aa r X i v : . [ m a t h . L O ] J un Even ordinals and the Kunen inconsistency ∗ Gabriel GoldbergEvans HallUniversity DriveBerkeley, CA 94720June 3, 2020
Abstract
This paper contributes to the theory of large cardinals beyond the Kuneninconsistency, or choiceless large cardinal axioms, in the context where theAxiom of Choice is not assumed. The first part of the paper investigatesa periodicity phenomenon: assuming choiceless large cardinal axioms, theproperties of the cumulative hierarchy turn out to alternate between even andodd ranks. The second part of the paper explores the structure of ultrafiltersunder choiceless large cardinal axioms, exploiting the fact that these axiomsimply a weak form of the author’s Ultrapower Axiom [1]. The third andfinal part of the paper examines the consistency strength of choiceless largecardinals, including a proof that assuming DC, the existence of an elementaryembedding j : V λ +3 → V λ +3 implies the consistency of ZFC + I . By a recentresult of Schlutzenberg [2], an elementary embedding from V λ +2 to V λ +2 doesnot suffice. Assuming the Axiom of Choice, the large cardinal hierarchy comes to an abrupt haltin the vicinity of an ω -huge cardinal. This is the content of Kunen’s InconsistencyTheorem. The anonymous referee of Kunen’s 1968 paper [3] raised the questionof whether this theorem can be proved without appealing to the Axiom of Choice.This question remains unanswered. If the answer is no, then dropping the Axiom ofChoice, a choiceless large cardinal hierarchy extends unimpeded beyond the Kunenbarrier. The consistency of these large cardinals beyond choice would raise profoundphilosophical problems, arguably undermining the status of ZFC as a foundationfor all of mathematics. (These problems will not be discussed further here.)Of course, G¨odel’s Incompleteness Theorem precludes a definitive positive an-swer to the question of the consistency of any large cardinal axiom, choiceless or ∗ This research was supported by NSF Grant DMS 1902884. The author thanks Peter Koellnerand Farmer Schlutzenberg for their comments on the paper. refute the choiceless large cardinalsin ZF. Many partial results towards this appear in Woodin’s
Suitable ExtenderModels II ; for example, [4, Section 7] and [5, Section 5]. In the other direction, thetheory of large cardinals just below the Kunen inconsistency has been developedquite extensively: for example, in [5] and [6]. The theory of choiceless large cardinalsfar beyond the Kunen inconsistency, especially Berkeley cardinals, is developed in[7] and [8]. Following Schlutzenberg [9], we take up the theory of choiceless largecardinals right at the level of the principle that Kunen refuted in ZFC. In particular,we will be concerned with the structure of nontrivial elementary embeddings from V λ + n to V λ + n where λ is a limit ordinal and n is a natural number. One of thegeneral themes of this work is that, assuming choiceless large cardinal axioms, thestructure of V λ +2 n is very different from that of V λ +2 n +1 .The underlying phenomenon here involves the definability properties of rank-into-rank embeddings, which is the subject of Section 3.2. An ordinal α is said to be even if α = λ + 2 n for some limit ordinal λ and some natural number n ; otherwise, α is odd. Theorem 3.3.
Suppose ǫ is an even ordinal.(1) No nontrivial elementary embedding from V ǫ to V ǫ is definable over V ǫ .(2) Every elementary embedding V ǫ +1 to V ǫ +1 is definable over V ǫ +1 . This theorem was the catalyst for most of this research. It was independentlydiscovered by Schlutzenberg, and is treated in greater detail in the joint paper [10].We will use the following notation:
Definition 1.1.
Suppose M and N are transitive classes and j : M → N is anelementary embedding. Then the critical point of j , denoted crit( j ), is the leastordinal moved by j . The critical supremum of j , denoted κ ω ( j ), is the least ordinalabove crit( j ) that is fixed by j .Most of the study of rank-to-rank embeddings has focused on embeddings j either from V κ ω ( j ) to V κ ω ( j ) or from V κ ω ( j )+1 to V κ ω ( j )+1 . The reason, of course,is that assuming the Axiom of Choice, these are the only rank-to-rank embeddingsthere are. (This well-known fact follows from the proof of Kunen’s theorem.) Part ofthe purpose of this paper is to use Theorem 3.3 to extend this theory to embeddingsof V ǫ and V ǫ +1 where ǫ is an arbitrary even ordinal.The mysterious analogy between the structure of the inner model L ( R ) assum-ing AD L ( R ) and that of L ( V λ +1 ) under the axiom I motivates much of the theory2f L ( V λ +1 ) developed in [5]. In Section 4, we attempt to develop a similar analogybetween the structure of arbitrary subsets of V ǫ +1 assuming that there is an ele-mentary embedding from V ǫ +2 to V ǫ +2 , and the structure of subsets of R assumingfull AD.Our main focus in Section 4 is the following sequence of cardinals: Definition 1.2.
We denote by θ α the supremum of all ordinals that are the sur-jective image of V β for some β < α .The problem of determining the structure of the cardinals θ α is a choiceless ana-log of the (generalized) Continuum Problem. Note that for any limit ordinal λ , θ λ is a strong limit cardinal and θ λ +1 = ( θ λ ) + . We conjecture that this phenomenongeneralizes periodically: Conjecture 4.1.
Suppose ǫ is an even ordinal and there is an elementary embeddingfrom V ǫ +1 to V ǫ +1 . Then θ ǫ is a strong limit cardinal and θ ǫ +1 = ( θ ǫ ) + . Under the Axiom of Determinacy, θ ω = ω is a strong limit cardinal, θ ω +1 = ω , θ ω +2 = Θ is a strong limit cardinal, and θ ω +3 = Θ + .In addition to this numerology, various partial results of Section 4 suggest thatConjecture 4.1 holds, or at least that θ ǫ is relatively large and θ ǫ +1 is relativelysmall. For example: Theorem 4.3.
Suppose ǫ is an even ordinal. Suppose j : V ǫ +2 → V ǫ +2 . Then thereis no surjection from P (( θ ǫ +1 ) + λ ) onto θ ǫ +2 where λ = κ ω ( j ) . Theorem 4.2.
Suppose ǫ is an even ordinal. Suppose j : V ǫ +3 → V ǫ +3 is anelementary embedding with critical point κ . Then the interval ( θ ǫ +2 , θ ǫ +3 ) containsfewer than κ regular cardinals. The attempt to prove Conjecture 4.1 leads to the following principle:
Definition 1.3.
We say V α +1 satisfies the Collection Principle if for every binaryrelation R ⊆ V α × V α +1 , there is a subrelation S ⊆ R such that dom( S ) = dom( R )and ran( S ) is the surjective image of V α .From one perspective, the Collection Principle is a weak choice principle. Itfollows from the Axiom of Choice, because one can take the subrelation S to bea uniformization of R . Another perspective is that the Collection Principle statesthat θ α is regular in a strong sense. In particular, the Collection Principle easilyimplies that θ α is a regular cardinal. Under AD, the converse holds at ω + 1: if θ ω +1 is regular, then V ω +2 satisfies the Collection Principle. Theorem 4.12.
Suppose ǫ is an even ordinal. Suppose j : V ǫ +2 → V ǫ +2 is anontrivial elementary embedding. Assume κ ω ( j ) - DC and that V ǫ +1 satisfies the The axiom I states that there is an elementary embedding from L ( V λ +1 ) to L ( V λ +1 ) withcritical point less than λ . In the context of ZF, a cardinal θ is a strong limit cardinal if θ is not the surjective image of P ( β ) for any ordinal β < θ . ollection Principle. Then θ ǫ +2 is a strong limit cardinal. Moreover, for all β <θ ǫ +2 , P ( β ) is the surjective image of V ǫ +1 . The proof of this theorem involves generalizing Woodin’s Coding Lemma. Thetheorem yields a new proof of the Kunen inconsistency theorem: assuming theAxiom of Choice, the hypotheses of Theorem 4.12 hold, yet θ ǫ +2 = | V ǫ +1 | + is nota strong limit cardinal, and it follows that there is no elementary embedding from V ǫ +2 to V ǫ +2 . (A slightly more detailed proof appears in Corollary 4.13.)To drive home the contrast between the even and odd levels, we show that thefinal conclusion of Theorem 4.12 fails at the even levels: Theorem 4.19.
Suppose ǫ is an even ordinal and there is an elementary embeddingfrom V ǫ +2 to V ǫ +2 . Then for any ordinal γ , there is no surjection from V ǫ × γ onto P ( θ ǫ ) . Section 5 concerns the theory of ultrafilters assuming choiceless large cardinals.Woodin proved that choiceless large cardinal axioms (combined with “ κ ω ( j )-DC”)imply the existence of measurable successor cardinals. The ultrafilters he producedbear a strong resemblance to the ultrafilters arising in the context of AD. Here weexpand upon that theme.First, we study the ordinal definability of ultrafilters on ordinals: Theorem 5.17.
Suppose j : V ǫ +3 → V ǫ +3 is an elementary embedding. Let λ = κ ω ( j ) . Assume λ - DC . Suppose U is a λ + -complete ultrafilter on an ordinal lessthan θ ǫ +2 . Then the following hold:(1) U ∩ HOD belongs to
HOD .(2) U belongs to an ordinal definable set of cardinality less than λ .(3) For an OD -cone of x ∈ V λ , the ultrapower embedding j U is amenable to HOD x . This result uses an analog of the Ultrapower Axiom of [1] that is provable fromchoiceless large cardinals (Theorem 5.12).Finally, we prove a form of strong compactness for κ ω ( j ) where j : V → V is anelementary embedding: Theorem 5.24.
Suppose j : V → V is a nontrivial elementary embedding. Let λ = κ ω ( j ) . Assume λ - DC holds. Then every λ + -complete filter on an ordinalextends to a λ + -complete ultrafilter. This result is an application of the Ketonen order on filters, a wellfounded partialorder on countably complete filters on ordinals that simultaneously generalizes theJech order on stationary sets and the Mitchell order on normal ultrafilters.Like many of the arguments of this paper (e.g., Theorem 5.17), the proof ofTheorem 5.24 is general enough that it yields a new consequence of I : The choice principle λ -DC is defined in Section 2.2 heorem 5.25 (ZFC) . Suppose λ is a cardinal and there is an elementary embed-ding from L ( V λ +1 ) to L ( V λ +1 ) with critical point less than λ . Then in L ( V λ +1 ) ,every λ + -complete filter on an ordinal extends to a λ + -complete ultrafilter. In the last section of this paper, Section 6, we turn to consistency results. Mostof these results predate the groundbreaking theorem of Schlutzenberg [2] that theexistence of an elementary embedding j : L ( V λ +1 ) → L ( V λ +1 ) with critical pointbelow λ is equiconsistent with the existence of an elementary embedding from V λ +2 to V λ +2 , but it is useful to keep this theorem in mind to appreciate the statementsof our theorems.We prove the equiconsistency of various choiceless large cardinals associatedwith the Kunen inconsistency: Theorem 6.8.
The following statements are equiconsistent over ZF :(1) For some λ , there is a nontrivial elementary embedding from V λ +2 to V λ +2 .(2) For some λ , there is an elementary embedding from L ( V λ +2 ) to L ( V λ +2 ) withcritical point below λ .(3) There is an elementary embedding j from V to an inner model M that is closedunder V κ ω ( j )+1 -sequences. Combined with Schlutzenberg’s Theorem, this shows that all of these principlesare equiconsistent with the the existence of an elementary embedding from L ( V λ +1 )to L ( V λ +1 ) with critical point below λ .Our next theorem shows that choiceless large cardinal axioms beyond an ele-mentary embedding from V λ +2 to V λ +2 are stronger than I : Theorem 6.19.
Suppose λ is an ordinal and there is a Σ -elementary embedding j : V λ +3 → V λ +3 with λ = κ ω ( j ) . Assume DC V λ +1 . Then there is a set genericextension N of V such that ( V λ ) N satisfies ZFC + I . The following result is an immediate corollary:
Corollary.
Over
ZF + DC , the existence of an elementary embedding from V λ +3 to V λ +3 implies the consistency of ZFC + I . By Schlutzenberg’s Theorem, the hypothesis of Theorem 6.19 cannot be reducedto the existence of an elementary embedding from V λ +2 to V λ +2 , or even the exis-tence of a Σ -elementary embedding j : V λ +3 → V λ +3 with j ( V λ +2 ) = V λ +2 .Schlutzenberg [2] poses the problem of calculating the exact consistency strengthover ZF of the existence of an elementary embedding from V λ +2 to V λ +2 in termsof large cardinal axioms compatible with the Axiom of Choice. We sketch how tocalculate the consistency strength of this assertion over ZF + DC: Theorem 6.20.
The following statements are equiconsistent over
ZF + DC :(1) For some ordinal λ , there is an elementary embedding from V λ +2 to V λ +2 .
2) The Axiom of Choice + I . We defer to the appendix some facts about countably complete filters and ultra-filters that are used in Section 3.3 and Section 5. This is accomplished by consideringa version of the Ketonen order studied in [1] that is applicable to countably com-plete filters on complete Boolean algebras in the context of ZF + DC. This level ofgenerality is overkill, but it makes the proofs slicker.
In this section, we lay out some of the notational conventions we will use in thispaper.
Most importantly, we work throughout this paper in ZF alone, without as-suming the Axiom of Choice, explicitly making note of any other choice principleswe use. Most of the notation discussed here is standard, with the notable exceptionof Section 2.4, which introduces a class of structures hH α i α ∈ Ord which will be veryuseful throughout the paper.
We use the following notation for elementary embeddings:
Definition 2.1.
Suppose M and N are structures in the same signature. Then E ( M, N ) denotes the set of elementary embeddings from M to N , and E ( M ) denotesthe set of elementary embeddings from M to itself.Typically the structures we consider are of the form ( M, ∈ ) where M is a tran-sitive set. We will always suppress the membership relation, writing E ( M ) whenwe mean E ( M, ∈ ).Following [5], we use the following notation for the critical sequence of an ele-mentary embedding: Definition 2.2.
Suppose M and N are transitive structures and j ∈ E ( M, N ). The critical point of j , denoted crit( j ), is the least ordinal moved by j . The critical se-quence of j is the sequence h κ n ( j ) | n < ω i defined by κ ( j ) = crit( j ) and κ n +1 ( j ) = j ( κ n ( j )). Finally, the critical supremum of j is the ordinal κ ω ( j ) = sup n<ω κ n ( j ).Of course, crit( j ) may not be defined since j may have no critical point. Evenif crit( j ) is defined, κ n +1 ( j ) may not be for some n < ω , since it is possible that κ n ( j ) / ∈ M . We say that T ⊆ X <λ is a tree if for all s ∈ T , for all α < dom( s ), s ↾ α ∈ T . A tree T ⊆ X <λ is λ -closed if for any s ∈ X <λ with s ↾ α ∈ T for all α < dom( s ), s ∈ T .A cofinal branch of a tree T ⊆ X <λ is a sequence s ∈ X λ such that x ↾ α ∈ T forall α < λ .Various weak choice principles will be used throughout the paper. The mostimportant are the following: 6 efinition 2.3. Suppose λ is a cardinal and X is a set. • λ -DC X denotes the principle asserting that every λ -closed tree of sequences T ⊆ X <λ with no maximal branches has a cofinal branch. • λ -DC denotes the principle asserting that λ -DC Y holds for all sets Y . • The Axiom of Dependent Choice, or DC, is the principle ω -DC.In the context of ZF, it may be that there is a surjection from X to Y but noinjection from Y to X . We therefore use the following notation: Definition 2.4. If X and Y are sets, then X (cid:22) ∗ Y if there is a partial surjectionfrom Y to X . We let [ X ] Y = { S ⊆ X | S (cid:22) ∗ Y } .We use partial surjections because these are what arise naturally in practice,but of course X (cid:22) ∗ Y if and only if either X = ∅ or there is a total surjection from Y to X . We use the following convention: a filter over a set X is a filter on the Booleanalgebra P ( X ). Filters on Boolean algebras do not come up until the appendix, sountil then, we use the word filter to refer to a filter over some set. Definition 2.5.
Suppose γ is an ordinal. A filter F is γ -saturated if there isno sequence h S α | α < γ i of F -positive sets such that S α ∩ S β is F -null for all α < β < γ ; F is weakly γ -saturated if there is no sequence h S α | α < γ i of pairwisedisjoint F -positive sets.If F is γ -complete, then F is γ -saturated if and only if F is weakly γ -saturated. Definition 2.6. If B is a set, a filter F is B -complete if for any b ∈ B and any D ⊆ F such that D (cid:22) ∗ b , T D ∈ F . A filter F is X -closed if F is { X } -complete.A filter is said to be countably complete if it is ω -complete.We will need the standard derived ultrafilter construction: Definition 2.7.
Suppose h : P ( X ) → P ( Y ) is a homomorphism of Boolean alge-bras and a ∈ Y . The ultrafilter on X derived from h using a is the ultrafilter over X defined by the formula { A ⊆ X : a ∈ h ( A ) } .Our notation for ultrapowers is standard in set theory. If U is an ultrafilterover a set X , then j U : V → M U denotes the associated ultrapower. If h M x i x ∈ X is a sequence of structures in the same signature, then Q x ∈ X M x /U denotes theirultraproduct. 7 .4 The structures H α Although the subject of this paper is rank-to-rank embeddings (i.e., elements of E ( V α ) for some ordinal α ), it is often convenient to lift these embeddings to act onlarger structures. The issue is that many sets are coded in V α but do not belongto V α . This is especially annoying when α is a successor ordinal, in which case V α fails to be closed under Kuratowski pairs. This motivates introducing the followingstructures: Definition 2.8.
For any set X , let H ( X ) denote the union of all transitive sets M such that M (cid:22) ∗ S for some S ∈ X . For any ordinal α , let H α = H ( V α ).If κ is a (wellordered) cardinal, then H ( κ ) is the usual structure H ( κ ). In ZF,however, there may be other structures of the form H ( X ). Notice that H α +1 is thecollection of sets that are the surjective image of V α . Definition 2.9.
For any set X , let θ ( X ) denote the least ordinal that is not thesurjective image of some set S ∈ X . Let θ α = θ ( V α ).Then θ ( X ) = H ( X ) ∩ Ord. The cardinals θ α are studied in Section 4.Note that for all α , V α ⊆ H α . We claim that every embedding in E ( V α ) extendsuniquely to an embedding in E ( H α ). This is a consequence of a coding of H α insidethe structure V α . We proceed to describe one such coding, and then we sketch howthis yields the unique extension of embeddings from E ( V α ) to E ( H α ).Let p : V → V × V denote some Quine-Rosser pairing function , which is abijection that is Σ -definable without parameters such that for all infinite ordinals α , p [ V α ] = V α × V α .Fix an infinite ordinal α . We will define a partial surjectionΦ α : V α → H α For each x ∈ V α , let R x = p [ x ] be the binary relation coded by x , and let D x denotethe field of R x . Note that D x ∈ V α . Let E α be the set of x ∈ V α such that R x isa wellfounded extensional relation on D x . For x ∈ E α , let π x : D x → M x be theMostowski collapse. LetC α = { ( x, y ) | x ∈ E α and y ∈ D x } For ( x, y ) ∈ C α , let Φ α ( x, y ) = π x ( y ). Then ran(Φ α ) = S x ∈ V α M x = H α .Since Φ α is definable over H α , one has that for any i : H α → H α , i (Φ α ( x, y )) =Φ α ( i ( x ) , i ( y )). Conversely, since the sets C α and Φ − α [ ∈ ] = { ( u, w ) ∈ C α × C α :Φ α ( u ) ∈ Φ α ( w ) } , and Φ − α [=] are definable over V α , one has that if j : V α → V α iselementary, then setting j ⋆ (Φ α ( x, y )) = Φ α ( j ( x ) , j ( y )) , the embedding j ⋆ : H α → H α is well-defined and elementary. Definition 2.10.
For any j ∈ E ( V α ), j ⋆ denotes the unique elementary embedding k : H α → H α such that k ↾ V α = j . 8he rest of the section contains an analysis of j ⋆ when j : V α → V α is onlyassumed to be Σ n -elementary for some n < ω . This is rarely relevant, and thereader can skip it for now.We claim that if i : H α → H α is a Σ -elementary embedding, then i (Φ α ( x, y )) =Φ α ( i ( x ) , i ( y )) for all ( x, y ) ∈ C α . To see this, fix ( x, y ) ∈ C α . Let a = Φ α ( x, y ).Let M be an admissible set in H α such that x, y ∈ M . Then M satisfies thestatement “ a = π x ( y ) where π x is the Mostowski collapse of ( D x , R x ).” Since i is Σ -elementary, it follows that i ( M ) is an admissible set that satisfies “ i ( a ) = π i ( x ) ( i ( y )).” This is expressible as a Σ statement, so is upwards absolute to V .Therefore i ( a ) = π i ( x ) ( i ( y )), or in other words, i (Φ α ( x, y )) = Φ α ( i ( x ) , i ( y )), asdesired.Conversely, suppose j : V α → V α is Σ -elementary, and let j ⋆ : H α → H α be defined by j ⋆ (Φ α ( x, y )) = Φ α ( j ( x ) , j ( y )). We claim j ⋆ : H α → H α is well-defined and Σ -elementary. Note that C α , Φ − α [=], and Φ − α [ ∈ ] are Π -definableover V α . This implies that j ⋆ is a well-defined ∈ -homomorphism. If M ∈ H α is atransitive set, then taking x ∈ E α such that M = M x , the satisfaction predicatefor ( D x , R x ) ∼ = M is ∆ -definable over V α from x , and hence j ↾ M : M → j ( M ) isfully elementary. Since every x ∈ H α belongs to some transitive M ∈ H α , and sucha set M is a Σ -elementary substructure of H α , it follows that j is Σ -elementary. Definition 2.11.
Suppose i : V α → V α is a Σ -elementary embedding. Then i ⋆ denotes the unique Σ -elementary embedding from H α to H α extending i .By a localization of the arguments above, one obtains the following fact aboutthe elementarity of j ⋆ : Lemma 2.12.
Suppose n < ω and i : V α → V α is a Σ n +1 -elementary embedding.Then i ⋆ : H α → H α is Σ n -elementary.Sketch. For example, take the case n = 1. Suppose ψ is a Σ -formula, a ∈ H α and H α satisfies ∃ v ψ ( j ( a ) , v ). Fix ( x , y ) ∈ C α with Φ α ( x , y ) = a . Then V α satisfies that there is some ( x , y ) ∈ C α such that ( D x , R x ) is an end-extensionof ( D j ( x ) , R j ( x ) ) and ( D x , R x ) satisfies ψ ( j ( y ) , y ). This is Σ -expressible in V α , so by elementarity, there is some ( x , y ) ∈ C α such that ( D x , R x ) is anend-extension of ( D x , R x ) and ( D x , R x ) satisfies ψ ( y , y ). It follows that M x satisfies ψ ( a, Φ α ( x , y )), and hence H α satisfies ∃ v ψ ( a, v ), as desired. The results of this section are inspired by the work of Schlutzenberg [9], whichgreatly expands upon the following theorem of Suzuki: Suzuki actually proved the slightly stronger schema that no elementary embedding from V to V is definable from parameters over V . heorem 3.1 (Suzuki) . If κ is an inaccessible cardinal, no nontrivial elementaryembedding from V κ to V κ is definable over V κ from parameters. Schlutzenberg [9] extended this to limit ranks: Theorem 3.2 (Schlutzenberg) . Suppose λ is a limit ordinal. Then no nontrivialelementary embedding from V λ to V λ is definable over V λ from parameters. Schlutzenberg noted that the situation for elementary embeddings from V λ +1 to V λ +1 , where λ is a limit ordinal, is completely different: every elementary embeddingfrom V λ +1 to V λ +1 is definable from parameters over V λ +1 . He then asked thecorresponding question for elementary embeddings of V λ + n for n >
1. The answeris given by the following theorem, which is established by the main results of thissection:
Theorem 3.3.
Suppose ǫ is an even ordinal. (1) No nontrivial elementary embedding from V ǫ to V ǫ is definable over V ǫ fromparameters.(2) Every elementary embedding V ǫ +1 to V ǫ +1 is definable over V ǫ +1 from parame-ters. This theorem is the first instance of a periodicity phenomenon in the hierarchyof choiceless large cardinal axioms, leading to a generalization to arbitrary ranks ofthe basic theory of rank-to-rank embeddings familiar from the ZFC context. (1) isproved as Proposition 3.20 of Section 3.3 and (2) as Theorem 3.13 of Section 3.2.We note that Schlutzenberg rediscovered Theorem 3.3, and this theorem is themain subject of the joint paper [10]. V ǫ +1 That elementary embeddings of odd ranks are definable (Theorem 3.3 (2)) came asquite a surprise to the author, but with hindsight emerges as a natural generalizationa well-known phenomenon from the standard theory of rank-to-rank embeddings.
Definition 3.4.
Suppose λ is a limit ordinal and j : V λ → V λ is an elementaryembedding. Then the canonical extension of j is the embedding j + : V λ +1 → V λ +1 defined by j + ( X ) = S Γ ∈ V λ j ( X ∩ Γ).While the canonical extension of an embedding in E ( V λ ) is not necessarily anelementary embedding, it is true that if i ∈ E ( V λ +1 ), then necessarily i = ( i ↾ V λ ) + .The proof is easy, but since it is relevant below, we give a detailed sketch. Fix X ∈ V λ +1 . Clearly i ( X ∩ Γ) ⊆ i ( X ) for all Γ ∈ V λ , and this easily implies theinclusion ( i ↾ V λ ) + ( X ) ⊆ i ( X ). For the reverse inclusion, suppose a ∈ i ( X ). Since Schlutzenberg also proved many other definability results for rank-to-rank embeddings, incor-porating, for example, constructibility and ordinal definability. An ordinal α is said to be even if for some limit ordinal λ and some natural number n , α = λ + 2 n ; otherwise, α is odd . is a limit ordinal, there is some Γ ∈ V λ such that a ∈ i (Γ); for example, onecan take Γ = V ξ +1 where ξ = rank( a ). Now a ∈ i ( X ) ∩ i (Γ) = i ( X ∩ Γ), so a ∈ ( i ↾ V λ ) + ( X ). The key property of λ that was used in this proof is that any i : V λ → V λ is a cofinal embedding in the sense that for all a ∈ V λ , there is someΓ ∈ V λ with a ∈ i (Γ).Suppose now that α is an arbitrary infinite ordinal. We want to generalize thecanonical extension operation to act on embeddings j ∈ E ( V α ). It is easy to see thatif α is a successor ordinal, the naive generalization (i.e., j + ( X ) = S Γ ∈ V α j ( X ∩ Γ) for X ∈ V α +1 ) does not have the desired effect. (For example, adopting this definition,one would have j + ( { V α − } ) = ∅ .) Instead, one must make the following tweak: Definition 3.5.
Suppose α is an infinite ordinal and j : V α → V α is an elementaryembedding. Then the canonical extension of j is the embedding j + : V α +1 → V α +1 defined by j + ( X ) = S Γ ∈H α j ⋆ ( X ∩ Γ).See Section 2.4 for the definition of the structure H α , and the basic facts aboutlifting elementary embeddings from V α to H α . Thus j + ( X ) is the union of all setsof the form j ⋆ ( X ∩ Γ) where Γ is coded in V α . The following easily verified lemmaclarifies the definition: Proposition 3.6.
Suppose α is an infinite ordinal and j ∈ E ( V α ) . Then j + ( X ) = (S Γ ∈ V α j ( X ∩ Γ) if α is a limit ordinal S Γ ∈ [ V α ] Vα − j ⋆ ( X ∩ Γ) if α is a successor ordinal While the definition of the canonical extension operation directly generalizesDefinition 3.4, a key new phenomenon arises at successor ranks: it is no longer clearthat every elementary embedding i : V α +1 → V α +1 satisfies i = ( i ↾ V α ) + It is easy to show that for all X ∈ V α +1 , ( i ↾ V α ) + ( X ) ⊆ i ( X ), but the reverseinclusion is no longer clear.In fact, the reverse inclusion is only true for even values of α . This is proved byan induction that simultaneously establishes the canonical extension property andthe cofinal embedding property , which we now define. Definition 3.7.
Suppose ǫ is an ordinal. Then ǫ has the canonical extension prop-erty if for any i ∈ E ( V ǫ +1 ), i = ( i ↾ V ǫ ) + .The terminology is motivated by equivalence of the canonical extension propertywith the statement that an elementary embedding in E ( V ǫ ) extends to at most oneembedding in E ( V ǫ +1 ). Definition 3.8.
Suppose ǫ is an ordinal. Then ǫ has the cofinal embedding property if for any i ∈ E ( V ǫ ), for any A ∈ H ǫ , there is some Γ ∈ H ǫ with A ∈ i ⋆ (Γ).11hus the cofinal embedding property states that every embedding in E ( V ǫ )induces a cofinal embedding in E ( H ǫ ). The proof that limit ordinals have thecanonical extension property generalizes to all ordinals with the cofinal embeddingproperty: Lemma 3.9.
Suppose ǫ is an ordinal. If ǫ has the cofinal embedding property, then ǫ has the canonical extension property.Proof. Fix an elementary embedding i : V ǫ +1 → V ǫ +1 . We must show i = ( i ↾ V ǫ ) + .Take X ∈ V ǫ +1 . The inclusion ( i ↾ V ǫ ) + ( X ) ⊆ i ( X ) is true regardless of parity:if Γ ∈ H ǫ , then i ⋆ ( X ∩ Γ) = i ( X ∩ Γ) ⊆ i ( X ) by the elementarity of i and theuniqueness of i ⋆ , so i + ( X ) = S Γ ∈H α i ⋆ ( X ∩ Γ) ⊆ i ( X ).To show i ( X ) ⊆ ( i ↾ V ǫ ) + ( X ), suppose A ∈ i ( X ). Since ǫ has the cofinalembedding property, there is some Γ ∈ H ǫ such that A ∈ i ⋆ (Γ). Now A ∈ i ( X ) ∩ i ⋆ (Γ) = i ( X ) ∩ i (Γ) = i ( X ∩ Γ) = i ⋆ ( X ∩ Γ) ⊆ i + ( X )This completes the proof.The periodicity phenomenon is a result of the following lemma: Lemma 3.10.
Suppose ǫ is an ordinal. If ǫ has the canonical extension property,then ǫ + 2 has the cofinal embedding property.Proof. Fix i ∈ E ( V ǫ +2 ) and A ∈ V ǫ +2 . LetΓ = { ( k + ) − [ A ] | k ∈ E ( V ǫ ) } Then Γ (cid:22) ∗ E ( V ǫ ) (cid:22) ∗ V ǫ +1 . Since Γ ∪ V ǫ +1 is transitive, it follows that Γ ∈ H ǫ +1 .This allows us to prove the cofinal embedding property and the canonical ex-tension property for even ordinals by induction: Corollary 3.11.
Every even ordinal has the cofinal embedding property.Proof.
Suppose λ is a limit ordinal. We show that λ + 2 n has the cofinal embeddingproperty by induction on n < ω .We first prove the base case, when n = 0. Suppose A ∈ H λ . Fix ξ < λ suchthat A ∈ H ξ . Then we have H ξ ∈ H λ and A ∈ j ⋆ ( H ξ ) since H ξ ⊆ H j ( ξ ) = j ⋆ ( H ξ ).This shows that λ has the cofinal embedding property.For the induction step, assume that λ + 2 n has the cofinal embedding property.Then by Lemma 3.10, λ + 2 n has the canonical extension property, and so byLemma 3.9, λ + 2 n + 2 has the cofinal embedding property. Corollary 3.12.
Every even ordinal has the canonical extension property.
As an immediate consequence, we have Theorem 3.3 (2):
Theorem 3.13.
Suppose ǫ is an infinite even ordinal and i : V ǫ +1 → V ǫ +1 is anelementary embedding. Then i is definable over V ǫ +1 from i [ V ǫ ] . roof. Clearly i is definable over H ǫ +1 from i ↾ V ǫ since i = ( i ↾ V ǫ ) + and thecanonical extension operation is explicitly defined over H ǫ +1 . Moreover, i ↾ V ǫ isdefinable over V ǫ +1 from i [ V ǫ ] as the inverse of the Mostowski collapse. It followsthat i is definable over H ǫ +1 from i [ V ǫ ]. But by coding elements of H ǫ +1 as elementsof V ǫ +1 as in Section 2, one can translate this into a definition of i over V ǫ +1 from i [ V ǫ ].Let us put down for safe-keeping the following version of the cofinal embeddingproperty that is often useful: Definition 3.14.
Suppose σ is a set such that ( σ, ∈ ) is wellfounded and extensional.Then j σ : M σ → σ denotes the inverse of the Mostowski collapse of σ .Suppose ǫ is an even ordinal. For any A ∈ V ǫ +2 , let f A : V ǫ +1 → V ǫ +2 be thepartial function defined by f A ( σ ) = ( j + σ ) − [ A ].We leave f A ( σ ) undefined if one of the following holds: • ( σ, ∈ ) is not wellfounded and extensional. • j σ is not an elementary embedding from V ǫ to V ǫ . Proposition 3.15.
Suppose j : V ǫ +2 → V ǫ +2 is an elementary embedding. Thenfor any A ∈ V ǫ +2 , A = j ⋆ ( f A )( j [ V ǫ ]) .Proof. Since f A is definable from A over H ǫ +2 , j ⋆ ( f A )( j [ V ǫ ]) = f j ( A ) ( j [ V ǫ ]). Notethat M j [ V ǫ ] = V ǫ and j j [ V ǫ ] = j ↾ V ǫ . Therefore by the elementarity of j ⋆ , f j ( A ) ( j [ V ǫ ]) = (( j ↾ V ǫ ) + ) − [ j ( A )] = ( j ↾ V ǫ +1 ) − [ j ( A )] = A Our original approach to Theorem 3.13 diverged from the one presented here inthat we used a superficially different definition of the canonical extension operationfrom Definition 3.5. This approach is not as clearly motivated by the canonicalextension operation for embeddings from V λ to V λ where λ is a limit ordinal (Defi-nition 3.4), but it might illuminate the underlying combinatorics. This constructionis described in more detail in [10].Suppose j : V ǫ +2 → V ǫ +2 is elementary, and let U be the ultrafilter derived from j using j [ V ǫ ]. Let j U : V → M U denote the ultrapower associated to U . It is easy tosee that Ult( V ǫ +1 , U ) ∼ = V ǫ +1 . Assume ǫ has the cofinal embedding property. Thenmoreover Ult( V ǫ +2 , U ) ∼ = V ǫ +2 . Therefore we identify Ult( V ǫ +2 , U ) with V ǫ +2 . Asa consequence, for every X ∈ V ǫ +3 , we can identify j U ( X ) with a subset of V ǫ +2 ;that is, we identify j U ( X ) with an element of V ǫ +3 . Given the cofinal embeddingproperty, it is not hard to show that j U ↾ V ǫ +3 is the only possible extension of j toan elementary embedding of V ǫ +3 . Therefore one can set j + = j U instead of usingDefinition 3.5, and then prove Corollary 3.11 and Corollary 3.12 by very similararguments to the ones given above.Obviously the two approaches to the canonical extension operation are verysimilar, but we just want to highlight that the canonical extension operation, likeeverything else in set theory, is really an ultrapower construction.13 .3 Undefinability over V ǫ In this section, we establish Theorem 3.3 (1). At this point, we have found threeproofs, increasing chronologically in complexity and generality.We begin by giving a sketch of the simplest of these proofs in the successorordinal case. (Note that the limit case is handled by Schlutzenberg’s Theorem 3.2.)Suppose ǫ is an even ordinal and j : V ǫ +2 → V ǫ +2 is a nontrivial elementary em-bedding. Let U be the ultrafilter over V ǫ +1 derived from j using j [ V ǫ ]. It turnsout that if j is definable over V ǫ +2 , then U belongs to the ultrapower of V by U .A fundamental fact from the ZFC theory of large cardinals, proved for example in[11], is that no countably complete ultrafilter belongs to its own ultrapower. Theidea of the proof of Proposition 3.20 is to try to push this through to the currentcontext.Recall that the Mitchell order is defined on countably complete ultrafilters U and W by setting U ⊳ W if U ∈ Ult(
V, W ). The “fundamental fact” mentioned abovesimply states that assuming the Axiom of Choice, the Mitchell order is irreflexive.(In fact, it is wellfounded.) In the context of ZF, especially given the failure of Lo´s’sTheorem, the Mitchell order is fairly intractable, and in particular, we do not knowhow to prove its irreflexivity. Instead, we use a variant of the Mitchell order calledthe internal relation (introduced in the author’s thesis [1]) that is more amenableto combinatorial arguments.
Definition 3.16.
Suppose U and W are countably complete ultrafilters over sets X and Y . We say U is internal to W , and write U < W , if there is a sequence ofcountably complete ultrafilters h U y | y ∈ Y i such that for any relation R ⊆ X × Y : ∀ U x ∀ W y R ( x, y ) ⇐⇒ ∀ W y ∀ U y x R ( x, y ) (1)Using notation that is standard in ultrafilter theory, (1) states that U × W iscanonically isomorphic to W - P y ∈ Y U y .Of course, from this combinatorial definition, it is not clear that the internalrelation is related to the Mitchell order at all. Using Lo´s’s Theorem, however, thefollowing is easy to verify: Definition 3.17. If U and W are ultrafilters, then the pushforward of U to M W is the M W -ultrafilter s W ( U ) = { A ∈ j W ( P ( X )) | ( j W ) − [ A ] ∈ U } . Proposition 3.18 (ZFC) . Suppose U and W are countably complete ultrafilters.Let X be the underlying set of U . Then the following are equivalent:(1) U < W .(2) s W ( U ) ∈ M W .(3) j U ↾ Ult(
V, W ) is definable from parameters over Ult(
V, W ) . (3) is not the right perspective when Lo´s’s Theorem does not hold for W . Theequivalence of (1) and (2), however, is essentially a consequence of ZF: For any predicate P , we write “ ∀ U x P ( x )” to mean that { x ∈ X : P ( x ) } ∈ U . emma 3.19. Suppose U and W are countably complete ultrafilters over X and Y . Then U < W if and only if there is a sequence of countably complete ultrafilters h U y i y ∈ Y such that [ h U y i y ∈ Y ] W = s W ( U ) .Proof. One shows that [ h U y i y ∈ Y ] W = s W ( U ) if and only if the equivalence (1) fromDefinition 3.16 holds.For the forwards direction, assume [ h U y i y ∈ Y ] W = s W ( U ). Fix R ⊆ X × Y . Weverify the equivalence (1) from Definition 3.16.Suppose that ∀ W y ∀ U y x R ( x, y ). Then R y ∈ U y for W -almost all y ∈ Y . By assumption, this means [ h R y i y ∈ Y ] W ∈ s W ( U ), so by the definition of s W ( U ),( j W ) − ([ h R y i y ∈ Y ] W ) ∈ U . In other words, { x ∈ X | ∀ W y x ∈ R y } ∈ U , whichmeans that ∀ U x ∀ W y R ( x, y ).The proof that ∀ U x ∀ W y R ( x, y ) implies ∀ W y ∀ U y x R ( x, y ).We omit the proof that (1) from Definition 3.16 implies [ h U y i y ∈ Y ] W = s W ( U ),since the proof is straightforward and the result is never actually cited.The key advantage of the internal relation over the Mitchell order is that itsirreflexivity can be proved in ZF by a combinatorial argument that will be given inthe appendix: Theorem 7.15.
Suppose U is a countably complete ultrafilter and crit( j U ) exists.Then U < U . With this in hand, we can prove the undefinability theorem.
Proposition 3.20.
Suppose ǫ is an even ordinal and j : V ǫ +2 → V ǫ +2 is a nontrivialelementary embedding. Then j is not definable from parameters over V ǫ +2 .Proof. Assume towards a contradiction that j is definable over V ǫ +2 from param-eters. Let U be the ultrafilter over V ǫ +1 derived from j using j [ V ǫ ]. Clearly U isdefinable over V ǫ +2 from j , and hence U is definable over V ǫ +2 from parameters.Let k : Ult( V ǫ +2 , U ) → V ǫ +2 be the canonical factor embedding defined by k ([ f ] U ) = j ⋆ ( f )( j [ V ǫ ]). The elementarity of j ⋆ implies that k is a well-definedinjective homomorphism of structures. Moreover, Proposition 3.15 implies that k issurjective. So Ult( V ǫ +2 , U ) is canonically isomorphic to V ǫ +2 , and we will identifythe two structures.Fix a formula ϕ ( v, w ) and a parameter B ∈ V ǫ +2 such that U = { A ∈ V ǫ +2 | V ǫ +2 (cid:15) ϕ ( A, B ) } Let U σ = { A ∈ V ǫ +2 | V ǫ +2 (cid:15) ϕ ( f A ( σ ) , f B ( σ )) } where f A and f B are as defined inDefinition 3.14.We claim that [ hU σ | σ ∈ V ǫ +1 i ] U = s U ( U ). This follows from the elementarityof j , which implies[ hU σ | σ ∈ V ǫ +1 i ] U = { A ∈ V ǫ +2 | V ǫ +2 (cid:15) ϕ ( f A ( j [ V λ ]) , B ) } = { A ⊆ V ǫ +1 | j − [ A ] ∈ U} = s U ( U ) For y ∈ Y , R y denotes the set { x ∈ X | ( x, y ) ∈ R } .
15t is also easy to show that U σ is a countably complete ultrafilter for U -almost all σ ∈ V ǫ +1 . By Lemma 3.19, hU σ | σ ∈ V ǫ +1 i witnesses U < U . Since j U ↾ ǫ = j ↾ ǫ , j U has a critical point. This contradicts Corollary 7.15. The proof of Proposition 3.20 raises a number of questions that also arise nat-urally in the study of choiceless cardinals. The one we will focus on concernsthe supercompactness properties of ultrapowers. Suppose ǫ is an even ordinal, j : V ǫ +2 → V ǫ +2 is an elementary embedding, and U is the ultrafilter over V ǫ +1 derived from j using j [ V ǫ ]. Motivated by the proof of Corollary 3.12, one might askwhen j U [ S ] ∈ M U for various sets S . This is related to the definability of elemen-tary embeddings because if S ∈ V β is transitive and j extends to an elementaryembedding i : V β → N , then j U [ S ] ∈ M U if and only if i ↾ S is definable over N from parameters in i [ V β ] ∪ V ǫ +2 .Of course, by Corollary 3.12, j U [ V ǫ +1 ] belongs to M U . It follows easily that j U [ S ] ∈ M U for all S (cid:22) ∗ V ǫ +1 . (See Definition 2.4 for this notation.) The converseremains open: if j U [ S ] ∈ M U , must S (cid:22) ∗ V ǫ +1 ? We still do not know the answerto this question, even in the following special case: with U as above, it is not hardto show that j U [ P ( ǫ + )] belongs to M U , yet it is far from clear whether one canprove P ( ǫ + ) (cid:22) ∗ V ǫ +1 without making further assumptions. Of course, assumingthe Axiom of Choice, if U is an ultrafilter over X , S is a set, and j U [ S ] ∈ M U ,then | S | ≤ | X | . (See [1, Proposition 4.2.31].) We are simply asking whether a veryspecial case of this fact can be proved in ZF.We begin by proving a general theorem that subsumes the ZFC result mentionedabove: Theorem 3.21.
Suppose X is a set such that X × X (cid:22) ∗ X . Suppose U is anultrafilter over X , κ = crit( j U ) , and α ≥ κ is an ordinal. If j U [ α ] ∈ M U , then α (cid:22) ∗ X . Note that we implicitly assume that U has a critical point. Definition 3.22.
Suppose U is an ultrafilter over a set X . A sequence h S x | x ∈ X i is an A -supercompactness sequence for U if it has the following properties: • For all x ∈ X , S x ⊆ A . • Every a ∈ A belongs to S x for U -almost all x ∈ X . • For any X -indexed sequence h a x | x ∈ X i with a x ∈ S x for U -almost all x ∈ X , there is some a ∈ A with a x = a for U -almost all x ∈ X . Supercompact ultrafilters are the combinatorial manifestation of supercompactembeddings: In the case that A is not wellorderable, the right concept seems to be that of a normal A -supercompactness sequence, which has the stronger property that for any X -indexed sequence h B x | x ∈ X i with ∅ 6 = B x ⊆ S x for U -almost all x ∈ X , there is some a ∈ A with a ∈ B x for U -almost all x ∈ X . emma 3.23. h S x | x ∈ X i is an A -supercompactness sequence for U if and onlyif [ h S x | x ∈ X i ] U = j U [ A ] . The first step of Theorem 3.21 is the following well-known fact:
Lemma 3.24.
Suppose U is an ultrafilter over a set X and κ = crit( j U ) . Thenthere is a surjection from X to κ .Proof. Since U is κ -complete but not κ + -complete, there is a strictly decreasingsequence h A α | α < κ i of sets A α ∈ U such that T α<κ A α = ∅ . Let f : X → κ bedefined by f ( x ) = min { α | x / ∈ A α +1 } Since the sequence h A α | α < κ i is strictly decreasing, f is a surjection.The second step of Theorem 3.21 is more involved: Lemma 3.25.
Suppose U is an ultrafilter over a set X , κ = crit( j U ) , γ ≥ κ is anordinal, and h S x | x ∈ X i is a γ -supercompactness sequence for U . Then for anysurjection p from X to κ , there is a cofinal function from X to γ that is ordinaldefinable from h S x | x ∈ X i and p . In fact, the function produced by Lemma 3.25 will be Σ -definable from h S x i x ∈ X and p , but ordinal definability will suffice for our applications. Proof.
There are two cases.
Case 1.
For U -almost all x ∈ X , sup S x < γ .It does no harm to assume that sup S x < γ for all x ∈ X . This is becausethe sequence obtained from h S x | x ∈ X i by replacing S x with the empty setwhenever sup S x ≥ γ is a supercompactness sequence (and is ordinal definable from h S x | x ∈ X i ).Let f : X → γ be the function defined by f ( x ) = sup( S x ∩ γ )Then f is cofinal in γ , which proves Lemma 3.25 in Case 1. To see that f is cofinal,fix an ordinal α < γ . The definition of a supercompactness sequence implies thatthe set B α = { x ∈ X | α ∈ S x } belongs to U , so we may fix an x ∈ X such that α ∈ S x . Then f ( x ) = sup S x > α . Thus f is cofinal in γ . Case 2.
For U -almost all x ∈ X , sup S x = γ .As in Case 1, it does no harm to assume sup S x = γ for all x ∈ X .The first step is to show that the sets S x cannot have a common limit point ofuniform cofinality κ in the following sense: Claim.
Suppose ν < γ . There is no sequence of sets h E x | x ∈ X i such that for all x ∈ X , E x ⊆ S x ∩ ν , E x has ordertype κ , and E x is cofinal in ν . roof. Fix one last cofinal set E ⊆ ν of ordertype κ . Let M x = L [ S x , E x , E ]Since there is a definable sequence h < x | x ∈ X i such that < x is a wellorder of M x , Lo´s’s Theorem holds for the ultraproduct M = Q x ∈ X M x /U . Thus M is a properclass model of ZFC, although it may be that M is illfounded. That being said, γ + 1is contained in the wellfounded part of M . (As usual, the wellfounded part of M is taken to be transitive.) Indeed, [ h S x | x ∈ X i ] U = j U [ γ ] by Lemma 3.23, so M contains a wellorder of ordertype γ and hence is wellfounded up to γ + 1.The purpose of including the set E in each M x is to ensure that M correctlycomputes the cofinality of sup j U [ ν ]. The argument is standard, at least in thecontext of the Axiom of Choice. Since E ∈ M x for every x ∈ X , j U [ E ] = j U ( E ) ∩ j U [ γ ] belongs to M . Since ot( E ) = κ , ot( j U [ E ]) = κ . Therefore M satisfies thatsup j U [ E ] has cofinality κ . Since E is a cofinal subset of ν , sup j U [ E ] = sup j U [ ν ].Thus: cf M (sup j U [ ν ]) = κ (2)Let E ∗ = [ h E x | x ∈ X i ] U . Then Lo´s’s Theorem implies that the following holdin M : • E ∗ ⊆ j U [ ν ]. • E ∗ has ordertype j U ( κ ). • E ∗ is cofinal in j U ( ν ).The first bullet point uses that j U [ γ ] ∩ j U ( ν ) = j U [ ν ].Since E ∗ ⊆ j U [ ν ], j U [ ν ] is cofinal in j U ( ν ), and hencesup j U [ ν ] = j U ( ν )Combining this with (2), cf M ( j U ( ν )) = κ But cf( ν ) = κ in M x for every x ∈ X , so by Lo´s’s Theoremcf M ( j U ( ν )) = j U ( κ )Thus κ = j U ( κ ). This contradicts that κ is the critical point of j U .We now sketch the last idea of the proof. Let C x denote the set of limit pointsof S x . Suppose towards a contradiction that there is no cofinal function from X to γ . Then the intersection T x ∈ X C x is a closed unbounded subset of γ , and henceshould contain a point ν of cofinality κ . This almost contradicts the claim. There-fore to finish, we carefully go through the standard proof that T x ∈ X C x is closedunbounded, checking that it either produces a cofinal function from X to γ that is(ordinal) definable from h S x i x ∈ X and p or else produces a genuine counterexampleto the claim. 18ow, the details. We define various objects by a transfinite recursion with stagesindexed by ordinals α . The first α stages of the construction produce ordinals h δ βx | ( x, β ) ∈ X × α i such that for each x ∈ X , h δ βx | β < α i is an increasing sequence of elements of S x .It remains to define δ αx for each x ∈ X . There are two possibilities: Subcase 1.
The set { δ βx + 1 | x ∈ X, β < α } is bounded below γ .In this case, set δ α = sup { δ βx + 1 : x ∈ X, β < α } and for each x ∈ X : δ αx = min( S x \ δ α ) Subcase 2.
The set { δ βx + 1 | x ∈ X, β < α } is cofinal in γ .If this case arises, the construction terminates.The construction must halt at some stage α ∗ ≤ κ . To see this, assume towardsa contradiction that it does not. Let E x = { δ βx | β < κ } . Since the construction didnot halt at stage κ , E x is bounded strictly below γ . The construction ensures thatif α < β < κ , then δ αx < δ βx for any x, y ∈ X . It follows that all the sets E x havethe same supremum, say ν . But for every x ∈ X , E x ⊆ S x ∩ ν , E x has ordertype κ ,and E x is cofinal in ν . This contradicts the claim.Suppose first that α ∗ is a limit ordinal. Then for each α < α ∗ , let δ α =sup β<α δ βx + 1. Since α ∗ is the first stage at which the construction halts, δ α < γ for all α < α ∗ . Let f ( x ) = δ p ( x ) for those x such that p ( x ) < α ∗ . Since p is a surjection from X to κ , the range of f is equal to { δ β | β < α ∗ } , which is cofinal in γ .Otherwise α = β + 1 for some ordinal β . Then of course the function f ( x ) = δ βx must be cofinal in γ .Theorem 3.21 is a consequence of Lemma 3.25 and the following elementary fact: Lemma 3.26.
Suppose X is a set, δ is an ordinal, and for each γ ≤ δ , f γ is acofinal function from X to γ . Suppose d : X → X × X is a surjection. Then thereis a surjection g from X to δ that is ordinal definable from d and h f γ | γ ≤ δ i .Proof. We will define a sequence h g γ | γ ≤ δ i by recursion and set g = g δ . For α ≤ δ ,suppose h g γ | γ < α i is given. Define h : X × X → δ by setting h ( x, y ) = g γ ( y )where γ = f α ( x ). Then let g α = h ◦ d . Proof of Theorem 3.21.
By Lemma 3.24, there is a surjection p : X → κ . For each γ ≤ δ , h S x ∩ γ | x ∈ X i is a γ -supercompactness sequence. Applying Lemma 3.25,let f γ : X → γ be the least cofinal function ordinal definable from p and h S x ∩ γ | x ∈ X i . The hypotheses of Lemma 3.26 are now satisfied by taking d : X × X → X to be any surjection. As a consequence, δ (cid:22) ∗ X , which proves the theorem.19 .5 Ordinal definability and ultrapowers We finally turn to a generalization of Theorem 3.21 that has no ZFC analog:
Theorem 3.27.
Suppose X is a set, γ is an ordinal, U is an ultrafilter over X × γ , κ is the critical point of j U , and θ ≥ κ is an ordinal. If j U [ θ ] ∈ M U and j U ( θ ) = θ ,then θ (cid:22) ∗ X . Very roughly, this theorem says that supercompactness up to a fixed point θ cannot be the result of the wellorderable part of an ultrafilter. In another sense,however, the proof of Theorem 3.21 is more general than that of Theorem 3.27. Aroutine modification of the proof of Theorem 3.21 shows the following fact: Theorem 3.28.
Suppose M is an inner model and X ∈ M is a set such that X × X (cid:22) ∗ X in M . Suppose U is an M -ultrafilter over X , κ is the critical pointof j U , and α ≥ κ is an ordinal. If j U [ α ] ∈ Ult(
M, U ) , then α (cid:22) ∗ X . The difference is that we do not require U ∈ M . It is not clear that it is possibleto modify the proof of Theorem 3.27 to obtain such a result.The proof of Theorem 3.27 uses the following version of Vopˇenka’s Theorem,which for reasons of citation we reduce to Bukovsky’s Theorem: Theorem 3.29 (Vopˇenka) . Suppose X and T are sets. Then for any x ∈ X , HOD
T,x is an ι -cc generic extension of HOD T where ι is the least regular cardinalof HOD
T,x greater than or equal to θ { X } .Proof. Recall that θ { X } is the least ordinal not the surjective image of X .By Bukovsky’s Theorem ([12], Fact 3.9), it suffices to verify that HOD T hasthe uniform ι -covering property in HOD T,x . This amounts to the following task.Suppose α and β are ordinals and f : α → β is a function in HOD T,x . We mustfind a function F : α → P ( β ) in HOD T such that for all ξ < α , F ( ξ ) is a set ofcardinality less than ι containing f ( ξ ) as an element.Since f is OD T,x , there is an OD T function g : α × X → β such that g ( ξ, x ) = f ( ξ )for all ξ < α . Let F ( ξ ) = { g ( ξ, u ) | u ∈ X } . Clearly F is OD T , so F ∈ HOD T . Fix ξ < α . By definition, f ( ξ ) = g ( ξ, x ) ∈ F ( ξ ). Finally, since F ( ξ ) (cid:22) ∗ X , ι ∗ X , and F ( ξ ) is wellorderable, | F ( ξ ) | < ι . Proof of Theorem 3.27.
Let j = j U .Fix a function S : X × γ → P ( θ ) such that [ S ] U = j [ θ ]. (That is, S is a θ -supercompactness sequence for U .) For any set T , let M T = Y ( x,ξ ) ∈ X × γ HOD
T,x /U Notice that j [ θ ] ∈ M S since S ( x, ξ ) ∈ HOD
S,x for all ( x, ξ ) ∈ X × γ . It follows thatfor any set T , P ( θ ) ∩ HOD
S,T ⊆ M S,T A ∈ P ( θ ) ∩ HOD
S,T , then j ( A ) and j ↾ θ both belong to M S,T , so A ∈ M S,T since A = ( j ↾ θ ) − [ j ( A )].The key idea of the proof is to construct a sequence T = h T ( ν ) | ν < β ∗ i ofsubsets of θ such that { T ( ν ) | ν < β ∗ } = P ( θ ) ∩ M S,T (3)The construction proceeds by recursion. Suppose T ↾ β = h T ( ν ) | ν < β i hasbeen defined. Assume that { T ( ν ) | ν < β } ( P ( θ ) ∩ M S,T ↾ β , and let T β ⊆ θ bethe least set in the canonical wellorder of P ( θ ) ∩ M S,T ↾ β that does not belong to { T ( ν ) | ν < β } .Eventually, one must reach an ordinal β such that { T ( ν ) | ν < β } = P ( θ ) ∩ M S,T ↾ β : otherwise one obtains a sequence h T ( ν ) | ν ∈ Ord i of distinct subsets of θ ,violating the Replacement and Powerset Axioms. At the least such ordinal β , theconstruction terminates, and one sets β ∗ = β and T = h T ( ν ) | ν < β ∗ i , securing(3).Let δ be the least ordinal such that (2 δ ) HOD
S,T > θ . We claim that j [ P ( δ ) ∩ HOD
S,T ] ∈ M S,T (4)Let P bd ( δ ) denote the set of bounded subsets of δ . Since (2 <δ ) HOD
S,T ≤ θ and j [ θ ] ∈ M S,T , j [ P bd ( δ ) ∩ HOD
S,T ] ∈ M S,T by a standard argument: letting f : θ → P bd ( δ ) ∩ HOD
S,T be a surjection with f ∈ HOD
S,T , j [ P bd ( δ ) ∩ HOD
S,T ] = j ( f )[ j [ θ ]] ∈ M S,T .We claim that j ( δ ) = sup j [ δ ]. This will imply (4), since then j [ P ( δ ) ∩ HOD
S,T ]is equal to the set of A ∈ P ( j ( δ )) ∩ M S,T such that A ∩ α ∈ j [ P bd ( δ ) ∩ HOD
S,T ] forall α < j ( δ ). Here we make essential use of the equality (3).Let δ ∗ = sup j [ δ ]. Let P = Ult(HOD S,T , U )so j restricts to an elementary embedding from HOD S,T to P . To prove that j ( δ ) = δ ∗ , it suffices by the minimality of δ and the elementarity of j to show that(2 δ ∗ ) P > j ( θ ), or, since j ( θ ) = θ , that (2 δ ∗ ) P > θ . Suppose towards a contradictionthat this is false, so (2 δ ∗ ) P ≤ θ .Since j ↾ P bd ( δ ) ∩ HOD
S,T ∈ M S,T , j ↾ P bd ( δ ) ∩ HOD
S,T ∈ HOD
S,T . Thereforethe one-to-one function h : P ( δ ) ∩ HOD
S,T → P ( δ ∗ ) ∩ P defined by h ( A ) = j ( A ) ∩ δ ∗ belongs to HOD S,T . Since (2 δ ∗ ) P ≤ θ , there is an injective function from ran( h ) to θ in P , hence in M S,T , and hence in HOD
S,T by (3). Therefore in HOD
S,T , P ( δ ) ∩ HOD
S,T (cid:22) ran( h ) (cid:22) ∗ θ Since HOD
S,T satisfies the Axiom of Choice, it follows that (2 δ ) HOD
S,T ≤ θ ,which contradicts the definition of δ . This contradiction establishes that δ ∗ =sup j [ δ ], finishing the proof of (4).Let ι = θ +HOD S,T . Then ι ≤ j ( ι ) = θ + P ≤ θ + M S,T ≤ ι j ( θ ) = θ , and the final inequality follows from (3). Hence j ( ι ) = ι . Since ι ≤ (2 δ ) HOD
S,T , j [ ι ] ∈ M S,T as an immediate consequence of (4). Weomit the proof, which is similar to the proof above that j [ P bd ( δ ) ∩ HOD
S,T ] ∈ M S,T .We finally show that θ (cid:22) ∗ X . Assume towards a contradiction that this fails.Then by Theorem 3.29, for every x ∈ X , HOD S,T,x is an ι -cc generic extension ofHOD S,T . By elementarity, it follows that M S,T is an ι -cc generic extension of P .In particular, P is stationary correct in M S,T at ι . This allows us to run Woodin’sproof [13] of the Kunen Inconsistency Theorem to reach our final contradiction.Let B = { ξ < ι | cf( ξ ) = ω } . Recall that κ denotes the critical point of j . SinceHOD S,T satisfies the Axiom of Choice, the Solovay Splitting Theorem [14] applied inHOD
S,T yields a partition h B ν | ν < κ i of ( B ) HOD
S,T into HOD
S,T -stationary sets.Let h B ′ ν | ν < j ( κ ) i = j ( h B ν | ν < κ i ). Then B ′ κ is P -stationary in ι . Therefore B ′ κ is M S,T -stationary in ι since P is stationary correct in M S,T at ι . Since j [ ι ] ∈ M S,T is an ω -closed unbounded set in M S,T and M S,T satisfies that B ′ κ is a stationaryset of ordinals of cofinality ω , the intersection j [ ι ] ∩ B ′ κ is nonempty. Fix ξ < ι suchthat j ( ξ ) ∈ B ′ κ . Clearly ξ ∈ B , so since h B ν | ν < κ i partitions B , there is some ν < ι such that ξ ∈ B ν . Now j ( ξ ) ∈ j ( B ν ) = B ′ j ( ν ) . Therefore the intersection B ′ j ( ν ) ∩ B ′ κ is nonempty, and so since h B ′ ν | ν < j ( κ ) i is a partition, j ( ν ) = κ . Thiscontradicts that κ is the critical point of j .As a corollary of Theorem 3.27, we answer the following question of Schlutzen-berg. Suppose j : V → V is an elementary embedding. Is every set ordinal definablefrom parameters in the range of j ? The question is motivated by the well-knownZFC fact that if j : V → M is an elementary embedding, then every element in M is ordinal definable in M from parameters in the range of j .The answer to Schlutzenberg’s question, however, is no. Theorem 3.30.
Suppose ǫ ≤ η ≤ η ′ are ordinals, ǫ is even, and j : V η → V η ′ is acofinal elementary embedding such that j ( ǫ ) = ǫ . Then j ⋆ ↾ θ ǫ is not definable over H η ′ from parameters in j ⋆ [ H η ] ∪ V ǫ ∪ θ η ′ . To put this theorem in a more familiar context, let us state a special case.
Corollary 3.31.
Suppose j : V → V is an elementary embedding. Let λ = κ ω ( j ) .Then j [ λ ] is not ordinal definable from parameters in j [ V ] ∪ V λ .Proof of Theorem 3.30. Let θ = θ ǫ . Since j ( ǫ ) = ǫ , j ⋆ ( θ ) = θ . Suppose towardsa contradiction that the theorem fails. Then there is a set p ∈ V η , a set a ∈ V ǫ ,an ordinal α < η ′ , and a formula ϕ such that j ⋆ [ θ ] is the unique set k ∈ H η ′ suchthat H η ′ satisfies ϕ ( k, j ( p ) , a, α ). By Proposition 3.15 (or trivially if ǫ is a limitordinal), there is an ordinal ξ such that ξ + 2 ≤ ǫ , a set x ∈ V ξ +1 , and a function f : V ξ +1 → V ǫ such that j ( f )( x ) = a .Let γ be an ordinal such that α < j ( γ ). Define g : V ξ +1 × γ → P ( θ ) so that j ⋆ ( g )( x, α ) = j ⋆ [ θ ]: let g ( u, β ) be the unique set k ∈ H η such that H η satisfies ϕ ( k, p, f ( u ) , β ). Let U be the ultrafilter over V ξ +1 × γ derived from j using ( x, α ).It is easy to check that [ g ] U = j U [ θ ]: [ h ] U ∈ [ g ] U if and only if j ( h )( x, α ) ∈ j ( g )( x, α )if and only if j ( h )( x, α ) = j ( ν ) for some ν < θ if and only if there is some ν < θ h ( u, β ) = ν for U -almost all ( u, β ). By Theorem 3.27, there is a surjectionfrom V ξ +1 to θ , and since V ξ +1 ∈ V ǫ , this contradicts the definition of θ .We include a final result AD R -like result about ordinal definability assuming anelementary embedding from V ǫ +3 to V ǫ +3 : Theorem 3.32.
Suppose ǫ is an even ordinal and there is a Σ -elementary em-bedding from V ǫ +3 to V ǫ +3 . Then there is no sequence of functions h f α | α < θ ǫ +2 i such that for all α < θ ǫ +2 , f α is a surjection from V ǫ +1 to α . This result cannot be proved from the existence of an elementary embedding j : V ǫ +2 → V ǫ +2 (if this hypothesis is consistent): the inner model L ( V ǫ +1 )[ j ]satisfies that there is an elementary embedding from V ǫ +2 to V ǫ +2 , but using that L ( V ǫ +1 )[ j ] satisfies that V = HOD V ǫ +1 ,i where i = j ↾ L ( V ǫ +1 )[ j ], one can easilyshow that in L ( V ǫ +1 )[ j ], there is a sequence h f α | α < θ ǫ +2 i such that for all α < θ ǫ +2 , f α is a surjection from V ǫ +2 to α . Proof of Theorem 3.32.
Suppose towards a contradiction that h f α | α < θ ǫ +2 i issuch a sequence. As a consequence of this assumption and Lemma 3.26, θ ǫ +2 isregular.Note that h f α | α < θ ǫ +2 i ∈ H ǫ +3 . Using the notation from Section 2, fix u ∈ C ǫ +3 such that Φ ǫ +3 ( u ) = h f α | α < θ ǫ +2 i . (Recall that C ǫ +3 is the set of codesin V ǫ +3 for elements of H ǫ +3 .)For ℓ = 0 ,
1, suppose j ℓ : ( V ǫ +2 , u ℓ ) → ( V ǫ +2 , u )is an elementary embedding. Necessarily, u ℓ = j − ℓ [ u ]. Assume j ↾ V ǫ = j ↾ V ǫ . Weclaim that j ⋆ [ θ ǫ +2 ] = j ⋆ [ θ ǫ +2 ]. First, let F ⊆ θ ǫ +2 be the set of common fixed pointsof j ↾ θ ǫ +2 and j ↾ θ ǫ +2 . Since θ ǫ +2 is regular, F is ω -closed unbounded. For each α ∈ F , there is some g ℓα ∈ H ǫ +2 such that j ⋆ℓ ( g ℓα ) = f α . Indeed, one can set g ℓα =Φ( u ℓ ) α . Now j ⋆ [ α ] = j ⋆ [ g ℓα [ V ǫ +1 ]] = f α [ j [ V ǫ +1 ]]. Similarly j ⋆ [ α ] = f α [ j [ V ǫ +1 ]].Since j ↾ V ǫ = j ↾ V ǫ , j ↾ V ǫ +1 = j ↾ V ǫ +1 by Corollary 3.11. It follows that j ⋆ [ α ] = j ⋆ [ α ]. Since F is unbounded in θ ǫ +2 , this implies j ⋆ [ θ ǫ +2 ] = j ⋆ [ θ ǫ +2 ].Let E = { k | k ∈ E (( V ǫ +2 , k − [ u ]) , ( V ǫ +2 , u )) }E = { k ↾ V ǫ | k ∈ E } Let C = T k ∈E k ⋆ [ θ ǫ +2 ]. For any i ∈ E , let A i ⊆ θ ǫ +2 be equal to k ⋆ [ θ ǫ +2 ] forany k ∈ E extending i . This is well-defined by the previous paragraph. Clearly C = T { A i | i ∈ E } . Since θ ǫ +2 is regular and E (cid:22) ∗ V ǫ +1 , it follows that C is ω -closed unbounded in θ ǫ +2 .Now suppose j : V ǫ +3 → V ǫ +3 is Σ -elementary. Then j ⋆ ( E ) = { k | k ∈ E (( V ǫ +2 , k − [ u ]) , ( V ǫ +2 , u )) } (5) j ⋆ ( C ) = \ k ∈ j ⋆ ( E ) k ⋆ [ θ ǫ +2 ] (6)23erifying these equalities is a bit tricky since we only know that j ⋆ : H ǫ +3 → H ǫ +3 is Σ -elementary. (See Lemma 2.12.) (5) is proved by writing E = T n<ω E n where E n = { k | k : ( V ǫ +2 , k − [ u ]) → Σ n ( V ǫ +2 , u ) } Notice that E n is Σ -definable over H ǫ +3 from u and V ǫ +2 , and j ⋆ ( E ) = T n<ω j ⋆ ( E n ).This easily yields (5). (6) is proved by checking that the ⋆ -operation on E ( V ǫ +2 ) isΣ -definable over H ǫ +3 from the parameter H ǫ +2 .It follows that j ↾ V ǫ +2 ∈ j ⋆ ( E ), and hence j ⋆ ( C ) ⊆ j ⋆ [ θ ǫ +2 ]. The only way thisis possible is if | C | < crit( j ). But C is unbounded in θ ǫ +2 . This contradicts that θ ǫ +2 is regular.The Axiom of Choice implies the existence of a sequence h f α | α < θ ǫ +2 i suchthat for all α < θ ǫ +2 , f α is a surjection from V ǫ +1 to α , so Theorem 3.32 yields anew proof of the Kunen Inconsistency Theorem. θ α sequence In this section we study the sequence of cardinals θ α (Definition 2.9). The resultswe will prove suggest that if ǫ is an even ordinal, then assuming choiceless largecardinal axioms, θ ǫ should be relatively large and θ ǫ +1 should be relatively small. Conjecture 4.1.
Suppose ǫ is an even ordinal and there is an elementary embeddingfrom V ǫ +1 to V ǫ +1 . • θ ǫ is a strong limit cardinal. • θ ǫ +1 = ( θ ǫ ) + . We note that if ǫ is a limit ordinal, then one can prove in ZF that θ ǫ is a stronglimit cardinal and θ ǫ +1 = ( θ ǫ ) + . Our first two theorems towards Conjecture 4.1 are the following:
Theorem 4.2.
Suppose ǫ is an even ordinal. Suppose j : V ǫ +3 → V ǫ +3 is anelementary embedding with critical point κ . Then the interval ( θ ǫ +2 , θ ǫ +3 ) containsfewer than κ regular cardinals. This theorem shows that θ ǫ +3 is not too much larger than θ ǫ +2 . Theorem 4.14shows that under further assumptions, θ ǫ +2 is an inaccessible limit of regular car-dinals, so Theorem 4.2 captures a genuine difference between the even and oddlevels. Recall that in the context of ZF, a cardinal is defined to be a strong limit cardinal if it is notthe surjective image of the powerset of any smaller cardinal. heorem 4.3. Suppose ǫ is an even ordinal. Suppose j : V ǫ +2 → V ǫ +2 . Then forany α < κ ω ( j ) , there is no surjection from P ( θ + αǫ +1 ) onto θ ǫ +2 . While we cannot show that θ ǫ +2 is a strong limit, this theorem shows that ithas some strong limit-like properties. Theorem 4.12 below proves the stronger factthat V ǫ +1 surjects onto P ( α ) for all α < θ ǫ +2 , but this theorem requires weak choiceassumptions.The following lemma, which is a key aspect of the proof of both Theorem 4.2and Theorem 4.3, roughly states that rank-to-rank embeddings have no generatorsin the interval ( θ ǫ +2 , θ ǫ +3 ). Lemma 4.4.
Suppose ǫ is an even ordinal and j : V ǫ +1 → V ǫ +1 is an elementaryembedding. Then for every ordinal ν < θ ǫ +1 , there are ordinals α, β < θ ǫ and afunction g : α → θ ǫ +1 such that ν = j ⋆ ( g )( β ) .Proof. Let R be a prewellorder of V ǫ of length ν + 1. Then j ( R ) is a prewellorder of V ǫ of length at least ν + 1. Fix a ∈ V ǫ with rank j ( R ) ( a ) = ν in R . By Corollary 3.12,find an ordinal ξ such that ξ + 2 ≤ ǫ and a = j ⋆ ( f )( x ) for some f : V ξ +1 → V ǫ and x ⊆ V ξ . Define d : V ξ +1 → θ ǫ +1 by setting d ( u ) = rank R ( f ( u )). Then j ⋆ ( d )( x ) = ν ,so ν ∈ ran( j ⋆ ( d )). Let g : α → θ ǫ +1 be the order-preserving enumeration of ran( d ),and note that α < θ ǫ since d witnesses ran( d ) (cid:22) ∗ V ξ +1 . By elementarity, j ⋆ ( g )enumerates ran( j ⋆ ( d )), and hence there is some β < θ ǫ such that j ⋆ ( g )( β ) = ν . Proof of Theorem 4.2.
Let η be the ordertype of the set of regular cardinals inthe interval ( θ ǫ +2 , θ ǫ +3 ). Then η is fixed by j ⋆ , so η = κ . Suppose towards acontradiction that η > κ . Let δ be the κ -th regular cardinal in ( θ ǫ +2 , θ ǫ +3 ). Then j ⋆ ( δ ) is a regular cardinal strictly above δ , so j ⋆ ( δ ) is not equal to sup j ⋆ [ δ ], whichhas cofinality δ . Therefore sup j ⋆ [ δ ] < j ⋆ ( δ ). By Lemma 4.4, there are ordinals α, β < θ ǫ +2 and a function g : α → δ such that j ⋆ ( g )( β ) = sup j ⋆ [ δ ]. Since δ isregular, there is some ordinal ρ < δ such that ran( g ) ⊆ ρ . Therefore j ⋆ ( g )( β ) Let E = h D ( j, a ) | a ∈ [ θ ǫ ] <ω i be the extender of length θ ǫ derived from j . Notice that E is definable over H ǫ +2 from j ⋆ ↾ P bd ( θ ǫ ), hence from j ↾ V ǫ +1 , and hence from j ↾ V ǫ by Corollary 3.11. In fact, there is a partial sequence h F ( σ ) | σ ∈ V ǫ +1 i definable without parameters over H ǫ +2 such that E = F ( j [ V ǫ ]):to be explicit, F ( σ ) is the extender of length θ ǫ derived from k where k = (( π σ ) + ) ⋆ for π σ : σ → M the Mostowski collapse of σ .Let j E : H ǫ +2 → N be the associated ultrapower embedding, and let k : N →H ǫ +2 be the associated factor embedding, defined by k ([ f, a ] E ) = j ⋆ ( f )( a ). Let ν = crit( k ). Note that ν must exist or else j E ↾ θ ǫ +2 = j ⋆ ↾ θ ǫ +2 , contrary tothe undefinability of j ⋆ ↾ θ ǫ +2 over H ǫ +3 from parameters, like E , that lie in V ǫ +2 (Lemma 3.23).Note that ν is a generator of j ⋆ , in the sense that for any ordinals α, β < ν andany function g : α → θ ǫ +2 , ν = j ⋆ ( g )( β ): otherwise ν ∈ ran( k ) by definition. As an25mmediate consequence of Lemma 4.4, it follows that ν ≥ θ ǫ +1 . Since j ⋆ fixes everycardinal in the interval ( θ ǫ +1 , θ + κǫ +1 ), it follows that ν ≥ θ + κǫ +1 .Notice that for any η < θ + κǫ +1 and any set A ⊆ η , j E ( A ) = j ⋆ ( A ). Indeed, j E ( A ) ∩ ν = j ⋆ ( A ) ∩ ν for any set of ordinals A . Suppose towards a contradictionthat p : P ( η ) → θ ǫ +2 is a surjection. Define g : V ǫ +1 → P ( θ ǫ +2 ) by setting g ( σ ) = p ◦ j F ( σ ) [ P ( η )].Let U be the ultrafilter over V ǫ +1 derived from j using j [ V ǫ ]. It is easy to checkthat j U ↾ H ǫ +2 = j ⋆ ↾ H ǫ +2 . Therefore [ g ] U = j U ( g )( j [ V ǫ ]) = j U ( p ) ◦ j E [ P ( η )] = j U ( p ) ◦ j U [ P ( η )] = j U ◦ p [ P ( η )] = j U [ θ ǫ +2 ]. This shows that j U [ θ ǫ +2 ] ∈ M U , so byTheorem 3.21, it follows that θ ǫ +2 (cid:22) ∗ V ǫ +1 , which is a contradiction.This shows that for any η < θ + κǫ +1 , P ( η ) does not surject onto θ ǫ +2 . The sameargument applied to the finite iterates of j shows that for any n < ω , η < θ + κ n ( j ) ǫ +1 , P ( η ) does not surject onto θ ǫ +2 . This proves the theorem.As a corollary of the proof of Theorem 4.3, we have the following fact, whichexhibits a difference between the even and odd levels with regard to Lemma 4.4: Proposition 4.5. Suppose ǫ is an even ordinal and j : V ǫ +2 → V ǫ +2 is an elemen-tary embedding. Then j has a generator in the interval ( θ ǫ +1 , θ ǫ +2 ) . One of the central theorems in the analysis of L ( V λ +1 ) assuming the axiom I is Woodin’s generalization of the Moschovakis Coding Lemma. Here we prove anew Coding Lemma. This Coding Lemma lifts Woodin’s to structures of the form L ( V ǫ +1 ) where ǫ is even and the appropriate generalization of I holds. But more-over, the proof adds a new twist to Woodin’s and as a consequence it applies to a hostof models beyond L ( V ǫ +1 ). For example, the Coding Lemma holds in HOD( V ǫ +1 ),and more interestingly, the Coding Lemma holds in V itself under what seem to bereasonable assumptions. Definition 4.6. Suppose ǫ and η are ordinals, ϕ : V ǫ +1 → η is a surjection, and R is a binary relation on V ǫ +1 . • A relation ¯ R ⊆ R is a ϕ -total subrelation of R if ϕ [dom( ¯ R )] = ϕ [dom( R )]. • A set of binary relations Γ on V ǫ +1 is a code-class for η if for any surjection ψ : V ǫ +1 → η , every binary relation on V ǫ +1 has a ψ -total subrelation in Γ. • The Coding Lemma holds at ǫ if every ordinal η < θ ǫ +2 has a code-class Γsuch that Γ (cid:22) ∗ V ǫ +1 .The Coding Lemma has a number of important consequences. For example: Proposition 4.7. Suppose ǫ is an ordinal at which the Coding Lemma holds. Then θ ǫ +2 is a strong limit cardinal. In fact, for any η < θ ǫ +2 , P ( η ) (cid:22) ∗ V ǫ +1 . roof. Fix a code-class Γ for η with Γ (cid:22) ∗ V ǫ +1 . Fix a surjection ϕ : V ǫ +1 → η .It is immediate that P ( η ) = { ϕ [dom( R )] | R ∈ Γ } . Therefore since Γ (cid:22) ∗ V ǫ +1 , P ( η ) (cid:22) ∗ V ǫ +1 . Definition 4.8. Then the Collection Principle states that every class binary re-lation R whose domain is a set has a set-sized subrelation ¯ R such that dom( ¯ R ) =dom( R ).It seems that one needs a local form of the Collection Principle to prove theCoding Lemma: Theorem 4.9. Suppose ǫ is an even ordinal and M is an inner model containing V ǫ +1 . Suppose there is an embedding j ∈ E ( V ǫ +2 ∩ M ) with crit( j ) = κ . Assume ( H ǫ +2 ) M satisfies κ -DC and the Collection Principle. Then M satisfies the CodingLemma at ǫ . Note that we only require the first-order Collection Principle to hold in ( H ǫ +2 ) M .We begin by proving a Weak Coding Lemma, which requires some more defini-tions. Definition 4.10. Suppose ǫ and η are ordinals, ϕ : V ǫ +1 → η is a surjection, and R is a binary relation on V ǫ +1 . • A relation ¯ R ⊆ R is a ϕ -cofinal subrelation of R if either ϕ [dom( R )] is notcofinal in η or ϕ [dom( ¯ R )] is cofinal in ϕ [dom( R )]. • A set of binary relations Γ on V ǫ +1 is a weak code-class for η if for any surjec-tion ψ : V ǫ +1 → η , every binary relation on V ǫ +1 has a ψ -cofinal subrelationin Γ. • The Weak Coding Lemma holds at ǫ if every ordinal η < θ ǫ +2 has a weakcode-class Γ such that Γ (cid:22) ∗ V ǫ +1 . Lemma 4.11. Suppose ǫ is an even ordinal and M is an inner model containing V ǫ +1 . Suppose there is an elementary embedding from V ǫ +2 ∩ M to V ǫ +2 ∩ M withcritical point κ . Assume ( H ǫ +2 ) M satisfies κ -DC . Then the Weak Coding Lemmaholds at ǫ in M .Proof. Assume towards a contradiction that M does not satisfy the Weak CodingLemma at ǫ . Let η be the least ordinal for which there is no weak code-class Γwith Γ (cid:22) ∗ V ǫ +1 . Notice that η is definable in ( H ǫ +2 ) M , and hence is fixed by anyembedding in E (( H ǫ +2 ) M ).We now use that ( H ǫ +2 ) M satisfies κ -DC to construct a sequence h ( A α , ϕ α ) | α < κ i by recursion. Each A α will be a binary relation on V ǫ +1 , and each ϕ α will be asurjection from V ǫ +1 to η . Suppose h ( A α , ϕ α ) | α < β i has been defined. LetΓ be the collection of binary relations on V ǫ +1 definable (from parameters) over27 V ǫ +1 , A α ) for some α < β . Obviously, Γ (cid:22) ∗ V ǫ +1 , so by choice of η , Γ is not aweak code-class for η . We can therefore choose a binary relation A β on V ǫ +1 and asurjection ϕ β from V ǫ +1 to η such that A β has no ϕ β -cofinal subrelation in Γ. Thiscompletes the construction.Fix an embedding j ∈ E ( V ǫ +2 ∩ M ) with critical point κ . Then j extendsuniquely to j ⋆ : ( H ǫ +2 ) M → ( H ǫ +2 ) M . Apply j ⋆ twice to h ( A α , ϕ α ) | α < κ i : h ( A α , ϕ α ) : α < κ i = j ⋆ ( h ( A α , ϕ α ) | α < κ i ) h ( A α , ϕ α ) | α < κ i = j ⋆ ( h ( A α , ϕ α ) | α < κ i )Notice the following equality: j ⋆ ( A κ , ϕ κ ) = ( A κ , ϕ κ ) (7)(7) implies: j [ A κ ] is a ϕ κ -cofinal subrelation of A κ .The point here is that j ⋆ ( η ) = η , so j ⋆ [ η ] is cofinal in η . Hence ϕ κ [dom( j [ A κ ])] = j ⋆ [ ϕ κ [dom( A κ )]]is cofinal in dom( A κ ). (Note that for every β < κ , the set dom( A β ) is cofinal in η :otherwise for all α < β , A α is vacuously a ϕ β -cofinal subrelation of A β .)Recall, however, that our construction ensured that for all β < κ , A β has no ϕ β -cofinal subrelation that is definable over ( V ǫ +1 , A α ) for some α < β . By elementarity, A κ has no ϕ κ -cofinal subrelation that is boldface definable over ( V ǫ +1 , A α ) forsome α < κ . We reach a contradiction by showing that j ⋆ [ A κ ] is definable over( V ǫ +1 , A κ ).Combinatorially, the following equation is the new ingredient in this proof: j ( j )( A κ ) = A κ (8)(More formally ( j ⋆ ( j ↾ V ǫ )) + ( A κ ) = A κ ; this notation is just too unwieldy.) (8)implies that A κ = j ( j ) − [ A κ ]. By Theorem 3.13, j ( j ) ↾ V ǫ +1 is definable over V ǫ +1 from its restriction to V ǫ , and therefore A κ is definable from parameters over( V ǫ +1 , A κ ). Similarly, j [ A κ ] is definable over ( V ǫ +1 , A κ ). It follows that j [ A κ ] isdefinable over ( V ǫ +1 , A κ ).The proof that the Weak Coding Lemma implies the Coding Lemma is a directgeneralization of Woodin’s: Proof of Theorem 4.9. Assume towards a contradiction that the Coding Lemmafails in M at ǫ . Let η be the least ordinal for which there is no code-class Γ suchthat Γ (cid:22) ∗ V ǫ +1 .Let ψ : V ǫ +1 → η be an arbitrary surjection. We begin by using the CollectionPrinciple to construct a set Λ (cid:22) ∗ V ǫ +1 that is a code-class for every α < η . Considerthe relation S ⊆ V ǫ +1 × H ǫ +2 defined by S ( a, Γ) ⇐⇒ Γ is a code-class for ψ ( a ) with Γ (cid:22) ∗ V ǫ +1 28y the minimality of η , S is a total relation. Since H ǫ +2 satisfies the CollectionPrinciple, there is a total relation ¯ S ⊆ S with ¯ S ∈ H ǫ +2 . LetΛ = [ Γ ∈ ran( ¯ S ) ΓClearly Λ is a code-class for every α < η . Since ¯ S ∈ H ǫ +2 , Λ (cid:22) ∗ V ǫ +1 , so there is asurjection π : V ǫ +1 → Λ.Applying the Weak Coding Lemma, fix a weak code-class Σ for η with Σ (cid:22) ∗ V ǫ +1 .Let Γ be the collection of binary relations on V ǫ +1 definable over ( H ǫ +2 , π ) usingparameters in Σ. Clearly Γ (cid:22) ∗ V ǫ +1 . We finish by showing that Γ is a code-classfor η .Fix a surjection ϕ : V ǫ +1 → η and a binary relation R on V ǫ +1 . We must find a ϕ -total subrelation ¯ R of R that belongs to Γ. First consider the relation S ( a, u ) ⇐⇒ π ( u ) is a ϕ -total subrelation of R ↾ ϕ ψ ( a )(Here R ↾ ϕ β = R ↾ { b | ϕ ( b ) < β } .) Notice that S is a total relation due to theconstruction of Λ. Let ¯ S ∈ Σ be a ψ -cofinal subrelation of S . Since ¯ S ⊆ S , for all u ∈ ran( ¯ S ), π ( u ) ⊆ R . Moreover since ¯ S is a ψ -cofinal subrelation of S , for cofinallymany β < η , there is some u ∈ ran( ¯ S ) such that π ( u ) is a ϕ -total subrelation of R ↾ β . Let ¯ R = [ u ∈ ran( ¯ S ) π ( u )Then ¯ R is a ϕ -total subrelation of R . Moreover ¯ R ∈ Γ since ¯ R is definable over( H ǫ +2 , π ) using the parameter ¯ S ∈ Σ. Theorem 4.12. Suppose ǫ is an even ordinal. Suppose there is an elementary em-bedding from V ǫ +2 to V ǫ +2 with critical point κ . Assume H ǫ +2 satisfies the CollectionPrinciple and κ - DC . Then θ ǫ +2 is a strong limit cardinal. As a corollary of Theorem 4.12, we have a proof of the Kunen InconsistencyTheorem that seems new: Corollary 4.13 (ZFC) . There is no elementary embedding from V λ +2 to V λ +2 .Proof. Assume towards a contradiction that there is an elementary embedding from V λ +2 to V λ +2 . The Axiom of Choice implies that all the hypotheses of Theorem 4.9are satisfied when M = V . Therefore by Theorem 4.12, θ λ +2 is a strong limitcardinal. On the other hand, the Axiom of Choice implies θ λ +2 = | V λ +1 | + , whichis not a strong limit cardinal.Another consequence of the Coding Lemma beyond Theorem 4.12 is the follow-ing theorem: Theorem 4.14. Suppose ǫ is an even ordinal. Suppose there is an elementary em-bedding from V ǫ +2 to V ǫ +2 with critical point κ . Assume H ǫ +2 satisfies the CollectionPrinciple and κ - DC . Then θ ǫ +2 is a limit of regular cardinals. M as in Theorem 4.9, buthere it seems one must require that j is a proper embedding in the sense of [5], andsince we do not want to introduce the notion of a proper embedding, we omit theproof.The proof uses the following lemma, which is a direct generalization of a con-struction due to Woodin: Lemma 4.15 ([5], Lemma 6) . Suppose ǫ is an even ordinal and j : V ǫ +2 → V ǫ +2 isan elementary embedding. Then for any set A ⊆ V ǫ +2 , there is some set B ∈ V ǫ +2 such that j ( B ) = B and A ∈ L ( V ǫ +1 , B ) . We warn that (2.2) in the proof of Lemma 6 of [5] contains a typo. We also needa routine generalization of another theorem of Woodin: Theorem 4.16 ([5], Lemma 22) . Suppose ǫ is an even ordinal, B is a subset of V ǫ +1 ,and j : L ( V ǫ +1 , B ) → L ( V ǫ +1 , B ) is an elementary embedding such that j ( B ) = B .Let θ be θ ǫ +2 as computed in L ( V ǫ +1 , B ) . Then θ is a limit of regular cardinals in L ( V ǫ +1 , B ) . We note that although Woodin’s proof seems to use λ -DC, this is not reallynecessary by the proof of Theorem 4.18 below. Proof of Theorem 4.14. Fix a cardinal η < θ ǫ +2 . We will show that there is aregular cardinal in the interval ( η, θ ǫ +2 ). By the Coding Lemma, there is a code-class Γ for η such that Γ (cid:22) ∗ V ǫ +1 . Let ϕ : V ǫ +1 → η be a surjection in M and let A ⊆ V ǫ +1 be a set such that Γ ⊆ L ( V ǫ +1 , A ) and such that L ( V ǫ +1 , A ) satisfies that η < θ ǫ +2 . Let B be a set such that j ( B ) = B and A ∈ L ( V ǫ +1 , B ).Let θ be θ ǫ +2 as computed in L ( V ǫ +1 , B ). We claim that for every ordinal γ < θ , γ η ⊆ L ( V ǫ +1 , B ). To see this, fix s : η → γ , and we will show that s ∈ L ( V ǫ +1 , B ).Let ψ : V ǫ +1 → γ be a surjection. Let R = { ( x, y ) | s ( ϕ ( x )) = ψ ( y ) } . Since Γ is acode-class for η , there is a subrelation ¯ R of R in Γ such that dom( ¯ R ) = dom( R ).But ¯ R ∈ L ( V ǫ +1 , B ) and s is clearly coded by ¯ R . Therefore s ∈ L ( V ǫ +1 , B ), asdesired.By Theorem 4.16, θ is a limit of regular cardinals in L ( V ǫ +1 , B ). Therefore let ι ∈ ( η, θ ) be a regular cardinal of L ( V ǫ +1 , B ). Since ι η ⊆ L ( V ǫ +1 , B ), cf( ι ) ∈ ( η, θ ).In particular, there is a regular cardinal in the interval ( η, θ ǫ +2 ). Question 4.17. Suppose ǫ is an even ordinal. Suppose there is an elementaryembedding from V ǫ +2 to V ǫ +2 with critical point κ . Assume H ǫ +2 satisfies theCollection Principle and κ -DC. Must θ ǫ +2 be a limit of measurable cardinals?Let us now show that for certain inner models, one can avoid the extra assump-tions in Theorem 4.9. Theorem 4.18. Suppose N is an inner model of ZFC , ǫ is an even ordinal, A ⊆ V ǫ +1 , and W is a set. Let M = N ( V ǫ +1 , A )[ W ] . Suppose there is an elementaryembedding from M ∩ V ǫ +2 to M ∩ V ǫ +2 . Then the Coding Lemma holds in M at ǫ . ketch. We just describe how to modify the proofs above to avoid assuming Collec-tion and Dependent Choice.The first point is that one can prove ( H ǫ +2 ) M satisfies the Collection Principle.To see this, note that by the definition of M , for any X ∈ M , there is an ordinal γ such that X (cid:22) ∗ V ǫ +1 × γ in M . Therefore fix a surjection f : V ǫ +1 × γ → ( H ǫ +2 ) M with f ∈ M . For each x ∈ V ǫ +1 , let Γ x = f [ { x } × γ ] and let < x be the wellorderof H x induced by f . Suppose R ⊆ ( H ǫ +2 ) M is a definable relation whose domainbelongs to ( H ǫ +2 ) M . Then R ∈ M . For each x ∈ V ǫ +1 , let r x ( a ) be the < x -least b in H x such that ( a, b ) ∈ R . The sequence h r x | x ∈ V ǫ +1 i belongs to M , andindeed it belongs to ( H ǫ +2 ) M . This is because one can define a partial surjectionfrom dom( R ) to r x uniformly in x . Therefore S x ∈ V ǫ +1 r x ∈ ( H ǫ +2 ) M , and letting¯ R = S x ∈ V ǫ +1 r x , we have dom( ¯ R ) = dom( R ). This verifies that ( H ǫ +2 ) M satisfiesthe Collection Principle.To finish the sketch, we describe how in this special case one can avoid the useof Dependent Choice in the proof of the Weak Coding Lemma (Lemma 4.11). Oneneed only modify the construction of the sequence h ( A α , ϕ α ) | α < κ i . One insteadconstructs a sequence h ( A x,α , ϕ x,α ) | ( x, α ) ∈ V ǫ +1 × κ i . The construction proceedsas follows. Suppose that h ( A x,α , ϕ x,α ) | ( x, α ) ∈ V ǫ +1 × β i has been defined. Bythe failure of the Weak Coding Lemma, there is some ( A, ϕ ) such that A has no ϕ -cofinal subrelation that is definable over ( V ǫ +1 , A x,α ) for some ( x, α ) ∈ V ǫ +1 × β .For each x ∈ V ǫ +1 , let ( A x,β , ϕ x,β ) be the < x -least such ( A, ϕ ) ∈ H x , if one exists.This completes the construction.Now one considers: h ( A x,α , ϕ x,α ) | ( x, α ) ∈ V ǫ +1 × κ i = j ( h ( A x,α , ϕ x,α ) | ( x, α ) ∈ V ǫ +1 × κ i )Fix any x ∈ V ǫ +1 such that ( A x,κ , ϕ x,κ ) is defined. One then uses ( A x,κ , ϕ x,κ ) inplace of ( A κ , ϕ κ ). The rest of the proof is unchanged.We conclude this section by showing that the Coding Lemma fails at odd or-dinals in a strong sense. For example, we show that if ǫ is even and there is anelementary embedding from V ǫ +3 to V ǫ +3 , then V ǫ +2 does not surject onto P ( θ ǫ +2 ).By Proposition 4.7, this implies that the Coding Lemma does not hold at ǫ + 1. Theorem 4.19. Suppose ǫ is an even ordinal and there is an elementary embeddingfrom V ǫ +2 to V ǫ +2 . Then for any ordinal γ , there is no surjection from V ǫ × γ onto P ( θ ǫ ) .Proof. Suppose towards a contradiction that the theorem fails. Let θ = θ ǫ . Let γ be the least ordinal such that P ( θ ) (cid:22) ∗ V ǫ × γ . We first show γ (cid:22) ∗ V ǫ × P ( θ ) (cid:22) ∗ V ǫ +1 (9)For the first inequality, fix a surjection f : V ǫ × γ → P ( θ ). Define g : V ǫ × P ( θ ) → γ 31y setting g ( A, S ) = min { ξ | f ( A, ξ ) = S } . Let T be the range of g . Then f [ V ǫ × T ] = P ( θ ), so by the minimality of γ , it must be that | T | = γ . Thus γ (cid:22) ∗ V ǫ × P ( θ ).For the second inequality, note that θ (cid:22) ∗ V ǫ so P ( θ ) (cid:22) ∗ P ( V ǫ ) = V ǫ +1 . It followseasily that V ǫ × P ( θ ) (cid:22) ∗ V ǫ +1 .By our large cardinal hypothesis, there is an elementary embedding j : H ǫ +2 →H ǫ +2 . Note that f ∈ H ǫ +2 by (9). By the elementarity of j , j ( f ) is a surjectionfrom V ǫ × j ( γ ) to P ( θ ); therefore, for some ( a, α ) ∈ V ǫ × j ( γ ), j ( f )( a, α ) = j [ θ ]. Bythe cofinal embedding property (Proposition 3.15), or trivially if ǫ is a limit ordinal,there is an ordinal ξ such that ξ + 2 ≤ ǫ , a set x ⊆ V ξ , and a function g : V ξ +1 → V ǫ such that j ( g )( x ) = a . Let U be the ultrafilter over V ξ +1 × γ derived from j using( x, α ). Let h : V ξ +1 × γ → P ( θ ) be defined by h ( u, β ) = f ( g ( u ) , β ). Then it is easyto see that [ h ] U = j U [ θ ]. This contradicts Theorem 3.27.It is a bit strange that for example we require an embedding from V ǫ +2 to V ǫ +2 to show this structural property of P ( θ ǫ ). The theorem implies that P ( θ ǫ ) cannotbe wellordered, so for example in the case “ ǫ = κ ω ( j ),” one cannot reduce the largecardinal hypothesis to an embedding j : V λ +1 → V λ +1 assuming the consistencyof ZFC plus I . (Similar results hold for j : V ǫ +2 → V ǫ +2 , considering the model L ( V ǫ +1 )[ j ].) Inspecting the proof, however, one obtains the following result: Theorem 4.20. Suppose ǫ is an even ordinal and there is an elementary embeddingfrom V ǫ +1 to V ǫ +1 . Then there is no surjection from V ǫ onto P ( θ ǫ ) . The following question is related to Theorem 4.19 and might be more tractablethan the question of whether θ ǫ +1 = ( θ ǫ ) + : Question 4.21. Suppose ǫ is an even ordinal and there is an elementary embeddingfrom V ǫ +2 to V ǫ +2 . Is there a surjection from P ( θ ǫ ) to θ ǫ +1 ? With the goal of refuting strong choiceless large cardinal axioms in mind, Woodin [4]showed that various consequences of the Axiom of Choice follow from the existenceof large cardinals at the level of supercompact and extendible cardinals. Whiledeveloping set theoretic geology in the choiceless context, Usuba realized that theapparently much weaker notion of a L¨owenheim-Skolem cardinal does just as wellas a supercompact. Definition 5.1. A cardinal κ is a L¨owenheim-Skolem cardinal if for all ordinals α < κ ≤ γ , for any a ∈ V γ , there is an elementary substructure X ≺ V γ +1 such that[ X ] V α ⊆ X , a ∈ X , and for some β < κ , X (cid:22) ∗ V β .Here we give a proof of Ulam’s theorem on the atomicity of saturated filters inZF assuming the existence of two strategically placed L¨owenheim-Skolem cardinals.32ur arguments are inspired by the ones in [8], and our result generalizes someof the theorems of that paper while simultaneously reducing their large cardinalhypotheses.Recall that the usual proof of Ulam’s theorem uses a splitting argument thatseems to make heavy use of a strong form of the Axiom of Dependent Choice.Here it is shown that this can be avoided if one is allowed to take two elementarysubstructures. Theorem 5.2. Suppose γ is a cardinal, κ < κ are L¨owenheim-Skolem cardinalsabove γ , and δ is an ordinal. Suppose F is a filter over δ that is V κ -complete andweakly γ -saturated. Then for some cardinal η < γ , there is a partition h S α | α < η i of δ such that F ↾ S α is an ultrafilter for all α < η .Proof. Since κ is a L¨owenheim-Skolem cardinal, we can fix an elementary sub-structure X ≺ V δ + ω +1 with the following properties: • X (cid:22) ∗ V β for some β < κ . • γ, κ , κ , δ, and F belong to X . • [ X ] V α ⊆ X for every α < κ .Let π : H X → V δ + ω +1 be the inverse of the Mostowski collapse of X , and let¯ γ, ¯ κ , ¯ κ , ¯ δ, and ¯ F be the preimages under π of γ, κ , κ , δ, and F respectively.For each ordinal ξ < δ , let U ξ denote the H X -ultrafilter derived from π using ξ . Since π has critical point above κ and H X is closed under V α -sequences forevery α < κ , for all ξ < δ , U ξ is V κ -complete. (More precisely, U ξ generates a V κ -complete filter.) Since { U ξ | ξ < δ } ⊆ P ( H X ) and H X (cid:22) ∗ V β , { U ξ | ξ < δ } (cid:22) ∗ V β +1 .For each ξ < δ , let B ξ = { ξ ′ | U ξ ′ = U ξ } . We make the obvious observation thatthe map sending B ξ to U ξ is a (well-defined) one-to-one correspondence. It followsthat { B ξ | ξ < δ } (cid:22) ∗ V β +1 . Let T ⊆ δ be the set of ξ < δ such that B ξ is F -positive.Since F is V κ -complete and κ > β + 1, T ∈ F . (This is a standard argument: { B ξ | ξ ∈ δ \ T } is a collection of F -null sets with { B ξ | ξ ∈ δ \ T } (cid:22) ∗ V β +1 , andhence T ξ ∈ δ \ T B ξ is F -null by the V κ -completeness of F . Therefore the complementof T ξ ∈ δ \ T B ξ belongs to F . Note that ξ ∈ T if and only if B ξ ⊆ T , and hence T = S ξ ∈ T B ξ . It follows that T is the complement of T ξ ∈ δ \ T B ξ , so T ∈ F asdesired.) Since F is weakly γ -saturated and { B ξ | ξ ∈ T } is a partition of δ intopositive sets, |{ B ξ | ξ ∈ T }| < γ . Given the one-to-one correspondence describedabove, it follows that |{ U ξ | ξ ∈ T }| < γ .Now since κ is a L¨owenheim-Skolem cardinal and γ < κ , we can fix an ele-mentary substructure Y ≺ V δ + ω +2 with { U ξ | ξ ∈ T } ∈ Y, X ∈ Y , γ ⊆ Y , and Y (cid:22) ∗ V β ′ for some β ′ < κ . Since |{ U ξ | ξ ∈ T }| < γ , it follows that U ξ ∈ Y for all ξ ∈ T . Notice, however, that U ξ ∩ Y ∈ X since [ X ] V β ′ ⊆ X . For ξ ∈ T , let A ξ = \ { A ∈ U ξ | A ∈ Y } Notice that A ξ ∈ U ξ for all ξ ∈ T since U ξ is V κ -complete.33e claim that A ξ ∩ A ξ = ∅ whenever U ξ = U ξ . (Obviously if U ξ = U ξ ,then A ξ = A ξ .) To see this, note that since U ξ = U ξ and Y ≺ V δ + ω +2 , thereis some A ∈ H X ∩ Y with A ∈ U ξ and ¯ δ \ A ∈ U ξ . It follows that A ξ ⊆ A and A ξ ⊆ ¯ δ \ A , and hence A ξ ∩ A ξ = ∅ , as desired.Let S = { ξ ∈ T | ¯ F ⊆ U ξ } . Thus S = T ∩ T { A ∈ F | A ∈ X } , so since T and T { A ∈ F | A ∈ X } belong to F , S ∈ F . We claim that for all ξ ∈ S , A ξ is an atom of ¯ F in H X . Fix ξ ∈ S , and suppose towards a contradiction that E and E are disjoint ¯ F -positive subsets of A ξ that belong to H X . Since π ( E ) is F -positive, π ( E ) ∩ S = ∅ , so fix ξ ∈ π ( E ) ∩ S . Note that E ∈ U ξ since U ξ isthe ultrafilter derived from π using ξ . Similarly fix ξ ∈ π ( E ) ∩ S , and note that E ∈ U ξ . Since E and E are disjoint, it follows that U ξ = U ξ . In particular, oneof them is not equal to U ξ . Assume without loss of generality that U ξ = U ξ . Since E ∈ U ξ , we have E ∩ A ξ = ∅ . It follows that A ξ ∩ A ξ = ∅ , and this contradictsthat A ξ ∩ A ξ = ∅ whenever U ξ = U ξ .Therefore { A ξ | ξ ∈ S } is a set of atoms for ¯ F . We claim S ξ ∈ S A ξ ∈ F . Supposenot, towards a contradiction. In other words, the set E = ¯ δ \ [ ξ ∈ S A ξ is F -positive. For every ξ ∈ S , since A ξ ∈ U ξ and ¯ F ⊆ U ξ for all ξ ∈ S , necessarily¯ F ↾ A ξ = U ξ . Note that E belongs to U ξ for some ξ ∈ S : indeed, since π ( E ) is F -positive and S ∈ F , π ( E ) ∩ S = ∅ ; now for any ξ ∈ π ( E ) ∩ S , E ∈ U ξ . But then E ∩ A ξ = ∅ , which contradicts that E = ¯ δ \ S ξ ∈ S A ξ .Finally, note that { A ξ | ξ ∈ T } ∈ H X since [ H X ] γ ⊆ H X . Hence H X satisfiesthat there is a partition of an ¯ F -large set into fewer than γ -many atoms. By theelementarity of π , V δ + ω +1 satisfies that there is a partition of an F -large set intofewer than γ -many atoms. Obviously this is absolute to V , which completes theproof. Corollary 5.3. Suppose η is a limit of L¨owenheim-Skolem cardinals and there isan elementary embedding from V η +2 to V η +2 . If η is regular, then η is measurable,and if η is singular, then η + is measurable. Recall the notion of an ultrafilter comparison (Definition 7.6) that played a rolein Proposition 3.20. One obtains an order on ultrafilters over ordinals by setting U < k W if there is an ultrafilter comparison from ( U, id) to ( W, id). Let us give amore concrete definition of this order. Definition 5.4. Suppose F is a filter over X and h G x | x ∈ X i is a sequence offilters over Y . Then the F -limit of h G x | x ∈ X i is the filter F -lim x ∈ X G x = { A ⊆ Y | { x ∈ X | A ∈ G x } ∈ F } efinition 5.5. Suppose δ is an ordinal. The Ketonen order is defined on countablycomplete ultrafilters over δ by setting U ≤ k W if U is of the form W -lim α<δ Z α where Z α is a countably complete ultrafilter over δ concentrating on α + 1.The following fact is easy to verify: Lemma 5.6. The Ketonen order is transitive and anti-symmetric. Definition 5.7. Set U < k W if U ≤ k W and W k U .Equivalently, U < k W if U ≤ k W and U = W . Also U < k W if and only if U is of the form W -lim α<δ Z α where Z α is a countably complete ultrafilter over δ concentrating on α . Also, as we mentioned above, U < k W if there is an ultrafiltercomparison from ( U, id) to ( W, id). In the appendix, we give a proof of the followingtheorem: Theorem 7.13 (DC) . The Ketonen order is wellfounded. Definition 5.8 (DC) . The Ketonen rank of U , denoted σ ( U ), is the rank of U inthe Ketonen order, and the Ketonen rank of δ , denoted σ ( δ ), is the rank of theKetonen order on δ .A straightforward alternate characterization of the Ketonen order turns out tobe important here. Definition 5.9. Suppose δ is an ordinal. A function h : P ( δ ) → P ( δ ) is Lipschitz if for all α < δ and all A, B ⊆ δ with A ∩ α = B ∩ α , h ( A ) ∩ α = h ( B ) ∩ α ,and h is strongly Lipschitz if for all α < δ and all A, B ⊆ δ with A ∩ α = B ∩ α , h ( A ) ∩ ( α + 1) = h ( B ) ∩ ( α + 1). Lemma 5.10. Suppose U and W are countably complete ultrafilters over δ . • U ≤ k W if and only if there is a countably complete Lipschitz homomorphism h : P ( δ ) → P ( δ ) such that h − [ W ] = U . • U < k W if and only if there is a countably complete strongly Lipschitz homo-morphism h : P ( δ ) → P ( δ ) such that h − [ W ] = U . Notice that an elementary embedding from P ( δ ) to P ( δ ) is a countably completeLipschitz homomorphism. The Ultrapower Axiom (UA) is an inner model principle introduced by the au-thor in [1] to develop the general theory of countably complete ultrafilters and inparticular the theory of strongly compact and supercompact cardinals. Implicit inthe statement of UA is the assumption of the Axiom of Choice, and dropping thatassumption, there are a number of inequivalent reformulations of the principle. InZFC, however, the Ultrapower Axiom is equivalent to the linearity of the Ketonenorder. The following theorem therefore shows that in one sense, the existence of aReinhardt cardinal almost implies the Ultrapower Axiom.35 heorem 5.11. Suppose j : V → V is an elementary embedding and κ ω ( j ) - DC holds. Then for any ordinals δ and ξ , the set of countably complete ultrafilters over δ of Ketonen rank ξ has cardinality strictly less than κ ω ( j ) . We will prove a more technical theorem that also applies to L ( V λ +1 ) and othersmall models. Theorem 5.12. Suppose ǫ is an ordinal, M is an inner model containing V ǫ +1 , and δ < θ Mǫ +2 is an ordinal. Suppose there is a nontrivial embedding j ∈ E ( V ǫ +3 ∩ M ) such that j ↾ P M ( δ ) belongs to M . Assume M satisfies λ - DC where λ = κ ω ( j ) .Then in M , for any ordinal ξ < θ ǫ +3 , the set of countably complete ultrafilters over δ of Ketonen rank ξ is wellorderable and has cardinality strictly less than λ . Why V ǫ +3 ? The point of this large cardinal hypothesis is that working in M ,every countably complete ultrafilter over an ordinal less than θ ǫ +2 belongs to H ǫ +3 ,and moreover the Ketonen order and its rank function are definable over H ǫ +3 .Therefore by the remarks following Definition 2.8, an embedding j ∈ E ( V ǫ +3 ∩ M )lifts to an embedding j ⋆ ∈ E (( H ǫ +3 ) M ) that is Ketonen order preserving and inaddition respects Ketonen ranks in the sense that σ ( j ⋆ ( U )) = j ⋆ ( σ ( U )) for anycountably complete ultrafilter U on an ordinal less than θ ǫ +2 . This is what isneeded for the proof of Theorem 5.12. Proof of Theorem 5.12. By considering the least counterexample, we may assumewithout loss of generality that j fixes δ and ξ . Since M satisfies λ -DC, it suffices toshow that in M , there is no λ -sequence of distinct countably complete ultrafiltersover δ of Ketonen rank ξ . Suppose towards a contradiction that h U α | α < λ i issuch a sequence. Note that h U α | α < λ i is coded by an element of V ǫ +3 ∩ M , so wecan apply j to it. This yields: h U α | α < λ i = j ( h U α | α < λ i ) h U α | α < λ i = j ( h U α | α < λ i )Let κ be the critical point of j . We claim that the following hold for A ∈ P M ( δ ): A ∈ U κ ⇐⇒ j ( A ) ∈ U j ( κ ) (10) A ∈ U κ ⇐⇒ j ( i )( A ) ∈ U κ (11)where i = j ↾ P M ( δ ). (10) is trivial since j ( U κ ) = U j ( κ ) . (11) is slightly more subtlebecause we are not assuming j ↾ P ( P ( δ )) belongs to M , and therefore j ( j )( U κ ) isnot obviously well-defined. Note however that for all α < κ , A ∈ U α if and onlyif i ( A ) ∈ U α . By the elementarity of j , for all α < j ( κ ), A ∈ U α if and only if j ( i )( A ) ∈ U α . In particular, A ∈ U κ if and only if j ( i )( A ) ∈ U κ , proving (11).Now notice that in M , i and j ( i ) are countably complete Lipschitz homomor-phisms from P ( δ ) to P ( δ ). Therefore by the characterization of the Ketonen orderin terms of Lipschitz homomorphisms (Lemma 5.10), (10) and (11) imply: U κ ≤ k U j ( κ ) U κ ≤ k U κ j fixes ξ , which implies that the three ultrafilters U κ , U κ , and U j ( κ ) have Ketonen rank ξ . Since the Ketonen order is wellfounded, it follows that theinequalities above cannot be strict, and so since the Ketonen order is antisymmetric, U j ( κ ) = U κ = U κ . This contradicts that h U α | α < λ i is a sequence of distinctultrafilters.Although Theorem 5.12 shows that the Ketonen order is almost linear underchoiceless large cardinal assumptions, true linearity is incompatible with κ ω ( j )-DC: Proposition 5.13 (DC) . Suppose j : V λ +2 → V λ +2 is an elementary embeddingwith λ = κ ω ( j ) . Suppose λ is a limit of L¨owenheim-Skolem cardinals. Assume therestriction of the Ketonen order to λ + -complete ultrafilters over λ + is linear. Then ω -DC is false.Proof. We begin by outlining the proof. Assume towards a contradiction that ω -DC is true. We first prove that the Ketonen least ultrafilter U on λ + , which existsby the linearity of the Ketonen order, extends the ω -closed unbounded filter. Next,we show that, assuming ω -DC, there is a normal ultrafilter W on λ + extendingthe ω -closed unbounded filter.Let us show that the existence of the ultrafilters U and W actually impliesthat ω is measurable, contradicting ω -DC. Clearly U < k W . As a consequence U = W -lim α<λ + U α where U α is a countably complete ultrafilter such that α ∈ U α for W -almost all α < λ + . For each α < λ + , let δ α be the least ordinal such that δ α ∈ U α . Then δ α = α for W -almost all α < λ + : otherwise, since W is normal,there is a δ < λ + such that δ α = δ for W -almost all α < λ + , which implies δ ∈ W -lim α<λ + U α = U , contradicting that U is a uniform ultrafilter over λ + . Fixan ordinal α < λ + of cofinality ω such that δ α = α . Then U α is a fine ultrafilterover α ; that is, every set in U is cofinal in α . This implies that cf( α ) carries auniform countably complete ultrafilter D : let f : α → ω be any monotone functionand let D = f ∗ ( U α ). This means that ω is measurable, which is a contradiction.The first step is to show that DC implies that there is a normal filter over λ + extending the ω -closed unbounded filter. For this, we show that the weak club filter is normal. This is the filter F generated by sets of the form { sup( σ ∩ λ + ) | σ ≺ M } where M is a structure in a countable language containing λ + . The normalityof this filter, given the L¨owenheim-Skolem hypothesis, is proved by Usuba as [15,Proposition 3.5]. By DC, the set S = { α < λ + | cf( α ) = ω } is F -positive. Hence F ↾ S is a normal filter extending the ω -closed unboundedfilter. By the Woodin argument and Theorem 5.2, F ↾ S is atomic. Therefore thereis some T ⊆ S such that F ↾ T is an ultrafilter. Of course F ↾ T is normal since F is, and hence we have obtained a normal ultrafilter U extending the ω -closedunbounded filter. 37e claim that no uniform ultrafilter over λ + lies below U in the Ketonen order.To see this, suppose Z < k U , and we will show that there is some δ < λ + such that δ ∈ Z . Suppose Z = U -lim ξ<λ + D ξ where D ξ is a countably complete ultrafilterover λ + with ξ ∈ D ξ for U -almost all ξ < λ + . For U -almost all ξ < λ + , ξ hascofinality ω and D ξ is a countably complete ultrafilter with ξ ∈ D ξ , so there is some δ ξ < ξ with δ ξ ∈ D ξ . Since U is normal, there is a fixed ordinal δ < λ + such that δ ξ = δ for U -almost all ξ < λ + . Hence δ ∈ D ξ for U -almost all ξ < λ + , so since Z = U -lim ξ<λ + D ξ , δ ∈ Z .Thus U is the Ketonen least ultrafilter over λ + .Using ω -DC, one can show that { α < λ + | cf( α ) = ω } is positive with respectto the weak club filter. It follows as above that the ω -closed unbounded filterextends to a normal ultrafilter. As explained in the first two paragraphs, this leadsto the conclusion that ω is measurable, which contradicts our assumption that ω -DC holds.It would not be that surprising if it turned out to be possible to refute thelinearity of the Ketonen order outright from choiceless large cardinals. If the lin-earity of the Ketonen order is consistent with choiceless large cardinals, however,then perhaps there is an interesting theory of choiceless large cardinals in whichchoice fails low down. We will not pursue this idea further here since it leads tohighly speculative territory. We do note that one can make do with a weaker choiceassumption in the proof of Theorem 5.12: Theorem 5.14. Suppose ǫ is an ordinal, M is an inner model of DC containing V ǫ +1 , and δ < θ Mǫ +2 is an ordinal. Suppose there is a nontrivial j ∈ E ( V ǫ +3 ∩ M ) suchthat j ↾ P M ( δ ) belongs to M . Assume λ = κ ω ( j ) is a limit of L¨owenheim-Skolemcardinals in M . Then the following hold in M :(1) For any ordinal ξ < θ ǫ +3 , the set of countably complete ultrafilters over δ ofKetonen rank ξ is the surjective image of V α for some α < λ .(2) The set of V λ -complete ultrafilters of Ketonen rank ξ is wellorderable and hascardinality strictly less than λ . The proof uses the following lemma, whose analog in the context of AC is wellknown and does not require the supercompactness assumption that we make below: Lemma 5.15. Assume M is a model of set theory and j : M → N is an elementaryembedding with critical point κ . Assume there is a L¨owenheim-Skolem cardinal η in M such that κ < η < j ( κ ) and j [ V ξ ∩ M ] ∈ N for all ξ < η . Then for any set S ∈ M such that j ( S ) = j [ S ] , there is some α < κ such that M satisfies S (cid:22) ∗ V α .Proof. We first observe that if there is a surjection f : S → S ′ in M , then j ( S ′ ) = j ( f )[ j ( S )] = j ( f )[ j [ S ]] = j [ S ′ ]Work in M , and assume towards a contradiction that S ∗ V α for any α < κ . Fix γ > β and an elementary substructure X ≺ V γ with S ∈ X , [ X ] V α ⊆ X , and for38ome ν < η , X (cid:22) ∗ V ν . Let S ′ = X ∩ S . Notice that there is no surjection from V α to S ′ for any α < κ : if there is, then S ′ ∈ X since X is closed under V α -sequences,and hence S ′ = S because S ∈ X and ( S \ S ′ ) ∩ X = ∅ ; but then S (cid:22) ∗ V α , which isa contradiction. Let ξ be the least rank of a set a that is in bijection with S ′ . Then κ ≤ ξ < η .We now leave M . On the one hand, j ( ξ ) > j ( κ ) > η > ξ . On the other hand, j ( S ′ ) = j [ S ′ ] by our first observation. Let f : a → S ′ be a bijection in M , andnotice that a ∈ N and j ( f ) ◦ j ↾ a ∈ N is a bijection between j [ S ′ ] and a thatbelongs to N . Therefore in N , | j ( S ′ ) | = | a | . It follows that in N , ξ is the least rankof a set in bijection with j ( S ′ ). This contradicts that j ( ξ ) > ξ . Proof of Theorem 5.14. Suppose towards a contradiction that the theorem fails.We work in M for the time being. Let ξ < θ ǫ +3 be the least ordinal such that theset of countably complete ultrafilters over δ of Ketonen rank ξ is not the surjectiveimage of V β for any β < λ . Let S be any set of countably complete ultrafilters over δ of Ketonen rank ξ . Leaving M , a generalization of the proof of Theorem 5.12 willshow that j ( S ) = j [ S ]. Assume otherwise, fix U ∈ j ( S ) \ j [ S ] and consider j ( U )and j ( j )( U ). On the one hand, these ultrafilters must be distinct: by elementarity j ( U ) / ∈ j ( j [ S ]) = j ( j )[ j ( S )], whereas evidently j ( j )( U ) ∈ j ( j )[ j ( S )]. On the otherhand, j ↾ P ( δ ) and j ( j ) ↾ P ( δ ) witness that U ≤ k j ( U ) , j ( j )( U ) in M , and therefore U = j ( U ) = j ( j )( U ) since σ ( U ) = σ ( j ( U )) = σ ( j ( j )( U )) = ξ since of course j and j ( j ) fix the definable ordinal ξ . This is a contradiction, so in fact j ( S ) = j [ S ].By Lemma 5.15, M satisfies that S (cid:22) ∗ V α for some α < crit( j ). This contradic-tion proves (1).Now consider the set S of V λ -complete ultrafilters over δ of Ketonen rank ξ . Byan argument similar to that of Theorem 5.2, one can use the L¨owenheim-Skolemassumption to find a “discretizing family” for S , or in other words a function f : S → P ( δ ) such that f ( U ) ∈ U \ W for all W ∈ S except for U . Then the function g ( U ) = min f ( U ) is an injection from S into δ , so S is wellorderable. Since θ α < λ for all α < λ , it follows that | S | < λ , proving (2).We remark that an argument similar to the proof of Theorem 5.14 can be used toestablish the Coding Lemma (Theorem 4.9) from a L¨owenheim-Skolem hypothesisrather than dependent choice.The semi-linearity of the Ketonen order given by Theorem 5.12 implies that V is in a sense “close to HOD.” (No such closeness result is known to be provablefrom large cardinal axioms consistent with the Axiom of Choice, so this perhapscomplicates the intuition that choiceless large cardinal axioms imply that HOD isa small model.) We first state the theorem in two special cases: Theorem 5.16. Suppose λ is a cardinal such that λ - DC holds. Assume M = L ( V λ +1 ) or M = V . Suppose there is an elementary embedding j : M → M with κ ω ( j ) = λ . Then M satisfies the following statements:(1) Every countably complete ultrafilter over an ordinal belongs to an ordinal defin-able set of size less than λ . 2) Every λ + -complete ultrafilter over an ordinal δ is ordinal definable from a subsetof δ .(3) For any set of ordinals S , every λ + -complete ultrafilter is amenable to HOD S .(4) For any λ + -complete ultrafilter U on an ordinal, the ultrapower embedding j U is amenable to HOD x for a cone of x ∈ V λ . We now prove a more technical result that immediately implies the previoustheorem. Theorem 5.17. Suppose ǫ is an even ordinal, M is an inner model of DC contain-ing V ǫ +1 . Assume there is an elementary embedding j from V ǫ +3 ∩ M to V ǫ +3 ∩ M such that j ↾ P ( δ ) ∈ M for all δ < θ Mǫ +2 . Assume λ = κ ω ( j ) is a limit of L¨owenheim-Skolem cardinals in M . Then the following hold in M :(1) Every countably complete ultrafilter over an ordinal below θ ǫ +2 belongs to anordinal definable set of size less than λ .(2) Every λ + -complete ultrafilter over an ordinal δ < θ ǫ +2 is ordinal definable froma subset of δ .(3) For any set of ordinals S , every λ + -complete ultrafilter over an ordinal δ < θ ǫ +2 is amenable to HOD S .(4) For any λ + -complete ultrafilter U on an ordinal less than θ ǫ +2 , the ultrapowerembedding j U is amenable to HOD x for a cone of x ∈ V λ .Proof. We work entirely in M , using only the conclusion of Theorem 5.14.(1) is clear from Theorem 5.12.For (2), suppose U is a countably complete ultrafilter over δ . Let ξ be theKetonen rank of U , and let h U α : α < η i enumerate the λ + -complete ultrafilters ofKetonen rank ξ . Choose a set A ⊆ δ such that A ∈ U and A / ∈ U α for any α < η ;this is possible because η < λ and the ultrafilters in question are λ + -complete. Since U is the unique λ + -complete ultrafilter over δ of Ketonen rank ξ such that A ∈ U , U is ordinal definable from A .We now prove (3). Let ¯ U = U ∩ HOD S . We must show that ¯ U ∈ HOD S . Fixan OD set P of cardinality less than λ such that U ∈ P . Note that F = T P isordinal definable. Let ¯ F = F ∩ HOD S . Then ¯ F ∈ HOD S . Using λ + -completeness,it is obvious that F is λ -saturated, and it follows that ¯ F is λ -saturated in HOD S .Applying the Ulam splitting theorem inside HOD S , there is some η < λ and apartition h A α | α < η i ∈ HOD S of δ into atoms of ¯ F . Since S A α = δ , there is some α < η such that A α ∈ U . It follows that ¯ F ↾ A α ⊆ U ∩ HOD S = ¯ U , and since ¯ F ↾ A α is a HOD S -ultrafilter, this implies that ¯ F ↾ A α = ¯ U . Clearly ¯ F ↾ A α ∈ HOD, andtherefore so is ¯ U .We only sketch the proof of (4), which requires knowledge of the proof ofVopˇenka’s Theorem. (This is the theorem stating that every set of ordinals isset-generic over HOD; see [13].) We first show that for any cardinal γ , there is some40 ∈ V λ such that j U ↾ P ( γ ) is amenable to HOD x . Let E be the HOD-extenderof length j U ( γ ) derived from j U . Notice that E ⊆ HOD by (3), and moreover E belongs to an ordinal definable set X of size less than λ since U does.The set X is (essentially) a condition in the Vopˇenka forcing to add E to HOD,and below this condition, the Vopˇenka algebra has cardinality less than λ , since it isisomorphic to P ( X ) ∩ OD. It follows that E belongs to HOD x where x ∈ V λ is thegeneric for this Vopˇenka forcing below the condition given by X . Each ultrafilter of E lifts uniquely to an ultrafilter of HOD x by the L´evy-Solovay Theorem [16]: theseultrafilters are λ + -complete and x is HOD-generic for a forcing of size less than λ .It follows that the HOD x -extender of j U of length j U ( γ ) can be computed from E inside HOD x , simply by lifting all the measures of E to HOD x . But from thisextender, one can decode j U ↾ P ( γ ) ∩ HOD x . This shows that there is some x ∈ V λ such that j U ↾ P ( γ ) is amenable to HOD x .If follows from the pigeonhole principle that there is some x ∈ V λ such that j U is amenable to HOD x . Now for any x ≥ OD x , j U is amenable to HOD x by exactlythe same argument we used above to show that E extends from HOD to HOD x .This proves (4).The fixed point filter associated to a set of elementary embeddings plays a keyrole in the theory developed in [5]: Definition 5.18. Suppose j is a function and X is a set. ThenFix( j, X ) = { x ∈ X | j ( x ) = x } Suppose σ is a set of functions. Then Fix( σ, X ) = T j ∈ σ Fix( j, X ).Suppose E is a set of elementary embeddings whose domains contain the set X .Suppose B is a set. Then the fixed point filter B -generated by E on X , denoted F B ( E , X ), is the filter over X generated by sets of the form Fix( σ, X ) where σ ⊆ E and σ (cid:22) ∗ b for some b ∈ B .The sort of techniques we have been using yield the following representation the-orem for ultrafilters over ordinals, which says that in the land of choiceless cardinals,every ultrafilter over an ordinal is one set away from a fixed point filter: Theorem 5.19 (DC) . Suppose ǫ is an ordinal and j : V ǫ +3 → V ǫ +3 is an elementaryembedding. Assume λ = κ ω ( j ) is a limit of L¨owenheim-Skolem cardinals. Suppose δ < θ ǫ +2 is an ordinal and U is a V λ -complete ultrafilter over δ . Then there isan ordinal definable set of elementary embeddings E and a set A ⊆ δ such that U = F V λ ( E , δ ) ↾ A .Proof. The proof is by contradiction. Suppose ξ is least possible Ketonen rank ofa V λ -complete ultrafilter over an ordinal for which the theorem fails. Obviously ξ < θ ǫ +3 . Let j : V ǫ +3 → V ǫ +3 be an elementary embedding. Note that ξ isdefinable in H ǫ +3 , and therefore j ( ξ ) = ξ . It follows that j ( U ) = U for any V λ -complete ultrafilter over an ordinal δ < θ ǫ +2 of Ketonen rank ξ : as in Theorem 5.12, j ↾ P ( δ ) is a countably complete Lipschitz homomorphism witnessing U ≤ k j ( U ),41hile j ( U ) has rank ξ since j ( ξ ) = ξ . Let E be the set of Σ -elementary embeddings k : V ǫ +3 → V ǫ +3 such that k ( ξ ) = ξ . The argument we have just given shows that k ( U ) = U for any k ∈ E .Let F = F V λ ( E , δ ). Clearly, F is ordinal definable: in fact, F is definable over H ǫ +3 from an ordinal parameter. We claim F is κ -saturated where κ = crit( j ).This follows from Woodin’s proof of the Kunen inconsistency theorem. Suppose F is not κ -saturated, so there is a partition h S α | α < κ i of δ into pairwise disjoint F -positive sets. Let h T α | α < j ( κ ) i = j ( h S α | α < κ i ). Since F is first-orderdefinable over H ǫ +3 , T α is F -positive for all α . In particular, T κ is F -positive, orin other words, T κ has nonempty intersection with every set in F . The set of fixedpoints of j below δ belongs to F , so T κ contains an ordinal η that is fixed by j .Now η ∈ S α for some α < κ , and therefore η = j ( η ) ∈ j ( S α ) = T j ( α ) . It followsthat T j ( α ) ∩ T κ = ∅ , so since the sets h T α | α < κ i are pairwise disjoint, j ( α ) = κ .This contradicts that κ is the critical point of j .Using the L¨owenheim-Skolem cardinals, it is easy to show that F is V λ -complete.Therefore by Theorem 5.2, F is atomic.We now show that F ⊆ U . Suppose k ∈ E . As we noted in the first paragraph, k ( U ) = U . Therefore k is a countably complete Lipschitz homomorphism with k − [ U ] = U . If U contains the set of ordinals that are not fixed by k , then k witnesses that U is strictly below U in the Ketonen order, which is impossible.Since U is an ultrafilter, U must instead concentrate on fixed points of k . Since U is V λ -complete, it follows that U contains the basis generating F as in Definition 5.18,so F ⊆ U .Since F is atomic, there is some atom A of F such that U = F ↾ A , and thiscompletes the proof.A number of interesting questions remain. We state them in the context of I ,which is arguably the simplest special case, but obviously the same questions arerelevant in the choiceless large cardinal context. Question 5.20. Assume I . In L ( V λ +1 ), is there a surjection from V λ +1 onto theset of λ + -complete ultrafilters over λ + ?The question is at least somewhat subtle, since one can show that in L ( V λ +1 ),there is a δ < θ λ +2 such that there is no surjection from V λ +1 onto the set of λ + -complete ultrafilters over δ . In fact, one can take δ = ( δ ) L ( V λ +1 ) . This isexactly parallel to the situation in L ( R ). An even more basic question is whetherthe ultrapower of λ + by the unique normal ultrafilter over λ + concentrating onordinals of cofinality ω is smaller than λ + λ .Another question, directly related to Theorem 5.12, concerns the size of an-tichains in the Ketonen order: Question 5.21. Assume I . In L ( V λ +1 ), if h U α | α < λ i is a sequence of λ + -complete ultrafilters over ordinals, must there be α ≤ β < λ such that U α ≤ k U β ?A positive answer would bring us even closer to a “proof of the UltrapowerAxiom” from choiceless cardinals. Actually one can prove a weak version of this for λ + -sequences of ultrafilters, whose statement and proof are omitted.42 .4 The filter canonical extension property We now turn to a different application of the Ketonen order: extending filters toultrafilters. For this, we need the Ketonen order on countably complete filters,which was introduced in [1]: Definition 5.22. Suppose δ is an ordinal. The Ketonen order on filters is definedon countably complete filters F and G on δ as follows: • F < k G if F ⊆ G -lim α<δ F α where for G -almost all α < δ , F α is a countablycomplete filter over δ with α ∈ F α . • F ≤ k G if F ⊆ G -lim α<δ F α where for G -almost all α < δ , F α is a countablycomplete filter over δ with α + 1 ∈ F α .The notation is a bit unfortunate since the Ketonen order on ultrafilters (Def-inition 5.5) need not be equal to the restriction of the Ketonen order on filters tothe class of ultrafilters (although the latter order is an extension of the former one).For example, assuming I , in L ( V λ +1 ), any ω -club ultrafilter over λ + lies below any ω -club ultrafilter over λ + in the Ketonen order on filters, but not in the Ketonenorder on ultrafilters. (We do not know whether the two Ketonen orders can divergeassuming ZFC, though it seems very likely that this can be forced.) The UltrapowerAxiom obviously implies that the Ketonen order on filters coincides with the Keto-nen order on ultrafilters. We will not make substantial use of the Ketonen order onultrafilters for the rest of the paper, so this ambiguity causes no real problem.A distinctive feature of the Ketonen order on filters is that ≤ k is not antisym-metric; similarly F ≤ k G but G k F does not imply F < k G . This makes it hardto generalize arguments like Theorem 5.12 from countably complete ultrafilters tocountably complete filters. Still, many of the key combinatorial properties of theKetonen order do generalize. For example, it is easy to see that the Ketonen orderon filters is transitive. Most importantly, we show in the appendix that this orderis wellfounded: Theorem 7.12 (DC) . The Ketonen order on countably complete filters is well-founded. In the choiceless context, we say a cardinal κ is strongly compact if for everyset X , there is a κ -complete fine ultrafilter over P κ ( X ). Suppose j : V → V is anelementary embedding and λ -DC holds where λ = κ ω ( j ). It seems possible that λ + is then strongly compact. While we do not know how to prove this, and expectit is not provable, we can establish a consequence of strong compactness that isequivalent to strong compactness in ZFC. The consequence we are referring to isthe filter canonical extension property , which is said to hold at κ if every κ -completefilter over an ordinal extends to a κ -complete ultrafilter. If κ is strongly compact,then a standard argument, which does not require the Axiom of Choice, shows thatthe filter canonical extension property holds at κ . (On the other hand, the proofthat every κ -complete filter extends to a κ -complete ultrafilter does use the Axiomof Choice, and in fact any cardinal with this stronger form of the filter canonicalextension property must be inaccessible.)43 heorem 5.23. Suppose j : V → V is an elementary embedding. Assume λ - DC holds where λ = κ ω ( j ) . Then every λ + -complete filter extends to a λ + -completeultrafilter. This is an immediate consequence of the following more local theorem: Theorem 5.24. Suppose ǫ is an even ordinal and ν ≤ ǫ is a limit of L¨owenheim-Skolem cardinals. Suppose there is an elementary embedding j : V ǫ +3 → V ǫ +3 with κ ω ( j ) ≤ ν . Then every V ν -complete filter over an ordinal less than θ ǫ +2 extends toa V ν -complete ultrafilter.Proof. For any elementary embedding k : V ǫ +2 → V ǫ +2 , let k ′ : H ǫ +3 → H ǫ +3 bedefined by k ′ = ( k + ) ⋆ , assuming that k + : V ǫ +3 → V ǫ +3 is Σ -elementary, so that( k + ) ⋆ is well-defined.Let λ = κ ω ( j ), where j is as in the statement of the theorem. We begin with abasic observation, whose proof is lifted from a claim in [7]: for any n < ω and any ξ ≤ ǫ , there is a Σ n -elementary embedding i : H ǫ +3 → H ǫ +3 such that κ ω ( i ) = λ and i ( ξ ) = ξ . To see this, suppose the claim fails for some n . Consider the least ξ for which there is no such embedding. Then ξ is first-order definable from λ over H ǫ +3 , so j ′ ( ξ ) = ξ . But then j ′ itself witnesses that ξ is not a counterexample toour basic observation, and this is a contradiction.Suppose towards a contradiction that the theorem fails. Fix an ordinal η < θ ǫ +2 and a filter F that is minimal in the Ketonen order among all V ν -complete filtersover η that do not extend to V ν -complete ultrafilters.Fix natural numbers n < n < n that are sufficiently far apart for the followingproof to work. For concreteness, one can take n = 10, n = 15, and n = 20.Let E be the set containing every elementary embedding k : V ǫ +2 → V ǫ +2 whoseextension k ′ : H ǫ +3 → H ǫ +3 is well-defined and Σ n -elementary, fixes ν and η ,and has F in its range. Note that E is Σ n -definable over V ǫ +3 . Since ν and η can be coded by a single ordinal ξ ≤ ǫ , we can fix a Σ n -elementary embedding i : H ǫ +3 → H ǫ +3 with κ ω ( i ) = λ , i ( ν ) = ν , and i ( η ) = η .Let G = F V ν ( D , η ) where D is the set of embeddings k ′ : H ǫ +2 → H ǫ +2 inducedby embeddings k ∈ E as in the previous paragraph. The filter G is λ -saturated,as a consequence of Woodin’s proof of the Kunen inconsistency theorem. Supposetowards a contradiction that there is a partition h S α : α < λ i of η into G -positivesets. Let h T α : α < λ i = i ( h S α : α < λ i )Since E is Σ n -definable over V ǫ +3 and i is Σ n -elementary on V ǫ +3 , i ( G ) = F V ν ( D , η ),where D is the set of embeddings k ′ : H ǫ +2 → H ǫ +2 induced by embeddings k ∈ i ( E ).Notice that i ↾ V ǫ +2 ∈ i ( E ): this follows easily by our choice of i and the fact that i ( F ) ∈ ran( i ). Let κ be the critical point of i . Since T κ is i ( G )-positive, it followsthat { ξ | i ( ξ ) = ξ } ∩ T κ is nonempty. Fix ξ such that i ( ξ ) = ξ and ξ ∈ T κ . Note that ξ ∈ S α for some α since h S α | α < λ i is a partition of η . Therefore since i ( ξ ) = ξ , ξ ∈ i ( S α ) = T i ( α ) . Since κ is the critical point of i , i ( α ) = κ . But ξ ∈ T κ ∩ T i ( α ) ,and this contradicts that h T α | α < λ i is a partition.44ince ν is a limit of L¨owenheim-Skolem cardinals, G is V ν -complete, and soTheorem 5.2 implies that there is some ρ < λ and a partition h A α | α < ρ i of η into G -positive sets such that G ↾ A α is an ultrafilter for all α < ρ .The main claim is that G ∪ F generates a proper filter. Granting the claim, theproof is completed as follows. Let H be the filter generated by G ∪ F . Since G and F are V ν -complete filters, given that H is proper, in fact H is V ν -complete. Inparticular, for some α < ρ , A α is H -positive. Let U = G ↾ A α , which is an ultrafilterby definition. Since H ↾ A α is a proper filter and the ultrafilter U = G ↾ A α iscontained in H ↾ A α , in fact, H ↾ A α = U . Since F ⊆ H ⊆ U , U is a V ν -completeextension of F . This contradicts our choice of F , and completes the proof modulothe claim.We finish by showing that G ∪ F generates a proper filter. Suppose it does not,so there is a set in G whose complement is in F . Since G = F V ν ( E , η ), this meansthat there is some β < ν and a sequence h i x | x ∈ V β i ⊆ E such that the set T = [ { α < η | i ′ x ( α ) > α } belongs to F . Let j x = i ′ x .Fix x ∈ V β for the rest of the paragraph. Since i x ∈ E , there is a filter F x such that j x ( F x ) = F . Moreover, j x : H ǫ +3 → H ǫ +3 is Σ n -elementary, j x ( ν ) = ν ,and j x ( η ) = η . It follows that F x does not extend to a V ν -complete ultrafilter:this is because j x is Σ n -elementary and it is a Σ n -expressible fact in H ǫ +3 that j x ( F x ) = F does not extend to a V ν -complete ultrafilter.Let D xα denote the ultrafilter over η derived from j x using α . For α ∈ T , let D α = \ { D xα | j x ( α ) > α } Thus for all α ∈ T , D α is a countably complete filter and α ∈ D α .Notice that \ x ∈ V β F x ⊆ F -lim α ∈ T D α (12)The proof is a matter of unwinding the definitions. Fix A ∈ T x ∈ V β F x . For each x ∈ V β , let S x = { α < η | A ∈ D xα } . In other words, S x = j x ( A ), and so since A ∈ F x , S x ∈ j x ( F x ) = F . Let S = T x ∈ V β S x . Since F is V ν -complete, S ∈ F . Bydefinition, for α ∈ S ∩ T , A ∈ T D xα ⊆ D α . Since S ∩ T ∈ F , this means that for F -almost all α , A ∈ D α . In other words, A ∈ F -lim α ∈ T D α , as desired.Since F x is V ν -complete for every x ∈ V β , T x ∈ V β F x is a V ν -complete filter.Since α ∈ D α for all α ∈ T , (12) implies that T x ∈ V β F x < k F . Since F is a minimalcounterexample to the theorem, it follows that there is a V ν -complete ultrafilter W that extends T x ∈ V β F x .Recall that for every x ∈ V β , F x does not extend to a V ν -complete ultrafilter.It follows that there is a set in W whose complement belongs to F x . Since ν is alimit of L¨owenheim-Skolem cardinals, for some ordinal γ > ǫ , there is an elementarysubstructure X ≺ V γ with V β ⊆ X , h F x | x ∈ V β i ∈ X , W ∈ X , and X (cid:22) ∗ V ζ forsome ζ < ν . Let S be the intersection of all W -large sets that belong to X . Since45 (cid:22) ∗ V ζ , ζ < ν , and W is V ν -complete, S ∈ W . We claim that the complement of S belongs to T x ∈ V β F x . To see this, fix an x ∈ V β . There is a W -large set A ∈ X whose complement belongs to F x since X is an elementary substructure of V γ thatcontains F x and W . Since S is the intersection of all W -large sets in X , S ⊆ A .Hence the complement of S contains the complement of A , and it follows that thecomplement of S belongs to F x .The existence of a set S ∈ W whose complement is in T x ∈ V β F x contradicts that W extends T x ∈ V β F x . This contradiction proves the claim that G ∪ F generates aproper filter, and thereby proves the theorem as explained above.By a similar argument, we also have the following consequence of I : Theorem 5.25 (ZFC) . Suppose there is an elementary embedding from L ( V λ +1 ) to L ( V λ +1 ) with critical point below λ . Then in L ( V λ +1 ) , every λ + -complete filterover an ordinal less than θ λ +2 extends to a λ + -complete ultrafilter. In a groundbreaking recent development, Schlutzenberg [2] has proved the consis-tency of the existence of an elementary embedding from V λ +2 to V λ +2 relative toZF + I : Theorem 6.1 (Schlutzenberg) . Assume λ is an even ordinal and j : L ( V λ +1 ) → L ( V λ +1 ) is an elementary embedding with crit( j ) < λ . Let M = L ( V λ +1 )[ j ↾ V λ +2 ] . Then V λ +2 ∩ M = V λ +2 ∩ L ( V λ +1 ) . Hence M satisfies that there is an elementary embed-ding from V λ +2 to V λ +2 . It follows that the existence of an elementary embedding from V ǫ +2 to V ǫ +2 isequiconsistent with I . Moreover, neither hypothesis implies that V ǫ +1 exists.If ǫ is even and H ǫ +2 satisfies the Collection Principle, every elementary em-bedding from V ǫ +2 to V ǫ +2 extends to a Σ -elementary embedding from V ǫ +3 to V ǫ +3 . Therefore the existence of a Σ -elementary embedding from V ǫ +3 to V ǫ +3 isin some sense the first rank-to-rank axiom beyond an elementary embedding from V ǫ +2 to V ǫ +2 . (Also see Theorem 6.8.) We can prove the existence of sharps fromthis principle: Theorem 6.2. Suppose ǫ is an even ordinal and there is a Σ -elementary embeddingfrom V ǫ +3 to V ǫ +3 . Then A exists for every A ⊆ V ǫ +1 . Recall the following theorem: Proposition 6.3. Suppose λ is a cardinal and there is an elementary embedding j : L ( V λ ) → L ( V λ ) such that κ ω ( j ) = λ . Then V λ exists and for some α < λ , thereis an elementary embedding from V α to V α . 46e will prove the following somewhat unexpected equiconsistency at the V λ +2 to V λ +2 level, which shows that Proposition 6.3 does not generalize to the othereven levels: Theorem 6.8. The following statements are equiconsistent over ZF :(1) For some λ , there is a nontrivial elementary embedding from V λ +2 to V λ +2 .(2) For some λ , there is an elementary embedding from L ( V λ +2 ) to L ( V λ +2 ) withcritical point below λ .(3) There is an elementary embedding j from V to an inner model M that is closedunder V κ ω ( j )+1 -sequences. Combined with Schlutzenberg’s theorem, all these principles are equiconsistentwith the existence of an elementary embedding from L ( V λ +1 ) to L ( V λ +1 ) withcritical point below λ . In particular, the existence of an elementary embedding from L ( V λ +1 ) to L ( V λ +1 ) with critical point below λ is equiconsistent with the existenceof an elementary embedding from L ( V λ +2 ) to L ( V λ +2 ) with critical point below λ . We then turn to some long-unpublished work of the author. The following istechnically an open question: Question 6.4. Does the existence of a nontrivial elementary embedding from V to V imply the consistency of ZFC + I ?Combining a forcing technique due to Woodin [4] and the Laver-Cramer theoryof inverse limits [6], we provide the following partial answer: Theorem 6.19. Suppose λ is an ordinal and there is a Σ -elementary embedding j : V λ +3 → V λ +3 with λ = κ ω ( j ) . Assume DC V λ +1 . Then there is a set genericextension N of V such that ( V λ ) N satisfies ZFC + I . In particular, in the presence of DC, the existence of a Σ -elementary embedding j : V λ +3 → V λ +3 with λ = κ ω ( j ) implies the consistency of ZFC + I .We also briefly outline a proof of the following theorem: Theorem 6.20. The following statements are equiconsistent over ZF + DC :(1) For some λ , E ( V λ +2 ) = { id } .(2) For some λ , λ -DC holds and E ( V λ +2 ) = { id } .(3) The Axiom of Choice + I . The equivalence of (2) and (3) is Schlutzenberg’s Theorem.47 .2 Equiconsistencies and sharps We begin with the equiconsistencies for embeddings of the even levels. Here weneed some basic observations about ultrapowers assuming weak choice principles,which we will later apply to inner models of the form L ( V ǫ +1 )[ C ], which satisfythese principles. Lemma 6.5. Suppose ǫ is an even ordinal and M is an inner model containing V ǫ +1 . Suppose j : V Mǫ +2 → V Mǫ +2 is a Σ -elementary embedding. Assume that for allrelations R ⊆ V ǫ +1 × M in M , there is some S ⊆ R in M such that dom( S ) =dom( R ) and, in M , ran ( S ) (cid:22) ∗ V ǫ +1 . Let U be the M -ultrafilter derived from j using j [ V ǫ ] . Then the ultrapower of M by U satisfies Lo´s’s Theorem. Moreover, if U ∈ M , then in M , Ult( M, U ) is closed under V ǫ +1 -sequences.Proof. To establish Lo´s’s Theorem, it suffices to show that if R ⊆ V ǫ +1 × M belongsto M and dom( R ) ∈ U , has a U -uniformization in M , which is just a f ⊆ R in M such that dom( f ) ∈ U . We can reduce to the case of relations on V ǫ +1 × V ǫ +1 .Given R ⊆ V ǫ +1 × M , take S ⊆ R with dom( S ) = dom( R ) and ran( S ) (cid:22) ∗ V ǫ +1 .Fix a surjection p : V ǫ +1 → ran( S ). Let R ′ = { ( x, y ) | ( x, p ( y )) ∈ S } If g is a U -uniformization of T , then p ◦ g is a U -uniformization of S , and therefore p ◦ g is a U -uniformization of R .Therefore fix R ⊆ V ǫ +1 × V ǫ +1 in M with dom( R ) ∈ U . We have that j [ V ǫ ] ∈ j (dom( R )) by the definition of a derived ultrafilter. Note that dom( j ( R )) = j (dom( R ))by the Σ -elementarity of j . (Here we extend j to act on R , which is essentially anelement of V ǫ +2 .) Therefore j [ V ǫ ] ∈ dom( j ( R )).Fix y ∈ V ǫ +1 such that ( j [ V ǫ ] , y ) ∈ R . Then the function f y given by Defini-tion 3.14 has the property that y = j ( f )( j [ V ǫ ]) (by the proof of Corollary 3.12), butalso f y ∈ M since f y is definable over V ǫ +1 from y . Let g = f ∩ R , so g ⊆ R . Notethat j ( f )( j [ V ǫ ]) = y has the property that ( j [ V ǫ ] , y ) ∈ j ( R ), and hence j ( g )( j [ V ǫ ])is defined and is equal to y . In other words, j [ V ǫ ] ∈ j ( { x ∈ V ǫ +1 | ( x, g ( x )) ∈ R } ),again using the Σ -elementarity of j on V ǫ +2 . This means that dom( g ) ∈ U , so g isa U -uniformization of R that belongs to M .We finally show that if U ∈ M , then Ult( M, U ) is closed under V ǫ +1 -sequencesin M . We might as well assume V = M , since what we are trying to prove isfirst-order over M . Let N = Ult( V, U ). We cannot assume N is transitive, but wewill abuse notation by identifying certain points in N with their extensions.We first show that every set X ∈ [ N ] V ǫ +1 is covered by a set Y ∈ N such that Y (cid:22) ∗ V ǫ +1 in N . Let p : V ǫ +1 → X be a surjection. Let R ⊆ V ǫ +1 × V be therelation defined by R ( x, f ) if p ( x ) = [ f ] U . Take S ⊆ R such that dom( S ) = dom( R )and ran( S ) (cid:22) ∗ V ǫ +1 . Then X ⊆ { j ( f )([id] U ) | f ∈ ran( S ) } ⊆ { g ([id] U ) | f ∈ j ( S ) } Let Y = { g ([id] U ) | f ∈ j ( S ) } . Then X ⊆ Y , Y ∈ N , and Y (cid:22) ∗ V ǫ +1 in N .48ow we show that N is closed under V ǫ +1 -sequences. It suffices to show that[ N ] V ǫ +1 ⊆ N . Fix X ∈ [ N ] V ǫ +1 . Take Y ∈ N with X ⊆ Y and Y (cid:22) ∗ V ǫ +1 in N .Let q : V ǫ +1 → Y be a surjection that belongs to N . Consider the set A = { x ∈ V ǫ +1 | q ( x ) ∈ X } By Corollary 3.12, A ∈ N . Hence q [ A ] = X belongs to N . This finishes theproof.To prove the wellfoundedness of the ultrapower seems to require a stronger hy-pothesis which is related to Schlutzenberg’s results on ultrapowers using L¨owenheim-Skolem cardinals. Lemma 6.6. Suppose ǫ is an even ordinal and j : V ǫ +2 → V ǫ +2 is a Σ -elementaryembedding. Assume that every transitive set N containing V ǫ +1 has an elementarysubstructure H containing V ǫ +1 such that H (cid:22) ∗ V ǫ +1 . Let U be the ultrafilter over V ǫ +1 derived from j using j [ V ǫ ] . Then Ult( V, U ) is wellfounded.Proof. Assume towards a contradiction that the lemma fails. Let α be an ordinalgreater than ǫ such that V α is a Σ -elementary substructure of V . Then Ult( V α , U )is illfounded. Let H be an elementary substructure of V α containing V ǫ +1 and U such that H (cid:22) ∗ V ǫ +1 . (Take H = H ′ ∩ V α where H ′ is an elementary substructureof a N = V α ∪ { V α × U} .)Let P be the Mostowski collapse of H . Let W = U ∩ P . Since V ǫ +1 ⊆ H , W is theimage of U under the Mostowski collapse map. Therefore by elementarity, Ult( P, W )is illfounded. Note that there is a Σ -elementary embedding k : Ult( P, W ) → j U ( P )defined by k ([ f ] W ) = [ f ] U . Therefore j U ( P ) is illfounded. Let E ⊆ V ǫ +1 × V ǫ +1 be awellfounded extensional relation whose Mostowski collapse is P . Then in Ult( V, U ), j U ( E ) has Mostowski collapse j U ( P ) since Lo´s’s Theorem holds by Lemma 6.5.(Note that the L¨owenheim-Skolem hypothesis of this lemma is stronger than thecollection hypothesis from Lemma 6.5.) It follows that j U ( E ) is illfounded. Since E ⊆ V ǫ +1 , j U ( E ) ∼ = j ( E ). Therefore j ( E ) is illfounded. (Here we must extend j slightly to act on binary relations.) This contradicts that j is a Σ -elementaryembedding from V ǫ +2 to V ǫ +2 , since such an embedding preserves wellfoundedness.The following lemma gives an example of a structure satisfying the hypothesesof Lemma 6.5 and Lemma 6.6: Lemma 6.7. Suppose j : V ǫ +2 → V ǫ +2 is a Σ -elementary embedding. Let U be theultrafilter over V ǫ +1 derived from j using j [ V ǫ ] . Then for any class C , the ultrapowerof L ( V ǫ +1 )[ C ] by U using functions in L ( V ǫ +1 )[ C ] is wellfounded and satisfies Lo´s’sTheorem.Proof. Let M = L ( V ǫ +1 )[ C ] . The L¨owenheim-Skolem hypothesis of Lemma 6.6 holdsinside M as an immediate consequence of the fact that M satisfies that everyset is ordinal definable from parameters in V ǫ +1 ∪ { C ∩ M } . This yields Skolemfunctions h f x | x ∈ V ǫ +1 i for any transitive structure N : if ϕ ( v , v ) is a formula,49 x ( ϕ, p ) is the least a ∈ OD C ∩ M,x in the canonical wellorder of OD C ∩ M,x suchthat N (cid:15) ϕ ( a, p ). (Obviously this would work for any structure N in a countablelanguage.) If V ǫ +1 ⊆ N , then closing under these Skolem functions, one obtains anelementary substructure H ≺ N containing V ǫ +1 such that H (cid:22) ∗ V ǫ +1 . Since thishypothesis implies the collection hypothesis from Lemma 6.5, Lo´s’s Theorem holdsfor the ultrapower in question.For the proof of wellfoundedness, we would like to apply Lemma 6.6 inside M ,but the problem arises that U ∩ M may not belong to M . Note, however, that itsuffices to show that Ult( M ′ , U ∩ M ′ ) is wellfounded where M ′ = L ( V ǫ +1 )[ C, U ]: ifthe ultrapower i : M ′ → Ult( N, U ∩ N ) is wellfounded, then since Ult( M, U ∩ M )elementarily embeds into i ( M ) via the canonical factor map, Ult( M, U ∩ M ) iswellfounded as well. Since M ′ is of the form L ( V ǫ +1 )[ C ′ ] for some class C ′ coding C and U , the previous paragraph yields that the hypothesis of Lemma 6.6 holdsinside M ′ . Therefore Lemma 6.6 yields the wellfoundedness of Ult( M ′ , U ∩ M ′ ),which completes the proof. Theorem 6.8. The following theories are equiconsistent:(1) For some λ , there is a Σ -elementary embedding from V λ +2 to V λ +2 .(2) For some λ , there is an elementary embedding from V λ +2 to V λ +2 .(3) For some λ , there is an elementary embedding from L ( V λ +2 ) to L ( V λ +2 ) withcritical point below λ .(4) There is an elementary embedding j : V → M where M is an inner model thatis closed under under V λ +1 -sequences for λ = κ ω ( j ) .Proof. Clearly each statement is implied by the next (except for (4)!), so it sufficesto show that (1) implies that (4) holds in an inner model. Assume (1). Let λ bethe least ordinal such that there is a Σ -elementary embedding j : V λ +2 → V λ +2 .Then λ is a limit ordinal. (1) still holds in L ( V λ +2 )[ U ], and so applying the proofof Lemma 6.7 and Lemma 6.5, we obtain that L ( V λ +2 )[ U ] satisfies (4). This provesthe theorem.The following is a proof, without requiring λ -DC or any choice principles, of [5,Lemma 28]: Theorem 6.9. Suppose ǫ is an even ordinal, A, B ⊆ V ǫ +1 , A ∈ L ( V ǫ +1 , B ) , and j : L ( V ǫ +1 , B ) → L ( V ǫ +1 , B ) is an elementary embedding that fixes B . Assume ( θ ǫ +2 ) L ( V ǫ +1 ,A ) < ( θ ǫ +2 ) L ( V ǫ +1 ,B ) Then A exists, and A ∈ L ( V ǫ +1 , B ) .Proof. For D ⊆ V ǫ +1 , let M D = L ( V ǫ +1 , D ) and let θ D = ( θ ǫ +2 ) L ( V ǫ +1 ,D ) . If D ∈ M B , let U D be the M D -ultrafilter over V ǫ +1 derived from j using j [ V ǫ ]. Let j D : M D → Ult( M D , U D ) be the ultrapower embedding.50ote that M A ∩ V ǫ +2 (cid:22) ∗ V ǫ +1 in M B . Therefore j ↾ M A ∩ V ǫ +2 ∈ M B This yields that U A ∈ M B . Thus within M B , one can compute the ultrapower j A : M A → Ult( M A , U A )In particular, j A ↾ θ B ∈ M B . By a standard argument, j B ↾ θ B / ∈ M B . ( Sketch: Assume not. Inside M B , compute first j B ↾ L θ B ( V ǫ +1 ), then U B , and finally j B : M B → M B . Now in M B , there is a definable embedding from V to V , contradictingTheorem 3.1.)Since j A ↾ θ B ∈ M B and j B ↾ θ B / ∈ M B , it must be that j A ↾ θ B = j B ↾ θ B .Let k : Ult( M A , U A ) → j B ( M A ) be the factor embedding. Note that k ↾ V ǫ +1 is the identity. Therefore j A ( A ) = j B ( A ) = j ( A ), and so by elementarity andwellfoundedness, Ult( M A , U A ) = j B ( M A ) = M j ( A ) . Since j A ↾ θ B = j B ↾ θ B , k hasa critical point, and crit( k ) < θ B . Clearly crit( k ) > ǫ since k ↾ V ǫ +1 is the identity.Thus we have produced an elementary embedding k : L ( V ǫ +1 , j ( A )) → L ( V ǫ +1 , j ( A ))with critical point between ǫ and θ B . This implies that j ( A ) exists. Moreover,since j B ↾ Z ∈ M B for all transitive sets such that Z (cid:22) ∗ V ǫ +1 in M B , the sameholds true of k . In particular, the normal M j ( A ) -ultrafilter W on crit( k ) derivedfrom k belongs to M B . Here we use the Coding Lemma (Theorem 4.18) to see that P (crit( k )) ∩ M A ⊆ P (crit( k )) ∩ M B (cid:22) ∗ V ǫ +1 in M B .Since W is V ǫ +1 -closed (Definition 2.5), it is easy to check that the ultrapower of M j ( A ) by W satisfies Lo´s’s Theorem. This ultrapower is wellfounded since it admitsa factor embedding into k . The elementary embedding j W : M j ( A ) → M j ( A ) is therefore definable over M B . Thus M B satisfies that j ( A ) exists. By elemen-tarity, M B satisfies that A exists. By absoluteness, A exists, and A ∈ M B . Corollary 6.10. Suppose ǫ is an even ordinal and there is a Σ -elementary em-bedding from V ǫ +2 to V ǫ +2 . Then A exists for every A ⊆ V ǫ +1 such that ( θ ǫ +2 ) L ( V ǫ +1 ,A ) < θ ǫ +1 Proof. Fix A ⊆ V ǫ +1 . By Lemma 4.15, there is a set B ⊆ V ǫ +1 such that A ∈ L ( V ǫ +1 , B ), j ( B ) = B , and( θ ǫ +2 ) L ( V ǫ +1 ,A ) < ( θ ǫ +2 ) L ( V ǫ +1 ,B ) Taking the ultrapower of L ( V ǫ +1 , B ) by the ultrafilter derived from j , Lemma 6.7shows that one obtains an elementary embedding i : L ( V ǫ +1 , B ) → L ( V ǫ +1 , B ) suchthat i ( B ) = B . By Theorem 6.9, this implies that A exists.51 orollary 6.11. Suppose ǫ is an even ordinal and there is a Σ -elementary em-bedding from V ǫ +3 to V ǫ +3 . Then A exists for every A ⊆ V ǫ +1 .Proof. We claim that for all A ⊆ V ǫ +1 , ( θ ǫ +2 ) L ( V ǫ +1 ,A ) < θ ǫ +2 . The corollary thenfollows by applying Corollary 6.10. To prove the claim, note that L ( V ǫ +1 , A ) satisfiesthat there is a sequence h f α | α < θ ǫ +2 i such that for all α < θ ǫ +2 , f α : V ǫ +1 → α isa surjection; this is immediate from the fact that L ( V ǫ +1 ) satisfies that every set isordinal definable from parameters in V ǫ +1 ∪ { A } . By Theorem 3.32, this does nothold in V , and therefore ( θ ǫ +2 ) L ( V ǫ +1 ,A ) < θ ǫ +2 .By a similar proof, we obtain the following consistency strength separation: Theorem 6.12. The existence of a Σ -elementary embedding from V λ +3 to V λ +3 implies the consistency of ZF plus the existence of an elementary embedding from V λ +2 to V λ +2 . This follows immediately from a more semantic fact: Proposition 6.13. Suppose ǫ is an even ordinal and there is a Σ -elementaryembedding from V ǫ +3 to V ǫ +3 . Then there is a set E ⊆ V ǫ +1 and an inner model M ⊆ L ( V ǫ +1 , E ) containing V ǫ +1 such that M satisfies that there is an elementaryembedding from V ǫ +2 to V ǫ +2 .Proof. Fix an elementary embedding j : V ǫ +2 → V ǫ +2 . Then the model M = L ( V ǫ +1 )[ j ] satisfies that there is an elementary embedding from V Mǫ +2 to V Mǫ +2 , namely j ↾ V Mǫ +2 . This model also satisfies that there is a sequence h f α | α < θ ǫ +2 i such thatfor all α < θ ǫ +2 , f α : V ǫ +1 → α is a surjection. Thus ( θ ǫ +2 ) M < θ ǫ +2 by the proofof Theorem 3.32. By condensation, this implies that V ǫ +2 ∩ M (cid:22) ∗ V ǫ +1 . Therefore V Mǫ +2 ∪ { j ↾ V Mǫ +2 } ∈ H ǫ +2 It follows that there is a wellfounded extensional relation E on V ǫ +1 whose Mostowskicollapse is V Mǫ +2 ∪ { j ↾ V Mǫ +2 } . Hence M ⊆ L ( V ǫ +1 , E ), as desired. Proof of Theorem 6.12. By minimizing, we may assume λ is a limit ordinal. ByProposition 6.13, there is a set E ⊆ V λ +1 and an inner model M ⊆ L ( V λ +1 , E )containing V λ +1 such that M satisfies that there is an elementary embedding from V λ +2 to V λ +2 . By Corollary 6.11, E exists. Therefore L ( V λ +1 , E ) has a properclass of inaccessible cardinals. Fix an inaccessible δ of L ( V λ +1 , E ) such that δ > λ .Then M ∩ V δ is a model satisfying ZF plus the existence of an elementary embeddingfrom V λ +2 to V λ +2 . It is natural to wonder whether choiceless large cardinal axioms really are strongerthan the traditional large cardinals in terms of the consistency hierarchy. Perhapsthe situation is analogous to the status of the full Axiom of Determinacy in that tra-ditional large cardinal axioms imply the existence of an inner model of ZF containing52hoiceless large cardinals. Could fairly weak traditional large cardinal axioms implythe consistency of the axioms we have been considering in this paper? Using thetechniques of inner model theory, one can show that the choiceless cardinals implythe existence of inner models with many Woodin cardinals. But what about largecardinal axioms currently out of reach of inner model theory?In fact, Woodin showed that one can prove that certain very large cardinals areequiconsistent with their choiceless analogs. For example: Theorem 6.14 (Woodin) . The following theories are equiconsistent: • ZF + there is a proper class of supercompact cardinals . • ZFC + there is a proper class of supercompact cardinals . In this context, we are using the following definition of a supercompact cardinal: Definition 6.15. A cardinal κ is supercompact if for all α ≥ κ , for some β ≥ α and some transitive set N with [ N ] V α ⊆ N , there is an elementary embedding j : V β → N such that crit( j ) = κ and j ( κ ) > α .The proof shows that if there is a proper class of supercompact cardinals, there isa class forcing extension preserving all supercompact cardinals in which the Axiomof Choice holds. (Not every countable model of ZF is an inner model of a model ofZFC, since for example every inner model of a model of ZFC has a proper class ofregular cardinals. More recently, Usuba showed that the existence of a proper classof L¨owenheim-Skolem cardinals suffices to carry out Woodin’s forcing construction.)In particular, this theorem implies that the existence of an elementary embed-ding from V λ to V λ , in ZF alone, implies the consistency of the existence of a properclass of supercompact cardinals in ZFC. Indeed, the same ideas produce modelsof ZFC with many n -huge cardinals from the same hypothesis. But the questionarises whether the weakest of the choiceless large cardinal axioms in fact impliesthe consistency (with ZFC) of all the traditional large cardinal axioms.In this section, we combine Woodin’s method of forcing choice and a reflectiontheorem due to Scott Cramer to prove the following theorem: Theorem 6.16. Over ZF + DC , the existence of a Σ -elementary embedding from V λ +3 to V λ +3 implies Con(ZFC + I ) . This will follow as an immediate consequence of Theorem 6.19 below.We appeal to the following result due to Scott Cramer: Theorem 6.17 (Cramer, [6]) . Suppose λ is a cardinal, V λ +1 exists, and there isa Σ -elementary embedding from ( V λ +1 , V λ +1 ) to ( V λ +1 , V λ +1 ) . Assume DC V λ +1 .Then there is a cardinal ¯ λ < λ such that V ¯ λ ≺ V λ and there is an elementaryembedding from L ( V ¯ λ +1 ) to L ( V ¯ λ +1 ) with critical point less than ¯ λ . This uses the method of inverse limit reflection , which is the technique usedto prove reflection results at the level of I . For smaller large cardinals, reflection53esults are typically not very deep, and tend to require no use of the Axiom ofChoice. It is not clear, however, whether inverse limit reflection can be carriedout without the use of DC. This is the underlying reason that DC is required as ahypothesis in Theorem 6.19.We also appeal to the following theorem of Woodin: Theorem 6.18 (Woodin, [4]) . Suppose δ is supercompact, ¯ λ < δ is such that V ¯ λ ≺ V δ , and j : V ¯ λ +1 → V ¯ λ +1 is an elementary embedding. Then there is a partialorder P ⊆ V λ +1 definable over V ¯ λ +1 without parameters, and a P -name ˙ Q such thatfor any V -generic filter G ⊆ P , the following hold: • V ¯ λ +1 [ G ] = V [ G ] ¯ λ +1 and j [ G ] ⊆ G . • V [ G ] satisfies ¯ λ -DC . • Q = ( ˙ Q ) G is a ¯ λ + -closed partial order in V [ G ] . • For any V [ G ] -generic filter H ⊆ Q , V [ G ][ H ] δ satisfies ZFC . Combining these two theorems, we show: Theorem 6.19. Suppose λ is an ordinal and there is a Σ -elementary embedding j : V λ +3 → V λ +3 with λ = κ ω ( j ) . Assume DC V λ +1 . Then there is a set genericextension N such that ( V λ ) N satisfies ZFC + I .Proof. By Corollary 6.11, V λ +1 exists. Since V λ +1 is definable without parametersin V λ +2 , any elementary embedding from V λ +2 to V λ +2 restricts to an elementaryembedding from ( V λ +1 , V λ +1 ) to ( V λ +1 , V λ +1 ). Therefore the hypotheses of Theo-rem 6.17 are satisfied. It follows that there is a cardinal ¯ λ < λ such that V ¯ λ ≺ V λ and there is an elementary embedding from j : L ( V ¯ λ +1 ) → L ( V ¯ λ +1 ) with criticalpoint less than ¯ λ .Now let δ < λ be a supercompact cardinal of V λ such that δ > ¯ λ and V δ ≺ V λ .(If k : V λ → V λ is elementary, then any point above ¯ λ on the critical sequence of k will do.)It suffices to show that V λ has a forcing extension satisfying ZFC + I . Thehypotheses of Theorem 6.18 hold in V λ , with j equal to j ↾ V ¯ λ +1 . Therefore take P and ˙ Q as in Theorem 6.18, relative to V λ . Let G ⊆ P be V -generic and H ⊆ ( ˙ Q ) G be V [ G ]-generic. We claim that V [ G ][ H ] δ satisfies ZFC + I . The fact that V [ G ][ H ] δ satisfies ZFC is immediate from Theorem 6.18 applied in V λ . Moreover j [ G ] ⊆ G , P ∈ L λ ( V ¯ λ +1 ), and j ( P ) = P , so by standard forcing theory, j extendsto an elementary embedding from L δ ( V ¯ λ +1 )[ G ] to L δ ( V ¯ λ +1 )[ G ]. Since V ¯ λ +1 [ G ] = V [ G ] ¯ λ +1 = V [ G ][ H ] ¯ λ +1 , it follows that j extends to an elementary embedding from L δ ( V [ G ][ H ] ¯ λ +1 ) to L δ ( V [ G ][ H ] ¯ λ +1 ).Therefore V [ G ][ H ] δ is a model of ZFC + I , completing the proof.We finish by very briefly sketching the following equiconsistency: Theorem 6.20. The following statements are equiconsistent over ZF + DC : 1) For some λ , E ( V λ +2 ) = { id } .(2) For some λ , λ -DC holds and E ( V λ +2 ) = { id } .(3) The Axiom of Choice + I . The equiconsistency of (2) and (3) is due to Schlutzenberg.The equiconsistency uses Schlutzenberg’s Theorem (Theorem 6.1) to reduce tothe situation where inverse limit reflection [6] can be applied. Theorem 6.21. Assume there is an embedding j ∈ E ( L ( V λ +1 )) with λ = κ ω ( j ) .Assume DC holds in L ( V λ +1 ) . Then for any infinite cardinal γ < λ , if γ -DC holdsin V λ , then γ -DC holds in L ( V λ +1 ) .Proof. Assume γ -DC holds in V λ . By a standard argument, it suffices to showthat γ -DC V λ +1 holds in L ( V λ +1 ). Suppose T is a γ -closed tree on V λ +1 with nomaximal branches. We must find a cofinal branch of T . Fix α < ( θ λ +2 ) L ( V λ +1 suchthat T ∈ L α ( V λ +1 ). By inverse limit reflection [6], there exist γ < ¯ λ < ¯ α < λ and an elementary embedding J : L ¯ α ( V ¯ λ +1 ) → L α ( V λ +1 ) with T ∈ ran( J ). Let¯ T = J − ( T ). Working in V λ , γ -DC yields a cofinal branch ¯ b ⊆ ¯ T . Since ¯ b is a γ -sequence of elements of V ¯ λ +1 , ¯ b ∈ L ( V ¯ λ +1 ). Therefore ¯ ∈ L ¯ α ( V ¯ λ +1 ). (We mayassume without loss of generality that ¯ α ≥ J (¯ b ) is a cofinal branch of T ,as desired. Corollary 6.22. Assume there is an embedding j ∈ E ( L ( V λ +1 )) with λ = κ ω ( j ) .Assume DC holds in L ( V λ +1 ) . Then for any infinite cardinal γ < λ , if γ -DC holdsin V λ , then γ -DC holds in L ( V λ +1 )[ j ↾ V λ +2 ] .Proof. Let M = L ( V λ +1 )[ j ↾ V λ +2 ]. Again, it suffices to show γ -DC V λ +1 holds in M . But by Schlutzenberg’s Theorem, V λ +2 ∩ M = V λ +2 ∩ L ( V λ +1 ), so M satisfies γ -DC V λ +1 if and only if L ( V λ +1 ) does. Applying Theorem 6.21 then yields thecorollary. Proof of Theorem 6.20. Assume (1). We may assume V = L ( V λ +1 )[ j ] for a non-trivial embedding j ∈ E ( V λ +2 ) with κ ω ( j ) = λ . We build a forcing extensionsatisfying (2). Let h Q α | α < λ i be Woodin’s class Easton iteration for forcingAC, as computed in V λ . (See [4, Theorem 226].) Let P be the inverse limit ofthe sequence h Q α i α<λ , and let ˙ P α,λ be the factor forcing, so Q α ∗ ˙ P α,λ ∼ = P . Byconstruction, there is an increasing sequence h κ α i α<λ such that Q α forces κ α -DCover V λ and P α,λ is κ + α -closed in V . But by Corollary 6.22, Q α forces κ α -DC over V . Therefore using that P α,λ is κ + α -closed in V , if G ⊆ P is V -generic, V [ G ∩ Q α ] isclosed under κ α -sequences in V [ G ] and V [ G ] satisfies κ α -DC. Since the cardinals κ α increase to λ , this shows that V [ G ] satisfies λ -DC. Moreover the standard mastercondition argument for I -embeddings, given for example in [17, Lemma 5.2], showsthat G can be chosen so that the embedding j lifts to an elementary embedding j ∗ : V [ G ] λ +2 → V [ G ] λ +2 .The equiconsistency of (2) and (3) is Schlutzenberg’s Theorem [2].55 Appendix In this appendix, we collect together the wellfoundedness proofs for the various Ke-tonen orders we have used throughout the paper. We take a more general approachby considering a Ketonen order on countably complete filters on complete Booleanalgebras. The orders we have considered so far belong to the special case wherethe Boolean algebras involved are atomic. In our view, the more abstract approachsignificantly clarifies the wellfoundedness proofs. For a more concrete approach, seethe treatment in the author’s thesis [1]. Definition 7.1. Suppose B and B are complete Boolean algebras. A σ -map from B to B is a function that preserves 0, 1, and countable meets.More abstractly, a σ -map is a homomorphism between B and B viewed ascountably complete bounded meet-semilattices. We work with σ -maps rather thancountably complete homomorphisms so that our results apply to the Ketonen orderon filters in addition to the Ketonen order on ultrafilters: if h F y | y ∈ Y i is asequence of countably complete filters over X , then the function h : P ( X ) → P ( Y )defined by h ( A ) = { y ∈ Y | A ∈ F y } is a σ -map, but is a countably completehomomorphism only if F y is an ultrafilter for all y ∈ Y .Given a σ -map from B to B , one can define a natural B -valued relation onnames for ordinals in V B and V B : Definition 7.2. Suppose B and B are Boolean algebras, ˙ α ∈ V B and ˙ α ∈ V B are names for ordinals, and h : B → B is a σ -map. Then J ˙ α < ˙ α K h = _ β ∈ Ord h ( J ˙ α < β K B ) · J ˙ α = β K B Note that “ ˙ α < ˙ α ” is not a formula in the forcing language associated to either B or B . The notation should be regarded as purely formal.This notation is motivated by the following considerations. Suppose h : B → B is a complete homomorphism. Then there is an embedding i : V B → V B definedby i ( ˙ x ) = h ◦ ˙ x . In this case, J ˙ α < ˙ α K h = J i ( ˙ α ) < α K B . More generally, J ˙ α < ˙ α K h = J h ( J ˙ α < ˙ α K B ) ∈ ˙ G B K B Recall that ˙ G B denotes the canonical name for a generic ultrafilter in the forcinglanguage associated with the complete Boolean algebra B . Given our assertionabove that “ ˙ α < ˙ α ” is not a formula in the forcing language associated to B , thereader may want to take some time interpreting the right-hand side of the formulaabove.The following lemma asserts a form of wellfoundedness for the relation given byDefinition 7.2: Lemma 7.3. Suppose h B n , h n,m : B n → B m | m ≤ n < ω i is an inverse system ofcomplete Boolean algebras and σ -maps. Suppose for each n < ω , ˙ α n is a B n -namefor an ordinal. Then V n<ω h n, ( J ˙ α n +1 < ˙ α n K h n +1 ,n ) = 0 . roof. Assume towards a contradiction that the lemma is false. Let β be the leastordinal such that for some h B n , h n,m , ˙ α n : m ≤ n < ω i witnessing the failure of thelemma, J ˙ α = β K B · V n<ω h n, ( J ˙ α n +1 < ˙ α n K h n +1 ,n ) = 0.The definition of J ˙ α < ˙ α K h , yields: J ˙ α = β K B · J ˙ α < ˙ α K h , = J ˙ α = β K B · h , ( J ˙ α < β K B ) ≤ h , ( J ˙ α < β K B ) (13)For each m < ω , let a m = V m ≤ n<ω h n,m ( J ˙ α n +1 < ˙ α n K h n +1 ,n ). Since h , is a σ -map, a = J ˙ α < ˙ α K h , · h , ( a )As a consequence of this and (13): J ˙ α = β K B · a = J ˙ α = β K B · J ˙ α < ˙ α K h , · h , ( a ) ≤ h , ( J ˙ α < β K B ) · h , ( a )= h , ( J ˙ α < β K B · a )By our choice of β , J ˙ α = β K B · a = 0, so we can conclude that J ˙ α < β K B · a = 0.Therefore there is some ξ < β such that J ˙ α = ξ K B · a = 0. This contradicts theminimality of β . Definition 7.4. Suppose F and F are countably complete filters on the completeBoolean algebras B and B . A σ -reduction h : F → F is a σ -map h : B → B such that F ⊆ h − [ F ]. Suppose ˙ α ∈ V B and ˙ α ∈ V B are names for ordinals.A σ -comparison h : ( F , ˙ α ) → ( F , ˙ α ) is a σ -reduction h : F → F such that J ˙ α < ˙ α K h ∈ F . Theorem 7.5. There is no infinite sequence of σ -comparisons and countably com-plete filters of the form · · · h , −→ ( F , ˙ α ) h , −→ ( F , ˙ α ) h , −→ ( F , ˙ α ) .Proof. Assume towards a contradiction that there is such a sequence. Fix n < ω .Since h n +1 ,n : ( F n +1 , ˙ α n +1 ) → ( F n , ˙ α n ) is a σ -comparison, J ˙ α n +1 < ˙ α n K h n +1 ,n ∈ F n Let h n, = h , ◦ · · · ◦ h n,n − . Clearly h n, : F n → F is a σ -reduction, and therefore h n, ( J ˙ α n +1 < ˙ α n K h n +1 ,n ) ∈ F Since F is a countably complete filter, V m<ω h m, ( J ˙ α m +1 < ˙ α m K h m +1 ,m ) ∈ F . Inparticular, this infinite meet is not 0, contrary to Lemma 7.3.We now use this to prove the wellfoundedness of the Ketonen order and theirreflexivity of the internal relation. This is a matter of specializing the theoremswe have proved to the case of atomic Boolean algebras. Definition 7.6. A pointed filter on a set X is a pair ( F, f ) where F is a countablycomplete filter over X and f : X → Ord is a function.57very function f : X → Ord can be associated to the P ( X )-name τ f for theordinal f ( x G ) where x G ∈ X is the point in X selected by the generic (i.e., principal)ultrafilter G ⊆ P ( X ). More concretely, the name τ f is defined by setting dom( τ f ) = S x ∈ X f ( x ) and τ f ( α ) = { x ∈ X | α < f ( x ) } for all α ∈ dom( τ f ). Identifying ( F, f ) and ( F, τ f ), Definition 7.4 is transformed asfollows: Definition 7.7. Suppose ( F, f ) and ( G, g ) are pointed filters over X and Y and Z = h Z y | y ∈ Y i is a sequence of countably complete filters over X . • Z is a filter reduction from F to G if F = G -lim y ∈ Y Z y . • Z is a filter comparison from ( F, f ) to ( G, g ) if Z is a filter reduction from F to G and for G -almost all y ∈ Y , for Z y -almost all x ∈ X , f ( x ) < g ( y ).We write Z : F → G to indicate that Z is a filter reduction from F to G . We write Z : ( F, f ) → ( G, g ) to indicate that Z is a filter comparison from ( F, f ) to ( G, g ).Clearly, σ -reductions and σ -comparisons generalize filter reductions and filtercomparisons. Let us state this more precisely. Definition 7.8. Let Φ be the function sending a σ -map h : P ( X ) → P ( Y ) to thesequence h Z y | y ∈ Y i where Z y = { A ⊆ X | y ∈ h ( A ) } is the filter over X derivedfrom h using y . Lemma 7.9. Suppose ( F, f ) and ( G, g ) are pointed filters over X and Y . Suppose h : P ( X ) → P ( Y ) is a σ -map. Then Φ( h ) is a filter reduction from F to G if andonly if h is a σ -reduction from F to G . Moreover Φ( h ) is a filter comparison from ( F, f ) to ( G, g ) if and only if h is a σ -comparison from ( F, τ f ) to ( G, τ g ) . As an immediate corollary of these lemmas and Theorem 7.5, we have the fol-lowing theorems: Theorem 7.10. There is no descending sequence of pointed filters and filter com-parisons of the form · · · Z −→ ( F , f ) Z −→ ( F , f ) Z −→ ( F , f ) . Of course, the Ketonen order on filters can be characterized in terms of thenotion of a filter reduction: Lemma 7.11. Suppose F and F are countably complete filters over ordinals. Then F < k F in the Ketonen order on filters if and only if there is a σ -comparison from ( F , id) to ( F , id) . Theorem 7.12 (DC) . The Ketonen order on filters is wellfounded. Whenever U < k W in the Ketonen order on ultrafilters, U < k W in the Ketonenorder on filters. Therefore Theorem 7.12 implies: Theorem 7.13 (DC) . The Ketonen order on ultrafilters is wellfounded. 58e finally prove the irreflexivity of the internal relation. Lemma 7.14. Suppose U and W are countably complete ultrafilters over sets X and Y . Suppose h Z y | y ∈ Y i is a sequence of countably complete ultrafilters witnessing U < W . Suppose κ is an ordinal and g : Y → κ is a function such that for any α < κ , g ( y ) > α for W -almost all y ∈ Y . Then for any function f : X → κ , Z : ( U, f ) → ( W, g ) .Proof. Let h U y | y ∈ Y i witness that U < W . Then easily U = W - lim y ∈ Y Z y So h Z y | y ∈ Y i is an ultrafilter reduction from U to W . Fix a function f : X → κ .We must now verify that for W -almost all y ∈ Y , for Z y -almost all x ∈ X , f ( x ) < g ( y ). Since h Z y | y ∈ Y i witnesses U < W , it suffices to show that for U -almost all x ∈ X , for W -almost all y ∈ Y , f ( x ) < g ( y ). But this is a trivialconsequence of our assumption on g and W . Corollary 7.15. Suppose U is a countably complete ultrafilter such that j U has acritical point. Then U < U .Proof. Let X be the underlying set of U . The fact that j U has a critical point κ implies that there is a function g : X → κ such that for any α < κ , g ( x ) > α for U -almost all x ∈ X . Assume U < U . Then by Lemma 7.14, there is an ultrafiltercomparison from ( U, g ) to ( U, g ). This contradicts Theorem 7.10. References [1] Gabriel Goldberg. The Ultrapower Axiom . PhD thesis, Harvard University,2019.[2] Farmer Schlutzenberg. On the consistency of ZF with an elementary embedding j : V λ +2 → V λ +2 . arXiv preprint, arXiv:2006.01077, 2020.[3] Kenneth Kunen. Elementary embeddings and infinitary combinatorics. J.Symbolic Logic , 36:407–413, 1971.[4] W. Hugh Woodin. Suitable extender models I. J. Math. Log. , 10(1-2):101–339,2010.[5] W. Hugh Woodin. Suitable extender models II: beyond ω -huge. J. Math. Log. ,11(2):115–436, 2011.[6] Scott S. Cramer. 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