THE MAXIMALITY of the CORE MODEL 1. Introduction This Paper

Home , Core model, Extender (set theory), Weakly compact cardinal, Woodin cardinal

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 8, Pages 3119–3141 S 0002-9947(99)02411-3 Article electronically published on March 29, 1999

THE MAXIMALITY OF THE CORE MODEL

E. SCHIMMERLING AND J. R. STEEL

Abstract. Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K,2)Kcom- putes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K, 4) (joint with W. J. Mitchell) K κ is universal for mice of height κ whenever κ 2,5)ifthereisaκ k ≤ ≥ℵ such that κ is either a singular countably closed cardinal or a weakly compact <ω cardinal, and κ fails, then there are inner models with Woodin cardinals, and 6) an ω-Erdös cardinal suffices to develop the basic theory of K.

1. Introduction This paper deals with the core model, or K, as defined in [St1]. Our main results are interesting as part of the pure theory of this model, but they also yield some new consistency strength lower bounds, as we shall point out. Of course, one cannot give a sensible definition of K, much less prove anything about it, without an anti-large-cardinal hypothesis. For the most part, the hypothesis of this nature, that we shall assume here, is that there is no inner model satisfying “There is a Woodin cardinal”. The results we shall prove about K under this hypothesis have all been proved by others under more restrictive anti-large-cardinal hypotheses, and our proofs build on those earlier proofs. The term “the core model” was introduced by Dodd and Jensen to mean, roughly, “the largest canonical inner model there is”. They gave a precise definition of the term which captures this intuition in the case where there is no inner model with a measurable cardinal in [DoJ]. The work of Mitchell, and then Martin, Mitchell and Steel, led to a precise definition which captures the intuition in the case where there is no inner model with a Woodin cardinal, and every set has a sharp. (Cf. [St1]. The hypothesis that every set has a sharp seems to be a weakness in the basic theory of the core model for a Woodin cardinal. Theorem 5.1 shows how aweakerErdös partition property suffices.) The canonicity of the core model is evidenced by the absoluteness of its definition: if G is set generic over V ,then KV =KV[G]. (Cf. [St1, 7.3].) The maximality of the core model manifests itself in several ways. Perhaps the most important of these is the weak covering property of K, that it computes successors of singular cardinals correctly. (Cf. [St1, 8.15], [MiSchSt], [MiSch], and [SchW].) Other manifestations of the maximality of K are 1 its Σ3 correctness (cf. [St1, 7.9]), and its rigidity (cf. [St1, 8.8]).

Received by the editors May 17, 1997 and, in revised form, October 25, 1997. 1991 Mathematics Subject Classiﬁcation. Primary 03E35, 03E45, 03E55. Key words and phrases. Large cardinals, core models. This research was partially supported by the NSF.

c 1999 American Mathematical Society

3119

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Our main result here is that K is maximal in another sense, namely, that every extender which “could be added” to K is already in K. More precisely, Theorem 2.3 says that K is maximal in the sense that any countably certified extender that coheres with K is actually on the extender sequence of K, and similarly for the iterates of K. Mitchell showed earlier that every countably complete K-ultrafilter is in K under the stronger hypothesis that there is no inner model with a measurable cardinal κ with o(κ)=κ++. (Cf. [Mi].) It is not known if Theorem 2.3 holds for countably closed extenders. Theorem 2.5 is a result similar to Theorem 2.3 for mice with sufficiently small projectum. Our first application of Theorem 2.3 is Theorem 3.1, which says that K computes successors of weakly compact cardinals correctly. This extends Kunen’s result for L and the weak covering properties of K mentioned above. Recall from [St1] that K is the transitive collapse of an elementary substructure of Kc, the background certified core model, and that Kc is the limit of the maximal, 1-small construction C. In Theorem 3.2, another application of Theorem 2.3, we c show that K and all the models on C are iterates of K. Theorem 3.2 is used to prove Theorem 3.3, which says, roughly, that a Mahlo cardinal is a kind of closure point of the construction C (although not in as strong a sense as we would like). The universality of initial segments of K is related to some open questions in descriptive set theory, such as the consistency strength of u2 = ω2.Jensenshowed that it is consistent for a countable mouse to iterate past K ω ,sothatKω k 1 k 1 need not be universal for mice of height ω1. Theorem 3.4, another application of maximality, shows that K κ is universal≤ for mice of height κ whenever κ is a k ≤ cardinal ω2. (This result is joint with W. J. Mitchell.) Using≥ Theorem 3.1 and [Sch2], we prove Theorem 4.1, which says that if κ is <ω a weakly compact cardinal and κ fails, then there is an inner model with a <ω Woodin cardinal. We recall the definition of κ in Section 4; it is weaker than Jensen’s principle κ, and stronger than his κ∗ . By combining our work with his core model induction, Woodin has shown that if there is a weakly compact cardinal <ω κ such that κ fails, then there is an inner model with ω Woodin cardinals. We conjecture that the existence of such a κ implies that there is an inner model with a superstrong cardinal. Less ambitiously, one might aim to construct an inner model with a cardinal which is strong past a Woodin cardinal, that being the frontier of basic inner model theory at the moment. The proof of Theorem 4.1, together with the results in [Sch2] and [MiSchSt], are used in the proof of Theorem 4.2, where we show that if κ is a singular countably <ω closed cardinal and κ fails, then there is an inner model with a Woodin cardinal. With little additional work, the conclusion can be strengthened to the existence of an inner model with n Woodin cardinals for any n<ω. Woodin’s core model induction does not seem to yield an inner model with ω Woodin cardinals in this situation. Again, we conjecture that the existence of such a κ implies that there is an inner model with a superstrong cardinal. Some of the theorems in this paper have the hypothesis that Ω is a measurable cardinal. This is used to see that the theory of [St1] applies (sharps for bounded subsets of Ω would do). For our theorems that have this hypothesis on Ω, The- <ω orem 5.1 shows how to reduce it to the partition property: Ω ( ω ) α for all α<Ω. For example, if it exists, the Erdös cardinal κ(ω) has this−→ property, and it is consistent with V = L that κ(ω) exists.

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2. Maximality Definition 2.1. Suppose that is a premouse and that F is an extender that coheres with .Letκ=crit(FM)andν=ν(F). Suppose that M n ([κ] )M. A⊆n<ω P [ Then a weak -certificate for ( ,F) is a pair (N,G) such that A M (a) N is a transitive, power admissible set, ωN N, ⊂ κ M N, ∪A∪{Jκ }⊂ and G is an extender over N, (b) F ([ν]<ω )=G ([ν]<ω ), ∩ ×A ∩ ×A (c) Vν ult(N,G), +1 ⊂ (d) if j : N ult(N,G) is the ultrapower map, then and j( κM) agree below lh(F),−→ and M J (e) (α) N, (α), whenever α<κ. hP ∩ ∈i ≺ hP ∈i Weak -certificates are weaker than the -certificates of [St1, 1.1] in two ways. A A First, we weakened the condition that Vκ N to just condition (e). Second, ult(N,G) is not required to be countably closed.⊆ The following definition should be compared with [St1, 1.2]. Definition 2.2. Suppose that is a premouse and F is an extender of length α that coheres with .Then(M ,F)isweakly countably certified iff for all countable M M n ([κ] ) JαM A⊆n<ω P ∩ [ there is a weak -certificate for ( M,F). A Jα When is an iteration tree, we shall sometimes write ( ,η)for ηT,the ηth modelT of , to avoid double superscripts. When hasM a finalT model,M we shall T T denote it either by T or by ( , ). The following is our main maximality theorem. M∞ M T ∞ Theorem 2.3. Suppose that Ω is a measurable cardinal and that there is no inner model with a Woodin cardinal. Let be a normal iteration tree on K of successor T length < Ω. Suppose that F is an extender that coheres with T and that M∞ lh(EηT ) < lh(F )

for all η

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Proof of Theorem 2.3. Fix Ω, and F as in the statement of Theorem 2.3 and let κ =crit(F), ν = ν(F ), and λ =lh(T F).

Let µ0 be an inaccessible cardinal < Ω such that F Vµ0 .LetWbe an A0- soundness witness for K .Then induces an iteration∈ tree on W with the same µ0 extenders, drops, degrees,J and treeT structure as , and it is enough to show that F is on the sequence of the final model of the inducedT tree. For the rest of the proof, we shall write for the induced iteration tree on W , and never again refer to the original iterationT tree on K. As in [St1, 9.7], we write Φ( ) for the phalanx derived from .TheF-extension of Φ( ) is defined in [St1, 8.5].T By [St1, 8.6], we are done ifT we show that the F -extensionT of Φ( ) is (Ω + 1)-iterable. (Here we use the hypothesis that is normal.) For contradiction,T suppose that is an ill-behaved iteration treeT on the F -extension of Φ( ). We interpret = U to mean that the ultrapower in the definition of the F -extensionT of Φ( ) isU ill-founded.∅ By the tree property at Ω, lh( ) < Ω. T ByU the argument of [St1, 6.14], we may fix a successor cardinal µ such that µ0 <µ<Ωand there is an iteration tree on W with the same extenders, drops, degrees, • S Jµ and tree structure as , such that ηS is an initial segment of ηT whenever η

σ(λ +1)=πX(λ+1).

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Say πY ( Y )= and πY ( Y )= .Notethat Y has the same extenders, drops, S S P P S degrees, and tree structure as ,andthat ( Y,0) is crit(πY )-strong. Let S MS n = ran(σ) ([κ] )PY A ∩ n<ω P [ and (N,G)beaweak -certificate for ( Y ,F). This is possible because ( T ,F) A P M∞ is weakly countably certified, and Y and agree with T on subsets of κ.Let j:N ult(N,G) be the ultrapowerP map.P M∞ −→ <ω Pick a coordinate b [ν] together with functions u νu ,u λu , and ∈ 7→ 7→ u σu in N such that 7→ N ν =[b, u νu] , 7→ G N λ =[b, u λu] , 7→ G and N σ =[b, u σu] . 7→ G This is possible by clause (c) in Definition 2.1. ByLo´ s’ theorem, there is a set B Gb such that for all u B, ∈ ∈ σu is an elementary embedding of into σu( ) for every initial seg- • |M| M M ment of a model on Φ( X ), M S νu = σu(νX ) <κ,and • λu =σu(λX)<κ. • Lemma 2.4.1. For Gb -almost every u B, there is a Σk+1-elementary, weak k- embedding ∈

τ :ultk( X,FX) Y P −→ P such that τλX = σuλX .

Proof. Let fn n<ω be a sequence of Σk( Y )-functions and h | i P bn n<ω h | i be a sequence of coordinates from [λ]<ω such that

Y ultk( Y ,F) ran(σ)= [bn,fn]P n<ω . | P |∩ F | n o We assume that b = b0 bn bn+1 for all n<ω. We assume that fn is chosen to be a constant function whenever⊂ ⊂ possible. We assume that if

Y [bn,fn]FP =σ(ξ) σ(ξ),a for some ξ<λX,thenσ(ξ) bn and fn is the projection function u u . ∈ 7→ For each ξ<λX and n ω with σ(ξ) bn,let ∈ ∈ n bn σ(ξ),bn C = u [κ]| | σ b ,bn (ξ)=u . ξ ∈ | u 0 n n o Then C Gb .SinceNis countably closed (or, just because the map u σu ξ ∈ n 7→ and all the reals are in N), it follows that n C ξ<λX σ(ξ) bn ξ | ∧ ∈ is an element of N.

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If n<ωand e is the G¨odel number for a Σk formula ϕ such that

Y ultk( Y ,F) =ϕ [bn,fn]P , P | F then let h i

n bn T = u [κ]| | Y =ϕ[fn(u)] . e ∈ |P | n n o Then, Te Fbn byLo´ s’ theorem and our choice of . Moreover, the partial function (n,∈ e) ∩AT n is an element of N since N is countablyA closed. 7→ e Let Bn n<ω be a sequence in N such that Bn Gb for all n<ω,withthe h | i ∈ n following three properties. First, B0 = B. Second, for every m, e < ω such that T m is deﬁned, there is an n m with e ≥ bm,bn m (Bn) T . ⊆ e Third, for every m<ωand ξ<λX, there is an n m such that ≥ bm,bn m (Bn) C . ⊆ ξ We need a “ﬁber” through the sets Bn. There is a tree A in N that searches for a function g : ω [ κ ] <ω such that −→ g(m) g(n) Bn ⊂ ∈ and g(m),g(n) (bn) = bm

whenever m

1 Y τ σ− [bn,fn]FP = fn (g (n)) . n Because we “met” the sets T , τ is a weak k-embedding of ultk( X ,FX)into Y e P P (recall that weak k-embeddings need not be Σk+1-elementary). Because fn was chosen to be the constant function whenever possible, if

iX : X ultk( X ,FX) P −→ P is the ultrapower map, then τ iX = σ.Sinceiis a k-embedding and σ is elementary, ◦ n τ is a Σk+1-elementary. Because we “met” the sets Cξ , it follows that

τλX = σuλX .

Lemma 2.4.2. There are functions u u and u τu in N such that for Gb- almost every u B,bothV and N satisfy7→ P 7→ ∈ (a) u is a conditionally (λu +1)-strong premouse, P (b) u and P agree below λu +1,and P Jκ (c) τu is a Σk+1-elementary, weak k-embedding of

ultk( X ,FX) P into u such that τu λX = σu λX . P

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Proof. First note that Y is conditionally (λ + 1)-strong as witnessed by an initial P segment of the iteration tree Y andtheidentitymapon Y. Given u B and a correspondingS τ as in Lemma 2.4.1,P let ∈ Y ∗ = P ((λu +1) ran (τ)) , Pu Hω ∪ and let

τ ∗ :ultk( X,FX) ∗ u P −→ P u be the corresponding collapse of τ. Then the functions u u∗ and u τu∗ satisfy conditions (a), (b), and (c) in V , but these functions are7→ not P necessarily7→ in N. By the elementarity given in clause (e) of Deﬁnition 2.1, for each appropriate u B, there is a premouse u and an embedding τu,bothinN, which satisfy conditions∈ (a), (b), and (c) ofP Lemma 2.4.2 in both N and V .SinceNsatisﬁes AC, there are appropriate functions u u and u τu in N. 7→ P 7→ Fix functions as in Lemma 2.4.2 and let N ∗ =[b, u u] P 7→ P G and N τ ∗ =[b, u τu] . 7→ G Lemma 2.4.3. τ ∗ is a Σk+1-elementary, weak k-embedding of

ultk( X ,FX) P into ∗,andτ∗ λX =σλX. P Proof. Immediate from the deﬁnitions of ∗ and τ, clause (c) of Lemma 2.4.2, and Lo´ s’ theorem. P

Lemma 2.4.4. ( Y , ) and ∗ agree below λ. M S ∞ P Proof. Clearly ( Y , ), ( , ), and ( , ) all agree beyond λ.Bycoher- M S ∞ M S ∞ M T ∞ ence, ( Y , ) and ult( Y ,F) agree below λ. Clearly ( Y , )and Y agree M S ∞ P M S ∞ P below κ. Hence, by clause (d) of Deﬁnition 2.1, ( Y , )andj( PY ) agree below M S ∞ Jκ λ. By clause (b) of Lemma 2.4.2, u agrees with P beyond λu for Gb-almost every P Jκ u B. It follows byLo´ s’ theorem that ( Y , ) agrees with ∗ below λ. ∈ M S ∞ P Lemma 2.4.5. ∗ is conditionally (λ +1)-strong. Moreover, if ∗ is an iteration P S tree which witnesses that ∗ is conditionally (λ +1)-strong, then ∗ uses the same P S extenders as Y below λ. S Proof. ByLo´ s’ theorem and clause (a) of Lemma 2.4.2, ∗ is conditionally (λ +1)- P strong in ult(N,G). By clause (c) of Deﬁnition 2.1, Vν ult(N,G). It follows +1 ⊂ that ∗ is conditionally (λ + 1)-strong in V . In light of Lemma 2.4.4, any itera- P tion tree ∗ witnessing that ∗ is conditionally (λ + 1)-strong must use the same S P extenders as Y below λ. S _ Lemma 2.4.6. Φ( Y ) ∗,k,ν is an iterable phalanx. S hP i _ Proof. Put A =Φ( Y) ∗,k,ν .ThatAis a phalanx follows from Lemma 2.4.4. S hP i Fix ∗ and an embedding ψ of ∗ into an initial segment of ( ∗, )which S P M S ∞ witness that ∗ is conditionally (λ + 1)-strong. P Let us write ψ( ∗) for the initial segment of ( ∗, )intowhichψelementarily P M S ∞ embeds ∗.Since ( ∗,0) is (λ + 1)-strong, there is P MS

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a successor-length iteration tree on W , such that lh(EηR) >λwhenever • ηλ. M S M∞

We may use ϕ to copy ∗ to an iteration tree ϕ ∗ on R .Let S S M∞ ϕ : ( ∗, ) ( ϕ ∗ , ) ∞ MS ∞ −→ M S ∞ be the ﬁnal map in the aforementioned copying construction. Then crit(ϕ ) >λ ∞ and ϕ ∗ uses the same extenders as Y below λ. Therefore ϕ ψ is an elementary S S ∞ ◦ embedding of ∗ into the initial segment (ϕ ψ)( ∗)of (ϕ ∗, )and P ∞ ◦ P M S ∞ crit(ϕ ψ) >λ. ∞ ◦ Recall that was our original iteration tree on W .Put T _ B=Φ( ) (ϕ ψ)( ∗),k,ν . T h ∞◦ P i The agreement described above implies that B is a phalanx. It is a phalanx “de- _ rived” from and ϕ ∗, both iteration trees on W . Such W -generated phalanxes are iterableT byR [St1,S 6.9]. But now, the pair of maps (πY , (ϕ ψ))canbeusedinthestandardwayto reduce the iterability of A to that of B∞. ◦

Recall that X is a simple, ill-behaved iteration tree on V _ Φ( X ) ultk( X ,FX),k,νX . S h P i Using the pair of maps (σ, τ ∗ )wecancopy X to an ill-behaved iteration tree _ V on Φ( Y ) ∗,k,ν . But now we have a contradiction of Lemma 2.4.6. This completesS thehP proofi of Theorem 2.3.

We conclude this section with a version of Theorem 2.3 for mice with suﬃciently small projectum. Unlike Theorem 2.3, Theorem 2.5 does not have as a hypothesis that there is no inner model with a Woodin cardinal, nor any such anti-large- cardinal hypothesis. For this reason, Theorem 2.5 is applicable to certain core models beyond one Woodin cardinal. Theorem 2.5. Let be a sound premouse such that M ρω( ) crit(E) M ≤ whenever E is an extender from the -sequence. Suppose that every countable phalanx which is realizable in an -basedM phalanx is (ω +1)-iterable. Let be a M 1 T normal iteration tree on . Suppose that F is an extender that coheres with T such that M M∞

lh(EηT ) < lh(F ) for all η

Suppose that ( T ,F) is countably certiﬁed. Then F is on the T -sequence. M∞ M∞

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Sketch. Put κ =crit(F), ν = ν(F ), and λ =lh(F). Say Φ( )_ ult( ,F),k,ν T h P i is the F -extension of Φ( ). Let θ be a large regular cardinal and πX : MX V θ T −→ be the uncollapse of a countable elementary substructure X of Vθ.SayπX( )= , T T πX(F)=F,πX(ν)=ν,πX(λ)=λ,andπX( )= . P P Let Y be a an elementary substructure of Vθ with X (λ +1) Y ∪ ⊂ and card(Y ) = card(ν). Let πY : MY V θ be the uncollapse of Y and let −→ 1 σ = π− πX . Y ◦ The hypotheses on soundness and the projectum of imply that πY ( )= ,so σ( )= . M T T TNow letT n = ran(σ) ([κ] )P A ∩ n<ω P [ and (N,G)bean -certificate for ( ,F). (At this point, we could, in fact, take a weak -certificateA for ( ,F)solongasP N.Notethat has cardinality κ.) TheA idea now is to replaceP “K”by“ P∈” in proof of TheoremP 2.3 to see that≤ the F -extension of Φ( ) is realizable in an M-based phalanx. More specifically, there is T M an iteration tree ∗ on extending and a Σk -elementary, weak k-embedding T M T +1 τ of ultk( , F )into ( ∗, ) with τλ = σλ. (We omit the details.) By theP iterabilityM hypothesisT ∞ on , there is a successful coiteration of Φ( ) and the F -extension of Φ( ). Just asM in the proof of [St1, 8.6], our hypothesesT on soundness and the projectumT of allow us to conclude that F is on the ( , )- M M T ∞ sequence. By the elementarity of π, F is on the T -sequence. M∞ 3. Applications of maximality Kunen showed that if κ is a weakly compact cardinal and (κ+)L <κ+, then 0# exists; see [Jech, p.384]. Our first application of Theorem 2.3 is an extension of Kunen’s result to the current generation of core models. Theorem 3.1. Suppose that Ω is a measurable cardinal, and that there is no inner model with a Woodin cardinal. Let κ be a weakly compact cardinal < Ω.Then (κ+)K=κ+. The result analogous to Theorem 3.1 was proved for measurable cardinals κ in [St1, 8.15], for singular countably closed cardinals in [MiSchSt], and for all singular cardinals in [MiSch]. From [MiSch], we already know that (κ+)K has cofinality at least κ whenever κ is a cardinal 2. Lemma 3.1.1 will be used in the≥ℵ proofs of Theorems 3.1, 4.1, and 4.2. We remark that if κ is a strong limit cardinal, as in Theorems 3.1 and 4.1, then any A coding Vκ satisfies the conclusion of Lemma 3.1.1. Lemma 3.1.1. Assume there is no proper class model with a Woodin cardinal, and let κ be a cardinal such that α<κimplies αω κ, and every subset of κ has a ≤

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sharp. Then there is an A κ such that whenever B and C are subsets of κ,and is a premouse with A, ⊆ L[B] L[C],then P P∈ ∩ ( is κ-strong)L[B] ( is κ-strong)L[C]. P ⇐⇒ P Moreover, if there is a measurable cardinal greater than κ,thenKand KB agree below (κ+)KL[B] for all B such that A L[B]. ∈ Proof. Let Ω = κ+. Then, for any A Ω, Ω is an A-indiscernible and the theory of K up to Ω of [St1] applies in L[A].⊆ (Cf. [St1, p.58] and 5ofthispaper.) § Inductively we build Aα,forα<κwhich are cardinals of a “stable K”we determine along the way; the ﬁnal A is just the amalgamation of the Aα’s. Suppose we have Aα such that for all B,C κ with Aα L[B] L[C], L[B]and L[C] agree on which mice in L[B] L[C] are <α⊆ -strong, and∈ on what∩ K up to α is, and on the fact that α is a cardinal∩ of K. Call a premouse “stably <α-strong” P if is <α-strong in L[B], for some B with Aα L[B]. For each countable X α andP countable premouse , if there is a stably∈ <α-strong premouse and⊆ an embedding π : withQ X = ran(π) α, then pick such a of sizeP κ and set (α, X, )=Q−→P. (It is possible to pick∩ of the same sizeP as α.) For≤ the α currentlyP underQ consideration,P since αω κ,P there are κ many such (α, X, ). ≤ ≤ P Q We take D κ to code the function (X, ) (α, X, ) together with Aα.It is easy to see⊆ that if D L[B] L[C], thenQ L7→[B P]andLQ[C] agree on which mice in L[B] L[C] are α-strong.∈ [If∩ L[B] L[C]and is not α-strong in L[B], then there∩ is a stably <α-strongR∈, an ill-behaved∩ countableR iteration tree on a countable phalanx (( , ),β), andP an elementary embedding π such that U Q S π (( , ),β) =(( , ),α). Q S P R Let X = ran(π) α. By absoluteness, there is an elementary embedding τ : in L[C] such that∩ ran(τ) α = X. Another application of absolutenessS−→R gives an elementary embedding σ :∩ (α, X, )inL[C] such that ran(σ) α = X.But then is not α-strong in LQ−→P[C] as witnessedQ by (σ, τ ) .] Thus we have∩ determined the “stablyR α-strong” mice, and “stable K up to the Uα+ of stable K”. Let γ = α+ of stable K,andAγ code D together with stable K up to γ. Now let A code the join of the Aα’s. It is clear from the construction that the ﬁrst conclusion of Lemma 3.1.1 holds with “<κ-strong” in place of “κ-strong”. Suppose that B,C κ, is a premouse, and A, L[B] L[C]. Say is not κ-strong in L[B]. We⊆ shallR show that is not κ-strongR∈ in L∩[C], from whichR the rest of Lemma 3.1.1 follows easily. R Since is not κ-strong in L[B], there is a stably <κ-strong premouse and and a countableR ill-behaved iteration tree on the phalanx (( , ),κ). LetP T P R α~ = αi i<ω h | i be an increasing sequence of cardinals of the stable K up to κ such that for every η

~ = i i<ω P hP | i such that, for each i<α, i embeds into with critical point >αi,so i is stably P P P αi-strong. Let us construe as an iteration tree on the phalanx (( ~ , ),~α). Thus T P R

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construed, is still ill-behaved. Let M be the transitive collapse of a countable T elementary substructure of VΩ and let π : M V Ω be the uncollapse. Assume that −→ π (( ~, ), β~) =((~, ),~α) Q S P R and π( )= . By elementarity and absoluteness, is an ill-behaved iteration tree U T U on (( ~, ), β~). Put Xi = ran(π) αi. InQL[SC], we have the premouse∩ , the countable ill-behaved iteration tree R U on the countable phalanx (( ~, ), β~), and each of the countable sets Xi’s, but not Q S necessarily α~ nor X~ = Xi i<ω .But,inL[C], there is a tree that simultaneously searches for h | i

an increasing sequence ~γ = γi i<ω of cardinals of stable K up to κ, • h | i a sequence Yi i<ω such that, for every i<ω, (γi,Yi, ) is deﬁned, and • h | i P Q an elementary embedding τ : such that ran(τ) γi = Yi and • S−→R ∩ τ(βi)=γi for i<ω. There is an inﬁnite branch through this tree which is determined by α~, X~ ,and π. By absoluteness, there is a branch through this tree in L[C]. So let ~γi, Y~ ,and τbe determined by such a branch. By absoluteness, in L[C], there is a sequence ~σ = σi i<ω such that σi is an elementary embedding of into (γi,Yi, )for eachhi<ω| . i Q P Q Let (~σ,τ) denote the iteration tree obtained by copying using (~σ,τ). Now U U (( P (γi,Yi, i) i<ω , ),~γ) h Q | i R is not iterable in L[C] as witnessed by the ill-behaved iteration tree (~σ,τ) .But, U for all i<ω,P(γi,Yi, i)isstably<γi-strong. Therefore is not κ-strong in L[C]. Q R

Proof of Theorem 3.1. Let κ be a weakly compact cardinal < Ω and put λ =(κ+)K. Assume, for contradiction, that λ<κ+. Let N be a transitive, power admissible set such that ωN N, ⊂ K Vκ N, ∪Jλ+1 ⊂ and card(N)=κ. Pick A κ so that N L[A]. We assume that A satisﬁes the conclusion of Lemma 3.1.1.⊆ (As remarked∈ earlier, this follows.) A We shall write KA for KL[A].Letλ=(κ+)K . By Lemma 3.1.1, KA and K agree below λ. Since A# exists, (κ+)L[A] <κ+, so card ( (κ) L[A]) = κ. Because κ is weakly compact, there is a nonprincipal, κ-completeP L[A∩]-ultraﬁlter U over κ; see [Jech, p.384]. Then ult(L[A],U) is well-founded. Let j : L[A] ult(L[A],U)=L[j(A)] −→ be the ultrapower map. Then crit(j)=κand A = j(A) κ L[j(A)], ∩ ∈

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so L[A] L[j(A)]. We remark that (κ) L[j(A)] L[A] is possible; see [Jech, ⊆ P ∩ 6⊂j(A) p.394]. It follows from Lemma 3.1.1 that λ =(κ+)K ,andthatKj(A) and K agree past λ.Thus (κ) Kj(A)= (κ) KA. P ∩ P ∩ <ω Let Ej be the superstrong extender derived from j. So, for every a [j(κ)] , ∈ a (Ej )a = x [κ]| | x L[A] a j(x) . ⊆ | ∈ ∧ ∈ n o We claim that for every sequence xα α<κ in L[A], h | i <ω Ej [j(κ)] xα α<κ L[j(A)]. ∩ ×{ | } ∈ The argument is due to Kunen, as in [MiSt, 1.1]: just note that for any a [j(κ)]<ω, ∈ a Ea = xα [κ]| | α<κ a j(xα) ⊆ | ∧ ∈ and n o

j(xα) α<κ L[j(A)]. h | i∈ Set <ω A F = Ej [j(κ)] K ∩ × and <ω G = Ej [j(κ)] N . ∩ × Then (Kj(A),F)and(N,G) are elements of L[j(A)]. Moreover, L[j(A)] satisﬁes the sentence, “F ξ coheres with K and (N,G)isan -certiﬁcate for (K, F ξ)where A = ([κ]n)K A n<ω P [ and unbounded many ξ

By deﬁnition, K is the transitive collapse of an elementary substructure of Kc; see [St1, 5]. Our next application of Theorem 2.3 shows that there is an iteration tree on K§ with last model Kc. For the statement of Theorem 3.2, recall that Kc is deﬁned as the limit of the maximal 1-small construction ξ ξ<Ω; see [St1, 1]. hN | i § Theorem 3.2. Assume that Ω is a measurable cardinal and that there is no inner model with a Woodin cardinal. Let α Ω and k<ω. Suppose that ( , ) is the coiteration of ≤ U V

(K, Ck( ξ)) . N Then is trivial. In particular, since Kc is universal, Kc is an iterate of K. V

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Proof. The proof is by induction. Fix ξ Ωandk<ω. Assume that C`( ζ )does not move in its coiteration with K whenever≤ N

(ζ,`)

For contradiction, assume that Ck( ξ) moves in its coiteration with K. Let ( , ) be the coiteration of N U V (K, Ck( ξ)) N and θ0. By the induction hypothesis,

Ck 1( ξ) = Ck( ξ). − N 6 N Thus, Ck 1( ξ)is(k 1)-sound but not k-sound. By the induction hypothesis, − N − there is an iteration tree on K such that the last model of is Ck 1( ξ). But S S − N then Ck( ξ) is a model along the main branch of .Namely, N S Ck( ξ)=( ∗ )S N Mγ+1 where γ + 1 is the last drop to an ultrapower of degree at most k 1 along the − main branch of .Butthen (γ+ 1) witnesses that the theorem holds for (ξ,k), a contradiction.S S

Recall that κ is a premouse of height κ whenever κ is a cardinal, but that κ N Kc N need not equal κ . Theorem 3.3 appears to be the ﬁrst step in showing that Kc J κ = when κ is a Mahlo cardinal, but we do not know if this is true. N Jκ Theorem 3.3. Suppose that there is no inner model with a Woodin cardinal and that Ω is a measurable cardinal. Let κ<Ωbe a Mahlo cardinal. Then there is an K iteration tree on κ such that T = κ. In particular, T J M∞ N + + (α )Nκ = α

for all singular cardinals α<κ,so κ is universal for mice of height κ. N ≤ Proof. By Theorem 3.2, there is an iteration tree on K with T = κ.Cer- tainly lh( ) κ + 1. Let us assume that lh( )=κT+ 1, the otherM case∞ beingN clear. T ≤ T It is enough if we show that does not drop and that i0T,κ(α) <κfor all α<κ. Suppose otherwise. FormT a continuous chain

Xα α<κ h | i of elementary submodels of Vκ , , such that card(Xα)=αand h +1 ∈ Ti ω Xα Xα ∈ +1 for all α<κ.Let

πα:Pα V κ −→ +1

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be the uncollapse of Xα and Gα be the length κ extender derived from πα whenever α<κ. There is a stationary set of inaccessible cardinals α<κsuch that if T (β +1)Tκ and α =pred (β+ 1), then α =crit(πα), πα(α)=κ,Xα is countably closed, T (α) Pα,and(Pα,Gα)isan -certificate for Mα ∩P ∈ A T ( Mα ,E ) lh(E ) βT J βT where n = ([α] ) αT . A n<ω P ∩|M | [ The proof is similar to the standard argument that coiterations of mice terminate. T Fix any such α<κ.Letβbe such that α =pred (β+1) and (β+1)Tκ. Clearly EβT coheres with κ and ( κ,ET ) is countably certified. Note that lh(ET ) is a cardinal N N β β in κ.Letγ<κbe the supremum of the η<κsuch that ρω( η) < lh(ET ). It N N β is not hard to see that γ is the passive initial segment of κ of height lh(EβT ), _ N N γ ET is a premouse, and that ( γ ,ET ) is certified as above. Therefore N h β i N β ρω( γ ) ν(ET ) < lh(ET ), N +1 ≤ β β a contradiction. Theorem 3.4 (with W. J. Mitchell). Suppose that there is no inner model with a Woodin cardinal and that Ω is a measurable cardinal. Let κ be a cardinal such that κ<Ω.Then K is universal for premice of height κ. ℵ2 ≤ Jκ Theorem 3.4 reduces quickly to the case in which κ is a successor cardinal. The authors proved Theorem 3.4 under the assumption that κ is a countably closed, successor cardinal. Then, Mitchell saw how to eliminate the countable closure, by adapting the methods of [MiSch].

Sketch of Theorem 3.4. Let κ be the least cardinal between 2 and Ω for which Theorem 3.4 fails. Then κ is a successor cardinal, say κ = µ+.ℵ Suppose that is a counterexample to the theorem and that ( , )isthe KM S T coiteration of ( , ). Then, there is club set C [0,κ]T such that Jκ M ⊆ T [α, κ]T = , D ∩ ∅

crit(iα,κT )=α, and

iα,κT (α)=κ

whenever α C. Let us also assume that C [0,κ]S and that ∈ ⊆ iS 00α α. 0,α ⊆ Deﬁne β(α)tobetheT-successor of α in [0,κ]T. First assume that µω = µ. Form a continuous chain

Xα α<κ h | i of elementary submodels of V ω , , such that card(Xα)=µ, h Ω+ ∈ Ti ω Xα Xα , ∈ +1

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and Xα κ κ for all α<κ.Let ∩ ∈ πα:Pα V ω −→ Ω+ be the uncollapse of Xα and let Gα be the length κ extender derived from πα. Then, as in the proof of Theorem 3.3, for a stationary set of α C,(Pα,Gα)isa weak -certiﬁcate for ∈ A T ( Mβ ,E ), lh(E ) βT J βT where β = β(α)and

n = ([α] ) αT . A n<ω P ∩|M | [ It follows that ( S ,ET ) is weakly countably certified. (Note that (Pα,Gα) cannot Mβ β be an -certificate since Vα Pα.) A 6⊆ K While is an iteration tree on κ , we may consider its action on all of K.Let be theS iteration tree on K withJ the same extenders, drops, and degrees as . R S Then ( R,ET ) is weakly countably certified. Hence, by Theorem 2.3, ET is on Mβ β β the βR-sequence. So EβT is on the βS-sequence, which is in obvious contradiction withM the comparison process. M ω ω Now drop the assumption that µ = µ and the condition Xα Xα+1 above. Instead, we assume that the elementary chain of models is internally∈ approachable; that is, we require that

Xγ γ α Xα h | ≤ i∈ +1 for all α<κ. Define Pα, πα,andGα as in the paragraph above. The difference is that Pα is not countably closed. But notice that in the proof of Theorem 2.3, countable closure was only used to see that the ET -extension of Φ( ) is iterable, β(α) R so that we could apply [St1, 8.6]. The methods of [MiSch] adapt to show that for a stationary set of α C,theET -extension of Φ( ) is iterable. Below we sketch ∈ β(α) R the main ideas of the proof. Since crit(ET )=αand ET measures all sets in R, ET is applied to all β(α) β(α) Mα β(α) of R in forming the ET -extension of Φ( ). Mα β(α) R Let us say that a mouse is α-good if replacing the starting model R of the Q Mα phalanx Φ( (α+1)) by yields an iterable phalanx. (This corresponds to [MiSch, Definition 2.7].)R Let usQ refer to the phalanx thus obtained by the abbreviated _ _ notation Φ( α) . In particular, R is α-good, since Φ( α) R is just R hQi Mα R hMα i the K-based phalanx Φ( (α + 1)). Let us say that an αR-good mouse lifts badly from α to κ if replacing the Q starting model ult( R,ET )oftheET -extension of Φ( ) by ult( ,Gα) yields Mα β(α) β(α) R Q a phalanx which is not iterable. (This corresponds to [MiSch, Definition 2.4].) Let us refer to the phalanx thus obtained by the abbreviated notation _ Φ( ) ult( ,Gα) . R h Q i Because Gα extends ET ,ifforsomeα C, R does not lift badly from α to β(α) ∈ Mα κ, then the ET -extension of Φ( ) is iterable. So, we assume that for all α C, β(α) R ∈ αR lifts badly from α to κ, and the proof will be complete when we reach a contradiction.M

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Suppose that is α-good and lifts badly from α to κ.Thenwesaythat is minimal at α if forQ every iteration tree on Φ( α)_ , Q U R hQi no proper initial segment of U lifts badly from α to κ,and • M∞ if U lifts badly from α to κ,then U lies above in the tree order of • M∞ M∞ Q U and there is no dropping along the branch from to U . Q M∞ (This deﬁnition corresponds to the conclusion of [MiSch, Lemma 3.2].) Starting from

= R, N0 Mα we can find an α-good mouse α which lifts badly from α to κ and is minimal at α Q by the following procedure. If i is not minimal at α,thenlet i witness that i N U N is not minimal at α and let i be the shortest initial segment of ( i, )which N +1 M U ∞ lifts badly from α to κ. The iterability of K implies that each i is α-good. [More precisely, it is the iterability of K-based phalanxes which is usedN here and proved _ in [St1, 9]. The phalanx Φ( α) i is K-based.] (The same argument proves § R hN i [MiSch, Fact 3.2.1].) Moreover, there is an i<ωsuch that i+1 is not defined, again by the iterability of K. [The idea is that we may, in a naturalN way, consider α_ _ _ R U0 U1 ··· as an iteration tree on K. Moreover, if n is defined for every n<ω, then this “concatenation” has no cofinal well-foundedU branch, either because the only can- didate for such a branch has infinitely many drops, or because the underlying tree has no cofinal branches.] (The same argument proves [MiSch, Fact 3.2.2].) Put α = i for this i. We may assume that α Xα . Q N Q ∈ +1 We can pick functions fk,α for k<ωand α C which represent in ult( αR,EαT ), _ ∈ M a“proof”thatΦ( ) ult( αR,Gα) is not iterable. [Take a countable elementary submodel R h M i

Y V ω ω ≺ Ω+ + with everything in it. Let the elements of

Y ult( R,Gα) ∩ Mα be represented by the fk,α’s.] Let α be the transitive collapse of the hull in αR of M M

α fk,α k<ω . ∪{ | } Then α is still α-good and lifts badly from α to κ. M For `<ω,let α,` be the transitive collapse of the hull in α of M M α fk,α k<` . ∪{ | } Each α,` is coded by a subset of α belonging to α. [Here we use that αR has heightM Ω.] So, if α is a limit, then M M

α,` ran(π ) M ∈ γ(α,`),α 1 for some γ(α, `) <α.Hereπγ,α is the elementary embedding π− πγ from Pγ into α ◦ Pα for all γ<α. By Fodor’s lemma, there is a stationary set S C and a ﬁxed γ<κsuch that γ(α, `) <γfor all α S and `<ω. [Here is where⊆ we use that <κ.] ∈ ℵ1 Fix α S.Sayπγ,α(Z`)= α,` for `<ω.Let ∗be the direct limit of the ∈ M M Z`’s under the collapses of the natural maps between the α,`’s. ∗ is called the M M

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pull-back of R to γ. (This corresponds to [MiSch, Lemma 2.1].) Since α is α- Mα M good, ∗ is γ-good. Moreover, the construction of ∗ guarantees that since α M M M lifts badly from α to κ, ∗ lifts badly from γ to κ. (This corresponds to [MiSch,

Corollary 2.5(b)].) BehindM the scenes is the following commutative diagram. /

αR / ult( αR,Gα)

O q

q M M

/ / ∗ / ult( ∗,Gαα) ult( ∗,Gα) M M M Let us say that a γ-good mouse lifts badly from γ to α if Q _ Φ( α) ult( ,Gγ α) R h Q i is a phalanx which is not iterable. Since ult( ∗,Gγ α)embedsinto R with critical point at least α,and R is M Mα Mα α-good, ∗ does not lift badly from γ to α.Infact, ∗hereditarily does not lift M M badly from γ to α in the sense that V does not lift badly from γ to α whenever _ M∞ is an iteration tree on Φ( γ) ∗ . [If is such a tree, then we may copy toV an iteration tree on R hM i V V W _ Φ( α) ult( ∗,Gγ α) , R h M i using the identity maps on models in Φ( γ) and the ultrapower map on ∗.We R M_ may in turn copy to an iteration tree on Φ( (α+1)). Since Φ( α) X W X_ R R _ hM∞i is K-based, it is iterable. Since Φ( α) W embeds into Φ( α) X ,it too is iterable.] R hM∞ i R hM∞i On the other hand, R and γ both lift badly from γ to α. This follows from Mγ Q the fact that they lift badly from γ to κ and are elements of Xα. [If is any mouse Q which lifts badly from γ to α and Xα,thenPα satisﬁes that the phalanx Q∈ 1 _ 1 _ 1 π− Φ( ) ult( ,Gγ) =Φ(π− ( )) ult(π− ( ),Gγ α) α R h Q i α R h α Q i 1 is not iterable. One can use this to show that πα− ( ) lifts badly from γ to α.So lifts badly from γ to α.] (This corresponds to [MiSch,Q Lemma 2.6].) Q _ _ Let ( , ) be the coiteration of Φ( γ) Qγ and Φ( γ) ∗ .Since is an iterationU V tree on K, using the hull andR deﬁnabilityh i propertiesR hM and soundnessi R we see that either

(a) U lies above γ in the tree order of ,or M∞ Q U (b) V lies above ∗ in the tree order of . M∞ M V For contradiction, assume that (a) does not hold. Then (b) holds. We claim that V is an initial segment of U and that there is no dropping on the branch from M∞ M∞ ∗ to V . The proof is like the proof that K is universal. (Recall that γ is Man iterationM∞ tree on K.) From the following commutative diagram, it is clearR that V lifts badly from γ to κ,since ∗ does.

M∞ M / ∗ / ult( ∗,Gγ)

M M

/ V / ult( V ,Gγ) M∞ M∞ (It is a general fact that if a γ-good mouse which lifts badly from γ to κ embeds M into a mouse 0 with critical point at least γ,then 0 also lifts badly from γ to M M

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κ. This is like [MiSch, Lemma 3.1(a)].) Since V is an initial segment of U , M∞ M∞ U also lifts badly from γ to κ. Since (a) fails, this contradicts the minimality of M∞ γ. Therefore, (a) holds. Q We next claim that U is an initial segment of V and that there is no M∞ M∞ dropping on the branch from γ to U . If (b) does not hold, then this is essentially the universality of K again.Q If (b)M∞ holds and our claim fails, then there is no dropping on the branch from ∗ to V . Therefore V is a proper initial M M∞ M∞ segment of U which lifts badly from γ to κ. This contradicts the minimality of M∞ γ. Q We have seen that there is an embedding of γ into U with critical point at Q M∞ least γ. From the following commutative diagram, it is clear that U lifts badly M∞ from γ to α,since γ does.

Q / γ / ult( ∗,Gγ α)

Q M

/ U / ult( V ,Gγα) M∞ M∞ (It is a general fact that if a γ-good mouse which lifts badly from γ to α embeds Q intoamouse 0 with critical point at least γ,then 0 also lifts badly from γ to α. This is like [MiSch,Q Lemma 3.1(b)].) Q Since U is an initial segment of V , V also lifts badly from γ to α.But M∞ M∞ M∞ V does not lift badly from γ to α, since, as the reader will recall, is an iteration M∞ _ V tree on Φ( γ) ∗ and ∗ hereditarily does not lift badly from γ to α.This is a contradiction.R hM i M

4. Square principles Recall from [Sch1, 5.1] or [Sch2] the following hierarchy of principles intermediate + <λ to Jensen’s well-known principles κ and κ∗ from [J1]. If 1 <λ κ ,thenκ +≤ is the principle asserting the existence of a sequence ν ν<κ such that for every limit ordinal ν (κ, κ+), hF | i ∈ (1) 1 card( ν) <λ,and ≤ F (2) for all C ν, (a) C is∈F a closed, unbounded subset of ν, (b) C has order type κ,and ≤ (c) C µ µ whenever µ<νis a limit point of C. ∩ ∈F λ <λ+ 1 κ We write κ for κ ;soκ κ and κ∗ κ . Using these forms of weak square, some lower bounds on the≡ large cardinal≡ consistency strength of the Proper Forcing Axiom and other principles were obtained in [Sch1, Sch2, SchSt]. Theorem 3.1 and its proof will be used to prove Theorem 4.1 below. Theorem 4.1. Suppose that there is no inner model with a Woodin cardinal. Then <ω κ holds whenever κ is a weakly compact cardinal. If there is a measurable cardinal Ω, but no inner model with a Woodin cardinal, <ω then κ holds for every weakly compact cardinal κ<Ω; this is immediate from Theorem 3.1 and [Sch2]. Theorem 4.1 says that the measurable cardinal is not necessary. Our proof of Theorem 4.1 can be modiﬁed to show that if κ is a weakly compact cardinal and <ω fails, then for every A κ and n<ωthere is an iterable κ ⊆

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inner model which contains A and has n Woodin cardinals. Combining this with <ω [MaSt1] shows that the failure of κ at such a κ implies PD. Using our proof of <ω Theorem 4.1, Woodin has since shown that if κ is weakly compact and κ fails, then L(R)-determinacy holds. Woodin’s proof is an induction on the levels of L(R), with Theorem 4.1 as the base case, and he makes the general case look like the base case. Proof of Theorem 4.1. Suppose that κ is a weakly compact cardinal and that there <ω is no inner model with a Woodin cardinal. Assume for contradiction that κ fails. Then A# exists whenever A is a bounded subset of κ. This follows from Jensen’s theorem that κ holds in L[A] (cf. [J1]) and Kunen’s theorem on successors of weakly compact cardinals relativized to L[A] (cf. [Jech, p.384]). Since κ is weakly compact, A# exists for all A κ.LetΩ=κ+.ThenΩisanA-indiscernible for every A κ. So, for every A ⊆κ, the theory of K up to Ω of [St1] applies in L[A]. Again, we⊆ shall write KA for ⊆KL[A]. Let Aα α<Ω be a sequence of subsets of κ, λα α<Ω be a sequence of h | i h | i ordinals, Uα α<Ω be a sequence of ﬁlters on κ,and Nα α<Ω be a sequence of transitive,h power| admissible,i countably closed sets ofh cardinality| iκ such that

Vκ L[Aα], ⊂ Aα and Uα are elements of L[Aβ],

A + K α λα = κ ,

Uα is a nonprincipal, κ-complete L[Aα]-ultrafilter over κ,and KAα Vκ N ∪Jλα+1 ⊂ whenever α<β<Ω. And, also, we assume that A0 satisfies the conclusion of Lemma 3.1.1. (As remarked before Lemma 3.1.1, this follows.) We claim that for unboundedly many α<Ω, λα α. Suppose otherwise, then ≥ there is a stationary set S Ωandafixedλ<Ω such that λα = λ whenever ⊂ α S.Fixα S.LetA=Aα,U=Uα, ∈ ∈ j:L[A] ult(L[A],U) −→ be the ultrapower map, Ej be the length j(κ) extender derived from j,andN =Nα. Note that j(A) (κ+)K = λ since j(A) (κ+)K λ ≥ and L[j(A)] L[Aβ ] for any β S α. But now a contradiction follows as in the proof of Theorem⊂ 3.1. [If ∈ − <ω A F = Ej ([j(κ] K ) ∩ × and <ω G = Ej ([j(κ] N), ∩ × then for unboundedly many ξ

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A A By Lemma 3.1.1, K α and K β agree below λα whenever α<β<Ω. Let ~ ˙ KAα E = E λα α< [Ω and = E~ . By the claim of the previous paragraph, M JΩ + (κ )M =Ω.

<ω Aα By [Sch2], κ holds in K for all α<Ω. Moreover, by [Sch2, Corollary 1.2.1], <ω the witnesses to κ can be chosen to be “local deﬁnable”, so that for each α<Ω, α α <ω Aα there is a sequence = ν ν<λα that witnesses κ in K , such that α β F hF | i α <ω = λα whenever α<β<Ω. So = α<Ω witnesses κ in ,and thereforeF F in V . This is a contradiction. F F M S If there is a measurable cardinal Ω, but no inner model with a Woodin cardinal, <ω then κ holds for every singular cardinal κ<Ω. This is immediate from [MiSch] and [Sch2]. The next theorem says that the measurable cardinal is not necessary, at least when κ is a singular countably closed cardinal. (Recall that κ is countably closed iﬀ αω <κwhenever α<κ.) Theorem 4.2. Suppose that there is no inner model with a Woodin cardinal. Then <ω κ holds whenever κ is a singular countably closed cardinal.

Jensen proved that if 0¶ does not exist, then κ holds whenever κ is a singular cardinal. (Cf. [J2] and [J3].) It is not known if Theorem 4.2 holds for all singular cardinals, nor if its conclusion can be strengthened to κ holding. The proof of Theorem 4.2 can be extended to show that if κ is a singular countably closed <ω cardinal and κ fails, then for every A κ and every n<ω, there is an inner model which contains A and has n Woodin⊆ cardinals. Proof of Theorem 4.2. Suppose that κ is a singular countably closed cardinal and that there is no inner model with a Woodin cardinal. Assume for contradiction <ω that κ fails.Jensenshowedthatκ holds in L[A] whenever A κ (cf. [J1]). Therefore (κ+)L[A] <κ+ whenever A κ. ⊆ ⊆ Lemma 4.2.1. A# exists whenever A κ. ⊆ Sketch. It is clear that Jensen’s covering lemma (cf. [DeJ]) can be adapted to show that A# exists whenever A is a bounded subset of κ. Suppose that A is an unbounded subset of κ and that λ =(κ+)L[A] <κ+. We shall get A# by piecing together lift-ups of sharps for bounded subsets of κ. Let π : N V θ for some large θ with N transitive, countably closed, card(N) < κ,andπcoﬁnal−→ in λ.Sayπ(A)=A,π(κ)=κ,andπ(λ)=λ.LetEbe the length κ extender derived from π.Ifλis not a cardinal of L[A], then as usual we let be the ﬁrst level of L[A]whichseesthis;notethat M

= M (κ pM ) M Hn+1 ∪ n+1 for an appropriate n<ω(ﬁne structure works above κ), set ∗ =ultn( ,E), M M and get that ∗ is a level of L[A] collapsing λ. So λ is a cardinalM of L[A], and thus E measures all subsets of κ in L[A]. Let π∗ : L[A] ult(L[A],E)=L[A] be the lift-up of π.LetDbe the class of strong limit cards−→ of coﬁnality κ+. Note the tuples from D are A-indiscernible with respect to parameters < κ.Also,π∗(α)=αfor α D. By the elementarity of π∗,the ∈

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tuples from D are A-indiscernible with respect to parameters from κ.SinceDis stationary, we are done. Let ~κ be an increasing sequence of cardinals coﬁnal in κ,Ω=κ+,andAbe as in Lemma 3.1.1. Choose Aα α<Ω and λα α<Ω so that h | i h | i ~κ, A L[A ] ∈ 0 and whenever α<β<Ω,

Aα L[Aβ], ∈ (κ+)L[Aα] < (κ+)L[Aβ], and + KAα λα =(κ ) .

Then, for every α<Ω, κ is singular in L[Aα], so, by the weak covering theorem of [MiSch] applied in L[Aα],

+ L[Aα] (κ ) = λα . The rest of the proof is just like that of Theorem 4.1.

5. An Erdos¨ property It has been observed that the existence of sharps for all sets (rather than a measurable cardinal Ω) suffices to develop the theory in [St1], and consequently, to prove a version of the results in this paper. (Cf. [St1, p.58].) Theorem 5.1 below, an improvement to Theorem 3.1, illustrates how an Erdös partition property on Ω consistent with V = L suffices in many core model applications. Let K be the premouse determined by the inductive definition of the core model in [St1, 6]. By [St1], if Ω is a measurable cardinal, then K = K up to Ω, but it is § not knowne if one can show in ZFC that K is a proper class. e Theorem 5.1. Suppose that there is no inner model with a Woodin cardinal and e that κ<Ωare cardinals satisfying the partition relation Ω ( ω ) <ω.Suppose −→ κ that κ is a weakly compact cardinal. Then K isapremouseofheight>κ+ and (κ+)K = κ+. e Off course, Theorem 3.1 is just a special case of Theorem 5.1. The same method of proof can be used to show that the hypothesis Ω ( ω ) <ω −→ κ for all κ<Ω, is more than enough for all applications in [MiSch], [MiSchSt], [Sch1], [Sch2], [St1], and [St3]. In fact, the proof uses only a model theoretic consequence of the partition relation. Proof of Theorem 5.1. Suppose that κ<Ω are cardinals such that κ is weakly compact and Ω ( ω ) <ω. −→ κ Assume that there is no inner model with a Woodin cardinal. We shall adapt an argument of Reinhardt and Silver; see [Jech, pp.394–395]. Let A be the structure V , ,κ expanded to include predicates for Skolem functions. h Ω ∈ i

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Let I be an infinite set of indiscernibles for A such that I (κ, Ω). The existence of I follows from our assumption that Ω ( ω ) <ω in the standard⊂ way. Let −→ κ π : B A −→ be the inverse transitive collapse of the closure of I under the Skolem functions for 1 1 A.Fixsomeα Iand put α = π− (α)andκ=π− (κ). Let j : B B be an elementary embedding∈ obtained by “shifting” α,sothatcrit(j) α−→.Witha judicious choice of I, we can arrange that crit(j) > κ. ≤ Now, using j just as the embedding from a normal measure was used in [St1], one can show that the inductive definition of the core model, as carried out in B, succeeds up to crit(j). (The details are just like those described in [St1, p.58]. Recall that [St1, 1– 6, 9] develops the theory of Kc and K in V uptoameasurable cardinal while working§ § § over a structure V,,U where U is a normal measure on h i the measurable cardinal. Let Uj be the B-ultrafilter on crit(j)givenbyX Uj iff c ∈ crit(j) j(X). Then Uj is weakly amenable to B. Now construct K and K up ∈ to crit(j) over the structure B ,,Uj . The proofs in [St1, 1–6, 9] go through with only obvious changes toh| show| thati the construction succeeds.)§§ § Since π is an elementary embedding, K has height j(crit(j)) >κ+. Arguing as in Theorem 3.1, we see that ≥ B =(κ+)eK =κ+ . | By elementarity again, we conclude that (κ+)K = κ+. f References [DeJ] K.I. Devlin and R.B. Jensen, Marginalia to a theorem of Silver,inLogic Conference, Kiel 1974, Lecture Notes in Mathematics, 499, Springer-Verlag, 1975, 115–142. MR 58:235 [Do1] A.J. Dodd, Strong Cardinals, circulated notes, 1981. [Do2] A.J. Dodd, Thecoremodel, London Mathematical Society Lecture Note Series, 61, Cambridge University Press, Cambridge-New York, 1982. MR 84a:03062 [DoJ] A.J. Dodd and R.B. Jensen, The core model, Ann. Math. Logic 20 (1981), no. 1, 43–75. MR 82i:03063 [Jech] T. Jech, Set theory, Academic Press, New York-London, 1978. MR 80a:03062 [J1] R.B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), no. 3, 229–308. MR 46:8834 [J2] R.B. Jensen, Non overlapping extenders, circulated notes. [J3] R.B. Jensen, Some remarks on below 0¶ , circulated notes. [MaSt1] D.A. Martin and J.R. Steel, A proof of projective determinacy,J.Amer.Math.Soc. 2(1989) 71–125. MR 89m:03042 [MaSt2] D.A. Martin and J.R. Steel, Iteration trees,J.Amer.Math.Soc.7(1994) 1–73. MR 94f:03062 [Mi] W.J. Mitchell, The core model for sequences of measures I, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 2, 229–260. MR 85i:03163 [MiSch] W.J. Mitchell and E. Schimmerling, Weak covering without countable closure, Math. Res. Lett. 2 (1995), no. 5, 595–609. MR 96k:03123 [MiSchSt] W.J. Mitchell, E. Schimmerling, and J.R. Steel, The covering lemma up to a Woodin cardinal, Ann. Pure Appl. Logic 84 (1997), no. 2, 219–255. MR 98b:03067 [MiSt] W.J. Mitchell and J.R. Steel, Fine structure and iteration trees, Lecture Notes in Logic 3, Springer-Verlag, Berlin, 1994. MR 95m:03099 [Sch1] E. Schimmerling, Combinatorial principles in the core model for one Woodin cardinal, Ann. Pure Appl. Logic 74 (1995), no. 2, 153–201. MR 96f:03041 [Sch2] E. Schimmerling, A finite family weak square principle, to appear in J. Symbolic Logic.

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[SchSt] E. Schimmerling and J.R. Steel, Fine structure for tame inner models, J. Symbolic Logic 61 (1996), no. 2, 621–639. MR 97c:03123 [SchW] E. Schimmerling and W.H. Woodin, The Jensen covering property,toappearinJ. Symbolic Logic. [St1] J.R. Steel, The core model iterability problem, Lecture Notes in Logic 8, Springer- Verlag, Berlin, 1996. CMP 98:04 [St2] J.R. Steel, Inner models with many Woodin cardinals, Ann. Pure Appl. Logic 65 (1993), no. 2, 185–209. MR 95c:03132 [St3] J.R. Steel, Core models with more Woodin cardinals,preprint.

Department of Mathematics, University of California, Irvine, Irvine, California 92697-3875 Current address: Mathematical Sciences Department, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 E-mail address: [email protected]

Department of Mathematics, University of California, Berkeley, Berkeley, Cali- fornia 94720 E-mail address: [email protected]

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