Extenders under ZF and constructibility of rank-to-rank embeddings
aa r X i v : . [ m a t h . L O ] J u l Extenders under ZF andconstructibility of rank-to-rank embeddings
Farmer Schlutzenberg ∗ [email protected] 10, 2020 Abstract
Assume ZF (without the Axiom of Choice). Let j : V δ → V δ be a non-trivial Σ -elementary embedding, where δ is a limit ordinal. We provesome basic restrictions on the constructibility of j from V δ ; in particular,if j ∈ L ( V δ ) then cof( δ ) > ω . We show that, however, assuming an I -embedding, with the appropriate δ, j , it is possible to have j ∈ L ( V δ ).Assuming Dependent Choice and that δ has countable cofinality (but notassuming V = L ( V δ )), and j is as above, we show that the collectionof such embeddings is of high complexity, and that there are “perfectlymany” such embeddings. We also show that a ZF theorem of Suzuki, thatno elementary j : V → V is definable from parameters, actually followsfrom a theory weaker than ZF. The main results rely on a developmentof extenders under ZF, which we also give. Large cardinal axioms are typically exhibited by a class elementary embedding j : V → M where V is the universe of all sets and M ⊆ V some transitive inner model.Stronger large cardinal notions are obtained by demanding more resemblancebetween M and V , and by demanding that more of j gets into M . WilliamReinhardt, in [8] and [9], took this notion to its extreme by setting M = V ;that is, he considered embeddings of the form j : V → V. The critical point of such an embedding became known as a Reinhardt cardinal .But Kunen showed in [7] that if V | = ZFC then there is no such j , and in factno ordinal λ and elementary embedding j : V λ +2 → V λ +2 . Given this violation ∗ Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda-tion) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics M¨unster:Dynamics-Geometry-Structure. The least ordinal κ such that j ( κ ) > κ . If there is j : V → V then there is a proper class of ordinals α such that j : V α → V α ,and then j : V α + n → V α + n for each n < ω . AC , most research in large cardinals following Kunen’s discovery has beenfocused below the AC threshold. In this paper, we drop the AC assumption,and investigate embeddings j : V δ → V δ , assuming only ZF in V . (We mentionit explicitly when we use some DC .) Thus, Kunen’s objections are not validhere, and moreover, no one has succeeded in refuting j : V λ +2 → V λ +2 , or even j : V → V , in ZF alone, except for in the sense we describe next.The following theorem is Suzuki [13, Theorem 2.1]: Fact 1.1 (Suzuki) . Assume ZF and let j ⊆ V be a class which is definablefrom parameters. Then it is not the case that j : V → V is a Σ -elementaryembedding. So if we interpret “class” as requiring definability from parameters – whichis one standard interpretation – then Suzuki’s theorem actually disproves theexistence of Reinhardt cardinals from ZF . However, there are other interpreta-tions of “class” which are more liberal, and for which this question is still open.(And significantly, note that this theorem doesn’t seem to say anything aboutthe existence of an elementary j : V λ +2 → V λ +2 .)What can “classes” be, if not definable from parameters? One way to con-sider this question, is to shift perspective: instead of considering V , let us con-sider some rank segment V δ of V , where δ is an ordinal. The sets X ⊆ V δ suchthat X is definable from parameters over V δ , are exactly those sets X ∈ L ( V δ ),the first step in G¨odel’s constructibility hierarchy above base set V δ . So bySuzuki’s theorem, if V δ | = ZF , then there is no Σ -elementary j : V δ → V δ with j ∈ L ( V δ ). Viewed this way, we now have the obvious question, as to whetherwe can have such a j ∈ L α ( V δ ), for larger α . And the subsets of V δ which are in L ( V δ ), for example, provide another reasonable notion of “class” with respectto V δ .So, wanting to examine these issues, this paper focuses mostly on the ques-tion of the constructibility of such j from V δ , looking for generalizations ofSuzuki’s theorem, and examining the consequences of having such j ∈ L ( V δ ).We remark that the papers [4], [2] develop some general analysis of rank-to-rank embeddings j : V δ → V δ , for arbitrary ordinals δ , where we include allΣ -elementary maps in this notion. But in this paper, we consider only thecase that δ is a limit. For convenience, we write E m ( V δ ) for the set of allΣ m -elementary j : V δ → V δ , and E ( V δ ) = E ω ( V δ ).The first basic result here is the following (in the case that δ is inaccessi-ble, the result is due independently and earlier to Goldberg, who used othermethods): Theorem (6.3) . Assume ZF + V = L ( V δ ) where cof( δ ) > ω . Then E ( V δ ) = ∅ ;that is, there is no Σ -elementary j : V δ → V δ . Here given a structure M and X ⊆ M , we say that X is definable from parameters over M iff there is ~p ∈ M <ω and a formula ϕ such that for all x ∈ M , we have x ∈ X iff M | = ϕ ( x, ~p ). Actually, because V | = ZF , Σ -elementarity is equivalent to full elementarity in this case. That is, we start with V δ as the first set in the hierarchy, instead of L = ∅ , and thenproceed level-by-level as for L . Most of the results in this paper first appeared in the author’s informal notes [12]. Thosenotes were broken into pieces according to theme, and this paper is one of those pieces. Thereare also some further observations here not present in [12], mainly in §
7. We also corrected anerror in the definition of extender given in [12]; see § δ ) = ω in L ( V δ ), we can still establish the following: Theorem (7.6) . Assume ZF , δ ∈ Lim and j ∈ E ( V δ ) . Let κ ∈ OR be leastwith L κ ( V δ ) admissible. Then j / ∈ L κ ( V δ ) , j is not Σ e L κ ( V δ )1 , and not Π e L κ ( V δ )1 . We also show in Theorem 7.8, assuming an I -embedding, that it is consistentthat j is I and j ∈ L κ +1 ( V δ ) (for the appropriate δ ).The question of constructibility of embeddings is also related to questions onuniqueness of rank-to-rank embeddings. That is, consider an I rank-to-rankembedding j (so j : V δ → V δ where δ is a limit and δ = κ ω ( j ) = lim n<ω κ n where κ = cr( j ) and κ n +1 = j ( κ n )). Each finite iterate j n is also such anembedding, and lim n<ω cr( j n ) = δ . It follows easily that for each α < δ , thereare multiple elementary k : V δ → V δ such that k ↾ V α = j ↾ V α (just consider j n ( j ) for some sufficiently large n ). But what if instead, j : V λ + ω → V λ + ω iselementary where λ = κ ω ( j )? Can there be n < ω such that j is the uniqueelementary k : V λ + ω → V λ + ω extending j ↾ V λ + n ? The I argument clearlydoesn’t work here, since an elementary k : V λ + ω → V λ + ω must have cr( k ) < λ .The answer is “no” under DC : Theorem (7.11) . Assume ZF and let δ be a limit ordinal where E ( V δ ) = ∅ , and m ∈ [1 , ω ] . Then:1. In a set-forcing extension of V , for each V -amenable j ∈ E m ( V δ ) and α < δ there is a V -amenable k ∈ E m ( V δ ) with k ↾ V α = j ↾ V α but k = j .2. If DC holds and cof( δ ) = ω then for each j ∈ E m ( V δ ) and α < δ there is k ∈ E m ( V δ ) with k ↾ V α = j ↾ V α but k = j . More generally, given a Σ -elementary j ∈ E ( V δ ) with limit δ , one can askwhether j ∈ HOD( V δ ). (The I example above does give an instance of this,since L ( V δ ) ⊆ HOD( V δ ).)We also prove the following related result, which is another kind of strength-ening of Suzuki’s theorem above, and also a strengthening of [4, Theorem 5.7part 1***?]. The theory RL ( Rank Limit , see Definition 2.2) is a sub-theory of ZF , which holds in V η for limit η : Theorem (2.3) . Assume RL . Then there is no Σ -elementary j : V → V whichis definable from parameters. As a useful tool, and also for more general use elsewhere, we develop thetheory of extenders and ultrapowers under ZF (see § AC . However,it turns out that a proper class of weak L¨owenheim-Skolem (wLS) cardinals (dueto Usubua [14]; see Definition 5.3) gives enough choice to secure Lo´s’ theoremfor the kinds of extenders over V we will consider here. A proper class of wLScardinals is not known to be inconsistent with choiceless large cardinals, andis in fact implied by a super Reinhardt; it also holds if AC is forceable with aset-forcing (see [14]). Under this large cardinal assumption, we will prove thefollowing generalization that measurability classifies critical points under ZFC ;we say that an ordinal κ is V -critical iff there is an elementary j : V → M withcr( j ) = κ : V -amenable means that j ↾ V α ∈ V for each α < δ . heorem (5.8) . Assume ZF + “there is a proper class of weak L¨owenheim-Skolem cardinals”. Then the class of V -critical ordinals is definable withoutparameters. We now list some background and terminology, which is pretty standard. Ourbasic background theory throughout is ZF (in fact, many of the results requiremuch less, but we leave that to the reader), with additional hypotheses statedas required. However, in §
2, we work in the weaker theory RL . In the ZF (orweaker) context, certain familiar definitions from the ZFC context need to bemodified appropriately.Work in ZF . OR denotes the class of ordinals and Lim the class of limitordinals. Let δ ∈ Lim. The cofinality of δ , regularity , singularity are definedas usual (in terms of cofinal functions between ordinals). We say δ (or V δ ) is inaccessible iff there is no ( γ, f ) such that γ < δ and f : V γ → δ is cofinal in δ .A norm on a set X is a surjective function π : X → α for some α ∈ OR. Theassociated prewellorder on X is the relation R ⊆ X where xRy iff π ( x ) ≤ π ( y ).This can of course be inverted. If δ ∈ OR is regular but non-inaccessible, thenthe
Scott ordertype scot( δ ) of δ is the set P of all prewellorders of V α +1 inordertype δ , where α is least admitting such. See [4, 5.2, 5.3***?] for somebasic facts about this. A partial function f from (some subset of) X to Y isdenoted f : p X → Y .Let M = ( U, ∈ M , A , . . . , A n ) be a first-order structure with universe U and A ⊆ U . (We normally abbreviate this by just writing A ⊆ M .) We say that A is definable over M from parameters if there is a first-order formula ϕ ∈ L ˙ A ,..., ˙ A n (with symbols ∈ , = , ˙ A , . . . , ˙ A n ) and some ~x ∈ M <ω such that for all y ∈ M ,we have y ∈ A iff M | = ϕ ( ~x, y ). This is naturally refined by Σ n -definable fromparameters , if we require ϕ to be Σ n , and by from parameters in X , if we restrictto ~x ∈ X <ω (where X ⊆ M ), and by from ~x , if we may use only ~x .For an extensional structure M = ( ⌊ M ⌋ , ∈ M , = M ), the wellfounded part wfp( M ) of M is the class of all transitive isomorphs of elements of M . Thatis, the class of all transitive sets x such that ( x, ∈ ↾ x, = ↾ x ) is isomorphicto ( y, ∈ M ↾ y, = M ↾ y ) for some y ∈ ⌊ M ⌋ . The illfounded part illfp( M ) of M is ⌊ M ⌋ \ wfp( M ). We have wfp( M ) , illfp( M ) ∈ L ( M ), which results by ranking theelements of M as far as is possible. Note that illfp( M ) is the largest X ⊆ ⌊ M ⌋ such that for every x ∈ X there is y ∈ X such that y ∈ M x . If M models enoughset theory that it has a standard rank function, but M is illfounded, then OR M (the collection of all x ∈ ⌊ M ⌋ such that M | =“ x is an ordinal”) is illfounded,because if x ∈ M y then rank M ( x ) ∈ M rank M ( y ).Given a structure M and k ∈ [1 , ω ], E k ( M ) denotes the set of all Σ k -elementary j : M → M , and E ( M ) denotes E ω ( M ). Let δ ∈ Lim and j ∈ E ( V δ ).For C ⊆ V δ , define j + ( C ) = S α<δ j ( C ∩ V α ). Let κ = cr( j ) and κ n +1 = j ( κ n )and κ ω ( j ) = sup n<ω κ n (note κ ω ( j ) ≤ δ ). We write j = j and j n +1 = ( j n ) + ( j n )for n < ω ; then j n ∈ E ( V δ ) (see [4, Theorem 5.6***?]) and κ n = cr( j n ). ZF denotes the two-sorted theory, with models of the form M = ( V, E, P ),where (
V, E ) | = ZF and P is a collection of “subsets” of V . The elements of V are the sets of M and the elements of P the classes . The axioms are likethose of ZF , but include Separation for sets and Collection for sets with respect4o all formulas of the two-sorted language, and also Separation for classes withrespect to such formulas. When we “work in ZF ”, we mean that we work insuch a model M , and all talk of proper classes refers to elements of P . Notethat the theory is first-order; there can be countable models of ZF . One has( V κ , ∈ , V κ +1 ) | = ZF iff κ is inaccessible.The language of set theory with predicate L ∈ ,A is the first order languagewith binary predicate symbol ∈ and predicate symbol A . The theory ZF ( A )is the theory in L ∈ ,A with all ZF axioms, allowing all formulas of L ∈ ,A in theSeparation and Collection schemes (so A represents a class). ZFR ( ZF + Rein-hardt) is the theory ZF ( j )+“ j : V → V is Σ -elementary”. By [6, Proposition5.1], ZFR proves that j : V → V is fully elementary (as a theorem scheme).Work in ZF . Then κ ∈ OR is
Reinhardt iff there is a class j such that( V, j ) | = ZFR and κ = cr( j ). Following [1], κ ∈ OR is super Reinhardt iff forevery λ ∈ OR there is a class j such that ( V, j ) | = ZFR and cr( j ) = κ and j ( κ ) ≥ λ . RL ⇒ no parameter-definable j : V → V Fact 2.1 (Suzuki) . Assume ZF . Then no class j which is definable from pa-rameters is an elementary j : V → V . Of course, the theorem is really a theorem scheme, giving one statement foreach possible formula ϕ being used to define j (from a parameter). We generalizehere the proof of [4, Theorems 5.6, 5.7***?], in order to show Suzuki’s fact aboveis actually a consequence of lesser theory RL , which is the basic first order theorymodelled by V λ for limit λ (without choice): Definition 2.2. RL (for Rank Limit ) is the theory in L consisting of EmptySet, Extensionality, Foundation, Pairing, Union, Power Set, Separation (for allformulas, from parameters), together with the statements “For every ordinal α , α + 1 exists, V α exists and h V β i β<α exists”, and “For every x , there is an ordinal α such that x ∈ V α ”. ⊣ Since RL lacks Collection, the two extra statements regarding the cumulativehierarchy at the end are important. Clearly a model M | = RL can contain objects R such that M | =“ R is a wellorder”, but such that there is no α ∈ OR M suchthat M | =“ α is the ordertype of R ”. Theorem 2.3.
Assume RL . Then there is no Σ -elementary j : V → V whichis definable from parameters.Proof Sketch. The proof is a refinement of those of [4, Theorems 5.6, 5.7***?],with which we assume that reader is familiar. By Suzuki’s fact, we may assumethat ZF fails. So either (i) OR is Σ e m -singular for some m ∈ N ; that is, there isan ordinal γ and Σ m formula ϕ and parameter p such that ϕ ( p, ξ, β ) defines acofinal map γ → OR, via ξ β , and we take then ( γ, k ) then lexicographicallyleast, so γ is regular; or (ii) otherwise, and there is η ∈ OR and m ∈ N and a Σ m formula ϕ and p ∈ V such that ϕ ( p, x, β ) defines a cofinal map f : V η → OR,via x β , and we take η least such, which as in [4, Remark 5.3***?] impliesthat η = α + 1 and that rg( f ) has ordertype OR, and then we may in facttake f to be surjective, and obtain prewellorders of V α +1 having ordertype OR,5nd so define P = scot(OR) as before. ( RL suffices here; for example, startingwith an arbtirary definable cofinal f : V α +1 → OR, for each β ∈ OR, rg( f ) ∩ β is a set, by Separation. And the fact that a prewellorder has ordertype OR isa first-order statement under RL .) Let x be this γ or P respectively. Notethat x is definable without parameters (but maybe not Σ -definable withoutparameters).Now as before, given a Σ -elementary j : V → V which is definable fromparameters, we can define finite iterate j n , and j n : ( V, j n ) → ( V, j n +1 )is ∈ -cofinal Σ -elementary. Note here that N above denotes the standard inte-gers, which might not be in V , so m is standard, but when we mention “ ω ” atpresent, we are talking about the ω of the model V of RL we are working in,which might be illfounded from outside. Now the sequence h j n i n<ω is in fact adefinable class (show by induction on n < ω that for each α ∈ OR V , there is β ∈ OR V such that j n ↾ V α is computable from j ↾ V β ). So basically as in [4], foreach α ∈ OR V , there is n < ω such that j n ( α ) = α (and hence j m ( α ) = α for m ≥ n ); however, we must use here the definability of h j n i n<ω from parameters,and Separation, to get that the relevant sequence of sets h A n i n<ω is a set; hencewe can consider A = T n<ω A n and j ( A ). Similarly, if OR is regular to classfunctions definable from parameters and P is as above, then j n ( P ) = P forsome n .Now given any Σ -elementary j : V → V , we have that j is fully elementaryiff j ( x ) = x . For if j is elementary then j ( x ) = x because x is outrightdefinable. The other direction is proved like in [4], but we must restrict to classes A ⊆ V which are definable from parameters over V (which clearly suffices forour purposes here), since we need to use Separation to get that the relevant x -indexed sequence of sets h A y i y is a set.Now assume there is some j : V → V which is Σ -elementary and definablefrom parameters, and fix a formula ϕ and parameter p which defines j . So thereis such a ( ϕ, p, j ) satisfying j ( x ) = x (of course, if we need to pass to j n with n non-standard, we can incorporate n into p ). For q ∈ V let j q = { ( x, y ) (cid:12)(cid:12) ϕ ( q, x, y ) } . Let κ ∈ OR be least such that for some q , j q : V → V is Σ -elementary andcr( j q ) = κ and j q ( x ) = x . Since x is outright definable, so is κ .Fix p witnessing this. Since j p is Σ -elementary and j p ( x ) = x , j p isfully elementary. But j p ( κ ) > κ , so κ / ∈ rg( j p ), a contradiction. L ( V δ ) We saw in § j ∈ E ( V δ ) where δ ∈ Lim, then j is not definable from parameters over V δ .The sets X ⊆ V δ which are definable from parameters over V δ are exactly thosewhich are in L ( V δ ), or equivalently, in terms of Jensen’s hierarchy, those whichare in J ( V δ ) (the rudimentary closure of V δ ∪ { V δ } ). Beyond this, it is naturalto consider whether one might get such a j ∈ E ( V δ ) which is constructible from V δ , i.e., in L ( V δ ). This also naturally ramifies: given an ordinal α , can there be6uch a j ∈ J α ( V δ ) (or ∈ L α ( V δ ))? The question can also of course be extendedbeyond L ( V δ ).Later in the paper, we will find some quite precise answers to some of thesequestions. But in this section, for a warm-up and for some motivation, weconsider the simplest instance not covered by [13] or [4]; that is, the question ofwhether there can be a j ∈ E ( V δ ) which is at the simplest level of definabilitybeyond definability from parameters over V δ . Remark 3.1.
We use Jensen’s refinement of the J -hierarchy into the S -hierarchy.Here is a summary of the features we need; the reader can refer to [5] or [10]for more details. Recall that for a transitive set X , J ( X ) = { f ( V λ , ~x ) (cid:12)(cid:12) f is a rudimentary function and ~x ∈ X <ω } = [ n<ω S n ( X ) , where S is Jensen’s S -operator, and S ( X ) = X and S n +1 ( X ) = S ( S n ( X )).Taking S defined appropriately (maybe not exactly how Jensen originally de-fined it), each S n ( X ) is transitive, S n ( X ) ∈ S n +1 ( X ) and so S n ( X ) ⊆ S n +1 ( X ).So S n ( X ) S n +1 ( X ) J ( X ) and for Σ formulas ϕ and y ∈ J ( X ), J ( X ) | = ϕ ( y ) ⇐⇒ ∃ n < ω [ y ∈ S n ( X ) | = ϕ ( y )] . The truth of Σ statements over J ( X ) also reduces uniformly recursively tocountable disjunctions of statements over X . That is, there is a recursive func-tion ( ϕ, ~f , n ) ψ = ψ ϕ, ~f,n such that for all Σ formulas ϕ , tuples ~f of (termsfor) rudimentary functions and n < ω , then ψ = ψ ϕ, ~f,n is a formula in L , andfor all transitive X and ~x ∈ X <ω , J ( X ) | = ϕ ( ~f ( X, ~x )) ⇐⇒ ∃ n < ω [ X | = ψ ϕ, ~f,n ( ~x )] . Also, for each rudimentary f , the graph { ( ~x, y ) (cid:12)(cid:12) ~x ∈ X <ω and y = f ( X, ~x ) } is Σ -definable over J ( X ) from the parameter X , uniformly in X . For A ⊆ X ,we have A ∈ J ( X ) iff A is definable from parameters over X . Lemma 3.2. ( ZF ) Let λ ∈ Lim and j ∈ E ( V λ ) . There is a unique j ′ ∈ E ( J ( V λ )) with j ⊆ j ′ .Proof. We must set j ′ ( λ ) = λ and j ′ ( V λ ) = V λ . Because J ( V λ ) = [ n<ω S n ( V λ ) = { f ( V λ , ~x ) (cid:12)(cid:12) ~x ∈ V <ωλ } , and since for rudimentary f , the graph { ( ~x, y ) (cid:12)(cid:12) ~x ∈ V <ωλ and y = f ( V λ , ~x ) } is Σ over J ( V λ ) in the parameter V λ , we must set j ′ ( f ( V λ , ~x )) = f ( V λ , j ( ~x )),giving uniqueness.But this definition gives a well-defined j ′ ∈ E ( J ( V λ )). This is by [5]: allrudimentary functions are simple, and hence for each Σ formula ϕ and all rudfunctions f , . . . , f k − , there is a formula ϕ ′ such that for all ~x ∈ V <ωλ and y i = f i ( V λ , ~x ), we have J ( V λ ) | = ϕ ( ~y ) ⇐⇒ V λ | = ϕ ′ ( ~x ) . j , V λ | = ϕ ′ ( ~x ) ⇐⇒ V λ | = ϕ ′ ( j ( ~x )) ⇐⇒ J ( V λ ) | = ϕ ( j ′ ( y ) , . . . , j ′ ( y k − )) . Note j ′ is also ∈ -cofinal, and hence Σ -elementary. And j ⊆ j ′ .Note that in the following, j ↾ V α ∈ V λ for each α < λ . The following theoremstrengthens [4, Theorem 5.7 part 1***?] (in a different manner than 2.3). Wewill actually prove more later (Theorems 6.3 and 7.6), but we start here forsome motivation: Theorem 3.3. ( ZF ) Let λ ∈ Lim and j ∈ E ( V λ ) . Then e j = { j ↾ V α (cid:12)(cid:12) α < λ } isnot Σ e J ( V λ )1 .Proof. Suppose otherwise. Note that each finite iterate j n is then likewise de-finable, so by [4, Theorem 5.6***?], we may assume that j ∈ E ( V λ ). Let κ bethe least critical point among all such fully elementary j , select j witnessing thechoice of κ , and choose p ∈ V λ and a Σ formula ϕ such that e j = { k ∈ V λ (cid:12)(cid:12) J ( V λ ) | = ϕ ( k, p, V λ ) } . For n < ω , let η n = S { α < λ (cid:12)(cid:12) S n ( V λ ) | = ϕ ( j ↾ V α , p, V λ ) } . By Remark 3.1and since j is not definable from parameters over V λ (by Theorem 2.3 or [4,Theorem 5.7***?]), it follows that η n < λ . Note η n ≤ η n +1 and sup n<ω η n = λ .Let ( q, µ ) ∈ V λ × λ and m < ω . Say that ( q, µ ) is m -good iff for each n ≤ m ,there are ( ξ n , ξ ′ n ) ∈ λ and ℓ n ∈ V λ such that:1. ℓ n : V ξ n → V ξ ′ n is Σ -elementary and cofinal (where if ξ n is a successor γ + 1, cofinality means that ξ ′ n is also a successor and ℓ n ( V ξ n − ) = V ξ ′ n − )2. for all Σ m formulas ψ and x ∈ V ξ n [ V λ | = ψ ( x ) iff V λ | = ψ ( ℓ n ( x ))],3. S n ( V λ ) | = “ ℓ n is the union of all k such that ϕ ( k, q, V λ )”,4. ℓ i ⊆ ℓ n for each i ≤ n ,5. cr( ℓ n ) = µ .If ( q, µ ) is n -good, write ℓ qn = ℓ n and ξ qn = ξ n .Let also n < ω . Say that ( q, µ ) is ( m, n ) -strong iff ( q, µ ) is m -good and η n ≤ ξ qm . Say that ( q, µ ) is n -strong iff ∃ m < ω [( q, µ ) is ( m, n )-strong].Recall j ∈ E ( V λ ) with cr( j ) = κ , etc. Let α < λ be such that p ∈ j ( V α ).Let A m = { ( q, µ ) ∈ V α × κ (cid:12)(cid:12) ( q, µ ) is m -good } . Let A ω = T m<ω A m . Note that h A m i m<ω ∈ V λ .By Lemma 3.2, because j ∈ E ( V λ ), it extends uniquely to a b j ∈ E ( J ( V λ ))with b j ( V λ ) = V λ . This gives that b j ( S n ( V λ )) = S n ( V λ ) for each n < ω , and wehave a fully elementary map j ∗ n = b j ↾ S n ( V λ ) : S n ( V λ ) → S n ( V λ ) . Claim 1. A ω = ∅ , and moreover, ( p, κ ) ∈ j ( A ω ) . roof. Because j ∗ n is fully elementary and A n ∈ V λ , note j ( A n ) = j ∗ n ( A n ) = { ( q, µ ) ∈ V j ( α ) × j ( κ ) (cid:12)(cid:12) ( q, µ ) is n -good } . But ( p, κ ) is n -good, so ( p, κ ) ∈ j ( A n ). Also h A n i n<ω ∈ V λ , and j ( A ω ) = j ( \ n<ω A n ) = \ n<ω j ( A n ) , so ( p, κ ) ∈ j ( A ω ) = ∅ , so A ω = ∅ .Let B ( m,n ) = { ( q, µ ) ∈ V α × κ (cid:12)(cid:12) ( q, µ ) is ( m, n )-strong } , and B n = { ( q, µ ) ∈ V α × κ (cid:12)(cid:12) ( q, µ ) is n -strong } and B ω = T n<ω B n . These sets are in V λ . Claim 2. A ω ∩ B ω = ∅ , and moreover, ( p, κ ) ∈ j ( A ω ∩ B ω ) .Proof. By the previous claim and like in its proof, it suffices to see that ( p, κ ) ∈ j ( B n ) for each n < ω . But B n = S m<ω B ( m,n ) , so j ( B n ) = [ m<ω j ( B ( m,n ) ) , and j ( B ( m,n ) ) = j ∗ m ( B ( m,n ) ) == { ( q, µ ) ∈ V j ( α ) × j ( κ ) (cid:12)(cid:12) ( q, µ ) is m -good, with j ( η n ) ≤ ξ qm } . But there is m < ω with j ( η n ) ≤ η m = ξ pm . Then ( p, κ ) ∈ j ( B ( m,n ) ), so( p, κ ) ∈ j ( B n ), as required.Now by the claims, we may pick ( q, µ ) ∈ A ω ∩ B ω . Let ℓ = S n<ω ℓ qn . Claim 3. ℓ : V λ → V λ is fully elementary, cr( ℓ ) = µ < κ , and e ℓ is Σ e J ( V λ )1 .Proof. Because ( q, µ ) ∈ A ω , for each n < ω , ℓ qn : V ξ qn → V ξ ′ n ] is cofinal Σ -elementary with cr( ℓ qn ) = µ , and ℓ qn ⊆ ℓ qn +1 . So ℓ is a function with domain V λ = S n<ω V ξ qn ; the equality is because ( q, µ ) is n -strong for each n < ω . So ℓ : V λ → V λ . But because ( q, µ ) is m -good for each m < ω , ℓ is fully elementary.And cr( ℓ ) = µ . Finally note that e ℓ is appropriately definable from the parameter( q , V λ ).But µ < κ , so the claim contradicts the minimality of κ , QED.One can directly generalize the foregoing argument, showing that an elemen-tary j : V δ → V δ cannot appear in J α ( V δ ), for some distance. But especiallyonce we get to α ≥ κ = cr( j ) (or worse, α ≥ κ ω ( j )), things are clearly moresubtle, because in order to extend j to ˆ j : J α ( V δ ) → J α ( V δ ), ˆ j must moveordinals ≥ δ . But a natural and general way to extend j is through taking anultrapower by the extender of j . So we treat this topic in the next section.9 Ultrapowers and extenders under ZF In order to help us analyse the model L ( V δ ) further, and for more generalpurposes, we want to be able to deal with ultrapowers via extenders. This will,for example, assist us in extending embeddings of the form j : V δ → V δ to largermodels, such as L ( V δ ).Without AC , there are some small technical difficulties here in the defini-tions, which we will work through first. The more significant issues are thoseof wellfoundedness of the ultrapower and (generalized) Lo´s’ theorem; when weapply extenders, we will usually want to know that these properties hold of theultrapowers we form. Definition 4.1. ( ZF ) Write ≈ rank for the equivalence relation determined byrank; that is, x ≈ rank y iff rank( x ) = rank( y ). A set r is rank-extensional iff forall x, y ∈ r with x = y but x ≈ rank y , there is z ∈ r with z ∈ x ⇔ z / ∈ y .Let r be rank-extensional. The membership-rank diagram of r is the struc-ture mr( r ) = ( r, ∈ ↾ r, ≈ rank ↾ r ). ⊣ An easy induction on rank gives:
Lemma 4.2. ( ZF ) Let r be rank-extensional. Then:– For all α ∈ OR , r ∩ V α is also rank-extensional.– There is no non-trivial automorphism of mr( r ) . Definition 4.3. ( ZF ) We say that r is an index iff r is finite and rank-extensional.Note that we have an equivalence relation on the class of all indices given by a ≈ b iff mr( a ) ∼ = mr( b ). The signature sig( a ) of an index a is the equivalenceclass of a with respect to ≈ . Note that every signature is represented by some el-ement of V ω . By selecting in some natural way the minimal such representative,we may consider sig( a ) as being this element of V ω . A signature is a set sig( a )for some index a . Given a transitive set X , we write h X i <ω for the set of indices r ⊆ X , and given an index b , h X i b denotes { r ∈ h X i <ω (cid:12)(cid:12) sig( r ) = sig( b ) } .If a ≈ b then π ab : mr( a ) → mr( b ) denotes the unique isomorphism. Let a, e a, e b be indices with a ⊆ e a and sig( e a ) = sig( e b ). Then e b e aa denotes π e a e b “ a . ⊣ Lemma 4.4. ( ZF ) For every finite set c there is an index b with c ⊆ b and rank( c ) = rank( b ) . One might have expected (as did the author initially) that for every finite extensional set a , there is a finite extensional set b with a ⊆ b (where extensional means that for all x, y ∈ a with x = y , there is z ∈ a with z ∈ x iff z / ∈ y ). In fact, in the first draft of the notes [12](v1 on arxiv.org), which contained the first version of the development of extenders here, wedefined index with extensionality replacing rank-extensionality, and we made precisely thatclaim. But that claim is false; here is a counterexample. Define sets n ′ as follows:– 0 ′ = ∅ ,– 1 ′ = { ′ } and 2 ′ = { ′ } ,– 3 ′ = { ′ , ′ } and 4 ′ = { ′ , ′ } ,– (2 n + 1) ′ = { ′ , ′ , . . . , (2 n ) ′ } and (2 n + 2) ′ = { ′ , ′ , . . . , (2 n + 1) ′ } .Let x = { (2 n ) ′ (cid:12)(cid:12) n ∈ ω } and y = { (2 n + 1) ′ (cid:12)(cid:12) n ∈ ω } . Then p = { x, y } is finite but thereis no finite extensional q with p ⊆ q . (Let p ⊆ q with q finite and consider the largest k ∈ ω such that k ′ ∈ q . Observe that either x ∩ q = k ′ ∩ q or y ∩ q = k ′ ∩ q , and hence q is notextensional.) roof. This is an induction on rank (recall though that we don’t assume AC ).Assume c = ∅ . Let α be the maximum rank of elements of c , so rank( c ) = α + 1.Let c max be the set of elements of c of rank α .First choose a finite set c ′ such that c ⊆ c ′ , all elements of c ′ \ c have rank < α , and with c ′ extensional with respect to c max (that is, for all x, y ∈ c max with x = y , there is z ∈ c ′ with z ∈ x ∆ y ). Note that rank( c ′ \ c max ) ≤ α . So byinduction we can fix an index b ′ with c ′ \ c max ⊆ b ′ and rank( b ′ ) ≤ α . Now notethat b = b ′ ∪ c max is as desired. Definition 4.5 (Extender, Ultrapower) . ( ZF ) Let M | = RL be transitive. Wesay that A is amenable to M , and write A ⊆ ambl M , iff A ⊆ M and A ∩ x ∈ M for each x ∈ M . We also write A ⊆ ambl h M i <ω to mean A ⊆ ambl M and A ⊆ h M i <ω .Let M, N be transitive, satisfying RL , and j : M → N be Σ -elementaryand ∈ -cofinal. The extender derived from j is the set E of all pairs ( A, a )such that a ∈ h N i <ω , A ⊆ ambl M and a ∈ j ( A ). Given a ∈ h N i <ω , let E a = { A (cid:12)(cid:12) ( A, a ) ∈ E } . Let A ⊆ ambl h M i <ω and a, e a be indices with a ⊆ e a . Then A a e a denotes theset of all u ∈ h M i e a such that u e aa ∈ A , and A e aa denotes the set of all u ∈ h M i a such that there is v ∈ A ∩ h M i e a and u = v e aa .Let f : h M i <ω → V . Let a, e a ∈ h N i <ω with a ⊆ e a . We define a function f a e a : h M i <ω → V as follows. Let u ∈ h M i <ω . If sig( u ) = sig( e a ) then f a e a ( u ) = ∅ , and if sig( u ) = sig( e a ) then f a e a ( u ) = f ( u e aa ).Given a transitive rudimentarily closed structure P in the language of settheory (or possibly in a larger language), we say that E is an extender over P iff OR M < OR P and V P OR M = M . Suppose E is over P . A P -relevantpair (with respect to E ) is a pair ( a, f ) such that a ∈ h N i <ω and f ∈ P and f : h M i <ω → P . We define the (internal) ultrapower Ult ( P, E ) of P by E .We first define an equivalence relation ≈ E on the class of P -relevant pairs, bysetting ( a, f ) ≈ E ( b, g ) iff for some/all c ∈ h N i <ω with a ∪ b ⊆ c , we have { u ∈ h M i <ω (cid:12)(cid:12) f ac ( u ) = g bc ( u ) } ∈ E c . We write [ a, f ] P, E for the equivalence class of ( a, f ). We define the relation ∈ E likewise, replacing the condition “ f ac ( u ) = g bc ( u )” with “ f ac ( u ) ∈ f bc ( u )”.Let ⌊ U ⌋ be the class of equivalence classes of P -relevant pairs with resepctto ≈ E , and ∈ ′ be the relation on ⌊ U ⌋ induced by ∈ E . Then the ( internal ) ultrapower . Ult ( P, E ) of P by E is the structure U = ( ⌊ U ⌋ , ∈ ′ ). If U isextensional and wellfounded then we identify it with its transitive collapse. Wedefine the associated ultrapower embedding i P, E : P → U by i P, E ( x ) = [( ∅ , c x )] P, E Clearly (
A, a ) ∈ E iff ( A ∩ V Mξ , a ) ∈ E where ξ is any ordinal in M such that a ∈ j ( V Mξ ).And given the manner in which E will be used, we could have actually added the extra demandthat A ∈ M to the requirements specifying when ( A, a ) ∈ E , and in terms of informationcontent and cardinality, it would be more natural to do so. But it is convenient in other waysto allow more arbitrary amenable sets A . The “sub-0” in “Ult ” and the “super-0” in “ i P, E ” denotes the internality of the ultra-power, i.e. that the functions f used in forming the ultrapower all belong to P . This is anartefact of related notation in inner model theory, where one can have Ult n for n ≤ ω . c x : h M i <ω → P is the constant function c x ( u ) = x . We often abbreviatethis by i PE or i E . For any set y , let ind( y ) be the unique index with universe { y } , and let elmt(ind( y )) = y and elmt( u ) = ∅ if u is not of form ind( y ). Wewrite spt( E ) = N . ⊣ Of course, if P = AC then the proof of Lo´s’ theorem does not go through inthe usual manner, so in general Ult ( P, E ) might not even be extensional.
Lemma 4.6. ( ZF ) With notation as in Definition 4.5, we have:1. Let a, e a ∈ h N i <ω with a ⊆ e a and A ⊆ ambl h M i <ω . Then:(a) If A ⊆ h M i a and B = h M i a \ A then B a e a = h M i e a \ A a e a .(b) E a is an ultrafilter over the set of all A ⊆ ambl h M i <ω and h M i a ∈ E a ,and in fact, D V Mξ E a ∈ E a for each ξ < OR M with a ⊆ j ( V Mξ ) .(c) A ∈ E a iff A a e a ∈ E e a .(d) If A ∈ E e a then A e aa ∈ E a . f ac = ( f ab ) bc for all a, b, c ∈ h N i <ω with a ⊆ b ⊆ c and all functions f .3. In the definition of ≈ E and ∈ E , the choice of c is irrelevant.4. ≈ E is an equivalence relation on the P -relevant pairs,5. ∈ E respects ≈ E .6. N ⊆ wfp( U ) and for each β < OR N we have V Uβ = V Nβ . Moreover, for x ∈ V Nβ , we have x = [ind( x ) , elmt] P, E .Proof. Parts 1 and 2 are straightforward. Part 3: Consider ≈ E , and pairs( a, f ) , ( b, g ). Take c, c ′ ∈ h N i <ω with a ∪ b ⊆ c, c ′ . Note we may assume c ⊆ c ′ .We must see A ∈ E c iff A ′ ∈ E c ′ where– A = { u ∈ h V Mα i c (cid:12)(cid:12) f ac ( u ) = g bc ( u ) } ,– A ′ = { u ∈ h V Mα i c ′ (cid:12)(cid:12) f ac ′ ( u ) = g bc ′ ( u ) } .As c ⊆ c ′ , part 2 gives that A ′ = A cc ′ , so A ∈ E c iff A ′ ∈ E c ′ by part 1. Therest of parts 3–5 is similar or follows easily.Part 6: Easily from the definitions, for x, y ∈ N we get(ind( x ) , elmt) ≈ E (ind( y ) , elmt) ⇐⇒ x = y, (ind( x ) , elmt) ∈ E (ind( y ) , elmt) ⇐⇒ x ∈ y. So let ( a, f ) be a P -relevant pair and x ∈ N be such that( a, f ) ∈ E (ind( x ) , elmt) . Note we may assume x ∈ a and there is ξ < OR M such that a ∈ j ( V Mξ ) andrg( f ) ⊆ V Mξ , and so g = f ↾ V Mξ ∈ M . But then ( a, f ) ≈ E ( a, g ) and j ( g )( a ) ∈ j (elmt ↾ V Mξ )(ind( x )) = x, so j ( g )( a ) = y for some y ∈ x . But then ( a, f ) ≈ E (ind( y ) , elmt), as desired. But note that here the converse does not have to hold. efinition 4.7. ( ZF ) Let E be an extender over a transitive rudimentarilyclosed structure M . We say that Σ - Lo´s’ criterion holds for Ult ( M, E ) iff forall n < ω , for all f , f , . . . , f n ∈ M , for all a ∈ h spt( E ) i <ω , and all Σ formulas ϕ , if there are E a -measure one many u ∈ h M i <ω such that M | = ∃ y ∈ f ( u )[ ϕ ( f ( u ) , . . . , f n ( u ) , y )]then there is b ∈ h spt( E ) i <ω and g ∈ M such that a ⊆ b and for E b -measureone many v , we have M | = g ( v ) ∈ f ab ( v ) and ϕ ( f ab ( v ) , . . . , f abn ( v ) , g ( v )) . We define
Lo´s’ criterion for Ult ( M, E ) analogously, but we allow arbitraryformulas ϕ , and the ∃ quantifier is unbounded. ⊣ Theorem 4.8 (Generalized Lo´s’ Theorem) . ( ZF ) Let M be a transitive rudi-mentarily closed structure and E be an extender over M . Suppose that Σ - Lo´s’criterion holds for Ult ( M, E ) . Then for all n < ω , all f , . . . , f n ∈ M , all a , . . . , a n ∈ h spt( E ) i <ω and all Σ formulas ϕ , letting a ∈ h spt( E ) i <ω be suchthat a i ⊆ a for each i , we have Ult ( M, E ) | = ϕ ([ a , f ] , . . . , [ a n , f n ]) iff there are E a -measure one many v ∈ h M i <ω such that M | = ϕ ( f a b ( v ) , . . . , f a n bn ( v )) . Therefore, the ultrapower embedding i M, E is ∈ -cofinal and Σ -elementary. More-over, if Lo´s’ criterion holds for Ult ( M, E ) , then the above equivalence holds forarbitrary formulas ϕ , and i M, E is fully elementary.Proof. This is basically the usual induction to prove Lo´s’ theorem under AC ,except that we use Lo´s’ requirement instead of appealing to AC . One difference,however, is that we need to allow enlarging a to b in order to find an element[ b, g ] of the ultrapower witnessing the a statement; in the usual proof of Lo´s’theorem, one can take a = b . V -criticality and wLS cardinals Under
ZFC , the fact that κ = cr( j ) for some elementary j : V → M with M transitive, is equivalent to the measurability of κ . Therefore this “ V -criticality”of κ is first-order. We make a brief digression to consider this question under ZF . Definition 5.1. ( ZF ) An ordinal κ is V - critical iff there is an elementary j : V → M with cr( j ) = κ , where M ⊆ V is transitive. ⊣ Lemma 5.2. ( ZF ) Let κ be V -critical. Then κ is inaccessible.Proof. Suppose not and let α < κ and f : V α → κ be cofinal. Let j : V → M beelementary with cr( j ) = κ . Then j ( f ) = j ◦ f = f , although by elementarity, j ( f ) : V α → j ( κ ) is cofinal, a contradiction.13e do not know whether ZF proves that V -criticality is a first-order prop-erty. But we will show that ZF +“There is a proper class of weak L¨owenheim-Skolem cardinals” does prove this. Recall this notion from [14, Definition 4]: Definition 5.3 (Usuba) . ( ZF ) A cardinal κ is weak L¨owenheim-Skolem ( wLS )if for every γ < κ and α ∈ [ κ, OR) and x ∈ V α , there is X V α with V γ ⊆ X , x ∈ X and the transitive collapse of X in V κ . ⊣ Remark 5.4.
Usuba also defines
L¨owenheim-Skolem (LS) cardinals, which isat least superficially stronger. As Usuba mentions in [14],
ZFC proves that thereis a proper class of LS cardinals, and that (a result of Woodin is that) underjust ZF , every supercompact cardinal is an LS cardinal. Thus, assuming ZF and that there is a super Reinhardt cardinal, then there is a proper class ofLS cardinals, and hence wLS cardinals. The next lemma is immediate, due toUsuba: Lemma 5.5. ( ZF ) We have:1. The class of wLS cardinals is closed.2. Suppose there is a proper class of wLS cardinals and let γ ∈ OR be regular.Then there is a proper class of wLS cardinals δ such that cof( δ ) = γ . Definition 5.6. ( ZF ) A V -criticality pre-witness is a tuple ( κ, δ, N, j ) such that δ is a weak L¨owenheim-Skolem cardinal, N | = RL is transitive, j : V δ → N is ∈ -cofinal and Σ -elementary and cr( j ) = κ . ⊣ Theorem 5.7. ( ZF ) Let ( κ, δ, N, j ) be a V -criticality pre-witness. Let U =Ult( V, E j ) . Then:1. Lo´s’ criterion holds for U , so U is extensional and i E is elementary. and2. If cof( δ ) > ω then U is wellfounded and j ⊆ i E and κ = crit ( i E ) is V -critical.Proof. Part 1: Let a ∈ h N i <ω and f : h V δ i <ω → V and ϕ be a formula and sup-pose that for E a -measure one many u ∈ h V δ i <ω , there is y such that ϕ ( f ( u ) , y ).Let n < ω be large and α ∈ OR be large and with V α n V . Let γ < δ besuch that a ⊆ j ( V γ ). So h V γ i <ω ∈ E a . Applying weak L¨owenheim-Skolemnessat δ , let X V α with V γ ∪ { γ, f, δ, j, N, E } ⊆ X and such that the transitive collapse ¯ X of X is in V δ . Let π : ¯ X → X be theuncollapse map. Let π ( ¯ f ) = f , etc.By the elementarity, for each u ∈ h V γ i <ω , and each v ∈ ¯ X , we have that¯ X | = ϕ ( ¯ f ( u ) , v ) iff V | = ϕ ( f ( u ) , π ( v )).Note we can fix y ∈ j ( ¯ X ) such that j ( ¯ X ) | = ϕ ( j ( ¯ f )( a ) , y ). Let ξ ∈ ( γ, δ )with ¯ X ∈ V ξ and b ∈ h j ( V ξ ) i <ω with a ∪ { a, y } ⊆ b . Then for E b -measure onemany w ∈ h V ξ i <ω , letting ( a w , y w ) = π bw ( a, y ), we have that a w ∈ h V γ i <ω and y w ∈ (cid:10) ¯ X (cid:11) <ω and ¯ X | = ϕ ( ¯ f ( a w ) , y w ), and hence V | = ϕ ( f ( a w ) , π ( y w )). In [12, v1], it asserted that ZF + a proper class of Reinhardt cardinals proves there is aproper class of LS cardinals, but this should have been a super Reinhardt (which easily impliesa proper class of the same).
14o define g : h V ξ i <ω → V by setting g ( w ) = π ( y w ) for all such w (and g ( w ) = ∅ otherwise). Then for E b -measure one many w ∈ h V ξ i <ω , we have V | = ϕ ( f ab ( w ) , g ( w )), as desired.Part 2: Suppose not. So cof( δ ) > ω . For limit ordinals ξ < δ , let E ξ be theextender derived from j ↾ V ξ : V ξ → V sup j “ ξ . Let U ξ = Ult ( V, E ξ ) and k ξ : U ξ → Ult ( V, E ) the natural factor map and k ξζ : U ξ → U ζ likewise. We have not verified Lo´s’ criterion for these partialultrapowers, so we do not claim elementarity of the maps; nor do we claim that U ξ is extensional. But note that k ξ and k ξζ are well-defined and respect “ ∈ ” and“=” (that is, in the sense of the ultrapowers, even if they fail extensionality),and commute; that is, k ξβ = k ζβ ◦ k ξζ .Let ξ ≤ δ be a limit. Let O ξ = OR U ξ = Ult V (OR , E ξ ), with the nota-tion meaning that we use all functions (in V ) which map into OR to form theultrapower. Each O ξ is a linear order. Now U ξ is wellfounded iff O ξ is well-founded (see § O δ is illfounded. By restricting k ξζ , we get acommuting system of order-preserving maps ℓ ξζ : O ξ → O ζ (so ℓ ξζ ⊆ k ξζ and ℓ ξβ = ℓ ζβ ◦ ℓ ξζ ). Note that the direct limit of the O ξ under the maps ℓ ξζ , for ξ ≤ ζ < δ , is isomorphic to O δ , ℓ ξδ is the direct limit map. Note that each ℓ ξζ is cofinal.We claim there is ξ < δ such that O ξ is illfounded (here we use cof( δ ) > ω ).For suppose not. Then O ξ ∼ = OR. Define a sequence h ξ n , η n i n<ω of pairs ofordinals. Let ξ = 0. Now ℓ δ is cofinal. Let η be the least η with ℓ δ ( η ) inthe illfounded part of O δ . Given ( ξ n , η n ) with i ξ n δ ( η n ) in the illfounded part,there is a pair ( ξ, η ) such that i ξ n ξ ( η n ) > η and i ξδ ( η ) in the illfounded part.Let ( ξ n +1 , η n +1 ) be lexicographically least such. Let ξ = sup n<ω ξ n . Becausecof( δ ) > ω , we have ξ < δ . But the sequence just constructed exhibits that O ξ is illfounded, a contradiction.So fix ξ < δ with O ξ illfounded. Let n < ω be large, let α ∈ OR be large with V α n V ; hence, for some β < α , we have V α | =“Ult ( V β , E ξ ) is illfounded”.Let ξ ′ = sup j “ ξ . Using the weak L¨owenheim-Skolemness of δ , let X V α with V ξ ∪ V Nξ ′ ∪ { N, j, ξ, E ξ , β } ⊆ X and the transitive collapse ¯ X of X in V δ . So letting π : ¯ X → X be the uncol-lapse map, we have π ( E ξ ) = E ξ , π ↾ V ξ = id, etc. And ¯ X | =“Ult ( V ¯ β , E ξ ) isillfounded”, where π ( ¯ β ) = β . As ¯ X is transitive and models enough of ZF , itfollows that ¯ U = Ult ( V ¯ X ¯ β , E ξ ) is illfounded.Now define σ : ¯ U → N by setting σ ([ a, f ]) = j ( f )( a ). (This makes sense,as f ∈ ¯ X ∈ V δ = dom( j ).) Then note that (since E ξ is derived from j ), σ is ∈ -preserving. But then N is illfounded, contradicting our assumption that N is transitive. So U is wellfounded, as desired. This completes the proof of thetheorem.Of course under ZFC , V -criticality is equivalent to measurability, and hasa first-order formulation. We can now generalize this result: Theorem 5.8. ( ZF ) Assume a proper class of wLS cardinals. Let κ ∈ OR .Then the following are equivalent:– κ is V -critical there is a V -criticality pre-witness ( κ, δ, N, j ) with cof( δ ) > ω ,– there is a V -criticality pre-witness ( κ, δ, N, j ) such that Ult ( V, E j ) is well-founded, where E j is the extender derived from j .In particular, V -criticality is first-order definable over V .Proof. Suppose first that κ is V -critical, and let k : V → M be elementary withcr( k ) = κ . By Lemma 5.2, κ is regular. So by hypothesis and Lemma 5.5, wecan fix a L¨owenheim-Skolem cardinal δ > κ with cof( δ ) > ω (in fact cof( δ ) = κ is possible). Let ξ = sup k “ δ and N = V Mξ and j = k ↾ V δ . Then ( κ, δ, N, j ) isa V -criticality pre-witness with cof( δ ) > ω . We will show below that it followsthat Ult ( V, E j ) is wellfounded, but here it is easier: define ℓ : Ult ( V, E j ) → M,ℓ ([ a, f ] VE j ) = k ( f )( a ) , which, directly from the definition of “ ∈ ” and “=” in the ultrapower, pre-serves membership and equality. But M is transitive, so Ult ( V, E j ) is well-founded. Note that we have not yet proved that Lo´s’ theorem holds, or eventhat Ult ( V, E j ) is extensional.Now suppose that ( κ, δ, N, j ) is any V -criticality pre-witness such that either U = Ult ( V, E ) is wellfounded, where E = E j , or cof( δ ) > ω . Then by Theorem5.7, U is wellfounded and Lo´s’ criterion holds for this ultrapower, and hence theultrapower map k : V → U is elementary by Generalized Lo´s’ Theorem, so U isextensional, so we take U transitive, and j ⊆ k , so cr( k ) = κ . Corollary 5.9. ( ZF ) Let κ ∈ OR . Then κ is “definably V -critical” (that is,witnessed by some definable-from-parameters class k : V → M ) iff there is a V -criticality pre-witness ( κ, δ, N, j ) such that cof( δ ) > ω .Proof. Just use the proof of Theorem 5.8, but now all relevant classes are de-finable from parameters.
Corollary 5.10. ( ZF ) If κ is super Reinhardt, then there is a normal measureon κ concentrating on V -critical ordinals.Proof. If there is a super Reinhardt cardinal then there is a proper class of weakL¨owenheim Skolem cardinals, by Remark 5.4, so the theorem applies, and easilyyields the corollary.
Question 5.11. ( ZF ) Suppose κ is Reinhardt. Is V -criticality first-order?Must there be a V -critical ordinal < κ ?Of course if V -criticality is first-order and κ is Reinhardt, then like before,there are unboundedly many V -critical ordinals < κ . Remark 5.12.
One can now easily observe that if (
V, j ) | = ZFR then there is noset X such that V = L ( X ). Actually much more than this is known, but here Assume ZFR. Goldberg showed a few years ago (unpublished at the time) that every sethas a sharp. The author showed in [12] that V = HOD( X ) for every set X . Goldberg [3] andUsuba [15] then both (independently) sent proofs to the author that there is no set-forcingextension satisfying AC . The author then showed that M n ( X ) exists for every set X , but theproof is not yet published. Much more local proofs of sharp existence can now be seen in [2,***where?] and [11, ***where?].
16s the proof of this simpler fact: Suppose otherwise. Then there is a proper classof wLS cardinals. Let δ ∈ OR be such that cof( δ ) > ω and X ∈ V δ and j “ δ ⊆ δ .Let E be the extender derived from j ↾ V δ : V δ → V δ . Then (cr( j ) , δ, V δ , j ↾ V δ ) isa V -criticality pre-witness. So by Theorem 5.8, U = Ult ( V, E ) is extensionaland wellfounded, the ultrapower map k : V → U is elementary, and V δ ⊆ U . Soin fact U = L ( V δ ) = V . But k is definable from the parameter E , contradictingSuzuki 2.1. L ( V δ ) and uncountable cofinality Lemma 6.1. ( ZF ) Assume δ is inaccessible and V = HOD( V δ ) . Then δ is wLS.In fact, for all α ∈ ( δ, OR) and p ∈ V α and β < δ there is ( X, ¯ δ, π ) such that δ, p ∈ X V α and β ≤ ¯ δ < δ and X ∩ V δ = V ¯ δ and π : V ¯ δ → X is a surjection.Proof. Given n < ω , we may assume that V α n V , by increasing α and thensubsuming the old α into the parameter p . So since V = HOD( V δ ), we mayassume V α = Hull V α Σ (OR ∪ V δ ) . (1)Let X ′ = Hull V α ( V β ∪ { p, δ } ) and ¯ δ = sup( X ′ ∩ δ ) and X = Hull V α ( V ¯ δ ∪ { p, δ } ).By the inaccessibility of δ , we have ¯ δ < δ . So it suffices to see that X V α and X ∩ V δ = V ¯ δ , as clearly we get an appropriate surjection π .We first show that X ∩ V δ = V ¯ δ . Given n < ω and ξ < δ , let ε n ( ξ ) = sup(Hull V α Σ n ( V ξ ∪ { p, δ } ) ∩ δ ) . By inaccessibility of δ , we have ε n ( ξ ) < δ . Note ε n ( ξ ) is definable over V α from the parameter ( ξ, p, δ ) (since n is fixed). Therefore ε n “¯ δ ⊆ ¯ δ , which gives X ∩ V δ = V ¯ δ .Now for elementarity. First note that for each n < ω and ξ < δ , there is η < δ such that for each Σ n formula and ~x ∈ V <ωξ , if V α | = ∃ yϕ ( y, ~x, p, δ ) thenthere is y ∈ V α with V α | = ϕ ( y, ~x, p, δ ) and y being Σ n +3 -definable from elementsof V η ∪ { p, δ } . This is an easy consequence of line (1) (using the usual trick ofminimzing on ordinal parameters to get rid of them) and the inaccessibility of δ . Let η n ( ξ ) be the least η that witnesses this for ξ . Note that η n “¯ δ ⊆ ¯ δ . Butthen X V α and X ∩ V δ = V ¯ δ , as desired. Theorem 6.2. ( ZF ) Assume V = HOD( V δ ) where cof( δ ) > ω . Let j ∈ E ( V δ ) .Then for all sufficiently large m < ω , Lo´s’ criterion holds for Ult(
V, E ) , where E = E j m , and this ultrapower is wellfounded. We can easily deduce the main result of this section:
Theorem 6.3. ( ZF ) Assume V = L ( V δ ) where cof( δ ) > ω . Then E ( V δ ) = ∅ .Proof. Suppose j ∈ E ( V δ ). By the theorem, we may assume Lo´s’ theorem andwellfoundedness for Ult( V, E j ). But then as in Remark 5.12, we get i E j : V → V is definable from E j , a contradiction. There is actually a more efficient argument available here, which we had used in versionv1 of [12]: Instead of arguing via Theorem 5.7, one can directly establish Lo´s’ theorem andwellfoundedness for the ultrapower, using that it is a factor of j ; for this, we don’t need totake cof( δ ) > ω . roof of Theorem 6.2. If δ is inaccessible then by Lemma 6.1, δ is wLS, so byTheorem 5.7, we can take j itself (i.e. m ≥ δ is non-inaccessible.Let γ = cof( δ ). Now by [4, Theorem 5.6***?], and replacing j with j m witha sufficiently large m , we may assume:– j : ( V δ , A ) → ( V δ , j + ( A )) is fully elementary for every A ⊆ V δ ,– if γ < δ then j ( γ ) = γ , and– if γ = δ is regular but non-inaccessible, then j ( P ) = P , where P = scot( δ ). Claim 4.
Lo´s’ criterion holds.Proof.
Let ϕ be a Σ k formula and let Ω ∈ OR be such that V Ω k +2 V ;in particular, V Ω = Hull V Ω Σ ( V δ ∪ Ω). Let α < δ and f : h V α i <ω → V and a ∈ (cid:10) V j ( α ) (cid:11) <ω be such that for E a -measure one many u ∈ h V α i <ω , we have V Ω | = ∃ y [ ϕ ( f ( u ) , y )] . For u ∈ h V α i <ω , let β u be the least β < δ such that there is z ∈ V β and y ∈ V Ω such that V Ω | = ϕ ( f ( u ) , y ) and y is Σ -definable from ordinals and z over V Ω .Given β ≤ δ , let A β be the set of all u ∈ h V α i <ω such that β u ≤ β . Note that A δ has E a -measure one, if β < β ≤ δ then A β ⊆ A β , and A δ = S β<δ A β . Sublaim 1.
There is β < δ such that A β is E a -measure one.Proof. Suppose not. Suppose first that δ is singular, so γ = cof( δ ) < δ . Let f : γ → δ be cofinal. Then A δ = [ ξ<γ A f ( ξ ) . The sequence (cid:10) A f ( ξ ) (cid:11) ξ<γ ∈ V δ . But recall j ( γ ) = γ , so it follows that j ( A δ ) = j ( [ ξ<γ A f ( ξ ) ) = [ ξ<γ j ( A f ( ξ ) ) . But a ∈ j ( A δ ), so there is ξ < γ such that a ∈ j ( A f ( ξ ) ), so A f ( ξ ) is E a -measureone, a contradiction.Now suppose instead that δ is regular. So P = scot( δ ) ∈ V δ and j ( P ) = P .(see § R ∈ P ,let π R : V α +1 → δ denote the corresponding norm (that is, π R is a surjectionand xRy iff π R ( x ) ≤ π R ( y )). Define an equivalence relation ≈ on P × V α +1 bysetting ( R, y ) ≈ ( R ′ , y ′ ) iff π R ( y ) = π R ′ ( y ′ ). Let [ R, y ] denote the equivalenceclass of (
R, y ). Let E be the set of equivalence classes. Define the prewellorder ≤ ∗ on E by ( R, y ) ≤ ∗ ( R ′ , y ′ ) iff π R ( y ) ≤ π R ′ ( y ′ ). So ≤ ∗ has ordertype δ . So ~A = (cid:10) A π R ( y ) (cid:11) [ R,y ] ∈ E ∈ V δ . Since j ( P ) = P , j also fixes α, E , ≤ ∗ , ~A , and j “ E is cofinal in ≤ ∗ . Since A δ = S ~A and this union is increasing in ≤ ∗ , it follows as in the previous case thatthere is [ R, y ] ∈ E such that a ∈ j ( A π R ( y ) ).18o fix β as in the subclaim. Let A ′ β be the set of pairs ( u, z ) such that u ∈ h V α i <ω and z ∈ V β and ( u, z ) are as above. Since a ∈ j ( A β ), we have some b with ( a, b ) ∈ j ( A ′ β ), and note we may assume that a ∪ { a } ⊆ b ∈ h V δ i <ω ,by increasing β if needed. Now for z ∈ h A δ i b with ( z ba , z ) ∈ A ′ β , let g ( z ) bethe least y which is Σ -definable over V Ω from ordinals and z , and such that V Ω | = ϕ ( f ( z ba ) , y ). (Here we minimize on the Σ formula and ordinal parametersin order to specify the least y .) Then note that for E b -measure one many z , wehave V Ω | = ϕ ( f ab ( z ) , g ( z )), as desired. Claim 5.
Ult(
V, E ) is wellfounded.Proof. We argue much as in the proof of Theorem 5.7. Fix some Ω ∈ ( δ, OR)and some large enough k < ω with V Ω k +2 V . As before, it suffices to see thatUlt V Ω (Ω , E ) is wellfounded, where the notation means we use all functions in V Ω to form the ultrapower.Now V Ω = Hull V Ω Σ ( V δ ∪ Ω) and (because k is large enough) V Ω computes theultrapower, and its wellfounded (and illfounded) parts. Therefore, this reflectsinto X = Hull V Ω ( V δ ∪ { E } ) (note we have dropped the parameters in Ω \ δ ), and X V Ω . Let H be the transitive collapse of X . Then it suffices to see that O = Ult H (OR H , E ) is wellfounded.Given η ≤ δ , let H η = Hull H ( V η ∪ { E } ). Note we do not claim that H η H .But we have H δ = H = S η<δ H η . Let O η be the substructure of O given byelements of the form [ a, f ] where a ∈ h V η i <ω and f ∈ H η . So if η < η then O η is just the restriction of O η to its domain, and O = S η<δ O η .Now suppose that O is illfounded. Then there is η < δ such that O η isillfounded. For suppose otherwise. We argue like in the proof of Theorem 5.7.Let η be the least η such that some illfp( O ) ∩ O η = ∅ , and let x be the < O η -least x ∈ illfp( O ) ∩ O η . Then given η n , x n , let η n +1 be the least η ∈ ( η n , δ ) suchthat there is x ∈ illfp( O ) ∩ O η with x < O η x n . Setting η = sup n<ω η n , we get acontradiction.Now fix such an η < δ and let γ be the ordertype of OR ∩ H η and π : γ → OR ∩ H η the uncollapse map. Given f : h V η i <ω → OR with f ∈ H η , note thatrg( f ) ⊆ rg( π ). Let ¯ f : h V η i <ω → γ be the natural collapse (so π ◦ ¯ f = f ). Wehave a surjection σ : V η → γ . Given f as above, let f ′ be the correspondingcollapse; that is, f ′ : h V η i <ω → V η +1 , f ′ ( u ) = { x ∈ V η (cid:12)(cid:12) σ ( x ) = ¯ f ( u ) } . Let F = { f ′ (cid:12)(cid:12) f ∈ H η and f : h V η i <ω → OR } . Let ≤ σ be the prewellorder of V η induced by σ . Clearly then U = Ult F ( ≤ σ , E )is illfounded, as it is in fact isomorphic to O η . But by the Σ -elementarity of j , j ( ≤ σ ) ∈ V δ is a prewellorder of V j ( η ) (in particular wellfounded), and we canabsorb U into j ( ≤ σ ) as usual, by mapping[ a, f ′ ] F E j ( f ′ )( a ) . So j ( ≤ σ ) is illfounded, a contradiction, proving the claim.By the two claims, we are done. 19 Admissible L κ ( V δ ) and countable cofinality Lemma 7.1. ( ZF ) Assume V = L ( V δ ) where δ ∈ Lim . Let j ∈ E ( V δ ) . Then:1. cof( δ ) = ω and j is fully elementary.2. For all α ≤ OR , Σ - Lo´s’ criterion holds for Ult ( J α ( V δ ) , E j ) .3. Ult ( L ( V δ ) , E ) is illfounded.Proof Sketch. We write J α for J α ( V δ ). Part 1 is by Theorem 6.3 and [4, Theo-rem 5.6***?]. Note that trivially then, j (cof( δ )) = cof( δ ).Part 2: This is much as in the proof of Theorem 6.2, using that J β = Hull J β Σ ( V δ ∪ ( δ + β ))for all β ≤ α . Suppose for example that α = β + 1. Let f, h ∈ J α with f, h : h V δ i <ω → V . Suppose that for E a -measure one many u , we have J α | = ∃ y ∈ h ( u ) [ ϕ ( f ( u ) , y )] , where ϕ is Σ . We have m < ω such that f, g ∈ S m ( J β ), where S denotesJensen’s S -operator (so J α = S k<ω S k ( J β )). Fix a surjection π : ( V δ × β <ω ) → S m ( J β )with π ∈ J α . Then arguing as before, using that cof( δ ) = ω , we can find ξ < δ such that for E a -measure one many u , there is y ∈ π “( V ξ × β <ω ) such that J α | = y ∈ h ( u ) & ϕ ( f ( u ) , y ) . (2)Now for pairs ( u, v ) ∈ h V δ i <ω × V ξ , let g ′ ( u, v ) be the least y ∈ π “( { v } × β <ω )such that line (2) holds, if there is such a y . Then we find an appropriate index b and convert g ′ into a function g , with ( b, g ) witnessing Σ - Lo´s’ criterion, likebefore.Part 3: Since V = L ( V δ ) | = ZF and by part 2, i VE : V → V is fully elementary,so this is like before. Remark 7.2.
Let M be a transitive set. Recall that M is admissible iff M satisfies Pairing, Infinity, Σ -Separation, and whenever d, p ∈ M and ϕ is aΣ formula and M | = ∀ x ∈ d ∃ y ϕ ( x, y, p ) then there is e ∈ M such that M | = ∀ x ∈ d ∃ y ∈ e ϕ ( x, y, p ). Definition 7.3.
Given a transitive set X , let κ X denote the least κ ∈ OR suchthat J κ ( X ) is admissible. ⊣ Recall the notation wfp and illfp from § Fact 7.4. ( ZF ) Let M be an extensional structure in the language of set theory,let X ∈ wfp( M ) ( and we assume M is transitive below X ) . Suppose M | = “ V = L ( X ) ” ( but M might not satisfy ZF ) . If M is illfounded then J κ X ( X ) ⊆ M . roof. Let λ = OR ∩ wfp( M ). Because λ ( OR M but λ / ∈ M , and M | =“ V = L ( X )”, and hence, M | =“I am rudimentarily closed”, it is easy to see that λ is closed under ordinal addition and multiplication. Moreover, it is easy to seethat J λ ( X ) ⊆ M , and hence, J λ ( X ) ⊆ wfp( M ).Now suppose that λ < κ X . Then we can fix a Σ formula ϕ and d, p ∈ J λ ( X )such that λ is the least λ ′ such that J λ ′ ( X ) | = ∀ x ∈ d ∃ y ϕ ( x, y, p ) . Note then that for all α ∈ OR M \ λ , M | = “ J α ( X ) | = ∀ x ∈ d ∃ y ϕ ( x, y, p )” . But then for such α , M | = “ { α ∈ OR (cid:12)(cid:12) J α ( X ) | = ¬∀ x ∈ d ∃ y ϕ ( x, y, p ) } ∈ J α +1 ( X )” . But note that this set is exactly λ , so λ ∈ M , a contradiction. Fact 7.5. ( ZF ) Let X be transitive. Then for every α ≤ κ X , we have J α ( V δ ) = Hull J α ( V δ )Σ ( V δ ∪ { V δ } ) . Therefore ( i ) P ( V δ ) ∩ J α +1 ( V δ )
6⊆ J α ( V δ ) and ( ii ) for every x ∈ J α ( V δ ) thereis a surjection π : V δ → x with π ∈ J α ( V δ ) .Proof. Let H = Hull J α ( V δ )1 ( V δ ∪ { V δ } ) and β = sup( H ∩ OR). Note then that H = J β ( V δ ). So it suffices to see that β = α , so suppose β < α . Then J β ( V δ )is inadmissible. So let p, d ∈ J β ( V δ ) and ϕ be Σ , such that β is least such that J β ( V δ ) | = ∀ x ∈ d ∃ y ϕ ( x, y, p ) . Then J α ( V δ ) | = ∃ β ′ ∈ OR [ J β ′ ( V δ ) | = ∀ x ∈ d ∃ y ϕ ( x, y, p )] . But β is the least such β ′ , and since p, d ∈ H , it follows that β ∈ H , a contra-diction.Part (i) of the “therefore” clause now follows by a standard diagonalization.For part (ii), if α = β + 1, use that J α = S n<ω S n ( J β ) (where S n is the n thiterate of Jensen’s S -operator) and for each n ∈ [1 , ω ), S n ( J β ) = Hull S n ( J β )Σ ( V δ ∪ { V δ , β } ) . We now prove the promised strengthening of Theorem 3.3:
Theorem 7.6. ( ZF ) Let δ ∈ Lim and j : V δ → V δ be Σ -elementary. Let θ = κ V δ ( see 7.3 ) . Then j / ∈ J θ ( V δ ) . In fact, j is not Σ e J θ ( V δ )1 , and not Π e J θ ( V δ )1 .Proof. We write J α for J α ( V δ ). Suppose first j / ∈ J θ , and we deduce the rest.Let ϕ be Σ and p ∈ J θ , and suppose for all x, y ∈ V δ , we have j ( x ) = y iff J θ | = ϕ ( x, y, p ). Then note that J θ | = ∀ x ∈ V δ ∃ α ∈ OR [ J α | = ∃ y ∈ V δ ϕ ( x, y, p )] , λ < θ such that J λ | = ∀ x ∈ V δ ∃ y ∈ V δ ϕ ( x, y, p ) . But then for x, y ∈ V δ , we have j ( x ) = y iff J λ | = ϕ ( x, y, p ), so j ∈ J θ ,contradiction.Now suppose that for all x, y ∈ V δ , we have j ( x ) = y iff J θ | = ¬ ϕ ( x, y, p ).Note for each x ∈ V δ , letting y = j ( x ), J θ | = ∀ y ′ ∈ V δ \{ y } ϕ ( x, y ′ , p ) , and so by admissibility, there is (a least) α x < θ such that J α x | = ∀ y ′ ∈ V δ \{ y } ϕ ( x, y ′ , p ) . Then J θ | = ∀ x ∈ V δ ∃ α ∈ OR [ J α | = ∃ y ∈ V δ ∀ y ′ ∈ V δ \{ y } ϕ ( x, y ′ , p )], but thenby admissibility, we get sup x ∈ V δ α x < θ , but then j ∈ J θ , a contradiction.So we need to see j / ∈ J θ . Suppose otherwise. By Lemma 7.1, cof L ( V δ ) ( δ ) = ω and j : V δ → V δ is fully elementary. Let E = E j and α < θ with E ∈ J α .Let κ = cr( E ). Let M = J α + κ +1 and U = Ult ( M, E ). By Lemma 7.1,Σ - Lo´s’ criterion holds for U , so i ME is ∈ -cofinal and Σ -elementary. Because M | =“ V = L ( V δ )”, therefore U | =“ V = L ( i ME ( V δ ))”. Claim 6. i ME ( V δ ) = V δ .Proof. It suffices to see that i ME is continuous at δ . So let f ∈ M and α < δ with f : h V α i <ω → δ , and let a ∈ (cid:10) V j ( α ) (cid:11) <ω . We want to see that [ a, f ] ME < δ .But cof( δ ) = ω , so fix g : ω → δ cofinal, and for n < ω let A n = { u ∈ h V α i <ω (cid:12)(cid:12) f ( u ) < g ( n ) } . Then h V α i <ω = S n<ω A n , so the usual argument gives A n ∈ E a for some n < ω ,which suffices.By the claim, V δ ∈ wfp( U ) and U | =“ V = L ( V δ )”. Claim 7. U is illfounded.Proof. Suppose U is wellfounded. Then note that i ME ( α + κ ) > α + κ , and bythe previous claim, that J i ME ( α + κ )+1 ⊆ U , so P ( V δ ) ∩ J α + κ +2 ⊆ U. But P ( V δ ) ∩ U ⊆ M , because given any A ∈ P ( V δ ) ∩ U , we can find some pair( a, f ) such that [ a, f ] ME = A , with f ∈ M and a ∈ V δ , and since E ∈ M , iteasily follows that A ∈ M . Putting the ⊆ -statements together, we contradictFact 7.5.By the above claim and Fact 7.4, we have J θ ⊆ U , so P ( V δ ) ∩ J θ ⊆ U . Butthen we reach a contradiction like in the proof of the claim. This completes theproof.We next observe that the preceding result is optimal, at least in the casewhere we have a lot of AC : 22 act 7.7 (Corazza) . ( ZFC ) Suppose j ∈ E ( V δ ) , where δ ∈ Lim ( so δ = κ ω ( j ) and V δ | = ZFC ) . Then there is a set-forcing P which forces (i) V δ | = ZFC + “ V =HOD ” and (ii) there is k ∈ E ( V δ ) with ˇ j ⊆ k . Theorem 7.8. ( ZF ) Let δ ∈ OR with V δ | = ZFC + “ V = HOD ” and E ( V δ ) = ∅ ,and take δ least such. Let θ = κ V δ and M = J θ ( V δ ) . Then there is j ∈ E ( V δ ) which is Σ M ( { V δ } ) ∧ Π M ( { V δ } ) , meaning there are Σ formulas ϕ, ψ such that ∀ x, y ∈ V δ [ j ( x ) = y ⇐⇒ M | = [ ϕ ( x, y, V δ ) ∧ ¬ ψ ( x, y, V δ )]] . Proof.
Note cof( δ ) = ω . The j satisfying these requirements is just the left-mostbranch through the natural tree searching for such an embedding. That is, let T be the tree whose nodes are finite sequences(( j , α , β ) , ( j , α , β ) , . . . , ( j n − , α n − , β n − ))such that for each i < n , j i : V α i → V β i is elementary and κ = cr( j i ) exists, V κ | =“ V = HOD”, and if i + 1 < n then β i < α i +1 and j i ⊆ j i +1 .Now any Σ -elementary j : V δ → V δ determines an infinite branch through T (we can take α = cr( j ) and β n = j ( α n ) and α n +1 = β n + 1). This is clearenough except for the fact that V κ | =“ V = HOD” where κ = cr( j ). But because V δ | = ZFC , we must have κ ω ( j ) = δ , and since V δ | =“ V = HOD”, it follows that V κ | =“ V = HOD” also. Conversely, let h ( j i , α i , β i ) i i<ω be an infinite branchthrough T , and λ = S i<ω α i = S i<ω β i . Then λ ∈ Lim and j ∈ E ( V λ ), andsince V κ | =“ V = HOD” where κ = cr( j ), therefore V λ | = ZFC +“ V = HOD”. Infact j is fully elementary (either since V λ | = ZF , or because cof( λ ) = ω and by[4, Theorem 5.6***?]), so λ = δ by minimality.Note that T is definable over V δ . Now the rank analysis of T is computedover M . That is, given a node t ∈ T , let T t = { s ∈ T (cid:12)(cid:12) t E s or s E t } . Then there is a rank function for T t (in V ) iff there is one in M ; this is a standardconsequence of admissibility. Let < ∗ be the standard wellorder of V δ resultingfrom the fact that V δ | =“ V = HOD”. Let b = h t i i i<ω be the left-most branchof T with respect to < ∗ . That is, t = ∅ , t = h ( j , α , β ) i is the < ∗ -least nodeof T of length 1 such that there is no rank function for T t (in M ), and then t = h ( j i , α i , β i ) i i< is the < ∗ -least node of T of length 2, extending t , suchthat there is no rank function for T t (in M ), etc. Note here that because T t n has no rank function (in M ), t n +1 does exist. This determines our branch b ,and hence a Σ -elementary j : V δ → V δ .Finally note that b is appropriately definable. Definition 7.9. ( ZF ) Let δ ∈ Lim and m ≤ ω . Then T = T δ,m denotes thefollowing tree of attempts to build a (possibly partial) Σ m -elementary j : p V δ → V δ . The nodes in T are finite sequences t = (( j , α , β ) , . . . , ( j n , α n , β n ))such that j i : V α i → V β i is Σ -elementary and cofinal, j i : p V δ → V δ is Σ m -elementary on its domain V α i , β i < α i +1 , and j α i ⊆ j α i +1 . Write j t = j n . In both statements here the V δ is in the sense of the forcing extension. α ∈ OR, let T α be the α th derivative of T , defined as follows. Set T = T , and for limit λ set T λ = T α<λ T α . Given T α , T α +1 is the set of all t ∈ T α such that for every β < δ there is an extension s of t with s ∈ T α suchthat β ∈ dom( j s ). Let T ∞ = T OR . We say that T ∞ is perfect iff for every t ∈ T ∞ there are s , s ∈ T ∞ , both extending t , such that j s j s j s .We write [ T ] for the set of infinite branches through T . Clearly each b ∈ [ T ]determines a Σ -elementary map j b : V λ → V λ for some limit λ ≤ δ , and if λ = δ then j is Σ m -elementary.We say that an embedding k : V λ → V λ , for limit λ , is V -amenable iff k ↾ V α ∈ V for each α < λ . Clearly j b above is V -amenable. ⊣ Note h T α i α ∈ OR ∈ L ( V δ ), where T = T δ,m . The following lemma shows thatif T ∞ = ∅ then T ∞ is, by a certain natural measure, of maximal complexity: Lemma 7.10. ( ZF ) Let δ ∈ Lim . Let γ be least such that T γ = T ∞ . Then:1. If E m ( V δ ) = ∅ then T ∞ = ∅ .2. If T ∞ = ∅ then T ∞ is perfect and γ = κ V δ .3. If T ∞ = ∅ then γ < κ V δ .Proof. If j ∈ E m ( V δ ) then easily T ∞ = ∅ , and in fact, if we force over V to collapse V δ to become countable, then in V [ G ], there is an infinite branch b ∈ [ T ∞ ] with j b = j .The fact that γ ≤ κ = κ V δ is standard: Suppose not and let t ∈ T κ \ T κ +1 .Then we can fix α < δ such that no s ∈ T κ extending t has α ∈ dom( j s ).But then J κ ( V δ ) | =“For every s ∈ T extending t with α ∈ dom( j s ) there is β ∈ OR such that s / ∈ T β ”. By admissibility, it follows that there is ξ < κ suchthat J ξ ( V δ ) satisfies this. But then note that t / ∈ T κ , contradiction. The sameargument (but slightly simpler) shows that if T ∞ = ∅ then γ < κ .Now suppose that T ∞ = ∅ but T ∞ is not perfect. Then fix t ∈ T ∞ such thatfor all s , s ∈ T ∞ extending t , we have j s ⊆ j s or j s ⊆ j s . Let S be theset of all s ∈ T ∞ extending t , and j = S s ∈ S j s . Then note that j : V δ → V δ isa well-defined function and is Σ m -elementary, and j ∈ L ( V δ ).In fact, j ∈ J κ , contradicting Theorem 7.6. For fix α < δ with α > dom( j t );so j ↾ V α ∈ V δ . Then for all nodes s ∈ T extending t with V α ⊆ dom( j s ) and j ↾ V α = j s ↾ V α , there is ξ < κ with s / ∈ T ξ . So by admissibility, there is ξ < κ such that s / ∈ T ξ for all such s . Therefore, for each α < δ such that α > dom( j t ),there is β < δ and a map k : V α → V β (actually k = j ↾ V α ) with j t ⊆ k andthere is ξ < κ such that s / ∈ T ξ for all s as above. So by admissibility, there is ξ < κ which works simultaneously for all α < δ . But then clearly j is definablefrom parameters over J ξ , so j ∈ J κ , contradicting Theorem 7.6.It remains to see that if T ∞ = ∅ , hence perfect, then γ = κ V δ . So supposeotherwise. Let γ < ξ < κ V δ . Then T ∞ ∈ J ξ , and we can force over L ( V δ ) with T ∞ , in the obvious manner, with the generic filter G being an infinite branch b through T ∞ , and note that by genericity, j b : V δ → V δ is Σ m -elementary (thatis, genericity ensures that dom( j b ) = V δ ). Note that j b is V -amenable.Now we will proceed through basically the argument from before, but justneed to see that things adapt alright to the generic embedding j b . We firstconsider the finite iterates ( j b ) n of j b and the eventual fixedness of ordinals(that is, whether ( j b ) n ( α ) = α for some n ). If α < δ is a limit and j b ( α ) = α j b ↾ V α determines ( j b ) n ↾ V α for each n < ω as usual, and this is all in V , so all ordinals < α are eventually fixed. So we may assume that there isa bound < δ on such ordinals α . If δ = α + ω for some limit α then clearly j b ( α + n ) = α + n for all n < ω , so we are also done in this case. So we are leftwith the case that δ is a limit of limits, and there is α < δ such that j b fixes noordinal in [ α, δ ), and take α least such. In particular, j b ( α ) > α . Letting α = α and α n +1 = j b ( α n ), note that sup n<ω α n = δ (for if η = sup n<ω α n < δ then j b ↾ V η ∈ V , so h α n i n<ω ∈ V , so cof V ( η ) = ω , so j b ( η ) = η , contradiction). But j b has unboundedly many fixed points < α . Therefore ( j b ) has unboundedlymany < α , etc, ( j b ) n has unboundedly many < α n . But then ( j b ) n ↾ V α n isenough to determine ( j b ) n + k ↾ V α n for k < ω (working in V ), so we get that allpoints < α n are eventually fixed. So if δ is singular in V , we can find n < ω suchthat ( j b ) n (cof V ( δ )) = cof V ( δ ). Similarly, if δ is regular but non-inaccessible, wecan find n < ω such that ( j b ) n (scot V ( δ )) = scot V ( δ ) (for this, argue as beforeto first find n < ω and a limit α < δ such that ( j b ) n has cofinally many fixedpoints < α and scot V ( δ ) ∈ V α , and then proceed in V ).So fix n < ω such that k = ( j b ) n is like this. Then for each η ∈ OR, Σ - Lo´s’theorem holds for Ult ( J η , E k ) The proof is just like before – the fact that k / ∈ V does not matter. (The ultrapower is formed using only functions in V , so allthe calculations with partitioning measure one sets is done in V , and becauseeither V δ is inaccessible in V or k fixes the relevant objects, the argument goesthrough.)So consider U = Ult ( J χ , E k ) where χ = ( ω ˙( δ + ξ )) + cr( k ) + 1 (we had P = T ∞ ∈ J ξ ). We claim that U is illfounded. For otherwise OR U > χ , so asbefore, we get t = Th J χ Σ ( V δ ∪ { V δ } ) ∈ J χ [ G ] . Let τ ∈ J χ be a T ∞ -name such that τ G = t . Now t ∈ L ( V δ ), and since G is L ( V δ )-generic, there is p ∈ G such that L ( V δ ) | =“ p forces ˇ t = τ ”. But then for( ϕ, x ) ∈ V δ we have ( ϕ, x ) ∈ t ⇐⇒ J χ | = p P ( ϕ, x ) ∈ τ, because for example if ( ϕ, x ) ∈ t but there is q ≤ p and J χ | = q P ( ϕ, x ) / ∈ τ ,then p cannot have forced ˇ t = τ ; moreover here this forcing relation is definableover J χ , and in fact, it is definable from parameters over J χ − . For the Σ forcing relation over J ω · ( δ + ξ ) is ∆ J ω · ( δ + ξ ) ( { P } ), because we have enough closureat this stage, and this is then maintained level by level, and the Σ forcing rela-tion over J χ − is ∆ J χ − ( { P } ), and the Σ n forcing relation for Σ e J χ − n -definablenames ⊆ J χ − is definable from P over J χ − , but τ can be taken to be such aname. Hence we get t ∈ J χ , which is a contradiction.So U is illfounded, and hence J κ Vδ ⊆ wfp( U ). But then we again get t ∈J χ [ G ], which is again a contradiction, completing the proof. Theorem 7.11. ( ZF ) Let δ ∈ Lim with E ( V δ ) = ∅ , where m ∈ [1 , ω ] . Then:1. In a set-forcing extension of V , for each V -amenable j ∈ E m ( V δ ) and α < δ there is a V -amenable k ∈ E m ( V δ ) with k ↾ V α = j ↾ V α but k = j . Since j b / ∈ V , it seems that δ might not have cofinality ω in V here. . If DC holds and cof( δ ) = ω then for each j ∈ E m ( V δ ) and α < δ there is k ∈ E m ( V δ ) with k ↾ V α = j ↾ V α but k = j .Proof. By Lemma 7.10, T ∞ is perfect (notation as there), which immediatelygives the theorem (of course we can take the generic extension to be V [ G ] where G collapses V δ to become countable).We now show that the kind of embedding defined in Theorem 7.8 cannot beextended to the whole of J θ ( V δ ): Theorem 7.12. ( ZF ) Let δ ∈ Lim and j ∈ E ( V δ ) . Let θ = κ V δ . Suppose that j is definable over J θ ( V δ ) . Then there is α < θ such that Ult ( J α ( V δ ) , E j ) isillfounded.Proof. Write J α for J α ( V δ ). Since j ∈ L ( V δ ), we have cof( δ ) = cof L ( V δ ) ( δ ) = ω . Claim 1.
Let α < θ with α either a successor or cof V ( α ) < δ and j continuousat cof( α ) . Let f : h V δ i <ω → J α and a ∈ h V δ i <ω . Then there is g ∈ J α suchthat g ( u ) = f ( u ) for E a -measure one many u .Proof. By the usual calculations using the continuity of j , we can assume thatrg( f ) ⊆ x for some x ∈ J α . But by Fact 7.5, there is a surjection π : V δ → x with π ∈ J α . Therefore, we can assume that x = V δ . (That is, given y ∈ x , let Z y = { w ∈ V δ (cid:12)(cid:12) π ( w ) = y } , and then let z y = Z y ∩ V β where β is least such thatthis intersection is = ∅ , and then let f ′ : h V δ i <ω → V δ be f ′ ( u ) = z f ( u ) . Thenclearly f ′ codes f modulo π .) But since cof( δ ) = ω , there is then β < δ suchthat f ( u ) ∈ V β for E a -measure one many u . But then restricting to this set, weget a function g ∈ V δ .So let M = J θ and U = Ult ( M, E ). Suppose now that the theorem fails.
Claim 2. U = M is wellfounded and j + = i ME : M → U is cofinal and Σ -elementary, with j + ( V δ ) = V δ and j ⊆ j + .Proof. We also have Σ - Lo´s’ criterion for the ultrapower, by Lemma 7.1. Thisgives the Σ -elementarity of i ME . And note that by the previous claim and ourcontradictory hypothesis, U is wellfounded. The fact that j + ( V δ ) = V δ followsfrom the continuity of j + at δ , which holds because cof( δ ) = ω , like in the proofof the previous claim.It remains to see that U = M . Note that U = J γ ( V δ ) for some γ , andcertainly θ ≤ γ . But M | =“There is no α > δ ∈ OR such that J α is admissible”,so by Σ -elementarity and as j + ( V δ ) = V δ , U satisfies this statement. Therefore γ ≤ θ , so we are done. Claim 3.
Given any Σ -elementary k : V δ → V δ , there is at most one extensionof k to a Σ -elemenentary k + : J θ → J θ with k + ( V δ ) = V δ . Moreover, if k isdefinable from parameters over J θ then so is k + .Proof. Let us first observe that k + ↾ θ is uniquely determined. Given α < θ ,there is a Σ formula ϕ and z ∈ V δ such that α is the least α ′ ∈ OR such that J α ′ | = ψ ( z, V δ ), where ψ ( ˙ z, ˙ v ) = “ ∀ x ∈ ˙ v ∃ y ϕ ( x, y, ˙ z, ˙ v )” . k + ( α ) must be the least α ′ ∈ OR such that J α ′ | = ψ ( k ( z ) , V δ ).But now J α = Hull J α ( V δ ∪ { V δ } ), and since k + ↾ J α : J α → J k + ( α ) must beΣ -elementary and have k + ( V δ ) = V δ , k determines k + ↾ J α .The “moreover” clause clearly follows from the manner in which we havecomputed k + above from k . Claim 4.
Let k + : J θ → J θ be Σ -elementary with k + ( V δ ) = V δ . Then k + isfully elementary.Proof. Given γ ≤ δ and α ≤ θ , let t αγ = Th J α Σ ( V γ ∪ { V δ } ). Let e t αγ code t αγ asa subset of V γ . We claim that k + ( e t θδ ) = e t θδ . For given γ < δ , since e t θγ ∈ V δ , byadmissibility there is α γ < θ such that t βγ = t θγ for all β ∈ [ α γ , θ ]. But then iteasily follows that k ( e t θγ ) = e t θk ( γ ) . But then it follows that k ( e t θδ ) = e t θδ .Also, if δ is singular in J θ then k is continuous at cof J θ ( δ ), because k ( δ ) = δ .Simlarly if J θ | =“ δ is regular but not inaccessible”.From here we can argue as in the proof of [4, Theorem 5.6***?].Using the preceding claims, we can now derive the usual kind of contradic-tion, considering the least critical point κ of any Σ -elementary k + : J θ → J θ such that k + ( δ ) = δ and k + is Σ n -definable from parameters over J θ (for someappropriate n ). This completes the proof. Remark 7.13.
The argument above shows that if δ is a limit and κ = κ V δ ,then cof L ( V δ ) ( κ ) = cof L ( V δ ) ( δ ), definably over J κ ( V δ ). It also shows that, with T ∞ as before, for each α < δ , since T ∞ ∩ V α ∈ V δ , there is γ < κ such that T γ ∩ V α = T ∞ ∩ V α . Also note that, for example, if δ = λ + ω , then T ∞ cannot bejust finitely splitting beyond some node t (counting here the number of nodesbeyond t in each V λ + n , for n < ω ). For otherwise (taking t with dom( j t ) = V λ + n for some n ), the tree structure of ( T ∞ ) t is coded by a real, hence is in V δ , andan admissibility argument easily gives that ( T ∞ ) t is computed by some γ < κ ,which leads to a contradiction as before. However, such an argument doesn’tseem to work in the case that ( T ∞ ) t is just ω -splitting. Remark 7.14.
In the only example we have where δ ∈ Lim and and j ∈ E ( V δ ) ∩ L ( V δ ), we have V δ | = ZFC +“ V = HOD”, so L ( V δ ) | = AC and δ = κ ω ( j ).Is it possible to have j ∈ L ( V δ ) ∩ E ( V δ ) with κ ω ( j ) < δ ? Or even with κ ω ( j ) = δ but L ( V δ ) | = ¬ AC ? Is it possible to have this with ( V δ , j ) | = ZFR? We knowthat we need cof L ( V δ ) = ω for this. We have the tree T ∞ ∈ L ( V δ ). But withoutleft-most branches, it is not clear to the author how to get an embedding in L ( V δ ) from this. Relatedly, is it possible for j to be generic over L ( V δ ) and have V δ = V L ( V δ )[ j ] δ ? References [1] Joan Bagaria, Peter Koellner, and W. Hugh Woodin. Large cardinals be-yond choice.
Bulletin of Symbolic Logic , 25, 2019.[2] Gabriel Goldberg. Even ordinals and the Kunen inconsistency.arXiv:2006.01084.[3] Gabriel Goldberg. Personal communication, 2020.274] Gabriel Goldberg and Farmer Schlutzenberg. Periodicity in the cumulativehierarchy. arXiv: 2006.01103.[5] R. Bj¨orn Jensen. The fine structure of the constructible hierarchy.
Annalsof Mathematical Logic , 4(3), 1972.[6] Akihiro Kanamori.
The higher infinite: large cardinals in set theory fromtheir beginnings . Springer monographs in mathematics. Springer-Verlag,second edition, 2005.[7] Kenneth Kunen. Elementary embeddings and infinitary combinatorics.
Journal of Symbolic Logic , 36(3), 1971.[8] W. N. Reinhardt.
Topics in the metamathematics of set theory . PhD thesis,UC Berkeley, 1967.[9] W. N. Reinhardt. Remarks on reflection principles, large cardinals, and el-ementary embeddings. In
Axiomatic set theory (Proc. Sympos. Pure Math.,Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967) , pages 189–205. Amer. Math. Soc., Providence, R. I., 1974.[10] Ralf Schindler and Martin Zeman. Fine structure. In