κ for all q ∈ Q . For any p ∈ Q ,if ( ∗ ) holds for Q ↾ p , then p (cid:13) Silver , and otherwise for some q ≤ p , q (cid:13) Levy .Thus (cid:13) Q Levy ∨ Silver . Suppose K is Q -generic over V , and X is A ( µ, κ )-genericover V . There are two further forcings R , R over V [ X ] that respectively get filters G, H such that V [ G ][ X ] is P ∗ Add( κ )-generic, and V [ H ][ X ] is P ∗ Add( κ )-generic.If V [ K ] | = Levy , then K V [ H ][ X ], and if V [ K ] | = Silver , then K / ∈ V [ G ][ X ].Thus V [ X ] has no Q -generics. (cid:3) Construction of a dense ideal. First we will define a useful strengtheningof “nicely layered.” Definition. P is ( µ, κ )-very nicely layered (with collapses) when there is a sequence h Q α : α < κ i = L such that:(1) L witnesses that P is ( µ, κ ) -nicely layered (with collapses),(2) L is ⊆ -increasing,(3) every subset of P of size < µ with a lower bound has an infimum, and(4) there is a system of continuous projection maps π α : P → Q α such that for each α , π α ↾ Q α = id , and for β < α < κ , π β = π β ◦ π α .(By continuous, we mean that for any X ⊆ P , if inf( X ) exists, then for all α < κ , inf( π α [ X ]) = π α (inf( X )) .) A typical example is the Levy collapse Col( µ, <κ ). In the general case, we willusually abbreviate the action of the projection maps π α ( q ) by q ↾ α . In applyingclause (3), we will use the next proposition, proof of which is left to the reader. Proposition 2.17. If P is a partial order such that every descending chain oflength < µ has an infimum, then every directed subset of size < µ has an infimum. Theorem 2.18. Assume κ carries an almost-huge tower of height δ , and let j : V → M be given by the tower. Let µ, λ be regular such that µ < κ ≤ λ < δ .Suppose (cid:13) A ( µ,κ ) “ ˙ P is ( κ, δ ) -very nicely layered and forces δ = λ + .” If X ∗ H is A ( µ, κ ) ∗ ˙ P -generic, then in V [ X ][ H ] , there is a normal, κ -complete, λ -dense idealon P κ ( λ ) .Proof. Let H X be the A ( µ, κ )-generic filter computed from X . Let K × C be B ( µ, κ ) /H X × Col( µ, λ )-generic over V [ X ][ H ], and for brevity let W = V [ X ][ H ][ K ][ C ].Note that V [ X ][ K ] = V [ G ][ X ], where G ∗ X is someCol( µ, <κ ) ∗ Add( κ )-generic filter over V . Let h Q α : α < δ i witness that P is( κ, δ )-nicely layered. By the distributivity of B ( µ, κ ) /H X in V [ X ], P and its layers Q α are still κ -closed in V [ G ][ X ]. For α < β , the relation (cid:13) Q α “ Q β / Q α is κ -closed”holds in V [ G ][ X ] because in V [ X ], B ( µ, κ ) /H X × Q α is κ -distributive. Furthermore,since no sequences of length < µ are added, the forcing given by the definition ofCol( µ, λ ) is the same between V , W , and intermediate models.The forcing to get from V [ G ] to W is equivalent to (Add( κ ) × Col( µ, λ )) ∗ P .Let L be the collection of subforcings of the form (Add( κ ) × Col( µ, λ )) ∗ Q α for α < δ . This sequence then witnesses the ( µ, δ )-NLC property in V [ G ]. The closureproperties are evident, and since the whole forcing has the δ -c.c., functions from µ to ordinals are indeed captured by these factors.Let P = P ( µ ) W , and consider the submodel M ( P ). In W , Q = P (Add( δ )) M ( P ) has cardinality δ . To show this, let Y ⊆ δ be Add( δ )-genericover W . By Theorem 2.11, Y is A ( µ, δ )-generic over V , and hence over M since(Col( µ, <δ ) ∗ Add( δ )) M = (Col( µ, <δ ) ∗ Add( δ )) V by the closure of M . Since M [ Y ]thinks j ( δ ) is inaccessible, M [ Y ] | = | Q | < j ( δ ), so W [ Y ] | = | Q | = δ since j ( δ ) < ( δ + ) V . Since W | = 2 µ = δ , W and W [ Y ] have the same cardinals, so W | = | Q | = δ .Therefore, working in W , we can inductively build a set ˆ X ⊆ δ that is Add( δ )-generic over M ( P ) with ˆ X ∩ κ = X . By Lemma 2.10, ˆ X is A ( µ, δ )-generic over M [ G ]. A further forcing produces G ′ ⊇ G , such that G ′ ∗ ˆ X is Col( µ, <δ ) ∗ Add( δ )-generic over M , so we have an elementary ˆ j : V [ G ][ X ] → M [ G ′ ][ ˆ X ] extending j .By elementarity, for the corresponding filters H X and H ˆ X on the respective alge-bras A ( µ, κ ) V and A ( µ, δ ) M , we have j [ H X ] ⊆ H ˆ X . Hence we can define in W therestricted elementary embedding ˆ j : V [ X ] → M [ ˆ X ]. ENSE IDEALS AND CARDINAL ARITHMETIC 13 Now we wish to extend ˆ j to have domain V [ X ][ H ]. As in the argument forLemma 2.9, every element of (Ord µ ) W is coded by some element of M and some y ⊆ µ coded in ˆ X , so M [ ˆ X ] is closed under <δ -sequences from W . Consequently, H ∩ Q α and ˆ j [ H ∩ Q α ] are in M [ ˆ X ] for all α < δ . Also, M [ ˆ X ] (cid:15) “ˆ j ( P ) is ( δ, j ( δ ))-very nicely layered.” Each ˆ j [ H ∩ Q α ] is a directed set of size µ in M [ ˆ X ], so it hasan infimum m α ∈ ˆ j ( Q α ).Let h A α : α < δ i ∈ W enumerate the maximal antichains of ˆ j ( P ) from M [ ˆ X ].(There are only δ many because M [ ˆ X ] thinks this partial order has inaccessiblesize j ( δ ) and is j ( δ )-c.c.) Inductively define an increasing sequence of ordinals h α i i i<δ ⊆ δ , and a corresponding decreasing sequence of conditions h p i i i<δ ⊆ ˆ j ( P )as follows.Assume as the induction hypothesis that we have defined the sequences up to i ,and for all ξ < i and all α < δ , p ξ is compatible with m α , and for all ξ < i , there issome a ∈ A ξ such that p ξ ≤ a . Let q i = inf ξ α i , m α ↾ j ( α i ) = m α i .This is because for any α < β < δ , m β ↾ j ( α ) =(inf { j ( p ) : p ∈ H ↾ β } ) ↾ j ( α ) = inf { j ( p ) ↾ j ( α ) : p ∈ H ↾ β } = inf { j ( p ↾ α ) : p ∈ H ↾ β } = inf { j ( p ) : p ∈ H ↾ α } = m α . The upward closure of the sequence h p i i i<δ is a filter ˆ H which is ˆ j ( P )-genericover M [ ˆ X ]. For all p ∈ H , ˆ j ( p ) ∈ ˆ H since there is some m α ≤ ˆ j ( p ). Thuswe get an extended elementary embedding ˆ j : V [ X ][ H ] → M [ ˆ X ][ ˆ H ]. In W , wedefine an ultrafilter U over ( P ( P κ λ )) V [ X ][ H ] : let A ∈ U iff j [ λ ] ∈ ˆ j ( A ). Note that j [ λ ] ∈ P j ( κ ) ( j ( λ )) M [ ˆ X ][ ˆ H ] . U is κ -complete and normal with respect to functionsin V [ X ][ H ]. If f : P κ ( λ ) → λ is a regressive function in V [ X ][ H ] on a set A ∈ U ,then ˆ j ( f )( j [ λ ]) = j ( α ) for some α < λ , so { z ∈ A : f ( z ) = α } ∈ U .Now the forcing to obtain U was Q = B ( µ, κ ) /H X × Col( µ, λ ), the product of a κ -dense and a λ -dense partial order. In V [ X ][ H ], let e : P ( P κ λ ) → B ( Q ) be definedby e ( A ) = || ˇ A ∈ ˙ U || . Let I be the kernel of e . I is clearly a normal, κ -completeideal. e lifts to a boolean embedding of P ( P κ λ ) /I into B ( Q ). Since Q is λ + -c.c., I is λ + -saturated. If h [ A α ] : α < λ i is a maximal antichain in P κ ( λ ) /I , then ∇ A α is the least upper bound and is in the dual filter to I . e ( ∇ A α ) = ||∇ A α ∈ ˙ U || = 1,and this is the least upper bound in B ( Q ) to { e ( A α ) : α < λ } . This is because ifthere were a generic extension in which all A α / ∈ U , then ∇ A α / ∈ U as well since U is normal with respect to sequences from V [ X ][ H ]. Therefore e is a completeembedding, and thus I is λ -dense. (cid:3) We can also characterize the exact structure of P ( P κ λ ) /I . First note the fol-lowing about the ground model embedding j : V → M . M is the direct limit ofthe coherent system of α -supercompactness embeddings j α : V → M α for α < δ .Every member of M α is represented as j α ( f )( j α [ α ]) for some function f ∈ V withdomain P κ ( α ). If k α : M α → M is the factor map such that j = k α ◦ j α , then thecritical point of k α is above α , so k α ( x ) = k α [ x ] when M α (cid:15) | x | ≤ | α | . Since M is the direct limit, for any x ∈ M , there is some α < δ and some f ∈ V such that x = k α ([ f ]) = k α ( j α ( f )( j α [ α ])) = j ( f )( k α ( j α [ α ])) = j ( f )( j [ α ])) . Let U ⊆ P ( P κ λ ) /I be generic over V [ X ][ H ], and let j U : V → N be the genericultrapower embedding. Since e : P ( P κ λ ) /I → B ( Q ) is a complete embedding,forcing with B ( Q ) /e [ U ] over V [ X ][ H ][ U ] produces a model W as above. Noticethat the definition of e and U makes A ∈ U iff j [ λ ] ∈ ˆ j ( A ). Hence we can definean elementary embedding k : N → M [ ˆ X ][ ˆ H ] by k ([ f ]) = ˆ j ( f )( j [ λ ]), and we haveˆ j = k ◦ j U .What is the critical point of k ? Since N (cid:15) µ + = δ , certainly it must be atleast δ . Let β be any ordinal. There is some α such that λ ≤ α < δ and some f ∈ V such that β = j ( f )( j [ α ]). Let b : λ → α be a bijection in V [ X ][ H ]. Then β = j ( f )(ˆ j ( b )[ j [ λ ]]). Furthermore, j [ λ ] = k ( j U [ λ ]). Therefore, β = k ( j U ( f )( j U ( b )[ j U [ λ ]])).Thus β ∈ ran( k ), and so k does not have a critical point.Therefore, N = M [ ˆ X ][ ˆ H ]. By the closure of M [ ˆ X ][ ˆ H ], the generic K × C for Q is in M [ ˆ X ][ ˆ H ] = N ⊆ V [ X ][ H ][ U ]. So the quotient B ( Q ) /e [ U ] is trivial and P ( P κ λ ) /I ∼ = B ( Q ) ↾ q for some q .The generic embeddings coming from I extend the original almost-hugeness em-bedding. In particular, j [ δ ] is cofinal in j ( δ ). This can also be deduced fromthe assumption that there is some A ∈ I ∗ of size λ , which of course follows from λ <κ = λ . In contrast, Burke and Matsubara [1] proved that if there is a normal,fine, κ -complete, λ + -saturated ideal on P κ ( λ ) and cf( λ ) < κ , then it is forced thatsup( j [ λ + ]) < j ( λ + ). It seems to be unknown whether it is consistent to have satu-rated ideals on P κ ( λ ) for successor κ and singular λ , and this result suggests thatquite different methods will be needed for an answer.2.2.1. Minimal generic supercompactness. Generalizing supercompactness, we willsay cardinal κ is generically supercompact when for every λ ≥ κ , there is a forcing P such that whenever G ⊆ P is generic, there is an elementary embedding j : V → M ,where M is a transitive class in V [ G ], crit( j ) = κ , j ( κ ) > λ , and M λ ∩ V [ G ] ⊆ M .We note that unlike in the case of non-generic supercompactness, the conditionthat j [ λ ] ∈ M does not imply that M is closed under λ -sequences from V [ G ].Whenever a supercompact κ is turned into a successor cardinal by a κ -c.c. forcing,we’ll have that for all λ ≥ κ , there is a normal, fine, precipitous ideal on P κ ( λ )whose generic embeddings always extend the original supercompactness embedding.But if j : V → M is an embedding coming from a normal ultrafilter on P κ ( λ ), then2 λ <κ < j ( κ ) < (2 λ <κ ) + . If κ = µ + in a generic extension V [ H ], and a furtherextension gives ˆ j : V [ H ] → M [ ˆ H ] ⊆ V [ H ][ G ] extending j , then M [ ˆ H ] is not closedunder λ -sequences from V [ H ][ G ]. This is because | λ | = | j ( κ ) | = µ in V [ H ][ G ],while M [ ˆ H ] thinks j ( κ ) is a cardinal.Stronger properties of ideals on P κ ( λ ) are needed to give genuine generic super-compactness. One such property is λ + -saturation, which is implied by λ -density.We now sketch how to get a model in which there is a successor cardinal κ suchthat for all regular λ ≥ κ , there is a normal, κ -complete, λ -dense ideal on P κ ( λ ).Start with a super-almost-huge cardinal κ and a regular µ < κ . The first partof the forcing is A ( µ, κ ). Then we do a proper class iteration, which we prefer todescribe instead as an iteration up to an inaccessible δ > κ such that V δ (cid:15) κ issuper-almost-huge. ENSE IDEALS AND CARDINAL ARITHMETIC 15 Let T = { α < δ : κ carries an almost-huge tower of height α } . Let C bethe closure of T , and let h α β i β<δ be its continuous increasing enumeration. Over V A ( µ,κ ) , let P δ be the Easton-support limit of the following: • Let P = Col( κ, < α ). • If β is zero or a successor ordinal, let P β +1 = P β ∗ Col( α β , < α β +1 ). • If β is a limit ordinal such that α β is singular, let P β +1 = Col( α + β , < α β +1 ). • If β is a limit ordinal such that α β is regular, let P β +1 = Col( α β , < α β +1 ).It is routine to verify that this iteration preserves the regularity of the membersof T , the successors of the singular limit points of T , and the regular limit points of T . Further, the set of non-limit-points of T becomes the set of successors of regularcardinals between κ and δ .Let X ⊆ κ be A ( µ, κ )-generic over V , and let H ⊆ P δ be generic over V [ X ].Suppose κ ≤ λ < δ , and λ is regular in V [ X ][ H ]. Then there is some successorordinal β < δ such that α β ∈ T and α β = λ + . Consider the subforcing A ( µ, κ ) ∗ P β =( A ( µ, κ ) ∗ P β − ) ∗ Col( λ, < α β ). The forcing P β is ( κ, α β )-very nicely layered in V [ X ].If j : V → M β is an almost-huge embedding with critical point κ and j ( κ ) = α β ,then by Theorem 2.18, there is a normal, κ -complete, λ -dense ideal on P κ ( λ ) in V [ X ][ H β ]. Now note that the tail-end forcing P β,δ is α β -closed. Since λ <κ = λ in V [ X ][ H β ], no new subsets of P κ ( λ ) are added by the tail. The collection { A α : α <λ } witnessing the λ -density of I retains this property, as this is a local property ofthe boolean algebra P κ ( λ ) /I and { A α : α < λ } . Normality and completeness of I are likewise preserved.This method is quite flexible, and can done by iterating collapsing posets otherthan the Levy collapse, or by using products rather than iterations.2.2.2. Dense ideals on successive cardinals? At the time of this writing, it is un-known whether there can exist simultaneously a normal κ -dense ideal on κ and anormal κ + -dense ideal on κ + . The following is the current best approximation.Suppose h κ n : n < ω i is a sequence of cardinals such that for all n , κ n carries analmost-huge tower of height κ n +1 . Such a sequence will be called an almost-hugechain . Obviously, extending this to sequences of length longer than ω requires anextra idea; perhaps we just stack one ω -chain above another, or maybe postulatesome relationship between the ω -chains. By Theorem 2.4, such chains occur quiteoften below a huge cardinal.Suppose h κ n : 0 < n < ω i is an almost-huge chain, and µ < κ is regular.Consider the full-support iteration P of h P n : n < ω i , where P = A ( µ, κ ), andfor all n < ω , P n +1 = P n ∗ A ( κ n , κ n +1 ). The stage P = A ( µ, κ ) ∗ A ( κ , κ )regularly embeds into A ( µ, κ ) ∗ (Col( κ , < κ ) ∗ Add( κ )). The first two stageshere add a normal κ -dense ideal on κ and make κ = µ + , κ = µ ++ . The thirdstage preserves this since it adds no subsets of κ . By Lemma 2.9, the quotientforcing Q to get from V P to this three-stage extension is κ -distributive. Now thetail-end forcing P / P is κ -strategically closed. Since Q does not add any plays ofthe relevant game of length < κ , P / P remains κ -strategically closed in V P ∗ Q , soforcing with it preserves the κ -dense ideal on κ . Also, Q remains κ -distributivein V P , since Q × ( P / P ) is κ -distributive in V P . It thus remains the case in V P that there is a κ -distributive forcing adding a normal κ -dense ideal on κ .Similarly, consider V P n for n > P n = P n − ∗ ( A ( κ n − , κ n ) ∗ A ( κ n , κ n +1 )). Since | P n − | = κ n − (or µ for n = 2), κ n retains an almost-huge tower of height κ n +1 in V P n − . Thus the same argument applies: In V P n , there is a κ n +1 -distributive forcingadding a normal κ n -dense ideal on κ n , and this remains true in V P . Therefore, weobtain a model in which for all n > 0, there is a µ + n +1 -distributive forcing addinga normal µ + n -dense ideal on µ + n . By repeating this with a tall enough stack ofalmost-huge chains, we obtain the consistency of ZFC with the statement, “For allregular cardinals κ , there is a κ ++ -distributive forcing adding a normal κ + -denseideal on κ + .” 3. Structural constraints Saturated ideals have a strong influence over the combinatorial structure of theuniverse in their vicinity. Phenomena of this type may also be viewed as the universeimposing constraints on the structural properties of ideals. Below are some of themost interesting known results to this effect. Proofs can be found in [7].(1) (Tarski) If I is a nowhere-prime ideal which is κ -complete and µ -saturated forsome µ < κ , then 2 <µ ≥ κ .(2) (Jech-Prikry) If κ = µ + , 2 µ = κ , and there is a κ -complete, κ + -saturated idealon κ , then 2 κ = κ + .(3) (Jech-Prikry) If κ = µ + , and there is a κ -complete, κ + -saturated ideal on κ ,then there are no κ -Kurepa trees.(4) (Woodin) If there is a countably complete, ω -dense ideal on ω , then there isa Suslin tree.(5) (Woodin) If there is a countably complete, uniform, ω -dense ideal on ω , then2 ω = ω . (Uniform means that all sets of size < ω are in the ideal–equivalentto fineness.)(6) (Shelah) If 2 ω < ω , then N S ω is not ω -dense.(7) (Gitik-Shelah) If I is a κ -complete, nowhere-prime ideal, then d( I ) ≥ κ .We note that result (2) easily generalizes to the following: If κ = µ + , 2 µ = κ ,and there is a normal, fine, κ -complete, λ + -saturated ideal on P κ ( λ ), then 2 λ = λ + .If no requirements are made for the ideal I and the set Z on which it lives, almostno structural constraints on quotient algebras remain. The following strengthensa folklore result, probably known to Sikorski. The argument was supplied by DonMonk in personal correspondence. Proposition 3.1. Let B be a complete boolean algebra, and let κ be a cardinal suchthat κ ≥ | B | . There is a uniform ideal I on κ such that B ∼ = P ( κ ) /I .Proof. Let κ , B be as hypothesized. By the theorem of Fichtenholz-Kantorovichand Hausdorff (see [10, Lemma 7.7]), there exists a family F of 2 κ many subsets of κ such that for any x , ..., x n , y , ..., y m ∈ F , x ∩ ... ∩ x n ∩ ( κ \ y ) ∩ ... ∩ ( κ \ y m )has size κ . F generates a free algebra: closing F under finitary set operations givesa family of sets G such that any equation holding between elements of G expressedas boolean combinations of elements of F holds in all boolean algebras. If we pickany surjection h : F → B and extend it to h : G → B in the obvious way, then h will be a well-defined homomorphism.Let I bd be the ideal of bounded subsets of κ . Since all elements of G are eitherempty or have cardinality κ , G ∼ = G/I bd , so h has an extension h from the algebragenerated by G ∪ I bd to B , where h ( x ) = 0 for all x ∈ I bd . Finally, by Sikorski’sextension theorem, there is a further extension to a homomorphism h : P ( κ ) → B .The kernel of h is an ideal I such that P ( κ ) /I ∼ = B . (cid:3) ENSE IDEALS AND CARDINAL ARITHMETIC 17 Cardinal arithmetic and ideal structure. A careful examination of theproof of Woodin’s theorem (5) shows that ω can be replaced by any ω n , 2 ≤ n < ω .Aside from that, Woodin’s argument is rather specific to the cardinals involved.In [7], Foreman asked (Open Question 27) whether the analogous statement holdsone level up: Question (Foreman) . Does the existence of an ω -complete, ω -dense, uniformideal on ω imply that ω = ω ? To answer this, we invoke an easy preservation lemma about ideals under smallforcing. If I is an ideal, P is a partial order, and G ⊆ P is generic, then ¯ I denotesthe ideal generated by I in V [ G ], i.e. { X : ( ∃ Y ∈ I ) X ⊆ Y } . Lemma 3.2. Suppose I is a κ -complete ideal on Z ⊆ P ( X ) , P is a partial order,and G is P -generic.(1) If sat( P ) ≤ κ , then ¯ I is κ -complete in V [ G ] .(2) If d( P ) < κ , then d( ¯ I ) V [ G ] ≤ d( I ) V .Proof. For (1), let ˙ s be a P -name for a sequence of elements of ¯ I of length less than κ . By κ -saturation, let β < κ be such that 1 (cid:13) dom ( ˙ s ) ≤ β . For each α < β , let A α be a maximal antichain such that for p ∈ A α , p (cid:13) ˙ s ( α ) ⊆ ˇ b pα , where b pα ∈ I .Then B = S p,α b pα ∈ I , and 1 (cid:13) S ˙ s ⊆ ˇ B .For (2), let D ⊆ P be a dense set of size less than κ , and let A ∈ ¯ I + . Then A = S d ∈ D ∩ G { z : d (cid:13) z ∈ ˙ A } . By (1), there is some d ∈ D such that { z : d (cid:13) z ∈ ˙ A } / ∈ I . This shows that ( P ( Z ) /I ) V is dense in ( P ( Z ) / ¯ I ) V [ G ] , and the conclusionfollows. (cid:3) Corollary 3.3. If there is a κ + -complete, κ + -dense, uniform ideal on κ ++ , then κ = κ + .Proof. Suppose for a contradiction that f : P ( κ ) → κ ++ is a surjection. Let P = Col( ω, κ ), and let G be P -generic. Since d( P ) = κ , Lemma 3.2 implies that ¯ I is κ + -complete and κ + -dense in V [ G ]. Furthermore, in V [ G ], κ + = ω and κ ++ = ω .Thus Woodin’s theorem implies that V [ G ] (cid:15) CH. However, f witnesses the failureof CH, a contradiction. (cid:3) Another interesting constraint can be derived from the following: Theorem 3.4 (Shelah [15]) . Suppose V ⊆ W are models of ZFC. If κ is a regularcardinal in V , and cf( κ ) = cf( | κ | ) in W , then ( κ + ) V is not a cardinal in W . Corollary 3.5 (Burke-Matsubara [2]) . If κ = µ + , λ ≥ κ is regular, and I is anormal, fine, κ -complete, λ + -saturated ideal on P κ ( λ ) , then { z : cf( z ) = cf( µ ) } ∈ I ∗ .Proof. Let G be a generic ultrafilter extending I ∗ . Since crit( j ) = κ and λ + ispreserved, j ( κ ) = λ + , and | λ | = µ in V [ G ]. By Shelah’s theorem, cf( λ ) = cf( µ ) in V [ G ] and in the ultrapower M since M µ ∩ V [ G ] ⊆ M . Since 1 (cid:13) [id] = j [ λ ], Lo´s’stheorem gives { z : cf( z ) = cf( µ ) } ∈ I ∗ . (cid:3) Theorem 3.6. Suppose κ = µ + , and I is a normal, fine, κ + -saturated ideal on κ .Then P ( κ ) /I is cf( µ ) -distributive iff µ < cf( µ ) = µ . Proof. Suppose P ( κ ) /I is cf( µ )-distributive, and let { f α : α < δ } be an enumera-tion of [ µ ] < cf( µ ) , where δ is a cardinal. If µ < δ , then for any P ( κ ) /I -generic G ,([ µ ] < cf( µ ) ) V is a proper subset of ([ µ ] < cf( µ ) ) V [ G ] , since j [ δ ] = j ( δ ). This contradictsthe distributivity of P ( κ ) /I .Since P ( κ ) /I is κ + -saturated, it is cf( µ )-distributive iff it is (cf( µ ) , κ )-distributive. Let G be P ( κ ) /I -generic and let M be the generic ultrapower. Let β < cf( µ ), and suppose f ∈ V [ G ] is a function from β to κ . By Theorem 1.13, f ∈ M . By Corollary 3.5, M (cid:15) cf( κ ) = cf([id]) = cf( µ ). Thus there is a γ < κ such that ran( f ) ⊆ γ . Observe that j ( β γ ) = ( β γ ) M = ( β γ ) V , since µ β < κ . Hence f ∈ V . (cid:3) Stationary reflection. A stationary subset S of a regular cardinal κ is saidto reflect if there is some α < κ of uncountable cofinality such that S ∩ α isstationary in α . A collection of stationary subsets { S i : i < δ } of κ is said to reflectsimultaneously if there is some α < κ if S i ∩ α is stationary for all i < δ . It iswell known that if κ = µ + and X is a set of regular cardinals below µ , then thestatement that every stationary subset of { α < κ : cf( α ) ∈ X } reflects contradicts (cid:3) µ , and the statement that every pair of stationary subsets of { α < κ : cf( α ) ∈ X } reflect simultaneously contradicts the weaker principle (cid:3) ( κ ). Theorem 3.7. Suppose there is a κ + -complete, κ ++ -saturated, uniform ideal on κ + n for some n ≥ . Then for ≤ m ≤ n , every collection of κ many stationarysubsets of κ + m contained in cof( ≤ κ ) reflects simultaneously.Proof. Suppose I is such an ideal and j : V → M ⊆ V [ G ] is a generic embeddingarising from the ideal. The critical point of j is κ + , and all cardinals above κ + arepreserved. Since I is uniform, and there is a family of κ + n +1 many almost-disjointfunctions from κ + n to κ + n , j ( κ + n ) ≥ ( κ + n +1 ) V . The first n − V above κ must map onto the first n − M above κ . But in M , thereare at least n − κ, ( κ + n +1 ) V ) since all cardinals above κ + are preserved. Thus if j ( κ + n ) > ( κ + n +1 ) V , then κ + n +1 would be collapsed. Sofor 1 ≤ m ≤ n , j ( κ + m ) = ( κ + m +1 ) V .Let { S α : α < κ } be stationary subsets of κ + m concentrating on cof( ≤ κ ), where2 ≤ m ≤ n . By the κ ++ -chain condition, these sets remain stationary in V [ G ]. Bythe above remarks, γ = sup( j [ κ + m ]) < j ( κ + m ). For each α , j ↾ S α is continuoussince κ < crit( j ). For each α , let C α be the closure of S α . In V [ G ], we can define acontinuous increasing function f : C α → γ extending j ↾ S α by sending sup( S α ∩ β )to sup( j [ S α ∩ β ]) when β is a limit point of S α . This shows that j [ S α ] is stationaryin γ . Now M may not have j [ S α ] as an element, but it satisfies that j ( S α ) ∩ γ isstationary in γ . Furthermore, j ( { S α : α < κ } ) = { j ( S α ) : α < κ } , and M sees thatthese all reflect at γ . By elementarity, the S α have a common reflection point. (cid:3) Proposition 3.8. Suppose µ, κ, λ are regular cardinals such that ω < µ < κ = µ + < λ , and I is an ideal on P κ ( λ ) as in Theorem 2.18. Then every collection { S i : i < µ } of stationary subsets of λ ∩ cof( ω ) reflects simultaneously.Proof. The algebra P ( P κ ( λ )) /I is isomorphic to B ( P × Q ), where P is κ -dense and (cid:13) P “ Q is µ -closed.” Forcing with P × Q thus preserves the stationarity of anysubset of λ ∩ cof( ω ). If j : V → M ⊆ V [ G ] is a generic embedding arising fromthe ideal, then since j [ λ ] ∈ M and M thinks j ( λ ) is regular, γ = sup( j [ λ ]) < j ( λ ).The restriction of j to each S i is continuous, and as above we may define in V [ G ] ENSE IDEALS AND CARDINAL ARITHMETIC 19 a continuous increasing function from the closure of S i into γ , showing j [ S i ] isstationary in γ for each i . Thus M | = ( ∀ i < µ ) j ( S i ) ∩ γ is stationary, so byelementarity, the collection reflects simultaneously. (cid:3) Nonregular ultrafilters. The computation of the cardinality of ultrapowersis an old problem of model theory. Originally, it was conjectured that if µ, κ areinfinite cardinals, and U is a countably incomplete uniform ultrafilter on κ , then | µ κ /U | = µ κ [3]. It was shown by Donder [5] that this conjecture holds in the coremodel below a measurable cardinal. A key tool in computing the size of ultrapowersis the notion of regularity: Definition. An ultrafilter U on Z is called ( µ, κ )-regular if there is a sequence h A α : α < κ i ⊆ U such that for any Y ⊆ κ of order type µ , T α ∈ Y A α = ∅ . Theorem 3.9 (Keisler [12]) . Suppose U is a ( µ, κ ) -regular ultrafilter on Z , wit-nessed by h A α : α < κ i . For each z ∈ Z , let β z = ot( { α : z ∈ A α } ) < µ . Then forany sequence of ordinals h γ z : z ∈ Z i , we have | Q γ β z z /U | ≥ | Q γ z /U | κ . Obviously any uniform ultrafilter on a cardinal κ is ( κ, κ )-regular. Also, any fineultrafilter on P κ ( λ ) is ( κ, λ )-regular, as witnessed by h ˆ α : α < λ i . Much can beseen by exploiting a connection between dense ideals and nonregular ultrafilters. Lemma 3.10 (Huberich [9]) . Suppose B is a complete boolean algebra of density κ ,where κ is regular. Then there is an ultrafilter U on B such that whenever X ⊆ B and P X ∈ U , then there is Y ⊆ X such that | Y | < κ and P Y ∈ U .Proof. Let D = { d α : α < κ } be dense in B . For any maximal antichain A ⊆ B , let γ A > α < γ A , there are β < γ A and a ∈ A such that d β ≤ d α ∧ a . Let C A = { d ∈ D ↾ γ A : ( ∃ a ∈ A ) d ≤ a } . Let F = { P C A : A is amaximal antichain } .We claim F has the finite intersection property. Let A , ..., A n be maximalantichains. We may assume γ A ≤ ... ≤ γ A n . Let d α ≤ d ∧ a for some a ∈ A ,where α < γ A . Let d α ≤ d α ∧ a for some a ∈ A , where α < γ A . Proceedinginductively, we get a descending chain d α ≥ ... ≥ d α n , where each d α i ∈ C A i . Thus d α n ≤ P C A ∧ ... ∧ P C A n .Let U ⊇ F be any ultrafilter. If P X ∈ U , then we can find an antichain A that is maximal below P X such that ( ∀ a ∈ A )( ∃ x ∈ X ) a ≤ x . Extending A itto a maximal antichain A ′ , we have P C A ′ ∈ F . Since | C A ′ | < κ , the conclusionfollows. (cid:3) If I is an ideal on Z and U ′ is an ultrafilter on P ( Z ) /I , then U ′ generates anultrafilter U ⊇ I ∗ on Z by taking U = { X : [ X ] I ∈ U ′ } . Lemma 3.11. Suppose κ = µ + , λ is regular, and I is a normal and fine, κ -complete, λ -dense ideal on Z ⊆ P κ ( λ ) . Then any ultrafilter U ⊇ I ∗ given byLemma 3.10 is (cf( µ ) + 1 , λ ) -regular.Proof. Let U ′ be an ultrafilter on P ( Z ) /I given by Lemma 3.10 and let U be thecorresponding ultrafilter on Z . By Corollary 3.5, { z : cf( z ) = cf( µ ) } ∈ I ∗ . For such z , choose A z ⊆ z of order type cf( µ ) that is cofinal in z . We will inductively builda sequence of intervals { ( x α , y α ) : α < λ } , each contained in λ , such that y α < x β when α < β , and such that for all α , { z : A z ∩ ( x α , y α ) = ∅} ∈ U .Suppose we have constructed the intervals up to β . Let λ > x β > sup { y α : α <β } . For z ∈ ˆ x β , let y β ( z ) ∈ z be such that A z ∩ ( x β , y β ( z )) = ∅ . Since I is normal, there is a maximal antichain A of I -positive sets such that for all a ∈ A , y z ( β ) isthe same for all z ∈ a . There is some A ′ ⊆ A of size < λ such that P A ′ ∈ U ′ . Let y β > x β be such that for z ∈ a ∈ A ′ , y β ( z ) < y β .For α < λ , let X α = { z : A z ∩ ( x α , y α ) = ∅} . Since each A z has order typecf( µ ) and the intervals ( x α , y α ) are disjoint and increasing, each A z cannot havenonempty intersection with all intervals in some sequence of length greater thancf( µ ). Thus if s ⊆ λ and z ∈ T α ∈ s X α , then ot( s ) ≤ cf( µ ). (cid:3) Lemma 3.12. Suppose I is a λ -dense, κ -complete ideal on Z , and P ( Z ) /I is acomplete boolean algebra. Then any U ⊇ I ∗ given by Lemma 3.10 has the propertythat for all α < κ , | α Z /U | ≤ <λ .Proof. Let U ′ be an ultrafilter on P ( Z ) /I given by Lemma 3.10 and let U be thecorresponding ultrafilter on Z . To compute a bound on | α Z /U | for α < κ , weidentify a small subset of α Z and show that it contains representative of everyequivalence class modulo U . Let D witness λ -density. Choose an antichain A ⊆ D of size < λ , and choose f : A → α . There are P γ<λ λ γ · α γ = 2 <λ many choices.Using κ -completeness, let { B β : β < α } be pairwise disjoint and such that each[ B β ] I = P f − ( β ). Let g f : Z → α be defined by g f ( z ) = β if z ∈ B β and g f ( z ) = 0if z / ∈ S β<α B β .Now let g : Z → α be arbitrary. By κ -completeness, A = { g − ( β ) : β < α and g − ( β ) ∈ I + } forms a maximal antichain. Let A ′ ⊆ D be a maximal antichainrefining A . There is some A ′′ ⊆ A ′ of size < λ such that P A ′′ ∈ U ′ . Let f : A ′′ → α be defined by f ( a ) = β iff a ≤ I g − ( β ). If [ B ] I = P A ′′ , then { z ∈ B : g ( z ) = g f ( z ) } ∈ I , so g = U g f . (cid:3) The following contrasts with the consistency results of Section 2: Corollary 3.13. Suppose µ is a singular cardinal such that cf( µ ) < µ , λ is reg-ular, and <λ < λ . Then there is no normal and fine, λ -dense ideal on P µ + ( λ ) .Furthermore, there is a proper class of such λ .Proof. Suppose such an ideal exists, and let U ⊇ I ∗ be given by Lemma 3.10.Then | Q µ/U | ≤ <λ , and U is (cf( µ ) + 1 , λ )-regular. Theorem 3.9 implies that | Q cf( µ ) /U | ≥ λ , a contradiction.Assume for a contradiction that α is such that 2 <λ = 2 λ for all regular λ ≥ α .Let κ = 2 α . We will show by induction the impossible conclusion that 2 β = κ for all β ≥ α . Suppose that this holds for all γ < β . If β is regular, 2 <β = 2 β by assumption, so 2 β = κ . If β is singular, then by [10, Theorem 5.16] 2 β =(2 <β ) cf( β ) = κ cf( β ) . If cf( β ) < γ < β , then κ = 2 γ = (2 γ ) cf( β ) = κ cf( β ) . (cid:3) Corollary 3.14. If κ is singular such that cf( κ ) < κ , then there is no uniform, κ + -complete, κ + -dense ideal on κ + n for n ≥ .Proof. Assume I is a uniform, κ + -complete, κ + -dense ideal on κ + n for some n ≥ φ : P ( κ + ) → P ( κ + n ) /I by X 7→ || κ + ∈ j ( X ) || P ( κ + n ) /I . Let J = ker φ . φ liftsto an embedding of P ( κ + ) /J into P ( κ + n ) /I . Since J is clearly normal and κ ++ -saturated, the embedding is regular, since for a maximal antichain { A α : α < κ + } , (cid:13) κ + ∈ j ( ∇ α<κ + A α ), so it is forced that for some α < κ + , φ ( A α ) is in thegeneric filter. Thus J is a normal κ + -dense ideal on κ + . We have 2 <κ + < κ + by Corollary 3.3, so κ cannot be singular such that 2 cf( κ ) < κ . (cid:3) ENSE IDEALS AND CARDINAL ARITHMETIC 21 These methods can also be used to deduce more cardinal arithmetic consequencesof dense ideals. First we need a few more lemmas: Theorem 3.15 (Kunen-Prikry [14]) . If κ is regular and U is a ( κ + , κ + ) -regularultrafilter, then U is ( κ, κ ) -regular. Lemma 3.16. Suppose ( L, < ) is a linear order such that for all x ∈ L , |{ y ∈ L : y < x }| ≤ κ . Then | L | ≤ κ + . Corollary 3.17. Suppose there is a κ + -complete, κ + -dense ideal on κ + n , where n ≥ . Then for ≤ m ≤ n , κ + m = κ + m +1 .Proof. Let I be such an ideal, and let U ⊇ I ∗ be given by Lemma 3.10. ByLemma 3.12, | κ κ + n /U | ≤ κ , which is κ + by Corollary 3.3. Note that for anycardinal µ , any ultrafilter V on a set Z , and any g : Z → µ + , { [ f ] V : f < V g } hascardinality at most | µ Z /V | . Thus, applying Lemma 3.16 inductively, we get that | ( κ + m ) κ + n /U | ≤ κ + m +1 for all m < ω . U is ( κ + n , κ + n )-regular, so by Theorem 3.15, it is ( κ + m , κ + m )-regular for m ≤ n .Assume for induction that 2 κ + r = κ + r +1 for r < m ≤ n ; note the base case m = 1holds. Let { X α : α < κ + m } witness ( κ + m , κ + m )-regularity, and let β z = ot( { α : z ∈ X α } ). By Theorem 3.9 and the above observations, we have:2 κ + m ≤ | Y β z /U | ≤ | Y κ + m − /U | = | Y κ + m /U | ≤ κ + m +1 . (cid:3) We note that if the hypothesis of Corollary 3.17 is consistent, then no cardinalarithmetic above κ + n can be deduced from it, since any forcing which adds nosubsets of κ + n will preserve the relevant properties of the ideal.By combining this technique with the results of Section 2, we can answer thefollowing, which was Open Question 16 from [7]: Question (Foreman) . Is it consistent that there is a uniform ultrafilter U on ω such that ω ω /U has cardinality ω ? Is it consistent that there is a uniform ultra-filter U on ℵ ω +1 such that ω ℵ ω +1 /U has cardinality ℵ ω +1 ? Give a characterizationof the possible cardinalities of ultrapowers. Theorem 3.18. Assume ZFC is consistent with a super-almost-huge cardinal.Then it is consistent that every regular uncountable cardinal κ carries a uniformultrafilter U such that | ω κ /U | = κ . This follows from Section 2 and the next result. Lemma 3.19. Suppose κ = µ + , GCH holds at cardinals ≥ µ , and for all regular λ ≥ κ , there is a normal and fine, κ -complete, λ -dense ideal on P κ ( λ ) . Then forevery regular λ , there is a uniform ultrafilter U on λ such that | µ λ /U | = λ .Proof. Let I be a normal and fine, κ -complete, λ -dense ideal on Z = P κ ( λ ), where κ = µ + and λ is regular. Note that | Z | = λ , and every Y ⊆ Z of size < λ isin I . Let U ⊇ I ∗ be given by Lemma 3.12, so that | µ Z /U | ≤ <λ . If 2 <λ = λ ,then | µ Z /U | ≤ λ . Since 2 µ = κ and any ultrafilter extending I ∗ is ( κ, λ )-regular,Theorem 3.9 implies that | κ Z /U | > λ , and Lemma 3.16 implies that | κ Z /U | ≤| µ Z /U | + . Thus | µ Z /U | = λ . (cid:3) The following extra conclusion can be immediately deduced in the case of µ < ℵ ω and λ = ρ + , where cf( ρ ) = ω . Suppose µ = ω n . Since | ω Zn +1 /U | > λ , we cannothave | ω Zm /U | < ρ for any m , since by Lemma 3.16, we would have | ω Zr /U | < ρ for all r < ω . Also, U is ( ω, ω )-regular, so Theorem 3.9 implies that | ω Z /U | ≥ | ω Z /U | ω .Thus | ω Zm /U | = λ for all m ≤ n .4. Compatibility with square Solovay [16] showed that (cid:3) δ fails when δ ≥ κ and κ is strongly compact. Incontrast, we will show ( ∀ δ ≥ κ ) (cid:3) δ is consistent with the kind of generically super-compact κ constructed in Section 2. The key difference is that nontrivial forcingsmay be absorbed into the quotient algebras of the ideals in the generic case.We start with a model given by Section 2, force (cid:3) , and show that dense idealsstill exist. We will use the following variation on Foreman’s duality theorem [8]. Lemma 4.1. Suppose I is a precipitous ideal on Z ⊆ P ( X ) and e : P → B ( P ( Z ) /I ) is a regular embedding. Suppose that for all generic G ⊆ B ( P ( Z ) /I ) , if j : V → M ⊆ V [ G ] is the associated embedding and H = e − [ G ] , there is a filter ˆ H ∈ V [ G ] that is j ( P ) -generic over M and such that j [ H ] ⊆ ˆ H . Then there is a P -name fora precipitous ideal J on Z such that B ( P ∗ ˙ P ( Z ) /J ) ∼ = B ( P ( Z ) /I ) . Furthermore, J has the same completeness and normality properties as I .Proof. Let H be P -generic over V , and let G be B ( P ( Z ) /I )-generic over V with e [ H ] ⊆ G . Let j : V → M be the generic ultrapower embedding, and let ˆ H beas hypothesized. Then j is uniquely extended to ˆ j : V [ H ] → M [ ˆ H ]. In V [ H ],let Q = B ( P ( Z ) /I ) /e [ H ], and let J = { X ⊆ Z : 1 (cid:13) Q [id] M / ∈ ˆ j ( X ) } . Let ι : P ∗ ˙ P ( Z ) /J be defined by ι ( p, ˙ X ) = e ( p ) ∧ || [id] ∈ ˆ j ( ˙ X ) || .It is easy to see that ι is order and antichain preserving. To see that the rangeof ι is dense, let A ∈ I + be arbitrary. Take G with A ∈ G , so that [id] ∈ j G ( A ).If H = e − [ G ], then some p ∈ H forces A ∈ J + . Let ˙ X be a P -name such that p (cid:13) ˙ X = ˇ A and q (cid:13) ˙ X = Z whenever q ⊥ p ; this makes sure ˙ X is forced to bein J + . Then ι ( p, ˙ X ) ≤ A , since any G with e ( p ) ∧ || [id] ∈ ˆ j ( ˙ X ) || ∈ G must have[id] ∈ j ( A ) and thus A ∈ G .Suppose H ∗ ¯ G ⊆ P ∗ P ( Z ) / ¯ I is generic, and let G ∗ ˆ H = ι [ H ∗ ¯ G ]. For A ∈ J + , A ∈ ¯ G iff [id] M ∈ ˆ j ( A ). If i : V [ H ] → N = V [ H ] Z / ¯ G is the canonicalultrapower embedding, then there is an elementary embedding k : N → M [ ˆ H ]given by k ([ f ] N ) = ˆ j ( f )([id] M ), and ˆ j = k ◦ i . Thus N is well-founded, so J isprecipitous. If f : Z → Ord is a function in V , then k ([ f ] N ) = j ( f )([id] M ) = [ f ] M .Thus k is surjective on ordinals, so it must be the identity, and N = M [ ˆ H ]. Since i = ˆ j and ˆ j extends j , i and j have the same critical point, so the completeness of J is the same as that of I . Finally, since [id] N = [id] M , I is normal in V iff J is normalin V [ H ], because j ↾ X = ˆ j ↾ X , and normality is equivalent to [id] = j [ X ]. (cid:3) For a cardinal δ , let S δ be the collection of bounded approximations to a (cid:3) δ sequence. That is, a condition is a sequence h C α : α ∈ η ∩ Lim i such that η < δ + isa successor ordinal, each C α is a club subset of α of order type ≤ δ , and whenever β is a limit point of C α , C α ∩ β = C β . For proof of the following lemma, we referthe reader to [4]. ENSE IDEALS AND CARDINAL ARITHMETIC 23 Lemma 4.2. For every cardinal δ , S δ is countably closed and ( δ + 1) -strategicallyclosed and adds a (cid:3) δ sequence h C α : α ∈ δ + ∩ Lim i = S G , where G ⊆ S δ is thegeneric filter. For every regular λ ≤ δ , there is a S δ -name for a “threading” partialorder T λδ that adds a club C ⊆ ( δ + ) V of order type λ and such that whenever α isa limit point of C , C ∩ α = C α . Furthermore, S δ ∗ T λδ has a λ -closed dense subsetof size δ . Theorem 4.3. Suppose κ is super-almost-huge and µ < κ is regular. Then thereis a µ -distributive forcing extension in which κ = µ + , (cid:3) λ holds for all cardinals λ ≥ κ , and for all regular λ ≥ κ there is a normal, fine, κ -complete, λ -dense idealon P κ ( λ ) .Proof. By Section 2, we may pass to a µ -distributive forcing extension in which κ = µ + and for all regular λ ≥ κ there is a normal, fine, κ -complete, λ -dense idealon P κ ( λ ), and GCH holds above µ . Over this model, force with P , the Eastonsupport product of S λ where λ ranges over all cardinals ≥ κ . For every cardinal λ , P naturally factors into P <λ × P ≥ λ . Note that if λ ≥ κ , P ≥ λ is ( λ + 1)-strategicallyclosed.First we show that for each regular λ ≥ κ , P ≥ λ is λ + -distributive in V P <λ .Suppose that H × H is ( P <λ × P ≥ λ )-generic, and f : λ → Ord is in V [ H ][ H ].Then in V [ H ], there is a P <λ -name τ for f . By GCH and the fact that we takeEaston support, | P <λ | = λ , so it is λ + -c.c. in V [ H ]. Thus τ may be assumedto be a subset of V of size λ . By the strategic closure of P ≥ λ , τ ∈ V . Thus f = τ H ∈ V [ H ], establishing the claim.Next we show that P preserves all regular cardinals. First note that since P is( κ + 1)-strategically closed, P cannot change the cofinality of any regular δ to some λ ≤ κ . If P does not preserve regular cardinals, then in some generic extension V [ G ],there are λ < δ which are regular in V with κ < λ , such that V [ G ] | = cf( δ ) = λ .Let H = H × H , where H ⊆ P <λ and H ⊆ P ≥ λ . By the λ + -c.c. of P <λ , V [ H ] | = cf( δ ) > λ , and by the λ + -distributivity of P ≥ λ in V [ H ], V [ H ] | = cf( δ ) >λ , a contradiction. Since a square sequence is upwards absolute to models with thesame cardinals and S λ regularly embeds into P for all λ ≥ κ , P forces ( ∀ λ ≥ κ ) (cid:3) λ .For each regular λ ≥ κ , let Z λ = P κ ( λ ). We want to show that in V P , for eachregular λ ≥ κ , there is a normal, fine, λ -dense ideal on Z λ . It suffices to show thatsuch an ideal exists in V P <λ , since P ≥ λ adds no subsets of λ , and | Z λ | = λ . Firstnote that by the strategic closure of P , the dense ideal on κ is unaffected.Let Q be the Easton support product of S λ ∗ T µλ , where λ ranges over all cardinals.There is a coordinate-wise regular embedding of P into Q . When λ is regular, Q <λ has a dense µ -closed subset of size λ . Hence it regularly embeds into B (Col( µ, λ )).The dense ideal I λ on Z λ in V has quotient algebra isomorphic to B ( R × Col( µ, λ ))for some small R , and so Q <λ regularly embeds into this forcing.If G ⊆ P ( Z λ ) /I λ is generic, let H be the induced generic for P <λ , and let j : V → M ⊆ V [ G ] be the ultrapower embedding. Recall that crit( j ) = κ , j ( κ ) = λ + , λ ++ is a fixed point of j , and j [ λ ] ∈ M . First note that j [ λ ] \ j ( κ ) is an Easton setin M . 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