double helix in large large cardinals and iteration of elementary embeddings

Annals of Pure and Applied Logic 146 (2007) 199–236www.elsevier.com/locate/apal

Double helix in large large cardinals and iteration ofelementary embeddings

Sato Kentaro∗

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USAGraduate School of Science and Technology, Kobe University, Rokkodai, Kobe 657-8501, Japan

Received 14 June 2006; accepted 15 February 2007Available online 21 February 2007

Communicated by T. Jech

Abstract

We consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wiseimplications between large large cardinals.

Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties andit has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly changethis situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters.

As results of our investigation of this helical structure, we have (i) new characterizations of extendible cardinals and ofVopenkan cardinals in terms of elementary embeddings from the universe V , (ii) a new large large cardinal property which lookslike Shelahness properly between Vopenkaness and almost hugeness, and (iii) a more direct relation between Vopenkaness andWoodinness than already known.

We also consider limits of the helices and relations with the axioms I1–I3.c© 2007 Elsevier B.V. All rights reserved.

Keywords: Iteration of elementary embeddings; Large large cardinal; Double helix; n-huge cardinal; Super-n-huge cardinal; Vopenka-n-subtlecardinal

1. Introduction

Large cardinals are cardinals with strong closure or reflecting properties (under some operations) that imply theconsistency of ZFC. Many large cardinal properties have been proposed, e.g., inaccessible, Mahlo, weakly compact,0#, measurable cardinals, and it is known that, quite interestingly, their consistency strengths are almost linearlyordered. These properties have been considered as transcendental properties beyond the properties below them andso it has been considered that, to introduce stronger cardinals than the given large cardinals, we have to invent acompletely new property, which cannot be accessed by the operations associated with the given cardinals. In thispaper, we point out that such a picture is not true at least for relatively stronger large cardinal properties, called large

∗ Corresponding address: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.E-mail address: [email protected].

0168-0072/$ - see front matter c© 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.apal.2007.02.003

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200 K. Sato / Annals of Pure and Applied Logic 146 (2007) 199–236

large cardinals. The sequence of these large cardinal properties occurs repeatedly in the sequence of large cardinalproperties, only changing parameters, as in the chart at the end of this paper. (It must be mentioned, however, thata small but similar structure has already been known among n-ineffability, n-almost ineffability and n-subtlety, asshown in [2].) We call this phenomenon the helix. Though properties with different parameters are different, they aresimilar to each other and behave quite similarly.

Typical large cardinal properties are characterized by the existence of various kinds of elementary embeddings,e.g., 0# exists iff there is a non-trivial elementary embedding between inner models, a measurable cardinal exists iffthere is a non-trivial elementary embedding from the universe V . Most large large cardinals, among which we willpoint out the helices, are characterized by the existence of elementary embedding from V with some strong properties(thus their strengths are beyond that of measurability), while some of them were previously not, e.g., Vopenkaness,extendibility. It is also known that these elementary embeddings can be approximated by those induced by some kindsof ultrapowers which share the large cardinal properties with the original embedding. (Thus large cardinal propertiescan be formulated in ZFC.)

To find the helical structure, the key tool is iteration of elementary embeddings. It is easy to define iterations forelementary embeddings whose domains are V , because the elementary embeddings (rigorously, any fragments ofthem) are in the domains. We extend the notion of iteration to general elementary embeddings so that this notionis applicable even in model theoretic situations. We also show that approximation works well for this kind ofiteration, i.e., iterations of elementary embeddings are approximated by iterations of those induced by some kindsof ultrapowers. This extension allows us to define iterated versions of the various known large cardinal propertiesincluding Vopenkaness and extendibility so that the original properties are seen as “1-time iterated” versions. Thuswe have many similar but different large cardinal properties, only by changing the number of iterations. Indeed, theparameters we mentioned above are essentially the numbers of iterations in this sense, i.e., indicate how many timesthe embeddings are iterated.

According to this recipe, we will have the first helix, involving large cardinal properties between superstrongnessand superhugeness. However, the naive iteration recipe does not work for supercompactness, for the naive iteratedversions of supercompactness are equivalent to usual (“1-time iterated” version of ) supercompactness. Thus we needto modify the naive iteration recipe in order to make the helix work.

This modification leads us to the second helix, which involves large large cardinal properties between strongnessand superstrongness, i.e., strongness, Woodinness, Shelahness and superstrongness. Then it is natural to ask whetherthese two helices can be unified in the expected way. The answer will turn out to be no, though they can be unified inan unexpected way, as explained below.

Actually, some parts of the helices are essentially already known, e.g., [5,1] and the comment just before[6, Proposition 26.12]. Nevertheless, these works were only local, i.e., involve only a few properties. Our investigationcombines and extends them, so that it reveals the global structure of the helices which involves almost all known largelarge cardinals beyond measurability.

With several exceptions, proofs of hierarchy results of the double helix seem to be straightforward generalizationsof those for 1-time iterated versions, from e.g., [6, Chap. 5], [13] and [10]. The significance of this paper is,however, the invention of appropriate definitions of multiple times iterated versions of large large cardinals making thestraightforward generalizations of the proofs work well. And this straightforwardness is what guarantees the strongsimilarity among the sequences with different parameters.

The large large cardinal properties among which we will investigate the structure of helices are those treated in(mentioned in the chart at the end of) [6] except Shelahness, because this book has been respected as a standardtextbook in the field. There have been several other large cardinal properties introduced by several authors, e.g.,A1–A7 from [13], and it is likely that we can similarly find helices among these large cardinals.

Moreover, we do have essentially new results, whose 1-fold version is not known, mainly on the relation betweentwo helices, for the relation between different level (more precisely, between n-th level of the first helix and n + 1-thlevel of the second) is of interest. Especially, since the two helices are linked with each other in a strange way, alinearity problem naturally occurs: does the consistency of the large cardinal properties in the double helix form alinear order? To answer this question, we will show that n-fold extendibility and n-fold Vopenkaness are equivalent ton+1-fold strongness and n+1-fold Woodinness respectively, and that n-fold almost hugeness strongly implies n+1-fold Shelahness. Thus, we will have obtained characterizations of (multiple) extendibility and Vopenkaness in termsof elementary embeddings whose domains are V with strong conditions, while they have seemed to be unusual among

K. Sato / Annals of Pure and Applied Logic 146 (2007) 199–236 201

large large cardinal properties since they have been defined not in terms of such elementary embeddings. Furthermore,by the strong implication from n-fold almost hugeness to n + 1-fold Shelahness, we will have a new large cardinalbetween (ordinary) almost hugeness and (ordinary) Vopenkaness.

It is well known that there is an analogy between Vopenkaness-extendibility relation and Woodinness-strongnessrelation, e.g., Vopenkaness and Woodinness can be characterized as some kinds of “boldface” extendibility andstrongness respectively. As we will see, this analogy still holds between their multiple versions. We will have more:the equivalence between strongness and extendibility (with +1-shift of levels) and that between Woodinness andVopenkaness (with +1-shift, again). Moreover, beyond the equivalence, we will show that the (ordinary, or n-fold)Vopenka filter is exactly the (2-fold, or n + 1-fold) Woodin filter, while it is well known that with these filers wecan investigate Vopenkaness and Woodiness more closely. Thus we will have the analogy (and equivalence) betweenVopenkaness and Woodinness, not only between Vopenkaness-extendibility and Woodinness-strongness relations,and this analogy will be analyzed as the strong connection among different levels of the helix (in other words,between ordinary Woodinness and 2-fold Woodinness) and the (filter-level) equivalence between 2-fold Woodinnessand ordinary Vopenkaness.

We will also define the limits of the double helix and investigate the relation with other known axioms, e.g., I1,I2 and I3 (see [6, Section 24]). As a result, our limit versions of large large cardinal properties are comparable withLaver’s E-series from [8], which refines I1–I3.

Let us explain the notation and terminology briefly. In this paper, all first order structures are relational exceptwhen otherwise mentioned. For a transitive class M and an ordinal α, Mα denotes M ∩ Vα . For a set X and a cardinalκ , κ[X ] denotes the set of all those subsets of X whose cardinality is κ . Pµλ = <µ

[λ] = {x ⊂ λ | |x | < µ}, and≤µ[λ] = {x ⊂ λ | |x | ≤ µ}. <ω↑κ is the set of finite increasing sequences of ordinals < κ . For a family 〈Xα |α < κ〉

with Xα ⊂ κ , 4α∈κ Xα = {α < κ | (∀ξ < α)(α ∈ Xξ )}. A filter F on a cardinal κ is normal iff, if 〈Xα |α < κ〉 ∈ κ Fthen 4α∈κ Xα ∈ F . For a tree T and t ∈ T , T/t = {s | t_s ∈ T }.

2. Partition space

As mentioned in the Introduction, large large cardinal properties are formalized in ZFC, by the method ofapproximation by ultrapowers. Nevertheless, some of them cannot be captured by ordinary ultrapowers, and theyneed some variants: limit ultrapowers or extenders.

In this paper, to make the discussions uniform, we use the notion of partition space, with which we can treatordinary ultrapowers, limit ultrapowers and extenders uniformly. As shown in [11] by the author, the family ofultrapowers induced by partition spaces enjoys the following nice properties: (1) it includes the family of ordinaryultrapowers; (2) it includes the family of ultrapowers by extenders; (3) it is closed under (finite and infinite) iterations,and so we can develop the iteration theory for ultrapowers by partition spaces, in a quite similar way to that of forcing(especially, finite support iteration) as in [11].

Definition 2.1. P ⊂ P(S) is a partition of S iff (∀B, C ∈ P)(B 6= C implies B ∩ C = ∅) and S =⋃

P . For twopartitions P1, P2 of S, P1 @ P2 iff P1 is finer than or equal to P2, i.e., (∀B ∈ P2)(B =

⋃{C ∈ P1 |C ⊂ B}), and we

let P1 u P2 = {B ∩ C | B ∈ P1, C ∈ P2}. T(S) denotes the set of all partitions of S.A partition space is a pair (S,A) of a set S and a nonempty family A of partitions of S such that

if P1, P2 ∈ A then P1 u P2 ∈ A.

For a partition P of S, P is an admissible partition iff there is Q ∈ A such that P A Q.∗ denotes the partition space ({∗}, {{∗}}), where ∗ is just a symbol that does not represent any particular set.For a cardinal λ, a partition space A is λ-complete iff, for ν < λ and 〈Pξ | ξ < ν〉 ∈ νA, uξ<ν Pξ =

{⋂

ξ<ν f (ξ) | f ∈∏

ξ<ν Pξ } is admissible with respect to A.For two partition spaces (S,A) and (T,B), a map σ : S → T is a pseudo-continuous map from (S,A) to (T,B)

iff P ∈ B implies the admissibility of σ ∗P with respect to A, where σ ∗P = {σ−1[B] | B ∈ P}.

Note that both of {S} and {S,∅} are partitions of S and that (S, T(S)) forms a partition space.In what follows, we confuse A and (S,A), for S can be recovered from A, i.e., S =

⋃ ⋃A. In this case, S is

denoted by D(A). Thus D(∗) = {∗}.


Note that a partition space A forms a filter-base on the poset of partitions with @. Actually, the filter generatedby A, i.e., the set of all admissible partitions, plays the central role. In the following sections, all partition spaces areclosed @-upward and so P is admissible with respect to A iff P ∈ A. However, if we postulate partition spaces to beclosed upward, then this notion becomes non-absolute (with respect to transitive models). This is why we define thisnotion in the present way.

Remark 2.1. For a partition space (S,A), A can be seen as a basis for a uniformity on S and the pseudo-continuityis exactly the uniform continuity between uniform spaces generated by the given partition spaces. Thus, the categoryParSp of partition spaces and pseudo-continuous maps is a full subcategory of that of uniform spaces and uniformlycontinuous maps.

Definition 2.2. For a partition space A, the Boolean algebra B(A) is defined as follows:

B(A) = {B ⊂ D(A) | {B, D(A) \ B} is admissible w.r.t. A}, ordered by the inclusion relation ⊂ .

It is easy to see that B(A) actually forms a Boolean algebra. For example, if B, C ∈ B(A), say {B,¬B} A P and{C,¬C} A Q, then {B ∩ C,¬(B ∩ C)}, {B ∪ C,¬(B ∪ C)} A {B ∩ C, B \ C, C \ B,¬(B ∪ C)} A P u Q ∈ A.

Definition 2.3. For a map f , define par( f ) = { f −1[{x}] | x ∈ Im( f )}.

For a partition space A,

Ult(A) = { f : D(A)→ V |par( f ) is admissible w.r.t. A}.

For a first order structure M, Ult(A, M) is { f : D(A)→ |M| |par( f ) is admissible w.r.t. A} with relation Ult(A, R)

defined by Ult(A, R)( Ef ) holds iff (∀s ∈ D(A)) R( Ef (s)). jA : V → Ult(A) (or |M| → Ult(A, M)) is defined byjA(x)(s) = x for all s ∈ D(A).

For a pseudo-continuous map σ : A → B, define Ult(σ ) : Ult(B) → Ult(A) (or Ult(σ ) : Ult(B, M) →

Ult(A, M)) by Ult(σ )( f ) = f ◦ σ .

Note that, in spite of the notation Ult, ultrafilters are not involved in the structure Ult(A).Since par( f ◦ σ) = σ ∗par( f ), if f ∈ Ult(B) then Ult(σ )( f ) ∈ Ult(A). It follows easily that Ult is a contravariant

functor from ParSp. Since Ult(∗, (V,∈)) ∼= (V,∈), jA can be seen as Ult(!A), where !A : A → ∗ is the uniquepseudo-continuous map from A to ∗.

Note also that, if we allow partition spaces to be proper classes, then Ult(A) is the class of all pseudo-continuousmaps from A to T(V ).

Definition 2.4. Let U be an ultrafilter of B(A).U is A-κ-complete iff, for any admissible P with respect to A, |P| < κ implies P ∩U 6= ∅.The equivalence relation ∼U on Ult(A) is defined by f ∼U g ⇔ {s ∈ D(A) | f (s) = g(s)} ∈ U .

Note {s ∈ D(A) | f (s) = g(s)} ∈ B(A) since {{s ∈ D(A) | f (s) = g(s)},¬{s ∈ D(A) | f (s) = g(s)}} Apar( f ) u par(g). Ult(A)/U is the structure whose domain is Ult(A)/ ∼U and whose interpretation E of ∈ is definedby [ f ]U E[g]U ⇔ {s ∈ D(A) | f (s) ∈ g(s)} ∈ U . For a first order structure M, Ult(A, M)/U is defined similarly.jA,U : V → Ult(A)/U (or M→ Ult(A, M)/U ) is defined by jA,U (x) = [ jA(x)]U .

For a pseudo-continuous map σ : A→ B, define the ultrafilter σ@U = {B ∈ B(B) | σ−1[B] ∈ U } of B(B).

Obviously, Ult(T(S))/U is a usual ultrapower on S. By the standard argument, we have the following version ofŁos’s theorem:

Lemma 2.5. For a formula ϕ and Ef ∈ Ult(A)<ω, Ult(A)/U |= ϕ([ Ef ]U ) iff {s ∈ D(A) |ϕ( Ef (s))} ∈ U.Thus a pseudo-continuous map σ : A → B induces the elementary embedding Ult(σ )/U : Ult(B)/σ@U →

Ult(A)/U, by (Ult(σ )/U )([ f ]σ@U ) = [Ult(σ )( f )]U .

In what follows, we identify a well-founded structure with its transitive collapse.

Lemma 2.6. Let κ be a cardinal. LetA be a partition space and U anA-κ-complete ultrafilter of B(A). If Ult(A)/Uis well-founded, then jA,U �Vκ = id.


Proof. We prove that jA,U (α) = α by induction on α < κ . Suppose that f ∈ Ult(A) is such that [ f ]U ∈ jA,U (α).We may assume Im( f ) ⊂ α. Then { f −1

[{ξ}] | ξ ∈ α} is admissible with respect to A and of cardinality less than κ .Thus f −1

[{ξ}] ∈ U for some ξ and so [ f ]U = jA,U (ξ) = ξ ∈ α. �

Let us turn to consider iterations. In the present paper, we restrict ourselves to consider only finite step “recycled”iterations, i.e., A ∗ jA(A) ∗ · · · , whereas in [11] more general iterations are considered.

Definition 2.7. Let n be a positive integer and A a partition space. Define An recursively: A1= A and

D(An+1) = D(An)× D(A) An+1= {P • F | P ∈ An, F ∈ PA}

where P • F = {X × Y | X ∈ P, Y ∈ F(X)}.For an ultrafilter U of B(A), define U n recursively as follows, where X(Es) = {t ∈ D(A) | Es_t ∈ X}:

U 1= U X ∈ U n+1

⇔ {Es ∈ D(An) | X(Es) ∈ U } ∈ U n

For i ≤ j , πi, j denotes the canonical pseudo-continuous map A j→ Ai , dropping the last j − i-components. Let

jA,i→ j = Ult(πi, j ) and jA,U,i→ j = Ult(πi, j )/U j .

As one can easily see, U n is an ultrafilter of B(An). By the standard argument, one can show the following keylemma. Note that we do not need to assume the well-foundedness of Ult(An)/U n .

Lemma 2.8. Let n be a positive integer, A a partition space and U an ultrafilter of B(A).

(Ult( jAn ,U n (A))/jAn ,U n (U ))Ult(An)/U n∼= Ult(An+1)/U n+1

A discussion which is similar to the proof of this lemma is given in [4, Section 31].Now let us turn to consider limits. The limits we will need are direct limits (with respect to the category of

elementary embeddings), i.e., filtered colimits in category theoretic terms, and so we can expect that inverse limits inthe category of partition spaces correspond to them.

We start with some categorical preliminaries. For a pseudo-continuous map σ : A → B, D(σ ) denotes the mapσ itself, typed as D(A) → D(B). Thus D is a functor from ParSp to Sets. Similarly, we can define a Booleanhomomorphism B(σ ) : B(B)→ B(A) by B(σ )(B) = σ−1

[B].

Definition 2.9. Let I be a filtered indexing category, i.e., (i) Ob(I) is not empty, (ii) for any I1, I2 ∈ Ob(I), there arearrows a : I3 → I1 and b : I3 → I2 in I and (iii) for any parallel arrows a, b : I1 → I2 there is an arrow c : I3 → I1in I such that a ◦ c = b ◦ c.

For a functor F : I → ParSp, define a partition space Lim F as follows:

D(Lim F) = Lim(D ◦ F) Lim F = {Q | (∃I ∈ Ob(I))(∃P ∈ F(I ))(Q A (pI )∗P)},

where pI : Lim(D ◦ F)→ D(F(I )) is the canonical projection.

It is easy to see that Lim gives filtered limits in ParSp. (As shown in [11], ParSp has arbitrary limits.) Note that,even if F(I )’s are discrete, i.e., F(I ) = T(SI ), Lim F is usually not.

Proposition 2.10. Let I be a filtered indexing category and F : I → ParSp a functor. Assume also that all D(pI )’sare surjections.

Then (i) there is one-to-one correspondence between ultrafilters U of B(Lim F) and sequences 〈UI 〉I∈Ob(I) ofultrafilters UI of B(F(I )) subject to the condition that, for a : I → J , F(a)@UI = UJ , such that (pI )@U = UIfor all I ∈ Ob(I), and (ii) the embeddings Ult(pI )/UI : Ult(F(I ))/UI → Ult(Lim F)/U form the filtered colimitof Ult(F(I ))/UI ’s with elementary embeddings Ult(F(a))/UI .

Proof. (i) is obvious. The assumption that the pI ’s are surjective guarantees that Lim F is non-empty and that U isnon-trivial. For (ii), it suffices to show that, for any f ∈ Ult(Lim F), we have I ∈ Ob(I) and g ∈ Ult(F(I )) suchthat f = Ult(pI )(g). Since par( f ) is admissible with respect to Lim F , we have I ∈ Ob(I) and P ∈ F(I ) such thatpar( f ) A (pI )

∗P , and hence if s, s′ ∈ (pI )−1 B for some B ∈ P then f (s) = f (s′). Thus pI (x) = pI (y) implies

f (x) = f (y), which gives us g such that g ◦ pI = f , defined by g(s) = f (t) if pI (t) = s. Obviously par(g) A Pand so g ∈ Ult(F(I )), with f = g ◦ pI . �

As shown in [11], the assumption that projections are surjective always holds if we consider only iteration.


3. Normality in partition spaces

Let us turn to define the partition spaces which are used to define large cardinal properties:

Definition 3.1. For ordinals µ ≤ λ, partition spaces Nµ,λ and Eλ,µ are defined as follows:

D(Nµ,λ) = Pµλ P ∈ Nµ,λ ⇔ (∃ν < λ)(P A {{s ∈ D(Nµ,λ) | s ∩ ν = t} | t ∈ <µ[ν]})

D(Eλ,µ) = λµ P ∈ Eλ,µ ⇔ (∃a ⊂ λ)(|a| < ω & P A {{s ∈ D(Eλ,µ) | s �a = f } | f ∈ aµ})

It is easy to see that Nµ,λ and Eλ,µ form partition spaces. Note that Nµ,λ is cf(λ)-complete.In order to capture large cardinal properties in terms of special kinds of ultrafilters, the notion of normality for

ultrafilters is important and this is the reason we define the notion of partition space A with a set D(A), rather than ina more abstract way in terms of algebras associated with partition spaces.

Definition 3.2. Let κ be a cardinal and λ ≥ µ be ordinals ≥ κ .(1) A κ-normal ultrafilter U of T(Pµλ) (or T(P(λ))) is a κ-complete ultrafilter on Pµλ (or P(λ)) such that

1. {s ∈ Pµλ | ot(s ∩ κ) < κ} ∈ U (or {s ∈ P(λ) | ot(s ∩ κ) < κ} ∈ U ),2. for any ξ < λ, {s ∈ Pµλ | ξ ∈ s} ∈ U (or {s ∈ P(λ) | ξ ∈ s} ∈ U ),3. for any f ∈ Ult(T(Pµλ)) (or ∈ Ult(T(P(λ)))), if f (s) ∈ s for all nonempty s ∈ Pµλ (or ∈ P(λ)), then

f −1[{x}] ∈ U for some x ∈ Im( f ).

(2) A κ-normal ultrafilter A of Nµ,λ is an Nµ,λ-κ-complete ultrafilter of B(Nµ,λ) such that

1. {s ∈ D(Nµ,λ) | ot(s ∩ κ) < κ} ∈ A,2. for any ξ < λ, {s ∈ D(Nµ,λ) | ξ ∈ s} ∈ A,3. for any f ∈ Ult(Nµ,λ), if f (s) ∈ s for all nonempty s ∈ D(Nµ,λ), then f −1

[{x}] ∈ A for some x ∈ Im( f ),4. for 〈xn | n ∈ ω〉 ∈ ω A,

⋂n∈ω xn 6= ∅.

(3) A κ-normal filter E of Eλ,µ is an Eλ,µ-κ-complete ultrafilter of B(Eλ,µ) such that

1. {s ∈ D(Eλ,µ) | s(κ) < κ} ∈ E ,2. for α < β < λ, {s ∈ D(Eλ,µ) | s(α) < s(β)} ∈ E ,3. for α < λ and f ∈ Ult(Eλ,µ) if f (s) < s(α) for all s ∈ D(Eλ,µ) with s(α) 6= ∅, then there is ξ < α such that{s ∈ D(Eλ,µ) | f (s) = s(ξ)} ∈ E ,

4. for 〈xn | n ∈ ω〉 ∈ ω E ,⋂

n∈ω xn 6= ∅.

Note that, because of the first condition in (2) and (3), the definition does not make sense if λ = κ .Intuitively, the fourth conditions guarantee the well-foundedness, the first guarantee j (κ) > κ and the second and

third guarantee [s 7→ s ∩ ν] = j“ν and [s 7→ s(α)] = α. The next lemma can easily be shown, but is quite useful.

Lemma 3.3. Let M be a transitive class with M |= ZFC. If Vλ+ω ⊂ M then the notion of κ-normality for Eλ,µ

defined above is absolute between M and V , i.e., if Eλ,µ, E ∈ M, then M |= “E is a κ-normal ultrafilter of Eλ,µ” isequivalent to V |= “E is a κ-normal ultrafilter of Eλ,µ”.

Proof. Note that Eλ,µ ∈ M implies (λµ)M=

λµ and so (µ<ω

µ)M=

µ<ωµ. Any f ∈ Ult(Eλ,µ) with Im( f ) ⊂ µ is

coded by a pair (a, f ) of a ∈ <ω[µ] and f : |a|µ→ µ with f (s) = f (s ∩ a). Then it is obvious. �

The next two lemmata are well known and easy to prove, in the same way as Lemmata 3.6 and 3.7:

Lemma 3.4. Let κ, µ, λ as in Definition 3.2 and U a κ-normal ultrafilter of T(Pµλ) (or T(P(λ))). Then

• Ult(T(Pµλ))/U (or Ult(T(P(λ)))/U) is well-founded,• crit( jT(Pµλ),U ) = κ (or crit( jT(P(λ)),U ) = κ),• [res(ν)]U = jT(Pµλ),U “ν (or = jT(P(λ)),U “ν) for any ordinal ν ≤ λ, where res(ν)(s) = s ∩ ν,•

λ(Ult(T(Pµλ))/U ) ⊂ Ult(T(Pµλ))/U (or λ(Ult(T(P(λ)))/U ) ⊂ Ult(T(P(λ)))/U, respectively), and


• for a cardinal ν > λ, (i) if ν(λµ)= ν then jT(Pµλ),U (ν) < ν+ (or if ν(2λ)

= ν then jT(P(λ)),U (ν) < ν+) and (ii) ifcf(ν) > λµ then jT(Pµλ),U (ν) =

⋃( jT(Pµλ),U )“ν (or if cf(ν) > 2λ then jT(P(λ)),U (ν) =

⋃( jT(P(λ)),U )“ν). Thus,

(iii) for inaccessible ν > λ, jT(Pµλ),U (ν) = ν (or jT(P(λ)),U (ν) = ν).

Lemma 3.5. Let j : V → M be a non-trivial elementary embedding with crit( j) = κ and λM ⊂ M. Let µ ≤ λ bea cardinal with j (µ) ≥ λ. Then, U = {x ⊂ Pµλ | j“λ ∈ j (x)} (or U = {x ⊂ P(λ) | j“λ ∈ j (x)}) is a κ-normalultrafilter of T(Pµλ) (or of T(P(λ))). Moreover, kU : Ult(T(Pµλ))/U → M (or Ult(T(P(λ))) → M) defined bykU ([ f ]U ) = j ( f )( j“λ) is elementary, kU ◦ jT(Pµλ),U = j (or kU ◦ jT(P(λ)),U = j ) and crit(kU ) > λ.

The next four lemmata are key lemmata to see how normality captures large cardinal properties.

Lemma 3.6. Let κ be a cardinal and λ ≥ µ two ordinals ≥ κ with λ > κ . For a κ-normal ultrafilter A of Nµ,λ,

• Ult(Nµ,λ)/A is well-founded,• crit( jNµ,λ,A) = κ ,• [res(ν)]A = jNµ,λ,A“ν for ν < λ, where res(ν)(s) = s ∩ ν,•

<cf(λ)(Ult(Nµ,λ)/A) ⊂ Ult(Nµ,λ)/A.• for a cardinal ν > λ, (i) if ν(λ<µ)

= ν then jNµ,λ,A(ν) < ν+ and (ii) if cf(ν) > λ<µ then jNµ,λ,A(ν) =⋃( jNµ,λ,A)“ν. Thus, (iii) for inaccessible ν > λ, jNµ,λ,A(ν) = ν.

Proof. The wellfoundedness of Ult(Nµ,λ)/A is immediate from clause 4 in the definition of κ-normality.Next, we prove that [res(ν)]A = jNµ,λ,A“ν. Note that obviously res(ν) ∈ Ult(Nµ,λ) for ν < λ. [res(ν)]A 3

jNµ,λ,A(ξ) for ξ < ν is immediate from the definition. Conversely, let f ∈ Ult(Nµ,λ) with [ f ]A ∈ [res(ν)]A.Then we may assume that f (s) ∈ s ∩ ν for all s ∈ D(Nµ,λ) \ {∅}. Then, by the normality of A, f −1

[{ξ}] ∈ Afor some ξ ∈ Im( f ). Let ξ = f (s) with s 6= ∅. Then ξ = f (s) ∈ s ∩ ν and so ξ ∈ ν. f −1

[{ξ}] ∈ A means{s ∈ D(Nµ,λ) | f (s) = ξ} ∈ A, i.e., [ f ]A = jNµ,λ,A(ξ) ∈ jNµ,λ,A“ν.

By Lemma 2.6, crit( jNµ,λ,A) ≥ κ . Conversely, by clause 1, jNµ,λ,A(κ) > ot([res(κ)]A) = ot( jNµ,λ,A“κ) = κ .Let 〈[ fξ ]A | ξ < ν〉 ∈ νUlt(Nµ,λ)/A, with fξ ∈ Ult(Nµ,λ) and ν < cf(λ). Define, for each s ∈ Pµ(λ),

g(s) : ν → V so that g(s)(ξ) = fξ (s). By cf(λ)-completeness of Nµ,λ, par(g) = uξ<νpar( fξ ) ∈ Nµ,λ andso g ∈ Ult(Nµ,λ). Now jNµ,λ,A(g)( jNµ,λ,A(ξ)) = [ fξ ]A and so jNµ,λ,A(g) � [res(ν)]A ∈ Ult(Nµ,λ)/A implies〈[ fξ ]A | ξ < ν〉 ∈ Ult(Nµ,λ)/A. Thus <cf(λ)Ult(Nµ,λ)/A ⊂ Ult(Nµ,λ)/A.

(i) Since |{ f ∈ Ult(Nµ,λ) | Im( f ) ⊂ ν}| ≤ ν(λ<µ)= ν, we have jNµ,λ,A(ν) < ν+. (ii) Suppose f ∈ Ult(Nµ,λ)

with [ f ]A < jNµ,λ,A(ν). We may assume Im( f ) ⊂ ν. Then |Pµλ| = λ<µ < cf(ν) implies Im( f ) ⊂ ν′ for some

ν′ < ν. Thus [ f ]A ∈ jNµ,λ,A(ν′) ∈ ( jNµ,λ,A)“ν. (iii) If ν is inaccessible, then {ν′ < ν | λ < ν′, ν′(λ<µ)

= ν′} is cofinal

in ν and so jNµ,λ,A(ν) = sup{ jNµ,λ,A(ν′) | ν′ < ν, ν′(λ<µ)

= ν′} = ν. �

Lemma 3.7. Let j : V → M be a non-trivial elementary embedding with crit( j) = κ and assume λ ≥ µ ≥ κ , λ > κ ,<λM ⊂ M and j (µ) ≥ λ. Define j in the following two ways:

(a) for x ∈ B(Nµ,λ), j(x) = {s ∈ ≤ j (µ)[ j (λ)] | s ∩ j (νx ) ∈ j (x � νx )}, where νx = min{ν < λ | (∀s ∈

D(Nµ,λ))(s ∈ x iff s ∩ ν ∈ x)} and where x �ν = {s ∩ ν | s ∈ x},(b) for f ∈ Ult(Nµ,λ), j( f ) is a map on ≤ j (µ)

[ j (λ)] with j( f )(s) = j ( f )(s ∩ j (ν f )) where ν f = min{ν′ <

λ | (∀s ∈ D(Nµ,λ))( f (s) = f (s ∩ ν′))} and where f is a map on ≤µ[ν f ] such that f (s ∩ ν f ) = f (s) for all

s ∈ D(Nµ,λ).Here, the existence of νx , ν f and f is guaranteed by the definition of Nµ,λ. Then, the following hold:(1) A = {x ∈ B(Nµ,λ) | j“λ ∈ j(x)} is a min(κ, cf(λ))-complete κ-normal ultrafilter of Nµ,λ.(2) If one defines kA([ f ]A) = j( f )( j“λ), then kA : Ult(Nµ,λ)/A → M is elementary, kA ◦ jNµ,λ,A = j and

crit(kA) ≥ λ.

Remark 3.1. In the case λ = j (µ), to make A = {x ∈ B(Nµ,λ) | j“λ ∈ j(x)} be non-trivial, we need to define j asin the statement, not as j(x) = {s ∈ < j (µ)

[ j (λ)] | · · · }.

Proof. Note that B(Nµ,λ) is closed under (< cf(λ)) intersections and unions.


We first claim that, to define A and kA, we can replace νx and ν f by any ν′ < λ with the condition that s ∈ xiff s ∩ ν′ ∈ x or with f (s) = f (s ∩ ν′) for all s ∈ Pµλ. I.e., j“λ ∩ j (νx ) ∈ j (x � νx ) iff j“λ ∩ j (ν′) ∈ j (x � ν′)

and j ( f )( j“λ ∩ j (ν f )) = j ( f′)( j“λ ∩ j (ν′)), where f

′(s ∩ ν′) = f (s). These are equivalent to the following: (i)

(∃s ∈ j (x))(s ∩ j (νx ) = j“νx ) iff (∃s ∈ j (x))(s ∩ j (ν′) = j“ν′) and (ii) j ( f )( j“ν f ) = j ( f′)( j“ν′).

Now, by the choice of ν′, s ∈ j (x) iff s∩ j (ν′) ∈ j (x) for all s ∈ (P j (µ) j (λ))M . Thus (∃s ∈ j (x))(s∩ j (ν′) = j“ν′)iff j“ν′ ∈ j (x) for such ν′. Therefore, (i) is equivalent to j“νx ∈ j (x) iff j“ν′ ∈ j (x), which is obvious.

Similarly, by the choice of ν f , f and ν′, f′, we have j ( f )(s ∩ j (ν f )) = j ( f )(s) = j ( f

′(s ∩ j (ν′)) for all

s ∈ (P j (µ) j (λ))M . By j“ν′ ∈ <λM ⊂ M and by | j“ν′|M = ν′ < λ ≤ j (µ), j“ν′ ∈ (P j (µ) j (λ))M , whichimplies (ii).

Now we show (2), assuming (1). By the claim we have just proved above, it is easy to see that M |=

ϕ( j( f1)( j“λ), . . . , j( fn)( j“λ)) iff {s ∈ D(Nµ,λ) |ϕ( f1(s), . . . , fn(s))} ∈ A, because we can take common ν′ forall fi ’s. This implies the well-definedness and elementariness of kA. Since ν = ot([res(ν)]A) by the last lemma, andsince kA([res(ν)]A) = j(res(ν))( j“λ) = j“λ ∩ j (ν) = j“ν, we have kA(ν) = kA(ot([res(ν)]A)) = ot( j“ν) = ν foreach ν < λ, i.e., crit(kA) ≥ λ.

Let us turn to (1). By <λM ⊂ M , we have j“ν′ ∈ M for all ν′ < λ, and so j“λ ∈ j(Pµλ). Therefore A 6= ∅.By the claim above, we can easily see that j preserves Boolean operations and < min(κ, cf(λ))-unions. Thus A isa min(κ, cf(λ))-complete ultrafilter of B(Nµ,λ). (Note that B(Nµ,λ) is closed under only < cf(λ) intersections.) IfP ∈ Nµ,λ, then we have ν < λ such that, for every x ∈ P , there is x ′ ∈ Pµν such that x = {s ∈ D(Nµ,λ) | s∩ν ∈ x ′}.Then, if |P| < κ = crit( j), j(

⋃P) =

⋃j“P . Thus, A is Nµ,λ-κ-complete. It is straightforward to check

{s ∈ D(Nµ,λ) | ot(s ∩ κ) < κ} ∈ A, and that, for every ξ < λ, j“λ ∈ j({s ∈ D(Nµ,λ) | ξ ∈ s}). Assume f (s) ∈ sfor all s ∈ D(Nµ,λ) \ {∅}, say f (s) = f (s ∩ ν) for some f and ν < λ. Then f (t) = f (t) ∈ t for all t ∈ Pµν andso j( f )( j“λ) = j ( f )( j“λ ∩ j (ν)) = j ( f )( j“ν) ∈ j“ν, which implies j( f )( j“λ) = j (ξ) for some ξ < λ, Thus{s ∈ D(Nµ,λ) | f (s) = ξ} ∈ A.

We can prove the clause 4 of κ-normality, by the standard tree argument. (Note that if cf(λ) > ω, completenessimplies this condition.) By the existence of kA : Ult(Nµ,λ)/A → M , Ult(Nµ,λ)/A is well-founded. Suppose, forcontradiction,

⋂n∈ω xn = ∅ for some 〈xn | n ∈ ω〉 ∈ ω A. Let νn < λ be such that s ∈ xn iff s ∩ νn ∈ xn

for all s ∈ D(Nµ,λ). We may assume that 〈xn | n ∈ ω〉 is decreasing and that 〈νn | n ∈ ω〉 is increasing. Fors ∈ D(Nµ,λ), s ∩ νm ∈ xm implies s ∩ νk ∈ xk if k ≤ m. Define bn(s) = 〈s ∩ νk | k ≤ n〉 for s ∈ D(Nµ,λ),and T = {bn(s) | s ∩ νn ∈ xn, n ∈ ω}. Then the tree (T,≤) is well founded, where ≤ is the extension relation,for otherwise an infinite path 〈sn | n ∈ ω〉 satisfies sn ⊂ νn , sn ∈ xn and sn ∩ νm = sm for all m ≤ n, and sos =

⋃k∈ω sk ∈

⋂n∈ω xn = ∅. Define fn ∈ Ult(Nµ,λ) by

fn(s) =

{rank(T/bn(s)) if bn(s) ∈ T0 otherwise.

Note that par( fn) A {{s ∈ D(Nµ,λ) | s ∩ νn = t} | t ∈ Pµνn} ∈ Nµ,λ. Then, {s ∈ D(Nµ,λ) | fn(s) 3 fn+1(s)} ⊃xn+1 ∈ A for all n ∈ ω and so 〈[ fn]A | n ∈ ω〉 shows the ill-foundedness of Ult(Nµ,λ)/A, a contradiction. �

Lemma 3.8. Let E be a κ-normal ultrafilter of Eλ,µ with λ, µ > κ . Then,

• Ult(Eλ,µ)/E is well-founded,• crit( jEλ,µ,E ) = κ .• [ev(α)]E = α for all α < λ, where ev(α)(s) = s(α), and• for a cardinal ν > λ, (i) if ν|µ| = ν, then jEλ,µ,E (ν) < ν+ and (ii) if cf(ν) > µ then jEλ,µ,E (ν) =

⋃jEλ,µ,E “ν.

Thus, (iii) for an inaccessible cardinal ν > λ, jEλ,µ,E (ν) = ν.

Proof. By clause 4 in the definition of κ-normality, Ult(Eλ,µ)/E is well-founded.We prove by induction on α that [ev(α)]E = α. By clause 2, we have ξ = [ev(ξ)]E ∈ [ev(α)]E for all ξ < α.

Conversely, if [ f ]E ∈ [ev(α)]E , i.e., {s ∈ D(Eλ,µ) | f (s) ∈ s(α)} ∈ E , then, by clause 3, we have ξ < α such that{s ∈ D(Eλ,µ) | f (s) = s(ξ)} ∈ E , i.e., [ f ]E = [ev(ξ)]E = ξ ∈ α. Thus [ev(α)]E = α.

By Lemma 2.6, crit( jEλ,µ,E ) ≥ κ . By clause 1 of κ-normality, we have κ = [ev(κ)]E < jEλ,µ,E (κ).Let ν > λ. (i) If ν|µ| = ν, by |{ f ∈ Ult(Eλ,µ) | (∀s ∈ D(Eλ,µ))( f (s) < ν)}| ≤ |<ω

[λ]| · |(<ωµ)ν| ≤ |λ| · ν|µ| =

|λ| · ν = ν, we have jEλ,µ,E (ν) < ν+. (ii) Suppose cf(ν) > µ and f ∈ Ult(Eλ,µ) with [ f ]E < jEλ,µ,E (ν). We may


assume Im( f ) ⊂ ν. Say f (s) = f ′(s �a) with a ∈ <ω[λ] and f ′ : aµ→ ν. Since |aµ| = |µ| < cf(ν), we have ν′ < ν

such that f (s) = f ′(s � a) < ν′ for all s ∈ D(Eλ,µ) and so [ f ]E < jEλ,µ,E (ν′). (iii) Suppose that ν is inaccessible.Then, by cf(ν) = ν > λ ≥ µ, since {θ |µ| | θ < ν} is cofinal in ν, (i) and (ii) imply (iii). �

Before the next lemma, we need some preliminaries:Suppose κ is regular cardinal and λ arnd µ are ordinals ≥ κ . Define a partition space Eκ

λ,µ as follows:

D(Eκλ,µ) = λµ P ∈ Eκ

λ,µ ⇔ (∃b ⊂ λ)(|b| < κ & P A {{s ∈ D(Eλ,µ) | s �b = f } | f ∈ bµ}).

Then it is easy to see that B(Eκλ,µ) is a κ-complete Boolean algebra generated by B(Eλ,µ). Note that, for each

x ∈ B(Eλ,µ), there are b ∈ <κ[λ] and x ′ ⊂ bλ such that x = {s ∈ λµ | s �b ∈ x ′}.

Lemma 3.9. Let j : V → M be a non-trivial elementary embedding with crit( j) = κ . Let λ and µ be ordinals ≥ κ

with j (µ) ≥ λ. Let r : j (λ)→ λ defined by r = j−1 � j (λ), i.e., r( j (ξ)) = ξ and r(ξ) is arbitrary if ξ /∈ j“λ.Define j in the following two ways:(a) For x ∈ B(Eκ

λ,µ), j(x) = {s ∈ j (λ) j (µ) | s � j (b) ∈ j (x ′)} where x = {s ∈ λµ | s � b ∈ x ′} with x ′ ⊂ bµ andb ∈ <κ

[λ].(b) For f ∈ Ult(Eλ,µ), j( f )(s) = j ( f ′)(s � j (b)) for s ∈ j (λ) j (µ) where f (s) = f ′(s �b) for all s ∈ D(Eλ,µ) and

where b ∈ <κ[λ].

Then (1) E = {x ∈ B(Eλ,µ) | r ∈ j(x)} is a κ-normal ultrafilter of Eλ,µ, and (2) kE : Ult(Eλ,µ)/E → M definedby kE ([ f ]E ) = j( f )(r) is elementary and kE ◦ jEλ,µ,E = j with crit(kE ) ≥ λ.

Proof. First we show that j is well defined, i.e., it does not depend on the choice of b. Assume that, for b1, b2 ∈<κ[λ],

s � b1 ∈ x ′1 iff s ∈ x iff s � b2 ∈ x ′2 for all s ∈ λµ. We need to show that, for all s ∈ j (λ) j (µ), s � j (b1) ∈ j (x ′1) iffs � j (b2) ∈ j (x ′2). We may assume b1 ⊂ b2. Then what we need to show is that, for all t ∈ j (b2) j (µ), t � j (b1) ∈ j (x ′1)iff t ∈ j (x ′2). By the definition of x ′1 and x ′2, we have t �b1 ∈ x ′1 iff t ∈ x ′2 for all t ∈ b2µ and hence t � j (b1) ∈ j (x ′1)iff t ∈ j (x ′2) for all t ∈ ( j (b2) j (µ))M . By | j (b2)| = | j“b2| < κ and κ M ⊂ M , this completes the claim for j(x).Similarly, we can show that j( f ) is well defined.

Since j(D(Eλ,µ)) = j (λ) j (µ) 3 r , E is an ultrafilter of B(Eλ,µ). Obviously j preserves Boolean operationsand < κ-unions, which immediately implies both the Eλ,µ-κ-completeness of E and clause 4 of κ-normality. It isstraightforward to see clauses 1 and 2.

To see clause 3, let f ∈ Ult(Eλ,µ) be such that f (s) < s(α) for all s ∈ D(Eλ,µ) with s(α) 6= 0. Then j( f )(r) <

r( j (α)) = α and so j( f )(r) = ξ = r( j (ξ)) for some ξ < α, which means {s ∈ D(Eλ,µ) | f (s) = s(ξ)} ∈ E .It is again straightforward to check the elementariness of kE , and that kE ◦ jEλ,µ,E = j . For ξ < λ, kE (ξ) =

kE ([ev(ξ)]E ) = j(ev(ξ))(r) = ev( j (ξ))(r) = ξ . Thus crit(kE ) ≥ λ. �

Remark 3.2. As one of the advantages of our notions of partition space and normality, we can point out that, in almostall cases (i.e., except for cf(λ) = ω for Nµ,λ), we can avoid the tree-argument as in the proofs of Lemma 3.7 and ofLemma 3.9 and we can capture large cardinal properties in terms of ultrafilters directly and easily.

Then, by the standard arguments, we can show the following characterization results of large cardinal properties.We omit proofs here, for we shall later prove general results (Lemmas 5.4 and 9.3) that immediately imply theseresults here.

• κ is almost huge iff there are an inaccessible cardinal λ > κ and a κ-normal ultrafilter A of Nκ,λ such that, forevery f ∈ Ult(Nκ,λ) with Im( f ) ⊂ κ , {s ∈ D(Nκ,λ) | f (s) < ot(s ∩ ν)} ∈ A for some ν < λ

• κ is Woodin iff, for any f : κ → κ , there are α, β, µ, λ < κ , with f “α ⊂ α and an α-normal ultrafilter E of Eλ,µ

with Vβ ⊂ Ult(Eλ,µ)/E and {s ∈ D(Eλ,µ) | s(β) = f (s(α))} ∈ E .

We can prove characterizations, similar to the latter, for superstrongness, Shelahness and strongness (seeLemma 5.5 and Section 9). In the proofs the next lemma plays an important role:

Lemma 3.10. Let j : V → M be a non-trivial elementary embedding with Vν ⊂ M for some cardinal ν. Assume thatU is an ultrafilter of B(A), where A is a partition space, and we have kU : Ult(A)/U → M with kU ◦ jA,U = j andcrit(kU ) ≥ λ for some λ. If |Vν |

M≤ λ then kU �Vν is identity and Vν ⊂ Ult(A)/U.


Proof. We have a bijection h : |Vν |∼= Vν such that h � |Vα| : |Vα|

∼= Vα for all ordinals α ≤ ν. Thenj (h) � |Mν |

M: |Mν |

M ∼= Mν = Vν . Thus, for every x ∈ Vν , we have ξ < |Vν |M such that x = j (h)(ξ) =

kU ( jA,U (h))(kU (ξ)) = kU ( jA,U (h)(ξ)) and so x ∈ Im(kU ). Thus Vν ⊂ Im(kU ). Since Vν is transitive andkU−1 � Im(kU ) must be Mostowski’s collapse, we have kU

−1 �Vν is identity and hence Vν ⊂ Ult(A)/U . �

4. Iteration of elementary embeddings and approximation

We first define the iteration of elementary embeddings whose domains are V , in a quite standard way:

Definition 4.1. Let j : V → M be an elementary embedding. For n ∈ ω, define M (n) and j (n) by

M (n)=

⋃α∈On

jn(Vα) j (n)=

⋃α∈On

jn−1( j �Vα).

We can easily see that jn= j (n)

◦ j (n−1)◦· · ·◦ j (1), e.g., j2(x) = j (( j �Vrank(x)+1)(x)) = j ( j �Vrank(x)+1)( j (x)) =

( j (2) � Vrank( j (x))+1)( j (x)) = ( j (2)◦ j)(x). One can easily see that, in the case where j is induced by an ordinary

ultrapower or by an extender, this definition of iteration coincides with the iteration of ultrapowers. Moreover, if j isinduced by any ultrafilter of a partition space, this definition coincides with the iteration U n of ultrafilter U defined inthe previous section.

However, this definition is applicable only to elementary embeddings whose domains are V . We extend thedefinition of iteration to any structures in more model theoretic situations.

For that purpose, we introduce the notion of iteration sequence, defined as follows.

Definition 4.2. Let 〈Mi | i ≤ n〉 be a sequence of first order structures. An iteration sequence through 〈Mi | i ≤ n〉is a sequence Ee = 〈e(i)

: |Mi−1| ∪ · · · ∪ |Mn−1| → |Mi | ∪ · · · ∪ |Mn| | 0 < i ≤ n〉 such that

e(i): (Mi−1, . . . ,Mn−1; e

(i)n−1, . . . , e(n−1)

n−1 ) ≺ (Mi , . . . ,Mn; e(i+1)n , . . . , e(n)

n ) for i ≤ n,

where (Mi1 , . . . ,Mik ; e(i1)ik

, . . . , e(ik )ik

) denotes the (ik − i1 + 1)-sorted first order structure which has every relation

in eachMi1 , . . . ,Mik , which also has function symbols e(i1)ik

, . . . , e(ik )ik

and which has the equality between sorts, andwhere

e(i)k = e(i) �(|Mi−1| ∪ · · · ∪ |Mk−1|) : (|Mi−1| ∪ · · · ∪ |Mk−1|)→ (|Mi | ∪ · · · ∪ |Mk |).

Define crit(Ee) = crit(e(1)), if Mi ’s are (fragments of) models of set theory and crit(e(1)) exists.

One can notice that, because of the equality relations between sorts, we are not free to replace each Mi by anisomorphic structure. One may think this contradicts the philosophy of model theory that we identify isomorphicstructures. However, theMi ’s must be considered as a result of iteration and this means only that the iteration processmust be considered in the on-the-nose way, not in the up-to-isomorphism way. Indeed, we can still identify isomorphiciteration sequences, i.e., two iteration sequences between which there exists a family of isomorphisms subject to anobvious coherence condition.

Actually, considering the iteration process in the on-the-nose way is quite natural. Let us see this in the caseof ultrapowers: for a first order structure M, if some element a ∈ |M| and some b ∈ |Ult(A, M)/U | accidentallycoincide with each other (without any model theoretic sense concerning the structures M and Ult(A, M)/U ), then thisrelation, i.e., equality, holds between jU (a) ∈ |Ult(A, M)/U | and jU (b) ∈ |Ult(A2, M)/U 2

|. With this example, wecan see that, in model theoretic constructions such as ultrapower, some “external” things beyond elementary propertiesof structures are usually assumed, and hence it is not unnatural to consider our notion of iteration, iteration sequence,in the on-the-nose way. Thus we can conclude that this does not conflict with the model theory philosophy.

First we check that this notion of iteration sequence extends the iteration of elementary embeddings.

Lemma 4.3. Let j : V → M be an elementary embedding. For any ordinal α and n ∈ ω, define

Ee( j;α; n)(i) = j (i) �(M (i−1)

j i−1(α)∪ · · · ∪ M (n−1)

jn−1(α)),

for 0 < i ≤ n. Then Ee( j;α; n) is an iteration sequence through 〈M (i)j i (α)| i ≤ n〉.


Proof. It suffices to show that

j (i) �(M (i−1)

j i−1(α)∪ · · · ∪ M (n−1)

jn−1(α)) :

(M (i−1)

j i−1(α), . . . , M (n−1)

jn−1(α); j (i) �(M (i−1)

j i−1(α)∪ · · · ∪ M (n−2)

jn−2(α)), . . . , j (n−1) � M (n−2)

jn−2(α))

≺ (M (i)j i (α)

, . . . , M (n)jn(α); j (i+1) �(M (i)

j i (α)∪ · · · ∪ M (n−1)

jn−1(α)), . . . , j (n) � M (n−1)

jn−1(α)).

Note that j (i) = j i−1( j), j (i+1)= j i−1( j (2)) and M (i)

= j i−1(M), M (i−1)= j i−1(V ) (precisely, the equations hold

for initial segments of the models) for all i and that hence the statement is reduced to

j �(Vα ∪ · · · ∪ M (n−i)jn−i (α)

) :(Vα, . . . , M (n−i)jn−i (α)

; j �(Vα ∪ · · · ∪ M (n−i−1)

jn−i−1(α)), . . . , j (n−i) � M (n−i−1)

jn−i−1(α))

≺ (M j (α), . . . , M (n−i+1)

jn−i+1(α); j (2) �(M j (α) ∪ · · · ∪ M (n−i)

jn−i (α)), . . . , j (n−i+1) � M (n−i)

jn−i (α))

which is obvious from the definition. �

Remark 4.1. If Mk = (Vζk ,∈) with limit ζk for k ≤ n, an iteration sequence Ee through 〈Mk | k ≤ n〉 is determinedby e(1) because e(k+1)

=⋃

α<ζn−2e(1)(e(k) �Vα) for k ≥ 2.

Definition 4.4. For an iteration sequence Ee, define ei,k : (Mi ∪ · · · ∪Mi+(n−k))→ (Mk ∪ · · · ∪Mn) for i ≤ k ≤ nby:

ei,i = id : (Mi ∪ · · · ∪Mn)→ (Mi ∪ · · · ∪Mn)

ei,k+1 = e(k+1)◦ (ei,k �(Mi ∪ · · · ∪Mi+(n−k)−1))

: (Mi ∪ · · · ∪Mi+(n−k)−1)→ (Mk ∪ · · · ∪Mn−1)→ (Mk+1 ∪ · · · ∪Mn).

Lemma 4.5. In the situation as in the last definition,

ei,k : (Mi , . . . ,Mi+(n−k); e(i+1)i+(n−k), . . . , e(i+(n−k))

i+(n−k) ) ≺ (Mk, . . . ,Mn; e(k+1)n , . . . , e(n)

n ).

Now we demonstrate that our definition of iteration of elementary embeddings, iteration sequence, is so strong thatthe approximation lemma holds. In the rest of this section, we assume thatMi ’s have ∈ as one of their relations.

Definition 4.6. Let Ee be an iteration sequence through 〈Mi | i ≤ n〉. For i ≤ n and r ∈ |M1|, define a sequencee(i)(r) of length i recursively:

e(1)(r) = 〈r〉 e(i + 1)(r) = (e(i+1)e(i)(r))_〈e0,i (r)〉.

For example, e(2)(r) = 〈e(2)(r), e0,1(r)〉 = 〈e1,2(r), e0,1(r)〉, e(3)(r) = 〈e1,3(r), e2,3(e0,1(r)), e0,2(r)〉, e(4)(r) =

〈e1,4(r), e2,4(e0,1(r)), e3,4(e0,2(r)), e0,3(r)〉. Thus the j-th component of e(i)(r) is e j+1,i (e0, j (r)).

Lemma 4.7. Let Ee be an iteration sequence through 〈Mi | i ≤ n〉. For r ∈M1, S ∈M0 with r ∈ e(1)(S) and withP(S) ⊂ |M0|, define U = {x ⊂ S | r ∈ e(1)(x)}. Then U n

= {x ⊂ Sn| e(n)(r) ∈ e0,n(x)}.

Proof. We prove that U i= {x ⊂ Si

| e(i)(r) ∈ e0,i (x)} by induction on i ≤ n. For i = 1, this is obvious, where weidentify the sequence 〈r〉 with the component r .

x ∈ U i+1 iff {Es ∈ Si| {t ∈ S | Es_

〈t〉 ∈ x} ∈ U } ∈ U i iff {Es ∈ Si| (e(1)

Es)_〈r〉 ∈ e(1)(x)} ∈ U i iff(e(i+1)e(i)(r))_〈e0,i (r)〉 ∈ e0,i (e(1)(x)) iff e(i + 1)(r) ∈ e0,i+1(x). �

Then, by the standard argument, we can obtain the following approximation lemma:

Theorem 4.8. Let Ee be an iteration sequence through 〈Mi | i ≤ n〉, with |M0| = Vζ for some inaccessible ordinal ζ .For r ∈M1, S ∈M0 with r ∈ e(1)(S), define U = {x ⊂ S | r ∈ e(1)(x)}.Then, for i ≤ n, we can define an elementary embedding k(i)

: Ult(T(S)i ,M0)/U i≺Mi with k(i)

◦ jT(S)i ,U i = e0,i .Such that the k(i)’s commute with the canonical embeddings: e(i+1)

◦ k(i)= k(i+1)

◦ jT(S),U,i→i+1, for i < n.


M0 · · · Mi Mi+1

...

Ult(T(S)i ,M0)/U i

Ult(T(S)i+1,M0)/U i+1

?jT(S),U

-e(1)

1 -e(i)

i -e(i+1)

i+1

?jT(S),U,i−1→i

?jT(S),U,i→i+1

��

��

��

��

��*

k(i)

��

��

��

��

��

��

��

��*

k(i+1)

Proof. Define k(i)([ f ]U i ) = e0,i ( f )(e(i)(r)). Note that Ult(T(S),M0) ⊂ Vζ = |M0| by the inaccessibility of ζ ,and hence that e0,i ( f ) is defined. It is immediate from the last lemma that k(i) is well-defined and elementary. (k(i+1)

◦

jT(S),U,i→i+1)([ f ]U i ) = k(i+1)([ f ◦ πi+1,i ]U i+1) = (e0,i+1( f ) ◦ πi+1,i )(e(i + 1)(r)) = e0,i+1( f )(e(i+1)e(i)(r)) =

e(i+1)(e0,i ( f )(e(i)(r))) = (e(i+1)◦ k(i))([ f ]U i ). �

One can see that the same kind of lemmata forNµ,λ’s and for Eλ,µ’s are also available, after one defines the multi-argument versions of j . These theorems show that the iteration process described by our notion of iteration sequenceand that described by iterated ultrapowers are compatible with each other. Thus, we can conclude that our notion ofiteration sequence is a quite appropriate generalization of iteration of ultrapowers.

Finally we mention a well-known property of elementary embeddings, essentially due to Kunen [7], which is basedon the following lemma:

Lemma 4.9 (E.g., [4]). For any γ ∈ On, there is an ω-Jonsson function f for γ , i.e., f : ω[γ ] → γ and

f “(ω[x]) = γ for every x ⊂ γ with |x | = |γ |.

Lemma 4.10. Let Ee be a non-trivial iteration sequence through 〈Vαi | i ≤ n〉 with limit αi ’s. Then there is no γ suchthat α0 > γ > crit(Ee), e(1)(γ ) = γ .

Proof. Suppose α0 > γ > crit(Ee) and e(1)(γ ) = γ . Let f be an ω-Jonsson function for γ , Then e(1)( f ) is ω-Jonssonfor e(1)(γ ) = γ . Thus, for every ξ ∈ γ , we have s ∈ ω

[e(1)“γ ] such that e(1)( f )(s) = ξ . Since crit(e(1)) > ω, we havet ∈ ω

[γ ] such that s = e(1)“t = e(1)(t) and so ξ = e(1)( f )(s) = e(1)( f )(e(1)(t)) = e(1)( f (t)) ∈ e(1)“γ . Thereforeγ ⊂ e(1)“γ , while crit(e(1)) ∈ γ \ e(1)“γ , a contradiction. �

5. Observations on multiple hugeness and superstrongness

Recall that a cardinal κ is n-huge (almost n-huge, n-superstrong) iff there is a non-trivial elementary embeddingj : V → M with crit( j) = κ such that jn(κ)M ⊂ M (< jn(κ)M ⊂ M , V jn(κ) ⊂ M , respectively), and that κ issuper-n-huge iff, for any ordinal ξ , there is a witness j of n-hugeness of κ with j (κ) > ξ . Now it is quite natural toask whether these properties can be formulated in terms of iterations.

Lemma 5.1. Let j : V → M be a non-trivial elementary embedding with crit( j) = κ , ζ an ordinal, and λ a cardinal.Then, for n ≥ 1, (1) Vζ ⊂ M (n) iff Vζ ⊂ M, and (2) λ(M (n)) ⊂ M (n) iff λM ⊂ M.

Proof. (1) Since M (n)⊂ M , Vζ ⊂ M (n) obviously implies Vζ ⊂ M . Conversely suppose Vζ ⊂ M . We prove by

induction on n that Vζ ⊂ M (n). By Vζ ⊂ M (n), we have M j (ζ ) ⊂ M (n+1) and so Vζ ⊂ M j (ζ ) ⊂ M (n+1).(2) First, we show for any inner model N that λN ⊂ N iff λOn ⊂ N . Let λOn ⊂ N . For f ∈ λN , fix µ so that

dom( f ) ⊂ Nµ and g : Nµ∼= |Nµ|

N for some g ∈ N . Then g ◦ f ∈ λOn ⊂ N and hence f ∈ N .Then λ(M (n)) ⊂ M (n) implies λOn ⊂ M (n)

⊂ M , which implies λM ⊂ M . Conversely, suppose λM ⊂ M . Weprove λOn ⊂ M (n) by induction on n. Since λOn ⊂ M (n) implies ( j (λ)On)M

⊂ M (n+1), we have (λOn)M⊂ M (n+1).

It remains to show that λOn ⊂ M , which is immediate from λM ⊂ M . �


Thus, we can re-formulate the large cardinal properties mentioned above in the following ways: κ is n-huge(almost n-huge, n-superstrong) iff there is a non-trivial elementary embedding j : V → M with crit( j) = κ

such that jn(κ)(M (n)) ⊂ M (n) (< jn(κ)(M (n)) ⊂ M (n), V jn(κ) ⊂ M (n), respectively). In other words, n-versions ofthese properties are equivalent to 1-versions of the same properties witnessed by n-times iterations of elementaryembeddings, e.g., κ is n-huge iff there is a non-trivial elementary embedding j : V → M whose n-times iteration jn

witnesses the (1-)hugeness of κ . This is the key to obtaining the helices below.Our first strategy is to define n-versions of large cardinal properties according to this observation. However, a

terminology for n-versions of some kind of properties may cause confusion, for example, n-extendible in the usualsense as in [6, Section 23] is not stronger than extendibility. For this reason, we employ the terminology “n-fold”. Ourofficial definition is as follows, although these definitions are not formulated in ZFC directly:

Definition 5.2. Let κ be a cardinal and n ≥ 1 and κ < κ1 < · · · < κn .1. κ is n-fold superstrong with targets κ1, . . . , κn iff there is a non-trivial elementary embedding j : V → M with

crit( j) = κ such that V jn(κ) ⊂ M (n) and j i (κ) = κi for i ≤ n.2. κ is n-fold almost huge with targets κ1, . . . , κn iff there is a non-trivial elementary embedding j : V → M with

crit( j) = κ such that < jn(κ)(M (n)) ⊂ M (n) and j i (κ) = κi for i ≤ n.3. κ is n-fold huge with targets κ1, . . . , κn iff there is a non-trivial elementary embedding j : V → M with crit( j) = κ

such that jn(κ)(M (n)) ⊂ M (n) and j i (κ) = κi for i ≤ n.4. κ is n-fold superstrong, almost huge or huge iff it is n-fold superstrong, almost huge or huge, respectively, with

some targets.5. κ is n-fold superhuge iff, for any γ , there is κ1 > γ such that κ is n-fold huge with the first target κ1.

By the discussions above, n-fold hugeness is just n-hugeness in the sense of [6, Section 24], and so on.Next, let us turn to characterizing these properties in terms of existence of ultrafilters. These characterizations will

guarantee that the definitions above can actually be formulated in ZFC. In [6, Section 24], such a characterization forn-fold hugeness is essentially already mentioned:

Lemma 5.3. κ is n-fold huge with targets κ1, . . . , κn iff there is a κ-normal ultrafilter U of T(P(κn)) such that, forevery i < n, {s ∈ P(κn) | ot(s ∩ κi+1) = κi } ∈ U.

Proof. This is easy from Lemmata 3.4 and 3.5. �

The characterizations of n-fold versions of almost hugeness and superstrongness can be obtained in a similar way,by using the notion of partition space.

Lemma 5.4. κ is n-fold almost huge with targets κ1, . . . , κn iff κ1, . . . , κn are inaccessible and there is a κ-normalultrafilter A of Nκn−1,κn such that

• {s ∈ D(Nκn−1,κn ) | ot(s ∩ κi+1) = κi } ∈ A for i < n − 1 where κ0 = κ and,• for any f ∈ Ult(Nκn−1,κn ) with Im( f ) ⊂ κn−1, there is ν < κn such that {s ∈ D(Nκn−1,κn ) | f (s) < ot(s ∩ ν)} ∈ A.

Proof. Let j : V → M be such that < jn(κ)M ⊂ M with crit( j) = κ , and κi = j i (κ) for i ≤ n. DefineA = {x ∈ B(Nκn−1,κn ) | j“κn ∈ j(x)}, where j is as in Lemma 3.7. Then, by Lemma 3.7, A is κ-normal ultrafilter ofNκn−1,κn . It is easy to see that {s ∈ D(Nκn−1,κn ) | ot(s ∩ κi+1) = κi } ∈ A for i < n − 1.

Let f ∈ Ult(Nκn−1,κn ) be such that Im( f ) ⊂ κn−1, say f (s) = f (s ∩ ν f ) for some map f on ≤κn−1 [ν f ]. Thenj( f )(t) = j ( f )(t ∩ j (ν f )) < j (κn−1) = κn for all t ∈ ≤κn [ j (κn)]. Thus j( f )( j“κn) < ν = ot(kA([res(ν)]A)) forsome ν < κn and so {s ∈ D(Nκn−1,κn ) | f (s) < ot(s ∩ ν)} ∈ A.

Conversely, suppose we have a κ-normal ultrafilter A of Nκn−1,κn as in the statement. By Lemma 3.6,crit( jNκn−1,κn ,A) = κ and <κn (Ult(Nκn−1,κn )/A) ⊂ Ult(Nκn−1,κn )/A. By assumption and Lemma 3.6, for i < n − 1,κi+1 = ot([res(κi+1)]A) = jNκn−1,κn ,A(κi ). It remains to show jNκn−1,κn ,A(κn−1) = κn . If ν < κn then, sinceot(res(ν)(s)) ≤ ot(s) < κn−1 for all s ∈ D(Nκn−1,κn ), ν = ot([res(ν)]A) < jNκn−1,κn ,A(κn−1). If [ f ]A <

jNκn−1,κn ,A(κn−1), then we may assume Im( f ) ⊂ κn−1 and so, by assumption, [ f ]A < ot([res(ν)]A) = ν for someν < κn . �

Lemma 5.5. Let κ be an inaccessible cardinal. κ is n-fold superstrong with targets κ1, . . . , κn iff there is a κ-normalultrafilter E of E = Eκn+1,κn−1+1 such that,


• Vκn ⊂ Ult(E),• for any i ≤ n − 1, {s ∈ D(E) | s(κi+1) = κi } ∈ E.

We omit the proof of this lemma here, since we will prove a more general result (Lemma 7.6) later.In the rest of this section, we investigate the implication relations among these large cardinal properties. By

definition the following implications are clear: n-fold superhugeness implies n-fold hugeness; n-fold hugeness impliesn-fold almost hugeness; n-fold almost hugeness implies n-fold superstrongness. Then it is quite natural to ask whetherthe converses hold. We show that the answer is no. It must be mentioned that the relation between n-fold superhugenessand n-fold hugeness has essentially already been obtained in [1].

Proposition 5.6. Let κ be supercompact and n-fold huge. Then there is a normal ultrafilter U on κ such that{α < κ |α is n-fold huge}, {α < κ | Vκ |= “α is n-fold huge”} ∈ U.

Proof. By Lemma 5.3, we have inaccessibles κ = κ0 < · · · < κn and a normal ultrafilter U ′ on P(κn) such that{s ∈ P(κn) | ot(s ∩ κi+1) = κi } ∈ U ′ for i ≤ n − 1. By supercompactness of κ , we have a non-trivial elementaryembedding j : V → M with crit( j) = κ such that 2κn < j (κ) and 2κn M ⊂ M . Thus P(κn)M

= P(κn) ∈ M ,U ′ ∈ M and (P(κn)κn)M

=P(κn)κn , which imply that the normality of U ′ is absolute between M and V . Thus M |=

“κ is n-fold huge”. Now P(κn)M , U ′ ∈ Mκn+ω ⊂ M j (κ), which implies M |= “V j (κ) |= “κ is n-fold huge” ”. LettingU = {x ∈ P(κ) | κ ∈ j (x)}, we get that {α < κ |α is n-fold huge}, {α < κ | Vκ |= “α is n-fold huge”} ∈ U . �

It is obvious that n-fold superhugeness implies supercompactness. Therefore,

Corollary 5.7. (1) An n-fold superhuge cardinal κ is the κ-th n-fold huge cardinal.(2) The assertion that “there is an n-fold superhuge cardinal” implies

Con(ZFC+ “n-fold huge cardinals form a proper class”).

Lemma 5.8. Let A abbreviate any of “huge”, “almost huge” or “superstrong”. For j : V → M with crit( j) = κ ,if M (n+1)

|= “V jn+1(κ) |= “κ is n-fold A with targets j1(κ), . . . , jn(κ)” ”, then there are a normal ultrafilter U on κ

and T ∈ U such that, for every σ ∈ (n+1)↑T ,

σ(0) is n-fold A with targets σ(1), . . . , σ (n), andVκ |= “σ(0) is n-fold A with targets σ(1), . . . , σ (n)”.

Proof. For σ ∈ <n+1κ , define Sσ by induction on |σ | = length(σ ) as follows: If σ(k) ∈ Sσ�k for all k < |σ |, thendefine Sσ as follows, and otherwise Sσ = κ .

Sσ =

α < κ |M (n−|σ |)|= “V jn−|σ |(κ) |=

“(σ_〈α〉)(0) is n-fold A with targets

(σ_〈α〉)(1), . . . , (σ_

〈α〉)(|σ |), κ, . . . , jn−1−|σ |(κ)” ”

.

Let U = {x ∈ P(κ) | κ ∈ j (x)}. Then, by induction on |σ |, we can prove that Sσ ∈ U . Define S(k)=

4α0∈κ · · · 4αk−1∈κ S〈α0,...,αk−1〉 for k ≤ n. Then, by normality, we have S(k)∈ U . Let T =

⋂k≤n S(k).

For σ ∈ (n+1)↑T , since σ(k) ∈ S(k), we have σ(k) ∈ Sσ�k for all k ≤ n, and hence σ(n) ∈ Sσ�n = {α < κ | Vκ |=

“σ(0) is n-fold A with targets σ(1), . . . , σ (n − 1), α”}. Thus,

Vκ |= “σ(0) is n-fold A with targets σ(1), . . . , σ (n)”.

By Lemmata 5.3, 5.4 and 5.5, we also have that σ(0) is n-fold A with targets σ(1), . . . , σ (n). �

Proposition 5.9. If κ is n-fold huge, then κ is n-fold almost huge and there are a normal ultrafilter U on κ and T ∈ Usuch that, for every σ ∈ (n+1)↑T ,

σ(0) is n-fold almost huge with targets σ(1), . . . , σ (n), andVκ |= “σ(0) is n-fold almost huge with targets σ(1), . . . , σ (n)”.

Thus, in particular, {α < κ |α is n-fold almost huge}, {α < κ | Vκ |= “α is n-fold almost huge”} ∈ U.


Proof. Let j : V → M witness the n-fold hugeness of κ , i.e., crit( j) = κ and jn(κ)M ⊂ M . Then j witnesses then-fold almost hugeness of κ . We show that the hypothesis of the last lemma holds for the case that A is “almost huge”,by Lemma 5.4.

We have inaccessibles κ = κ0 < · · · < κn and a κ-normal ultrafilter A of Nκn−1,κn with conditions mentionedthere, where κi = j i (κ). Since M (n+1) is closed under κn = jn(κ)-sequences by Lemma 5.1,Nκn−1,κn , A ∈ M (n+1)

jn+1(κ).

And the κ-normality and the conditions in Lemma 5.4 are absolute between M (n+1)

jn+1(κ)and V . Thus the hypothesis of

the last lemma holds. �

Corollary 5.10. (1) An n-fold huge cardinal κ is the κ-th n-fold almost huge cardinal.(2) The assertion that “there is an n-fold huge cardinal” implies

Con(ZFC+ “n-fold almost huge cardinals form a proper class”).

We postpone the investigation of the relation between n-fold almost hugeness and n-fold superstrongness, becauseit is natural to consider also n-fold Vopenkaness, n-fold extendibility and n-fold supercompactness, and these actuallylie between n-fold almost hugeness and n-fold supercompactness as shown later.

It is mentioned in [6, Section 26] that (n + 1)-fold superstrongness implies n-fold hugeness. We investigate thisimplication more closely, so that this is strengthened as follows.

Proposition 5.11. Let κ be (n + 1)-fold superstrong. Then κ is n-fold huge and there are a normal ultrafilter U on κ

and T ∈ U such that, for every σ ∈ (n+1)↑T ,

σ(0) is n-fold huge with targets σ(1), . . . , σ (n), andVκ |= “σ(0) is n-fold huge with targets σ(1), . . . , σ (n)”.

Thus, in particular, {α < κ | Vκ |= “α is n-fold superhuge”}, {α < κ |α is n-fold huge} ∈ U.

Proof. Let j : V → M witness the (n + 1)-fold superstrongness of κ , i.e., crit( j) = κ and V jn+1(κ) ⊂ M . We showthat the hypothesis of the Lemma 5.8 holds, by Lemma 5.3. Define U = {x ∈ P(κ) | κ ∈ j (x)}.

Define κi = j i (κ) and U ′ = {x ⊂ P(κn) | j“κn ∈ j (x)}. Since jn+1(κ) = j (κn) is inaccessible, j“κn ∈

V jn+1(κ) ⊂ M and so U ′ is a κ-normal ultrafilter on P(κn). It is straightforward to see {s ∈ P(κn) | ot(s ∩ κi+1) =

κi } ∈ U ′ for i ≤ n − 1. Thus by Lemma 5.3, κ is n-fold huge. Now U ′,P(κn) ∈ V jn+1(κ). To see

V jn+1(κ) |= “P(κn) and U ′witness the n-fold hugeness of κ with targets j1(κ), . . . , jn(κ)”,

it remains to show that the normality of U ′ is absolute between V and V jn+1(κ). However, P(κn)κn ∈ V jn+1(κ) impliesthis absoluteness. Then Lemma 5.8 completes the proof. �

Corollary 5.12. (1) An (n + 1)-fold superstrong cardinal κ is the κ-th n-fold huge cardinal.(2) The assertion that “there is an (n + 1)-fold superstrong cardinal” implies

Con(ZFC+ “n-fold superhuge cardinals form a proper class”).

6. Multiple Vopenkaness

Definition 6.1. A first order structure M is κ-natural iff its universe is Vξ for some limit ordinal ξ < κ and has onlyfinitely many relations including ∈ and only one constant.

For a cardinal κ , a κ-sequence 〈Mα |α < κ〉 is κ-natural iff there is a strictly increasing map f : κ → κ withα < f (α) for all α < κ such that, for every α < κ ,Mα is a κ-natural structure whose domain is V f (α) and which hasα as the interpretation of the constant.

Note that, for any homomorphism e :Mα →Mβ , e(α) = β must hold and so α 6= β implies the well-definednessof crit(e).


Definition 6.2. Let κ be a cardinal and n ≥ 1.A set X is n-fold Vopenka for κ iff, for every κ-natural sequence 〈Mα |α < κ〉, there are α0 < · · · <

αn < κ and an iteration sequence Ee through 〈Mαi | i ≤ n〉 such that crit(Ee) ∈ X . F (n)Vop,κ = {X ∈ P(κ) | κ \

X is not n-fold Vopenka for κ}.κ is n-fold Vopenka iff κ is regular and κ is n-fold Vopenka for κ .

One may notice that our multiple Vopenkaness is quite similar to Kanamori’s Vopenka-n-subtlety [5]. Indeed, if weconsider only first order structures of the form (V f (ζ ); ∈; ζ ), then both notions are exactly the same, by Remark 4.1.For this reason, all the results in this section are analogous to those in [5] with similar proofs. However, for theconvenience of the readers unfamiliar with the notion of iteration sequence, we investigate our multiple Vopenkanesswith rather detailed proofs to demonstrate how well this notion works.

The significant difference between Kanamori’s Vopenka-n-subtlety and our n-fold Vopenkaness is the directapplicability to model theoretical settings. While Vopenka-n-subtlety cannot directly be applied to families of general“proper classes” of first order structures, for n-fold Vopenkaness, the assumption of κ-naturalness of sequences canbe omitted and this assumption is added only for technical convenience. By virtue of our notion of iteration sequence,n-fold Vopenkaness can be seen as a model theoretic property, i.e., κ is n-fold Vopenka iff (Vκ ,P(Vκ)) satisfies thefollowing statement:

for any proper class of first order structures, we can pick a n-sequence consisting of elements of the class andwe have an iteration sequence through the n-sequence.

Since Vopenka’s principle is originally such a model theoretic assertion, our definition of n-fold Vopenkaness seemsto inherit the philosophy of the original motivation of Vopenka’s principle.

Moreover, it must be mentioned that Kanamori’s Vopenka-n-subtlety does not seem to work well in the laterdiscussion for helices, e.g., Theorem 7.2.

Lemma 6.3. For a cardinal κ , (1) X ∈ F (n)Vop,κ iff there is a κ-natural sequence such that, for every α0 < · · · < αn < κ

and every iteration sequence Ee through 〈Mαi | i ≤ n〉, crit(Ee) ∈ X; (2) F (n)Vop,κ is a fine filter; (3) F (n)

Vop,κ is a properfilter iff κ is n-fold Vopenka for κ .

Proof. (1) is obvious from definition. For (3), note that κ is n-fold Vopenka for κ iff ∅ /∈ F (n)Vop,κ .

(2) It is obvious from (1) that F (n)Vop,κ is closed upward. If 〈Mα |α < κ〉, 〈M′α |α < κ〉 witness X, Y ∈ F (n)

Vop,κ ,

then the sequence of M′′α = (Mα,M′α) witnesses X ∩ Y ∈ F (n)Vop,κ , where |M′′α| = |Mα| ∪ |M′α| and where M′′α

has all the relations of Mα and of M′α .Finally, to show the fineness, let γ < κ . Let Mα = (Vα+γ+ω; (α + γ ) \ γ ;α) . Then 〈Mα |α < κ〉 is κ-natural.

We claim that 〈Mα |α < κ〉 witnesses κ \ γ ∈ F (n)Vop,κ in the sense of (1). Suppose not, say α0 < · · · < αn < κ and Ee

is an iteration sequence through 〈Mαi | i ≤ n〉 with crit(Ee) = crit(e(1)) < γ . Since Mα |= “γ = min((α + γ ) \ γ )”,we have e(1)(γ ) = γ , contradicting Lemma 4.10. �

Lemma 6.4. For regular κ , if 〈Mγα |α < κ〉 witnesses Xγ ∈ F (n)

Vop,κ for γ < κ , then 〈Nα |α < κ〉 defined as follows

witnesses 4γ∈κ Xγ ∈ F (n)Vop,κ , where Nα = (Vα′; 〈M

γα | γ < α〉;α) and α′ = sup{rank(Mγ

α )+ ω | γ < α}.

Thus, if κ is regular, F (n)Vop,κ is a normal filter.

Proof. Let Ee be an iteration sequence through 〈Nαi | i ≤ n〉. Then, by κ-naturalness, crit(Ee) ≤ α0 ≤ αi for all i ≤ nand so, for any γ < crit(Ee), Ee restricted suitably is an iteration sequence through 〈Mγ

αi | i ≤ n〉, where note thate0,i (γ ) = γ . Thus crit(Ee) ∈ Xγ for every γ < crit(Ee), i.e., crit(Ee) ∈ 4γ∈κ Xγ . �

Lemma 6.5. Let κ be a regular cardinal and X ∈ F (n)Vop,κ . Then

{α < κ | there is a normal measure on α which contains α ∩ X} ∈ F (n)Vop,κ .

Therefore, if κ is n-fold Vopenka, then κ is κ-Mahlo, and in particular it is inaccessible.


Proof. Let Mα = (Vα+ω; ∈, X ∩ α;α). Let Y ⊂ κ be such that ξ ∈ Y iff ξ < κ and there are α0 < · · · < αn < κ

and an iteration sequence Ee through 〈Mαi | i ≤ n〉 with crit(Ee) = ξ . Then Y ∈ F (n)Vop,κ by Lemma 6.3(1). It suffices to

show that X ∩ Y ⊂ {α < κ | there is a normal measure on α which contains α ∩ X}.Let ξ ∈ X ∩ Y and say ξ = crit(Ee) for an iteration sequence Ee through 〈Mαi | i ≤ n〉 with α0 < · · · < αn < κ .

Define U = {x ∈ P(ξ) | ξ ∈ e0,n(x)}. Then it remains to show ξ ∩ X ∈ U , i.e., ξ ∈ e0,n(ξ ∩ X). Since ξ ≤ α0 wehave e0,n(ξ ∩ X) = e0,n(ξ ∩ (X ∩ α0)) = e0,n(ξ) ∩ (X ∩ αn) 3 ξ . �

Theorem 6.6. If κ is n-fold almost huge witnessed by j : V → M, then the following hold, where U = {x ∈P(κ) | κ ∈ j (x)}:

1. for every κ-natural sequence 〈Mα |α < κ〉, there is Y ∈ U such that, if σ ∈ (n+1)↑Y then there is an iterationsequence Ee through 〈Mσ(i) | i ≤ n〉 with the critical point σ(0).

2. {α < κ |α is n-fold Vopenka} = {α < κ | Vκ |= “α is n-fold Vopenka”} ∈ U.

Proof. Let j : V → M be such that < jn(κ)(M (n)) ⊂ M (n) and crit( j) = κ . Let U = {x ∈ P(κ) | κ ∈ j (x)}.To show 1, let 〈Mα |α < κ〉 be a κ-natural sequence, and define M(i)

α for α < j i (κ) and i ∈ ω by〈M(i)

α |α < j i (κ)〉 = j i (〈Mα |α < κ〉). Note that, for α < j i (κ) and k ≥ i , M(k)α =M(i)

α .We define Sσ ∈ U for σ ∈ ≤nκ and prove the following statement inductively:

for k ≤ n + 1 and σ ∈ kκ , if σ(i) ∈ Sσ�i for every i < k, then

M (n−k+1)|=

“there is an iteration sequence through 〈Mσ(i) | i < k〉_〈M(n−k+1)

j i (κ)| i ≤ n − k〉

with the critical point (σ_〈κ〉)(0)”

.

Define e(i)= j (i) �(|M(i)

j i−1(κ)| ∪ · · · ∪ |M(n)

jn−1(κ)|). Since 〈M(i+1)

j i (κ)| i ≤ n〉 = 〈M(n+1)

j i (κ)| i ≤ n〉 ∈ M (n+1), both of

|M(i)j i−1(κ)

|∪ · · ·∪ |M(n)

jn−1(κ)| and |M(i)

j i (κ)|∪ · · ·∪ |M(n)

jn(κ)| are in M (n+1). SinceM(k)

jk−1(κ)∈ M (k)

jk (κ)for all k ≤ n, the

closure of M (n+1) under < jn(κ)-sequences implies e(i)∈ M (n+1) for all i with 0 < i ≤ n. Obviously, crit(e(1)) =

κ = (〈κ〉)(0). By the absoluteness, to show that M (n+1)|= “Ee is an iteration sequence through 〈M(i+1)

j i (κ)| i ≤ n〉”, it

suffices to show the following:

j (i) �(M(i)j i−1(κ)

∪ · · · ∪M(n)

jn−1(κ)) :

(M(i)j i−1(κ)

, . . . ,M(n)

jn−1(κ); j (i) �(M(i)

j i−1(κ)∪ · · · ∪M(n−1)

jn−2(κ)), . . . , j (n−1) �M(n−1)

jn−2(κ))

≺ (M(i+1)

j i (κ), . . . ,M(n+1)

jn(κ) ; j (i+1) �(M(i+1)

j i (κ)∪ · · · ∪M(n)

jn−1(κ)), . . . , j (n) �M(n)

jn−1(κ)),

This follows from the following, by applying j i−1:

j �(M(1)κ ∪ · · · ∪M

(n−i+1)

jn−i (κ)) :

(M(1)κ , . . . ,M(n−i+1)

jn−i (κ); j �(M(1)

κ ∪ · · · ∪M(n−i)jn−i−1(κ)

), . . . , j (n−i) �M(n−i)jn−i−1(κ)

)

≺ (M(2)j (κ), . . . ,M

(n−i+2)

jn−i+1(κ); j (2) �(M(2)

j (κ) ∪ · · · ∪M(n−i+1)

jn−i (κ)), . . . , j (n−i+1) �M(n−i+1)

jn−i (κ)),

which is obvious from the definition. Thus, we have proved the statement for k = 0.Assume Sσ has been defined for length(σ ) < k and the statement holds for k. For σ ∈ kκ , if σ(i) ∈

Sσ�i for every i < k, then {α < κ |M (n−(k+1)+1)|= “there is an iteration sequence through 〈Mσ(i) | i <

k〉_〈Mα〉_〈M(n−k+1)

j i (κ)| i ≤ n − (k + 1)〉 with the critical point (σ_

〈α〉)(0)”} ∈ U . So, let Sσ be this set for everysuch σ and Sσ = κ for the other σ ’s. Then the statement holds for k + 1.

Now define S(k)= 4ξ0∈κ · · · 4ξk−1∈κ S〈ξ0,...,ξk−1〉 ∈ U for k ≤ n, and Y =

⋂k≤n S(k)

∈ U . Let σ ∈(n+1)↑Y . For k ≤ n, since σ(k) ∈ S(k), we have σ(k) ∈ Sσ�k . Thus, by the statement for k = n + 1,V |= “there is an iteration sequence through 〈Mσ(i) | i ≤ n〉 with crit(Ee) = σ(0)”.

To show 2, it suffices to show M |= “V j (κ) |= “κ is n-fold Vopenka” ”. Let 〈Mα |α < κ〉 be a κ-natural sequencein (V j (κ))

M . Then 〈Mα |α < κ〉 is a κ-natural sequence in the sense of V and so, by 1, there are α0 < · · · < αn < κ


and an iteration sequence Ee through 〈Mαi | i ≤ n〉. What we must show is Ee ∈ (V j (κ))M= V j (κ), which is obvious

from Mαi ∈ Vκ for all i ≤ n. �

Corollary 6.7. (1) An n-fold almost huge cardinal κ is the κ-th n-fold Vopenka cardinal.(2) The assertion that “there is an n-fold almost huge cardinal” implies

Con(ZFC+ “n-fold Vopenka cardinals form a proper class”).

7. Multiple extendibility

Now it is natural to define the n-fold versions of extendibility as follows:

Definition 7.1. Let κ be a cardinal, η an ordinal, F a class and n ≥ 1, κ1 < · · · < κn with κ1 > κ + η.

1. κ is n-fold η-extendible for F with targets κ1, . . . , κn iff there are κ + η = ζ0 < ζ1 < · · · < ζn and an iterationsequence Ee through 〈(Vζi , F ∩ Vζi ) | i ≤ n〉 with crit(Ee) = κ , and e0,i (κ) = κi .

2. κ is n-fold η-extendible for F iff κ is n-fold η-extendible for F with some targets κ1, . . . , κn with κ + η < κ1.3. κ is n-fold extendible for F iff, for every ordinal η, κ is n-fold η-extendible for F .4. κ is n-fold extendible iff it is n-fold extendible for ∅.

By Remark 4.1, n-fold extendibility is equivalent to the existence, for every η, of e : Vζn−1 ≺ Vζn with crit(e) = κ ,ζ0 = κ + η, e(κ) > κ + η and e(ζi ) = ζi+1 for i < n − 1.

Although the next theorem can be proved in the same way as for the 1-fold versions, e.g., [6, 24.15], we give thedetails of the proof because we will need them later.

Theorem 7.2. For regular κ and F ⊂ Vκ , {α < κ | (Vκ , F) |= “α is n-fold extendible for F”} ∈ F (n)Vop,κ .

Proof. Define g : κ → κ as follows, and C = {ρ < κ | g“ρ ⊂ ρ}.

g(ξ) =

{ξ, if (Vκ , F) |= “ξ is n-fold extendible for F”ξ + ηξ , otherwise, where ηξ = min{η ≥ 1 | (Vκ , F) |= “ξ is not n-fold η-extendible for F”}

Then C is closed and unbounded in κ and hence, by Lemma 6.4, C ∈ F (n)Vop,κ . Let 〈Mα |α < κ〉 witness this. For

α < κ , define γα = min{γ ∈ C | γ ≥ sup(Mα ∩ On)}. Then γα < κ by regularity. Define a κ-natural sequenceNα = (Vγα ; ∈, F ∩ Vγα ,Mα, C ∩ γα;α). We prove that, for α0 < · · · < αn < κ and an iteration sequence Ee through〈Nαi | i ≤ n〉, crit(Ee) is n-fold extendible for F in (Vκ , F).

Since being an iteration sequence is absolute, we have Vκ |= “Ee is an iteration sequence of length n”. Suppose(Vκ , F) 6|= “crit(Ee) is n-fold extendible for F” and let ξ = crit(Ee). Since ξ ∈ Vγα0

, ξ < γα0 and so g(ξ) < γα0

because γα0 ∈ C . Therefore e(1) �Vg(ξ) : Vg(ξ) ≺ Ve(1)(g(ξ)) has the critical point ξ because g(ξ) > ξ .Since Ee′ defined by (e′)(i) = ei � (|Mαi−1 | ∪ · · · ∪ |Mαn−1 |) is an iteration sequence through 〈Mαi | i ≤ n〉,

ξ = crit(Ee′) ∈ C . NowNα0 |= “ξ ∈ C∩γα0 ”, which impliesNα1 |= “e(1)(ξ) ∈ C∩γα1”. Thus ξ < e(1)(ξ) ∈ C , whichmeans ξ + ηξ = g(ξ) < e(1)(ξ). Therefore Ee′′ defined by (e′′)(i) = e(i) � (Ve0,i−1(g(ξ)) ∪ · · · ∪ Ve0,n−1(g(ξ))) witnessesthat ξ is n-fold ηξ -extendible for F . Since Ve0,i (g(ξ)), Ee′′ ∈ Vκ , we have (Vκ , F) |= “ξ is n-fold ηξ -extendible for F”,a contradiction. �

Corollary 7.3. The assertion that “there is an n-fold Vopenka cardinal” implies

Con(ZFC+ “n-fold extendible cardinals form a proper class”).

The next theorem also can be proved in the same way as in the 1-fold case.

Theorem 7.4. For a regular cardinal κ , κ is n-fold Vopenka iff, for any F : κ → Vκ , there is α < κ such that(Vκ , F) |= “α is n-fold extendible for F”.

Proof. The direction from left to right is immediate from the last theorem. Conversely, for a κ-natural sequence〈Mξ | ξ < κ〉, let C = {ζ < κ | (∀ξ < ζ)(rank(Mξ ) + ω < ζ)}. Take κ + η = ζ0 ∈ C . By η-extendibility forF , we have an iteration sequence Ee through 〈(Vζi ; ∈, F ∩ Vζi ) | i ≤ n〉 which provides an iteration sequence through〈Mξ | ξ < κ〉, where F = {〈ξ,Mξ 〉 | ξ < κ}, by Me0,i (α) = e0,i (Mα) for i ≤ n. �


Next, we investigate the relation between multiple extendibility and multiple superstrongness. It is known [6, 26.11]that 1-extendibility implies superstrongness. This implication can actually be extended (even in case of the 1-foldversions). To explain the extension, we introduce a new notion:

Definition 7.5. Let F be a class, κ a cardinal, γ an ordinal and n an integer ≥ 0. Let κ = κ0 < κ1 < · · · < κn withκ + γ < κ1. κ enjoys the property n-S-γ for F with targets κ1, . . . , κn iff there exists a non-trivial elementaryembedding j : V → M with crit( j) = κ and j i (κ) = κi for i ≤ n such that V jn(κ+γ ) ⊂ M and thatF ∩ V jn(κ+γ ) = j+(F) ∩ V jn(κ+γ ), where j+(F) =

⋃ξ∈On j (F ∩ Vξ ).

Note that κ is n-fold superstrong iff κ enjoys the property n-S-0 for ∅ and that κ is γ -strong iff κ enjoys property0-S-γ for ∅. Now we check that the property k-S-γ can be formulated in ZFC.

Lemma 7.6. For an ordinal γ , n ≥ 0 and a class F, κ enjoys the property n-S-γ for F with targets κ1, . . . , κn iffthere are cardinals κ + γ = λ0 < · · · < λn with λ0 < κ1 and a κ-normal ultrafilter E of E = Eλ,µ for some λ and µ

with λ ≥ µ and λ > λn, µ > λn−1 such that,

• Vλn ⊂ Ult(E)/E and F ∩ Vλn = jE,E+(F) ∩ Vλn ,

• for any i ≤ n − 1, {s ∈ D(E) | s(κi+1) = κi }, {s ∈ D(E) | s(λi+1) = λi } ∈ E (where κ0 = κ),• {s ∈ D(E) | s(κ + γ ) < κ} ∈ E.

Proof. (⇒) Let j : V → M be such that crit( j) = κ , j i (κ) = κi for i ≤ n, κ1 = j (κ) > κ + η, V jn(κ+γ ) ⊂ Mand F ∩ V jn(κ+γ ) = j+(F) ∩ V jn(κ+γ ). Define λi = j i (κ + γ ) and define r = j−1 � j (λn). Let λ = |Mλn |

M+ 1

and µ = |Vκn | + 1, i.e., E = E|Mλn |M+1,|Vκn |+1. Let E = {x ∈ B(E) | r ∈ j(x)}, where j is as in Lemma 3.9. Then,

by Lemma 3.9, E is a κ-normal ultrafilter of E . By Lemma 3.9, crit(kE ) ≥ |Vλn |M+ 1 > λn and so, by Lemma 3.10,

Vλn ⊂ Ult(E)/E and kE �Vλn = id which implies F ∩ Vλn = jE,E+(F) ∩ Vλn . Now it is easy to check the rest of the

conditions above.(⇐) Suppose that κ + γ = λ0 < · · · < λn and a normal ultrafilter E of E = Eλ,µ satisfy the conditions above. Byassumption, it remains to show that λn ≥ jE,E

n(κ+γ ) and κ+γ < jE,E (κ). The latter, i.e., [ev(κ+γ )]E < jE,E (κ)

is immediate from the assumption. For the former, we claim jEκn ,κn ,E (κi ) = κi+1, jEκn ,κn ,E (λi ) = λi+1 for i ≤ n − 1,which is immediate from the second condition in the statement. �

Theorem 7.7. Let γ be an ordinal such that |Vκ+γ | = κ+γ with cf(κ+γ ) > ω, and n ≥ 1. (1) If κ is n-fold (γ +1)-extendible for F with targets κ1, . . . , κn , then κ enjoys the property n-S-γ for F with the same targets. Conversely,(2) if κ enjoys the property n-S-γ for F with targets κ1, . . . , κn , then κ is n-fold γ -extendible for F with the sametargets.

In particular, if κ is n-fold 1-extendible, then κ is n-fold superstrong.

(2) is easy, because, if j : V → M witnesses the property n-S-γ for F with targets κ1, . . . , κn , then Ee( j; κ + γ ; n)

is an iteration sequence through 〈V j i (κ+γ ), F ∩ V j i (κ+γ ) | i ≤ n〉 with κi = e( j; κ + γ ; n)0,i (κ).In the rest of this section, we prove (1) of this theorem. The idea is to show that the “partial” elementary embedding

Ee is enough to produce a witness of property n-S-γ . Let κ be an n-fold γ + 1-extendible cardinal for F with targetsκ1, . . . , κn and let Ee = 〈e(i)

: Vζn−1 ≺ Vζn | 0 < i ≤ n〉 be an iteration sequence through 〈Vζi , F ∩ Vζi | i ≤ n〉, withcrit(Ee) = κ , ζ0 = κ + γ + 1 and e0,i (κ) = κi for i ≤ n. Define λi = e0,i (κ + γ ). Then Vλi+1 ⊂ Vζi . In particular,λn−1 ∈ Vζn−1 = dom(e(1)). Note also that cf(λi ) > ω for i ≤ n.

Remark 7.7.1. We can take ρ(i): λi + 1 ∼= λi so that e( j)(ρ(i)(λi )) = ρ(i+1)(λi+1) for all j − 1 ≤ i . To avoid heavy

notations, we drop ρ(i)’s and treat λi as if it is in Vλi in the following discussion. In particular, we treat λn−1 + 1 as ifit is in Vζn−1 = dom(e(1)).

Definition 7.7.2. Fix a bijection enua : ω → a between a and ω, for each a ∈ ω[λn + 1]. For y ⊂ a(λn−1 + 1), let

(enua)Ď(y) ⊂ ω(λn−1 + 1) be such that t ∈ y iff t ◦ enua ∈ (enua)Ďy.

Definition 7.7.3. For x ⊂ λn+1(λn−1 + 1) and a ∈ ω[λn + 1] such that x = {s ∈ D(Eλn+1,λn−1+1) | s � a ∈ x � a}

(where x �a = {s �a | s ∈ x}), define e(x, a) = (e(1)((enua)Ď(x �a))) ⊂ ωe(1)(λn−1 + 1) = ω(λn + 1) and

E = {x ∈ B(Eλn+1,λn−1+1) | enua ∈ e(x, a) for some a ∈ ω[λn + 1]}.


It does not depend on the choice of a whether enua ∈ e(x, a) or not. Indeed, for a ⊂ b, we can see thatt ∈ (enub)

Ď(x �b) iff t◦(enub)−1◦enua ∈ (enua)Ď(x �a), from which we can deduce t ∈ e(x, b) = e(1)((enub)

Ď(x �b))

iff t ◦ ((enub)−1◦ enua) ∈ e(1)((enua)Ď(x �a)) = e(x, a), since (enub)

−1◦ enua : ω→ ω is fixed by e(1).

In what follows, E = Eλn+1,λn−1+1. Obviously x 7→ e(x, a) preserves ∅, complements, < κ intersections, < κ

unions (for a fixed a). Thus it preserves the order ⊂ and so E is an E-κ-complete ultrafilter of B(E).

Lemma 7.7.4. Ult(E)/E is well-founded.

Proof. It suffices to show that⋂

n∈ω xn 6= ∅ for 〈xn | n ∈ ω〉 ∈ ω E . Say xn = {s ∈ D(E) | s �a ∈ x ′n} for a fixed a andx ′n ∈

a(λn−1+ 1). By definition, we can take such a with |a| = ω. Then, since enua ∈ e(xn, a) for all n ∈ ω, we haveenua ∈

⋂n∈ω e(xn, a) = e(

⋂n∈ω xn, a) which implies

⋂n∈ω xn 6= ∅. �

Definition 7.7.5. For f ∈ Ult(E), fa:

ω(λn−1 + 1)→ V is such that fa(s ◦ enua) = f (s), where a is a countable

subset of λn + 1 such that f (s1) = f (s2) whenever s1 �a = s1 �a.If additionally [ f ]E ∈ (Ult(E)/E) jE,E (λn−1), we may assume Im( f ) ⊂ Vλn−1 and so, because cf(λn−1) > ω, we

may treat fa

as if it is in Vλn−1+1 by the coding. Thus we can define e( f, a) = e(1)( fa).

Define k : (Ult(E)/E) jE,E (λn−1)→ Vζn by k([ f ]E ) = e( f, a)(enua).

Since fa(t ◦ enub

−1◦ enua) = f

b(t), e( f, a)(enua) does not depend on the choice of a and so k is well defined.

Moreover, by the standard argument, we can prove the following lemma:

Lemma 7.7.6. For F1, . . . , Fm ⊂ Vλn−1 ,

k : (Ult(E/E) jE,E (λn−1), jE,E (F1), . . . , jE,E (Fm)) ≺ (Vλn , e(1)(F1), . . . , e(1)(Fm))

with k ◦ jE,E �Vλn−1 = e(1) �Vλn−1 and k([ev(α)]E ) = α for α < λn−1 + 1.

Indeed, ev(α)a(t) = t (0), where α = enua(0), and k([ev(α)]E ) = e(1)(ev(α)

a)(enua) = enua(0) = α.

To see the elementariness of k, compute that

(Ult(E/E) jE,E (λn−1), jE,E (F1), . . . , jE,E (Fm)) |= ϕ([ Ef ]E , jE,E ( EF))

iff {s ∈ D(E) | (Vλn−1 ,EF) |= ϕ( Ef (s), EF)} ∈ E

iff enua ∈ e({s ∈ D(E) | (Vλn−1 ,EF) |= ϕ( Ef (s), EF)}, a)

= e(1)((enua)Ď{t ∈ a(λn−1 + 1) | (Vλn−1 ,EF) |= ϕ( Ef

a(t ◦ enua), EF)})

= e(1)({t ∈ ω(λn−1 + 1) | (Vλn−1 ,EF) |= ϕ( Ef

a(t), EF)})

= {t ∈ ω(λn + 1) | (Vλn , e(1)( EF)) |= ϕ(e(1)( Efa)(t), e(1)( EF))}

iff (Vλn , e(1)( EF)) |= ϕ(e(1)( Efa)(enua), e(1)( EF))

iff (Vλn , e(1)(F1), . . . , e(1)(Fm)) |= ϕ(k([ Ef ]E ), e(1)( EF)).

Proposition 7.7.7. E is κ-normal.

Proof. The E-κ-completeness and clause 4 of κ-normality are already shown. By using k, it is easy to see {s ∈D(E) | s(κ) < κ} ∈ E and {s ∈ D(E) | s(α) < s(β)} ∈ E for α < β.

Let f ∈ Ult(E) and α < λn + 1 be such that f (s) ∈ s(α) for any s ∈ D(E) with s(α) 6= ∅. Since k([ f ]E ) ∈

k([ev(α)]E ) = α, we have ξ < α such that k([ f ]E ) = ξ = k([ev(ξ)]E ), i.e., {s ∈ D(E) | f (s) = s(ξ)} ∈ E . �

Proposition 7.7.8. E satisfies the conditions in Lemma 7.6.

Proof. Since E is a κ-normal ultrafilter of E , [ev(α)]E = α for all α < λn + 1, and crit( jE,E ) = κ .By assumption, we have G ′ : λ0+1 ∼= Vλ0 in Vλ0+1. (Note that, by the remark above, we can treat λ0 as if λ0 ∈ Vλ0 .)

Applying e0,n−1, we get G = e0,n−1(G ′) : λn−1+1 ∼= Vλn−1 in Vλn−1+1 ⊂ Vζn−1 and so e(1)(G) : λn+1 ∼= Vλn . Thus,for any z ∈ Vλn , we have ξ ∈ λn+1 such that z = e(1)(G)(ξ). Define f ∈ Ult(E) by f (s) = G(s(ξ)) = (G◦ev(ξ))(s).


Then, since f (s) ∈ Vλn−1 , [ f ]E ∈ (Ult(E)/E) jE,E (λn−1) and k([ f ]E ) = e(1)( fa)(enua) = e(1)(G ◦ ev(ξ)

a)(enua) =

(e(1)(G) ◦ e(1)(ev(ξ)a))(enua) = e(1)(G)(k([ev(ξ)]E )) = e(1)(G)(ξ) = z. Thus we have Vλn ⊂ Im(k) and so

Vλn ⊂ Ult(E)/E .Thus k �Vλn is identity. Since e(1)

: (Vζn−1 , F ∩Vζn−1) ≺ (Vζn , F ∩Vζn ), (Vζn−1 , F) |= “∀x ∈ Vλn−1(x ∈ F ↔ x ∈F ∩ Vλn−1)” implies (Vζn , F) |= “∀x ∈ Vλn (x ∈ F ↔ x ∈ e(1)(F ∩ Vλn−1))” and hence F ∩ Vλn = jE,E

+(F)∩ Vλn .By kE ( jE,E (λi )) = e(1)(λi ) = λi+1 = kE ([ev(λi+1)]E ), we have [ev(λi+1)]E = jE,E (λi ) and so {s ∈

D(E) | s(λi+1) = λi } ∈ E , for i ≤ n − 1. Similarly we have {s ∈ D(E) | s(κi+1) = κi } ∈ E , for i ≤ n − 1,and {s ∈ D(E) | s(κ + γ ) < κ} ∈ E . �

Remark 7.1. Note that, in this proof, for any f : κ → κ , if κ + γ = e(1)( f )(κ) then κ + γ ≥ jE,E ( f )(κ) andλn = e0,n(e(1)( f )(κ)) ≥ jE,E

n( jE,E ( f )(κ)). Indeed, jE,E ( f )(κ) ≤ kE ( jE,E ( f )(κ)) = (kE ◦ jE,E )( f )(kE (κ)) =

e(1)( f )(κ) = λ0 and we can show by induction that

( jE,E )k+1( jE,E ( f )(κ)) ≤ (kE ◦ jE,E )(( jE,E )k( jE,E ( f )(κ))) ≤ e(1)(λk) = λk+1.

Therefore, the following two properties of κ are equivalent:

• for any f : κ → κ , there is an iteration sequence Ee through 〈Vζi | i ≤ n〉 with crit(Ee) = κ and e(1)( f )(κ) ≤ ζ0, and• for any f : κ → κ , there is j : V → M with crit( j) = κ with V jn( j ( f )(κ)) ⊂ M .

8. Multiple supercompactness

We have obtained the “helix” in large large cardinals, involving superhugeness, hugeness, almost hugeness,Vopenkaness, extendibility and superstrongness. In the hierarchy of large cardinals, one large cardinal property isusually put between extendibility and superstrongness, namely supercompactness. In this section, we investigatemultiple versions of supercompactness.

In the preceding sections, we have defined multiple versions of large cardinals and have shown the results whose1-fold versions are well-known in the same way as in the 1-fold cases, according to the principle that an n-fold versionis equivalent to the 1-fold version but witnessed by an n-step iteration of an elementary embedding. However, thisprinciple does not work well for supercompactness, for j : V → M satisfies λM ⊂ M iff it satisfies λ(M (n)) ⊂ M (n),by Lemma 5.1(2).

Thus we need to look for an appropriate definition of multiple supercompactness so that the known results anddiscussions for 1-fold supercompactness also hold and work in the multiple case, in the same way. As shown in thissection, the following definition works well to some extent and so seems to be reasonable.

Definition 8.1. Let κ and λ be cardinals and n ≥ 1.

1. κ is n-fold λ-supercompact iff there is a non-trivial elementary embedding j : V → M with crit( j) = κ andλ < j (κ) such that jn−1(λ)(M (n)) ⊂ M (n).

2. κ is n-fold supercompact iff, for every λ > κ , κ is n-fold λ-supercompact.

Note that jn−1(λ)(M (n)) ⊂ M (n) is equivalent to jn−1(λ)M ⊂ M by Lemma 5.1.Let us check that this property can actually be formulated in ZFC.

Lemma 8.2. Let κ and λ > κ be cardinals. κ is n-fold λ-supercompact iff there are λ = λ0 < · · · < λn−1 and aκ-normal ultrafilter U of Pκ ′λn−1 for some κ ′ with λn−1 ≥ κ ′ such that

• {s ∈ Pκ ′λn−1 | ot(s ∩ λi+1) = λi } ∈ U for all i ≤ n − 2 and• {s ∈ Pκ ′λn−1 | ot(s ∩ λ) < κ} ∈ U.

Proof. For a witness j : V → M of the n-fold λ-supercompactness of κ , define λi = j i (λ) for i ≤ n − 1,A = T(P jn−1(κ)λn−1) and U = {x ∈ B(A) | r ∈ j (x)} where r = j“λn−1. Note that, by jn−1(λ)M ⊂ M , r ∈ M andj (D(A)) = j (P jn−1(κ)λn−1) = (P jn(κ)( j (λn−1)))

M3 r , since |r |M = |r | = λn−1 = jn−1(λ) < jn(κ). It is easy to

see the rest of conditions above by Lemma 3.5.The converse is immediate from Lemma 3.4. �


Theorem 8.3. Let κ be an n-fold η-extendible cardinal. If λ+ ω ≤ κ + η then κ is n-fold λ-supercompact.

Proof. Let Ee = 〈e(i): Vζn−1 ≺ Vζn | 0 < i ≤ n〉 be an iteration sequence through 〈Vζi | i ≤ n〉 with crit(Ee) = κ ,

κ + η < e(1)(κ) and ζ0 = κ + η. Define κi = e0,i (κ) = (e(1))i (κ) and λi = e0,i (λ) = (e(1))i (λ) for i ≤ n. Noteλn−1 ∈ dom(e(1)).

Let A = T(Pκn−1λn−1) and U = {x ⊂ Pκn−1λn−1 | e(1)“λn−1 ∈ e(1)(x)}. We show that U is a κ-normal ultrafilterof T(Pκn−1λn−1). Note that, since e(1)(Pκn−1λn−1) = Pe(1)(κn−1)

e(1)(λn−1) 3 e(1)“λn−1, U is a proper filter. Bycrit(e(1)) = κ , U is κ-complete.

Suppose f ∈ Ult(A) with f (s) ∈ s for all s ∈ D(A) \ {∅}. Then e(1)( f )(e(1)“λn−1) ∈ e(1)“λn−1 and soe(1)( f )(e(1)“λn−1) = e(1)(ξ) for some ξ < λn−1, which means {s ∈ Pκn−1λn−1 | f (s) = ξ} ∈ U .

It remains to see {s ∈ Pκn−1λn−1 | ot(s ∩ λi+1) = λi } ∈ U for i < n − 1, and {s ∈ Pκn−1λn−1 | ot(s ∩ λ) < κ} ∈ U(which implies {s ∈ Pκn−1λn−1 | ot(s ∩ κ) < κ} ∈ U ), which are easy. �

Corollary 8.4. If κ is an n-fold extendible cardinal then it is n-fold supercompact.

Proposition 8.5. Let κ be n-fold |Vκ+η|-supercompact with η < κ . Then there are a normal ultrafilter U on κ andT ∈ U such that, for any σ ∈ (n+1)↑T ,

σ(0) is n-fold η-extendible with targets σ(1), . . . , σ (n), andVκ |= “σ(0) is n-fold η-extendible with targets σ(1), . . . , σ (n)”.

Thus, in particular, {α < κ |α is n-fold η-extendible}, {α < κ | Vκ |= “α is n-fold η-extendible”} ∈ U.

Proof. Let j : V → M witness the n-fold |Vκ+η|-supercompactness of κ . We prove the following with a witnessEe( j; κ + η; n) (see Lemma 4.3), which implies the statement in the same way as in the proof of 5.8:

M (n+1)|= “V jn+1(κ) |= “κ is n-fold η-extendible with targets j1(κ), . . . , jn(κ)” ”.

We claim that M (i)j i (κ)+η

∈ M (n+1) for all i and Ee( j; κ + η; n) ∈ M (n+1). By Vκ+η ∈ M (n+1), we have

M (i)j i (κ)+η

∈ M (n+1+i)⊂ M (n+1) for all i . By |M (k−i)

jk−i (κ)+η| ≤ |M (k−i)

jk−i (κ)+η|M(k−i)

= jk−i (|Vκ+η|) ≤ jn−1(|Vκ+η|),

we have j � M (k−i)jk−i (κ)+η

∈ M (n−i+2) and hence j (i) � M (k−1)

jk−1(κ)+η∈ M (n+1), if n − 1 ≥ k ≥ i . This implies

e( j; κ + η; n)(i) = j (i) �(M (i−1)

j i−1(κ)+η∪ · · · ∪ M (n−1)

jn−1(κ)+η) ∈ M (n+1).

By the absoluteness of iteration sequence, M (n+1)|= “κ is n-fold η-extendible with targets j1(κ), · · · jn(κ)”

witnessed by Ee( j; κ + η; n). Moreover, M (i)j i (κ)+η

∈ M (n+1)

jn+1(κ)and Ee ∈ M (n+1)

jn+1(κ), which complete the proof. �

Combining this result and Theorem 7.7, we obtain the following corollary. Moreover, if we define the “super”version of superstrongness, just as one define superhugeness from hugeness, then this result shows that n-fold supercompactness implies n-fold “super”-superstrongness, again by Theorem 7.7, because n-fold “super”-superstrongness is the property n-S-0 with arbitrarily large first targets.

Corollary 8.6. (1) Below an n-fold (2κ -)supercompact cardinal κ , there are κ many n-fold superstrong cardinals.(2) The assertion that “there is an n-fold supercompact cardinal” implies

Con(ZFC+ “n-fold superstrong cardinals form a proper class”).

Whereas all the results above (and those in the preceding sections) have been obtained by the same discussionsas in the cases of their 1-fold versions, several results known for ordinary supercompactness do not seem to carryover to n-fold versions in the same way, e.g., the statement that if λ is supercompact, then κ < λ is supercompactiff Vλ |= “κ is supercompact”. Likewise, several results are generalized in an unexpected way, e.g., Magidor’scharacterization (see [6, 22.10]) of supercompactness:

Proposition 8.7. Let κ be a cardinal. κ is n-fold supercompact iff, for every λ > κ , there are an iterationsequence Ee through 〈Vζi | i ≤ n〉 with λ + ω = ζn and with e(1)(crit(Ee)) > ζ0, e0,n(crit(Ee)) = κ and a family〈e(i): Vηi−1 ≺ Vηi | 0 < i ≤ n〉 such that e(i) is an extension of e(i) and ζn−1 ≤ η0.


Proof. We say that 〈e(i)| 0 < i ≤ n〉 is an elementary sequence through 〈Mηi | i ≤ n〉 iff, for all i ≤ n,

e(i):Mηi−1 ≺Mηi .

(“only if”-part) Assume that κ is n-fold supercompact and λ > κ . Let λ′ > λ be such that |Vλ′ | = λ′ and letj : V → M witness the n-fold λ′-supercompactness of κ . Then, since M (n−1) is closed under jn−1(λ′)-sequences,M (n−1)

|= “ jn−1(λ′) is a fixed by i” implies that jn−1(λ′) is fixed by i and, in particular, |V j i (λ+ω)| ≤ |V jn−1(λ′)| =

jn−1(λ′) for i ≤ n − 1. Thus we have e(i)= j (i) � (V j (i−1)(λ+ω) ∪ · · · ∪ V jn−1(λ+ω)) ∈ M (n) for all i ≤ n and so

M (n)|= “Ee is an iteration sequence through 〈Vζi | i ≤ n〉 with ζn = jn(λ) + ω, e(1)(crit(Ee)) > ζ0 and e0,n(crit(Ee)) =

jn(κ)”, where ζi = j i (λ+ ω).By j � V jn−1(λ+ω) ∈ M (n), we have j (i) � M (i−1)

jn+i−2(λ+ω)∈ M (n+i−1)

⊂ M (n) for 1 ≤ i ≤ n. M (n)|=

“Ee is an elementary sequence through 〈M (i)j (n+i−1)(λ+ω)

| i ≤ n〉 extending Ee”, where e(i)= j (i) � M (i−1)

jn+i−2(λ+ω). Since

M (i)j (n+i−1)(λ+ω)

= M (n)

j (n+i−1)(λ+ω), the elementariness of jn gives us the desired statement.

(“if”-part) Let Ee be an iteration sequence through 〈Vζi | i ≤ n〉 with ζn = λ + ω, and Ee an elementary sequenceextending Ee through 〈Vηi | i ≤ n〉 as above. Let α = crit(e) and let β be such that Vζ0 |= “β is the largest limit ordinal”.Then e0,n(β) = λ. We need e0,n to transfer the witness produced by Ee.

Let βi = e0,i (β) and U = {x ⊂ P(βn−1) | e(1)“βn−1 ∈ e(1)(x)}. Note that, since dom(e(1)) = Vζn−1 ⊃ Vβn−1+ω,e(1)(x) is defined for x ⊂ P(βn−1). One can easily see that U is an α-normal ultrafilter U of T(P(βn−1)) with{s ∈ P(βn−1) | ot(s ∩ β0) < α}, {s ∈ P(βn−1) | ot(s ∩ βi+1) = βi } ∈ U for i < n − 1. Since this statement canbe relativized into dom(e(1)) = Vη0 ⊃ Vζn−1 , applying e0,n , we have a κ-normal ultrafilter U ′ of T(P(λn−1)) whichwitnesses the n-fold λ0-supercompactness of κ in the sense of Vηn and so in the sense of V , where λi = e0,n(βi ) fori ≤ n − 1. Note that λ0 = e0,n(β0) = e0,n(β) = λ. �

Whereas, in the case of the 1-fold version, Magidor’s characterization provides the transcendence of extendibilityover supercompactness, the same discussion does not seem to work in the general case of multiple versions,because the generalization goes in a different direction. The transcendence of multiple extendibility over multiplesupercompactness has not been established.

9. The lower helix

Now we found the definition for multiple supercompactness. Then it is natural to define the multiple versions ofShelahness, Woodinness and strongness in the same way.

Definition 9.1. Let κ be a cardinal, γ an ordinal, F a class and n ≥ 1.

1. κ is n-fold Shelah iff, for each map f : κ → κ , there is a non-trivial elementary embedding j : V → M withcrit( j) = κ such that V jn( f )( jn−1(κ)) ⊂ M (n).

2. κ is n-fold Woodin iff, for each map f : κ → κ , there are α < κ with f “α ⊂ α and a non-trivial elementaryembedding j : V → M with crit( j) = α and j (κ) = κ such that V jn( f )( jn−1(α)) ⊂ M (n).

3. κ is n-fold γ -strong for F iff there is a non-trivial elementary embedding j : V → M with crit( j) = κ andκ + γ < j (κ) such that V jn−1(κ+γ ) ⊂ M (n) and j+(F) ∩ V jn−1(κ+γ ) = F ∩ V jn−1(κ+γ ).

4. κ is n-fold strong iff, for each γ ∈ On, κ is n-fold γ -strong for ∅.

Remark 9.1. κ is n + 1-fold γ -strong for F iff κ enjoys the property n-S-γ for F with some targets. Also, byTheorem 7.7, n + 1-fold strongness and n-fold extendibility are equivalent.

The usual formulation of Woodinness does not have the clause “ j (κ) = κ”. However, this has no effect in thecase of n = 1: By the same argument as in Theorem 9.9 it is known that we may assume j ( f )(α) = f (α).j ( f )(α) = f (α) < κ implies j (κ) = κ (if j is induced by an extender) by Lemma 3.8.

One may wonder why we do not define n-fold Shelahness and Woodinness by jn( f )(κ) and by jn( f )(α),instead of jn( f )( jn−1(κ)) and jn( f )( jn−1(α)). However, e.g., in the case of Shelahness, since crit( j1,n) = j (κ) >

max(κ, j ( f )(κ)), we have jn( f )(κ) = ( j1,n( j ( f )))(κ) = j1,n( j ( f )(κ)) = j ( f )(κ) and so the definition withjn( f )(κ) is equivalent to the 1-fold version.

We must check that these definitions can be formulated in ZFC. Lemma 7.6 tells us that n-fold γ -strongness canbe formulated. Thus we consider Shelahness and Woodinness:


Lemma 9.2. Let κ be a cardinal. If, for every f : κ → κ , there is a measurable cardinal α with f “α ⊂ α, then κ isMahlo.

Proof. If cf(κ) < κ then a cofinal map f : cf(κ)→ κ yields g : κ → κ such that g(0) = cf(κ) and g(1+ ξ) = f (ξ)

for ξ < cf(κ), which has no non-zero α < κ with g“α ⊂ α. Thus κ is regular. Let C ⊂ κ be club and let f : κ → Cenumerate C increasingly. We have a measurable α < κ with f “α ⊂ α and hence α ∈ C . �

Lemma 9.3. Let κ be a cardinal and n ≥ 1.

(1) κ is n-fold Shelah iff, for any strictly increasing map f : κ → κ , there are λ0 < · · · < λn−1 and a κ-normalultrafilter E of E = Eλ,µ for some λ, µ with λ > λn−1 such that• Vλn−1 ⊂ Ult(E)/E,• {s ∈ D(E) | s(λ0) = f (s(κ))} ∈ E,• for any i < n − 1, {s ∈ D(E, E) | s(λi+1) = λi } ∈ E.

(2) κ is n-fold Woodin iff, for any strictly increasing map f : κ → κ , there are α < κ with f “α ⊂ α,β0 < · · · < βn−1 < κ , and an α-normal ultrafilter E of E = Eλ,µ for some λ, µ < κ with λ > βn−1 suchthat• Vβn−1 ⊂ Ult(E)/E,• {s ∈ D(E) | s(β0) = f (s(α))} ∈ E,• for any i < n − 1, {s ∈ D(E, E) | s(βi+1) = βi } ∈ E.

Proof. In both cases, the direction from right to left is immediate from Lemma 3.8. Here, note that, in (2), since κ isinaccessible by Lemma 9.2, jE,E (κ) = κ . Conversely, let κ be n-fold Shelah or n-fold Woodin, and let f : κ → κ bestrictly increasing. Then f (ξ) ≥ ξ for every ξ < κ . Let j : V → M witness the n-fold Shelahness or Woodinness(with respect to f , with crit( j) = α) of κ . Define λi = j i ( j ( f )(κ)) and βi = j i ( j ( f )(α)) for i ≤ n − 1 andλ = |Vλn−1 |

M+ 1, |Vβn−1 |

M+ 1. Let µ = |Vλn−2 | + 1, |Vβn−2 | + 1 if n ≥ 2 and µ = |Vκ |, |Vα| otherwise. In (2), since

j (κ) = κ is inaccessible both in V and M , λ, µ < κ . Then j (µ) ≥ λ ≥ µ. Let E = {x ∈ B(Eλ,µ) | j−1 � j“λ ∈ j(x)},where j is as in Lemma 3.9. Then E is κ-normal or α-normal. By Lemma 3.10, we have Vλn−1 ⊂ Ult(E)/E andVβn−1 ⊂ Ult(E)/E . The rest is straightforward. �

Theorem 9.4. Let κ be n-fold superstrong. Then κ is n-fold Shelah and there is a normal ultrafilter U on κ such that{α < κ |α is n-fold Shelah}, {α < κ | Vκ |= “α is n-fold Shelah”} ∈ U.

Proof. Let j : V → M be such that crit( j) = κ and V jn(κ) ⊂ M . For f : κ → κ , since j ( f )(κ) ∈ j (κ), and sincejn( f )( jn−1(κ)) < jn(κ), we have V jn( f )( jn−1(κ)) ⊂ V jn(κ) ⊂ M . This means that κ is n-fold Shelah.

Let λ < jn(κ) be an inaccessible cardinal with λ ≥ jn( f )( jn−1(κ)) and µ = jn−1(κ). We may assume thatf is strictly increasing and so λ ≥ µ. Let r = j−1 � j (λ + 1) and define E = {x ∈ B(Eλ+1,µ+1) | r ∈ j(x)},where j is as in Lemma 3.9. Then D(Eλ+1,µ+1) =

(λ+1)(µ + 1) ∈ V jn(κ) and so, by the limitness of jn(κ),Eλ+1,µ+1, E ∈ V jn(κ) ⊂ M (n).

By Lemma 3.9, E is a κ-normal ultrafilter of Eλ,µ. By Lemma 3.10 and |Vλ|M= |Vλ| = λ < λ + 1, we have

Vλ ⊂ Ult(Eλ+1,µ+1)/E . Now it is easy to see that (Eλ+1,µ+1, E) witnesses the n-fold Shelahness for f in the senseof Lemma 9.3 and this can be relativized to V jn(κ). Thus V jn(κ) |= “κ is n-fold Shelah”, which is equivalent toM (n)

|= “V jn(κ) |= “κ is n-fold Shelah” ”. Obviously we also have M (n)|= “κ is n-fold Shelah”. �

Corollary 9.5. (1) An n-fold superstrong cardinal κ is the κ-th n-fold Shelah cardinal.(2) The assertion that “there is an n-fold superstrong cardinal” implies

Con(ZFC+ “n-fold Shelah cardinals form a proper class”).

Theorem 9.6. Let κ be n-fold Shelah. Then κ is n-fold Woodin and there is a normal ultrafilter U on κ such that{α < κ |α is n-fold Woodin}, {α < κ | Vκ |= “α is n-fold Woodin”} ∈ U.

This theorem can be proved in a similar way to the proof of the 1-fold version in [10], with a few non-trivialmodifications. We need to be careful of the additional clause “ j (κ) = κ”. Here we omit the proof, since later we willprove theorems (Propositions A.3 and A.5) which imply this theorem immediately.


Corollary 9.7. (1) An n-fold Shelah cardinal κ is the κ-th n-fold Woodin cardinal.(2) The assertion that “there is an n-fold Shelah cardinal” implies

Con(ZFC+ “n-fold Woodin cardinals form a proper class”).

Let us turn to the investigation of Woodinness.

Lemma 9.8. For a cardinal κ , f : κ → κ , F ⊂ Vκ and α < κ , assume (Vκ , f ×F) |= “α is n-fold strong for f ×F”.Then f “α ⊂ α and (Vκ , F) |= “there is a non-trivial elementary embedding j : V → M with crit( j) = α such thatV jn−1( j ( f )(α)) ⊂ M and j+(F) ∩ V jn−1( j ( f )(α)) = F ∩ V jn−1( j ( f )(α))”. In particular,

Vκ |= “there is a witness of the Shelahness of α with respect to f �α”.

Proof. Let us work in (Vκ , f × F). Let jξ : V → M witness the n-fold ((max{ξ, f (ξ)} + ω) − α)-strongness of α

with respect to f × F for ξ ≤ α. In particular, jξ ( f )(ξ) = f (ξ). For ξ < α, jξ ( f (ξ)) = jξ ( f )(ξ) = f (ξ) < jξ (α)

and so f (ξ) < α, i.e., f “α ⊂ α. For ξ = α, since jαn−1(max{α, f (α)} + ω) > jαn−1( f (α)) = jαn−1( jα( f )(α)),we have V( jα)n−1( jα( f )(α)) = V( jα)n−1( f (α)) ⊂ M and j+(F) ∩ V( jα)n−1( jα( f )(α)) = F ∩ V( jα)n−1( jα( f )(α)). �

The next theorem is among the results whose multiple versions are complicated in an unexpected way.

Theorem 9.9. Let κ be a cardinal. Then the following are equivalent:

1. κ is n-fold Woodin.2. for any F ⊂ Vκ , {α < κ | (Vκ , F) |= “α is n-fold strong for F”} is stationary in κ .3. for any F ⊂ Vκ , there is α < κ such that (Vκ , F) |= “α is n-fold strong for F”.4. for any f : κ → κ and any F ⊂ Vκ , there are α < κ with f “α ⊂ α and a non-trivial elementary embedding

j : V → M with crit( j) = α and j (κ) = κ such that V jn−1( j ( f )(α)) ⊂ M and that j (F) ∩ V jn−1( j ( f )(α)) =

F ∩ V jn−1( j ( f )(α)).

Proof. We only need to show that 1 implies 2 and that 3 implies 4.(3 ⇒ 4) By the last lemma, we can obtain the required statement relativized in Vκ , which, by Lemmata 9.3, 3.3, 3.8and 3.9, implies the required statement itself.(1⇒ 2) By Lemma 9.2, κ is Mahlo. Let C be club in κ and define g : κ → κ as follows:

g(ξ) =

{ξ, if (Vκ , F) |= “ξ is n-fold strong for F”ξ + γξ , otherwise, where γξ = min{γ < κ | (Vκ , F) 6|= “ξ is n-fold γ -strong for F”}.

Let f (ξ) ∈ C be an inaccessible cardinal > max(g(ξ), ξ)+ ω.By assumption, we have α < κ with f “α ⊂ α and a non-trivial elementary embedding j : V → M with

crit( j) = α and j (κ) = κ such that V jn( f )( jn−1(α)) ⊂ M . Since α is a limit of C∩α = jn(C)∩α, we have α ∈ jn(C).We claim M (n)

|= “(Vκ , jn(F)) |= “α is n-fold strong for jn(F)” ”.Let ν = jn−1( j (g)(α)) and λ = |Vν+1|

M and µ = |V jn−1(α)+1|. Define E = Eλ,µ and E = {x ∈ B(E) | j−1 �

j (λ) ∈ j(x)} where j is as in Lemma 3.9. Then E, E ∈ V jn( f )( jn−1(α)) ⊂ M (n). Lemma 3.9 shows that E is aκ-normal ultrafilter of E and, by Lemma 3.3, M (n)

|= “E is a κ-normal ultrafilter of E”.Since {h ∈ Ult(E) | Im(h) ⊂ Vν+1} = {h ∈ Ult(E)M(n)

| Im(h) ⊂ Vν+1}, we have jE,E (x) ∩ Vν+1 =

jE,EM(n)

(x)∩Vν+1 for all x . Moreover, by Lemma 3.10, kE �Vν+1 = id and so j (x)∩Vν+1 = jE,EM(n)

(x)∩Vν+1 forall x . In particular, jE,E

M(n)(g)∩ Vν+1 = j (g)∩ Vν+1 and so ( jE,E

M(n))n−1( jE,E

M(n)(g)(α)) = jn−1( j (g)(α)) = ν.

Since Vν ⊂ (Ult(E)/E)ν = (Ult(E)M(n)/E)ν by Lemma 3.10, one can see the following by Lemma 7.6:

M (n)|= “E witnesses the n-fold ( j (g)(α)− α)-strongness of α”.

For every y, since y ∩ Vα = j (y)∩ Vα we have ( jE,EM(n)

)n(y)∩ (Ult(E/E)M(n))( j M(n)

E,E )n(α)= ( jE,E

M(n))n( j (y))∩

(Ult(E/E)M(n))( j M(n)

E,E )n(α)and so

( jE,EM(n)

)n(y) ∩ Vν+1 = ( jE,EM(n)

)n( j (y)) ∩ Vν+1,


by ( jE,EM(n)

)n(α) > ( jE,EM(n)

)n−1( jE,EM(n)

(g)(α)) = ν. By the previous paragraph, we have jn(y) ∩ Vν+1 =

( jE,EM(n)

)n(y) ∩ Vν+1 = ( jE,EM(n)

)n( j (y)) ∩ Vν+1 = ( jE,EM(n)

)( jn(y)) ∩ Vν+1. In particular:

jn(g)( jn−1(α)) = ( jE,EM(n)

( jn(g)))( jn−1(α)) and jn(F) ∩ Vν = ( jE,EM(n)

)( jn(F)) ∩ Vν .

Since ν = jn(g)( jn−1(α)) = ( jE,EM(n)

)n−1( jn(g))(( jE,EM(n)

)n−1(α)) = ( jE,EM(n)

)n−1( jn(g)(α)), and sincejE,E

M(n)(α) jE,E

M(n)(g)(α) = j (g)(α) = jn(g)(α), we have (M (n), jn(F)) |= “E witnesses the n-fold ( jn(g)(α) −

α)-strongness for jn(F) of α”. By the definition of g, this means M (n)|= “(Vκ , jn(F)) |= “α ∈

jn(C) is n-fold strong for jn(F)” ”.Thus {ξ < α | (Vκ , F) |= “ξ ∈ C is n-fold strong for F”} ∈ U , where U = {x ⊂ α |α ∈ jn(x)}. �

Corollary 9.10. The assertion that “there is an n-fold Woodin cardinal” implies

Con(ZFC+ “n-fold strong cardinals form a proper class”).

Note that (n + 1)-fold strongness is equivalent to n-fold extendibility. Moreover, by Theorem 7.7, (n + 1)-fold 1-strongness implies n-fold 1-extendibility, which implies n-fold superstrongness. Regarding the transcendencerelation, (n + 1)-fold ω-strongness implies n-fold ω-extendibility whose transcendence over n-fold 1-extendibilitycan easily be established.

Thus, we have obtained the second helix, involving multiple extendibility, supercompactness, superstrongness,Shelahness, Woodinness and strongness, while the first helix involves large cardinal properties from multiplesuperhugeness to multiple superstrongness. (See the next chart in the next section.) Now it is natural to ask whetheror not these two helices can be unified, in the following way: the (n + 1)-fold strongness implies n-superhugeness.However, Theorem 7.7 answers ‘no’ to this question, because of the transcendence of n-fold superhugeness overn-fold extendibility.

10. Relation between two helices

In the preceding sections, we already have two helices which share some large large cardinal properties and, sincen + 1-fold strongness is equivalent to n-fold extendibility, it cannot be the case that n + 1-fold strongness impliesn-fold superhugeness. Thus, the two helices cannot be unified in the expected way.

......

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n + 1-fold(super-, almost)

hugen + 1-foldVopenka

n + 1-foldextendible

n + 1-foldsupercompact

n + 1-foldsuperstrong

n + 1-foldShelah, Woodin

n + 1-foldstrong

n-fold(super-, almost)

hugen-fold

Vopenkan-fold

extendiblen-fold

supercompactn-fold

superstrongn-fold

Shelah, Woodinn-foldstrong

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On the other hand, as mentioned in the Introduction, it is known that large cardinal properties with consistency-wise implications form an almost linear order. Then it is natural to ask whether or not the two helices can be unified inan unexpected way, or, in other words, whether the large large cardinal properties in the double helix can be arrangedin a linear order.

Since we already know the analogous relations between Vopenkaness and extendibility (Theorem 7.4), and betweenWoodinness and strongness (3 in Theorem 9.9), and since n-fold extendibility is exactly n + 1-fold strongness, it isnatural to consider whether n-fold Vopenkaness is n + 1-fold Woodinness. We show this in a quite strong sense:F (n+1)

Woo,κ = F (n)Vop,κ , where F (n+1)

Woo,κ is the filter for Woodinness defined below, corresponding to the Vopenka filter F (n)Vop,κ :


Definition 10.1. Let κ be a cardinal and n ≥ 1. A set X ⊂ κ is n-fold Woodin for κ iff, for any f : κ → κ

and any F ⊂ Vκ , there are α ∈ X with f “α ⊂ α and j : V → M with crit( j) = α and j (κ) = κ such thatV jn( f )( jn−1(α)) ⊂ M (n) and j (F) ∩ V jn( f )( jn−1(α)) = F ∩ V jn( f )( jn−1(α)).

F (n)Woo,κ = {X ∈ P(κ) | κ \ X is not n-fold Woodin for κ}.

Notice that our definition of Woodin filter F (n)Woo,κ is slightly different from that in [6, Section 26], though in [6] it is

defined only in the case n = 1. Our definition depends on characterization 4 of Woodinness in Theorem 9.9, while thedefinition in [6] depends on characterization 1. As shown below, our definition is easier to deal with than Kanamori’sdefinition.

Let us start by checking some fundamental results about this filter (including the fact that this is a filter).

Lemma 10.2. For a cardinal κ , (1) X ∈ F (n)Woo,κ iff there are f : κ → κ and F ⊂ Vκ such that, for every α < κ

with f “α ⊂ α, if there is a non-trivial elementary embedding j : V → M with crit( j) = α and j (κ) = κ such thatV jn( f )( jn−1(α)) ⊂ M and j (F) ∩ V jn( f )( jn−1(α)) = F ∩ V jn( f )( jn−1(α)), then α ∈ X; (2) F (n)

Woo,κ forms a fine filter; (3)

F (n)Woo,κ forms a proper filter iff κ is n-fold Woodin.

Proof. (1) and (3) are obvious by definition. We can see that F (n)Woo,κ forms a filter, because if ( f, F) witnesses

X ∈ F (n)Vop,κ and if (g, G) witnesses Y ∈ F (n)

Vop,κ , then ( f + g, F × G) witnesses X ∩ Y ∈ F (n)Vop,κ . For β ∈ κ ,

the constant map cβ with value β is such that cβ“α ⊂ α implies α ∈ κ \ β for α 6= 0. Thus κ \ β ∈ F (n)Woo,κ . �

Lemma 10.3. F (n)Woo,κ is a normal filter.

Proof. If ∅ ∈ F (n)Woo,κ , then this is obvious. Thus, assume ∅ /∈ F (n)

Woo,κ , then, by Lemma 9.2, κ is inaccessible.

Let (hγ , Hγ ) witness Xγ ∈ F (n)Woo,κ for γ < κ . We may assume that hγ is non-decreasing. Define g(ξ) =

sup{hγ (ξ) | γ < ξ} + ω ∈ κ and G =⋃

γ<κ{γ } × Hγ . We show that (g, G) witnesses 4γ<κ Xγ ∈ F (n)Woo,κ .

Let α < κ be such that g“α ⊂ α and let j : V → M be such that crit( j) = α, j (κ) = κ , V jn(g)( jn−1(α)) ⊂ M andj (G)∩ V jn(g)( jn−1(α)) = G ∩ V jn(g)( jn−1(α)). Let ν = jn(g)( jn−1(α)). For γ < α, to show α ∈ Xγ , we must show (i)hγ “α ⊂ α and (ii) V jn(hγ )( jn−1(α)) ⊂ M and (iii) j (Hγ ) ∩ V jn(hγ )( jn−1(α)) = Hγ ∩ V jn(hγ )( jn−1(α)).

(i) For ξ < α, hγ (ξ) ≤ hγ (max{ξ, γ } + 1) ≤ g(max{ξ, γ } + 1) < α. (ii) Since hγ (ξ) ≤ g(ξ) for ξ > γ ,j (hγ )(ξ) ≤ j (g)(ξ) for ξ > j (γ ) = γ and so j (hγ )(α) ≤ j (g)(α), which implies jn(hγ )( jn−1(α)) ≤ ν. (iii)j (Hγ ) ∩ Vν = ({γ } × Vκ) ∩ j (G) ∩ Vν = ({γ } × Vκ) ∩ G ∩ Vν = Hγ ∩ Vν , again because j (γ ) = γ . �

Theorem 10.4. For any F ⊂ Vκ , {α < κ | (Vκ , F) |= “α is n-fold strong for F”} ∈ F (n)Woo,κ .

Proof. Define f : κ → κ as follows and define g(ξ) = f (ξ)+ ω.

f (ξ) =

{ξ, if (Vκ , F) |= “ξ is n-fold strong for F”ξ + γξ , otherwise, where γξ = min{γ < κ | (Vκ , F) 6|= “ξ is n-fold γ -strong for F”}.

It suffices to show that, for α < κ with g“α ⊂ α and j : V → M with crit( j) = α and j (κ) = κ , ifV jn−1( j (g)(α)) ⊂ M and j ( f ×F)∩V jn−1( j (g)(α)) = ( f ×F)∩V jn−1( j (g)(α)) then (Vκ , F) |= “α is n-fold strong for F”.

Let α and j be such. Since j (g)(α) > j ( f )(α) = f (α), and since j (α) > j (g)(α) > f (α), α is n-fold ( f (α)−α)-strong for F . By Lemma 9.3, we also have (Vκ , F) |= “α is n-fold ( f (α)− α)-strong for F”. Thus, by the definitionof f , (Vκ , F) |= “α is n-fold strong for F”. �

Theorem 10.5. For any cardinal κ , F (n+1)Woo,κ = F (n)

Vop,κ .

Proof. We may assume ∅ 6∈ F (n+1)Woo,κ or ∅ 6∈ F (n)

Vop,κ . Then κ is inaccessible.

Let X ∈ F (n+1)Woo,κ ; say f : κ → κ and F are the witness. We may assume that f is strictly increasing. By

Theorem 7.2, it suffices to show that, if (Vκ , f × F) |= “α is n-fold extendible for f × F” then α ∈ X . Let g : κ → κ

be such that |Vg(ξ)| = g(ξ) > f (ξ) is inaccessible for ξ < κ . Suppose that α is n-fold (g(α) + 1 − α)-extendible


for f × F within Vκ . By Theorem 7.7, α is n + 1-fold ( f (α) + 1 − α)-strong for f × F within Vκ and so we havej : V → M with crit( j) = α, j (κ) = κ such that V jn( f (α)) ⊂ M and that j ( f ) ∩ V jn( f (α))+1 = f ∩ V jn( f (α))+1,j (F)∩V jn( f (α)) = F∩V jn( f (α)). Thus f (α) = j ( f )(α). For γ < α, since j ( f (γ )) = j ( f )(γ ) = f (γ ) < jn( f (α)),Lemma 4.10 implies f (γ ) < crit( j) = α, i.e., f “α ⊂ α. By the assumption that ( f, F) witnesses X ∈ F (n+1)

Woo,κ , wehave α ∈ X .

Conversely, suppose X ∈ F (n)Vop,κ ; say a κ-natural sequence 〈Mξ | ξ < κ〉 is the witness. Let F = {〈ξ,Mξ 〉 | ξ < κ}

and f (ξ) = sup{ξ + 1, rank(Mζ ) + 1 | ζ ≤ ξ}. We prove that f + 1 and f × F witness X ∈ F (n+1)Woo,κ . Let α and

j : V → M be such that f “α ⊂ α, crit( j) = α, V jn( j ( f )(α)) ⊂ M , j ( f ) ∩ V jn( j ( f )(α))+1 = f ∩ V jn( j ( f )(α))+1and j (F) ∩ V jn( j ( f )(α)) = F ∩ V jn( j ( f )(α)). Then j ( f )(ξ) = f (ξ) for ξ ≤ jn(α) and hence jn( f )(ξ) = f (ξ) forξ ≤ jn(α). For k < n, since rank(M jk (α)) < f ( jk(α)) ≤ f ( jn−1(α)) = jn−1( f (α)) = jn−1( j ( f )(α)), we have〈 jk(α),M jk (α)〉 ∈ F ∩ V jn−1( j ( f )(α)) and 〈 jk+1(α), j (M jk (α))〉 ∈ j (F) ∩ V jn( j ( f )(α)) = F ∩ V jn( j ( f )(α)) whichimply j (M jk (α)) =M jk+1(α). Therefore, Ee defined by e(i)

= j (i) �(|M j i−1(α)| ∪ · · · ∪ |M jn−1(α)|) forms an iterationsequence through 〈M j i (α) | i ≤ n〉. By assumption, α = crit( j) = crit(Ee) ∈ X . �

Corollary 10.6. A cardinal κ is n-fold Vopenka iff it is n + 1-fold Woodin.

This corollary can also be deduced from Theorems 7.4, 7.7 and 9.9 (see Remark 9.1).Then, to solve the “linearity problem”, it remains to decide the relation between n+1-fold Shelahness and variants

of n-fold hugeness. As one can see in the proof of Theorem 9.6, Shelahness is only slightly stronger than Woodinness,and so the (strong) implication from n-fold almost hugeness to n + 1-fold Woodinness can be strengthened to thatfrom n-fold almost hugeness to n + 1-fold Shelahness:

Theorem 10.7. Let κ be n-fold almost huge. Then κ is n+ 1-fold Shelah and there is a normal ultrafilter U on κ suchthat {α < κ |α is n + 1-fold Shelah}, {α < κ | Vκ |= “α is n + 1-fold Shelah”} ∈ U.

Proof. Let j : V → M witness the n-fold almost hugeness of κ and U = {x ⊂ κ | κ ∈ j (x)}.We first prove that F (n)

Vop,κ ⊂ U . For X ∈ F (n)Vop,κ , fix 〈Mα |α < κ〉witnessing this in the sense of Lemma 6.3(1). Let

〈M∗α |α < j (κ)〉 = j (〈Mα |α < κ〉). Then 〈M∗α |α < j (κ)〉 witnesses j (X) ∈ (F (n)Vop, j (κ))

M . By Theorem 6.6(1),{crit(Ee) | there is an iteration sequence Ee through 〈Mαi | i ≤ n〉 for some α0 < · · · < αn < κ} ∈ U , which impliesM |= “there is an iteration sequence Ee through 〈M∗αi

| i ≤ n〉 for some α0 < · · · < αn < j (κ) such that crit(Ee) = κ”.Therefore κ ∈ j (X), i.e., X ∈ U .

Let f : κ → κ . By Theorem 10.4, Lemma 9.8 and Theorem 10.5,

{α < κ | Vκ |= “there is a witness for the n + 1-fold Shelahness of α w.r.t. f �α”} ∈ F (n+1)Woo,κ = F (n)

Vop,κ ⊂ U,

i.e., M |= “V j (κ) |= “there is a witness for the n + 1-fold Shelahness of κ w.r.t. j ( f ) � κ = f ” ”. By M j (κ) = V j (κ)

and by Lemma 9.3(1), there is a κ-normal ultrafilter E ∈ V j (κ) of E = Eλ,µ for some λ, µ < j (κ) such thatV( jE,E )n( jE,E ( f )(κ)) ⊂ Ult(E)/E (in the sense of V ).

Since f is arbitrary, κ is n + 1-fold Shelah. Then, by Lemma 9.3(1), V j (κ) |= “κ is n + 1-fold Shelah”. ThenM |= “V j (κ) |= “κ is n + 1-fold Shelah” ” and hence M |= “κ is n + 1-fold Shelah”, which imply the rest of thestatement. �

Corollary 10.8. (1) An n-fold almost huge cardinal κ is the κ-th n + 1-fold Shelah cardinal.(2) The assertion that “there is an n-fold almost huge cardinal” implies

Con(ZFC+ “n + 1-fold Shelah cardinals form a proper class”).

11. The limit of the double helix

Now it seems natural to consider the limit stages of multiple versions of large large cardinals. According to theprinciple by which we defined multiple versions, it is natural to define ω-fold versions of them by using ω-timesiterations of elementary embeddings. To consider the possibility of this recipe, we first introduce the ω-times iterationof an elementary embedding:

M (ω)= dir.lim.〈M (n), jn,m | n ≤ m ∈ ω〉.


However, the definitions based on ω-times iterations are too strong, as explained below.By the standard argument (e.g., [4, Section 31]), one can prove that M (ω) is well-founded. Let jω denote the

canonical injection from V = M (0) to M (ω). Then jω(κ) = sup{ jn(κ) | n ∈ ω}, since every x ∈ jω(κ) has theform x = jn,ω(α) for some n ∈ ω and α < jn(κ), where crit( jn,ω) = jn(κ). Thus neither ω(M (ω)) ⊂ M (ω) norV jω(κ+1) ⊂ M (ω) is possible, since 〈 jn(κ) | n ∈ ω〉 ∈ V jω(κ)+1 and since M (ω)

|= “ jω(κ) is inaccessible”.The next natural candidate for a definition is to require n-fold versions with a shared witness, e.g., j : V → M

witnesses the ω-fold hugeness (or the ω-fold almost hugeness) of κ = crit( j) iff, for every n ∈ ω, jn(κ)(M (n)) ⊂ M (n)

(or < jn(κ)(M (n)) ⊂ M (n), respectively). Both of them are equivalent to < jω(κ)M ⊂ M , where jω(κ) = sup{ jn(κ) | n ∈ω}. According to this recipe, we introduce the following notions.

Definition 11.1. Let κ be a cardinal.

• κ is ω-fold superstrong (or ω-fold Shelah) iff there exists a non-trivial elementary embedding j : V → M withcrit( j) = κ such that V jω(κ) ⊂ M .• κ is ω-fold Woodin iff, for every f : κ → κ , there exist α < κ with f “α ⊂ α and a non-trivial elementary

embedding j : V → M with crit( j) = α and j (κ) = κ such that V jω(α) ⊂ M .• For an ordinal γ , κ is ω-fold γ -strong iff there exists a non-trivial elementary embedding j : V → M with

crit( j) = κ such that j (κ) > κ + γ and that V jω(κ) ⊂ M .κ is ω-fold strong iff it is ω-fold γ -strong for every γ .

Obviously, an ω-fold strong cardinal is also ω-fold superstrong.Note that the definitions of ω-fold hugeness, almost hugeness and supercompactness according to this recipe are

still too strong, i.e., they yield contradictions. Indeed, < jω(κ)M ⊂ M implies jω(κ)M ⊂ M as follows: for f ∈ jω(κ)M ,since f � jn(κ) ∈ M for n ∈ ω, we have 〈 f � jn(κ) | n ∈ ω〉 ∈ ω M ⊂ M . As is well known, or by Lemma 4.10,jω(κ)M ⊂ M implies a contradiction.

Note also that I2 (see [6, Section 24]) is equivalent to the existence of an ω-fold superstrong (Shelah) cardinal.By our preparations in earlier sections, it is easy to check that we can formalize these properties in ZFC, though thischaracterization is essentially known (e.g., [6, Proposition 24.2]).

Lemma 11.2. (1) For a cardinal κ and 〈κn | n ∈ ω〉 ∈ ωOn, the following are equivalent.

1. There is a non-trivial elementary embedding j : V → M with crit( j) = κ and jn(κ) = κn for n ∈ ω such thatV jω(κ) ⊂ M.

2. there exists a κ-normal ultrafilter A of Nδ,δ with δ = sup{κn | n ∈ ω} such that {s ∈ D(Nδ,δ) | ot(s ∩ κn+1) =

κn} ∈ A for all n ∈ ω and Vδ ⊂ Ult(Nδ,δ)/A.3. There is a non-trivial elementary embedding j : V → M with crit( j) = κ and jn(κ) = κn for n ∈ ω such that

V jω(κ) ⊂ M and that, for every inaccessible ν > jω(κ), j (ν) = ν.

(2) κ is ω-fold Woodin iff κ is regular and {α < κ | Vκ |= “α is ω-fold superstrong”} is stationary in κ .

Proof. (1) By Lemmas 3.7 and 3.10, 1 implies 2, since |Vδ|M= δ ≤ crit(kA). The implication from 2 to 3 is

immediate from Lemma 3.6. (2) is easily deduced from (1). �

The standard argument establishes the transcendence of ω-fold strongness over ω-fold Woodinness:

Proposition 11.3. A strong and ω-fold superstrong cardinal κ is ω-fold Woodin and there is a normal ultrafilter U onκ such that {α < κ |α is ω-fold Woodin}, {α < κ | Vκ |= “α is ω-fold Woodin”} ∈ U.

Corollary 11.4. (1) An ω-fold strong cardinal κ is the κ-th ω-fold Woodin cardinal.(2) The assertion that “there is an ω-fold strong cardinal” implies

Con(ZFC+ “ω-fold Woodin cardinals form a proper class”).

Corollary 11.5. (1) Below an ω-fold Woodin cardinal κ , there are κ many ω-fold superstrong cardinals.(2) The assertion that “there is an ω-fold Woodin cardinal” implies

Con(ZFC+ “ω-fold superstrong cardinals form a proper class”).


Next, let us consider the ω-fold versions of Vopenkaness and extendibility. It is easy to extend the notion of iterationsequence to length ω.

Definition 11.6. For a sequence 〈Mi | i ∈ ω〉 of first order structures, an iteration sequence through 〈Mi | i ∈ ω〉 is asequence Ee = 〈e(n)

:⋃

i≥n−1 |Mi | →⋃

i≥n |Mi | | 0 < n ∈ ω〉 such that

e(n): (Mi−1, · · · ; e(i), · · · ) ≺ (Mi , · · · ; e(i+1), · · · ) for i ∈ ω,

where (Mi ′ , · · · ; e(i ′), · · · ) denotes the ω-sorted first order structure defined similarly to Definition 4.2.

Definition 11.7. Let κ be a cardinal.(1) A set X is ω-fold Vopenka for κ iff, for every κ-natural sequence 〈Mα |α < κ〉, there are an increasing sequence〈αn | n ∈ ω〉 with αn < κ and an iteration sequence Ee through 〈Mαn | n ∈ ω〉 such that crit(Ee) ∈ X .

κ is ω-fold Vopenka iff κ is regular and κ is ω-fold Vopenka for κ .

F (ω)Vop,κ = {X ∈ P(κ) | κ \ X is not ω-fold Vopenka for κ}.

(2) Let F be a class.

1. For an ordinal η, κ is ω-fold η-extendible for F iff there are an increasing sequence 〈ζn | n ∈ ω〉 of ordinals withκ + η = ζ0 and an iteration sequence Ee through 〈(Vζn , Vζn ∩ F) | n ∈ ω〉 with crit(Ee) = κ and e(1)(κ) > κ + η.

2. κ is ω-fold extendible for F iff, for every ordinal η, κ is ω-fold η-extendible for F .3. κ is ω-fold extendible iff κ is ω-fold extendible for ∅.

Here, if we adjoin the direct limit of Ee, then the sequence of structures has length ω + 1. Thus the definition of theω-version above is natural, in which the limit is not involved. This corresponds to the recipe that we do not considerthe direct limit M (ω) but a family of large cardinal properties with one shared witness in Definition 11.1.

By the same proofs as those of Lemmas 6.3 and 6.4, we can conclude that F (ω)Vop,κ is a normal filter on κ . Since an

ω-fold Vopenka cardinal is also n-fold Vopenka, Lemma 6.5 yields that κ is inaccessible.Then, by the same proof as that of Theorem 7.2, we can establish the transcendence of ω-fold Vopenkaness over

ω-fold extendibility.

Theorem 11.8. For regular κ and F ⊂ Vκ , {α < κ | (Vκ , F) |= “α is ω-fold extendible for F”} ∈ F (ω)Vop,κ .

Corollary 11.9. The assertion that “there is an ω-fold Vopenka cardinal” implies

Con(ZFC+ “ω-fold extendible cardinals form a proper class”).

Next, we establish the transcendence of ω-fold superstrongness over ω-fold Vopenkaness.

Theorem 11.10. Let j : V → M witness the ω-fold superstrongness of κ and U = {x ⊂ κ | κ ∈ j (x)}.

1. For a κ-natural sequence 〈Mα |α < κ〉, there exists T ∈ U such that, for every 〈αn | n ∈ ω〉 ∈ ω↑T , there existsan iteration sequence through 〈Mαn | n ∈ ω〉 with the critical point α0.

In particular, an ω-fold superstrong cardinal is ω-fold Vopenka.2. {α < κ |α is ω-fold Vopenka}, {α < κ | Vκ |= “α is ω-fold Vopenka”} ∈ U.

Proof. Let 〈Mα |α < κ〉 be a κ-natural sequence. We prove that there are a sequence 〈Sσ | σ ∈<ω↑κ〉 of elements of

U , and 〈 f (n)σ | σ ∈

<ω↑κ, 0 < n < |σ |〉 such that

1. f (n)σ ⊂ f (n)

τ whenever σ ⊂ τ

2. if σ(n) ∈ Sσ�n for all n < |σ |, then(a) 〈 f (i)

σ | 0 < i < |σ |〉 is an iteration sequence through 〈Mσ(i) | i ≤ |σ | − 1〉(b) there is an iteration sequence Ee through 〈Mσ(i) | i < |σ |〉_〈M(n+1)

jn(κ) | n ∈ ω〉 with crit(Ee) = (σ_〈κ〉)(0) and

with e(n)⊃ f (n)

σ for all 0 < n < |σ |, where 〈M(n)α |α < jn(κ)〉 = jn(〈Mα |α < κ〉).

By 〈M(n+1)jn(κ) | n ≥ 1〉 = j (〈M(n+1)

jn(κ) | n ∈ ω〉) ∈ M and by M(1)κ ∈ V jn(κ) ⊂ M , we have 〈M(n+1)

jn(κ) | n ∈ ω〉 ∈ M .


Note that the statement “there is an iteration sequence Ee through 〈Mσ(i) | i < |σ |〉_〈M(n+1)jn(κ) | n ∈ ω〉, with

e(n)⊃ f (n)

σ for all 0 < n < |σ | and with crit(Ee) = (σ_〈κ〉)(0)” is absolute between M and V , since this is the

ill-foundedness of the tree (FIS(〈Mσ(i) | i < |σ |〉_〈M(n+1)jn(κ) | n ∈ ω〉, Efσ ), <FIS) ∈ M , where

FIS(〈Nn | n ∈ ω〉, Efσ ) =

Ee | Ee is an iteration sequence through 〈Nk | k < n〉 for some n ≥ |σ |with e(k)

⊃ f (k)σ for all 0 < k < |σ | and crit(Ee) = (σ_

〈κ〉)(0)

Ee <FIS Ee′ iff length(Ee) > length(Ee′) and e(i)

⊃ e′(i) for all 0 < i ≤ length(Ee′).

We construct the families and prove the statement inductively. For σ = 〈〉, no f (n)〈〉

is defined and there is an iteration

sequence 〈 j (n) � Xn−1 | 0 < n ∈ ω〉 through 〈M(n+1)jn(κ) | n ∈ ω〉, where Xn =

⋃i≥n |M

(i+1)

j i (κ)|.

Assume the statement for σ ∈ <ωκ . For σ ’s with σ(n) /∈ Sσ�n for some n, define Sσ = κ and define f (n)σ_〈α〉

arbitrarily subject to f (n)σ ⊂ f (n)

σ_〈α〉. Consider σ such that σ(n) ∈ Sσ�n for all n < |σ |. By the absoluteness above,

M |=“there is an iteration sequence Ee through 〈Mσ(i) | i < |σ |〉_〈M(n+1)

jn(κ) | n ∈ ω〉

with e(n)⊃ f (n)

σ for 0 < n < |σ | and crit(Ee) = (σ_〈κ〉)(0)”

.

By j ( fσ ) = fσ , we can easily see Sσ ∈ U , where

Sσ =

α < κ |there is an iteration sequence Ee through 〈Mσ(i) | i < |σ |〉_〈Mα〉

_〈M(n+1)

jn(κ) | n ∈ ω〉

with e(n)⊃ f (n)

σ for 0 < n < |σ | and crit(Ee) = (σ_〈α〉)(0)

.

For α ∈ Sσ , take such an iteration sequence Ee, and define f (n)σ_〈α〉 = f (n)

σ ∪ (e(n) � |Mσ(|σ |−1)|) for n ≤ |σ |. For

α /∈ Sσ , define f (n)σ_〈α〉 arbitrarily.

Define S(n)= 4ξn−1∈κ · · · 4ξ0∈κ S〈ξ0,...,ξn−1〉 ∈ U , and let T =

⋂n∈ω S(n)

∈ U . For g ∈ ω↑T , by g(n) ∈ T ⊂ S(n),we have g(n) ∈ Sg�n for all n ∈ ω. Define e(n)

=⋃

i>n f (n)

g�i :⋃

i≥n−1 |Mg(i)| →⋃

i≥n |Mg(i)|. Then, by definition,Ee is an iteration sequence through 〈Mg(n) | n ∈ ω〉.

Since the Vopenkaness of κ is absolute for every N with Vκ+1 ⊂ N , we have M |= “κ is ω-fold Vopenka” andM |= “V j (κ) |= “κ is ω-fold Vopenka” ”. �

Corollary 11.11. (1) An ω-fold superstrong cardinal is the κ-th ω-fold Vopenka cardinal.(2) The assertion that “there is an ω-fold superstrong cardinal” implies

Con(ZFC+ “ω-fold Vopenka cardinals form a proper class”).

This result suggests that ω-fold superstrongness plays the role of ω-fold almost hugeness, while our ω-fold almosthugeness is contradictory. Indeed, Lemma 11.2 shows that ω-fold superstrongness looks like “ω-fold almost hugeness”in the following sense: the property is equivalent to the existence of a κ-normal ultrafilter A of Nµ, jω(κ) for some µ

with j (µ) ≥ jω(κ) such that V jω(κ) ⊂ Ult(Nµ, jω(κ))/A.

Remark 11.1. The proof of the theorem also shows that (〈〉, T ) Pr(U ) “{α < κ | I3(α, κ)} is cofinal in κ”, wherePr(U ) is the Prikry forcing notion determined by U , where T is as in the proof for Mα = (Vα+1, {α},∈), and whereI3(β, δ) is the statement that “there is e : Vδ ≺ Vδ with crit(e) = β”.

Indeed, letting g : κ → κ be a Prikry sequence with g ∈ ω↑T , since g(n) ∈ Sg�n for all n ∈ ω,〈 f (i)

g�n | i < n〉 is an iteration sequence through 〈Vg(i) | i < n〉 for n ∈ ω. Thus, if one defines e(n)=

⋃i≥n f (n)

g�i

for n ∈ ω, then Ee = 〈e(n)| n ∈ ω〉 is an iteration sequence through 〈Vg(n) | n ∈ ω〉. One can easily check that

e(1)(g(0)) = f (1)〈g(0)〉(g(0)) > g(0). By V [g]κ = Vκ = Vsup{g(n) | n∈ω}, and by Lemma 12.1 below, we have I3(g(n), κ)

for n ∈ ω.

Corollary 11.12. A critical point of a witness of I2 can be forced to be the ω-time target of the critical point of awitness of I3.


Here, the ω-time target of crit(e) of e : Vδ ≺ Vδ is δ = sup{en(crit(e)) | n ∈ ω}. For many large large cardinalproperties, say A and B, between which the implication holds, if κ is of property n-fold A, then κ is the n-time targetof a cardinal of property n-fold B. E.g., an n-fold huge cardinal κ is jn(α) = κ for some j : V → M that witnessesthe n-fold almost hugeness of α. The corollary seems to be an analogy to this phenomenon. Note that, since the ω-timetarget is of cofinality ω, the phrase “can be forced” is inevitable.

12. Relation with Laver’s E-series

In this section, we consider the relation between the limit of our double helix and other large cardinal propertiesabove n-hugeness, namely, I0–I3 and Laver’s E-series.

The relation with I3 is easy, since I3 is the existence of an ω-fold 1-extendible cardinal:

Lemma 12.1. Let κ be a cardinal and η an ordinal. κ is ω-fold η-extendible iff there is a non-trivial elementaryembedding e′ : Vδ ≺ Vδ for some δ > κ with crit(e′) = κ and e′(κ) > κ + η.

Thus, for any n ∈ ω, I3 is equivalent to the existence of an ω-fold n-extendible cardinal.

Proof. Suppose that Ee is an iteration sequence through 〈Vζn | n ∈ ω〉 with κ = crit(Ee) and e(1)(κ) > κ + η. Sincee(n) �κn = id : Vκn ≺ Vκn+1 , where κn = e0,n(κ), we have id : Vκn ≺ Vδ , where δ = sup{κn | n ∈ ω}.

Let e′ = e(1): Vδ → Vδ . If Vδ |= ∃xϕ(e′Ea, x), then, for large enough n, Vκn+1 |= ∃xϕ(e′Ea, x). Thus

Vκn |= ∃xϕ(Ea, x), by e′ �Vκn : Vκn ≺ Vκn+1 , and so Vδ |= ∃xϕ(Ea, x).Conversely, for e′ : Vδ ≺ Vδ with crit(e′) = κ and e′(κ) > κ + η, define e(1)

= e′ and e(n+1)=

⋃ξ<δ e′(e(n) �Vξ )

by recursion. If we let ζn = en(κ + η), then Ee is an iteration sequence through 〈Vζn | n ∈ ω〉 with e(1)(κ) > κ + η.Thus κ is ω-fold η-extendible. �

The transcendence of ω-fold extendibility over ω-fold 1-extendibility can be established in the standard way. Notethat ω-fold extendibility is the weakest among our limit versions.

Corollary 12.2. (1) An ω-fold extendible cardinal κ is the κ-th element of the class of the critical points of allwitnesses of I3.(2) The assertion that “there is an ω-fold extendible cardinal” implies

Con(ZFC+ “the critical point of witnesses of I3 form a proper class”).

Next we deal with I1. However, it must be mentioned that Laver [8] discovered a finer hierarchy:

Definition 12.3. For e : Vβ → Vδ with limit β, δ ∈ On, define e+ : P(Vβ)→ P(Vδ) by e+(X) =⋃

ζ<β e(X ∩ Vζ ).(1) If (e, e+) : (Vβ ,P(Vβ)) → (Vδ,P(Vδ)) is a Σ 1

n -embedding, we say, for brevity, e is a Σ 1n -embedding. (2) For

n ≤ ω, En is the statement that there is a Σ 12n-embedding e : Vδ → Vδ for some δ.

Here, by Lemma 4.10, δ = sup{e0,n(crit(e)) | n ∈ ω}. It is known [8, Theorem 1.3] that I2 is E1 and I1 is Eω. Weshow the transcendence of E2 over ω-fold strongness, the strongest among our limit versions. This transcendence hasbeen established already implicitly in Laver’s paper [8]. However, he emphasized the set of ω-time targets δ of criticalpoints (which form an ω-club set) and did not mention the set of critical points or first targets of them. Therefore, weneed to review Laver’s discussion there closely.

The following fact is proved in [8, Lemma 2.1 and Theorems 2.3, 3.3].

Fact 12.4. Let n ≥ 1. (1) The statement “e : Vδ ≺ Vδ is a Σ 12n−1-embedding” is Π 1

2n in e, and if e : Vδ ≺ Vδ is aΣ 1

2n−1-embedding then it is a Σ 12n-embedding.

(2) Let 〈i e | i ∈ ω〉 be a sequence of Σ 1n -embeddings from Vδ to Vδ with crit(i e) < crit(i+1e) for i ∈ ω. Then the

inverse limit k of 〈i e | i ∈ ω〉, defined as follows, is a Σ 1n -embedding:

k : Vα ≺ Vδ with α = sup{crit(i e) | i ∈ ω}, and k �Vcrit(m e) = (0e ◦ · · · ◦ m−1e)�Vcrit(m e) for m ∈ ω.

Refining [8, Lemma 2.2, Theorem 3.4], we can obtain the following two results:


Lemma 12.5. Let n be odd and n ≥ 3. Let e : Vδ ≺ Vδ be Σ 1n and crit(e) = κ . For A, B ⊂ Vδ and m ∈ ω,crit(e′) < κ |

there are a Σ 1n−2-embedding e′ : Vδ ≺ Vδ and A′ ⊂ Vδ such that

e′i+1(crit(e′)) = ei (κ) for all i ≤ m, e′+(A′) = A and e′+(B) = e+(B)

∈ {x ⊂ κ | κ ∈ e(x)}.

Proof. By (1) of the last fact, the following statement, witnessed by e, A, is Σ 1n :

There are a Σ 1n−2-embedding e′ : Vδ ≺ Vδ and Y ⊂ Vδ such that e′i+1

(crit(e′)) = ei+1(κ) for all i ≤ m,e′+(Y ) = e+(A) and e′+(e+(B)) = e+(e+(B)). �

Proposition 12.6. Let e : Vδ → Vδ be Σ 1n+2 with crit(e) = κ and define U = {x ⊂ κ | x ∈ e(x)}. For C ⊂ Vδ , there

is T ∈ U such that, for every g ∈ ω↑T ,

there is a Σ 1n -embedding k : Vsup Im(g) ≺ Vδ with crit(k) = g(0) such that k(g(0)) = κ, k+(Cg) = C for some

Cg ⊂ Vsup Im(g)

Proof. Define κ j = e j (κ) for j ∈ ω. For m ∈ ω, let Gm be as follows:

Gm =

〈0e, . . . , m−1e〉 |i e : Vδ ≺ Vδ is Σ 1

n such that crit(i e) < crit(i+1e),i e(crit(i e)) = κ and i e(κ j ) = κ j+1 for all i < m − 1, j ∈ ω

.

We prove that there are 〈Sσ | σ ∈<ω↑κ〉, 〈Cσ | σ ∈

<ω↑κ〉 and 〈←−e (σ ) | σ ∈ <ω↑κ〉 such that

• Sσ ∈ U , Cσ ⊂ Vδ for any σ , and, whenever σ ⊂ τ ,←−e (σ ) ⊂←−e (τ ), and

• for any σ ∈ <ω↑κ if σ(k) ∈ Sσ�k for k < |σ |, then ←−e (σ ) ∈ G|σ |, i (e(σ ))+(Cσ�i+1) = Cσ�i andcrit(i (e(σ ))) = σ(i) for i < |σ |.

Let C〈〉 = C ,←−e (〈〉) = 〈〉. Assume that Sσ ∈ U is defined for σ ∈ <n↑κ and that both←−e (σ ) and Cσ are definedand satisfy the statement for all σ ∈ <(n+1)↑κ . For σ ∈ n↑κ , if σ(k) ∈ Sσ�k for all k < n, define

Sσ = {α < κ | there are C ′ ⊂ Vδ and e′ such that←−e (σ )_〈e′〉 ∈ G|σ |+1, (e′)+(C ′) = Cσ and crit(e′) = α}.

By the last lemma, we have Sσ ∈ U . For the other σ ∈ n↑κ , define Sσ = κ .For α ∈ Sσ , choose the witnesses C ′ ⊂ Vδ and e′. Define←−e (σ_

〈α〉) =←−e (σ )_〈e′〉 and Cσ_〈α〉 = C ′.

Then T =⋂

m∈ω4ξ0∈κ · · · 4ξm−1∈κ S〈ξ0,...,ξm−1〉 ∈ U . Now, for any g ∈ ω↑T , we have 〈i e | i ∈ ω〉 such that〈i e | i < m〉 ∈ Gm for m ∈ ω and crit(i e) = g(i). Let k : Vβ → Vδ be the inverse limit of 〈i e | i ∈ ω〉. Then, by (2) of

the fact, k is Σ 1n and β = sup{crit(i e) | i ∈ ω} = sup Im(g) and crit(k) = crit(0e). Since k �Vcrit(1e) =

0e �Vcrit(1e), wehave k(crit(k)) = 0e(crit(0e)) = κ . Let Cg =

⋃m∈ω(Cg�m+1 ∩ Vg(m)). Then

k+(Cg) =⋃m∈ω

(0e ◦ · · · ◦ me)(Cg�m+1 ∩ Vg(m)) =⋃m∈ω

C ∩ V(0e◦···◦m e)(crit(m e)) = C,

since sup{(0e ◦ · · · ◦ me)(crit(me)) |m ∈ ω} = sup{κm |m ∈ ω} = δ. �

Theorem 12.7. Let e : Vδ → Vδ be Σ 12n+2 with crit(e) = κ . Then, letting U = {x ⊂ κ | κ ∈ e(x)}, we have

{α < κ | for all γ < κ , there are β < κ and a witness e′ : Vβ ≺ Vβ of En with crit(e′)= α and e′(α) > γ } ∈ U.

Proof. Let T be as in the statement of the last proposition with C = e. Then T ∈ U .Fix arbitrary α ∈ T and any γ ∈ κ . Then we have g ∈ ω↑T with g(0) = α and g(1) > γ , and so there is a

Σ 12n-embedding k : Vβ → Vδ with crit(k) = α and k(α) = κ , and e′ : Vβ → Vβ such that k+(e′) = e, where


β = sup Im(g) > γ . Since the statement “e : Vδ ≺ Vδ is Σ 12n−1” is Π 1

2n , we have e′ : Vβ ≺ Vβ is Σ 12n−1 and so Σ 1

2n .By k(α) = κ = crit(e), we have crit(e′) = α.

By Lemma 4.10, β = sup{e′0,m(α) |m ∈ ω} > γ and so e′0,m(α) > γ for some m ∈ ω. e′0,m is Σ 12n with

crit(e′0,m) = α. �

This shows that En+1 strongly implies strengthened En by adding “with an arbitrarily large first target”.

Corollary 12.8. E2 implies Con(ZFC+ “ω-fold strong cardinals form a proper class”).

Between I2 and I3, another well known property exists, Martin’s I E originally from [9]. I E asserts that there ise : Vδ ≺ Vδ whose α-th iteration is well-founded for all α ∈ On. We consider the relation between I E and the limitof our helices, i.e., ω-fold Vopenkaness and ω-fold extendibility.

Definition 12.9. I Eω is the statement that there is a non-trivial elementary embedding e : Vδ ≺ Vδ with crit(e) = κ

such that the direct limit of 〈e(n): Vδ ≺ Vδ | n ∈ ω〉 is well founded.

Proposition 12.10. Let e : Vδ ≺ Vδ witness I Eω with crit(e) = κ . Then κ is ω-fold Vopenka and {α < κ | Vκ |=

“α is ω-fold Vopenka”}, {α < κ |α is ω-fold Vopenka} ∈ {x ⊂ κ | κ ∈ e(x)}.

Proof. Let e0,ω : Vδ → N be the direct limit of 〈e(i)| i ∈ ω〉. Fix a κ-natural sequence 〈Mα |α < κ〉.

Let 〈M(n)α |α < κ〉 = en(〈Mα |α < κ〉) and 〈M(ω)

α |α < δ〉 = e0,ω(〈Mα |α < κ〉) ∈ N . Then M(ω)α =M(n)

α

for α < κn . Define

FIS(〈Nα |α < κ〉) = {(Ee, σ ) | σ ∈ <ω↑κ and Ee is an iteration sequence through 〈Nσ(i) | i < |σ |〉},

(Ee, σ ) < (Ee′, σ ′) ⇔ σ ) σ ′ and e′(i) ⊂ e(i) for all 0 < i ≤ |σ ′|.

Then FIS(〈M(ω)α |α < δ〉) ∈ N . By the existence of an iteration sequence through 〈M(ω)

e0,n(κ) | n ∈ ω〉 =

〈M(n+1)e0,n(κ) | n ∈ ω〉 induced by e, FIS(〈M(ω)

α |α < δ〉) is ill-founded in V and so in N . By elementariness of

e0,ω : Vδ → N , FIS(〈Mα |α < κ〉) is ill-founded, i.e., there are g ∈ ω↑κ and an iteration sequence through〈Mg(n) | n ∈ ω〉. The rest can easily be proved. �

Corollary 12.11. (1) The critical point κ of a witness of I Eω is the κ-th ω-fold Vopenka cardinal.(2) I Eω implies Con(ZFC+ “ω-fold Vopenka cardinals form a proper class”).

Acknowledgements

The research for this paper is partially supported by Japan Associates for Mathematical Sciences. The author isvery grateful to Andreas Blass for giving him invaluable suggestions and comments throughout the research. Theauthor also would like to thank Akihiro Kanamori for giving them useful information, Toshimichi Usuba for tellingthe author about some papers referred to in this paper and the anonymous referee for careful reading and pointing outseveral minor mistakes.

Appendix A. Alternative numberings and classification

In this paper, we employ the numbering system of multiple versions of large large cardinals such that the 1-foldversions of the properties coincide with the ordinary (or classical) properties. However, with this numbering, there areseveral unnatural phenomena, e.g., n+1-fold Shelahness is between n-fold almost hugeness and n-fold Vopenkaness,or V jn−1(κ+γ ) ⊂ M seems to be better to be called “n − 1-fold” γ -strongness rather than “n-fold”. Actually, thereseem to be three distinct plausible criteria how to number the properties:


1. the 1-fold versions of large large cardinal properties are ordinary (or classical) ones.

2. the weakest property of the n + 1-th level strongly implies the strongest one of the n-th level.

3. the numbers of powers of j in the definitions coincide with the number of the level of the helix.

Unfortunately, these three criteria define distinct numberings. We employ the first one, while the other criteria arenot satisfied as mentioned above. According to the second criterion, n-th Shelahness, Woodinness, and strongness are(n + 1)-fold versions of them in our official numbering and so the third criterion is also satisfied for these properties,while n − 1 is still referred to in the definition of n-fold λ-supercompactness jn−1(λ)M ⊂ M . According to the thirdcriterion, n-th λ-supercompactness is jn(λ)M ⊂ M , i.e., n + 1-fold supercompactness in our official numbering, andsupercompactness and strongness are not weaker than hugeness and superstrongness but they are so strong that eventheir 0-th versions are meaningful, while n-th supercompactness strongly implies n+1-th superstrongness. Moreover,according to the second or third criterion, n-th superstrongness is weaker than n-th strongness; more precisely, it isjust n-th 0-strongness.

Thus, whichever numbering we choose, there must be unnatural phenomena. To let readers familiar with classicalversions of large large cardinals avoid confusion (especially to prevent “superstrongness” being weaker than“strongness”), we employ the first criterion in this paper. Then, if we allow the helix to be partitioned, to makethe second criterion, the most important feature of the helical structure, be satisfied, we need two helices as in thechart at the beginning of Section 10, i.e., one involving cardinals from superstrongness to superhugeness and the otherinvolving those from strongness to extendibility. This is why we call our helical structure “double helix”.

On the other hand, the third criterion seems to become appropriate when we try to classify many largelarge cardinals, as follows. As mentioned above, n-fold supercompactness jn−1(λ)M ⊂ M and n-fold strongnessV jn−1(κ+γ ) ⊂ M seem to be strengthened versions of (n − 1)-fold hugeness jn−1(κ)M ⊂ M and of (n − 1)-fold superstrongness V jn−1(κ) ⊂ M respectively, and the way to strengthen hugeness to superhugeness is alsoapplicable to almost hugeness and superstrongness. Moreover, as defining Shelahness V j ( f )(κ) ⊂ M and WoodinnessV j ( f )(α) ⊂ M , we can define j ( f )(κ)M ⊂ M and j ( f )(α)M ⊂ M . To express these strengthening operationssystematically, we introduce a new way of naming and numbering the properties:

Definition A.1. Let κ be a cardinal and n ≥ 0.

• κ is n-B-huge, n-B-almost huge or n-B-strong iff there is j : V → M with crit( j) = κ such that jn(κ)M ⊂ M ,that < jn(κ)M ⊂ M or that V jn(κ) ⊂ M respectively.

• κ is n-A-huge, n-A-almost huge or n-A-strong iff, for every γ , there is j : V → M with crit( j) = κ andj (κ) > κ + γ such that jn(κ)M ⊂ M , that < jn(κ)M ⊂ M or that V jn(κ) ⊂ M respectively.

• κ is n-P-huge or n-P-strong iff, for every γ , there is j : V → M with crit( j) = κ and j (κ) > κ + γ such thatjn(κ+γ )M ⊂ M or that V jn(κ+γ ) ⊂ M respectively.

• κ is n-W-huge or n-W-strong iff, for any f : κ → κ , there are α < κ with f “α ⊂ α and j : V → M withcrit( j) = α, j (κ) = κ such that jn( j ( f )(α))M ⊂ M or that V jn( j ( f )(α)) ⊂ M respectively.

• κ is n-S-huge or n-S-strong iff, for any f : κ → κ , there is j : V → M with crit( j) = κ such thatjn( j ( f )(κ))M ⊂ M or that V jn( j ( f )(κ)) ⊂ M respectively.

Note that if we had defined n-P-almost hugeness, n-W-almost hugeness and n-S-almost hugeness in the natural way,then they would be equivalent to n-P-hugeness, n-W-hugeness and n-S-hugeness respectively.

Thus, n-P-hugeness and n-P-strongness are n + 1-fold supercompactness and n + 1-fold strongness respectively.We have not formalized yet the n-S-hugeness and n-W-hugeness in ZFC. Now it is easy:

Proposition A.2. For a cardinal κ and n ≥ 0, there is j : V → M with crit( j) = κ such that jn( j ( f )(κ))M ⊂ M iffthere are λ0 < · · · < λn and a κ-normal ultrafilter U of T(P(λn)) such that {s ∈ P(λn) | f (ot(s ∩ κ)) = λ0} ∈ Uand that {s ∈ P(λn) | ot(s ∩ λi+1) = λi } ∈ U for all i ≤ n − 1.


· · · n + 2-S-strong

n + 1-S-huge

n + 1-S-strong

n-S-huge · · ·

· · · n + 2-W-strong

n + 1-W-huge

n + 1-W-strong

n-W-huge · · ·

· · · n + 2-P-strong

n + 1-P-huge

n + 1-P-strong

n-P-huge · · ·

· · · n + 2-A-strong

n + 1-A-huge

n + 1-A-almost huge

n + 1-A-strong

n-A-huge · · ·

· · · n + 2-B-strong

n + 1-B-huge

n + 1-B-almost huge

n + 1-B-strong

n-B-huge · · ·

p p p-

��+

p p p p p-

��+

��+

��+

��

��

��7

p p p-

��+

p p p p p-

��+

? ? ? ? ?��3

��3

��

��

��

��

��

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Then, the implications among these properties are as above, where the dotted arrows mean that the strong implication(i.e., the negation of the reverse consistency-wise implication,) has not been established.

Propositions 5.9 and 8.5 show the implications from n-B-hugeness to n-A-almost hugeness and from n-P-hugenessto n + 1-A-strongness. What we have not proved are the implications among n + 1-S-strongness, n-S-hugeness andn + 1-W-strongness and the equivalence between n + 1-W-strongness and n-W-hugeness.

Proposition A.3. Let n ≥ 0. Let j : V → M be such that crit( j) = κ and V jn+1( j ( f )(κ))+1 ⊂ M. Then we have

j ′ : V → N such that crit( j ′) = κ and j ′n( j ′( f )(κ))N ⊂ N.Thus, (1) n + 1-S-strongness implies n-S-hugeness and (2) n + 1-W-strongness implies n-W-hugeness.

Proof. If V jn+1( j ( f )(κ))+1 ⊂ M , then j“ jn( j ( f )(κ)) ∈ M . Let λi = j i ( j ( f )(κ)) for i ≤ n. Then U = {x ⊂P(λn) | j“λn ∈ j (x)} is a κ-normal ultrafilter of T(P(λn)) witnessing n-S-hugeness with respect to f . �

Corollary A.4. An n + 1-S-strong cardinal is n-S-huge.

Proposition A.5. Let n ≥ 0. Let f, g : κ → κ be such that g(ξ) = 2 f (ξ) and f (ξ) is inaccessible > ξ + 1 for allξ < κ . If there is j : V → M with crit( j) = κ such that jn( j (g)(κ))M ⊂ M, then we have α < κ with f “α ⊂ α andk : V → N with crit(k) = α such that Vkn+1(k( f )(α)) ⊂ N and k(ν) = ν for inaccessible ν ≥ κ . Moreover, thesestatements can be relativized into Vκ .

(1) Thus, n-S-hugeness implies n+1-W-strongness. Then obviously, if κ is n-S-huge then there is a normal ultrafilterU on κ such that {α < κ |α is n + 1-W-strong}, {α < κ | Vκ |= “α is n + 1-W-strong”} ∈ U.

(2) Moreover, n-W-hugeness implies n + 1-W-strongness.

Proof. By jn( j (g)(κ))M ⊂ M , we have P( jn( j ( f )(κ)))M(n+1)= P( jn( j ( f )(κ))). Therefore jn( j (g)(κ)) =

(2 jn( j ( f )(κ)))M(n+1)≥ 2 jn( j ( f )(κ)). Since M (n+1)

|= “ jn( j ( f )(κ)) is inaccessible” implies the inaccessibilityof jn( j ( f )(κ)), we have |V jn( j ( f )(κ))|

V= jn( j ( f )(κ)) and so |V jn( j ( f )(κ))+1|

V≤ jn( j (g)(κ)). Thus Ee =

Ee( j; j ( f )(κ)+1; n+1) ∈ M (n+1), which is an iteration sequence through 〈M (i)j i ( j ( f )(κ))+1 | i ≤ n+1〉 by Lemma 4.3.

jn+1( f )�κ = f ∈ Vκ+1 ⊂ dom(e(1)) implies j ( f )(κ) = e(1)( f )(κ) = e(1)( jn+1( f )�κ)(κ) and

M (n+1)|= “V jn+1(κ) |=

“there is an iteration sequence Ee through 〈Ve0,i (e(1)( jn+1( f )�κ)(κ))+1 | i ≤ n + 1〉

with crit(Ee) = κ and jn+1( f )”κ ⊂ κ” ”.

Therefore we have S = {α < κ | Vκ |= “there is an iteration sequence Ee through 〈Ve0,i (e(1)( f �α)(α))+1 | i ≤ n + 1〉such that crit(Ee) = α and f “α ⊂ α”} ∈ U ′, where U ′ = {x ⊂ κ | κ ∈ jn+1(x)}.


Thus, by Remark 7.1, for each α ∈ S, we have k : V → N with crit(k) = α such that Vkn+1(k( f )(α)) ⊂ N , that isinduced by a κ-normal ultrafilter lying in Vκ . Then, by Lemma 3.8, k(ν) = ν for all inaccessible ν ≥ κ .

(1) Thus, if κ is n-S-huge then it is n + 1-W-strong. By Lemma 9.3, if Vκ+1 ⊂ M ′, then the n + 1-W-strongnessof κ is absolute between V and M ′. Now, by n-S-hugeness with respect to ξ 7→ 2|ξ |, we have j ′ : V → M ′ withcrit( j ′) = κ such that V j ′n(κ+1) ⊂ M ′. Then M ′ |= “V j ′(κ) |= “κ is n + 1-W-strong” ”, which yields the rest of thestatements.

(2) For f : κ → {α < κ |α is inaccessible}, define g(ξ) = 2 f (ξ) for ξ < κ . If g“α ⊂ α, crit( j) = α andjn( j (g))(α)M ⊂ M , then, by applying the statement to α, f � α, g � α, we have β < α(< κ) with f “β ⊂ β andj ′ : V → N with crit( j ′) = β such that V j ′n+1( j ′( f )(β))

⊂ N with j ′(κ) = κ . �

Corollary A.6. (1) An n-S-huge cardinal κ is the κ-th n + 1-W-strong cardinal.(2) The assertion that “there is an n-S-huge cardinal” implies

Con(ZFC+ “n + 1-W-strong cardinals form a proper class”).

(2)’s of the two propositions above imply the following:

Corollary A.7. A cardinal is n + 1-W-strong iff it is n-W-huge.

The charts for ω-th and 0-th levels are as follows, where ω-versions are defined similarly to Definition 11.1:

ω-S-huge

ω-S-strong

ω-W-huge

ω-W-strong

ω-P-huge 0 = 1 ω-P-

strong I 2

ω-A-huge

ω-A-almost huge

ω-A-strong

ω-B-huge

ω-B-almost huge

ω-B-strong

QQQs-

6

1-S-strong

0-S-huge

0-S-strong

1-W-strong

0-W-huge

0-W-strong

1-P-strong

0-P-huge

0-P-strong

1-A-strong

0-A-huge

0-A-almost huge

0-A-strong

1-B-strong measurable 0-B-

huge0-B-

almost huge0-B-

strong

p p p p p p p p p p p p p p p p p p p p-��9

?

��9?

-��9

��

��

?

?

��:

?��*

Moreover, if we define B-En , W-En and P-En similarly, then we also have a sequence as follows:

B-Eω = I1 · · · B-En+1 P-En W-En B-En · · ·- - - - - --

Further problems

• Are “there is an n-fold extendible cardinal” and “there is an n-fold supercompact cardinal” equiconsistent, forn ≥ 2?• Are “there is an n + 1-S-strong cardinal” and “there is an n-S-huge cardinal” equiconsistent?

Chart of large cardinals

By our investigations in this paper, the hierarchy of large cardinal properties has become as in the followingchart. All arrows denote relative consistency (consistency-wise) implications, most of which also denote logicalimplications.


0 = 1 I0 = L I1 = Eω · · · En · · · E2

ω-foldWoodin

ω-foldstrong

ω-foldShelah

I2 = E1ω-fold

superstrong

I Eω I E

ω-foldVopenka

ω-foldextendible I3

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n + 1-foldsuperhuge

n + 1-foldhuge

n + 1-foldalmost huge

n + 2-foldShelah

n + 1-foldVopenka

n + 2-foldWoodin

n + 1-foldextendible

n + 2-foldstrong

n + 1-foldsupercompact

n + 1-foldsuperstrong

n-foldsuperhuge

n-foldhuge

n-foldalmost huge

n + 1-foldShelah

n-foldVopenka

n + 1-foldWoodin

n-foldextendible

n + 1-foldstrong

n-foldsupercompact

n-foldsuperstrong

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superhuge huge almost huge 2-foldShelah Vopenka 2-fold

Woodin extendible 2-foldstrong supercompact superstrong

Shelah Woodin strong

remarkable ω-Erdos 0# exists(∀a ∈ ωω)

a# existsω1-Erdos almost

Ramsey Ramsey ineffablyRamsey measurable 0Ď exists

totallyineffable

· · · n + 1-subtle n-ineffable n-almostineffable n-subtle · · · ineffable almost

ineffable subtle

ZFC inaccessible hyperinaccessible

Σ1-reflecting Mahlo hyperMahlo

weaklycompact

Πmn -

indescribabletotally

indescribable unfoldable

- - - - - -��9��9

�

��

��9

�

- - XXXXXXXXXz

?- - - - - - -

?

?

�

- - - - - - -

?

? - - - - - - -

?

?

�

- -XXXXXXXz

?

� � � � � � � � �

- - - - - - - - -

?� � � � � � � � �

For the implications between large cardinal properties below our double helix, one can refer to [6] and [4] where α-Erdos is denoted by κ → (α)<ω, except remarkable (see [12]) unfoldable (see [14]), Σ1-reflecting (see [3]) cardinals,and ineffability-subtlety “helix” (see [2]). For those above our double helix (i.e., E-series) and L , one can refer to [8].

References

[1] J. Barbanel, C. Diprisco, I. Tan, Many-times huge and superhuge cardinals, J. Symbolic Logic 49 (1984) 112–123.[2] H. Friedman, Subtle cardinals and linear orderings, Ann. Pure Appl. Logic 107 (2001) 1–34.[3] M. Goldstern, S. Shelah, The bounded proper forcing axiom, J. Symbolic Logic 60 (1995) 58–73.[4] T. Jech, Set Theory, 2nd ed., Springer-Verlag, Berlin, 1997.[5] A. Kanamori, On Vopenka’s and related principles, in: Logic Colloquium’77, North-Holland, Amsterdam, New York, 1978, pp. 145–153.[6] A. Kanamori, The Higher Infinite — Large Cardinals in Set Theory from their Beginning, Springer-Verlag, Berlin, 1994.[7] K. Kunen, Elementary embeddings and infinitary combinatorics, J. Symbolic Logic 36 (1971) 407–413.[8] R. Laver, Implications between strong large cardinal axioms, Ann. Pure Appl. Logic 90 (1997) 79–90.[9] D. Martin, Infinite games, in: Proc. ICM, Helsinki, 1978, Acad. Sci. Fennica, Helsinki, 1980, pp. 269–273.

[10] D. Martin, J. Steel, A proof of projective determinacy, J. Amer. Math. Soc. 2 (1989) 71–125.[11] K. Sato, Iteration theory for ultrapowers (in preparation).[12] R. Schindler, Proper forcing and remarkable cardinals II, J. Symbolic Logic 66 (2001) 1481–1492.[13] R. Solovay, W. Reinhardt, A. Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978) 73–116.[14] A. Villaveces, Chains of end elementary extensions of models of set theory, J. Symbolic Logic 63 (1998) 1116–1136.

double helix in large large cardinals and iteration of elementary embeddings

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