smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals

Advances in Mathematics 222 (2009) 565–595www.elsevier.com/locate/aim

Smoke and mirrors: Combinatorial properties of smallcardinals equiconsistent with huge cardinals ✩

Matthew Foreman

Mathematics Department, University of California, Irvine, CA, United States

Received 10 March 2006; accepted 6 May 2009

Available online 16 June 2009

Communicated by the Managing Editors of AIM

Abstract

Since the work of Godel and Cohen many questions in infinite combinatorics have been shown to beindependent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Axiom of Choice(ZFC). Attempts to strengthen the axioms to settle these problems have converged on a system of principlescollectively known as Large Cardinal Axioms.

These principles are linearly ordered in terms of consistency strength. As far as is currently known, allnatural independent combinatorial statements are equiconsistent with some large cardinal axiom. The stan-dard techniques for showing this use forcing in one direction and inner model theory in the other direction.

The conspicuous open problems that remain are suspected to involve combinatorial principles muchstronger than the large cardinals for which there is a current fine-structural inner model theory for.

The main results in this paper show that many standard constructions give objects with combinatorialproperties that are, in turn, strong enough to show the existence of models with large cardinals are largerthan any cardinal for which there is a standard inner model theory.© 2009 Elsevier Inc. All rights reserved.

Keywords: Large cardinals; Inner models; Forcing; Chang’s Conjecture; Non-stationary ideal

1. Introduction

Since the work of Godel and Cohen many questions in infinite combinatorics have been shownto be independent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Ax-

✩ This research was partially supported by NSF grant number DMS 0701030.E-mail address: [email protected].

0001-8708/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.aim.2009.05.006

566 M. Foreman / Advances in Mathematics 222 (2009) 565–595

iom of Choice (ZFC). Attempts to strengthen the axioms to settle these problems have convergedon a system of principles collectively known as Large Cardinal Axioms.

These principles are linearly ordered in terms of consistency strength. As far as is currentlyknown, all natural independent combinatorial statements are equiconsistent with some large car-dinal axiom. The standard techniques for showing this use forcing in one direction and innermodel theory in the other direction.

The conspicuous open problems that remain are suspected to involve combinatorial principlesmuch stronger than the large cardinals for which there is currently a fine-structural inner modeltheory. The main results in this paper show that many standard constructions give objects withcombinatorial properties that in turn, are strong enough to show the existence of models withlarge cardinals that are larger than any cardinal for which there is a standard inner model theory.

This paper presents several combinatorial properties of “small” cardinals (such as the ℵn’sfor n ∈ ω) which are equiconsistent with huge cardinals. These properties are sufficient to yieldgeneric elementary embeddings that coincide with large cardinal embeddings from inner models,and can be used to characterize filters that are the remnants of large cardinal ultrafilters after thelarge cardinal has been collapsed by forcing to be accessible.

The main theme of this paper is that if one carefully chooses a stationary set and constructsrelative to it and the non-stationary ideal then one can get an inner model for a very large cardinal.The crux of this approach is to describe the appropriate set and show that it is stationary.

In the beginning of Section 2, it is shown that every normal ideal is the projection of thenon-stationary ideal restricted to a stationary set. In particular, the duals of large cardinal ultra-filters are of this form. This gives an easy example of models of very large cardinals gotten byconstructing relative to a stationary set and the non-stationary ideal.

A more subtle application of this technique is described next. In Section 2.1.1, a property ofideals called decisiveness is defined. This property is strong enough to produce inner models withvery large cardinals. It is remarked that the typical ideals that are produced in consistency resultsare decisive, and hence these results give “reversals” of consistency results whose proofs start bycollapsing large cardinals to be small cardinals, in many cases making them equiconsistencies.A typical argument that induced ideals are decisive is given in Section 2.1.2, along with anexample of an indecisive ideal.

The second type of result giving inner models with huge cardinals involves a form of Chang’sConjecture. The spirit of Chang’s Conjecture is to ask that each structure on a given cardinalhave an elementary substructure with certain second order properties that are stronger than thosegiven by the ordinary Lowenheim–Skolem theorem. In Section 2.2, a strengthening of Chang’sConjecture is introduced, that asks that the Chang elementary submodel have a certain condensa-tion property (namely that its transitive collapse be correct about the closed unbounded filter). Itis shown that this strong Chang’s Conjecture yields the existence of an inner model with a hugecardinal. The technique presented is clearly very flexible and can be extended to many otherChang-type situations.

The strong Chang’s Conjecture that yields inner models with huge cardinals would be uninter-esting if it were not consistent. The main result in Section 3, is that the strong Chang’s Conjectureis consistent, modulo the consistency of two-huge cardinals.

Important consequences of supercompact cardinals include strong reflection results that canbe phrased as saying that certain sets are stationary. In the presence of supercompact cardinals,constructing relative to these stationary sets gives back an inner model for a supercompact car-dinal. In Section 4 it is shown that versions of these sets can be proved stationary assuming

M. Foreman / Advances in Mathematics 222 (2009) 565–595 567

Martin’s Maximum. It is conjectured that constructing relative to these analogous sets will giveinner models for supercompact cardinals.

1.1. The non-stationary ideal and Chang’s Conjecture

The main results of this paper show that in certain circumstances the closed unbounded filteron P(X) can instantiate properties of large cardinal ultrafilters. One way this can happen is ifvarious forms of Chang’s Conjecture hold. We now define the closed unbounded filter and itsdual, the non-stationary ideal, and explain the relationship between the closed unbounded filterand Chang’s Conjecture.

If X is an uncountable set and A is a structure with domain X in a countable language,we define CA to be {z ⊆ X: z ≺ A}. The collection {CA: A is a structure in a countablelanguage} generates a normal, fine, countably complete filter on P(X). We call this filter theclosed unbounded filter. We define the non-stationary ideal on P(X) to be the dual of this filter.Positive sets with respect to the non-stationary ideal are called stationary. We will denote thenon-stationary ideal by NS.

Note that a collection Z ⊂ P(X) is stationary iff for all structures A in a countable languagewith domain X there is a z≺A with z ∈Z.

Given cardinals λ0 < λ1 < · · ·< λn and κ0 < · · ·< κn with λi � κi , we say that the Chang’sConjecture (κn, κn−1, . . . , κ0) � (λn,λn−1, . . . , λ0) holds iff every structure A in a countablelanguage having domain κn has an elementary substructure B ≺ A with |B ∩ κi | = λi . Thereare several variants of Chang’s Conjecture that we will consider. For example we will write(κn, . . . , κ1, κ0) � (λn, . . . , λ1,<λ0) to mean that every structure A in a countable languagehaving domain κn has an elementary substructure B with |B ∩ κi | = λi for 1 � i � n and |B ∩κ0|< λ0.

Using the Downwards Lowenheim–Skolem theorem, the instance of Chang’s Conjecturegiven by (κn, κn−1, . . . , κ0) � (λn,λn−1, . . . , λ0) is easily seen to be equivalent to the appar-ently stronger property that if A is a structure in a countable language with domain some setX ⊇ κn, then there is a B≺A with |B| = λn and |B∩ κi | = λi for 0 � i � n. Thus we see:

Remark 1. The Chang’s Conjecture (κn, κn−1, . . . , κ0) � (λn,λn−1, . . . , λ0) is equivalent to thestatement that for all X ⊇ κn:

Sκ,λ ={z⊆X: for all i, |z ∩ κi | = λi

}is stationary.

Definition 2. We write CC(κ, λ) for the non-stationary ideal restricted to Sκ,λ, which we will

also write, NS � Sκ,λ. We will call CC(κ, λ) the Chang ideal for κ, λ.

The degree of completeness of the ideal CC(κ, λ) is ambiguous. For this reason we con-sider another closely related ideal. For a given set X ⊇ κn, let Sκn,...,κ0,λn,...,λ1,<λ0 = {z ⊆ X:for all 1 � i � n, z ∩ κi = λi and z ∩ λ0 ∈ λ0}. It is easy to check that the non-stationaryideal restricted to Sκn,...,κ0,λn,...,λ1,<λ0 is a normal, fine, λ0-complete ideal. We will denoteNS � Sκn,...,κ0,λn,...,λ1,<λ0 by CC((κn, . . . , κ0), (λn, . . . , λ1,<λ0)). Proposition 19 gives sufficientconditions for this to be a proper ideal.

When |H(λ)| = λ (e.g. at an inaccessible cardinal, or at every regular cardinal under GCH),normal, fine ideals I on P(H(λ)) and normal fine ideals on P(λ) can be canonically identified


via a bijection between λ and H(λ). This identification is independent of the bijection. Thisallows us to speak of an I -positive set S ⊆ P(λ) as being a collection of elementary substructuresof H(λ). Similarly we can speak of having almost all members of S contain a given element w ∈H(λ). To provide precision (if necessary) we can fix in advance a structure A= 〈H(λ),∈,Δ,b〉where Δ is a well-ordering of H(λ) in order-type λ and b is the induced bijection. We then workon the set of z such that skA(z ∩ λ)= z and identify z with z ∩ λ.

In the context of forcing, we associate subsets of λ that lie in V with elements of P(H(λ)V )V .

1.2. Canonically well-ordered sets

In [11], Solovay showed the following theorem

Theorem. If U is a supercompact (or strongly compact) ultrafilter on [λ]<κ , then there is a setA ∈U such that if x, y ∈A are distinct, then sup(x) �= sup(y).

Solovay’s proof also shows that if U is a huge ultrafilter on [λ]κ then there is a set A ∈U suchthat if x, y ∈A are distinct then sup(x) �= sup(y).

It follows immediately from Solovay’s theorem that every supercompact ultrafilter contains acanonically well-ordered set: the given A ∈U can be well-ordered by setting x < y iff sup(x) <

sup(y).There are many other situations where stationary sets are canonically well-ordered. This is

typically done by considering their characteristic functions. If A is a set of cardinals and z is aset, the characteristic function of z is the function

χz :A→OR

defined by χz(κ) = sup(z ∩ κ). In many cases z is completely determined by its characteristicfunction (see [3] for details).

A special case of this phenomenon is relevant to this paper. Let n ∈ ω, κ be a regular cardinaland θ be a regular cardinal much bigger than κ . Suppose that y, z≺ 〈H(θ),∈,Δ〉 (where Δ is awell-ordering of H(θ)) have the properties that:

(1) supy ∩ κ+i = sup z ∩ κ+i for 1 � i � n, and(2) y ∩ κ = z ∩ κ and y ∩ κ ∈ κ ,(3) for 0 � i � n, cf (y ∩ κ+i ) > ω,

then y ∩ κ+n = z ∩ κ+n.Define S to be the collection of z⊆ κ+n such that

(1) z ∩ κ ∈ κ ,(2) cf (z ∩ κ+i ) > ω for 0 � i � n,(3) if A= 〈H(θ),∈,Δ〉 then skA(z)∩ κ+n = z.

Then the map z �→ (z ∩ κ, sup(z ∩ κ+), . . . , sup(z ∩ κ+n)) is one to one on S. It follows thatthere is an absolute well-ordering of S corresponding to the lexicographical well-ordering ofκ × κ+ × · · · × κ+n.


1.3. Notation and conventions

The paper will use standard set theoretic notation wherever possible. In this section we high-light some potential exceptions.

We will write θ � κ to mean that θ is a regular cardinal at least (2κ)+. We will use the notationπN to be the unique isomorphism between an extensive set N and its transitive collapse N .

If I ⊂ P(Z) is an ideal (U ⊂ P(Z) a filter), then we let I (U ) be the dual filter (dual ideal).The word “projection” will be overworked in the paper, as it can mean several different things

according to context. Frequently, if X ⊂ Y , we will say that the function π :P(Y )→ P(X) de-fined by setting π(z)= z ∩ x is the “projection” of P(Y ) to P(X). This map extends to another“projection” map π :PP(Y )→ PP(X) given by π(A)= {π(z): z ∈ A}. Our notation will fre-quently not distinguish between π and π . If π :P(Y ) → P(X) is a projection map, then anideal I on P(X) is the projection of an ideal J on P(Y ) if for all A⊆ P(X) we have A ∈ I iffπ−1(A) ∈ J .

In the context of forcing, a projection from P to Q will be an order preserving map π : P→Q

with the property that for all p ∈ P and all q ∈Q with q � π(p) there is a p′ � p with π(p′) � q .The relevant consequence of being a forcing projection is that if G⊆ P is generic, then π“G⊂Q

is generic. We will say a partial ordering P is κ-closed iff every decreasing sequence of elementsof P has a lower bound.

We will use the notations ℵα(κ) and κ+α to mean the αth cardinal successor of κ .For A a structure with domain some set X and z ⊆ X we will write z ≺ A to mean that z is

the domain of an elementary substructure of A. If z⊆X we denote the Skolem hull of z in A byskA(z) or simply sk(z) if A is implicit. We will often take our structures to have domain H(θ)

the collection of sets of hereditary cardinality less than θ . When we write 〈H(θ),∈,Δ, . . .〉 wewill be assuming that Δ=Δ(θ) is a fixed-in-advance well-ordering of H(θ).

2. Constructing from stationary sets and the non-stationary ideal

In this section we foreshadow later results by building inner models with arbitrarily largecardinals by constructing relative to the non-stationary ideal.

We begin by recalling a general result of D. Burke [2] that had been proved earlier by theauthor in the special case that I is the dual of an ultrafilter:

Theorem 3 (D. Burke). Let λ be an uncountable cardinal and I a normal fine countably completeideal on P(λ) (so I ⊂ PP(λ)). Then for all λ′ > 2λ there is a stationary set A⊂ P(H(λ′)) suchthat the projection of the non-stationary ideal on A to λ is I .

Proof. We outline a short proof. Let N ≺ 〈H(λ′),∈,Δ, {λ, I }, . . .〉. We say that N is good ifffor all C ∈ I ∩N , N ∩ λ ∈ C. It is shown in [8], that the collection of good N is stationary; werepeat that argument. Let A be a structure in a countable language with domain H(λ′). We claimthat there is a set D ∈ I such that all z ∈D, skA(z) is good. Otherwise, countable completenessand normality gives a Skolem function τ for A and a fixed finite sequence γ such that for an I -positive set A of z, γ ⊂ z, τ ( γ ) ∈ I and z /∈ τ( γ ). Since A is I -positive, A∩ τ( γ ) �= ∅. Howeverany z ∈A∩ τ( γ ) contradicts the choice of τ, γ .

Let A⊂ P(H(λ′)) be the collection of good N . We claim that the projection to λ of NS � A

is I . Let B be a closed unbounded subset determined by a structure A on H(λ′). Then, by theprevious claim there is a set D ∈ I such that for all N ∈D, skA(N) is good. Since I is normal


we can find a set D′ ∈ I such that for all N ∈D′, skA(N) ∩ λ=N . Then D ∩D′ is a subset ofthe projection to λ of {N : N is good and N ≺A}. Thus the projection to λ of the non-stationaryideal on H(λ′) restricted to A is included in I .

To see the other direction, let D ∈ I . Then the projection of {N : D ∈N and N is good} to λ

is a subset of D. �Some inessential remarks. The choice to make A ⊂ H(λ′) was somewhat arbitrary. If X ≺H(λ′) has cardinality λ′ and P(λ) ⊂ X, then we get a stationary A ⊂ P(X) that projects to I .Since X has cardinality λ′ we can “copy over” A to a stationary subset T of P(λ′) that projectsto I . Note also that we can take the elements of A to have the same cardinality as typical elementsz of a set of measure one C for I . Even more is true: if there is a set C ∈ I such that every elementz ∈ C is ω-closed (as a set of ordinals), then we can find an A (and so also a T ) such that for allN ∈ A,N ∩OR is ω-closed. This has technical advantages in that the collection of N that havethe property that N ∩OR is ω-closed can be canonically well-ordered.

An examination of the argument given by Theorem 3 also yields the following observation:If I is an ultrafilter and U is a normal and fine ultrafilter on P(λ′) projecting to I , and A is theset constructed in Theorem 3, then A ∈U . Hence if U is (say) a supercompact ultrafilter, we cantake A to be canonically well-ordered.

Many large cardinal properties of a cardinal κ can be defined in terms of the existence ofcertain kinds of ultrafilters U on a subset of P(λ) for λ � κ . In attempting to construct a canonicalinner model theory for such cardinals, a major obstacle is finding a suitable set of measure oneto build into the model. Typically, doing simple relative constructibility, one finds that L[U ] isa very small model and the method fails. One can also simply “throw in” a canonically well-ordered set of measure one A ∈U and construct L[A,U ]. We describe briefly how to do this foran arbitrary filter or ideal.

Let A ⊂ [λ]<κ be a set such that the supremum function sup :A→ λ is 1–1. For β in therange of the sup function, let aβ ∈ A be such that sup(aβ) = β . Define A∗ ⊂ λ× λ by setting(α,β) ∈A∗ iff α ∈ aβ . Then A ∈ L[A∗]. Moreover,

L[A∗,U

] |� ZFC+ κ is a supercompact cardinal.

The point of the next corollary is that we are constructing from a definable ideal rather than asupercompact ultrafilter:

Corollary 4. Suppose that κ is [2λ]<κ -supercompact. Then there is a stationary set A⊂ [2λ]<κ

such that L[A∗,NS � [2λ]<κ ] is a model of “ZFC+ κ is λ supercompact.”

We note that there are many stationary sets A we could have chosen in the proof of Corollary 4.Another definition of such an A is:

{x ∈ [

H((

2λ)+)]<κ : x ∩ κ = κ and t.c.(x)⊆H(λx)

}

(where λx is the order type of λ∩ x).Moreover, using arguments similar to Theorem 1.3 of [8], one sees that if U is a supercompact

ultrafilter on [λ]<κ then there is always a stationary set A of “good” structures in H((2λ)+) suchthat the sup function is one to one on A. This is done by using the technique discussed in the


“inessential remarks.” The set A is taken so that its intersection with (2λ)+ is ω-closed. Hence,the assumption of Corollary 4 can be reduced to λ-supercompactness. Indeed one sees:

Corollary 5. Let κ < λ be regular cardinals. Then κ is λ-supercompact iff there is a stationaryset A⊂ [2λ]<κ such that the supremum function is one to one on A and L[NS � [2λ]<κ,A∗] is amodel of “ZFC+ κ is λ-supercompact.”

There is nothing in the previous arguments specific to supercompact cardinals in the previousarguments. They generalizes easily to very large cardinals such as huge cardinals or towers ofn-huge cardinals.

2.1. Getting models of very large cardinals

In this section we deduce the existence of inner models for very large cardinals from thecombinatorial properties of small cardinals. For the purposes of this section the words innermodel will mean a transitive class model of ZFC.

2.1.1. Decisive idealsWe begin with a definition that seems to adequately distinguish between ideals that arise as in-

duced ideals in models built after collapsing large cardinals by forcing, and those “natural ideals”whose generic embeddings are not the traces of large cardinal embeddings in inner models.

Definition 6. Let Z ⊂ P(X) and J be an ideal on Z. Let X′ ⊂ X and I be the projection of J

to an ideal on P(X′) via the map π(z)= z ∩X′. Then J decides I iff there is a set A ∈ I and awell-ordering W of A and sets A′, W ′, O ′ and I ′ such that for all generic G⊂ P(Z)/J :

(1) An initial segment of the ordinals of V Z/G is well-founded and isomorphic to (|A′|+)V and(2) if j : V →M is the canonical elementary embedding determined by replacing the ultraprod-

uct V Z/G by an isomorphic model M transitive up to |A′|+, then

j (A)=A′, j (W)=W ′, j“|A| =O ′, I ′ = j (I )∩ P(A′)V .

We will say that J is decisive if J decides itself; in other words, we take π to be the identitymap and I = J .

Some of the hypotheses in Definition 6 are easily satisfied. For example, by Lemma 9, if J

is normal and fine, and |X|� |A′|, then the first clause is automatically satisfied. Moreover, if A

has a ΔZF−1 well-ordering in H(|A|+) then W and W ′ automatically exist. This occurs if A is

simply well-ordered by properties of the characteristic function of its members.

Remark 7. It is possible to check that essentially every induced ideal J produced by collaps-ing a large cardinal and extending the large cardinal embedding is densely often decisive. (SeeTheorem 14 for one such argument.)

Moreover, the definition can easily be modified to account for arbitrary generic embeddings:We replace the j : V → V Z/G by a j : V →M where M is well-founded up to |A′|+. In thiscase we say that P decides I . The more general definition has the same consequences as the


definition in the case of generic embeddings that arise when forcing with the quotient of theform P(Z)/I .

The next well-known example was worked out by the author with A. Sharon, and gives theunderlying reason for the disjunctive conclusion of Theorem 10.

Example 8. Let M be a well-founded model of “V = L” and suppose that G⊂ (P (ω1)/NSω1)M

is generic over M . Let N be the generic ultraproduct of M by G. Then N is well-founded to ωM2 .

Moreover, if α = ωM1 , N |� “α is countable.” Hence there is an x ∈ N with such that N |� “x

is the least countable ordinal such that there is a subset of ω constructed at stage x that codes abijection between α and ω.” Clearly x must be bigger than the first ωM

2 many ordinals of N .This example shows more: if f :ω1 → ω1 is the function defined in L by setting f (α) to be

the least β > α such that there is a new subset of ω constructed at stage β , then f dominatesevery canonical function on a closed unbounded set.

The next lemma allows us to get arbitrarily large degrees of well-foundedness by taking theunderlying set for the ideal to be large.

Lemma 9. Let I be a normal, fine, countably complete ideal on P(X). Let G ⊂ PP(X)/I begeneric. Then there is an initial segment of the ordinals of V P(X)/G which is well-founded andhas order type at least |X|+.

Proof. Let α < |X|+ be an ordinal of cardinality |X|. Since we can copy I over to an idealon P(α), without loss of generality we can take X = α. Let id :P(α)→ P(α) be the identityfunction. Normality and fineness show that for all f :P(α)→ V , {z: f (z) ∈ id(z)} ∈G iff thereis a β ∈ α such that {z: f (z)= β} ∈G.

If i :V → V P(α)/G is the canonical ultrapower embedding, then

i � α :α→ V P(α)/G

is an isomorphism of α onto [id]G. Hence V P(α)/G contains a well-ordered set of type α andtherefore has a well founded ordinal of type α. �Theorem 10. Let μ � λ be cardinals. Let π :P(λ)→ P(μ) be defined by π(z)= z∩μ. Supposethat J is a normal, fine, ideal on a set Z ⊂ P(λ) that decides a countably complete ideal I ⊂P(Z′) for some Z′ ⊂ P(μ). Suppose that A,W witness the fact that J decides I and W wellorders A as 〈aβ : β < γ < |A|+〉. Let A∗ = {(α,β): α ∈ aβ} ⊂ μ× γ . Then either

L[A∗, I

] |� I is an ultrafilter on A

or for some generic G⊂ P(Z)/J if j :V → V Z/G is the ultrapower embedding, then

L[j(A∗

), I ′

] |� I ′ is an ultrafilter on j (A).

Proof. Suppose not. Since L[A∗, I ] has a canonical well-ordering, there is a formula φ(w,u, v)

and an ordinal δ < |A|+ that is definable in L[A∗, I ] such that B = {z′ ∈ A: Lδ[A∗, I ] |�


φ(z′,A∗, I )} does not belong to I or I . (The formula φ describes the least such B .) We canrestate this by saying that δ is the least ordinal such that Lδ+1[A∗, I ] |� “I is not an ultrafilter.”

Let G ⊂ P(Z)/J be generic and j :V → M be the canonical elementary embeddinginto a model M isomorphic to V Z/G that has ordinals well-founded up to (|A′|+)V . Thenj � L[A∗, I ] :L[A∗, I ]→ L[j (A∗), j (I )]M is an elementary embedding.

Since j (A) = A′, I ′ = j (I ) ∩ V and j (W) =W ′, we know that j (A∗) ∈ V and if ξ is anordinal in the well-founded part of M , (Lξ [j (A∗), I ′])V = (Lξ [j (A∗), j (I )])M .

Case 1. For some generic G⊆ P(Z)/J , j (δ) is not in the well-founded part of M .

Then L|A′|+ [j (A∗), I ′] |� “the dual of I ′ ∩ L|A′|+ [j (A∗), I ′] is an ultrafilter.” Since |A′| =|j (A∗)|, this implies that L[j (A∗), I ′] |� “the dual of I ′ ∩ L[j (A∗), I ′] is an ultrafilter,” whichis a contradiction.

Case 2. For all generic G⊆ P(Z)/J, j (δ) is in the well-founded part of M .

In M , let δ′ be the least ordinal such that Lδ′+1[j (A∗), j (I )] |� “j (I ) is not an ultrafilter.”Then δ′ = j (δ) so δ′ belongs to the well-founded part of M . Since (Lδ′+1[j (A∗), j (I )])M =(Lδ′+1[j (A∗), I ′])V , the ordinal δ′ can be characterized in V by saying that it is the least ordinalγ for which Lγ+1[j (A∗), I ′] |� “I ′ is not an ultrafilter.” Thus there is a fixed δ′ such that forall G,j (δ)= δ′ and a set B ′ such that for all generic G, j (B)= B ′. Since I is normal and fine,|A| � μ. Let i be the initial segment of O ′ of length μ. Then for all generic G, j“μ = i. Butthen either for all generic G, i ∈ j (B) or for all generic G, i /∈ B . It follows that either B ∈ I orB ∈ I . This is a contradiction. �

We can replace hypothesis (1) in the definition of “decisive” by the demand that if a =|j (A)|M and b = (a+)M , then M is well-founded up to b. This change in the hypothesis yieldsthe stronger conclusion that L[A∗, I ] |� I is an ultrafilter, as Case 1 of the proof does not arise.

Corollary 11. Suppose that I is a normal fine κ-complete decisive ideal on P(λ) with witnessesA, W . Then:

(1) If α < κ and A ⊂ [κ+α]<κ then there is an inner model of V with a cardinal μ that isμ+α-supercompact.

(2) If A⊂ [λ]κ then there is an inner model of V with a huge cardinal.(3) If A ⊂ {X: o.t.(X) = λ1 and o.t.(X ∩ λ1) = κ}, then there is an inner model of V with a

2-huge cardinal.Moreover if I is precipitous then:

(4) If A⊂ [λ]<κ then κ is λ-supercompact in an inner model of V .(5) If A⊂ [λ]κ then κ is huge in an inner model.(6) If A⊂ {X: o.t.(X)= λ1 and o.t.(X ∩ λ1)= κ}, then κ is 2-huge in an inner model.

In particular the following are equiconsistent:

(a) For 1 < n < m ∈ ω there is normal, fine, decisive ideal on [ωm]ωn .(b) There is a huge cardinal.


as are:

(a) For 1 < n < m ∈ ω there is normal, fine, decisive ideal on [ωm]<ωn .(b) There is a κ+(m−n) supercompact cardinal κ .

The reader is left to provide the obvious analogous corollaries for n-huge cardinals.

2.1.2. A sample argumentThe meta-claim is that essentially all ideals induced from large cardinal embeddings are de-

cisive. In support of this meta-claim, we give an example of an argument showing that in theKunen model [10] the natural precipitous ideal on [ωm]ωn is decisive. Of the consistency resultsgiven in Corollary 11, this is the most involved.

For the reader’s convenience we review the definition of the Silver collapse.

Definition 12. Let μ be a regular cardinal and κ an inaccessible cardinal. The Silver collapseS(μ, κ) consists of functions p : μ× κ → κ such that:

(1) p(α,β) < β .(2) |p|� μ.(3) There is a ξ ∈ μ such that dom(p)⊂ ξ × κ .

The ordering on S(μ, κ) is reverse inclusion.

Standard arguments show that the Silver collapse is μ-closed, κ-c.c. and collapses κ to be μ+.The advantage of the Silver collapse over the more standard Levy collapse is that if A⊂ S(μ,λ)

is a collection of compatible conditions which has cardinality μ and there is a ξ such that for allp ∈A, the domain of p is a subset of ξ × κ , then

⋃A is a condition. This allows us to construct

generic conditions for models of size μ.In the proof of Theorem 20 we will use the following property of the Silver collapse:

Remark 13. Suppose that A ⊆ μ× κ is a set in V of cardinality μ such that for some ξ,A ⊆ξ × κ . If G is generic, then there is a condition in G with domain A. In particular, if H ⊆G is acollection of conditions p with dom(p)⊆A that lies in V [G] there is an m ∈G with m � p forall p ∈H .

We note that there are many variations on the Silver collapse: one could for example allowarbitrary (fixed) cardinality � μ on the “vertical axis” and maintain the chain condition, or allowconditions p whose vertical support is an Easton support.

We summarize the Kunen construction of a normal fine precipitous ideal on [ωm]ωn . Thesummary here is given in a non-specific way in an attempt to describe the relevant features of thevarious versions of the construction. We note that the only feature of the Silver collapse we areusing in the proof of Theorem 14 is the existence of a master condition as described in Remark 13,hence the argument given below works for the variations of the Silver collapse. Before we beginwe recall the remarks in the introduction identifying H(λ) with λ for inaccessible λ’s (modulonormal ideals) and hence subsets of H(λ) with subsets of λ.

Let j : V →M be a huge embedding with critical point κ0 and j (κ0)= κ1. We will take M

to be the ultrapower of V by a normal fine ultrafilter U on [κ1]κ0 . The first step is to construct a


partial ordering P⊂ Vκ0 that is ωn−1-closed and collapses κ0 to be ωn and is κ0-c.c. The partialordering P will be an iteration with < ωn−1-supports of length κ0 and such that each coordinateis ωn−1-closed. The critical property of P is that if Q is a partial ordering of cardinality less thanκ0 and ι :Q→ P is a regular embedding then there is a coordinate α = α(Q) such that the rangeof ι is included in Pα and Pα+1 is Pα ∗SQ(|Q|+m−n−1, κ0). Hence ι can be extended canonicallyto a regular embedding of Q ∗ SQ(|Q|+m−n−1, κ0) into Pα+1.

This gives a construction P(κ0) that is definable using a well-ordering of H(κ0) as a parameter.Since κ0 is a large cardinal there are many α that reflect this definition and give partial orderingsP(α) such that P(α) = P(κ) ∩ Vα . Similarly, by applying j to prolong the well-ordering to awell-ordering of H(κ1), we see that we get many well-defined P(β) for β � κ1.

Since P is κ0-c.c. and a subset of Vκ0 , j is the identity map on P, and so P⊂ j (P) is a regularsubordering. Thus P(κ0)= P(κ1) ∩ Vκ0 and is a regular subordering of P(κ1). Hence there is anα(P(κ0)) such that P(κ0) ∗ SP(κ0)(κ+m−n−1

0 , κ1) is canonically embedded into P(κ1)α(P(κ0))+1.From this we see for U -almost all z ∈ [κ1]κ0 (identifying z with a subset of H(κ1) as described

in the introduction):

(1) If κz = z ∩ κ0 then (P ∗ SP(κ+m−n−10 , κ1)) ∩ z is isomorphic to the partial ordering P(κz) ∗

SP(κz)(κ+m−n−1z , κ0) via the transitive collapse map.

(2) There is an α � κz (depending only on κz) such that at α, Pα+1 = Pα ∗SP(κz)(κ+m−n−1z , κ0).

We will write this ordinal as α(κz), rather than α(P(κz)).

Let G ∗H be generic for P ∗ SP(κ+m−n−10 , κ1). Then κ0 = (ωn)

V [G] and κ1 = (ωm)V [G]. Theimage of G ∗H under the canonical embedding into j (P) lies in j (P)κ ∩ Vκ and the coordinateα(κ0). Let m be the j (P) term for

⋃j“H . Then (as argued by Kunen), m is a P(κ1)-term for an

element of SP(κ1)(κ+m−n−11 , j (κ1)). If G ∗ H ⊂ P(κ1) ∗ SP(κ1)(κ+m−n−1

1 , j (κ1)) is generic andextends the image of G ∗H and contains (1,m) then there is an extension of j to an elementaryembedding j :V [G ∗H ]→M[G ∗ H ].

There is a normal, fine ωn-complete ideal I definable in V [G ∗ H ] on [ωm]ωn by settingX ∈ I iff ‖j“κ1 ∈ j (X)‖ = 0 where the Boolean value is taken in the completion of j (P) ∗j (S(κ+m−n−1

0 , κ1))/(G ∗H ∪ {(1,m)}).The following theorem shows that the Kunen ideal is decisive. The technique of proof illus-

trates the ubiquity of decisive ideals.

Theorem 14. The ideal I is precipitous and decisive.

Hence we have proved the corollary stated above:

Corollary 15. The following are equiconsistent:

(a) For 1 < n < m ∈ ω there is normal, fine, decisive ideal on [ωm]ωn .(b) There is a huge cardinal.

Proof of Theorem 14. In an abuse of notation we will write UV to denote the ultrafilter U in V

and U to denote the filter it generates in V [G ∗H ]. Let U be the ideal generated by the dual ofUV in V [G ∗H ].

The Duality Theorem and its corollaries in [6], show immediately that I is precipitous andcharacterizes I as follows:


Let f : [κ1]κ0 → V be a function such that [f ]M = m, where m is the V j(P)-term for⋃

j“H .Then for almost all z, f (z) is a P(κ0)-term for an element of SP(κ0)(κ+m−n−1

0 , κ0). The DualityTheorem implies that I is the ideal defined in V [G ∗H ] by U � {z: f (z)G ∈H } and moreoverthat I is precipitous.

By elementarity, for U almost all z, if G(κz)=G ∩ P(κz) and Hz is the generic object givenby the canonical embedding of SP(κz)(κ+m−n−1

z , κ0) into the α(κz)th coordinate of P, and jz is

the inverse of the collapse map of z, then f (z) is the canonical V P(κ0) term for⋃

jz“Hz.We now check that I is decisive. By the Duality Theorem, if K ⊂ P([κ1]κ0)/I is generic, then

there is a generic G∗ H ⊂ j (P)∗S(κ+m−n−11 , j (κ1)) such that the generic ultrapower V [κ1]κ0

/K

is isomorphic to M[G∗ H ] and the ultrapower embedding k : V [G∗H ]→M[G∗ H ] extends j .In particular, the ultrapower embedding coincides with j on the ordinals, hence its restriction toany given ordinal is in V [G ∗H ]. Thus we can take O ′ = j“|κ1|+, which belongs to V .

Since UV is a huge ultrafilter there is a set A⊂ [κ1]κ0 belonging to U such that the supremumfunction on A is one to one. Since the generic ultrapower embedding coincides with j on V wecan take A′ = j (A) and W,W ′ to be the orderings given by the supremum functions.

What is left to show is that if k is the generic ultrapower embedding from V [G ∗ H ] →M[G ∗ H ], then k(I ) ∩ V [G ∗H ] ∈ V [G ∗H ]. It suffices to show that k(I ) ∩ V [G ∗H ] is theideal on P([j (κ1)]κ1)V [G∗H ] generated by the dual of j (UV ).

By the Duality Theorem, k(I ) is the ideal j (U) � M on P([j (κ1)]κ1)M[G∗H ] where

M = {z: j (f )(z)M[G] ∈ H }. We note that another description of M is that it is {z ∈ [κ2]κ1 :⋃jz“Hz ∈ H }. If B ∈ k(I ) ∩ V [G ∗ H ] then there are (p, s) ∈ j (P) ∗ Sj (P)(κ+m−n−1

1 , j (κ1))

compatible with G ∗H and (1,m) and a set S in the dual of j (U) such that:

(p, s) �j (P)∗S(κ+m−n−1

1 ,j (κ1))/G∗H B ⊂ S ∪ {z: j (f )(z) /∈ H

}.

Without loss of generality we can assume that p � s � m. Towards a contradiction we assumethat B is not in the ideal generated by j (U) in V [G ∗H ]. The fact that (p, s) is compatible withG∗H means that for all α < κ0, p(α) �Vκ0 belongs to the αth coordinate of G and if α∗ = α(κ0)

is the coordinate that is determined by P (as in the description above), then p(α∗) ∈H .Since B is not in the ideal generated by j (U) over V [G ∗H ] we can find a z ∈ B such that:

(1) z /∈ S,(2) the support of p ⊂ κz and α∗ < κz,(3) s ∈ z (viewing z as a subset of H(j (κ1))).

If π is the map taking z to its transitive collapse, we have π(p, s) ∈ P(κz) ∗ SP(κz)(κ+m−n−1z ,

κ1) and moreover π((p, s)) = (p, sz) for some P(κz)-term sz. The coordinate α(κz) in j (P) isabove the support of p so we can extend p to a condition p′ with support equal to the support ofp together with the single additional point α(κz) and where p′(α(κz))= sz. Clearly p′ � p.

Let G ⊂ j (P) be generic extending G ∗ H and containing the condition p′. Let Hz ⊂SP(κz)(κ+m−n−1

z , κ1) be the generic object given by G in the coordinate α(κz). In M[G], let

s′ =⋃jz“Hz where jz is the inverse of the collapse map. Then s′ = j (f )(z)G. Since the support

of each jz(p) ⊂ κz × j (κ1), s′ is an element of SG(κ+m−n−11 , j (κ1)). Since p′ � sz ∈ Hz, and

jz(sz)= s, we know that p′ � s′ � s. Let H be generic over V [G] with s′ ∈ H .Then (p, s) ∈ G ∗ H and in V [G ∗ H ] we know that z ∈ B\S and j (f )(z) ∈ H . This is a

contradiction establishing the theorem. �


An indecisive ideal. For completeness we finish this section with an example of a precipitousbut indecisive ideal. An indecisive ideal on [ω2]ω1 can be constructed by starting in a modelwith a Woodin cardinal. Collapse the first Ramsey cardinal to be ω2 so that (ω2,ω1) � (ω1,ω).Now collapse a Woodin cardinal to be ω3. Then results of [9], show that the Chang ideal for((ω2,ω1), (ω1,ω)) is precipitous. Were it decisive it would give an inner model with a hugecardinal. However if the model we start in is a suitably small inner model this is impossible.

2.2. Chang’s Conjecture and huge cardinals

In this section we show that a strong version of Chang’s Conjecture at small cardinals suchas ω4 implies that there is an inner model with a huge cardinal. In the next section we provethat this strong Chang’s Conjecture is consistent relative to a 2-huge cardinal. This locates theconsistency strength of this Chang’s Conjecture on small cardinals between a huge and a 2-hugecardinal.

Our combinatorial properties of the ωn’s will be asserting that a certain set is stationary. Thesecan be easily restated in terms of Chang’s Conjecture using Remark 1.

2.2.1. Chang substructures with ordinal intersectionFrom the Chang Conjecture (κ3, κ2, κ1) � (λ3, λ2, λ1) we immediately see that both

(κ3, κ2) � (λ3,< λ+2 ) and (κ2, κ1) � (λ2,< λ+1 ). We need a more refined version of the lat-ter properties.

We begin our discussion with a well-known lemma.

Lemma 16. Let λ � κ � θ be cardinals with λ and θ regular and with cf (κ) � λ. Let A be thestructure expanding 〈H(θ),∈,Δ, {κ,λ}〉 and N0 ≺A. Let N1 = skA(N0 ∪ sup(N0 ∩ λ)). Then

sup(N1 ∩ κ)= sup(N0 ∩ κ).

Proof. Let τ be a Skolem function. Since we are concerned only with the intersection of aSkolem hull with κ , without loss of generality we can assume that

τ :H(θ)× λ→ κ.

We must show that for each a ∈N0 and δ ∈ sup(N0 ∩ λ) there is a β ∈N0 ∩ κ with β � τ(a, δ).Fix such an a ∈N0 and δ. Choose a γ ∈N0 ∩ λ with δ < γ . Let

β = sup{τ(a,α): α < γ

}.

Then β is definable in N0 and is clearly at least τ(a, δ), as required. �The next proposition can be proved with a variety of hypotheses. We give two versions, prov-

ing the more difficult one.

Proposition 17. Suppose the GCH holds. Let λ � κ � θ be cardinals with λ and θ regular andwith cf (κ) � λ. Let A be a structure expanding 〈H(θ),∈,Δ, {κ,λ}〉 and N0 ≺ A. Let N1 =skA(N0 ∪ sup(N0 ∩ λ)) and let ρ = | sup(N0 ∩ λ)|. Then N1 ∩ λ= sup(N0 ∩ λ) and

|N1 ∩ κ| = |N0 ∩ κ| · ρ.


Proof. It follows from Lemma 16 that sup(N1 ∩ λ)= sup(N0 ∩ λ). Since sup(N0 ∩ λ)⊆N1 wesee that N1 ∩ λ= sup(N0 ∩ λ).

What remains are the cardinality assertions. We show that for every κ ∈ N0 with κ � θ andcf (κ) � λ we have |N1 ∩ κ| = |N0 ∩ κ| · ρ. It is clear that |N1 ∩ κ|� |N0 ∩ κ| · ρ. So we mustshow that |N1 ∩ κ|� |N0 ∩ κ| ·ρ. We will separate into cases so that Case 4 is the only place thatwe use the GCH.

If the proposition is false, we let κ be the least counterexample occurring for any N0 and letN0 be the corresponding model. We argue by contradiction. The case that κ = λ is immediatefrom Lemma 16, so we assume that κ > λ.

Let C ⊂ sup(N1 ∩ κ) be unbounded and have order type cf (N1 ∩ κ). By Lemma 16, we canassume that C ⊆N0. Note that

|N1 ∩ κ| = |C| · sup{|N1 ∩ α|: α ∈ C

}. (1)

Since C ⊆ N0, |C| � |N0 ∩ κ|. Hence is suffices to show that for all α ∈ C, |N1 ∩ α| �|N0 ∩ α| · ρ.

Case 1. κ is a limit cardinal.

In this case we can take C to consist of regular cardinals above λ. By induction, for eachα ∈ C, |N1 ∩ α|� |N0 ∩ α| · ρ, which is what we needed.

In the remaining cases κ is a successor cardinal. Let μ be such that κ = μ+. Since κ ∈ N0,we know μ ∈N0. For both i = 0 and i = 1 and all α ∈ C we have |Ni ∩ α|� |Ni ∩μ|. Hence itsuffices to see that |N1 ∩μ|� |N0 ∩μ| · ρ.

Case 2. κ = μ+ and cf (μ) � λ.

In this case it is immediate that |N1 ∩μ|� |N0 ∩μ| · ρ, since κ was the minimal counterex-ample.

Case 3. κ = μ+, μ is a limit cardinal and cf (μ)⊆N0.

Since cf (μ) ⊆ N0, we can choose a set D ⊂ N0 that is cofinal in μ, consists of regular car-dinals above λ and has order type cf (μ). By the minimality of κ for each δ ∈ D, |N1 ∩ δ| �|N0 ∩ δ| · ρ. It follows that:

|N1 ∩μ| = sup{|N1 ∩ δ|: δ ∈D

}� sup

{|N0 ∩ δ| · ρ: δ ∈D}= |N0 ∩μ| · ρ.

Case 4. Otherwise.

In this case, κ = μ+ and cf (μ) < λ. We argue slightly differently. We show that |N1 ∩ μ|�|N0 ∩ κ| · ρ. This suffices, since it shows that |N1 ∩ κ|� |C||N1 ∩μ|� |N0 ∩ κ||N0 ∩ κ| · ρ.

Since κ ∈N0, we see that μ ∈N0 and hence cf (μ) ∈N0. In particular, cf (μ) < sup(N0 ∩ λ).Let M = skA(N0 ∪ cf (μ)). We first argue that |M ∩μ|� |N0 ∩ κ| · ρ.By definition:

M ∩μ= {τ(a, η): a ∈N0, η < cf (μ), τ is a Skolem function and τ(a, η) < μ

}. (2)

Let S be the set on the right hand side of Eq. (2). We need to show that |S| is at most |N0 ∩ κ| ·ρ.


For each a ∈ N0 and each Skolem function τ :H(θ) × cf (μ)→ μ there is a function f ∈N0 ∩H(κ) such that for all η ∈ cf (μ), τ(a, η)= f (η). Hence

S = {f (η)

∣∣ f : cf (μ)→ μ, f ∈H(κ)∩N0}.

By the GCH, |H(κ)∩N0| = |N0 ∩ κ|. It follows that |S|� |N0 ∩ κ| · cf (μ).We now have N0 ≺M ≺ N1, cf (μ) ⊆M , and N1 = skA(M ∪ sup(M ∩ λ)). We can apply

Case 3 with M playing the role of N0. The result is that |N1 ∩μ|� |M ∩μ| ·ρ. Since |M ∩μ|�|N0 ∩ κ| · ρ we see that |N1 ∩μ|� |N0 ∩ κ| · ρ. �

The next proposition replaces the hypothesis of the GCH in Proposition 17, with a hypothesisstating that κ is not too far from λ. We note that the hypothesis always holds if there are at mostcountably many cardinals between κ and λ.

Proposition 18. Let λ � κ � θ be cardinals with λ and θ regular and with cf (κ) � λ. Let A

be a structure expanding 〈H(θ),∈,Δ, {κ,λ}〉 and N0 ≺A. Let ρ = | sup(N0 ∩ λ)|. Suppose thatξ ⊆N0 and κ � λ+ξ , the ξ th successor of λ. Let N1 = skA(N0 ∪ sup(N0 ∩ λ)). Then N1 ∩ λ=sup(N0 ∩ λ) and

|N1 ∩ κ| = |N0 ∩ κ| · ρ.

Proof. The proof is exactly the same as Lemma 17, except that Case 4 does not arise. �We can now show

Proposition 19. Let κn > κn−1 > · · ·> κ0 and λn > λn−1 > · · ·> λ0 be regular cardinals. Sup-pose that either:

(1) the GCH holds or(2) there are at most countably many cardinals between λ0 and κn

and that (κn, . . . , κ0) � (λn, . . . ,< λ0). Let X ⊇ κn be a set. Then the non-stationary idealrestricted to Z = def {z ∈ P(X): |z∩κi | = λi for i > 0, |z∩κ0|< λ0 and z∩λ0 ∈ λ0} is a proper,λ0-complete, normal, fine ideal.

Proof. Since the non-stationary ideal restricted to {z ∈ P(X): z ∩ λ0 ∈ λ0} is λ0-complete, nor-mal and fine, the only issue is the properness of the ideal. To show that the ideal is proper weneed to show that if B is a structure with domain X, then there is an elementary substructure z

of B with the property that |z ∩ κi | = λi for i > 0 and z ∩ λ0 ∈ λ0.Let θ � κn and A= 〈H(θ),∈,Δ,B, {κn, . . . , κ0, λn, . . . , λ0}〉. By the Chang Conjecture as-

sumption there is an N0 ≺A such that

(1) |N0| = λn,(2) |N0 ∩ κi | = λi for all i > 0, and(3) |N0 ∩ κ0|< λ0.

Applying either Proposition 17 or Proposition 18, we see that N1 =def skA(N0 ∪ sup(N0 ∩ λ0))

has the same cardinality intersection with each κi for i > 0, |N1 ∩ κ0| < λ0 and N1 ∩ λ0 = λ0.


The structure z defined as N1 ∩X is an elementary substructure of B with the desired cardinalstructure. �

If (κ2, κ1, κ0) � (λ2, λ1, λ0) then both (κ2, κ1) � (λ2, λ1) and (κ1, κ0) � (λ1, λ0). Proposi-tion 19 implies (under GCH or an assumption on the distance between λ0 and κn) that the idealsCC((κ2, κ1), (λ2,<λ+1 )) and CC((κ1, κ0), (λ1,<λ+0 )) (defined in Section 1.1) are proper ideals.

Getting huge cardinals from Chang’s Conjecture. As we remarked earlier, if κ > μ and thereare finitely many cardinals between κ and μ then any ideal I concentrating on {N ∈ P(κ): N ∩μ ∈ μ and for all cardinals λ,μ � λ � κ we have cf (N ∩λ) > ω}, has a canonically well-orderedset A ∈ I of cardinality κ . Thus any Chang ideal involving only cardinals bigger than or equal toω1 has a canonically well-ordered set of measure one. Hence some of the hypotheses of the nexttheorem are satisfied by most Chang Ideals. Hypotheses (c) and (d) are condensation propertiesthat hold in huge type embeddings. In the statement of the next theorem we will use πN to denotethe isomorphism between N and its transitive collapse N .

Theorem 20. Suppose that κ2 > κ1 > κ0 are cardinals and there is a regular θ and a stationaryset S ⊂ P(H(θ)) and sets A⊂ [κ1]κ0 ,A′ ⊂ [κ2]κ1 ,O ′, I ′ such that for all N ∈ S:

(a) N ∩ κ0 ∈ κ0 and N ∩ κ1 ∈A (so |N ∩ κ1| = κ0),(b) A,A′ ∈N and πN(A′)=A, and the map z �→ (z ∩ κ0, sup z) is one to one on A,(c) O ′ ∈N and N ∩ κ1 = πN(O ′),(d) I ′ ∈N and πN(I ′)= (NS � A) ∩ N .

Then there is an inner model with a huge cardinal.

Since CC((κ1, κ0), (κ0,<κ0)) is definable in N , hypothesis (d) can be restated as saying that

N ∩ (CC

((κ1, κ0), (κ0,<κ0)

)� A

)= πN

(I ′ � {z: z ∩ κ1 ∈ κ1}

).

Proof. Since A contains the projection of S to κ1, A is stationary; i.e. it is positive forthe ideal CC((κ1, κ0), (κ0,< κ0)). We claim that the non-stationary ideal restricted to S de-cides the ideal I = CC((κ1, κ0), (κ0,< κ0))�A = NS � A, with A,A′,O ′ and the canonicalwell-orderings of A and A′ as witnesses. This suffices since Theorem 10 implies that eitherthe filter dual to CC((κ1, κ0), (κ0,< κ0)) is a normal, fine, κ0-complete ultrafilter on A inL[A∗,CC((κ1, κ0), (κ0,< κ0)) � A] or the filter dual to CC((κ2, κ1), (κ1,< κ1)) is a normal,fine κ1-complete ultrafilter on A′ in L[(A′)∗,CC((κ2, κ1), (κ1,< κ1)) � A′].

If S ⊂ H(θ) is stationary then for all θ ′ > θ, {N ⊂ H(θ ′): N ∩ H(θ) ∈ S} is station-

ary. Hence, without loss of generality we can assume that θ � (22|A′ |)+. By Lemma 9, if

G⊆ PP(H(θ))/(NS �S) is generic then the ultrapower V S/G is well-founded beyond (22|A′ |)+.

Let M ∼= V S/G be transitive up to H((22|A′ |)+), and j :V →M be the canonical embedding.We now verify the conditions in the definition of decisive.

• In M :

π 2|A|′ +(j (A′)

)=A′
j“H((2 ) )


and for all N ∈ S,πN(A′)=A. Since j“H((22A′)+) ∈ j (S) hypothesis (b) shows that A′ =

πj“H((22A′

)+)(j (A′))= j (A).

• Since the well-orderings W and W ′ are defined absolutely it is automatic that j (W)=W ′.• Arguing as for A and A′, but using hypothesis (c), we have

O ′ = πj“H((22A′

)+)

(j (O ′)

)= j“H((

22A′ )+)∩ j (κ1)= j“κ1.

• The verification that I ′ = j (NS � A)∩ V is done the same way:

πj“H((22A′

)+)

(j (I ′)

)= I ′.

By hypothesis (d),

πj“H((22A′

)+)

(j (I ′)

)= (NS � A′)∩H(22A′ )V

. �The hypotheses (b)–(d) of Theorem 20 are condensation properties. They say that the tran-

sitive collapse of N fits well with elements of N . The most difficult to satisfy is condition (d),which says that the transitive collapse of N is big enough that it correctly computes the non-stationary ideal.

With this remark in mind we note that we can replace condition (d) of Theorem 20 with thedemand that all N ∈ S be correct, which we define to be the following condensation property:

Definition 21. Let A,A′, κ0, κ1 and κ2 be as in Theorem 20. Let

N ≺ ⟨H(θ),∈, {κ0, κ1, κ2},Δ, {A,A′}⟩

be a structure such that o.t.(N ∩ κ2)= κ1, o.t.(N ∩ κ1)= κ0,N ∩ κ0 ∈ κ0. We will say that N iscorrect for NS � A iff whenever πN :N → N is the transitive collapse map, we have

πN(NS � A′)= (NS�A)∩ N . (3)

Since being a set in the dual of NS � A is witnessed by an algebra on⋃

A, if πN(A′)= A itis always true that πN(NS � A′) ⊆ (NS � A) ∩ N . The force of the definition is that N containssufficiently many algebras to show the other inclusion.

To make our results concrete we prove a corollary. By choosing n= 0 the corollary providesa consistent statement about H(ω4) that gives an inner model with a huge cardinal:

Corollary 22. Suppose that n ∈ ω. Suppose that there are

A⊂ {z ∈ [ωn+2]ωn+1 : z ∩ωn+1 ∈ ωn+1

}

A′ ⊂ [ωn+3]ωn+2 , O ′ ∈ [ωn+3]ωn+2 , such that for all structures A with domain H(ωn+4) there isan N ≺A with A,A′ ∈N such that:

(a) N ∩ωn+2 ∈A,(b) πN(A′)=A,


(c) the map z �→ (z ∩ωn+1, sup(z)) is one to one on A,(d) O ′ ∈N and N ∩ωn+2 = πN(O ′),(e) N is correct for NS � A.

Then there is an inner model with a huge cardinal.

We remark that stating Corollary 22 in terms of the non-stationary ideal should not mask thefact that it is a form of Chang’s Conjecture. Explicitly, it is the usual (ωn+2,ωn) � (ωn+1,ωn−1)

augmented by condensation properties.Note that if n � 1 in the previous two corollaries, then there is always a canonically well-

ordered set of measure one for the Chang ideal (namely those Chang N that uniformly havecofinality bigger than ω). The issue in satisfying the hypotheses becomes correctness and decid-ing where that set goes.

We finish this discussion with two conjectures:

Conjectures.

(1) If the non-stationary ideal on ω1 is ω1-centered and the generic embedding restricted to theordinals is definable in V , then there is an inner model with an almost huge cardinal.

(2) If n > 1, (ωn+1,ωn) � (ωn,ωn−1) and CC((ωn+1,ωn), (ωn,ωn−1)) is ωn+1-saturated, thenthere is an inner model with a huge cardinal.

3. The consistency proof

We now outline the proof of the following theorem:

Theorem 23. If there is a 2-huge cardinal then there is a generic extension in which the GCHholds and the hypotheses of Corollary 22 hold.

Unfortunately, understanding the proof of Theorem 23 requires familiarity with [4].We review some forcing technology.

Lemma 24. Suppose that Z ⊂ P(X), and P is a partial ordering. Let θ be a regular cardinalwith 2|P| < θ and X,P ∈ H(θ). If Y ∈ P(Z)V , and for all closed unbounded sets C ⊂ H(θ)V

that lie in V , and all p ∈ P there is an N ∈ C with N ∩X ∈ Y and an (N,P)-generic conditionm stronger than p, then Y is positive for the closed unbounded filter in V [G] restricted to Z.

Moreover, if P is homogeneous, then a stronger result holds: for all N ≺ 〈H(θ),∈,Δ,

P, Y,Z〉, with N ∩X ∈ Y , if m is an (N,P)-generic condition, then

m � Y is positive for the closed unbounded filter.

To prove Lemma 24, we use the following easy forcing fact:

Fact 25. N ≺ 〈H(θ)V ,∈,Δ,P〉 and G ⊂ P be generic. Then the following are equivalent inV [G]:(1) There is an M ≺ 〈H(θ)V [G],∈,Δ,P,G〉 with M ∩H(θ)=N .(2) There is an (N,P)-generic condition m ∈G.


Proof of Lemma 24. Suppose that p � Y is non-stationary. (If P is homogeneous, we can takep to be the trivial condition.) Let N ≺ 〈H(θ),∈,Δ,P, Y,Z, {p}〉 be such that N ∩ X ∈ Y andthere is an (N,P)-generic condition m stronger than p. Then N has a term D for a closedunbounded subset of P(X) such that p � D ∩ Y = ∅. But m � there is an M ≺H(θ)V [G] suchthat M ∩H(θ)V =N . So m � N ∩X ∈ D, a contradiction. �Termspace Forcing. We will need some ad hoc facts about termspace forcings. This tech-nique was originally developed independently by Abraham and Laver and further explored in[5] and [7].

Let P be a partial ordering. Suppose that Q ∈ V P is a partial ordering. For τ1, τ2 ∈ V P suchthat ‖τi ∈Q‖ = 1, we will say that τ1 ∼ τ2 iff ‖τ1 = τ2‖ = 1. Note that we can choose a set ofrepresentatives for the equivalence classes of this relation.

Definition 26. The termspace forcing A(P,Q) for Q is the collection of equivalence classes ofP-terms τ with ‖τ ∈Q‖ = 1, with the ordering that τ1 � τ2 just in case ‖τ1 � τ2‖ = 1. When P

is clear from context we will write Q∗ for A(P,Q).

The following lemma is due independently to Abraham and Laver (see [1] for a proof):

Lemma 27. Suppose that Q is partial ordering in V P. Then:

(1) There is a canonical forcing projection from P × A(P,Q) onto a dense subset of P ∗ Q,induced by the identity map (p, τ ) �→ (p, τ ).

(2) If κ is a regular cardinal in V P and ‖Q is κ-closed‖= 1, then A(P,Q) is κ-closed.

We prove here the following slight generalization of Lemma 27:

Lemma 28. Suppose that P is an iteration 〈(Pα,Qα): α ∈ β〉 with support in an ideal K. DefineP∗ to be the partial ordering

∏K-supports Q∗α . Then B(P) is isomorphic to a regular subalgebra

of B(P∗) (i.e. forcing with P∗ gives a generic object for P).

Proof. We view p ∈ P to be a sequence of terms such that:

(1) The domain of p is in K.(2) For all β ∈ supp(p), ‖p(β) ∈Qβ‖Pβ

= 1.

The collection of such p is dense in every standard realization of P as an iteration.We claim that the identity map from P∗ to P is a projection. For this we must see that if p ∈ P∗

and q �P p, then there is an r �P∗ p such that r �P q . To define r we take a sequence of terms〈τα: α ∈ supp(q)〉 such that ‖τα ∈ Qα‖Pα

= 1, q � α � τα = q(α) and for all q ′ ∈ Pα , if q ′ isincompatible with q � α, then q ′ � τα = p(α). Setting r(α) = τα for α ∈ supp(q), we get thedesired condition. �

We note that the analogue of Lemma 27, item 2 is also true: If K is a κ-complete ideal andPα � Qα is κ-closed, then P∗ is κ-closed.

The next proposition follows immediately from results in [5]:


Proposition 29. Let κ be a 2-Mahlo cardinal. Then:

(1) If |P|< κ and Q ∈ V P is the Silver collapse S(μ,κ), then Q∗ has the κ-c.c.(2) If P is an iteration given by 〈(Pα,Qα): α < κ〉 where

(a) P has support of size μ,(b) P0 = S(μ,κ),(c) each non-trivial Qα is of the form SRα (μα, κ) where μ � μα < κ and Rα is a regular

subordering of Pα of cardinality μα ,then P∗ is κ-c.c., μ-closed and forcing with P∗ gives a generic object for P.

Subiterations. We now give a brief discussion of subiterations. If R is an iteration of length κ

then we can construct many other iterations that have R as a subiteration. To wit, if P is anotheriteration of length δ at least κ and ψ :κ → δ is an order preserving map, we will say that R isa subiteration of P iff R consists (exactly) of those functions r such that there is a p ∈ P for allα ∈ κ,p(ψ(α))= r(α).

The idea is that P is an iteration such that from time to time (i.e. at ordinals in the image of ψ )the forcing R is used, coordinate by coordinate.

It follows from the definition that if R is a subiteration of P with map ψ , K and K′ are idealsof supports of R and P, then X ∈ K iff ψ“X ∈ K′.

The only subtlety in the definition is that in a coordinate ψ(α), the terms allowed in Pψ(α)+1come from the same Boolean valued universe as they do in Rα . The many routine details involvedin subiterations were explored in [4], including the inductive proof that if R is a subiteration of P,then the canonical map induced by ψ from R to P is a regular embedding.

A small elaboration on the results in [4] shows the following (which is used to establishproperty P5 of the partial orderings P(α) ∗R(α,β) later in this section):

Lemma 30. Suppose that R0 and R1 are subiterations of P along functions ψ0 and ψ1, and thatthe ranges of ψ0 and ψ1 are disjoint.

(1) If G ⊂ P is generic and induces generic Hi ⊂ Ri (i = 0,1), then H0 × H1 ⊂ R0 × R1 isgeneric.

(2) Forcing with R0 ×R∗1 yields a generic object H0 ×H1 for R0 ×R1 such that the image ofH0 ×H1 can be extended to a generic G⊆ P.

The proof of Theorem 23. Let j : V →M be a 2-huge embedding with critical point κ0, andj i(κ0) = κi for i = 1,2,3. In [4] it was shown how to make a forcing extension by a partialordering of the form P ∗R ∗ S where:

(i) P is κ0-c.c., ωn-closed and makes κ0 into ωn+1.(ii) R⊂ Vκ1 is a definable (κ0,∞)-distributive partial ordering that is κ1-c.c., and makes κ1 into

κ+0 , i.e. ωn+2.(iii) S is the Silver collapse; a κ1-closed, κ2-c.c. partial ordering that makes κ2 = κ+1 , i.e. ωn+3.(iv) There is a condition m ∈ j (P)∗ j (R)∗ j (S) such that whenever G∗ H ∗ K ⊂ j (P)∗ j (R)∗

j (S) is generic with m ∈ G ∗ H ∗ K , there is a G ∗H ∗K in V [G ∗ H ∗ K] that is genericfor P ∗R ∗ S, and such that j (see Fig. 1) can be extended to a

j :V [G ∗H ∗K]→M[G ∗ H ∗ K].


Fig. 1. A diagram of where j takes cardinals.

(v) There is a canonical regular embedding ι : P ∗R ∗ S→ j (P) ∗ j (R) such that V j(P)∗j (R) isa κ2-c.c. homogeneous extension of V P∗R∗S. Moreover, if G ∗ H ⊂ j (P) ∗ j (R) is genericthen the G ∗H ∗K in condition (iv) is determined from G ∗ H via ι.

We will make a small modification of the forcing in [4] so that it also has the followingproperty:

(vi) In V [G ∗H ∗K] for all N ≺ 〈H(κ+22 )V [G∗H∗K],∈,Δ,H(κ+2

2 )V〉 with {P ∗ R ∗ S, j (P) ∗j (R), ι,G ∗H ∗K} ⊂N , if(a) |N | = κ1, N ∩ κ1 ∈ κ1 and N ∩ κ1 is 2-Mahlo in V ,(b) N ∩ κ2 ∈ V , and(c) the transitive collapse of N ∩HV (κ2) is HV (κ1),then there is a condition mN in the partial ordering j (P) ∗ j (R)/ι“(G ∗H ∗K) that forces(G ∗ H )∩N is generic over N (see Fig. 2) for j (P) ∗ j (R)/ι“(G ∗H ∗K).

We will describe the modification below, but first show:

Claim 31. Suppose that properties (i)–(vi) hold of P∗R∗S. Then in V [G∗H ∗K] the hypothesesof Corollary 22 hold.

Fig. 2. The N from item (vi).


Proof. We start by working in V . Standard arguments show that there is a set A⊂ [κ1]κ0 suchthat in V the following properties hold:

(1) the map z �→ (z ∩ κ0, sup z) is one to one on A and z ∩ κ0 is 2-Mahlo.(2) j“κ1 ∈ j (A) and for all N ∈A,N ∩ κ0 ∈ κ0.

(3) For all z ∈ A, skH(κ+21 )(z) ∩ κ1 = z and the transitive collapse of skH(κ+2

1 )(z) ∩ H(κ1) isH(κ0).

We note once more that a set A⊆ [κ1]κ0 corresponds canonically with a set A′ ⊆ [H(κ1)]κ0 viaour fixed-in-advance well-ordering Δ.

Returning to V [G ∗H ∗K], we show the following result that is sufficient to see Claim 31.

Main Claim. If we define A′ = j (A) and O ′ = j“κ1 and S ∈ V [G ∗H ∗K] is the collection ofN ≺H(ωn+4) such that A′,O ′ ∈N and (a)–(e) of Corollary 22 hold for N , then S is stationary.

Fix an algebra A in V [G ∗ H ∗ K] that expands 〈H(ωn+4),∈,Δ, j,H(κ+2 )V , {A,A′}〉. Toprove the main claim we show that

skj (A)(j“H(κ2)

V [G∗H∗K]) ∈ j (S). (4)

Hence M[G ∗ H ∗ K] |� “there is an N ≺ j (A) with N ∈ j (S).” By elementarity we see thatthere is an N ≺A with N ∈ S, thus establishing the Main Claim.

Claim 32. Let A be as above. Then skj (A)(j“H(ωn+3))∩H(j(ωn+3))= j“H(ωn+3).

Proof. Without loss of generality we can assume that A has definable Skolem functions thatare closed under composition. Let τ be a Skolem function. Without loss of generality we cantake τ :H(ωn+3)→ H(ωn+3). Then for all x ∈ H(ωn+3), j (τ )(j (x)) = j (τ (x)). In particularwe see that the range of j (τ ) applied to j“H(ωn+3) is a subset of j“H(ωn+3). Thus Claim 32holds. �

Claim 32 implies more: since (j � V )= j ,

j“H(ωn+3)V [G∗H∗K] ∩ j

(H(κ2)

V)= j“H(κ2)

V .

Since j (H(κ2)V ) = H(ωn+3)

M[G∗H∗K] ∩ M , we see that skj (A)(j“H(κ2)V ) ∩ M =

j“H(κ2)V ∩M = j“H(κ2)

V . Similarly,

j“(skA

(H(κ2)

V [G∗H∗K]))= skj (A)(j“H(κ2)

V [G∗H∗K]).

From Claim 32, we see that A is a positive set for CC((κ1, κ0), (κ0,< κ0)), since it shows thatany algebra A has an elementary substructure whose intersection with κ1 is in A.

To show that S is stationary we need to establish the relation in Eq. (4). Let N =skj (A)(j“H(κ2)

V [G∗H∗K]). Since we are working in M and arguing about j (S) we need to verifyclauses (a)–(e) of Corollary 22, with the subscripts of the κi ’s incremented by 1, A replaced byj (A), A′ replaced by j (A′), O ′ replaced by j (O ′) and so on. (See Fig. 1.)


• To see clause (a) we need to show that

∣∣skj (A)(j“H(κ2)

V [G∗H∗K])∩ωM[G∗H∗K]i+1

∣∣= ωM[G∗H∗K]i

for i = n,n+ 1, n+ 2 and ωM[G∗H∗K]n ⊂N . Since j“H(κ2)

V ∩ j (κi) has order type κi , and

κ0 ⊂ j“H(κ2)V , this follows immediately from Claim 32 after noticing that ω

M[G∗H∗K]n+i = κi

for i = 1,2 and 3, and that |κ0|M[G∗H∗K] = ωn. Finally N ∩ωMn+2 = j“κ1 ∈ j (A).

• To see that clause (b) holds: We must see that πN(j (A′))= j (A). This is immediate, sinceA′ ∈ skA(H(κ2)), and πN(j (A′))=A′, and A′ is defined as j (A).

• Clause (c) is immediate from the definition of A.• Clause (d) is similar to Clause (b). O ′ ∈ skA(H(κ2)), and so j (O ′) ∈N . Thus πN(j (O ′))=

O ′ = j“κ1 =N ∩ κ2.

This leaves the heart of the matter, verifying clause (e). We need to show that N is correct

for the Chang ideal NSM[G∗H∗K] � j (A). Correctness translates into showing that M[G ∗ H ∗ K]satisfies the statement

πN

(NS � j (A′)

)= (NS � j (A)

)∩ N .

By elementarity, πN(NS � j (A′)) is the ideal NSN � A′. Since

N = t.c.(N)= t.c.(skA

(H(κ2)

V [G∗H∗K])),

and skA(H(κ2)V [G∗H∗K]) is transitive up to κ2, we see that NSN � A′ = (NSV [G∗H∗K]�A′)∩ N .

Summarizing:

πN

(NS � j (A′)

)= (NSV [G∗H∗K]�A′

)∩ N .

Thus, since j (A)=A′, we need to show that

(NSM[G∗H∗K] � A′

)∩ N = (NSV [G∗H∗K] � A′

)∩ N .

Since N ⊆ V [G ∗H ∗K], it suffices to show that

NSV [G∗H∗K] � A′

is equal to

(NS � A′)M[G∗H∗K] ∩ V [G ∗H ∗K]. (5)

Since M[G ∗ H ∗ K] is closed under κ2 sequences from V [G ∗ H ∗ K] (and even fromV [G ∗ H ∗ K]) and A′ ⊆ [κ2]κ1 , it is clear that the first ideal is a subset of the second ideal.

To see the other inclusion let Y ⊆ A′ be a positive set for NSV [G∗H∗K] � A′ with Y ∈ V [G ∗H ∗K]. We must show that Y is positive for NSM[G∗H∗K].


Assume not. Since j (S) is homogeneous over j (P) ∗ j (R), we know that j (P) ∗ j (R) ∗ j (S)

is homogeneous over P∗R∗S. Since Y ∈ V [G∗H ∗K] and the non-stationary ideal is definablein V [G ∗ H ∗ K] from parameters in V , we see that Y is non-stationary in M[G ∗ H ∗ K] forevery generic G ∗ H ∗ K . In other words there is a j (P) ∗ j (R) ∗ j (S)/G ∗H ∗K-term F for afunction from κ<ω

2 to κ2 such that

1 �j (P)∗j (R)∗j (S)/G∗H∗K (∀z ∈ Y ) (z is not closed under F ).

The forcing j (S) is κ2-closed in M[G ∗ H ], so we can build a pseudo-generic tower of con-ditions in j (S) deciding F . Explicitly this is a sequence of conditions 〈sα: α < κ2〉 ⊂ j (S) thatare increasing in strength and for all γ ∈ κ<ω

2 there are α, δ < κ2 such that sα � F ( γ ) = δ. InM[G ∗ H ] define F ∗( γ )= δ iff there is an α, sα � F ( γ )= δ.

Then Y does not contain any z closed under F ∗, for if it did, there would be an α for allγ ∈ z<ω , there is a δ, sα � F ( γ )= δ. But then sα � “z is closed under F ,” a contradiction.

Thus Y ∈ NS�A′ in V [G ∗ H ] (and is forced to be by every condition in j (P) ∗ j (R)/G ∗H ∗K). We now use Lemma 24. Since Y is positive for NSV [G∗H∗K] � A′, for any structure C

expanding the structure 〈H(κ+22 )V [G∗H∗K],∈,Δ,H(κ+2

2 )V 〉 we can find N ≺ C with

(1) {P ∗R ∗ S, j (P) ∗ j (R), ι,G ∗H ∗K, F ∗} ⊆N ,(2) N ∩ κ2 ∈ Y (and hence N ∩ κ2 ∈ V ), and(3) the transitive collapse of N ∩H(κ2)=H(κ1).

Applying property (vi) of the forcing we can find an mN that forces G ∗ H is generic over N for

the partial ordering j (P)∗j (R)/ι“(G∗H ∗K). Hence by Lemma 24, Y is positive for NSV [G∗H ],a contradiction.

This completes the proof of the main claim and thus of Claim 31. �The modification. To finish the proof of Theorem 20, it remains to describe how to modify thepartial ordering P∗R∗S in [4] so that property (vi) of the forcing holds in the resulting the partialordering. To do this we need to describe the partial ordering P ∗ R in more detail. Note that inthe description we will identify a condition p in an iteration with the function whose domain isthe support p.

In [4] we build a family of partial orderings P(α) and R(α,β) for α and β 2-Mahlo cardinals.Here are some of the properties of these partial orderings (for a given fixed n):

P1 The partial orderings P(β) and R(α,β) are iterations of length β of β-c.c. partial orderings,with supports contained in the ideal of Easton Sets. In particular, the partial orderings areβ-c.c., and can be canonically viewed as subsets of Vβ . The partial ordering P(β) is ωn-closed over V and the partial orderings R(α,β) are (α,∞)-distributive over the modelsV P(α) where they are constructed.

The partial orderings R(α,β) and P(β) are defined simultaneously with the parameters α,β

in a complex inductive description. For notational simplicity we will write RP(α)(α,β) to meanthe partial ordering R(α,β) as constructed in V P(α).


P2 If α < β are 2-Mahlo, then P(α) is P(β)∩Vα and is a regular subordering of P(β). Similarly,if γ < β is 2-Mahlo, then R(α, γ ) is R(α,β)∩V

P(α)γ and is a regular subordering of R(α,β).

The partial ordering RP(α)(α,β) is a subiteration of P(β) defined with the parameters α,β ,and hence there is a canonical embedding

ια,β : P(α) ∗R(α,β)→ P(β).

We note that P(β)∩Vα = Pα(β)∩Vα , the αth stage of the iteration P(β). Hence P2 implies thatPα is a regular subordering of Pα(β). Thus P(α)-terms can be evaluated in Pδ(β) for any δ > α

and can be used to force at stage δ + 1. This is what allows RP(α)(α,β) to be a subiteration ofP(β).

P3 The partial orderings that appear in coordinates of either of the partial orderings P(β) orRP(α)(α,β) are of the form SP(γ )∗RP(γ )(γ,δ)(δ, β). The coordinates that appear in RP(α)(α,β)

are of two types:P3a Type 0 coordinates: These are coordinates of the form SP(γ )∗RP(γ )(γ,α)(α,β) for some

γ < α, where SP(γ )∗RP(γ )(γ,α)(α,β) is defined in the submodel V P(γ )∗RP(γ )(γ,α) of V P(α)

determined by the embedding ιγ,α . The result is that for all 2-Mahlo γ < α we canextend ιγ,α to a canonical embedding ι+γ,α from P(γ ) ∗ R(γ,α) ∗ SP(γ )∗R(γ,α)(α,β)

into P(α) ∗Rδ+1(α,β) and hence into P(α) ∗R(α,β).P3b Type 1 coordinates: These are coordinates of the form SP(α)∗R(α,γ )(γ,β) for some γ

between α and β .Note that the partial orderings in type 1 coordinates are γ -closed in the modelV P(α)∗R(α,γ ), but type 0 coordinates are not closed, even over P(α).

The type 0 coordinates are distinguished in that they come from submodels of P(α), andhence are typically not α-closed in V P(α). The type 1 coordinates come from models of the formP(α) ∗R(α, γ ) as it lies inside P(α) ∗R(α,β). They are γ -closed at the stage with which theyare forced in R(α,β).

The distinction between “type 0” and “type 1” coordinates is not well defined in P(β) sincethe forcing at a typical coordinate δ is SP(γ )∗R(γ,α)(α,β). This is a type 1 coordinate relative tothe subiteration corresponding to RP(γ )(γ,β) and a type 0 coordinate relative to the subiterationcorresponding to RP(α)(α,β).

P4 The partial ordering RP(β)(β, δ) has two phases in its iteration. The first phase is the <β-support iteration of the type 0 coordinates. These are the partial orderings SP(α)∗R(α,β)(β, δ)

for the 2-Mahlo α < β . In this phase, at a coordinate for SP(α)∗R(α,β)(β, δ), we useP(α) ∗R(α,β)-terms. Thus conditions consist of <β-sequences of terms from variousSP(α)∗R(α,β)(β, δ). This phase has length β .The second phase is an iteration with β supports of partial orderings of the formSP(β)∗R(β,γ )(γ, δ) for 2-Mahlo γ between β and δ. These are the type 1 ordinals. Thisphase has length δ.The second phase is organized so that if γ is a 2-Mahlo cardinal between β and δ, the γ thcoordinate of RP(β)(β, δ) is SP(β)∗R(β,γ )(γ, δ). In this way there is a natural regular embed-ding ι# from P(β) ∗ RP(β)(β, γ ) ∗ SP(β)∗R(β,γ )(γ, δ) into P(β) ∗ RP(β)(β, δ). The iterationuses P(β) ∗RP(β)(β, γ )-terms, in these coordinates.


P5 For each 2-Mahlo α, P(α) ∗R(α,β) is embedded into P(β) via ια,β , and the first phase ofRP(β)(β, δ) can be canonically factored into SP(α)∗R(α,β)(β, δ)×Q, where Q consists of the<β sequences of terms for elements of other partial orderings in the first phase.

P6 If α < γ < δ are 2-Mahlo, then there are subiterations of P(δ) corresponding to RP(α)(α, δ)

and RP(γ )(γ, δ). The two sets of coordinates of P(δ) used in these subiterations are notdisjoint. Indeed they will have exactly one coordinate in common that corresponds to a type 0coordinate for P(γ ) ∗R(γ, δ) and a type 1 coordinate for P(α) ∗R(α, δ).In this coordinate of P(δ) both subiterations use terms for SP(α)∗R(α,γ )(γ, δ) taken fromV P(α)∗R(α,γ ). The subiterations of P(δ) are synchronized so that the following diagram com-mutes:

P(α) ∗RP(α)(α, γ ) ∗ SP(α)∗R(α,γ )(γ, δ)

ι#

ι+α,γ

P(γ ) ∗RP(γ )(γ, δ)

ιγ,δ

P(α) ∗RP(α)(α, δ)ια,δ

P(δ)

If R(α,β) is defined in V P(α) by the procedure outlined above, let R1(α,β) be the secondphase of R(α,β). The coordinates of R1(α,β) are those that come after forcing with P(α) andthe first phase of R(α,β). We modify our partial ordering P(δ) in the following way:

The modificationWe start by reserving some thin class F of formerly irrelevant ordinals (such as the successorsof limit points of cofinality ω) for adding more partial orderings to the iteration. For 2-Mahloγ < δ with P(γ ) sitting as a regular subordering of P(δ), we wait until the first phase of theiteration RP(γ )(γ, δ) has been embedded into P(δ). We then force with termspace R

P(γ )

1 (γ, δ)∗at the next coordinate in F .

Note that this modification does not affect the other coordinates dedicated to the subiterationcorresponding to the second phase of R(γ, δ). The net effect is that there will be two differentembeddings of P(γ ) ∗ R(γ, δ) into P(δ)-one via the subiteration map ιγ,δ , the other via thetermspace for the second phase of the iteration.

The point of this modification is the following lemma:

Lemma 33. Suppose that N ⊃ M are transitive models of ZF− belonging to an arbitrary forcingextension V [G∗] of V such that

(1) α < β < γ are 2-Mahlo in V ,(2) H(γ )V ∈ N and M ∩ Vγ =H(γ )V ,(3) G(α)∗H(α,β)∗K(β,γ ) is generic over M for P(α)∗RP(α)(α,β)∗SP(α)∗R(α,β)(β, γ ) and

lies in V [G∗],(4) N = M[G(α) ∗H(α,β) ∗K(β,γ )].

Then:


(1) G(α) ∗H(α,β) ∗K(β,γ ) is generic over V for P(α) ∗R(α,β) ∗ S(β, γ ).(2) If G ⊂ P(γ ) is generic over V [G(α) ∗ H(α,β) ∗ K(β,γ )] for the partial ordering

P(γ )/ια,β“(G(α) ∗ H(α,β)) then in V [G,G(α) ∗ H(α,β) ∗ K(β,γ )] there is a filterG(β) ∗H(β,γ ) extending the image of G(α) ∗H(α,β) ∗K(β,γ ) under the map ι+α,β that

is N -generic for (P(β) ∗R(β, γ ))/ι+α,β“(G(α) ∗H(α,β) ∗K(β,γ )).

(3) In the first phase coordinates of RP(β)(β, γ ), except for the one corresponding to α, thegeneric object H(β,γ ) is given by ιβ,γ and G.

We note that hypothesis (1) implies that P(α) ∗ RP(α)(α,β) is embedded via ια,β into P(β)

and hence into P(γ ), since P(β) sits as a regular subordering of P(γ ).It follows from conclusion (2) that K(β,γ ) is the generic object determined by H(β,γ ) in

the first phase coordinate of RP(β)(β, γ ) assigned to α.

Proof of Lemma 33. Since H(γ )V ⊂ M , all of the partial orderings in question are subsetsof M . Since γ > α is inaccessible and P(α) ∗ R(α,β) ∗ S(β, γ ) is γ -c.c., the first item is im-mediate. By the γ -chain condition any maximal antichain of P(γ )/ια,β“(G(α) ∗H(α,β)) in N

belongs to H(γ )V [G(α)∗H(α,β)∗K(β,γ )], and the similar statement holds for P(β) ∗ RP(β)(β, γ ).Hence M[G(α) ∗H(α,β) ∗K(β,γ )]-generic objects are N -generic.

To see the second and third conclusion, we first note that P(β)⊂ P(γ ) and is a regular sub-ordering. Using hypothesis (4) we see that G induces an ultrafilter G(β)⊂ P(β) that is both N

and V generic.By property P5 of the forcing, we can factor the first phase of R(β, γ ) into a product

SP(α)∗R(α,β)(β, γ )×Q.

The map ιβ,γ and G give an N [G(β)]-generic object H0 for Q via the map ιβ,γ . Since G isgeneric over V [G(α) ∗ H(α,β) ∗ K(β,γ )], the product lemma shows that K(β,γ ) × H0 isgeneric over N [G(β)] for the first phase of the forcing RP(β)(β, γ ).

By the modification of the forcing P(γ ), there is a coordinate ξ < γ that is after all of thetype 0 ordinals for β for which we forced with the termspace R

P(β)

1 (β, γ )∗. Since this forc-ing is a subset of M[G(β)], and is forced with after the type 0 ordinals, the generic objectH ∗

1 ⊂ RP(β)

1 (β, γ )∗ is generic over N [G(β) ∗H0]. By Corollary 29, we see that H ∗1 induces an

N [G(β)∗H0] generic object H1 for the second phase of RP(β)(β, γ ). Letting H(β,γ )=H0∗H1we see Lemma 33. �

We are now ready to argue for property (vi). The partial ordering P ∗ R ∗ S referred to inproperty (vi) is the modified

P(κ0) ∗RP(κ0)(κ0, κ1) ∗ SP(κ0)∗R(κ0,κ1)(κ1, κ2).

The map ι � P(κ0) ∗R(κ0, κ1) ∗ S(κ1, κ2) is the map ι+κ0,κ1and sends SP(κ0)∗R(κ0,κ1)(κ1, κ2) to the

type 0 coordinate of R(κ1, κ2) corresponding to κ0.Our data consists of

• an N ≺ 〈H(κ+22 )V [G∗H∗K],∈,Δ,H(κ+2

2 )V 〉 with• {P ∗R ∗ S, j (P) ∗ j (R), ι,G ∗H ∗K} ⊂N ,


that lies in V [G ∗H ∗K] for some V -generic

G ∗H ∗K ⊂ P(κ0) ∗RP(κ0)(κ0, κ1) ∗ SP(κ0)∗R(κ0,κ1)(κ1, κ2).

It has the additional properties that N ∩H(κ2)V ∈ V , |N | = κ1, N ∩κ1 ∈ κ1 and is 2-Mahlo in V ,

and the transitive collapse of N ∩HV (κ2) is HV (κ1).With this data we will find a master condition mN in the partial ordering j (P) ∗ j (R)/ι“(G ∗

H ∗K) that forces G ∗ H is generic over N .Let N be the transitive collapse of N , πN be the collapse map, and jN be its inverse. Then:

(1) By the definability properties of the partial orderings we see that πN takes P(κ1) ∗R(κ1, κ2)

to P(κN0 ) ∗R(κN

0 , κ1), where κN0 =N ∩ κ1, and

(2) (G ∗H ∗K)∩N is generic over N ∩ V for (P(κ0) ∗R(κ0, κ1) ∗ S(κ1, κ2))∩N , so(3) πN(G ∗ H ∗ K) = G(κ0) ∗ H(κ0, κ

N0 ) ∗ K(κN

0 , κ1) is generic over M = πN(N ∩ V ) for

P(κ0) ∗RP(κ0)(κ0, κN0 ) ∗ SP(κ0)∗R(κ0,κ

N0 )(κN

0 , κ1).(4) We also note that πN(N ∩ V )⊃H(κ1)

V .

Force over V [G ∗ H ∗ K] with P(κ1)/ι“(G ∗ H) to get G. We now are in a position to applyLemma 33 with α = κ0, β = κN

0 , γ = κ1, and N and M defined as above. From this we conclude

that in V [K,G], there is an N ∩V -generic G(κN0 ) ∗H(κN

0 , κ1)⊂ P(κN0 ) ∗RP(κN

0 )(κN0 , κ1) as in

the conclusion of Lemma 33.We now construct a master condition m ∈ RP(κ1)(κ1, κ2) with the property that for all r ∈

H(κN0 , κ1), m � jN(r). It is a standard argument that such an m is generic over N .

We start by building an mf that is a master condition for the first phase of RP(κN0 )(κN

0 , κ1). Indoing this we must take special care at the type 0 coordinate corresponding to κ0, since in thiscoordinate the generic object given by H(κN

0 , κ1) is K .By clause (3) of the conclusion of Lemma 33, if ξ �= κ0 and Hξ is the generic object for

SP(ξ)∗R(ξ,κN0 )(κN

0 , κ1) in the first phase of the iteration RP(κN0 )(κN

0 , κ1), then Hξ is the genericobject given by ικN

0 ,κ1. By commutativity of the appropriate diagram Hξ is the generic object

given by ιξ,κ1 . Hence Hξ belongs to the model V [G(ξ)∗H(ξ, κ1)] (where G(ξ)∗H(ξ, κ1) is thegeneric object induced by ιξ,κ1 ) and in this model we can form the condition mξ =⋃

jN “Hξ ∈SP(ξ)∗R(ξ,κ1)(κ1, κ2).

Since πN � H(κ2)V ∈ V , K(κN

0 , κ1) = πN “(K ∩ N) ∈ V [G ∗ H ∗ K]. The domainof

⋃jN “K(κN

0 , κ1) has cardinality κ1 and is contained in κN0 × κ2. We also know that

jN “K(κN0 , κ1)⊆K , since jN “K(κN

0 , κ1)=N ∩K . By Remark 13, there is an mκ0 ∈K(κ1, κ2)

such that for all s ∈K(κN0 , κ1) we have mκ0 � jN(p).

We can conclude that the condition mf defined by setting mf (δ)=mδ , for δ a type 0 coordi-

nate, is a master condition for the first phase of the forcing RP(κN0 )(κN

0 , κ1).Finally, the second phase of the forcing in R(κ1, κ2) is κ+1 -closed, hence there is a condition

ms whose support is in the second phase of R(κ1, κ2) that is below each jN(p) restricted to thesecond phase of the forcing. The condition ms is formed by taking a union of the jN images ofcondition in H1(κ

N0 , κ1) in each second phase coordinate.

Let mN be the condition that is equal to mf on the first phase of the forcing and is equal toms on the second phase of the forcing.


Then for all conditions p ∈H(κN0 , κ1) we have jN(p) � mN . If mN ∈ H then the intersection

of G ∗ H with N is jN “(G(κN0 ) ∗H(κN

0 , κ1)). Since G(κN0 ) ∗H(κN

0 , κ1) is generic over N (for

the quotient forcing), we see that (G ∗ H )∩N is generic over N , as required.This finishes the proof of Theorem 23. �

4. An MM result

It is a standard fact that if κ is supercompact, λ > κ is regular, and 〈Sα: α < λ〉 is a partitionof λ∩ cof (ω) then for all supercompact measures U on [λ]<κ ,

{z ∈ [λ]<κ : z ∩ κ ∈ κ and α ∈ z iff Sα ∩ sup z is stationary in sup z

} ∈U.

Assuming that |H(λ)| = λ we can index the partition of λ ∩ cof (ω) by elements of H(λ),

〈Sx : x ∈H(λ)〉 and hence we can state that for all supercompact measures U on [H(λ)]<κ ,

{z ∈ [

H(λ)]<κ : z ∩ κ ∈ κ and x ∈ z iff Sx ∩ sup z is stationary in sup z

} ∈U.

By the results of Section 2, if κ is supercompact we can take a stationary subset of [2λ]<κ

with this property and construct from it with the non-stationary ideal as a predicate to get amodel where κ is λ-supercompact.

In this section we show that, assuming Martin’s Maximum, one can build stationary sets of theform above. It is a natural conjecture that constructing from one of these stationary sets togetherwith the non-stationary ideal would yield an inner model with some degree of supercompactness.We note that the next theorem is a variation of Theorem 10 of [9].

Theorem 34. Assume Martin’s Maximum. Suppose that λ � ω2 is regular and H ⊂ H(λ) hascardinality λ and λ⊂H . Let 〈Sx : x ∈H 〉 be a partition of λ∩ cof (ω) into stationary sets. Thenthere is a stationary subset A⊂ [H ]<ω2 such that for all N ∈A:

x ∈N iff Sx is stationary in sup(N ∩ λ).

Proof. Let 〈Tα: α < ω1〉 be a partition of the limit ordinals of ω1 into stationary sets. We canassume that Tα contains all limit ordinals in ω1. Fix an algebra A on H . Define a partialordering PA by setting p ∈ PA iff

(1) p is a function from δ+ 1 to H for some countable limit ordinal δ.(2) The range of p � δ is closed under the Skolem functions for A.(3) The restriction of p to limit ordinals less than or equal to δ is an order preserving, continuous

map into λ.(4) For all x ∈ range(p � δ), if β < δ is such that p(β)= x, then for all limit β ′ ∈ (β, δ + 1]

β ′ ∈ Tβ iff p(β ′) ∈ Sx.

The proof of the next two claims uses the standard lemma (see [9]) that if S ⊂ λ∩ cof (ω) andT ⊂ ω1 are stationary then the collection of N ⊂H such that


• N ∩ω1 ∈ T and• supN ∩ λ ∈ S

forms a stationary subset of [H ]<ω1 .

Claim 35. For all limit ordinals δ ∈ ω1, q ∈ PA, x ∈ H if δ > dom(q) then there is a p ∈ PA

stronger than q such that δ + 1= dom(p) and x ∈ range(p).

Proof. We show the conclusion for all q and δ > dom(q) by induction on limit ordinals δ. Sup-pose that δ ∈ Tβ where β < δ. If δ is of the form ω× (ξ + 1) then by induction we can assumethat dom(q)= (ω× ξ)+ 1. Let q ′ be a function with domain included in dom(q)+ 1 such thatβ ∈ dom(q ′). We extend q ′ to a condition. Let N ≺ 〈H(θ),∈,Δ,q ′,A, {δ, x,λ}〉 be a countablemodel such that sup(N∩λ) ∈ Sq ′(β). Extend q ′ to a function q∗ with domain δ = ξ×ω+ω so thatthe q∗ is a surjection onto N ∩H . Set p = q∗ ∪ {(δ, sup(N))}. Since δ ∈ Tβ and sup(N) ∈ Sp(β)

this is a condition.Now suppose that δ is a limit of limit ordinals. Choose a sequence of limit ordinals 〈δn: n ∈ ω〉

increasing to δ above the domain of q . By inductively applying the claim we can assume thatthere is a β ∈ dom(q) such that δ ∈ Tβ . Let N ≺ 〈H(θ),∈,Δ,q,A, {δ, x,λ}〉 be such thatsup(N ∩λ) ∈ Sq(β). Applying the induction hypothesis inside N we can build a sequence of con-ditions 〈pn: n ∈ ω〉 such that dom(pn)= δn + 1 and

⋃n pn is a surjection from δ onto N ∩H .

Letting p =⋃n pn ∪ {(δ, sup(N)∩ γ )} completes the proof of the claim. �

Claim 36. The partial ordering P preserves stationary subsets of ω1.

Proof. To see this, let S ⊂ ω1 be stationary and suppose that C ∈ V PA is a term for a closedunbounded subset of ω1. Let p ∈ PA. We will be done if we can find a condition q � p suchthat q � C ∩ S �= ∅. Choose an α such that S ∩ Tα is stationary. Without loss of generality wecan assume that α ∈ dom(p). Let N ≺ 〈H(θ),∈,Δ,p,A, {C}〉 be such that N ∩ ω1 ∈ S ∩ Tα

and sup(N ∩ λ) ∈ Sp(α). Choose a sequence of conditions 〈qn: n ∈ ω〉 ⊂ PA ∩N such that forall dense open sets D ∈ N , there is an n ∈ ω,qn ∈D. Let q =⋃

qn ∪ {(N ∩ ω1, sup(N ∩ λ))}.Then q is a condition that forces C is unbounded in N ∩ ω1, and hence q � N ∩ ω1 ∈ C ∩ S, asdesired.

To finish the proof of the theorem we set D = 〈Dα: α < ω1〉 where Dα is the dense openset of conditions p such that α ∈ dom(p). Then if G⊂ PA is D generic, and N = range(

⋃G),

then N ≺A and⋃

G restricted to the limit ordinals is a continuous order preserving map onto aclosed unbounded subset of sup(N ∩ λ) that is partitioned by {Sx : x ∈N}. �

Let κ be a supercompact cardinal and suppose that P is some standard iteration for cre-ating model V [H ] that satisfies MM . It follows from the results of Section 2, that if westart with an appropriate set A ∈ U where U is a supercompact filter on 2λ, then in V [H ],L[NS � [2λ]<ω2,A] |� ω

V [H ]2 is λ supercompact. For this reason it is natural to conjecture that a

set A of the form produced works in any model of MM:

Question. Assume MM and let κ = ω2. Is there a set A⊂H(ω4) such that L[A,NS�[ω4]ω2] |� κ

is κ+ supercompact?

A somewhat less ambitious question is: What is the consistency strength of the statement:


For all partitions 〈Sα: α ∈ ω3〉 of ω3 ∩ cof (ω), there is a stationary set of N ∈ [ω3]ω1 suchthat for all α ∈ ω3, α ∈N iff Sα is stationary in sup(N)?

What happens if we change the parameters to be different from ω1 and ω3?

5. Conclusion

In conclusion this paper exhibits various combinatorial properties of small cardinals whoseconsistency strengths are those of very large cardinals. These cardinals are so large that no tradi-tional inner model theory exists for them at this time.

In the process we were able to give criteria that distinguish between those precipitous idealsinduced by the remains of collapsed large cardinals and those natural ideals whose precipitous-ness is due to other reasons.

The models for large cardinals that are built in this paper are not elaborate – simply the resultof doing constructibility relative to a well-ordered set of measure one and the non-stationaryideal. What is novel are the choices of sets of measure one used in the constructions and thearguments that show that they work.

References

[1] Uri Abraham, Aronszajn trees on ℵ2 and ℵ3, Ann. Pure Appl. Logic 24 (3) (1983) 213–230.[2] D.R. Burke, Precipitous towers of normal filters, J. Symbolic Logic 62 (3) (1997) 741–754.[3] J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory. I, Ann. Pure Appl.

Logic 129 (1–3) (2004) 211–243.[4] M. Foreman, Large cardinals and strong model theoretic transfer properties, Trans. Amer. Math. Soc. 272 (2) (1982)

427–463.[5] M. Foreman, More saturated ideals, in: Cabal Seminar 79–81, Springer, Berlin, 1983, pp. 1–27.[6] M. Foreman, Calculating quotient algebras of generic embeddings, preprint, 15 pages.[7] M. Foreman, P. Komjath, The club guessing ideal (Commentary on a theorem of Gitik and Shelah), J. Math.

Log. 5 (1) (2005) 99–147.[8] M. Foreman, M. Magidor, Large cardinals and definable counterexamples to the continuum hypothesis, Ann. Pure

Appl. Logic 76 (1) (1995) 47–97.[9] M. Foreman, M. Magidor, S. Shelah, Martin’s maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of

Math. (2) 127 (1) (1988) 1–47.[10] K. Kunen, Saturated ideals, J. Symbolic Logic 43 (1) (1978) 65–76.[11] R. Solovay, Strongly compact cardinals and the GCH, in: Proceedings of the Tarski Symposium, Univ. California,

Berkeley, CA, 1971, in: Proc. Sympos. Pure Math., vol. XXV, Amer. Math. Soc., Providence, RI, 1974, pp. 365–372.

Further reading

[12] M. Foreman, Ideals and generic elementary embeddings, in: M. Foreman, A. Kanamori (Eds.), Handbook of SetTheory, in press.

[13] M. Gitik, On generic elementary embeddings, J. Symbolic Logic 54 (3) (1989) 700–707.

smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals

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