HHOD
IN INNER MODELS WITH WOODIN CARDINALS
SANDRA M ¨ULLER AND GRIGOR SARGSYAN
Abstract.
We analyze the hereditarily ordinal definable sets HOD in M n ( x )[ g ] for a Turing cone of reals x , where M n ( x ) is the canonicalinner model with n Woodin cardinals build over x and g is generic over M n ( x ) for the L´evy collapse up to its bottom inaccessible cardinal. Weprove that assuming Π n +2 -determinacy, for a Turing cone of reals x ,HOD M n ( x )[ g ] = M n ( M ∞ | κ ∞ , Λ) , where M ∞ is a direct limit of iteratesof M n +1 , δ ∞ is the least Woodin cardinal in M ∞ , κ ∞ is the least inacces-sible cardinal in M ∞ above δ ∞ , and Λ is a partial iteration strategy for M ∞ . It will also be shown that under the same hypothesis HOD M n ( x )[ g ] satisfies GCH. Introduction
An essential question regarding the theory of inner models is the analysis ofthe class of all hereditarily ordinal definable sets HOD inside various innermodels M of the set theoretic universe V under appropriate determinacyhypotheses. Examples for such inner models M are L ( R ), L [ x ], and thecanonical proper class x -mouse with n Woodin cardinals M n ( x ), but nowa-days also larger models of determinacy M are considered.One motivation for analyzing the internal structure of these models HOD M is given by Woodin’s results in [KW10] that under determinacy hypothesesthese models contain large cardinals. He showed in [KW10] for examplethat assuming ∆ determinacy there is a Turing cone of reals x such that ω L [ x ]2 is a Woodin cardinal in the model HOD L [ x ] . This result generalizes tohigher levels in the projective hierarchy. That means for n ≥ Π n +1 determinacy and Π n +2 determinacy there is a cone of reals x suchthat ω M n ( x )2 is a Woodin cardinal in the model HOD M n ( x ) | δ x , where M n ( x )denotes the canonical proper class x -mouse with n Woodin cardinals and δ x is the least Woodin cardinal in M n ( x ). Moreover, Woodin showed a similarresult for HOD L ( R ) . If we let Θ denote the supremum of all ordinals α Date : April 21, 2020.Part of this work was done whilst the authors were visiting fellows at the IsaacNewton Institute for Mathematical Sciences in the programme
Mathematical, Founda-tional and Computational Aspects of the Higher Infinite (HIF) funded by EPSRC grantEP/K032208/1. The first author, formerly known as Sandra Uhlenbrock, was partiallysupported by FWF grant number P 28157. The second author was partially supportedby the NSF Career Award DMS-1352034. a r X i v : . [ m a t h . L O ] A p r SANDRA M ¨ULLER AND GRIGOR SARGSYAN such that there exists a surjection π : R → α , then assuming ZF + AD, heshowed that Θ L ( R ) is a Woodin cardinal in HOD L ( R ) (see [KW10]). Thefact that these models of the form HOD M can have large cardinals as forexample Woodin cardinals motivates the question if they are in some sensefine structural as for example the models L [ x ] , M n ( x ), and L ( R ) are. A goodtest question for this is whether these models HOD M satisfy the generalizedcontinuum hypothesis GCH. If it turns out that HOD M is in fact a finestructural model, it would follow that it satisfies the GCH and even strongercombinatorial principles as for example the ♦ principle.The first model which was analyzed in this sense was HOD L ( R ) under theassumption that every set of reals in L ( R ) is determined (short: AD L ( R ) ).Using purely descriptive set theoretic methods Becker showed in [Be80] un-der this hypothesis that GCH α , i.e. 2 α = α + , holds in HOD L ( R ) for all α < ω L ( R )1 . Later J. R. Steel and W. H. Woodin were able to push the anal-ysis of HOD L ( R ) forward using more recent advances in inner model theory.In 1993 they first showed independently that the reals in HOD L ( R ) are thesame as the reals in M ω , the least proper class iterable premouse with ω Woodin cardinals. Then they showed in § L ( R ) in factagrees with the inner model N up to P ( ω L ( R )1 ), where N denotes the ω L ( R )1 -th linear iterate of M ω by its least measure and its images. Building on this,John R. Steel was able to show in [St95] that HOD L ( R ) agrees with the innermodel M ∞ up to ( δ ) L ( R ) , where M ∞ is a direct limit of iterates of M ω and( δ ) L ( R ) is the supremum of all ordinals α such that there exists a surjection π : R → α which is ∆ L ( R )1 definable. Finally, in 1996 W. Hugh Woodinextended this (see [StW16]) and showed that in fact HOD L ( R ) = L [ M ∞ , Λ],where Λ is a partial iteration strategy for M ∞ . For even larger models of de-terminacy M the corresponding model HOD M was first analyzed in [Sa09],where the second author showed that it is fine structural using a layeredhierarchy. Models of this form are nowadays called hod mice . A differentapproach for the fine structure of hod mice called the least branch hierarchyis studied in [St16].The question if HOD L [ x ] is a model of GCH or even a fine structural modelfor a Turing cone of reals x under a suitable determinacy hypothesis re-mains open until today. What has been done is the analysis of the modelHOD L [ x ][ G ] , where G is Col( ω, <κ x )-generic over HOD L [ x ] for the least in-accessible cardinal κ x in L [ x ]. Woodin showed in the 1990’s (see [StW16])that assuming ∆ determinacy there is a Turing cone of reals x such thatHOD L [ x ][ G ] = L [ M ∞ , Λ], where M ∞ is a direct limit of mice (which areiterates of M ) and Λ is a partial iteration strategy for M ∞ .In this article, we analyze HOD in the model M n ( x )[ g ] for any real x ofsufficiently high Turing degree under the assumption that every Π n +2 setof reals is determined. Here g is Col( ω, <κ )-generic over M n ( x ), where κ denotes the least inaccessible cardinal in M n ( x ). We first show that the OD IN INNER MODELS WITH WOODIN CARDINALS 3 direct limit model M ∞ , obtained from iterates of suitable premice, agreesup to its bottom Woodin cardinal δ ∞ with HOD M n ( x )[ g ] . In a second step, weshow that the full model HOD M n ( x )[ g ] is in fact of the form M n ( ˆ M ∞ | κ ∞ , Λ),where ˆ M ∞ = M n ( M ∞ | δ ∞ ), κ ∞ is the least inaccessible cardinal of ˆ M ∞ above δ ∞ , and Λ is a partial iteration strategy for M ∞ . Here and below M n ( ˆ M ∞ | κ ∞ , Λ) denotes the canonical fine structural model with n Woodincardinals build over the coarse objects ˆ M ∞ | κ ∞ and Λ. Our proof in factshows that HOD M n ( x )[ g ] is a model of GCH, ♦ , and other combinatorialprinciples which are consequences of fine structure.In the statement of the following main theorem and in fact everywhere in thisarticle whenever we write HOD M for some premouse M we mean HOD (cid:98) M (cid:99) ,where (cid:98) M (cid:99) denotes the universe of the model M . In particular, we do notallow the extender sequence of M as a parameter in the definition of HOD.It will be clear from the context if we consider the model M or the universe (cid:98) M (cid:99) of M , therefore we decided for the sake of readability to not distinguishthe notation for these two objects.The main result of this paper is the following theorem. Theorem 1.1.
Let n < ω and assume Π n +2 -determinacy. Then for aTuring cone of reals x , HOD M n ( x )[ g ] = M n ( ˆ M ∞ | κ ∞ , Λ) , where g is Col( ω, <κ ) -generic over M n ( x ) , κ denotes the least inaccessiblecardinal in M n ( x ) , ˆ M ∞ is a direct limit of iterates of M n +1 , δ ∞ is the leastWoodin cardinal in ˆ M ∞ , κ ∞ is the least inaccessible cardinal of ˆ M ∞ above δ ∞ , and Λ is a partial iteration strategy for M ∞ . Our proof in fact shows the following corollary.
Corollary 1.2.
Assume Π n +2 -determinacy. Then for a Turing cone ofreals x , HOD M n ( x )[ g ] (cid:15) GCH , where g is Col( ω, <κ ) -generic over M n ( x ) and κ denotes the least inacces-sible cardinal in M n ( x ) .Remark. In fact the full strength of Π n +2 -determinacy is not needed forthese results. It suffices to assume that M n ( x ) exists and is ω -iterable forall reals x (or equivalently Π n +1 -determinacy, see [MSW] and [Ne02]) andthat M n +1 exists and is ω -iterable. This is all we will use in the proof.Finally, we summarize some open questions related to these results. Thefollowing question already appears in [StW16]. Question 1.
Assume ∆ determinacy. Is HOD L [ x ] for a cone of reals x afine structural model? SANDRA M ¨ULLER AND GRIGOR SARGSYAN
Question 2.
Assume Π n +2 determinacy. Is HOD M n ( x ) for a cone of reals x a fine structural model? This article is structured as follows. In Section 2 we recall some preliminariesand fix the basic notation. In Section 3 we recall the relevant notions from[Sa13] and define the direct limit system converging to M ∞ , before we com-pute HOD M n ( x )[ g ] up to its Woodin cardinal in Section 4. In Section 5 wethen show how this can be used to compute the full model HOD M n ( x )[ g ] , i.e.,we finish the proof of Theorem 1.1. The authors thank Farmer Schlutzen-berg for the helpful discussions during the 4th M¨unster conference on innermodel theory in the summer of 2017. Finally, the authors thank the refereefor carefully reading the paper and making several helpful comments andsuggestions. 2. Preliminaries and notation
Whenever we say reals we mean elements of the Baire space ω ω . We alsowrite R for ω ω . HOD denotes the class of all hereditarily ordinal definablesets. Moreover HOD x for any x ∈ ω ω denotes the class of all sets whichare hereditarily ordinal definable over { x } . That means we let A ∈ OD x iffthere is a formula ϕ such that A = { v | ϕ ( v, α , . . . , α n , x ) } for some ordinals α , . . . , α n . Then A ∈ HOD x iff TC( { A } ) ⊂ OD x , where TC( { A } ) denotesthe transitive closure of the set { A } .We use the notions of premice and iterability from [St10, § −
4] and assumethat the reader is familiar with the basic concepts defined there. In mostcases we will demand ( ω, ω , ω )-iterability in the sense of Definition 4 . cutpoint of a premouse M is an infinite ordinal γ such that there is no extender E on the M -sequence with crit( E ) ≤ γ ≤ lh( E ). For some ZFC model M and some real x ∈ M we write L [ E ]( x ) M for theresult of a fully backgrounded extender construction above x inside M in thesense of [MS94], with the minimality condition relaxed to ω -small premice.Moreover, we let for a premouse M with M (cid:15) ZFC, a cardinal cutpoint η of M , and a premouse N of height η such that N ∈ P ( M| η ) ∩ M| ( η + ω ), P M ( N ) denote the result of a P -construction over N inside the model M in the sense of [SchSt09] or [Sa13, Proposition 2.3 and Definition 2.4].For x ∈ ω ω and n ≤ ω we let M n ( x ), if it exists, denote a countable,sound, ω -iterable x -premouse which is not n -small but all of whose properinitial segments are n -small. In fact, ω -iterability suffices to show thatsuch an M n ( x ) is unique. If M n ( x ) exists, we let M n ( x ) be the proper In the literature this is sometimes also called HOD { x } . Such a cutpoint γ is often also called a strong cutpoint. OD IN INNER MODELS WITH WOODIN CARDINALS 5 class premouse obtained by iterating the top extender of M n ( x ) out of theuniverse. 3. The direct limit system
To show that HOD M n ( x )[ g ] is a fine structural inner model, we will use an ex-tension of the direct limit system introduced in [Sa13]. For the reader’s con-venience we will first recall the relevant definitions and results from [Sa13],obtaining a direct limit system which is definable in M n ( x ). We use thechance to correct some minor errors in the presentation of that direct limitsystem in [Sa13]. Then we discuss the changes we need to make to obtaina direct limit system definable in M n ( x )[ g ]. Another application of a sim-ilar but slightly different direct limit system as in [Sa13] can be found in[SaSch18].Fix an arbitrary natural number n . Throughout the rest of this article wewill assume that M n +1 exists and is ( ω, ω , ω )-iterable and fix a real x thatcodes M n +1 . This implies Π n +1 determinacy or equivalently that M n ( z )exists and is ( ω, ω , ω )-iterable for all reals z (see [Ne95] and [MSW] for aproof of this equivalence due to Itay Neeman and W. Hugh Woodin). Finally,we fix a Col( ω, <κ )-generic g over M n ( x ), where κ is the least inaccessiblecardinal in M n ( x ). The first direct limit system.
We first recall the definition of a lowerpart premouse.
Definition 3.1.
Let a be a countable, transitive, self-wellordered set. Thenwe define the lower part model Lp n ( a ) as the model theoretic union of allcountable a -premice M with ρ ω ( M ) = a which are n -small, sound, and ( ω, ω , ω ) -iterable. If N is a countable premouse, we also use Lp n ( N ) to denote the premouseextending N which is defined similarly as the model theoretic union of pre-mice M (cid:68) N with ρ ω ( M ) ≤ N ∩ Ord which have
N ∩
Ord as a cutpoint,are n -small above N ∩
Ord, sound above
N ∩
Ord, and ( ω, ω , ω )-iterableabove N ∩
Ord.
Definition 3.2.
A countable premouse N is n -suitable iff there is an ordinal δ such that(1) N (cid:15) “ ZFC − Replacement” and
N ∩
Ord = sup i<ω ( δ + i ) N ,(2) N (cid:15) “ δ is a Woodin cardinal”,(3) N is ( n + 1) -small,(4) for every cutpoint γ < δ of N , γ is not Woodin in Lp n ( N | γ ) ,(5) N | ( δ +( i +1) ) N = Lp n ( N | ( δ + i ) N ) for all i < ω , and(6) for all η < δ , N (cid:15) “ N | δ is ( ω, η, η ) -iterable”. We say a transitive set a is self-wellordered iff a is wellordered in L ω [ a ]. SANDRA M ¨ULLER AND GRIGOR SARGSYAN If N is an n -suitable premouse we denote the ordinal δ from Definition 3.2by δ N . Moreover, we write ˆ N = M n ( N | δ N ) for any n -suitable premouse N . Then N = ˆ N | (( δ N ) + ω ) ˆ N for every n -suitable premouse N by well-known properties of the lower part model Lp n . We now give some definitionsindicating how n -suitable premice can be iterated. Definition 3.3.
Let N be an arbitrary premouse and let T be an iterationtree on N of limit length.(1) We say a premouse Q = Q ( T ) is a Q -structure for T iff M ( T ) (cid:69) Q , Q is sound, δ ( T ) is a cutpoint of Q , Q is ( ω, ω , ω ) -iterable above δ ( T ) ,and if Q (cid:54) = M ( T ) Q (cid:15) “ δ ( T ) is a Woodin cardinal” , and ( i ) over Q there exists an r Σ n -definable set A ⊂ δ ( T ) such that thereis no κ < δ ( T ) such that κ is strong up to δ ( T ) with respect to A as being witnessed by extenders on the sequence of Q for some n < ω , or ( ii ) ρ n ( Q ) < δ ( T ) for some n < ω .(2) Let b be a cofinal well-founded branch through T . Then we say a pre-mouse Q = Q ( b, T ) is a Q -structure for b in T iff Q is sound and Q = M T b | γ , where γ ≤ M T b ∩ Ord is the least ordinal such that either γ < M T b ∩ Ord and M T b | ( γ + 1) (cid:15) “ δ ( T ) is not Woodin”,or γ = M T b ∩ Ord and ρ n ( M T b ) < δ ( T ) for some n < ω or over M T b there exists an r Σ n -definable set A ⊂ δ ( T ) such that there is no κ < δ ( T ) such that κ is strong up to δ ( T ) withrespect to A as being witnessed by extenders on the sequence of M T b forsome n < ω .If no such ordinal γ ≤ M T b ∩ Ord exists, we let Q ( b, T ) be undefined.Remark. If it exists, M n +1 | ( δ + ω ) M n +1 is n -suitable, where δ is the leastWoodin cardinal in M n +1 . We denote this premouse by M − n +1 and writeΣ M − n +1 for its iteration strategy induced by the canonical Q -structure guidediteration strategy Σ M n +1 for M n +1 for countable stacks of normal trees with-out drops on the main branches.Our goal is to approximate the iteration strategy Σ M − n +1 inside HOD M n ( x )[ g ] .Analogous to [SchlTr, Definition 5.32] we define the following requirement,which will be used in Definition 3.6 to make the proof of Lemmas 3.8 and3.9 work. Definition 3.4.
Let N be an n -suitable premouse and let T be a normaliteration tree on N of length < ω V . Then we say that T is suitability strict iff for all α < lh( T ) , OD IN INNER MODELS WITH WOODIN CARDINALS 7 ( i ) if [0 , α ] T does not drop then M T α is n -suitable, and ( ii ) if [0 , α ] T drops then no R (cid:69) M T α is n -suitable. Definition 3.5.
Let N be an n -suitable premouse and let T be a normaliteration tree on N of length < ω V .(1) T is correctly guided iff for every limit ordinal λ < lh( T ) , if b isthe branch choosen for T (cid:22) λ in T , then Q ( b, T (cid:22) λ ) exists and Q ( b, T (cid:22) λ ) (cid:69) M n ( M ( T (cid:22) λ )) .(2) T is short iff T is correctly guided and in case T has limit length Q ( T ) exists and Q ( T ) (cid:69) M n ( M ( T )) .(3) T is maximal iff T is correctly guided and not short. Definition 3.6.
Let N be an n -suitable premouse. We say N is short treeiterable iff whenever T is a short tree on N , ( i ) T is suitability strict, ( ii ) if T has a last model, then every putative iteration tree U extending T such that lh( U ) = lh( T ) + 1 has a well-founded last model, and ( iii ) if T has limit length, then there exists a cofinal well-founded branch b through T such that Q ( b, T ) = Q ( T ) . This can be generalized to stacks of correctly guided normal trees.
Definition 3.7.
Let N be an n -suitable premouse and m < ω . Then we say ( T i , N i | i ≤ m ) is a correctly guided finite stack on N iff ( i ) N = N , ( ii ) N i is n -suitable and T i is a correctly guided normal iteration tree on N i which acts below δ N i for all i ≤ m , ( iii ) for every i < m either T i has a last model which is equal to N i +1 andthe iteration embedding i T i : N i → N i +1 exists or T i is maximal and N i +1 = M n ( M ( T i )) | ( δ ( T i ) + ω ) M n ( M ( T i )) .Moreover, we say that M is the last model of ( T i , N i | i ≤ m ) iff either ( i ) T m has a last model which is equal to M and the iteration embedding i T m : N m → M exists, ( ii ) T m is of limit length and short and there is a non-dropping cofinalwell-founded branch b through T m such that Q ( b, T ) exists, T m (cid:97) b iscorrectly guided, and M = M T b , or ( iii ) T m is maximal and M = M n ( M ( T m )) | ( δ ( T m ) + ω ) M n ( M ( T m )) .Finally, we say that M is a correct iterate of N iff there is a correctly guidedfinite stack on N with last model M . In case there is a correctly guided finitestack on N with last model M of length , i.e., such that m = 0 , we saythat M is a pseudo-normal iterate (or just pseudo-iterate ) of N . An iteration tree U is a putative iteration tree if U satisfies all properties of an iterationtree, but in case U has a last model we allow this last model to be ill-founded. SANDRA M ¨ULLER AND GRIGOR SARGSYAN
Analogous to Theorem 3 .
14 in [StW16] we also have a version of the com-parison lemma for short tree iterable premice and pseudo-normal iterates.
Lemma 3.8 (Pseudo-comparison lemma) . Let N and M be n -suitable pre-mice which are short tree iterable. Then there is a common pseudo-normaliterate R ∈ M n ( y ) such that δ R ≤ ω M n ( y )1 , where y is a real coding N and M . The proof of Lemma 3.8 is similar to the proof of Theorem 3 .
14 in [StW16],so we omit it. Similarly, we have an analogue to the pseudo-genericityiteration (see Theorem 3 .
16 in [StW16]).
Lemma 3.9 (Pseudo-genericity iterations) . Let N be an n -suitable pre-mouse which is short tree iterable and let z be a real. Then there is apseudo-normal iterate R of N in M n ( y, z ) such that z is B R -generic over R and δ R ≤ ω M n ( y,z )1 , where y is a real coding N and B R denotes Woodin’sextender algebra inside R . For the definition of the direct limit system converging to HOD we need thenotion of s -iterability. To define this, we first introduce some notation. Foran n -suitable premouse N , a finite sequence of ordinals s , and some k < ω let T N s,k = { ( t, (cid:112) φ (cid:113) ) ∈ [(( δ N ) + k ) N ] <ω × ω | φ is a Σ -formula and M n ( N | δ N ) (cid:15) φ [ t, s ] } , where (cid:112) φ (cid:113) denotes the G¨odel number of φ . Let Hull N denote an uncollapsedΣ hull in N . Then we let γ N s = sup( Hull N ( { T N s,k | k < ω } ) ∩ δ N )and H N s = Hull N ( γ N s ∪ { T N s,k | k < ω } ) . Then γ N s = H N s ∩ δ N . For s m = ( u , . . . , u m ) the sequence of the first m uniform indiscernibles, we write γ N m = γ N s m and H N m = H N s m . Then we havethat sup m ∈ ω γ N m = δ N (see Lemma 5.3 in [Sa13]). Definition 3.10.
Let N be an n -suitable premouse and s a finite sequenceof ordinals. Then N is s -iterable iff every correct iterate of N is short treeiterable and for every correctly guided finite stack ( T i , N i | i ≤ m ) on N withlast model M there is a sequence of non-dropping branches ( b i | i ≤ m ) anda sequence of embeddings ( π i | i ≤ m ) such that ( i ) if T i has successor length α + 1 , then b i = [0 , α ] T i and π i = i T i ,α is thecorresponding iteration embedding for i ≤ m , ( ii ) if T m is short, then b m is the unique cofinal well-founded branch through T m such that Q ( b m , T m ) exists and T m (cid:97) b m is correctly guided and π m = i T m b m is the corresponding iteration embedding, OD IN INNER MODELS WITH WOODIN CARDINALS 9 ( iii ) if T i is maximal, then b i is a cofinal well-founded branch through T i such that M T i b i = N i +1 if i < m or M T i b i = M if i = m , and π i = i T i b i isthe corresponding iteration embedding for i ≤ m , and ( iv ) if we let π = π m ◦ π m − ◦ · · · ◦ π then for every k < ω , π ( T N s,k ) = T M s,k . In this case we say that the sequence (cid:126)b = ( b i | i ≤ m ) witnesses s -iterability for (cid:126) T = ( T i , N i | i ≤ m ) or that (cid:126)b is an s -iterability branch for (cid:126) T and wewrite π (cid:126) T ,(cid:126)b = π . Now for every two s -iterability branches for (cid:126) T on N their correspondingiteration embeddings agree on H N s . Lemma 3.11 (Uniqueness of s -iterability embeddings, Lemma 5 . . Let N be an n -suitable premouse, s a finite sequence of ordinals, and (cid:126) T acorrectly guided finite stack on N . Moreover let (cid:126)b and (cid:126)c be s -iterabilitybranches for (cid:126) T . Then π (cid:126) T ,(cid:126)b (cid:22) H N s = π (cid:126) T ,(cid:126)c (cid:22) H N s . The uniqueness of s -iterability embeddings yields that for every n -suitable, s -iterable N , every correctly guided finite stack (cid:126) T on N and every s -iterability branch (cid:126)b for (cid:126) T , the embedding π (cid:126) T ,(cid:126)b (cid:22) H N s is independent ofthe choice of (cid:126)b , but it might still depend on (cid:126) T . This motivates the followingdefinition. Definition 3.12.
Let N be an n -suitable premouse and s a finite sequenceof ordinals. Then N is strongly s -iterable iff for every correct iterate R of N , R is s -iterable and for every two correctly guided finite stacks (cid:126) T and (cid:126) U on R with common last model M and s -iterability witnesses (cid:126)b and (cid:126)c for (cid:126) T and (cid:126) U respectively, we have that π (cid:126) T ,(cid:126)b (cid:22) H R s = π (cid:126) U ,(cid:126)c (cid:22) H R s . A so-called bad sequence argument shows the following lemma, which yieldsthe existence of strongly s -iterable premice. Lemma 3.13 (Lemma 5 . . For every finite sequence of ordinals s and any short tree iterable n -suitable premouse N there is a pseudo-normaliterate M of N such that M is strongly s -iterable. If N is strongly s -iterable and (cid:126) T is a correctly guided finite stack on N withlast model M , let π N , M ,s : H N s → H M s denote the embedding given byany s -iterability branch (cid:126)b for (cid:126) T . As N is strongly s -iterable, the embedding π N , M ,s does not depend on the choice of (cid:126) T and (cid:126)b .Recall that we write M − n +1 = M n +1 | ( δ + ω ) M n +1 , where δ is the least Woodincardinal in M n +1 , and Σ M − n +1 for the canonical iteration strategy for M − n +10 SANDRA M ¨ULLER AND GRIGOR SARGSYAN induced by Σ M n +1 . Moreover, recall that for m < ω , we write s m for the setof the first m uniform indiscernibles. Then M − n +1 is n -suitable and strongly s m -iterable for every m . Moreover, if (cid:126) T is a correctly guided finite stackon M − n +1 with last model M , then π M − n +1 , M ,s m agrees with the iterationembedding according to Σ M − n +1 on H M − n +1 s m . The first direct limit system wedefine will consist of iterates of M − n +1 . Definition 3.14.
Let ˜ F + = {N | N is an iterate of M − n +1 via Σ M − n +1 by a finite stack of trees } and for N , M ∈ ˜ F + let N ≤ + M iff M is an iterate of N via the tailstrategy Σ N as witnessed by some finite stack of iteration trees. Then we let ˜ M + ∞ be the direct limit of ( ˜ F + , ≤ + ) under the iteration maps.Remark. The prewellordering ≤ + on ˜ F + is directed and the direct limit˜ M + ∞ is well-founded as the limit system ( ˜ F + , ≤ + ) only consists of iteratesof M − n +1 via the canonical iteration strategy Σ M − n +1 .Since ˜ F + is not definable enough for our purposes, we now introduce anotherdirect limit system which has the same direct limit ˜ M + ∞ . Definition 3.15.
Let ˜ I = { ( N , s ) | N is n -suitable , s ∈ [Ord] <ω , and N is strongly s -iterable } and ˜ F = { H N s | ( N , s ) ∈ ˜ I} . For ( N , s ) , ( M , t ) ∈ ˜ I we let ( N , s ) ≤ ˜ I ( M , t ) iff there is a correctlyguided finite stack on N with last model M and s ⊆ t . In this case welet π ( N ,s ) , ( M ,t ) : H N s → H M t denote the canonical corresponding embedding.Remark. The prewellordering ≤ ˜ I on ˜ I is directed: Let ( N , s ) , ( M , t ) ∈ ˜ I .By Lemma 3.13 there exists an n -suitable premouse R which is strongly( s ∪ t )-iterable. Let S be the result of simultaneously comparing N , M and R in the sense of Lemma 3.8. Then ( S , s ∪ t ) ∈ ˜ I , ( N , s ) ≤ ˜ I ( S , s ∪ t ), and( M , t ) ≤ ˜ I ( S , s ∪ t ), as desired. Definition 3.16.
Let ˜ M ∞ be the direct limit of ( ˜ F , ≤ ˜ I ) under the embed-dings π ( N ,s ) , ( M ,t ) . For ( N , s ) ∈ ˜ I let π ( N ,s ) , ∞ : H N s → ˜ M ∞ denote thecorresponding direct limit embedding. The fact that ˜ M ∞ is well-founded follows from the next lemma. Lemma 3.17 (Lemma 5 .
10 in [Sa13]) . ˜ M ∞ = ˜ M + ∞ . OD IN INNER MODELS WITH WOODIN CARDINALS 11
The second direct limit system.
To obtain HOD of some inner modelfrom the direct limit, we in particular need to show that the direct limit isin fact contained in HOD of that inner model. In our setting we thereforeneed to internalize the direct limit system into the inner model M n ( x )[ g ]fixed above. We first aim to define a direct limit system similar to ( ˜ F , ≤ ˜ I )in M n ( x ) analogous to [Sa13]. In a second step, we then modify the systemto obtain direct limit systems with the same direct limit which are definablein M n ( x )[ g ].The notion of n -suitability from Definition 3.2 is already internal to M n ( x )and M n ( x )[ g ], i.e., if N ∈ M n ( x ) | κ then N is n -suitable in V iff N is n -suitable in M n ( x ) by the following lemma. Lemma 3.18.
Let δ denote the least Woodin cardinal in M n ( x ) .(1) For all y ∈ V M n ( x )[ g ] δ , Lp n ( y ) ∈ HOD M n ( x )[ g ] y .(2) V M n ( x ) δ and V M n ( x )[ g ] δ are closed under the operation y (cid:55)→ Lp n ( y ) .Proof. Let y ∈ V M n ( x )[ g ] δ be arbitrary. The model M n ( x )[ g ] can be organizedas a V M n ( x )[ g ] κ -premouse and as such it inherits the iterability from M n ( x )and is in fact equal to M n ( V M n ( x )[ g ] κ ). Consider L [ E ]( y ) M n ( x )[ g ] , the resultof a fully backgrounded extender construction above y using extenders fromthe sequence of M n ( x )[ g ] organized as a V M n ( x )[ g ] κ -premouse, and compareit with Lp n ( y ). First, we argue that Lp n ( y ) does not move. If it wouldmove, the Lp n ( y )-side of the coiteration would have to drop because Lp n ( y )does not have any total extenders. Moreover, it would have to iterate toa proper class model which is equal to an iterate of L [ E ]( y ) M n ( x )[ g ] . As L [ E ]( y ) M n ( x )[ g ] has n Woodin cardinals, this would imply that Lp n ( y ) has alevel which is not n -small, contradicting the definition of Lp n ( y ).Therefore, Lp n ( y ) (cid:67) R for some iterate R of L [ E ]( y ) M n ( x )[ g ] . The iterationfrom L [ E ]( y ) M n ( x )[ g ] to R resulting from the comparison process can bedefined over L [ E ]( y ) M n ( x )[ g ] from the extender sequence of L [ E ]( y ) M n ( x )[ g ] and a finite sequence of ordinals as it cannot leave any total measures behindand thus can only use measures of order 0. The extender sequence of the V M n ( x )[ g ] κ -premouse M n ( x )[ g ] is in HOD M n ( x )[ g ] V Mn ( x )[ g ] κ = HOD M n ( x )[ g ] . Therefore Lp n ( y ) ∈ HOD M n ( x )[ g ] by the definability of the L [ E ]-construction.For (2), the closure of V M n ( x )[ g ] δ follows immediately from (1). For V M n ( x ) δ notice that for y ∈ V M n ( x ) δ , Lp n ( y ) ∈ HOD M n ( x )[ g ] y ⊆ HOD M n ( x ) y by homo-geneity of the forcing. (cid:3) By stacking the Lp n -operation, this lemma in fact shows that for all y ∈ V M n ( x ) δ , M n ( y ) | κ ∈ HOD M n ( x )[ g ] y , where κ denotes the least measurablecardinal in M n ( y ). The definitions of short tree, maximal tree, and correctly guided finite stackwe gave above are internal to M n ( x ) and M n ( x )[ g ] as well, as they can bedefined only using the Lp n -operation. The only notion we have to take careof is s -iterability since it is not even clear how the sets T N s,k can be identifiedinside M n ( x ). This obstacle is solved by shrinking the direct limit system( ˜ F , ≤ ˜ I ) to a dense subset as follows. Definition 3.19.
Let G = {N ∈ M n ( x ) | κ | N is n -suitable and M n ( x ) (cid:15) “for some cardinalcutpoint η, δ N = η + , N | δ N ∈ P ( M n ( x ) | η + ) ∩ M n ( x ) | ( η + + ω ) , and M n ( x ) | η is generic over N for the δ N -generator version ofthe extender algebra at δ N ” } . See for example Section 4 . δ -generatorversion of the extender algebra at some Woodin cardinal δ . The followinglemma shows how we can use the fact that N ∈ G to detect M n ( N | δ N )inside M n ( x ). For some premouse R ∈ G we denote the last model of a P -construction above R| δ R performed inside M n ( x ) as introduced in [SchSt09](see also Proposition 2 . . P M n ( x ) ( R| δ R ). Lemma 3.20 (Lemma 5 .
11 in [Sa13]) . Let
N ∈ M n ( x ) | κ be an n -suitablepremouse such that for some cardinal cutpoint η < δ N of M n ( x ) , we havethat N | δ N ∈ P ( M n ( x ) | η + ) ∩ M n ( x ) | ( η + + ω ) and M n ( x ) | η is generic over N for the δ N -generator version of the extender algebra at δ N . Then N ∈ G and P M n ( x ) ( N | δ N ) = M n ( N | δ N ) . In particular, M n ( N | δ N )[ M n ( x ) | η ] = M n ( x ) . Using pseudo-genericity iterations (see Lemma 3.9) we can obtain the fol-lowing corollary.
Corollary 3.21.
Let N be a short tree iterable n -suitable premouse suchthat N ∈ M n ( x ) | κ . Then there is a correctly guided finite stack on N withlast model M such that M ∈ G and P M n ( x ) ( M| δ M ) = M n ( M| δ M ) . Now the following definition of s -iterability agrees with the previous onegiven in Definition 3.10 for n -suitable premice in G . Definition 3.22.
For
N ∈ G , s ∈ [Ord] <ω , and k < ω let T N , ∗ s,k = { ( t, (cid:112) φ (cid:113) ) ∈ [(( δ N ) + k ) N ] <ω × ω | φ is a Σ -formula and P M n ( x ) ( N | δ N ) (cid:15) φ [ t, s ] } . Then we say for
N ∈ G and s ∈ [Ord] <ω that M n ( x ) (cid:15) “ N is s -iterablebelow κ ” iff for every Col( ω, <κ ) -generic G over M n ( x ) and every correctlyguided finite stack (cid:126) T = ( T i , N i | i ≤ m ) ∈ HC M n ( x )[ G ] on N with last model OD IN INNER MODELS WITH WOODIN CARDINALS 13
M ∈ G , there is a sequence of branches (cid:126)b = ( b i | i ≤ m ) ∈ M n ( x )[ G ] and asequence of embeddings ( π i | i ≤ m ) satisfying ( i ) − ( iii ) in Definition 3.10such that if we let π (cid:126) T ,(cid:126)b = π m ◦ π m − ◦ · · · ◦ π , then for every k < ω , π (cid:126) T ,(cid:126)b ( T N , ∗ s,k ) = T M , ∗ s,k . In addition, we define M n ( x ) (cid:15) “ N is strongly s -iterable below κ ” analogousto Definition 3.12 for all Col( ω, <κ )-generic G and stacks (cid:126) T , (cid:126) U ∈ M n ( x )[ G ].For N ∈ G , s ∈ [Ord] <ω , and k < ω , we have T N , ∗ s,k = T N s,k , so we will omitthe ∗ for N ∈ G . Using this, γ N s and H N s are defined as before. Then wecan define the internal direct limit system as follows. Definition 3.23.
Let I = { ( N , s ) | N ∈ G , s ∈ [Ord] <ω , and M n ( x ) (cid:15) “ N is strongly s -iterable below κ ” } and F = { H N s | ( N , s ) ∈ I} . Moreover, for ( N , s ) , ( M , t ) ∈ I we let ( N , s ) ≤ ( M , t ) iff there is a correctlyguided finite stack on N with last model M and s ⊆ t . In this case welet as before π ( N ,s ) , ( M ,t ) : H N s → H M t denote the canonical correspondingembedding. For clarity, we sometimes write ≤ I for ≤ . Similar as before we have thatfor every N ∈ G and s ∈ [Ord] <ω there is a normal correct iterate M of N such that ( M , s ) ∈ I . Using the fact that κ is inaccessible and a limit ofcutpoints in M n ( x ) we can obtain the following lemma. Lemma 3.24 (Lemma 5 .
14 in [Sa13]) . ≤ is directed. Therefore we can again define the direct limit.
Definition 3.25.
Let M ∞ be the direct limit of ( F , ≤ ) under the embeddings π ( N ,s ) , ( M ,t ) . Moreover, let δ ∞ = δ M ∞ be the Woodin cardinal in M ∞ and π ( N ,s ) , ∞ : H N s → M ∞ be the direct limit embedding for all ( N , s ) ∈ I . An argument similar to the one for Lemma 3.17 shows that this direct limitis well-founded as well. As we will use ideas from this proof in the nextsection, we will give some details here. We again first define another directlimit system which consists of iterates of M − n +1 and then show that its directlimit M + ∞ is equal to M ∞ . Definition 3.26.
Let F + = {Q ∈ G | Q is the last model of a correctly guidedfinite stack on M − n +1 via Σ M − n +1 } . Moreover, let
P ≤ + Q for P , Q ∈ F + iff there is a correctly guided finitestack on P according to the tail strategy Σ P with last model Q . In this casewe let i P , Q : P → Q denote the corresponding iteration embedding.
Then ≤ + on F + is directed, so we can define the direct limit. Definition 3.27.
Let M + ∞ be the direct limit of ( F + , ≤ + ) under the embed-dings i P , Q . Moreover, let i Q , ∞ : Q → M + ∞ denote the direct limit embeddingfor all Q ∈ F + . Then it is easy to see that M + ∞ is well-founded as F + only consists of iteratesof M − n +1 according to the canonical iteration strategy Σ M − n +1 . Lemma 3.28 (Lemma 5 .
15 in [Sa13]) . M + ∞ = M ∞ and hence M ∞ iswell-founded.Proof. We construct a sequence ( Q i | i < ω ) of iterates of M − n +1 such that Q i ∈ F + for every i < ω and ( Q i | i < ω ) is cofinal in G , i.e., for every N ∈ G there is an i < ω such that Q i is the last model of a correctly guidedfinite stack on N .In V , fix some sequence ( ξ i | i < ω ) of ordinals cofinal in κ . We define( Q i | i < ω ) together with a strictly increasing sequence ( η i | i < ω ) ofcardinal cutpoints of M n ( x ) | κ by induction on i < ω . So let Q = M − n +1 and let η < κ be a cardinal cutpoint of M n ( x ). Moreover assume that wealready constructed ( Q i | i ≤ j ) and ( η i | i ≤ j ) with the above mentionedproperties such that in addition ( Q i | i ≤ j ) ∈ M n ( x ) | η j . Let Q ∗ j +1 be theresult of simultaneously pseudo-comparing (in the sense of Lemma 3.8) all n -suitable premice M such that M ∈ G ∩ M n ( x ) | η j . Then in particular Q ∗ j +1 is a normal iterate of Q j according to the canonical tail iteration strategyΣ Q j , but Q ∗ j +1 might not be in G . Let ν be a cardinal cutpoint of M n ( x )such that η j < ν < κ and Q ∗ j +1 ∈ M n ( x ) | ν . Note that such a ν exists as κ is inaccessible and a limit of cardinal cutpoints in M n ( x ). Let Q j +1 bethe normal iterate of Q ∗ j +1 according to the canonical tail strategy Σ Q ∗ j +1 ofΣ Q j obtained by Woodin’s genericity iteration such that M n ( x ) | ν is genericover Q j +1 for the δ Q j +1 -generator version of the extender algebra (see forexample Section 4 . Q j +1 ∈ G is as desired. Finally choose η j +1 < κ such that η j +1 > max( η j , ξ j ), η j +1 is a cardinal cutpoint in M n ( x )and ( Q i | i ≤ j + 1) ∈ M n ( x ) | η j +1 .Now we define an embedding σ : M ∞ → M + ∞ as follows. Let x ∈ M ∞ .Since ( Q i | i < ω ) is cofinal in G , there are i, m < ω such that ( Q i , s m ) ∈ I and x = π ( Q i ,s m ) , ∞ (¯ x ) for some ¯ x ∈ H Q i s m ⊆ Q i . Then we let σ ( x ) = i Q i , ∞ (¯ x ).It follows as in the proof of Lemma 5 .
10 in [Sa13] that the definition of σ does not depend on the choice of i, m < ω and in fact σ = id. (cid:3) Moreover, it is possible to compute δ ∞ . Lemma 3.29 (Lemma 5 .
16 in [Sa13]) . δ ∞ = ( κ + ) M n ( x ) . OD IN INNER MODELS WITH WOODIN CARDINALS 15
Direct limit systems in
HOD M n ( x )[ g ] . Finally, we will argue that M ∞ ∈ HOD M n ( x )[ g ] by first defining direct limit systems in various premice M ( y )satisfying certain properties definable in M n ( x )[ g ] and then showing thatthe direct limits M M ( y ) ∞ are equal to M ∞ . A similar approach but in acompletely different setting can be found in [SaSch18].In what follows, we will let ( K ( z )) N denote the core model constructedabove a real z inside some n -small model N with n Woodin cardinals in thesense of [Sch06], i.e., the core model K ( z ) is constructed between consecutiveWoodin cardinals. Lemma 1 . K ( x )) M n ( x ) = M n ( x ). We will use this fact and consider more arbitrarypremice with this property in what follows. We state the following definitionsin V , but we will later apply them inside M n ( x )[ g ]. Definition 3.30.
Let y ∈ ω ω ∩ M n ( x )[ g ] . Then we say y is pre-dlm-suitable iff there is a proper class y -premouse M ( y ) satisfying the following proper-ties. ( i ) M ( y ) is n -small and has n Woodin cardinals, ( ii ) the least inaccessible cardinal in M ( y ) is κ , ( iii ) M ( y ) = ( K ( y )) M ( y ) , and ( iv ) there is a Col( ω, <κ ) -generic h over M ( y ) such that M ( y )[ h ] = M n ( x )[ g ] . We also call such a y -premouse M ( y ) pre-dlm-suitable and say that M ( y )witnesses that y is pre-dlm-suitable . Using this, we can define a version of the direct limit system F inside arbi-trary pre-dlm-suitable y -premice M ( y ). Definition 3.31.
Let y ∈ ω ω be pre-dlm-suitable as witnessed by M ( y ) .Then we let G M ( y ) = {N ∈ M ( y ) | κ | N is n -suitable and M ( y ) (cid:15) “for some cardinalcutpoint η, δ N = η + , N | δ N ∈ P ( M ( y ) | η + ) ∩ M ( y ) | ( η + + ω ) , and M ( y ) | η is generic over N for the δ N -generator version ofthe extender algebra at δ N ” } . Analogous as before, we can now define when for an n -suitable premouse N , M ( y ) (cid:15) “ N is strongly s -iterable below κ ” by referring to P M ( y ) ( N | δ N ) inthe definition of ( T N , ∗ s,k ) M ( y ) . Let γ N ,M ( y ) s and H N ,M ( y ) s be defined analogousto γ N s and H N s inside M ( y ) using ( T N , ∗ s,k ) M ( y ) . For M ( y ) = M n ( x ) and N ∈ G this agrees with our previous definition of strong s -iterability. Definition 3.32.
Let y ∈ ω ω be pre-dlm-suitable as witnessed by M ( y ) .Then we let I M ( y ) = { ( N , s ) | N ∈ G M ( y ) , s ∈ [Ord] <ω , and M ( y ) (cid:15) “ N is strongly s -iterable below κ ” } and F M ( y ) = { H N ,M ( y ) s | ( N , s ) ∈ I M ( y ) } . Moreover, for ( N , s ) , ( M , t ) ∈ I M ( y ) we let ( N , s ) ≤ I M ( y ) ( M , t ) iff there isa correctly guided finite stack on N with last model M and s ⊆ t . In this casewe let π M ( y )( N ,s ) , ( M ,t ) : H N ,M ( y ) s → H M ,M ( y ) t denote the canonical correspondingembedding. Finally, let M M ( y ) ∞ denote the direct limit of ( F M ( y ) , ≤ I M ( y ) ) under these embeddings. We will now strengthen this and define when a real y ∈ ω ω (or a y -premouse M ( y )) is dlm-suitable. Definition 3.33.
Let y ∈ ω ω ∩ M n ( x )[ g ] be pre-dlm-suitable as witnessed bysome y -premouse M ( y ) . We say that y is dlm-suitable (witnessed by M ( y ) )iff ( i ) for every s ∈ [Ord] <ω there is a premouse N such that ( N , s ) ∈ I M ( y ) ,and ( ii ) for every N ∈ G M ( y ) , P M ( y ) ( N | δ N ) = K M n ( x )[ g ] ( N | δ N ) . Lemma 3.34. M n ( x ) witnesses that x is dlm-suitable.Proof. The fact that M n ( x ) satisfies ( i ) follows from Lemma 3.13 and Corol-lary 3.21, so we only have to show ( ii ). Let N ∈ G . Then P M n ( x ) ( N | δ N ) = M n ( N | δ N ) by Lemma 3.20. Moreover, there is some G generic over theresult of the P -construction P M n ( x ) ( N | δ N ) for the δ N -generator version ofthe extender algebra at δ N with P M n ( x ) ( N | δ N )[ G ] = M n ( x ). That means M n ( N | δ N )[ G ] = M n ( x ) . Now, K M n ( x )[ g ] ( N | δ N ) = K M n ( N | δ N )[ G ][ g ] ( N | δ N ) = K M n ( N | δ N ) ( N | δ N )= M n ( N | δ N ) = P M n ( x ) ( N | δ N ) , by generic absoluteness of the core model and Lemma 1.1 in [Sch06] (due toSteel). (cid:3) Condition ( ii ) in Definition 3.33 will ensure that for any dlm-suitable y -premouse M ( y ) and ( N , s ) , ( M , t ) ∈ I ∩ I M ( y ) with ( N , s ) ≤ I ( M , t ) and( N , s ) ≤ I M ( y ) ( M , t ), the induced embeddings π ( N ,s ) , ( M ,t ) and π M ( y )( N ,s ) , ( M ,t ) agree. Hence we can show in the following lemma that the direct limit OD IN INNER MODELS WITH WOODIN CARDINALS 17 M M ( y ) ∞ defined inside some dlm-suitable M ( y ) will in fact be the same asthe direct limit M ∞ defined inside M n ( x ). Lemma 3.35.
Let y ∈ ω ω be dlm-suitable as witnessed by M ( y ) . Then F and F M ( y ) have cofinally many points in common and M ∞ = M M ( y ) ∞ .Proof. Let h be Col( ω, <κ )-generic over M ( y ) such that M ( y )[ h ] = M n ( x )[ g ].Let ( N , s ) ∈ I and ( N (cid:48) , s (cid:48) ) ∈ I M ( y ) . We aim to show that there is some( M , t ) ∈ I ∩ I M ( y ) such that ( N , s ) ≤ I ( M , t ) and ( N (cid:48) , s (cid:48) ) ≤ I M ( y ) ( M , t ).As condition ( ii ) in Definition 3.33 yields that the embeddings associatedto F and F M ( y ) agree, this suffices to show that M ∞ = M M ( y ) ∞ .Let t = s ∪ s (cid:48) . By assumption, there is a t -iterable premouse R in M n ( x )and a t -iterable premouse R (cid:48) in M ( y ). Therefore we can assume that N and N (cid:48) are both t -iterable in M n ( x ) and M ( y ) respectively as we can replacethem by the result of their coiteration with R and R (cid:48) respectively.By the choice of M ( y ) and generic absoluteness of the core model we have M ( y ) = ( K ( y )) M ( y ) = ( K ( y )) M ( y )[ h ] (1) = ( K ( y )) M n ( x )[ g ] = ( K ( y )) M n ( x )[ g (cid:22) ξ ] , where ξ < κ is such that y ∈ M n ( x )[ g (cid:22) ξ ]. Analogously, using Lemma 1 . M n ( x ) = ( K ( x )) M n ( x ) = ( K ( x )) M n ( x )[ g ] (2) = ( K ( x )) M ( y )[ h ] = ( K ( x )) M ( y )[ h (cid:22) ξ (cid:48) ] , where ξ (cid:48) < κ is such that x ∈ M ( y )[ h (cid:22) ξ (cid:48) ]. Now we can obtain the followingclaim. Claim 1. M ( y ) and M n ( x ) have cofinally many common cardinal cutpointsbelow κ .Proof. As M ( y ) = K ( y ) M n ( x )[ g (cid:22) ξ ] is an inner model of M n ( x )[ g (cid:22) ξ ], everycardinal above ξ in M n ( x ) is a cardinal in M ( y ). Now let η > ξ, ξ (cid:48) be acutpoint of M n ( x ) which is large enough such that in M ( y ) there is somecutpoint between ξ (cid:48) and η . Suppose η is not a cutpoint of M ( y ), say thereis an extender E overlapping η in M ( y ). Since there is some cutpoint in M ( y ) between ξ (cid:48) and η , it follows that crit( E ) > ξ (cid:48) . Then, as M n ( x ) = K ( x ) M ( y )[ h (cid:22) ξ (cid:48) ] , by maximality of the core model there is also an extender onthe M n ( x )-sequence overlapping η , contradicting the assumption that η is acutpoint of M n ( x ). (cid:3) Moreover, Equations (1) and (2) yield M ( y ) ⊆ M n ( x )[ g (cid:22) ξ ] ⊆ M ( y )[ h (cid:22) ζ ] , where ξ (cid:48) < ζ < κ is such that g (cid:22) ξ ∈ M ( y )[ h (cid:22) ζ ]. By the intermediatemodel theorem (see for example Lemma 15 .
43 in [Je03]) this implies that M n ( x )[ g (cid:22) ξ ] is a generic extension of M ( y ) for a forcing of size less than κ . Since M n ( x )[ g (cid:22) ξ ] is a generic extension of M n ( x ) for a forcing of sizeless than κ as well, this implies by Theorem 1 . W ⊆ M n ( x ) ∩ M ( y ) such that M n ( x )[ g (cid:22) ξ ] is a genericextension of W for a forcing of size less than κ .As every generic extension via a forcing of size less than κ can be absorbedby the collapse of some ordinal β < κ , this yields that we can fix someordinal β < κ and some Col( ω, β )-generic b ∈ M n ( x )[ g ] over W such that x, y, N , N (cid:48) ∈ W [ b ]. Then M n ( x ) and M ( y ) exist in W [ b ] as definable sub-classes because( K ( x )) W [ b ] = ( K ( x )) M n ( x )[ g (cid:22) ξ ] = ( K ( x )) M n ( x ) = M n ( x )and similarly( K ( y )) W [ b ] = ( K ( y )) M n ( x )[ g (cid:22) ξ ] = ( K ( y )) M n ( x )[ g ] = ( K ( y )) M ( y )[ h ] = M ( y )by generic absoluteness of the core model again. Let ˙ x, ˙ y, ˙ N and ˙ N (cid:48) beCol( ω, β )-names for x, y, N and N (cid:48) in W . Moreover, let p ∈ Col( ω, β ) forceall properties we need about ˙ x, ˙ y, ˙ N and ˙ N (cid:48) . For q ≤ Col( ω,β ) p let b q be theCol( ω, β )-generic filter over W such that (cid:83) b q agrees with q on dom( q ) andwith (cid:83) b everywhere else.Now we construct ( M , t ) ∈ I ∩ I M ( y ) . Let η < κ be a cardinal cutpoint ofboth M ( y ) and M n ( x ) such that ξ, ξ (cid:48) < η , which exists by Claim 1. Thenin fact ( η + ) M n ( x ) = ( η + ) M ( y ) as by Equations (1) and (2) at the beginningof the proof( η + ) M ( y ) ≤ ( η + ) M n ( x )[ g (cid:22) ξ ] = ( η + ) M n ( x ) ≤ ( η + ) M ( y )[ h (cid:22) ξ (cid:48) ] = ( η + ) M ( y ) . By the same argument, ( η + ) K ( ˙ x bq ) = ( η + ) K ( ˙ y bq ) for all q ≤ Col( ω,β ) p .Work in W [ b ]. Using Lemmas 3.8 and 3.9, we obtain an inner model M bypseudo-comparing all ( ˙ N ) b q and ( ˙ N (cid:48) ) b q for q ≤ Col( ω,β ) p and simultaneouslypseudo-genericity iterating such that K ( ˙ x b q ) | η and K ( ˙ y b q ) | η are generic over M and δ M = ( η + ) K ( ˙ x bq ) = ( η + ) K ( ˙ y bq ) . Since M is definable in W [ b ] from { b q | q ≤ Col( ω,β ) p } and parameters from W , we have that in fact M ∈ W ⊆ M n ( x ) ∩ M ( y ), as M does not depend on the choice of the generic b .Moreover, M is a correct iterate of N in M n ( x ) and a correct iterate of N (cid:48) in M ( y ).As argued above, we can assume that N and N (cid:48) are t -iterable in M n ( x ) and M ( y ) respectively for t = s ∪ s (cid:48) . Therefore M is t -iterable in both, M n ( x )and M ( y ). Hence, ( M , t ) ∈ I ∩ I M ( y ) , ( N , s ) ≤ I ( M , t ), and ( N (cid:48) , s (cid:48) ) ≤ I M ( y ) ( M , t ), as desired. (cid:3) This yields that M ∞ ∈ HOD M n ( x )[ g ] . I.e. M ( y ) is a ground of M n ( x )[ g (cid:22) ξ ]. See for example [FHR15] or [Us17] for anintroduction to the theory of grounds. OD IN INNER MODELS WITH WOODIN CARDINALS 19
4. HOD below δ ∞ In this section we will show that HOD M n ( x )[ g ] and M ∞ agree up to δ ∞ bygeneralizing the arguments in Section 3 . M ( y )[ h ] and work with M ( y )directly instead. Choose for any ordinal α an arbitrary ( N , s ) ∈ I such that α ∈ s and let α ∗ = π ( N ,s ) , ∞ ( α ). Note that the value of α ∗ does not dependon the choice of ( N , s ). We also let t ∗ = { α ∗ | α ∈ t } for t ∈ [Ord] <ω . Lemma 4.1.
Let N be an n -suitable premouse and s ∈ [Ord] <ω such that ( N , s ) ∈ I . Let ¯ ξ < γ N s , ξ = π ( N ,s ) , ∞ ( ¯ ξ ) and t ∈ [Ord] <ω . Moreover, let ϕ ( v , v ) be a formula in the language of premice, i.e., we allow the extendersequence as a predicate. Then the following are equivalent. ( a ) M n ( M ∞ | δ ∞ ) (cid:15) ϕ ( ξ, t ∗ ) , ( b ) in M n ( x )[ g ] , there is some dlm-suitable y ∈ ω ω witnessed by M ( y ) with ( N , s ) ∈ I M ( y ) and a correctly guided finite stack on N with last model M ∈ M ( y ) such that whenever R ∈ G M ( y ) is the last model of a correctlyguided finite stack on M , then P M ( y ) ( R| δ R ) (cid:15) ϕ ( π M ( y )( N ,s ) , ( R ,s ) ( ¯ ξ ) , t ) .Proof. To prove that ( a ) implies ( b ) we assume toward a contradiction that( b ) is false. So in M n ( x )[ g ] for all dlm-suitable y ∈ ω ω and M ( y ) witnessingthis with ( N , s ) ∈ I M ( y ) and all correctly guided finite stacks on N withlast model M ∈ M ( y ), there is a correctly guided finite stack on M withlast model R ∈ G M ( y ) such that P M ( y ) ( R| δ R ) (cid:15) ¬ ϕ ( π M ( y )( N ,s ) , ( R ,s ) ( ¯ ξ ) , t ).We can assume without loss of generality that N ∈ M n ( x ) is the last modelof a correctly guided finite stack on M − n +1 via the canonical iteration strategyΣ M − n +1 and strongly s -iterable below κ with respect to branches choosen byΣ M − n +1 . Moreover, we can assume that max ( s ) is a uniform indiscernible. Ifthis is not already the case, we replace N by a pseudo-iterate of the resultof the pseudo-comparison of N with M − n +1 using Lemma 3.8 and Corollary3.21. Claim 1.
There are n -suitable premice N k ∈ F + for k < ω which are cofinalin F + such that N = N and for all k < ω , M n ( N k | δ N k ) (cid:15) ¬ ϕ ( ¯ ξ k , t ) , where ¯ ξ k = i N , N k ( ¯ ξ ) is the image of ¯ ξ under the iteration map induced by Σ M − n +1 .Proof. Let ( Q i | i < ω ) be an enumeration of F + and N = N . Then weconstruct N k +1 inductively. So assume that we already constructed N k andpseudo-coiterate N k with Q k to some model N ∗ k (see Lemma 3.8). By as-sumption ( b ) is false, so let R be a counterexample witnessing this for N ∗ k andthe dlm-suitable premouse M n ( x ). That means R ∈ G is the last model of a correctly guided finite stack on N ∗ k such that M n ( R| δ R ) = P M n ( x ) ( R| δ R ) (cid:15) ¬ ϕ ( i N , R ( ¯ ξ ) , t ) as i N , R (cid:22) H N s = π ( N ,s ) , ( R ,s ) . But R ∈ F + since R ∈ G and itis a correct iterate of Q k . Thus we can let N k +1 = R . (cid:3) Since ( N k | k < ω ) is cofinal in F + , it follows that the direct limit of( N k , i N k , N l | k < l < ω ) is equal to M + ∞ . Let ˆ N k = M n ( N k | δ N k ) and letˆ i ˆ N k , ∞ : ˆ N k → M n ( M + ∞ | δ M + ∞ ) = ˆ M + ∞ be the corresponding extension of thedirect limit map i N k , ∞ . Then we have for all sufficiently large k that M n ( M + ∞ | δ M + ∞ ) (cid:15) ¬ ϕ ( i N k , ∞ ( ¯ ξ k ) , ˆ i ˆ N k , ∞ [ t ]) . Since we assumed that N is strongly s -iterable below κ with respect tobranches choosen by Σ M − n +1 and ¯ ξ < γ N s , it follows that i N k , ∞ ( ¯ ξ k ) = i N , ∞ ( ¯ ξ ) = π ( N ,s ) , ∞ ( ¯ ξ ) = ξ as ¯ ξ k = i N , N k ( ¯ ξ ).Let k < ω be large enough such that ( N k , s ∪ t ) ∈ I and ˆ i ˆ N l , ˆ N l +1 ( s ) = s forall l ≥ k . Such a k exists by a so-called bad sequence argument similar to theone in the proof of Lemma 5 . l ≥ k and s − = s \ { max ( s ) } ,let γ ˆ N l s = sup( Hull ˆ N l | max ( s )1 ( s − ) ∩ δ N l ) , and H ˆ N l s = Hull ˆ N l | max ( s )1 ( γ ˆ N l s ∪ s − ) . Now we let for l < j , ˆ π ( N l ,s ) , ( N j ,t ) : H ˆ N l s → H ˆ N j t denote the canonicalcorresponding embedding extending π ( N l ,s − ) , ( N j ,t − ) given by the iterationembedding via a tail of the iteration strategy Σ M n +1 . Let ˆ M ∞ be the directlimit under these embeddings and let ˆ π ( N l ,s ) , ∞ : H ˆ N l s → ˆ M ∞ for l ≥ k denote the direct limit embedding.Now consider the mapˆ σ : ˆ M ∞ → ˆ M + ∞ = M n ( M + ∞ | δ M + ∞ )which is the canonical extension of the map σ : M ∞ → M + ∞ defined inthe proof of Lemma 3.28, i.e., for x ∈ ˆ M ∞ , say x = ˆ π ( N l ,s ) , ∞ (¯ x ) for some¯ x ∈ H ˆ N l s and k ≤ l < ω , let ˆ σ ( x ) = ˆ i ˆ N l , ∞ (¯ x ). Then it follows as in the proofof Lemma 5.10 in [Sa13] that ˆ σ = id and ˆ M ∞ = M n ( M ∞ | δ ∞ ). Moreover,we have that ˆ σ [ t ∗ ] = ˆ σ (ˆ π ( N k ,s ∪ t ) , ∞ [ t ]) = ˆ i ˆ N k , ∞ [ t ]. Therefore pulling backunder ˆ σ yields that M n ( M ∞ | δ ∞ ) (cid:15) ¬ ϕ ( ξ, t ∗ ) . This is the desired contradiction to ( a ).To show that ( b ) implies ( a ) we now assume that ( b ) is true. Let M ( y ) bethe dlm-suitable premouse with ( N , s ) ∈ I M ( y ) given by ( b ). As before wecan assume without loss of generality that N is the last model of a correctlyguided finite stack on M − n +1 via the canonical iteration strategy Σ M − n +1 , that N is strongly s -iterable below κ with respect to branches choosen by Σ M − n +1 , OD IN INNER MODELS WITH WOODIN CARDINALS 21 that max ( s ) is a uniform indiscernible, and that N ∈ G ∩G M ( y ) using Lemma3.35. Claim 2.
There are n -suitable premice N k ∈ F + for k < ω which are cofinalin F + such that N = N and for all k < ω , M n ( N k | δ N k ) (cid:15) ϕ ( ¯ ξ k , t ) , where ¯ ξ k = i N , N k ( ¯ ξ ) is the image of ¯ ξ under the iteration map induced by Σ M − n +1 .Proof. By the proof of Lemma 3.35, we can pick a sequence ( Q i | i < ω ) ofpremice cofinal in F + such that Q i ∈ F M ( y ) for all i < ω . Let N = N andconstruct N k +1 ∈ M ( y ) inductively. Assume that we already constructed N k and let M ∈ M ( y ) be the last model of a correctly guided finite stackon N witnessing that ( b ) is true. Simultaneously pseudo-coiterate M with N k and Q k to some premouse N ∗ k . Using genericity iterations and Lemma3.35, there is a pseudo-iterate R of N ∗ k such that R ∈ G ∩ G M ( y ) (see alsoCorollary 3.21). In particular, we have by dlm-suitability of M ( y ) that P M ( y ) ( R| δ R ) = K M n ( x )[ g ] ( R| δ R ) = K M n ( R| δ R )[ G ][ g ] ( R| δ R )= K M n ( R| δ R ) ( R| δ R ) = M n ( R| δ R )for some G generic over M n ( R| δ R ) for the extender algebra and there-fore M n ( R| δ R ) (cid:15) ϕ ( i N , R ( ¯ ξ ) , t ) using ( b ) as i N , R ( ¯ ξ ) = π ( N ,s ) , ( R ,s ) ( ¯ ξ ) = π M ( y )( N ,s ) , ( R ,s ) ( ¯ ξ ). Moreover, R is the last model of a correctly guided finitestack on Q k and thus R ∈ F + , so we can let N k +1 = R . (cid:3) As before we can use this claim to obtain that M n ( M ∞ | δ ∞ ) (cid:15) ϕ ( ξ, t ∗ ) , which proves ( a ). (cid:3) Let κ ∞ be the least inaccessible cardinal above δ ∞ in ˆ M ∞ = M n ( M ∞ | δ ∞ )and fix some H which is Col( ω, <κ ∞ )-generic over ˆ M ∞ . Then Lemma 4.1implies for example that ˆ M ∞ [ H ] and M n ( x )[ g ] are elementary equivalent(for formulae in the language of set theory) as for R as in the statementof Lemma 4.1, there is some Col( ω, <κ R )-generic G , where κ R is the leastinaccessible cardinal above δ R in P M ( y ) ( R| δ R ), such that P M ( y ) ( R| δ R )[ G ] = M n ( x )[ g ].We defined a direct limit system F M ( y ) for all dlm-suitable M ( y ) in M n ( x )[ g ].Therefore, there is a direct limit system F ∗ ,M ( y ) with the same properties foreach dlm-suitable M ( y ) in ˆ M ∞ [ H ] (adapting the definition of dlm-suitableto ˆ M ∞ [ H ]). It is easy to see that Lemma 4.1 implies that M ∞ is strongly s ∗ -iterable in all dlm-suitable M ( y ) in ˆ M ∞ [ H ] for all s ∈ [Ord] <ω . So wecan consider its direct limit embedding π M ( y ) ∞ = (cid:91) { ( π M ( y )( M ∞ ,s ∗ ) , ∞ ) F ∗ ,M ( y ) | s ∈ [Ord] <ω } in the system F ∗ ,M ( y ) . But in fact, π M ( y ) ∞ = π M ( z ) ∞ for two dlm-suitablemodels M ( y ) and M ( z ), so we call this unique embedding π ∞ . Lemma 4.2.
For all η < δ ∞ we have that π ∞ ( η ) = η ∗ .Proof. This is again a consequence of Lemma 4.1. Consider the dlm-suitablepremouse M n ( x ). Let η = π ( N ,s ) , ∞ (¯ η ) for some ( N , s ) ∈ I and ¯ η < γ N s andconsider the formula ϕ ( v , v , v , v ) = “1 (cid:13) Col( ω,<κ ∗ ) for all dlm-suitable y with v ∈ G M ( y ) , we have ( v , v ) ∈ I M ( y ) , v < γ v ,M ( y ) v , and π M ( y )( v ,v ) , ∞ ( v ) = v ” , where κ ∗ refers to the least inaccessible cardinal above the least Woodincardinal of the current model. Recall that for any dlm-suitable y and z witnessed by M ( y ) and M ( z ), for any ( N , s ) , ( M , t ) ∈ I M ( y ) ∩ I M ( z ) with ( N , s ) ≤ I M ( y ) ( M , t ) and ( N , s ) ≤ I M ( z ) ( M , t ) the induced embed-dings π M ( y )( N ,s ) , ( M ,t ) and π M ( z )( N ,s ) , ( M ,t ) agree. Hence, in M n ( x )[ g ], we have forevery R ∈ G which is the last model of a correctly guided finite stackon N that for all dlm-suitable y such that R ∈ G M ( y ) , in fact ( R , s ) ∈I M ( y ) , π ( N ,s ) , ( R ,s ) (¯ η ) < γ R ,M ( y ) s , and π M ( y )( R ,s ) , ∞ ( π ( N ,s ) , ( R ,s ) (¯ η )) = η . There-fore, P M ( y ) ( R| δ R ) (cid:15) ϕ ( R , s, π ( N ,s ) , ( R ,s ) (¯ η ) , η ). So Lemma 4.1 yields thatˆ M ∞ (cid:15) ϕ ( M ∞ , s ∗ , η, η ∗ ). So in ˆ M ∞ [ H ], for all dlm-suitable y , we have( π M ( y )( M ∞ ,s ∗ ) , ∞ ) F ∗ ,M ( y ) ( η ) = η ∗ , as desired. (cid:3) Theorem 4.3. V HOD Mn ( x )[ g ] δ ∞ = V M ∞ δ ∞ .Proof. By the internal definition of M ∞ from Lemma 3.35 we have that V M ∞ δ ∞ ⊆ V HOD Mn ( x )[ g ] δ ∞ . For the other inclusion we first show the followingclaim. Claim 1. π ∞ (cid:22) α ∈ ˆ M ∞ for all α < δ ∞ .Proof. As α < δ ∞ , there exists an s ∈ [Ord] <ω such that for all dlm-suitable M ( y ) in ˆ M ∞ [ H ], α < γ M ∞ ,M ( y ) s ∗ . For any such s and M ( y ) we have bydefinition that π ∞ (cid:22) α = ( π M ( y )( M ∞ ,s ∗ ) , ∞ ) F ∗ ,M ( y ) (cid:22) α . Therefore π ∞ (cid:22) α ∈ HOD ˆ M ∞ [ H ] M ∞ and thus π ∞ (cid:22) α ∈ ˆ M ∞ by homogeneity of the forcing P =Col( ω, <κ ∞ ). (cid:3) Now let A ∈ V HOD Mn ( x )[ g ] δ ∞ be arbitrary. Let α < δ ∞ be such that A ⊂ α is defined over M n ( x )[ g ] by a formula ϕ with ordinal parameters from t ∈ [Ord] <ω and let β < α be arbitrary. That means β ∈ A iff M n ( x )[ g ] (cid:15) OD IN INNER MODELS WITH WOODIN CARDINALS 23 ϕ ( β, t ). Lemma 4.1 yields that this is the case iff ˆ M ∞ [ H ] (cid:15) ϕ ( β ∗ , t ∗ ). Since β < α < δ ∞ , we have that β ∗ = π ∞ ( β ) by Lemma 4.2. Moreover, we haveby Claim 1 that π ∞ (cid:22) α ∈ ˆ M ∞ . Therefore, it follows by homogeneity ofthe forcing P = Col( ω, <κ ∞ ) that A ∈ ˆ M ∞ since t ∗ is a fixed parameter inˆ M ∞ . Thus A ∈ V M ∞ δ ∞ , as desired. (cid:3) The full
HOD in M n ( x )[ g ]To compute the full model HOD M n ( x )[ g ] , i.e., prove Theorem 1.1, we firstshow the following lemma. Lemma 5.1.
HOD M n ( x )[ g ] = M n ( A ) for some set A ⊆ ω M n ( x )[ g ]2 with A ∈ HOD M n ( x )[ g ] .Proof. Let V denote the Vopˇenka algebra in M n ( x )[ g ] for making a realgeneric over HOD M n ( x )[ g ] . By Vopˇenka’s theorem (see for example Theorem15 .
46 in [Je03] or Theorem 9 . . V -generic G x overHOD M n ( x )[ g ] such that x ∈ HOD M n ( x )[ g ] [ G x ] and in fact HOD M n ( x )[ g ] [ G x ] =HOD M n ( x )[ g ] x . Claim 1.
There is some ˜ V ∈ HOD M n ( x )[ g ] which is isomorphic to V and asubset of ω M n ( x )[ g ]2 .Proof. Work in M n ( x )[ g ]. Each real, i.e., element of P ( ω ), can be coded bya countable ordinal and each set of reals can be coded by an ordinal < ω .Forcing with the Vopˇenka algebra V is ω -c.c. in M n ( x )[ g ] as otherwise therewould be an ω sequence of pairwise distinct non-empty sets of reals, contra-dicting CH. The Vopˇenka algebra is in HOD M n ( x )[ g ] and when consideringHOD M n ( x )[ g ] [ G x ] cardinals ≥ ( κ + ) M n ( x ) are preserved. Since ( κ + ω ) M n ( x ) isbelow the least measurable cardinal of M n ( x ), M n ( x ) | ( κ + ω ) M n ( x ) can bewritten as the Lp n -stack of height ( κ + ω ) M n ( x ) above x and is therefore bythe argument in Lemma 3.18 an element of HOD M n ( x )[ g ] x = HOD M n ( x )[ g ] [ G x ].But the Vopˇenka algebra is a subset of some ordinal α < ( κ ++ ) M n ( x ) =( κ ++ ) HOD Mn ( x )[ g ] [ G x ] = ( κ ++ ) HOD Mn ( x )[ g ] , so there is some ˜ V ∈ HOD M n ( x )[ g ] which is isomorphic to V and a subset of ( κ + ) M n ( x ) . (cid:3) For the rest of this proof we write V for the ˜ V from the previous claim andshow that M n ( V ) = HOD M n ( x )[ g ] . Claim 2. G x is V -generic over M n ( V ) .Proof. The dense sets in question are elements of P ( V ) M n ( V ) and henceelements of Lp n ( V ) = (cid:83) { N | N is a countable V -premouse with ρ ω ( N ) = V which is n -small, sound, and ( ω, ω , ω )-iterable } . As V ∈ HOD M n ( x )[ g ] ,Lemma 3.18 yields that Lp n ( V ) ∈ HOD M n ( x )[ g ] , which implies the claim. (cid:3) Claim 3. M n ( x ) and M n ( V )[ G x ] have the same least measurable cardinal κ . Moreover, V M n ( x ) κ = V M n ( V )[ G x ] κ .Proof. Write κ M n ( x )0 and κ M n ( V )0 for the least measurable cardinal of M n ( x )and M n ( V ) respectively. M n ( V ) and M n ( V )[ G x ] have the same least mea-surable cardinal. The proof of Lemma 3.18 shows that Lp n ( z ) ∈ M n ( V )[ G x ]for any z ∈ V M n ( V )[ G x ] δ , where δ denotes the least Woodin cardinal in M n ( V )[ G x ]. As M n ( x ) | κ M n ( x )0 is equal to the Lp n -stack of height κ M n ( x )0 over x , it follows that M n ( x ) | κ M n ( x )0 ⊆ M n ( V )[ G x ]. Analogously, M n ( V ) | κ M n ( V )0 ⊆ M n ( x ) and in fact M n ( V )[ G x ] | κ M n ( V )0 ⊆ M n ( x ).Suppose κ M n ( V )0 < κ M n ( x )0 and let U be the measure on κ M n ( V )0 in M n ( V )(which we identify with the lift of this measure to M n ( V )[ G x ]). In particular, U measures all subsets of κ M n ( V )0 in M n ( x ) as well and κ M n ( V )0 is a cardinal in M n ( x ) because otherwise the function witnessing this would be an elementof M n ( x ) | κ M n ( x )0 and hence in M n ( V )[ G x ]. So V M n ( V )[ G x ] κ Mn ( V )0 = V M n ( x ) κ Mn ( V )0 . Now M n ( x ) | ( κ M n ( V )0 ) + = Lp n ( M n ( x ) | κ M n ( V )0 ) and hence it contains a mousewith the measure U on κ M n ( V )0 . This contradicts the fact that κ M n ( x )0 is theleast measurable cardinal in M n ( x ). The argument in the case κ M n ( V )0 >κ M n ( x )0 is analogous. Therefore, M n ( x ) and M n ( V )[ G x ] have the same leastmeasurable cardinal κ and the argument above shows V M n ( x ) κ = V M n ( V )[ G x ] κ . (cid:3) For what follows, it suffices to work with the least inaccessible λ < κ of M n ( x ) which is above κ , so we restrict ourselves to this situation. Claim 4. V M n ( V ) λ ⊆ HOD M n ( x )[ g ] .Proof. This follows from the proof of Lemma 3.18 as M n ( V ) | λ can be ob-tained as the Lp n -stack of height λ over V and V ∈ HOD M n ( x )[ g ] . (cid:3) Now we can show that the lemma holds below λ . Claim 5. V M n ( V ) λ = V HOD Mn ( x )[ g ] λ .Proof. We first show that V M n ( V ) λ [ G x ] = V HOD Mn ( x )[ g ] λ [ G x ]. The inclusion ⊆ follows from Claim 4. For the other inclusion we have thatHOD M n ( x )[ g ] [ G x ] = HOD M n ( x )[ g ] x ⊆ HOD M n ( x ) x ⊆ M n ( x ) , using the homogeneity and ordinal definability of the forcing Col( ω, <κ ).Therefore by Claim 3 V HOD Mn ( x )[ g ] λ [ G x ] ⊆ V M n ( x ) λ = V M n ( V ) λ [ G x ] . OD IN INNER MODELS WITH WOODIN CARDINALS 25
Finally, we argue that the equality V M n ( V ) λ [ G x ] = V HOD Mn ( x )[ g ] λ [ G x ] also holdstrue without adding the generic G x . As by Claim 4 we have V M n ( V ) λ ⊆ V HOD Mn ( x )[ g ] λ , we are again left with proving the other inclusion. Let P = V × Col( ω, <κ ). Then ( G x , g ) is P -generic over both V M n ( V ) λ and V HOD Mn ( x )[ g ] λ ,and V M n ( V ) λ [ G x , g ] = V HOD Mn ( x )[ g ] λ [ G x , g ]. Let a ∈ V HOD Mn ( x )[ g ] λ be a set ofordinals. Then there is a P -name σ ∈ V M n ( V ) λ such that σ ( G x ,g ) = a . Thisis forced over V HOD Mn ( x )[ g ] λ , i.e., there is a p ∈ P such that V HOD Mn ( x )[ g ] λ (cid:15) “ p (cid:13) σ = ˇ a ”. Thus V M n ( V ) λ can compute the elements of a using the forcingrelation for P below p . Hence a ∈ V M n ( V ) λ , as desired. (cid:3) Now we are able to extend Claim 3 to the full models.
Claim 6. M n ( V )[ G x ] = M n ( x ) .Proof. Consider M n ( x )[ g ] as a V M n ( x )[ g ] λ -premouse and note that it equals M n ( V M n ( x )[ g ] λ ). We use P M n ( x )[ g ] ( M n ( V ) | λ ) to denote the result of a P -construction in the sense of [SchSt09] above M n ( V ) | λ inside the V M n ( x )[ g ] λ -premouse M n ( x )[ g ]. By Claim 3, V M n ( V ) λ [ G x ] = V M n ( x ) λ , so V M n ( V ) λ [ G x ][ g ] = V M n ( x )[ g ] λ and this P -construction is well-defined. Moreover, the followingargument shows that the construction never projects across λ .Assume toward a contradiction that there is a level P of the P -constructionabove M n ( V ) | λ inside M n ( x )[ g ] such that ρ ω ( P ) = ρ < λ . That means thereis an r Σ k +1 ( P )-definable set a ⊆ ρ for some k < ω such that a / ∈ P . Asby the proof of Claim 4, M n ( V ) | λ ∈ HOD M n ( x )[ g ] it follows by definabilityof the P -construction and of the extender sequence of M n ( V M n ( x )[ g ] λ ) (seeLemma 1 . P ∈
HOD M n ( x )[ g ] . This meansthat in particular a ∈ HOD M n ( x )[ g ] . But a ⊆ ρ < λ and by Claim 5, V HOD Mn ( x )[ g ] λ = V M n ( V ) λ = V P λ , so a ∈ P . Contradiction.Now it follows by construction (see [SchSt09]) that P M n ( x )[ g ] ( M n ( V ) | λ )[ G x ][ g ] = M n ( x )[ g ] . But this yields that P M n ( x )[ g ] ( M n ( V ) | λ )[ G x ] = M n ( x ), without adding thegeneric g , by an argument as the one at the end of the previous claim.Moreover, P M n ( x )[ g ] ( M n ( V ) | λ ) = M n ( V ) and thus M n ( V )[ G x ] = M n ( x ), asdesired. (cid:3) This argument also shows the following claim.
Claim 7. M n ( V ) ⊆ HOD M n ( x )[ g ] . Now, the next claim follows from the first half of the proof of Claim 5.
Claim 8. M n ( V )[ G x ] = HOD M n ( x )[ g ] [ G x ] . Finally, the statement of Claim 8 also holds true without adding the generic G x by the argument at the end of the proof of Claim 5. Hence M n ( V ) =HOD M n ( x )[ g ] , as desired. (cid:3) Corollary 5.2.
Let F ( s ) = s ∗ for s ∈ [Ord] <ω . Then HOD M n ( x )[ g ] = M n ( M ∞ | δ ∞ , F (cid:22) δ ∞ ) . Proof.
Note that M ∞ | δ ∞ and F (cid:22) δ ∞ are elements of HOD M n ( x )[ g ] by con-struction. Let η = sup F ” δ ∞ and let γ be the least inaccessible cardinal of M n ( M ∞ | δ ∞ , F (cid:22) δ ∞ ) above η . Let A ⊆ ω M n ( x )[ g ]2 be as in the statementof Lemma 5.1, i.e., such that HOD M n ( x )[ g ] = M n ( A ). Moreover, let ϕ be aformula defining A , i.e., ξ ∈ A iff M n ( x )[ g ] (cid:15) ϕ ( ξ ). Then, as F ( ξ ) = π ∞ ( ξ )for ξ < δ ∞ by Lemma 4.2, ξ ∈ A iff M n ( M ∞ | δ ∞ ) (cid:15) “1 (cid:13) P ϕ ( π ∞ ( ξ )) , where π ∞ is the direct limitembedding from the systems on M ∞ ”for P = Col( ω, <κ ∞ ). Consider L [ E ]( M ∞ | δ ∞ ) M n ( M ∞ | δ ∞ ,F (cid:22) δ ∞ ) , the resultof a fully backgrounded extender construction in the sense of [MS94] in-side M n ( M ∞ | δ ∞ , F (cid:22) δ ∞ ) above M ∞ | δ ∞ . The premice M n ( M ∞ | δ ∞ ) and L [ E ]( M ∞ | δ ∞ ) M n ( M ∞ | δ ∞ ,F (cid:22) δ ∞ ) successfully compare to a common properclass premouse without drops on the main branches. Since the iterationstake place above δ ∞ , ξ < δ ∞ is not moved and we have by elementarity ξ ∈ A iff L [ E ]( M ∞ | δ ∞ ) M n ( M ∞ | δ ∞ ,F (cid:22) δ ∞ ) (cid:15) “1 (cid:13) P ϕ ( π ∞ ( ξ )) , where π ∞ is the direct limit embedding from the systems on M ∞ ” . Therefore it follows that A ∈ M n ( M ∞ | δ ∞ , F (cid:22) δ ∞ ).By the same argument as in the proof of Claim 3 in the proof of Lemma5.1 we now obtain that the universes of M n ( M ∞ | δ ∞ , F (cid:22) δ ∞ ) and M n ( A )agree up to their common least measurable cardinal. In particular, γ isalso the least inaccessible cardinal above η of M n ( A ) and we can rearrange M n ( M ∞ | δ ∞ , F (cid:22) δ ∞ ) and M n ( A ) as V M n ( M ∞ | δ ∞ ,F (cid:22) δ ∞ ) γ -premice. As such itfollows that the following equalities for classes (not structures) hold: M n ( M ∞ | δ ∞ , F (cid:22) δ ∞ ) = M n ( V M n ( M ∞ | δ ∞ ,F (cid:22) δ ) γ )= M n ( A ) = HOD M n ( x )[ g ] . (cid:3) The following corollary follows immediately from Lemma 4.2 and Corollary5.2.
Corollary 5.3.
HOD M n ( x )[ g ] = M n ( M ∞ | δ ∞ , π ∞ (cid:22) δ ∞ ) . We now consider the iteration strategy for M ∞ . Let Λ be the restrictionof Σ M − n +1 to correctly guided finite stacks (cid:126) T on M ∞ | δ ∞ such that (cid:126) T ∈ ˆ M ∞ | κ ∞ , where κ ∞ is the least inaccessible cardinal in ˆ M ∞ above δ ∞ . OD IN INNER MODELS WITH WOODIN CARDINALS 27
Lemma 5.4. Λ ∈ HOD M n ( x )[ g ] .Proof. Let T be a maximal tree on M ∞ | δ ∞ with T ∈ ˆ M ∞ | κ ∞ . More-over, let b = Λ( T ). Let R = M T b be the last model of T (cid:97) b . Then R ∈
HOD M n ( x )[ g ] . Moreover, let δ F ∗ ∞ be the least Woodin cardinal in M F ∗ ∞ ,the direct limit of the system F ∗ ,M ( y ) for some/all dlm-suitable M ( y ) inˆ M ∞ [ H ]. Then M F ∗ ∞ | δ F ∗ ∞ is an iterate of R . As π ∞ (cid:22) δ ∞ ∈ HOD M n ( x )[ g ] ,we can identify b inside M n ( x )[ g ] as the unique branch through T whichis ( π ∞ (cid:22) δ ∞ )-realizable, i.e., such that there is an elementary embedding σ : M T b → M F ∗ ∞ | δ F ∗ ∞ with π ∞ (cid:22) δ ∞ = σ ◦ i T b .The same argument applies to pseudo-normal iterates N of M ∞ with N | δ N ∈ ˆ M ∞ | κ ∞ and maximal iteration trees T on N | δ N such that T ∈ ˆ M ∞ | κ ∞ ,hence Λ ∈ HOD M n ( x )[ g ] . (cid:3) Similarly to Lemma 3 .
47 in [StW16] we finally need a method of Boolean-valued comparison. As the proof is analogous we omit it.
Lemma 5.5.
Let H be Col( ω, <κ ∞ ) -generic over ˆ M ∞ , and let Q be suchthat ˆ M ∞ [ H ] (cid:15) “ Q is countable and n -suitable”. Then there is an R suchthat(1) R is a pseudo-normal iterate of Q ,(2) R is a Σ M − n +1 -iterate of M ∞ , and(3) R ∈ ˆ M ∞ . Finally, we can finish the proof of Theorem 1.1.
Theorem 5.6.
HOD M n ( x )[ g ] = M n ( M ∞ | δ ∞ , π ∞ (cid:22) δ ∞ ) = M n ( ˆ M ∞ | κ ∞ , Λ) . Proof.
HOD M n ( x )[ g ] = M n ( M ∞ | δ ∞ , π ∞ (cid:22) δ ∞ ) is Corollary 5.3. Moreover, M n ( M ∞ | δ ∞ , π ∞ (cid:22) δ ∞ ) = M n ( ˆ M ∞ | κ ∞ , Λ) follows from Lemma 5.5 as fol-lows. First, Λ ∈ M n ( M ∞ | δ ∞ , π ∞ (cid:22) δ ∞ ) = HOD M n ( x )[ g ] and ˆ M ∞ | κ ∞ ∈ M n ( M ∞ | δ ∞ , π ∞ (cid:22) δ ∞ ) by considering the Lp n -stack on M ∞ | δ ∞ . The directlimit of F ∗ ,M ( y ) for some M ( y ) is the same as the direct limit of all Λ-iteratesof M ∞ which are an element of ˆ M ∞ | κ ∞ via the comparison maps. More-over, we have that π ∞ is the canonical direct limit map of this system andtherefore definable from ˆ M ∞ | κ ∞ and Λ. So π ∞ (cid:22) δ ∞ ∈ M n ( ˆ M ∞ | κ ∞ , Λ).Now, M n ( M ∞ | δ ∞ , π ∞ (cid:22) δ ∞ ) = M n ( ˆ M ∞ | κ ∞ , Λ) follows analogous to theproof of Corollary 5.2. (cid:3)
Note that Theorem 4.3 and Lemma 5.1 together imply Corollary 1.2, i.e.,that the GCH holds in HOD M n ( x )[ g ] . Finally, most of the arguments we gavein this and the previous sections generalize with only small changes to morearbitrary canonical self-iterable inner models, e.g. M ω , M ω +42 . We leavethe details to the reader. References [Be80] H. S. Becker. Thin collections of sets of projective ordinals andanalogs of L . Annals of Mathematical Logic , 19(3):205–241, 1980.[FHR15] Gunter Fuchs, Joel David Hamkins, and Jonas Reitz. Set-theoreticgeology.
Annals of Pure and Applied Logic , 166(4):464–501, 2015.[Fa] I. Farah. The extender algebra and Σ -absoluteness. In A. S.Kechris, B. L¨owe, and J. R. Steel, editors, The Cabal Seminar,Volume IV . To appear. Preprint available at https://arxiv.org/abs/1005.4193 .[Je03] T. J. Jech.
Set Theory . Springer Monographs in Mathematics.Springer, 2003.[KW10] P. Koellner and W. H. Woodin. Large Cardinals from Determinacy.In M. Foreman and A. Kanamori, editors,
Handbook of Set Theory .Springer, 2010.[La17] P. B. Larson.
Extensions of the Axiom of Determinacy . 10 2017.Preprint available at .[MS94] W. J. Mitchell and J. R. Steel.
Fine structure and iteration trees .Lecture notes in logic. Springer-Verlag, Berlin, New York, 1994.[MSW] S. M¨uller, R. Schindler, and W. H. Woodin. Mice with Finitelymany Woodin Cardinals from Optimal Determinacy Hypotheses.2019. Accepted at Journal of Mathematical Logic.[Ne02] I. Neeman. Optimal Proofs of Determinacy II.
Journal of Mathe-matical Logic , 2(2):227–258, 11 2002.[Ne95] I. Neeman. Optimal Proofs of Determinacy.
Bulletin of SymbolicLogic , 1(3):327–339, 09 1995.[Sa09] G. Sargsyan.
A tale of hybrid mice . PhD thesis, University ofCalifornia, Berkeley, 2009.[Sa13] G. Sargsyan. On the prewellorderings associated with directed sys-tems of mice.
J. Symb. Log. , 78(3):735–763, 2013.[SaSch18] G. Sargsyan and R. Schindler. Varsovian models I.
The Journalof Symbolic Logic , 83(2):496–528, 2018.[Sch06] R. Schindler. Core models in the presence of Woodin cardinals.
J.Symb. Log. , 71(4):1145–1154, 2006.[SchSt09] R. Schindler and J. R. Steel. The self-iterability of L [ E ]. J. Symb.Log. , 74(3):751–779, 2009.[SchlTr] F. Schlutzenberg and N. Trang. Scales in hybrid mice over R . Submitted , 2016. Preprint available at https://arxiv.org/abs/1210.7258 .[St10] J. R. Steel. An Outline of Inner Model Theory. In M. Foreman andA. Kanamori, editors,
Handbook of Set Theory . Springer, 2010.[St16] J. R. Steel. Normalizing iteration trees and comparing iterationstrategies. 2016. Preprint available at https://math.berkeley.edu/~steel/papers . OD IN INNER MODELS WITH WOODIN CARDINALS 29 [St93] J. R. Steel. Inner models with many Woodin cardinals.
Annals ofPure and Applied Logic , 65:185–209, 1993.[St95] J. R. Steel. HOD L ( R ) is a Core Model below Θ. The Bulletin ofSymbolic Logic , 1(1):75–84, 1995.[StW16] J. R. Steel and W. H. Woodin. HOD as a core model. In A. S.Kechris, B. L¨owe, and J. R. Steel, editors,
Ordinal definability andrecursion theory, The Cabal Seminar, Volume III . Cambridge Uni-versity Press, 2016.[Us17] T. Usuba. The downward directed grounds hypothesis and verylarge cardinals.
Journal of Mathematical Logic , 17(02), 2017.
Sandra M¨uller, Institut f¨ur Mathematik, UZA 1, Universit¨at Wien. Augasse2-6, 1090 Wien, Austria.
E-mail address : [email protected] Grigor Sargsyan, Department of Mathematics, Rutgers University, Hill Cen-ter for the Mathematical Sciences, 110 Frelinghuysen Rd., Pisacataway, NJ08854, USA
E-mail address ::