aa r X i v : . [ m a t h . L O ] J a n SOME MUTUALLY INCONSISTENT GENERIC LARGECARDINALS
MONROE ESKEW
In [2], a method was developed to show some non-implications between certainstrong combinatorial properties of ideals. It was shown that one may force to rid theuniverse of ideals of minimal density on a given successor cardinal above ω , whilepreserving “nonregular” ideals. Taylor proved that these are actually equivalentproperties for ideals on ω [8]. The author realized that the method was also ableto show that a broad class of “generic large cardinals” are individually consistent(relative to conventional large cardinals), yet mutually contradictory.In [3] and [4], Foreman proposed generic large cardinals as new axioms for settheory. The main virtues of these axioms are their conceptual similarity to conven-tional large cardinals, combined with their ability to settle classical questions likethe continuum hypothesis (CH). Foreman noted in [4] that some trouble was raisedfor this proposal by the discovery of a few pairs of these axioms that were mutu-ally inconsistent. For example, Woodin showed that under CH, one cannot haveboth an ω -dense ideal on ω and a normal ideal on [ ω ] ω with quotient algebraisomorphic to Col( ω, < ω ). An ω -dense ideal on ω is known to be consistent withCH from large cardinals, but no consistency proof is known for Woodin’s secondideal. However, Foreman noted that Woodin’s argument works equally well if wereplace ω by an inaccessible λ , and the consistency of this kind of ideal on [ λ ] ω can be established from a huge cardinal. Foreman remarked that such examplesof individually consistent but mutually inconsistent generic large cardinals seem tobe relatively rare. This paper brings to light a new class of examples through astrengthening of Woodin’s theorem: Theorem 0.1.
Suppose κ is a successor cardinal and there is a κ -complete, κ -denseideal on κ . Then neither of the following hold:(1) There is a weakly inaccessible λ > κ and a normal, κ -complete, λ + -saturatedideal on [ λ ] κ .(2) There is a successor cardinal λ > κ and a normal, κ -complete, λ -saturatedon [ λ ] κ , where the forcing associated to the ideal has uniform density λ andpreserves λ + -saturated ideals on λ . The ideal on [ ω ] ω that Woodin hypothesized will be seen to be a special caseof (2), and we will not need cardinal arithmetic assumptions in our argument. Wewill also give a relatively simple proof of the individual consistency of (1) and (2)from a huge cardinal, and even their joint consistency from a huge cardinal withtwo targets. For the individual consistency of dense ideals on general successors ofregular cardinals, we refer to [2].Let us fix some terminology and notation. For any partial order P , the saturationof P , sat( P ), is the least cardinal κ such that every antichain in P has size less than κ . The density of P , d( P ), is the least cardinality of a dense subset of P . Clearlyat( P ) ≤ d( P ) + for any P . We say P is κ -saturated if sat( P ) ≤ κ , and P is κ -denseif d( P ) ≤ κ . A synonym for κ -saturation is the κ chain condition ( κ -c.c.).For an ideal I on a set Z , we let I + denote subsets of Z not in I . P ( Z ) /I denotesthe boolean algebra of subsets of Z modulo I under the usual operations, wheretwo sets are equivalent if their symmetric difference is in I . The saturation anddensity of I refer to that of P ( Z ) /I . For preliminaries on forcing with ideals, werefer the reader to [4]. The following is the key backdrop for this paper: Proposition 0.2.
Suppose I is a normal, λ + -saturated ideal on Z ⊆ P ( λ ) . If G ⊆ P ( Z ) /I is generic, the ultrapower V Z /G is isomorphic to a transitive class M ⊆ V [ G ] . Suppose I is κ -complete but I ↾ A is not κ + -complete for any A ∈ I + .If j : V → M is the canonical elementary embedding, then crit( j ) = κ , [ id ] G = j [ λ ] ,and M λ ∩ V [ G ] ⊆ M . Recall that [ λ ] κ denotes { z ⊆ λ : ot( z ) = κ } . If j : V → M is an embeddingarising from a normal λ + -saturated ideal on [ λ ] κ , then by Lo´s’s theorem, λ =ot( j [ λ ]) = ot([ id ]) = j ( κ ). 1. Duality
In this section we present a generalization of Foreman’s Duality Theorem [5]that will be used for both the mutual inconsistency and individual consistency ofthe relevant principles.
Theorem 1.1.
Suppose I is a precipitous ideal on Z and P is a boolean algebra.Let j : V → M ⊆ V [ G ] denote a generic ultrapower embedding arising from I .Suppose ˙ K is a P ( Z ) /I -name for an ideal on j ( P ) such that whenever G ∗ h is P ( Z ) /I ∗ j ( P ) /K -generic and ˆ H = { p : [ p ] K ∈ h } , we have:(1) (cid:13) P ( Z ) /I ∗ j ( P ) /K ˆ H is j ( P ) -generic over M ,(2) (cid:13) P ( Z ) /I ∗ j ( P ) /K j − [ ˆ H ] is P -generic over V , and(3) for all p ∈ P , P ( Z ) /I j ( p ) ∈ K .Then there is P -name for an ideal J on Z and a canonical isomorphism ι : B ( P ∗ ˙ P ( Z ) /J ) ∼ = B ( P ( Z ) /I ∗ ˙ j ( P ) /K ) . Proof.
Let e : P → B ( P ( Z ) /I ∗ j ( P ) /K ) be defined by p
7→ || j ( p ) ∈ ˆ H || . By (3),this map has trivial kernel. By elementarity, it is an order and antichain preservingmap. If A ⊆ P is a maximal antichain, then it is forced that j − [ ˆ H ] ∩ A = ∅ . Thus e is regular.Whenever H ⊆ P is generic, there is a further forcing yielding a generic G ∗ h ⊆P ( Z ) /I ∗ j ( P ) /K such that j [ H ] ⊆ ˆ H . Thus there is an embedding ˆ j : V [ H ] → M [ ˆ H ] extending j . In V [ H ], let J = { A ⊆ Z : 1 (cid:13) ( P ( Z ) /I ∗ j ( P ) /K ) /e [ H ] [ id ] M / ∈ ˆ j ( A ) } . In V , define a map ι : P ∗ ˙ P ( Z ) /J → B ( P ( Z ) /I ∗ ˙ j ( P ) /K ) by ( p, ˙ A ) e ( p ) ∧ || [ id ] M ∈ ˆ j ( ˙ A ) || . It is easy to check that ι is order and antichain preserving.We want to show the range of ι is dense. Let ( B, ˙ q ) ∈ P ( Z ) /I ∗ ˙ j ( P ) /K , andWLOG, we may assume there is some f : Z → V in V such that B (cid:13) ˙ q = [[ f ] M ] K .By the regularity of e , let p ∈ P be such that for all p ′ ≤ p , e ( p ′ ) ∧ ( B, ˙ q ) = 0. Let ˙ A be a P -name such that p (cid:13) ˙ A = { z ∈ B : f ( z ) ∈ H } , and ¬ p (cid:13) ˙ A = Z . 1 (cid:13) P ˙ A ∈ J + because for any p ′ ≤ p , we can take a generic G ∗ h such that e ( p ′ ) ∧ ( B, ˙ q ) ∈ G ∗ h .Here we have [ id ] M ∈ j ( B ) and [ f ] M ∈ ˆ H , so [ id ] M ∈ ˆ j ( A ). Furthermore, ι ( p, ˙ A )forces B ∈ G and q ∈ h , showing ι is a dense embedding. (cid:3) roposition 1.2. If Z, I, P , J, K, ι are as in Theorem 1.1, then whenever H ⊆ P isgeneric, J is precipitous and has the same completeness and normality that I has in V . Also, if ¯ G ⊆ P ( Z ) /J is generic and G ∗ h = ι [ H ∗ ¯ G ] , then if ˆ j : V [ H ] → M [ ˆ H ] is as above, M [ ˆ H ] = V [ H ] Z / ¯ G and ˆ j is the canonical ultrapower embedding.Proof. Suppose H ∗ ¯ G ⊆ P ∗ P ( Z ) /J is generic, and let G ∗ h = ι [ H ∗ ¯ G ] andˆ H = { p : [ p ] K ∈ h } . For A ∈ J + , A ∈ ¯ G iff [ id ] M ∈ ˆ j ( A ). If i : V [ H ] → N = V [ H ] Z / ¯ G is the canonical ultrapower embedding, then there is an elementaryembedding 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 is precipitous. 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 theidentity, and N = M [ ˆ H ]. Since i = ˆ j and ˆ j extends j , i and j have the samecritical 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 normal in V [ H ], because j ↾ S Z = ˆ j ↾ S Z ,and normality is equivalent to [ id ] = j [ S Z ]. (cid:3) Theorem 1.1 is optimal in the sense that it characterizes exactly when an ele-mentary embedding coming from a precipitous ideal can have its domain enlargedvia forcing:
Proposition 1.3.
Let I be a precipitous ideal on Z and P a boolean algebra. Thefollowing are equivalent.(1) In some generic extension of a P ( Z ) /I -generic extension, there is an elemen-tary embedding ˆ j : V [ H ] → M [ ˆ H ] , where j : V → M is the elementary embed-ding arising from I and H is P -generic over V .(2) There are p ∈ P , A ∈ I + , and a P ( A ) /I -name for an ideal K on j ( P ↾ p ) suchthat P ( A ) /I ∗ j ( P ↾ p ) /K satisfies the hypothesis of Theorem 1.1.Proof. (2) ⇒ (1) is trivial. To show (1) ⇒ (2), let ˙ Q be a P ( Z ) /I -name for apartial order, and suppose A ∈ I + and ˙ H are such that (cid:13) P ( A ) /I ∗ ˙ Q “ ˙ H is j ( P )-generic over M and j − [ ˙ H ] is P -generic over V .” By the genericity of j − [ ˙ H ],the set of p ∈ P such that (cid:13) P ( A ) /I ∗ ˙ Q j ( p ) / ∈ ˙ H is not dense. So let p be suchthat for all p ≤ p , || j ( p ) ∈ ˙ H || 6 = 0. In V P ( A ) /I , define an ideal K on j ( P ↾ p )by K = { p ∈ j ( P ↾ p ) : 1 (cid:13) Q p / ∈ ˙ H } . We claim K satisfies the hypothesesof Theorem 1.1. Let G ∗ h be P ( A ) /I ∗ j ( P ↾ p ) /K -generic. In V [ G ∗ h ], letˆ H = { p ∈ j ( P ) : [ p ] K ∈ h } .(1) If D ∈ M is open and dense in j ( P ), then { [ d ] K : d ∈ D and d / ∈ K } is densein j ( P ) /K . For otherwise, there is p ∈ j ( P ) \ K such that p ∧ d ∈ K for all d ∈ D . By the definition of K , we can force with Q over V [ G ] to obtain a filter H ⊆ j ( P ) with p ∈ H . But H cannot contain any elements of D , so it isnot generic over M , a contradiction. Thus if h ⊆ j ( P ) /K is generic over V [ G ],then ˆ H is j ( P )-generic over M .(2) If A ∈ V is a maximal antichain in P , then { [ j ( a )] K : a ∈ A and j ( a ) / ∈ K } is a maximal antichain in j ( P ) /K . For otherwise, there is p ∈ j ( P ) \ K suchthat p ∧ j ( a ) ∈ K for all a ∈ A . We can force with Q over V [ G ] to obtain afilter H ⊆ j ( P ) with p ∈ H . But H cannot contain any elements of j [ A ], so j − [ H ] is not generic over V , a contradiction.(3) If p ∈ P ↾ p , || j ( p ) ∈ ˙ H || P ( A ) /I ∗ ˙ Q = 0, so 1 P ( Z ) /I j ( p ) ∈ K . (cid:3) emma 1.4. Suppose the ideal K in Theorem 1.1 is forced to be principal. Let ˙ m be such that (cid:13) P ( Z ) /I ˙ K = { p ∈ j ( P ) : p ≤ ¬ ˙ m } . Suppose f and A are suchthat A (cid:13) ˙ m = [ f ] , and ˙ B is a P -name for { z ∈ A : f ( z ) ∈ H } . Let ¯ I be the idealgenerated by I in V [ H ] . Then ¯ I ↾ B = J ↾ B , where J is given by Theorem 1.1.Proof. Clearly J ⊇ ¯ I . Suppose that p (cid:13) “ ˙ C ⊆ ˙ B and ˙ C ∈ ¯ I + ,” and let p ≤ p be arbitrary. WLOG P is a complete boolean algebra. For each z ∈ Z , let b z = || z ∈ ˙ C || . In V , define C ′ = { z : p ∧ b z ∧ f ( z ) = 0 } . p (cid:13) ˙ C ⊆ C ′ , so C ′ ∈ I + .If G ⊆ P ( Z ) /I is generic with C ′ ∈ G , then j ( p ) ∧ b [ id ] ∧ ˙ m = 0. Take ˆ H ⊆ j ( P )generic over V [ G ] with j ( p ) ∧ b [ id ] ∧ ˙ m ∈ ˆ H . Since b [ id ] (cid:13) Mj ( P ) [ id ] ∈ ˆ j ( C ), p ˙ C ∈ J as p ∈ H = j − [ ˆ H ]. Thus p (cid:13) ˙ C ∈ J + . (cid:3) Corollary 1.5. If I is a κ -complete precipitous ideal on Z and P is κ -c.c., thenthere is a canonical isomorphism ι : P ∗ P ( Z ) / ¯ I ∼ = P ( Z ) /I ∗ j ( P ) .Proof. If G ∗ ˆ H ⊆ P ( Z ) /I ∗ j ( P ) is generic, then for any maximal antichain A ⊆ P in V , j [ A ] = j ( A ), and M | = j ( A ) is a maximal antichain in j ( P ). Thus j − [ ˆ H ] is P -generic over V , and clearly for each p ∈ P , we can take ˆ H with j ( p ) ∈ ˆ H . Taking a P ( Z ) /I -name ˙ K for the trivial ideal on j ( P ), Theorem 1.1 implies that there is a P -name ˙ J for an ideal on Z and an isomorphism ι : B ( P ∗P ( Z ) /J ) → B ( P ( Z ) /I ∗ j ( P )),and Proposition 1.4 implies that J = ¯ I . (cid:3) Mutual inconsistency
Lemma 2.1.
Suppose I is a normal ideal on Z ⊆ P ( X ) and | Z | = | X | . Thefollowing are equivalent:(1) I is | X | + -saturated.(2) Every normal J ⊇ I on Z is equal to I ↾ A for some A ⊆ Z .(3) If [ A ] I (cid:13) τ ∈ V Z /G , then there is some function f : Z → V in V such that [ A ] I (cid:13) τ = [ ˇ f ] G .Proof. We only use | Z | = | X | for (3) ⇒ (1). To show (1) ⇒ (2), suppose I is | X | + -saturated. Let { A x : x ∈ X } be a maximal antichain in J ∩ I + . Then [ ∇ A x ]is the largest element of P ( Z ) /I whose elements are in J . Thus J = I ↾ ( Z \ ∇ A x ).For (2) ⇒ (1), suppose I is not | X | + -saturated, and let { A α : α < δ } be a maximalantichain where δ ≥ | X | + . Let J be the ideal generated by S { Σ α ∈ Y [ A α ] : Y ∈P | X | + ( δ ) } . Then J is a normal, proper ideal extending I . J cannot be equal I ↾ A for some A ∈ I + because if so, there is some α where A ∩ A α ∈ I + . A ∩ A α ∈ J byconstruction, but every I -positive subset of A is ( I ↾ A )-positive.For the implication (1) ⇒ (3), see Propositions 2.12 and 2.23 of [4]. For (3) ⇒ (1)under the assumption | Z | = | X | , use Remark 2.13. (cid:3) Suppose κ = µ + and I is a normal, κ -complete, λ + -saturated ideal on Z ⊆P κ ( λ ). If j : V → M ⊆ V [ G ] is a generic embedding arising from I , then by Lo´s’s theorem, [ id ] < j ( κ ), and thus | λ | = µ in V [ G ]. Since λ + is preserved, j ( κ ) = λ + . Now suppose P is a κ -c.c. partial order. By Corollary 1.5, P preservesthe λ + -saturation of I just in case P ( Z ) /I forces j ( P ) is λ + -c.c. The notion ofa c.c.c.-indestructible ideal on ω has been considered before, for example in [1]and [6], and a straightforward generalization would be to say that for successorcardinals κ , a normal κ -complete λ + -saturated ideal on Z ⊆ P κ ( λ ) is indestructiblef its saturation is preserved by every κ -c.c. forcing. For the next lemma however,we consider the dual notion by fixing a κ -c.c. partial order and quantifying oversaturated ideals on a set Z . Definition. P is Z -absolutely κ -c.c. when for all normal, | S Z | + -saturated ideals I on Z , (cid:13) P ( Z ) /I j ( P ) is j ( κ ) -c.c. Such partial orders abound; for example Add( ω, α ) is Z -absolutely c.c.c. for all Z and α , since Cohen forcing has an absolute definition. Also, if κ = µ + and Z is such that every normal ideal on Z is κ -complete, then all µ -centered or µ -c.c.partial orders are Z -absolutely κ -c.c.Proof of the following basic lemma is left to the reader: Lemma 2.2.
Suppose e : P → Q is a regular embedding and κ is a cardinal. Q is κ -dense iff P is κ -dense and (cid:13) P Q / ˙ G is κ -dense. Lemma 2.3.
Suppose κ = µ + and Z ⊆ { z ∈ P κ ( λ ) : z ∩ κ ∈ κ } . Suppose P is Z -absolutely κ -c.c. and κ ≤ d( P ↾ p ) ≤ λ for all p ∈ P . Then in V P , there are nonormal λ -dense ideals on Z .Proof. Suppose p (cid:13) ˙ J is a normal, fine, κ -complete, λ + -saturated ideal on Z . Let I = { X ⊆ Z : p (cid:13) X ∈ ˙ J } . It is easy to check that I is normal. The map σ : P ( Z ) /I → B ( P ↾ p ∗ P ( Z ) / ˙ J ) that sends X to ( || ˇ X ∈ ˙ J + || , ˙[ X ] J ) is an order-preserving and antichain-preserving map, so I is λ + -saturated.Let H be P -generic over V with p ∈ H . By Corollary 1.5, P V [ H ] ( Z ) / ¯ I ∼ =( P V ( Z ) /I ∗ j ( P )) /e [ H ], where e is the regular embedding from Theorem 1.1, so ¯ I is λ + = j ( κ )-saturated by the Z -absolute κ -c.c. By Proposition 1.2, ¯ I is normal,and clearly ¯ I ⊆ J . By Lemma 2.1, there is A ∈ ¯ I + such that J = ¯ I ↾ A . Since j ( P ) is forced to be nowhere < j ( κ )-dense, P ( Z ) /I ∗ j ( P ) is nowhere λ -dense. Sinced( P ) ≤ λ , ( P V ( Z ) /I ∗ j ( P )) /e [ H ] is nowhere λ -dense. Thus J is not λ -dense. (cid:3) We can now prove the mutual inconsistency of the following hypotheses when µ + = κ < λ = η + :(1) There is a κ -complete, κ -dense ideal I on κ .(2) There is a normal, κ -complete, λ -absolutely λ -saturated J ideal on [ λ ] κ , suchthat d( P ( A ) /J ) = λ for all A ∈ J + .Solovay observed that if there is a κ -complete, κ + -saturated ideal on κ , thenthere is a normal one as well. This also applies to dense ideals. For suppose I is a κ -complete, κ -dense ideal on some set Z . Let J = { X ⊆ κ : 1 (cid:13) P ( Z ) /I κ / ∈ j ( X ) } .It is easy to verify that J is normal and that the map X
7→ || κ ∈ j ( X ) || is an orderand antichain preserving map of P ( κ ) /J into P ( Z ) /I . If { X α : α < κ } is a maximalantichain in P ( κ ) /I , then [ ∇ X α ] J = [ κ ] J , so 1 (cid:13) P ( Z ) /I ( ∃ α < κ ) κ ∈ j ( X α ). Thus { e ( X α ) : α < κ } is maximal in P ( Z ) /I and J is κ -dense.Suppose (1) and (2) hold. If j : V → M ⊆ V [ G ] is a generic embedding arisingfrom P ([ λ ] κ ) /J , then j ( κ ) = λ and M λ ∩ V [ G ] ⊆ M . M thinks j ( I ) is a normal, λ -dense ideal on λ , and this holds in V [ G ] as well by the closure of M . But sincethe forcing to produce G is λ -absolutely λ -c.c. and of uniform density λ , Lemma 2.3implies that no such ideals can exist in V [ G ], a contradiction.Under GCH, this part of the result can be strengthened slightly. Suppose λ <λ = λ and P is λ -c.c. If θ is sufficiently large, then we can take an elementary M ≺ H θ with P ∈ M , | M | = λ , M <λ ⊆ M . Then any maximal antichain in P ∩ M is aember of M , and by elementarity is also a maximal antichain in P . It is clearfrom the definition that in general, if P is Z -absolutely κ -c.c., then so is any regularsuborder. So if P is also nowhere < λ -dense and λ -absolutely λ -c.c., forcing with P ∩ M will destroy all dense ideals on λ .To see that the same holds for P , suppose ˙ J is a P -name for a normal ideal on λ and D = { ˙ A α : α < λ } is a sequence of names for subsets of λ witnessing that˙ J is λ -dense. Let I = { X ⊆ λ : 1 (cid:13) P X ∈ ˙ J } . As in the proof of Lemma 2.3, I is a normal λ + -saturated ideal on λ , and it is forced that ¯ I = ˙ J ↾ ˙ A for some˙ A ∈ ˙ J + . Let θ be sufficiently large, and take M ≺ H θ with | M | = λ , λ ⊆ M , M <λ ⊆ M , { P , ˙ J, ˙ B, D } ∈ M . We may assume that each ˙ A α and ˙ B take theform ∪ β<λ S β × { ˇ β } , where each S β is a maximal antichain, so that they are all P ∩ M -names.Let G ⊆ P be generic and let G = G ∩ M . Let I be the ideal generated by I in V [ G ], and let I be the ideal generated by I in V [ G ]. In V [ G ], if C is an I -positive subseteq of ˙ B G , then in V [ G ], there is some A α such that A α \ C ∈ J .But this just means that some Y ∈ I covers A α \ C , and this is absolute to V [ G ].Thus I ↾ B is λ -dense in V [ G ], contradicting Lemma 2.3. Thus we have: Proposition 2.4. If µ + = κ < λ = η + and η = λ , then there cannot exist both a κ -complete, κ -dense ideal on κ and a normal, κ -complete, λ -absolutely λ -saturated,nowhere η -dense ideal on [ λ ] κ . Now we turn to the weakly inaccessible case. Suppose that κ = µ + , there is a κ -dense normal ideal I on κ , λ > κ is weakly inaccessible, and there is a normal, κ -complete, λ + -saturated ideal J on [ λ ] κ . If j : V → M ⊆ V [ G ] is an embeddingarising from J , then M | = j ( I ) is a normal, λ -dense ideal on λ , and V [ G ] satisfiesthe same by the closure of M . In V , we may define a normal ideal I ′ = { X ⊆ λ :1 (cid:13) P ([ λ ] κ ) /J X ∈ j ( I ) } . As before, the map X ( || ˇ X ∈ j ( I ) + || , ˙[ X ] j ( I ) ) preservesantichains, implying I ′ is λ + -saturated.Now since λ is weakly inaccessible in V , whenever i : V → M is an embeddingarising from I ′ , i ( λ ) > λ + since M has the same λ + and thinks i ( λ ) is a limitcardinal. By Lemma 2.1, for each α ≤ λ + , there is a function f α : λ → λ such that1 (cid:13) P ( λ ) /I ′ ˇ α = [ f α ]. This means that for α < β ≤ λ + , { γ : f α ( γ ) ≥ f β ( γ ) } ∈ I ′ .In V [ G ], I ′ ⊆ j ( I ). If H ⊆ P ( λ ) /j ( I ) is generic over V [ G ], and k : V [ G ] → N is the associated embedding, then N | = [ f α ] H < [ f β ] H < k ( λ ) for α < β ≤ λ + .Thus the ordertype of k ( λ ) is greater than λ + . But since j ( I ) is λ + -saturated and λ = µ + in V [ G ], k ( λ ) = λ + , a contradiction.The κ -density of the ideal on κ was only relevant to have a saturation property ofan ideal on λ that is upwards-absolute to a model with the same P ( λ ). Furthermore,if we replace the ideal J on [ λ ] κ by one on { z ⊆ λ + : ot( z ∩ λ ) = κ } , then we getenough closure of the generic ultrapower to derive a contradiction from simply a κ + -saturated ideal on κ . (This kind of ideal on P ( λ + ) will also be shown individuallyconsistent from a large cardinal assumption in the next section.) The commonthread is captured by the following: Proposition 2.5.
Suppose κ is a successor cardinal, λ > κ is a limit cardinal, and Z is such that λ ⊆ S Z , and for all z ∈ Z , z ∩ κ ∈ κ and ot( z ∩ λ ) = κ . Then thefollowing are mutually inconsistent:(1) There is a κ -complete, Z -absolutely κ + -saturated ideal on κ .(2) There is a normal λ + -saturated ideal on Z . . Consistency
First we introduce a simple partial order. The basic idea is due to Shioya [7]. If µ < κ are in Reg (the class of regular cardinals), we define: P ( µ, κ ) = <µ supp Y α ∈ ( µ,κ ) ∩ Reg
Col( α, < κ )The ordering is reverse inclusion. It is a bit more convenient to view P ( µ, κ ) as thecollection of partial functions with domain contained in κ such that: • ∀ ( α, β, γ ) ∈ dom p , α is regular, µ < α < κ , γ < α , and p ( α, β, γ ) < β . • |{ α : ∃ β ∃ γ ( α, β, γ ) ∈ dom p }| < µ . • ∀ α |{ ( β, γ ) : ( α, β, γ ) ∈ dom p }| < α .Proof of the next two lemmas is standard. Lemma 3.1. If µ is regular and κ > µ is inaccessible, then P ( µ, κ ) is κ -c.c., µ -closed, and of size κ . Lemma 3.2. If µ ≤ κ ≤ λ , then there is a projection σ : P ( µ, λ ) → P ( µ, κ ) givenby σ ( p ) = p ↾ κ . Suppose that there is elementary embedding j : V → M with crit( j ) = κ and M λ ⊆ M . If λ = j ( κ ), then κ is called huge . A slightly stronger hypothesis isthat λ = j ( κ ) + . This is substantially weaker than a cardinal, which is when λ = j ( κ ). The hugeness of κ with target λ is equivalent to the existence of a normal κ -complete ultrafilter on [ λ ] κ , and a shorthand for this is “ κ is λ -huge.” The slightlystronger hypothesis is equivalent to the existence of a normal κ -complete ultrafilteron { z ⊆ λ + : ot( z ∩ λ ) = κ } . The next result shows the relative consistency ofthe type of ideals mentioned in (1) of Theorem 0.1 and in the discussion precedingProposition 2.5. Proposition 3.3.
Suppose that λ > κ and there is a κ -complete normal ultrafilteron Z such that ∀ z ∈ Z (ot( z ∩ λ ) = κ ) . If µ < κ is regular and G ⊆ P ( µ, κ ) isgeneric, then in V [ G ] , κ = µ + , λ is inaccessible, and there is a normal, κ -complete, λ -saturated ideal J on Z , such that P ( Z ) /J ∼ = P ( µ, λ ) /G .Proof. Let I be the dual ideal to a κ -complete normal ultrafilter on Z and j the as-sociated embedding. By Corollary 1.5, P ( µ, κ ) ∗ P ( Z ) / ¯ I ∼ = P ( µ, λ ). Since j ◦ σ = id , P ( µ, λ ) /σ − [ G ] = P ( µ, λ ) /j [ G ], so the map given by Theorem 1.1 shows the desiredisomorphism. The normality and completeness claims follow from Proposition 1.2.It is clear that P ( µ, λ ) forces κ = µ + and preserves the inaccessbility of λ . (cid:3) Lemma 3.4 (Shioya [7]) . Assume GCH, κ is regular, λ ≤ κ is inaccessible, and P is κ -c.c. and of size ≤ κ . If G ⊆ P is generic, then in V [ G ] there is a projection π : Col( κ, < λ ) V → Col( κ, < λ ) V [ G ] .Proof. First note that Col( κ, λ ) is isomorphic to the subset of conditions q suchthat dom( q ) ⊆ Reg × κ , since Col( κ, η + ) ∼ = Q <κ supp α ∈ [ η,η + ] Col( κ, α ) for all η . So wework with this partial order instead.Inductively choose a sequence of P -names for ordinals below λ , h τ α : α < λ i , suchthat for every regular η ∈ [ κ, λ ] and every P -name σ , if (cid:13) σ < ˇ η , then there is α < η such that (cid:13) σ = τ α . For a given q ∈ Col( κ, λ ) V , let τ q be the P -name for a functionuch that (cid:13) dom( τ q ) = dom(ˇ q ), and for all ( α, β ) ∈ dom( q ), (cid:13) τ q ( α, β ) = τ q ( α,β ) .In V [ G ], let π ( q ) = τ Gq .To show π is a projection, suppose p ∈ G and p (cid:13) ˙ q ≤ π ( q ). By the κ -c.c., there is some d ⊆ λ × κ such that | d | < κ and (cid:13) dom ˙ q ⊆ ˇ d . Working in V , if ( α, β ) ∈ d \ dom( q ), then there is a name r ( α, β ) for an ordinal < α suchthat p (cid:13) ( α, β ) ∈ dom( ˙ q ) → ˙ q ( α, β ) = r ( α, β ). Then r ∪ q = q ≤ q , and p (cid:13) τ q ≤ ˙ q . (cid:3) Now we show the relative consistency of the type of ideal mentioned in (2) ofTheorem 0.1:
Theorem 3.5.
Assume GCH, µ < κ < δ < λ are regular and κ is λ -huge. P ( µ, κ ) ∗ ˙Col( δ, < λ ) forces that κ = µ + , λ = δ + , and there is a normal, κ -complete, λ -absolutely λ -saturated ideal J on [ λ ] κ of uniform density λ .Proof. Let I be the dual ideal to a κ -complete normal ultrafilter on [ λ ] κ , and j : V → M the associated embedding. It is easy to show that under 2 λ = λ + , λ + < j ( λ ) < λ ++ . Let G ⊆ P ( µ, κ ) be generic over V . By Proposition 3.3, in V [ G ],¯ I is a κ -complete normal ideal on [ λ ] κ such that P ([ λ ] κ ) / ¯ I ∼ = P ( µ, λ ) /σ − [ G ].If ˆ G ⊆ P ( µ, λ ) is generic extending G , then we can extend the embedding toˆ j : V [ G ] → M [ ˆ G ], and by Proposition 1.2, ˆ j is also a generic ultrapower embeddingarising from ¯ I . There is a projection from P ( µ, λ ) /G to Col( δ, < λ ) V , and byShioya’s lemma, there is in V [ G ] a projection from Col( δ, < λ ) V to Col( δ, < λ ) V [ G ] .Let H ⊆ Col( δ, λ ) V [ G ] be the generic thus projected from ˆ G . Since M [ ˆ G ] λ ∩ V [ ˆ G ] ⊆ M [ ˆ G ], and Col( δ, < λ ) is δ -directed closed and of size λ , we may take in M [ G ]a condition m ∈ Col( j ( δ ) , < j ( λ )) M [ ˆ G ] that is stronger than j ( q ) for all q ∈ H .Since j ( λ ) is inaccessible in M [ ˆ G ], and j ( λ ) < λ ++ , we may list all the dense opensubsets of Col( j ( δ ) , < j ( λ )) M [ ˆ G ] that live in M [ ˆ G ] in ordertype λ + and build a filterˆ H generic over M [ ˆ G ] with m ∈ ˆ H .Thus we have a P ([ λ ] κ ) / ¯ I -name for a filter ˆ H ⊆ Col( j ( δ ) , < j ( λ )) that is genericover M [ ˆ G ], and such that ˆ j − [ ˆ H ] is Col( δ, < λ )-generic over V [ ˆ G ]. Theorem 1.1implies that in V [ G ][ H ], there is a normal, κ -complete ideal J on [ λ ] κ with quotientalgebra isomorphic to P ( µ, λ ) / ( G ∗ H ). Because coordinates in [ κ, λ ) \{ δ } are ignoredby the projection, this forcing has uniform density λ .It remains to show the absoluteness of the λ -c.c. First we note the followingproperties of P ( µ, λ ) in V :(a) For all η < λ , |{ p ↾ η : p ∈ P ( µ, λ ) }| < λ .(b) If η < λ is regular and dom p ⊆ η × λ , then the ordertype of dom p in theG¨odel ordering on triples is less than η . This is simply because | p | < η .Now work in V ′ = V [ G ][ H ]. Suppose K is a normal λ + -saturated ideal on λ ,let A ⊆ P ( λ ) /K be generic, let i : V ′ → N be the associated embedding. Itsuffices to show that i ( P ( µ, λ ) V ) is λ + -c.c. in V ′ [ A ]. This is because the natureof the projection from P ( µ, λ ) to P ( µ, κ ) ∗ ˙Col( δ, < λ ) gives that two functions in P ( µ, λ ) / ( G ∗ H ) are incompatible just when they disagree at some point in theircommon domain, which is the same criterion for incompatibility in P ( µ, λ ). We willmake a ∆-system argument, but using the more absolute properties (a) and (b)above instead of pure cardinality considerations.et { p ξ : ξ < λ + } be a set of conditions in i ( P ( µ, λ ) V ). Since µ < λ + and we haveGCH, we can take some X ∈ [ λ + ] λ + such that {{ α : ∃ β ∃ γ ( α, β, γ ) ∈ p ξ } : ξ ∈ X } forms a ∆-system with root r . Denote dom( p ξ ↾ r × ( λ + ) ) by d ξ . There areunboundedly many η < λ satisfying (b), so let sup r < η < λ + be such that for all ξ ∈ X , the ordertype of d ξ in the G¨odel ordering on ( λ + ) is less than η . WLOGwe may assume ot( d ξ ) = η for all ξ ∈ X . Let h d ξ ( ν ) : ν < η i list the elements of d ξ in increasing order. If there is no ν such that { d ξ ( ν ) : ξ ∈ X } is unbounded inthe G¨odel ordering, then there is some ζ < λ + such that d ξ ⊆ ζ for all ξ ∈ X .But by property (a) and elementarity, there are < λ + many distinct elements of i ( P ( µ, λ ) V ) ↾ η , so we do not have an antichain. Otherwise, let ν be the leastordinal < η such that { d ξ ( ν ) : ξ ∈ X } is unbounded. We can recursively choose h ξ α : α < λ + i such that for α < β , d ξ α ( ν ) is above all elements of d ξ β ( ν ) in theG¨odel ordering. Let ζ be such that { d ξ α ( ν ) : α < λ + and ν < ν } ⊆ ζ . Againby property (a), there must be α < β such that p ξ α is compatible with p ξ β . Thisproves the λ -absolute λ -c.c. (cid:3) Using a technique of Magidor described in [4], one can show that if κ is λ -huge(or just λ -almost huge) and µ < κ is regular, then there is a κ + -saturated ideal on κ after forcing with P ( µ, κ ) ∗ ˙Col( κ, < λ ). It is unknown whether there can exist asuccessor cardinal κ and a normal, κ -complete, κ + -saturated ideal on [ κ + ] κ . In theabove construction, the closest we can come is a κ ++ -saturated ideal on [ κ ++ ] κ . Theessential reason for this is not in getting the master condition m , as this problemcan be overcome by constructing P ( µ, κ ) with the Silver collapse instead of the Levycollapse. Rather, the construction of the filter ˆ H seems blocked when we collapse λ to κ + .Finally, we note the mutual consistency of these different “generic hugeness”properties of a successor cardinal: Proposition 3.6.
Assume GCH, µ < κ < δ < λ < λ are regular, and κ is both λ -huge and λ -huge. Then there is a generic extension in which κ = µ + , λ = δ + , λ is inaccessible, and:(1) There is a normal, κ -complete, λ -saturated ideal on [ λ ] κ .(2) There is a normal, κ -complete, λ -saturated ideal on [ λ ] κ .Proof. Let G ∗ H be P ( µ, κ ) ∗ ˙Col( δ, < λ )-generic. Then (1) holds by Theorem 3.5.To show (2), we argue almost exactly the same as in the proof of Theorem 3.5,noting that Col( δ, < λ ) V projects to Col( δ, < λ ) V . (cid:3) References [1] James Baumgartner and Alan Taylor,
Saturation properties of ideals in generic extensions ii ,Trans. Amer. Math. Soc. (1982), no. 2, 587–609.[2] Monroe Eskew,
Measurability properties on small cardinals , Ph.D. thesis, UC Irvine, 2014.[3] Matthew Foreman,
Potent axioms , Trans. Amer. Math. Soc. (1986), no. 1, 1–28.[4] ,
Ideals and generic elementary embeddings , Handbook of set theory (Matthew Foremanand Akihiro Kanamori, eds.), vol. 2, Springer, Dordrecht, 2010, pp. 885–1147.[5] ,
Calculating quotient algebras of generic embeddings , Israel J. Math. (2013), no. 1,309–341.[6] Matthew Foreman, Menachem Magidor, and Saharon Shelah,
Martin’s maximum, saturatedideals and nonregular ultrafilters, i , Ann. of Math. (2) (1988), no. 1, 1–47.[7] Masahiro Shioya,
A new saturated filter , Research Institute for Mathematical SciencesKokyuroku (2008), 63–69.8] Alan Taylor,
Regularity properties of ideals and ultrafilters , Ann. Math. Logic (1979), no. 1,33–55.(1979), no. 1,33–55.