Identity Crises and Strong Compactness III: Woodin Cardinals ∗† Arthur W. Apter‡ Department of Mathematics Baruch College of CUNY New York, New York 10010 USA http://faculty.baruch.cuny.edu/apter [email protected] Grigor Sargsyan Group in Logic and the Methodology of Science University of California Berkeley, California 94720 USA http://math.berkeley.edu/∼grigor [email protected] October 25, 2004 (revised February 25, 2005)

Abstract We show that it is consistent, relative to n ∈ ω supercompact cardinals, for the strongly compact and measurable Woodin cardinals to coincide precisely. In particular, it is consistent for the first n strongly compact cardinals to be the first n measurable Woodin cardinals, with no cardinal above the nth strongly compact cardinal being measurable. In addition, we show that it is consistent, relative to a proper class of supercompact cardinals, for the strongly compact cardinals and the cardinals which are both strong cardinals and Woodin cardinals to coincide precisely. We also show how the techniques employed can be used to prove additional theorems about possible relationships between Woodin cardinals and strongly compact cardinals. ∗

2000 Mathematics Subject Classifications: 03E35, 03E55 Keywords: Woodin cardinal, strongly compact cardinal, strong cardinal, supercompact cardinal, non-reflecting stationary set of ordinals ‡ The first author’s research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant. †

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Introduction and Preliminaries

As is well-known, the class of strongly compact cardinals suffers from a severe identity crisis. Readers may consult [22], [4], [19], [3], [6], [7], and [2] for results obtained in this regard. We mention only that none of these articles says anything concerning possible relationships between strongly compact cardinals and Woodin cardinals. In particular, since it is known that the first, second, etc. Woodin cardinals can’t be measurable (and in fact, can’t even be weakly compact — see [17], page 384), it is impossible for the classes of strongly compact and Woodin cardinals to coincide precisely. One may still ask, however, whether there are other possible relationships between strongly compact and Woodin cardinals. In particular, is it possible for the classes of strongly compact and measurable Woodin cardinals to coincide precisely? Is it possible for the class of strongly compact cardinals to coincide precisely with the class of cardinals all of whose members are both strong cardinals and Woodin cardinals? The purpose of this paper is to provide affirmative answers to the above questions. Specifically, we prove the following theorems. Theorem 1 Let n ∈ ω be fixed but arbitrary. Suppose V  “ZFC + κ1 < · · · < κn are supercompact”. There is then a partial ordering P ⊆ V such that V P  “ZFC + κ1 , . . . , κn are both the first n measurable Woodin and strongly compact cardinals + The strongly compact and measurable Woodin cardinals coincide precisely”. Theorem 2 Let V  “ZFC + There is a proper class of supercompact cardinals”. There is then a partial ordering P ⊆ V such that V P  “ZFC + GCH + There is a proper class of strongly compact cardinals + No strongly compact cardinal κ is 2κ = κ+ supercompact + For all cardinals κ, κ is strongly compact iff κ is both a strong cardinal and a Woodin cardinal”. In fact, our methods are general enough that they will yield further results concerning the possible relationships between strongly compact and Woodin cardinals. We will return to this issue at the end of Section 2.

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We conclude Section 1 with some preliminary information. We mention that our notation and terminology are standard. Exceptions to this will be duly noted. For α < β ordinals, [α, β], [α, β), (α, β], and (α, β) are as in standard interval notation. When forcing, q ≥ p will mean that q is stronger than p. If G is V -generic over P, we will abuse notation somewhat and use both V [G] and V P to indicate the universe obtained by forcing with ˙ α i : α < κi is an iteration of length κ so P. If we also have that κ is inaccessible and P = hhPα , Q that at stage α, a nontrivial forcing is done based on the ordinal δα , then we will say that δα is in the field of P. If x ∈ V [G], then x˙ will be a term in V for x, and iG (x) ˙ will be the interpretation of x˙ using G. We may, from time to time, confuse terms with the sets they denote and write x when we actually mean x˙ or xˇ, especially when x is some variant of the generic set G, or x is in the ground model V . If κ is a regular cardinal, Add(κ, 1) is the standard partial ordering for adding a single Cohen subset of κ. If P is an arbitrary partial ordering, P is κ-distributive if for every sequence hDα : α < κi T of dense open subsets of P, α<κ Dα is dense open. P is κ-directed closed if for every cardinal δ < κ and every directed set hpα : α < δi of elements of P (where hpα : α < δi is directed if every two elements pρ and pν have a common upper bound of the form pσ ) there is an upper bound p ∈ P. P is κ-strategically closed if in the two person game in which the players construct an increasing sequence hpα : α ≤ κi, where player I plays odd stages and player II plays even and limit stages (choosing the trivial condition at stage 0), then player II has a strategy which ensures the game can always be continued. Note that if P is κ+ -directed closed, then P is κ-strategically closed. Also, if P is κ-strategically closed and f : κ → V is a function in V P , then f ∈ V . P is ≺κ-strategically closed if in the two person game in which the players construct an increasing sequence hpα : α < κi, where player I plays odd stages and player II plays even and limit stages (again choosing the trivial condition at stage 0), then player II has a strategy which ensures the game can always be continued. Suppose κ < λ are regular cardinals. A partial ordering Q that will be used in this paper is the partial ordering for adding a non-reflecting stationary set of ordinals of cofinality κ to λ. Specifically, Q = {s : s is a bounded subset of λ consisting of ordinals of cofinality κ so that for

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every α < λ, s ∩ α is non-stationary in α}, ordered by end-extension. Two things which can be shown (see [8]) are that Q is δ-strategically closed for every δ < λ, and if G is V -generic over Q, S in V [G], a non-reflecting stationary set S = S[G] = {Sp : p ∈ G} ⊆ λ of ordinals of cofinality κ has been introduced. It is also virtually immediate that Q is κ-directed closed. We mention that we are assuming familiarity with the large cardinal notions of measurability, strongness, strong compactness, and supercompactness. Interested readers may consult [18] or [23] for further details. Also, unlike [18], we will say that the cardinal κ is λ strong for λ > κ if there is j : V → M an elementary embedding having critical point κ so that j(κ) > |Vλ | and Vλ ⊆ M . As always, κ is strong if κ is λ strong for every λ > κ. Since the notion of Woodin cardinal is perhaps not quite as familiar as the large cardinals mentioned in the preceding paragraph, we finish this section with the definition of this concept and a brief discussion of some of the relevant facts. We refer readers to Section 26 of [18], pages 360–365 for additional details. The cardinal κ is Woodin iff for any f : κ → κ, there is an α < κ with f 00 α ⊆ α such that there is an elementary embedding j : V → M having critical point α with the additional property that Vj(f )(α) ⊆ M . By Exercise 26.10 of [18], κ must be a regular cardinal (and in fact must actually be a Mahlo cardinal). Also, by Theorem 26.14 of [18] (due to Woodin), the elementary embedding j in the above definition can be assumed to be witnessed by an extender E ∈ Vκ having the additional property that j(f )(α) = f (α). This has as an immediate consequence that the Woodinness of κ is witnessed in any model containing the true Vκ+1 , so Theorem 26.14 of [18] actually implies that if κ is 2κ supercompact, κ is a measurable Woodin cardinal and {δ < κ : δ is both a measurable and a Woodin cardinal} is unbounded in κ. However, by Lemma 2.1 of [7] and the succeeding remarks, if V  “κ is a strong cardinal” and j : V → M witnesses (at least) the 2κ supercompactness of κ, M  “κ is a strong cardinal”. This means that if κ is supercompact, {δ < κ : δ is both a strong and a Woodin cardinal} is unbounded in κ. Therefore, if κ is supercompact, by increasing the strength of the embedding j to witness larger degrees of supercompactness, the same sort of reflection argument actually shows that {δ < κ : δ is a cardinal which is measurable, Woodin, and δ +ω strong}, {δ < κ : δ is a cardinal which is δ +4

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supercompact and δ +17 strong}, etc., is unbounded in κ.

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The Proof of Theorem 1

We begin with a discussion of the proof of Theorem 1 when n = 1. We will treat this as a special case of the more general situation of arbitrary finite n. As such, we will actually be proving the following theorem when n = 1. Theorem 3 Let V  “ZFC + κ is supercompact”. There is then a partial ordering P ⊆ V such that V P  “ZFC + κ is both the least strongly compact and least measurable Woodin cardinal”. We will give two proofs of Theorem 3, one which is valid if the large cardinal structure of both our ground model and generic extension is restricted, and one which is valid regardless of the large cardinal structure of either our ground model or generic extension. Each of these proofs will have its advantages and disadvantages. The advantage of the proof to be given when there is a limited number of large cardinals is that it can be iterated and will yield a proof of Theorem 1 for arbitrary n ∈ ω. The disadvantage is that we must deal with a severely limited large cardinal structure. The proof that works regardless of the large cardinal structure of either our ground model or generic extension unfortunately can’t be iterated, and thus has this as its major disadvantage. Keeping this in mind, we begin with the proof when there is only one supercompact cardinal in our ground model with no measurable cardinals above it. Towards this end, let V  “ZFC + κ is supercompact + No cardinal λ > κ is measurable”. Without loss of generality, by doing a preliminary forcing if necessary, we assume in addition that V  GCH. We present now our partial ordering P, which will be a reverse Easton iteration of length κ that begins with P0 = Add(ω, 1). Let δ−1 = ω, and assume next that Pα for 0 ≤ α < κ has been defined. Take as an inductive hypothesis that |Pα | < κ. Let δα < κ be the least cardinal greater than sup({δβ : β < α}) such that Pα “δα is a measurable Woodin cardinal”. Note that δα always exists, since as we mentioned earlier, any cardinal δ which is 2δ supercompact has unboundedly many in δ measurable Woodin cardinals below it, which means by our inductive hypothesis and the results of [21] and [16] that Pα “There are unboundedly many in κ measurable Woodin cardinals 5

˙ α , where Pα “Q ˙ α adds a non-reflecting stationary below κ”. We may now define Pα+1 = Pα ∗ Q set of ordinals of cofinality ω to δα ”. It is easily verified that this definition preserves the inductive hypothesis. Lemma 2.1 V P  “No cardinal δ < κ is a measurable Woodin cardinal”. Proof: Suppose to the contrary that δ < κ is such that V P  “δ is a measurable Woodin cardinal”. Write P = Pα ∗ P˙ α , where Pα is the portion of P which adds non-reflecting stationary sets of ordinals of cofinality ω to cardinals at or below δ. By the definition of P, it must be the case that Pα “P˙ α is 2δ -strategically closed”, from which we may infer that Pα “δ is a measurable Woodin cardinal”. Since any cardinal containing a non-reflecting stationary set of ordinals of cofinality ω can’t be measurable (in fact, such a cardinal can’t even be weakly compact), it must be the case that Pα doesn’t add a non-reflecting stationary set of ordinals of cofinality ω to δ. Therefore, by the definition of Pα , it must be true that δ = δα , from which we may immediately infer that Pα does indeed add a non-reflecting stationary set of ordinals of cofinality ω to δ. This contradiction completes the proof of Lemma 2.1. 

Lemma 2.2 V P  “No cardinal δ < κ is strongly compact”. Proof: If V P  “Unboundedly in κ many cardinals δ < κ contain non-reflecting stationary sets of ordinals of cofinality ω”, then we may infer that V P  “No cardinal δ < κ is strongly compact”. This is since by Theorem 4.8 of [23] and the succeeding remarks, if δ contains a non-reflecting stationary set of ordinals of cofinality η < δ, then no cardinal in the half-open interval (η, δ] is δ strongly compact. Thus, to prove Lemma 2.2, it suffices to show that V P  “Unboundedly in κ many cardinals δ < κ contain non-reflecting stationary sets of ordinals of cofinality ω”. To demonstrate this last fact, fix a cardinal δ < κ. As in Lemma 2.1, write P = Pα ∗ P˙ α , where Pα is the portion of P which adds non-reflecting stationary sets of ordinals of cofinality ω at or below δ. Again as in Lemma 2.1, it is the case that Pα “P˙ α is 2δ -strategically closed”, which 6

˙α

means that V P = V Pα ∗P  “δ contains a non-reflecting stationary set of ordinals of cofinality ω” iff V Pα  “δ contains a non-reflecting stationary set of ordinals of cofinality ω”. This means that the proof of Lemma 2.2 will be complete once we have shown that P adds a non-reflecting stationary set of ordinals of cofinality ω to unboundedly in κ many cardinals δ < κ. However, for any β < κ, by the definition of P, |Pβ | < κ. Hence, by the results of [21] and [16], since in V there are unboundedly in κ many measurable Woodin cardinals below κ, Pβ “There are unboundedly in κ many measurable Woodin cardinals in the open interval (β, κ)”. This means by the definition of P that for any β < κ, there is a cardinal at or above β to which P adds a non-reflecting stationary set of ordinals of cofinality ω. This completes the proof of Lemma 2.2.  We note that the arguments given in Lemmas 2.1 and 2.2 show that if the cofinality of the ordinals in the non-reflecting stationary set of ordinals added by P is changed to some uncountable δ > ω, then V P  “No cardinal in the open interval (δ, κ) is either a measurable Woodin cardinal or is strongly compact”. This observation will be key in the proof of Theorem 1 for arbitrary finite n. Lemma 2.3 V P  “κ is a measurable Woodin cardinal”. Proof: Let j : V → M be an elementary embedding witnessing the 2κ = κ+ supercompactness of ˙ where the first ordinal in the field of Q ˙ is at or above κ. κ. This allows us to write j(P) = P ∗ Q, κ

If M P  “κ is a measurable Woodin cardinal”, then since M 2 ⊆ M , it is also true that V P  “κ is a measurable Woodin cardinal”, and the proof of Lemma 2.3 is complete. We suppose therefore ˙ is above κ++ . that this is not the case. This means that the first ordinal in the field of Q Let G be V -generic over P. Using GCH, standard arguments, as given, e.g., in the construction of the generic object G1 in the proof of Lemma 2.4 of [7], allow us to build in V [G] an M [G]-generic object H for Q. We may hence lift j in V [G] to j : V [G] → M [G][H], which means that κ is 2κ supercompact in V [G] (and consequently is also a measurable Woodin cardinal in V [G]). This completes the proof of Lemma 2.3.  7

Lemma 2.4 V P  “κ is strongly compact”. Proof: Let λ ≥ 2κ be a fixed but arbitrary regular cardinal, and let j : V → M be an elementary embedding witnessing the λ supercompactness of κ generated by a supercompact ultrafilter over Pκ (λ) such that M  “κ isn’t λ supercompact”. By the choice of λ, there is U ∈ M a normal measure over κ and k : M → N an elementary embedding generated via the ultrapower by U such that N  “κ isn’t measurable”. ˙ ∗ R˙ and j(P) = P ∗ S˙ ∗ T, ˙ where the field of Q ˙ is a subset of the Let i = k ◦ j. Write i(P) = P ∗ Q closed interval [κ, k(κ)], the field of R˙ is composed of ordinals in the open interval (k(κ), k(j(κ)), S˙ is a term for the forcing done at stage κ in M , and T˙ is a term for the forcing done between stages κ and j(κ) in M . By Lemma 2.3, in V , P “κ is a measurable Woodin cardinal”. Therefore, since M λ ⊆ M and λ ≥ 2κ , it is the case that in M , P “κ is a measurable Woodin cardinal” as well. This means that S˙ is a term for the partial ordering adding a non-reflecting stationary set of ordinals of cofinality ω to κ. Further, since V  “No cardinal above κ is measurable”, M  “No ˙ = κ, the results of [21] and cardinal in the half-open interval (κ, λ] is measurable”. Hence, as |P ∗ S| ˙

[16] imply that no cardinal in the half-open interval (κ, λ] is both measurable and Woodin in M P∗S . We may thus infer that T˙ is a term for a partial ordering that does only trivial forcing for cardinals in the half-open interval (κ, λ], since the first cardinal in the field of T˙ to which a non-reflecting ˙

stationary set of ordinals of cofinality ω is added must be both measurable and Woodin in M P∗S . Work now in N . By the definition of P, write P = P0 ∗ P˙ 00 , where |P0 | = ω, P0 is nontrivial, and P0 “P˙ 00 is ℵ1 -strategically closed”. In the terminology of [13] and [14], P “admits a gap at ℵ1 ”, so by Hamkins’ gap forcing results of [13] and [14], any cardinal measurable in N P had to have been measurable in N . Thus, since N  “κ isn’t measurable”, N P  “κ isn’t measurable” as well. This means that in N P , the partial ordering Q does not add a non-reflecting stationary set of ordinals of cofinality ω to κ. However, since by elementarity and the fact that in M , P “κ is a measurable Woodin cardinal”, in N , k(P) “k(κ) is a measurable Woodin cardinal”. Because in N , ˙ is actually composed of ordinals in the half-open k(P) = (i(P))k(κ) , this means that the field of Q interval (κ, k(κ)]. 8

We can now use the argument given in Lemma 4 of [6], Lemma 2.4 of [7], and Lemma 2.3 of [2] (originally due to Magidor but unpublished by him) to show that V P  “κ is λ strongly compact”. For a complete proof, we refer readers of this paper to one of the aforementioned articles. However, for completeness and comprehensibility, we provide an outline of the method of reasoning used. Let G0 be V -generic over P. Since N  “κ isn’t measurable”, N is an ultrapower of M via a normal measure over κ, and GCH holds at κ in V , it is possible once again to use the standard diagonalization techniques employed in the proof of Lemma 2.4 of [7] in the construction of the generic object G1 to build in V [G0 ] an N [G0 ]-generic object G1 for Q. Since GCH holds at λ in V ˙ we can once again use the standard diagonalization and no cardinal δ ∈ (κ, λ] is in the field of T, techniques to construct in V [G0 ] an M -generic object H for the term forcing partial ordering T∗ ˙ transfer it using k to an N -generic object associated with T˙ (which is defined with respect to P ∗ S), G∗2 for k(T∗ ) generated by k 00 H, and realize the transferred generic object using G0 ∗ G1 to obtain an N [G0 ][G1 ]-generic object G2 for R. i then lifts to i : V [G0 ] → N [G0 ][G1 ][G2 ], which witnesses the λ strong compactness of κ in V [G0 ]. Since λ was arbitrary, this completes the proof of Lemma 2.4.  Lemmas 2.1 - 2.4 complete the proof of Theorem 3 in the case when there is only one supercompact cardinal with no measurable cardinals above it.1  We continue with our discussion of the proof of Theorem 3 when n = 1. We turn now to the proof of this theorem in a universe in which the large cardinal structure can be arbitrary. Towards this end, let V  “ZFC + κ is supercompact”. We define our partial ordering P, which will be a Magidor style iteration [22] of length κ of Prikry forcing that begins with P0 as trivial forcing. As before, let δ−1 = ω. Assume next that Pα for 0 ≤ α < κ has been defined. Take as an inductive hypothesis that |Pα | < κ. Let δα < κ be the 1

We note that under the circumstances just described, Theorem 1 and Theorem 3 actually mean the same thing. This is since by the results of [21] and [16], V P  “No cardinal λ > κ is both measurable and Woodin”.

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least cardinal greater than sup({δβ : β < α}) such that Pα “δα is a measurable Woodin cardinal”. ˙ α , where Pα “Q ˙ α is Prikry As we had earlier, δα always exists. We may now define Pα+1 = Pα ∗ Q forcing over δα defined with respect to some normal measure µα over δα ”. Once again, it is easily verified that this definition preserves the inductive hypothesis. Lemma 2.5 V P  “No cardinal δ < κ is a measurable Woodin cardinal”. Proof: We mimic the proof of Lemma 2.1. Suppose to the contrary that δ < κ is such that V P  “δ is a measurable Woodin cardinal”. Write P = Pα ∗ P˙ α , where Pα is the portion of P which adds Prikry sequences to cardinals at or below δ. By the definition of P, it must be the case that

Pα “P˙ α doesn’t add any new subsets of 2δ ”, from which we may infer that Pα “δ is a measurable Woodin cardinal”. Since any cardinal containing a Prikry sequence must have cofinality ω and hence can’t be either measurable or Woodin, it must be the case that Pα doesn’t add a Prikry sequence to δ. Therefore, by the definition of Pα , it must be true that δ = δα , from which we may immediately infer that Pα does indeed add a Prikry sequence to δ. This contradiction completes the proof of Lemma 2.5. 

Lemma 2.6 V P  “No cardinal δ < κ is strongly compact”. Proof: As in the proof of Lemma 2.2, there are unboundedly in κ many cardinals δ < κ which are in the field of P, which in this case means that in V P , there are unboundedly in κ many cardinals δ < κ which contain Prikry sequences. However, since by Theorem 11.1 of [9], adding a Prikry sequence also adds a non-reflecting stationary set of ordinals of cofinality ω, again as in the proof of Lemma 2.2, in V P , there are no strongly compact cardinals below κ. This completes the proof of Lemma 2.6. 

Lemma 2.7 V P  “κ is a measurable Woodin cardinal”. 10

Proof: We begin by mimicing the proof of Lemma 2.3. Let j : V → M be an elementary ˙ where embedding witnessing the 2κ supercompactness of κ. This allows us to write j(P) = P ∗ Q, ˙ is at or above κ. the first ordinal in the field of Q κ

If M P  “κ is a measurable Woodin cardinal”, then since M 2 ⊆ M , it is also true that V P  “κ is a measurable Woodin cardinal”, and the proof of Lemma 2.7 is complete. We suppose therefore ˙ is above 2κ . that this is not the case. This means that the first ordinal in the field of Q Let now |

˙ iff | be the distance function of [22]. Define a term U˙ in V by p “A˙ ∈ U” P

p “A˙ ⊆ (Pκ (2κ ))V ” and there is q ∈ j(P) such that q ≥ j(p), |j(p) − q| = 0, j(p)  κ = q  κ = p, ˙ and q “hj(α) : α < 2κ i ∈ j(A)”. By the same proof as in the Lemma of [1], U˙ is a well-defined term for a supercompact ultrafilter over Pκ (2κ ) in V P . Thus, since V P  “κ is 2κ supercompact”, V P  “κ is a measurable Woodin cardinal”. This completes the proof of Lemma 2.7.  Magidor’s methods of [22] show that V P  “κ is strongly compact”. (This is since the work of [22] demonstrates that if any set of measurable cardinals below a strongly compact cardinal λ is destroyed via Magidor’s notion of iterated Prikry forcing, then the strong compactness of λ is preserved.) This observation, together with Lemmas 2.5 - 2.7, complete the proof of Theorem 3 when n = 1 and the large cardinal structure of both the ground model and the generic extension is arbitrary.  We note that the proof just presented requires no GCH assumptions on our ground model. This is in contrast to the earlier proof of Theorem 3 given for a universe containing a restricted number of large cardinals, which requires instances of GCH in order to show that strong compactness is preserved. We are now ready to give the proof of Theorem 1 for arbitrary finite n. Towards this end, fix n ∈ ω, n > 1. Suppose that V  “ZFC + κ1 < · · · < κn are supercompact + No cardinal λ > κn is measurable”. Without loss of generality, as in the proof given in [6] for arbitrary finite n of the Theorem 1 of that paper, we also assume that V  “Each κi for i = 1, . . . , n is Laver indestructible 11

[20] + 2κi = κ+ i for i = 1, . . . , n”. Take κ0 = ω. For i = 1, . . . , n, we define a partial ordering Pi in a manner analogous to the definition of the partial ordering P given at the beginning of this section. Pi will be a reverse Easton iteration of length κi that begins with Pi,0 = Add(κi−1 , 1) and δi,−1 = κi−1 . Then, as earlier, assume that Pi,α for 0 ≤ α < κi has been defined. Take as an inductive hypothesis that |Pi,α | < κi . Let δi,α < κi be the least cardinal greater than sup({δi,β : β < α}) such that Pi,α “δi,α is a measurable ˙ i,α , where Woodin cardinal”. As before, δi,α always exists. We may now define Pi,α+1 = Pi,α ∗ Q ˙ i,α adds a non-reflecting stationary set of ordinals of cofinality κi−1 to δi,α ”. It is easily

Pi,α “Q verified that this definition preserves the inductive hypothesis. Our partial ordering P used in the proof of Theorem 1 for arbitrary finite n is then the product partial ordering P1 × · · · × Pn . We do now a “downwards induction” to show that V P is as desired. By the arguments given in the proofs of Lemmas 2.1 - 2.4, which remain valid regardless of the cofinality of the ordinals in the non-reflecting stationary sets added by the appropriate forcing, V Pn  “κn is both a strongly compact and a Woodin cardinal, and no cardinal in the open interval (κn−1 , κn ) is either strongly compact or both measurable and Woodin”. Since by its definition, |P1 × · · · × Pn−1 | < κn , by the results of [21] and [16], these facts remain true in V Pn ×Pn−1 ×···×P1 = V P . Suppose now by induction that 1 ≤ i < n and that V Pn ×···×Pi+1  “κi+1 , . . . , κn are both strongly compact and Woodin cardinals, and no cardinal in the open intervals (κi , κi+1 ), . . . , (κn−1 , κn ) is either strongly compact or both measurable and Woodin”. Since by its definition, Pn × · · · × Pi+1 is κi -directed closed, by indestructibility, V Pn ×···×Pi+1  “κi is supercompact”. Also, the fact Pn × · · · × Pi+1 is κi -directed closed implies that V and V Pn ×···×Pi+1 have the same bounded subsets of κi . However, the definition of Pi is such that it retains in V Pn ×···×Pi+1 the forcing properties it had in V . This means that the arguments of Lemmas 2.1 - 2.3 remain valid and show that V Pn ×···×Pi+1 ×Pi  “κi is a measurable Woodin cardinal, and no cardinal in the open interval (κi−1 , κi ) is either strongly compact or both measurable and Woodin”. Further, for any j such that 1 ≤ j ≤ n, ˙ since Pj,0 adds a Cohen subset of κj−1 , P “Q ˙ is κ+ -strategically by writing Pj = Pj,0 ∗ Q, j,0 j−1 closed”, and |Pj | = κj , it is easy to see that forcing with Pj preserves GCH at both κj−1 and

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κj . It is then inductively the case that V Pn ×···×Pi+1  “GCH holds at κi+1 ”, so since in addition, V Pn ×···×Pi+1  “There are no measurable Woodin cardinals in the open interval (κi , κi+1 )”, as above, the argument given in Lemma 2.4 remains valid and shows that V Pn ×···×Pi+1 ×Pi  “κi is κi+1 strongly compact”.2 Therefore, since |Pi | < κi+1 in both V and V Pn ×···×Pi+1 , by the results of [21], V Pn ×···×Pi+1 ×Pi  “κi+1 is strongly compact”. Hence, by a theorem of DiPrisco [10], as V Pn ×···×Pi+1 ×Pi  “κi is κi+1 strongly compact and κi+1 is strongly compact”, V Pn ×···×Pi+1 ×Pi  “κi is strongly compact”. Thus, since |Pi−1 × · · · × P1 | < κi in both V and V Pn ×···×Pi+1 ×Pi , the results of [21] and [16] allow us to infer that V Pn ×···×Pi+1 ×Pi ×Pi−1 ×···×P1 = V P  “κi , . . . , κn are both strongly compact and Woodin cardinals, and no cardinal in the open intervals (κi−1 , κi ), . . . , (κn−1 , κn ) is either strongly compact or both measurable and Woodin”. From this, we may now infer that V P  “κ1 , . . . , κn are both the first n strongly compact and measurable Woodin cardinals”. Since |P| = κn and V  “No cardinal λ > κn is measurable”, by the results of [21], V P  “No cardinal λ > κn is measurable” as well. We therefore now know that V P  “The strongly compact and measurable Woodin cardinals coincide precisely”. This completes our discussion of the proof of Theorem 1 for arbitrary finite n.  We conclude Section 2 by making several remarks. The first is that the techniques of this section are quite general and allow for a diverse number of different theorems in the spirit of Theorems 1 and 3. For instance, if we wished to prove versions of Theorems 1 and 3 with the property “measurable Woodin cardinal” replaced with the property “measurable Woodin cardinal κ which is also κ+ω strong”3 , then this is easily accomplished by a slight modification of the definitions of our partial 2 To expand on this further, suppose that j : V Pn ×···×Pi+1 → M is in analogy to Lemma 2.4, i.e., j is an elementary embedding witnessing the κi+1 supercompactness of κi such that M  “κi isn’t κi+1 supercompact”. As before, suppose that k : M → N is an elementary embedding generated by a normal measure over κi such that N  “κi isn’t measurable”. The arguments of Lemma 2.4 then proceed almost virtually without change. We do explicitly note, however, that the argument given in Lemma 2.4 requires that for the version of T˙ appropriate to the current ˙ This is true if in M , there situation, no ordinal in the half-open interval (κi , κi+1 ] is an element of the field of T. are no measurable Woodin cardinals in the half-open interval (κi , κi+1 ]. By the κi+1 closure of M , there are no measurable Woodin cardinals in M in the open interval (κi , κi+1 ). However, since M  “κi isn’t κi+1 supercompact”, M  “κi+1 isn’t measurable”. This is since otherwise, M  “κi is δ supercompact for every δ < κi+1 and κi+1 is measurable”, so M  “κi is κi+1 supercompact”, a contradiction. Thus, in M , there are no measurable Woodin cardinals in the half-open interval (κi , κi+1 ]. 3 To be more explicit, we wish as before to construct three models. In the first two, the least strongly compact

13

orderings. Instead of destroying cardinals which are forced to be both measurable and Woodin, we destroy cardinals δ which are forced to be Woodin and δ +ω strong. Simple reworkings of our arguments then yield the desired results. In particular, when proving the appropriate analogues of Lemmas 2.3 and 2.7, we choose supercompactness embeddings into transitive inner models M +ω

which are at least 2κ

closed, and then essentially follow almost verbatim the remainder of the

arguments presented in each of the relevant lemmas. What is key to this analysis is that the property we have given in the preceding paragraph can be determined “locally” in two senses. One is that if the property is forced by the partial ordering to hold at a cardinal below the length of the iteration, then an initial segment of the partial ordering forces this property to hold as well (which is critical to the proofs of Lemmas 2.1 and 2.5). The other is that if the particular property is witnessed in a sufficiently closed transitive inner model M , then it must be true in V as well (which is critical to the proofs of Lemmas 2.3 and 2.7). Since this sort of “localness” is witnessed by many other potential properties (another example would be given by a cardinal δ being both δ +4 supercompact and δ +17 strong), there are a number of different versions of Theorems 1 and 3 which consequently can be proven. Another interesting aspect of the methods of this section is that they can be used when Hamkins’ gap forcing results of [13] and [14] (and the generalizations thereof given in [12]) fail. For instance, since Hamkins’ techniques of [13], [14], and [12] do not tell us that a gap forcing as defined in any of these papers creates no new cardinals δ which are δ +ω strong, we are forced to rely on the ideas used in the proof of Lemma 2.1 if we want to change the property from “measurable Woodin cardinal” to something on the order of “measurable Woodin cardinal δ which is also δ +ω strong”. This is critical in showing that all cardinals δ below the length of the iteration which are measurable, Woodin, and δ +ω strong are destroyed. (Since Hamkins’ gap forcing results do not hold for Prikry iterations, we must use the ideas of this paper, specifically those given in Lemma 2.5, when proving either Theorem 3 as originally stated or any generalizations thereof when there are no restrictions on the large cardinal structure of the universe.) We will comment on this further cardinal κ is also the least cardinal which has the additional properties of being measurable, Woodin, and κ+ω strong. In the last model, for some n ∈ ω, there are exactly n strongly compact cardinals, which are precisely those cardinals κ which are also measurable, Woodin, and κ+ω strong.

14

towards the end of the paper. Finally, for the same reasons as given in [3] and [6], we are restricted to proving Theorem 1 only for finite values of n. A more detailed discussion of why this is so may be found at the end of [3].

3

The Proof of Theorem 2

We turn now to the proof of Theorem 2. Proof: Let V  “ZFC + There is a proper class of supercompact cardinals”. Take K as the class of supercompact cardinals in V . As in [7], by “cutting off” the universe if necessary, we assume in addition without loss of generality that there are no inaccessible limits of supercompact cardinals. Further, as in [7], we also assume that for R = Add(ω, 1) ∗ R˙ 0 , V has been generically extended to a model V1 = V R in which K is the class of supercompact cardinals4 such that V1  “GCH + Every supercompact cardinal κ has its supercompactness indestructible under κ-directed closed set or class partial orderings preserving GCH”. Work now in V1 . For each supercompact cardinal κ, let γκ be the successor of the supremum of the supercompact cardinals below κ (with γκ0 = ω for κ0 the least supercompact cardinal). By our assumptions on V and their corresponding implications for V1 , γκ < κ. This allows us to define the partial ordering Pκ as the reverse Easton iteration of length κ which begins by adding a Cohen subset of γκ and then adds a non-reflecting stationary set of ordinals of cofinality γκ to each cardinal in the open interval (γκ , κ) which is in V (the original ground model) both a strong cardinal and a Woodin cardinal. The partial ordering P which is used to construct the model Q witnessing the conclusions of Theorem 2 is then defined as the Easton support product κ∈K Pκ . Standard arguments show that V1P  “ZFC + GCH”. We establish now some basic properties of each partial ordering Pκ . We take as definitions that Q Q P>κ = λ∈K,λ>κ Pλ and P<κ = λ∈K,λ<κ Pλ , and as a notational convention that in Lemmas 3.1 4

The fact that K remains the class of supercompact cardinals in V1 follows from the gap forcing results of [13] and [14] and the fact that all V -supercompact cardinals remain supercompact in V1 . This is in direct analogy to the argument given in [7].

15

3.4 that follow, κ always denotes a V - or V1 -supercompact cardinal. Note that once again, standard arguments show that for any κ ∈ K, V1P

>κ

 “ZFC + GCH” and V1P

>κ ×Pκ

 “ZFC + GCH”.

Lemma 3.1 V1P  “No cardinal in the open interval (γκ , κ) is both a strong cardinal and a Woodin cardinal”. Proof: Write R ∗ P˙ = R ∗ (P˙ >κ × P˙ κ × P˙ <κ ) in the form Add(ω, 1) ∗ (R˙ 0 ∗ (P˙ >κ × P˙ κ × P˙ <κ )) = S0 ∗ S˙ 1 . Since |S0 | = ω, S0 is nontrivial, and S0 “S˙ 1 is ℵ1 -strategically closed”, by the results of [13] and ˙

[14], any cardinal in V S0 ∗S1 = V1P which is both a strong cardinal and a Woodin cardinal had to have been both a strong cardinal and a Woodin cardinal in V . Further, since by its definition, P>κ >κ

is κ-directed closed in V1 , the partial ordering Pκ retains the same forcing properties in V1P >κ ×Pκ

had in V1 . This means that in V1P

as it

, no cardinal lying in the open interval (γκ , κ) which was

in V both a strong cardinal and a Woodin cardinal remains both a strong cardinal and a Woodin cardinal, since each such cardinal contains a non-reflecting stationary set of ordinals of cofinality γκ and hence is no longer weakly compact. As |P<κ | < γκ in both V1 and V1P

>κ ×Pκ

, by the results

of [16], no cardinal lying in the open interval (γκ , κ) which was in V both a strong cardinal and >κ ×Pκ ×P<κ

a Woodin cardinal remains both a strong cardinal and a Woodin cardinal in V1P

= V1P .

Thus, there can be no cardinals in the open interval (γκ , κ) which are in V1P both a strong cardinal and a Woodin cardinal. This completes the proof of Lemma 3.1. 

Lemma 3.2 V1P  “κ isn’t 2κ = κ+ supercompact”. Proof: By Lemma 3.1, in V1P , no cardinal in the open interval (γκ , κ) is both a strong cardinal and a Woodin cardinal. However, if κ were 2κ supercompact, then there would have to be unboundedly many in κ cardinals below κ which are both Woodin cardinals and strong cardinals. This completes the proof of Lemma 3.2. 

Lemma 3.3 V1P  “No cardinal in the open interval (γκ , κ) is strongly compact”. 16

>κ

Proof: As we remarked in the proof of Lemma 3.1, Pκ retains in V1P

the forcing properties it

had in V1 . Further, since κ is supercompact in V , there are unboundedly many in κ cardinals in the open interval (γκ , κ) which are both strong cardinals and Woodin cardinals in V . Therefore, >κ ×Pκ

by the definition of Pκ , V1P

 “Unboundedly many in κ cardinals in the open interval (γκ , κ)

contain non-reflecting stationary sets of ordinals of cofinality γκ ”. Consequently, as in the proof >κ ×Pκ

of Lemma 2.2 and the succeeding remarks, V1P

 “No cardinal in the open interval (γκ , κ) is

strongly compact”. Since |P<κ | < γκ , by the results of [21], V1P

>κ ×Pκ ×P<κ

= V1P  “No cardinal in

the open interval (γκ , κ) is strongly compact”. This completes the proof of Lemma 3.3.  Lemma 3.4 V1P  “κ is a Woodin cardinal”. Proof: Since P>κ is κ-directed closed in V1 and preserves GCH, by the indestructibility of κ in V1 >κ

under κ-directed closed partial orderings preserving GCH, V1P

 “κ is supercompact and hence

is a Woodin cardinal”. Consequently, to establish Lemma 3.4, we show that V1P Woodin cardinal”. This will suffice, since |P<κ | < κ in both V1 and V1P >κ ×Pκ

[16], if V1P

 “κ is a Woodin cardinal”, then V1P

>κ ×Pκ ×P<κ

>κ ×Pκ

>κ ×Pκ

 “κ is a

, so by the results of

= V1P  “κ is a Woodin cardinal”

as well. Towards this end, we begin by defining for notational simplicity V2 = V1P

>κ

and Q = Pκ . Take

V2 as our ground model. Let f1 ∈ V2Q be a function such that f1 : κ → κ. As Q retains in V2 the same forcing properties it had in V1 , Q is in V2 a reverse Easton iteration of length κ in V2 and hence is κ-c.c. This means the function f2 : κ → κ defined in V2 as f2 (α) = max(α, sup({β : ∃p ∈ Q[p “f1 (α) = β”]})) + 1 is well-defined and is such that for all α < κ, V2Q  “f1 (α) < f2 (α)”. Let f : κ → κ be given by f (α) = β where β is the least V2 -strong cardinal above f2 (α). Note that β is not a Woodin cardinal, since if it were, then by Proposition 26.13 of [18], A = {γ < β : γ is δ strong for every δ < β} is a stationary subset of β. However, by the proof of Lemma 2.1 of [7], every γ ∈ A must be a strong cardinal, since V2  “γ is δ strong for every δ < β and β is a strong cardinal”. This means there must be a V2 -strong cardinal in the open interval (f2 (α), β), which contradicts the fact that β is the least V2 -strong cardinal above f2 (α). 17

Since κ is a Woodin cardinal in V2 , by Theorem 26.14 of [18], we can let λ < κ be the least cardinal such that f 00 λ ⊆ λ and there is an extender E ∈ Vκ and an elementary embedding j : V2 → M generated by E having critical point λ such that Vj(f )(λ) ⊆ M and j(f )(λ) = f (λ). It is then the case that λ is not a Woodin cardinal in V2 . This is because if λ were a Woodin cardinal in V2 , then since f  λ : λ → λ, again by Theorem 26.14 of [18], there is some β < λ such that f 00 β ⊆ β and there is an extender E 0 ∈ Vλ and an elementary embedding i : V2 → N generated by E 0 having critical point β such that Vi(f )(β) ⊆ N and i(f )(β) = f (β). As λ < κ, E 0 ∈ Vκ . This, however, contradicts the minimality of λ. Let j : V2 → M be an elementary embedding generated by an extender E ∈ Vκ having critical point λ such that Vj(f )(λ) ⊆ M and j(f )(λ) = f (λ). Let λ0 = f (λ), and let Q∗ be the portion of Q defined between λ and λ0 . We present now an argument using ideas of Woodin and Gitik-Shelah [11], found in the proofs of Lemma 4.2 of [5] and Theorem 4.10 of [15], which shows that j lifts to ˙∗

˙ ∗)

j : V2Qλ ∗Q → M j(Qλ ∗Q

˙∗ Qλ ∗Q

in a way such that (Vj(f )(λ) )V2

˙∗

⊆ M j(Qλ ∗Q ) .

To begin, since V2  “λ isn’t a Woodin cardinal”, a fact which is absolute between transitive models containing the same Vλ+1 , the facts λ0 > λ and Vλ0 ⊆ M imply that M  “λ isn’t a Woodin ˙ ∗ ) and λ 6∈ field(j(Q)). Further, because cardinal” as well. We thus know that both λ 6∈ field(Q V2  “Every member of the range of f is not a Woodin cardinal”, by elementarity, M  “Every member of the range of j(f ) is not a Woodin cardinal”. In particular, since j(f )(λ) = f (λ) = λ0 , ˙ ∗ ) and λ0 6∈ M  “λ0 isn’t a Woodin cardinal”. We therefore also know that both λ0 6∈ field(Q field(j(Q)). Note now that for any cardinal δ ∈ (λ, λ0 ), V2  “δ is a strong cardinal” iff M  “δ is a strong cardinal”. To see this, suppose first that V2  “δ is a strong cardinal”. Since V2  “Every member of the range of f is a strong cardinal”, by elementarity, M  “Every member of the range of j(f ) is a strong cardinal”. Consequently, as j(f )(λ) = f (λ) = λ0 , M  “λ0 is a strong cardinal”. Thus, as Vλ0 ⊆ M and V2  “δ is a strong cardinal”, M  “δ is γ strong for every γ < λ0 and λ0 is a strong cardinal”, so as before, M  “δ is a strong cardinal”. Then, if M  “δ is a strong cardinal”, since Vλ0 ⊆ M , V2  “δ is γ strong for every γ < λ0 and λ0 is a strong cardinal”, so again, V2  “δ

18

is a strong cardinal”. Hence, as Vλ0 ⊆ M implies that the Woodin cardinals in the open interval (λ, λ0 ) are precisely the same in both V2 and M , we therefore have that the cardinals which are both strong cardinals and Woodin cardinals and which lie in the open interval (λ, λ0 ) are precisely the same in both V2 and M . Putting the work of the preceding two paragraphs together, we can now infer that j(Qλ ) = ˙ ∗ ∗ R, ˙ and that the first ordinal in the field of R˙ is above λ0 . Since we may assume by the Qλ ∗ Q regularity of λ0 that M λ ⊆ M , this means that if G is V2 -generic over Qλ and H is V2 [G]-generic over Q∗ , R is ≺λ+ -strategically closed in both V2 [G][H] and M [G][H], and R is λ0 -strategically closed in M [G][H]. We may assume that M = {j(h)(a) : a ∈ [λ0 ]<ω , h ∈ V2 , and dom(h) = [λ]|a| }. Therefore, as in both [15] and [5], by using a suitable coding which allows us to identify finite subsets of λ0 with elements of λ0 , the definition of M allows us to find some α < λ0 and function g so that ˙ ∗ = j(g)(α). Let N = {iG∗H (z) Q ˙ : z˙ = j(h)(λ, α, λ0 ) for some function h ∈ V2 }. It is easy to verify that N ≺ M [G][H], that N is closed under λ sequences in V2 [G][H], and that λ, α, λ0 , Q∗ , and R are all elements of N . Further, since R is j(λ)-c.c. in M [G][H] and by GCH at λ in V2 , there are only 2λ = λ+ many functions h : [λ]3 → Vλ in V2 , there are at most λ+ many dense open subsets of R in N . Therefore, since R is ≺λ+ -strategically closed in both M [G][H] and V2 [G][H], we can build H 0 in V2 [G][H] as follows. Let hDσ : σ < λ+ i enumerate in V2 [G][H] the dense open subsets of R present in N so that every dense open subset of R occurring in N appears at an odd stage at least once in the enumeration. If σ is an odd ordinal, σ = τ + 1 for some τ . Player I picks pσ ∈ Dσ extending qτ (initially, q0 is the empty condition), and player II responds by picking qσ ≥ pσ according to a fixed strategy S (so qσ ∈ Dσ ). If σ is a limit ordinal, player II uses S to pick qσ extending each q ∈ hqβ : β < σi. By the ≺λ+ -strategic closure of R in V2 [G][H], player II’s strategy can be assumed to be a winning one, so hqσ : σ < λ+ i can be taken as an increasing sequence of conditions with qσ ∈ Dσ for σ < λ+ . Let H 0 = {p ∈ R : ∃σ < λ+ [qσ ≥ p]}. We show now that H 0 is actually M [G][H]-generic ˙ for some name D˙ ∈ M . over R. If D is a dense open subset of R in M [G][H], then D = iG∗H (D)

19

Consequently, D˙ = j(h0 )(λ, λ1 , . . . , λn ) for some fixed function h0 ∈ V2 and λ < λ1 < · · · < λn < λ0 . Let D be a name for the intersection of all iG∗H (j(h0 )(λ, α1 , . . . , αn )), where λ < α1 < · · · < αn < λ0 is such that j(h0 )(λ, α1 , . . . , αn ) yields a name for a dense open subset of R. Since this name can be given in M and R is λ0 -strategically closed in M [G][H] and therefore λ0 -distributive in M [G][H], D is a name for a dense open subset of R which is definable without the parameters λ1 , . . . , λn . Hence, by its definition, iG∗H (D) ∈ N . Thus, since H 0 meets every dense open subset of R present ˙ H 0 ∩ iG∗H (D) ˙ 6= ∅. This means in N , H 0 ∩ iG∗H (D) 6= ∅, so since D is forced to be a subset of D, H 0 is M [G][H]-generic over R, so in V2 [G][H], j lifts to j : V2 [G] → M [G][H][H 0 ]. It remains to lift j through the forcing Q∗ while working in V2 [G][H]. To do this, it suffices to show that j 00 H ⊆ j(Q∗ ) generates an M [G][H][H 0 ]-generic object H 00 over j(Q∗ ). Given a dense ˙ for some name D˙ = j(D)(a), ~ open subset D ⊆ j(Q∗ ), D ∈ M [G][H][H 0 ], D = iG∗H∗H 0 (D) where ~ = hDσ : σ ∈ [λ]|a| i is a function. We may assume that every Dσ is a dense open a ∈ [λ0 ]<ω and D T subset of Q∗ . Since Q∗ is λ-distributive, it follows that D0 = σ∈[λ]|a| Dσ is also a dense open subset of Q∗ . As j(D0 ) ⊆ D and H ∩ D0 6= ∅, j 00 H ∩ D 6= ∅. Thus, H 00 = {p ∈ j(Q∗ ) : ∃q ∈ j 00 H[q ≥ p]} is our desired generic object, and j lifts to j : V2 [G][H] → M [G][H][H 0 ][H 00 ]. Since Vλ0 ⊆ M , ˙∗

˙ ∗)

(Vλ0 )V2 [G][H] ⊆ M [G][H] ⊆ M [G][H][H 0 ][H 00 ]. Therefore, j lifts to j : V2Qλ ∗Q → M j(Qλ ∗Q ˙∗ Q ∗Q V2 λ

such that (Vj(f )(λ) )

in a way

˙∗

⊆ M j(Qλ ∗Q ) .

˙ ∗∗Q ˙ ∗∗ . Since ˙ ∗ “Q ˙ ∗∗ is ζ-strategically closed for ζ the least inaccessible Write Q = Qλ ∗ Q Qλ ∗Q ˙∗

cardinal above λ0 ”, the extender F witnessing the aforementioned lift of j in V2Qλ ∗Q generates ˙ ∗ ∗Q ˙ ∗∗

in V2Qλ ∗Q

= V2Q an elementary embedding j : V2Q → N ∗ having critical point λ such that

Q

(Vj(f )(λ) )V2 ⊆ N ∗ . As for all α < κ, V2Q  “f1 (α) < f2 (α) < f (α)”, in V2Q , f100 λ ⊆ λ, and Q

(Vj(f1 )(λ) )V2 ⊆ N ∗ . Since f1 ∈ V2Q was arbitrary, this means that V2Q  “κ is a Woodin cardinal”. This completes the proof of Lemma 3.4.  Lemmas 3.1 - 3.4, the proof of Theorem 1 of [7] for a proper class of cardinals, and the proofs of Lemmas 2.4 and 2.5 of [7] now allow us to infer that V1P  “Each V1 -supercompact cardinal κ isn’t 2κ = κ+ supercompact but is strongly compact, strong, and Woodin, and there are no strongly 20

compact cardinals or cardinals which are both strong cardinals and Woodin cardinals in the open interval (γκ , κ) if κ is a V1 -supercompact cardinal”. Further, the fact there are no inaccessible limits of V1 -supercompact cardinals tells us that any cardinal (other than a V1 -supercompact cardinal) which is in V1P either strongly compact or both a strong cardinal and a Woodin cardinal would have to lie in an open interval of the form (γκ , κ) for some V1 -supercompact cardinal κ. Since this is impossible, this completes the proof of Theorem 2.  In conclusion to this paper, we note that unlike in Section 2, we may rely on the gap forcing results of [13] and [14] to show that after forcing with each component partial ordering Pκ , all cardinals in the open interval (γκ , κ) which are both strong cardinals and Woodin cardinals are destroyed. However, the “localness” that was present in the proofs found in Section 2, particularly for the preservation of Woodinness, is not available in the proof of Theorem 2 given, since strongness is a global and not a local property. It is for this reason that the ideas used in Lemma 3.4 are quite different from the ones employed in Lemmas 2.3 and 2.7.

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