“Identity Crises and Strong Compactness”

by

Arthur W. Apter^{∗,∗∗∗} Department of Mathematics Baruch College of CUNY New York, New York 10010

and

James Cummings^{∗∗,∗∗∗} Department of Mathematics Carnegie-Mellon University Pittsburgh, Pennsylvania 15213

February 24, 1998 (revised September 7, 1998)

Abstract: Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals κ_{1}, …, κ_{n} are so that κ_{i} for i = 1, …, n is both the ith measurable cardinal and κ_{i}^{+} supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.

∗Supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach). In addition, this research was partially supported by PSC-CUNY Grant 667379.

∗∗Supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach). In addition, this research was partially supported by NSF Grant DMS-9703945.

∗∗∗Both authors wish to express their gratitude to Menachem Magidor for his explanations to them given at the January 7-13, 1996 meeting in Set Theory held at the Mathematics Research Institute, Oberwolfach on his method of forcing to make the first n measurable and strongly compact cardinals coincide, for any finite n.

1991 Mathematics Subject Classification: Primary 03E35, 03E55.

Key Words and Phrases: Strongly compact cardinal, supercompact cardinal, measurable cardinal, identity crisis, reverse Easton iteration.

§0 Introduction and Preliminaries

As is well-known, the notion of strong compactness is a singularity in the hierarchy of large cardinals. The fundamental work of Magidor [Ma] shows that the least strongly compact cardinal κ can either be the least supercompact cardinal or the least measurable cardinal, in which case κ isn’t even 2^{κ} supercompact. A generalization of this result by Kimchi and Magidor [KiM] shows that the (possibly proper) classes of supercompact and strongly compact cardinals can coincide except at measurable limit points or that the first n (for n ∈ ω) strongly compact cardinals can be the first n measurable cardinals.

The purpose of this paper is to show that the techniques of [AS97a] and [AS97b] can be combined with unpublished ideas of Magidor to produce a model in which the first n (for n ∈ ω) strongly compact cardinals are not only the first n measurable cardinals, but each is a little supercompact. Specifically, we prove the following. Theorem 1. Con(ZFC + There are n ∈ ω supercompact cardinals) ⟹ Con(ZFC + The first n strongly compact cardinals κ_{1}, …, κ_{n} are the first n measurable cardinals + 2^{κ_{i}} = κ_{i}^{++} for i = 1, …, n + Each κ_{i} is κ_{i}^{+} supercompact for i = 1, … n).

A bit of history is perhaps in order now. As was just noted, in the early 1970s, Magidor in [Ma] showed the consistency, relative to the existence of a strongly compact cardinal, of the least strongly compact cardinal being the least measurable cardinal. In the spring of 1983, Woodin, in response to a question put to him by the first author, showed the consistency (see [CW]), relative to the existence of a cardinal κ which is κ^{+3} supercompact, of the least measurable cardinal κ being so that 2^{κ} = κ^{++} and κ is κ^{+} supercompact. In the mid 1980s, Kimchi and Magidor did the work of [KiM]. In late 1992, Shelah and the first author began the research leading to the results of [AS97a] and [AS97b]. The main theorem of [AS97a] showed, roughly speaking, the relative consistency of the classes of strongly compact and supercompact cardinals coinciding level by level, except where explicitly prohibited by ZFC. This strengthened the work of [KiM]. The main theorem of [AS97b] showed that Menas’ result of [Me] that the least measurable limit κ of either strongly compact or supercompact cardinals isn’t 2^{κ} supercompact is best possible by constructing, starting from a supercompact limit of supercompact cardinals, a model in which the least measurable limit κ of both strongly compact and supercompact cardinals is so that 2^{κ} = κ^{++} and κ is κ^{+} supercompact. The forcing conditions of [AS97b] were generalizations of the forcing conditions of [AS97a], and both provided, among other things, an alternate way of proving Woodin’s aforementioned 1983 theorem. This still left open the question of combining Woodin’s results with the results of Magidor and Kimchi and Magidor, i.e., obtaining a model in which the least measurable cardinal κ is both the least strongly compact cardinal and is κ^{+} supercompact, or in general, obtaining a model in which the first n measurable cardinals κ_{1}, …, κ_{n} (for n ∈ ω) are the first n strongly compact cardinals, with each measurable cardinal κ_{i} being κ_{i}^{+} supercompact. This question went unresolved for a number of years, despite several attempts at solving it made by Shelah and the first author. Then, during the January 7-13, 1996 meeting in Set Theory held at the Mathematics Research Institute, Oberwolfach, the first author proved (see [A97a]) Theorem 1 for n = 1. His proof, however, was non-iterative, and the question remained of proving Theorem 1 for arbitrary finite n. This question was finally resolved by the two authors of this paper during their stay as Research in Pairs fellows at the Mathematics Research Institute, Oberwolfach, June 8-21, 1997.

We take the opportunity here to make two remarks about Theorem 1. The first is that there is nothing special about each κ_{i} being κ_{i}^{+} supercompact in Theorem 1. In fact, each κ_{i} can be κ_{i}^{++}, κ_{i}^{+++}, κ_{i}^{+4}, etc. supercompact, so long as 2^{κ_{i}} > κ_{i}^{++}, 2^{κ_{i}} > κ_{i}^{+++}, 2^{κ_{i}} > κ_{i}^{+4}, etc. After completing the proof of Theorem 1, interested readers are invited to look at the statement of Theorem 3 of [AS97b] in order to determine for themselves what variants are possible. The second is that no proof is currently known, in both Theorem 1 and the corresponding result of [KiM], when n is infinite. This will be discussed further at the end of the paper.

The structure of this paper is as follows. Section 0 contains our introductory comments and preliminary remarks concerning notation, terminology, etc. Section 1 contains a discussion of a certain modification of the main forcing notion of [AS97a] (given at the end of [AS97b]) that will be critical in the proof of Theorem 1. Section 1 also contains a discussion of the main forcing notion of [AS97b], which can be used as an alternative in the proof of Theorem 1. Section 2 shows how the forcing notions discussed in Section 1 can be used to force a supercompact cardinal to have a certain special kind of supercompactness embedding that will be key in the proof of Theorem 1. Section 3 then gives a proof of Theorem 1 for n = 1. Section 4 contains a proof of Theorem 1 for arbitrary finite n. Section 5 has our concluding remarks.

We give now some preliminary information concerning notation and terminology. Essentially, our notation and terminology are standard, and when this is not the case, this will be clearly noted. For α < β ordinals, [α, β], [α, β), (α, β], and (α, β) are as in standard interval notation. If x is a set, then TC(x) is the transitive closure of x.

When forcing, q ≥ p will mean that q is stronger than p. For P a partial ordering, φ a formula in the forcing language with respect to P, and p ∈ P, p∥φ will mean p decides φ. For G V -generic over P, we will use both V [G] and V^{P} to indicate the universe obtained by forcing with P. If x ∈ V [G], then ẋ will be a term in V for x. We may, from time to time, confuse terms with the sets they denote and write x when we actually mean ẋ, especially when x is some variant of the generic set G, or x is in the ground model V.

If κ is a cardinal and P is a partial ordering, P is κ-closed if given a sequence ⟨p_{α} : α < κ⟩ of elements of P so that β < γ < κ implies p_{β} ≤ p_{γ} (an increasing chain of length κ), then there is some p ∈ P (an upper bound to this chain) so that p_{α} ≤ p for all α < κ. P is < κ-closed if P is δ-closed for all cardinals δ < κ. P is κ-directed closed if for every cardinal δ < κ and every directed set ⟨p_{α} : α < δ⟩ of elements of P (where ⟨p_{α} : α < δ⟩ is directed if for every two distinct elements p_{ρ}, p_{ν} ∈ ⟨p_{α} : α < δ⟩, p_{ρ} and p_{ν} have a common upper bound p_{σ}) 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 ⟨p_{α} : α ≤ κ⟩, where player I plays odd stages and player II plays even and limit stages, then player II has a strategy which ensures the game can always be continued. Note that 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 P is δ-strategically closed for all cardinals δ < κ. P is ≺ κ-strategically closed if in the two person game in which the players construct an increasing sequence ⟨p_{α} : α < κ⟩, where player I plays odd stages and player II plays even and limit stages, then player II has a strategy which ensures the game can always be continued. Note that trivially, if P is < κ-closed, then P is < κ-strategically closed and ≺ κ-strategically closed. The converse of both of these facts is false.

The usual partial ordering for adding λ Cohen subsets to a regular cardinal κ will be written as Add(κ, λ). Standard arguments show Add(κ, λ) is κ-directed closed. See [J], Lemmas 19.7 and 19.8, pages 181-182, for further details.

We mention that we are assuming complete familiarity with the notions of measurability, strong compactness, and supercompactness. Interested readers may consult [SRK], [Ka], or [KaM] for further details. We note only that all elementary embeddings witnessing the λ supercompactness of κ are presumed to come from some fine, κ-complete, normal ultrafilter U over P_{κ}(λ) = {x ⊆ λ : |x| < κ}, and all elementary embeddings witnessing the λ strong compactness of κ are presumed to come from some fine, κ-complete ultrafilter U over P_{κ}(λ). An equivalent definition for κ being λ strongly compact is that there is an elementary embedding j : V → M having critical point κ so that for any x ⊆ M with |x| ≤ λ, there is some y ∈ M such that x ⊆ y and M ⊧ “|y| < j(κ)”.

Finally, we mention that since ideas and notions from [AS97a] and [AS97b] are used throughout the course of this paper, it would be most helpful to readers if copies of these papers were kept close at hand. In fact, at many instances during our exposition, we will refer to proofs not given here but found in either [AS97a] or [AS97b]. These papers, however, need not be read in their entirety to follow this paper. In order to facilitate the understanding of readers, though, we will keep as much as possible to the notations and terminology of [AS97a] and [AS97b]. §1 The Forcing Notions of [AS97a] and [AS97b]

Fix γ < δ < λ, λ > δ^{+} regular cardinals in our ground model V, with δ inaccessible and λ either inaccessible or the successor of a cardinal of cofinality > δ. We recall now the partial orderings P_{δ,λ}^{0} and P_{δ,λ}^{2}[S] of [AS97a] and [AS97b] and both the version of P_{δ,λ}^{1}[S] given in Section 4 of [AS97b] (which is a modification of the partial ordering P_{δ,λ}^{1}[S] of [AS97a]) and the version of P_{δ,λ}^{1}[S] of [AS97b] used in the proof of Theorem 3 of that paper.

We assume GCH holds for all cardinals κ ≥ δ. As in Section 4 of [AS97b], the first notion of forcing P_{δ,λ}^{0} is just the standard notion of forcing for adding a non-reflecting stationary set of ordinals of cofinality γ to λ. Specifically, P_{δ,λ}^{0} = {p : For some α < λ, p : α → {0, 1} is a characteristic function of S_{p}, a subset of α not stationary at its supremum nor having any initial segment which is stationary at its supremum, so that β ∈ S_{p} implies β > δ and cof(β) = γ}, ordered by q ≥ p iff q ⊇ p and S_{p} = S_{q} ∩ sup(S_{p}), i.e., S_{q} is an end extension of S_{p}. It is well-known that for G V -generic over P_{δ,λ}^{0} (see [B] or [KiM]), in V [G], since GCH holds in V for all cardinals κ ≥ δ, a non-reflecting stationary set S = S[G] = ∪{S_{p} : p ∈ G} ⊆ λ of ordinals of cofinality γ has been introduced, the bounded subsets of λ are the same as those in V, and cardinals, cofinalities, and GCH at cardinals κ ≥ δ have been preserved. It is also virtually immediate that P_{δ,λ}^{0} is γdirected closed, and it can be shown (see [B], Lemma 4.15, page 436 or [KiM]) that P_{δ,λ}^{0} is ≺ λ-strategically closed.

Work now in V_{1} = V^{P_{δ,λ}^{0}}, letting Ṡ be a term always forced to denote the above set S. P_{δ,λ}^{2}[S] is the standard notion of forcing for introducing a club set C which is disjoint to S (and therefore makes S non-stationary). Specifically, P_{δ,λ}^{2}[S] = {p : For some successor ordinal α < λ, p : α → {0, 1} is a characteristic function of C_{p}, a club subset of α, so that C_{p} ∩ S = ∅}, ordered by q ≥ p iff C_{q} is an end extension of C_{p}. It is again well-known (see [MS]) that for H V_{1}-generic over P_{δ,λ}^{2}[S], a club set C = C[H] = ∪{C_{p} : p ∈ H} ⊆ λ which is disjoint to S has been introduced, the bounded subsets of λ are the same as those in V_{1}, and cardinals, cofinalities, and GCH for cardinals κ ≥ δ have been preserved.

The following lemma is proven in both [AS97a] and [AS97b]. Lemma 1 (Lemma 1 of [AS97a] and [AS97b]). ∥–_{P_{δ,λ}^{0}}“♣(Ṡ)”, i.e., V_{1} ⊧ “There is a sequence ⟨x_{α} : α ∈ S⟩ so that for each α ∈ S, x_{α} ⊆ α is cofinal in α, and for any A ∈ [λ]^{λ}, {α ∈ S : x_{α} ⊆ A} is stationary”.

We fix now in V_{1} a ♣(S) sequence X = ⟨x_{α} : α ∈ S⟩. We are ready to define in V_{1} in the same manner as was done in Section 4 of [AS97b] the partial ordering P_{δ,λ}^{1}[S]. First, since each element of S has cofinality γ, each x ∈ X can be assumed to be so that order type(x) = γ. Then, P_{δ,λ}^{1}[S] is defined as the set of all 4-tuples ⟨w, α, r̄, Z⟩ satisfying the following properties.

1. w ∈ [λ]^{<δ}.

2. α < δ.

3. r̄ = ⟨r_{i} : i ∈ w⟩ is a sequence of functions from α to {0, 1}, i.e., a sequence of subsets of α.

4. Z ⊆ {x_{β} : β ∈ S} is a set so that if z ∈ Z, then for some y ∈ [w]^{γ}, y ⊆ z and z − y is bounded in the β so that z = x_{β}. In other words, for every x_{β} ∈ Z, w ∩ x_{β} is cobounded in x_{β}.

As in [AS97a], the definition of Z implies |Z| < δ.

The ordering on P_{δ,λ}^{1}[S] is given by ⟨w^{1}, α^{1}, r̄^{1}, Z^{1}⟩ ≤ ⟨w^{2}, α^{2}, r̄^{2}, Z^{2}⟩ iff the following hold.

1. w^{1} ⊆ w^{2}.

2. α^{1} ≤ α^{2}.

3. If i ∈ w^{1}, then r_{i}^{1} ⊆ r_{i}^{2}.

4. Z^{1} ⊆ Z^{2}.

5. If z ∈ Z^{1} ∩ [w^{1}]^{γ} and α^{1} ≤ α < α^{2}, then |{i ∈ z : r_{i}^{2}(α) = 0}| = |{i ∈ z : r_{i}^{2}(α) = 1}| = γ. The intuition behind the definition of P_{δ,λ}^{1}[S] just given is essentially the same as

in [AS97a]. Specifically, we wish to be able simultaneously to make 2^{δ} = λ, destroy the measurability of δ, and be able to resurrect the < λ supercompactness of δ if necessary. P_{δ,λ}^{1}[S] has been designed so as to allow us to do all of these things.

The proof that V_{1}^{P_{δ,λ}^{1}[S]} ⊧ “δ is non-measurable” is as in Lemma 3 of [AS97a]. In particular, the argument of Lemma 3 of [AS97a] will show that δ can’t carry a γ-additive uniform ultrafilter. We can then carry through the proof of Lemma 4 of [AS97a] to show P_{δ,λ}^{0} ∗ (P_{δ,λ}^{1}[Ṡ] × P_{δ,λ}^{2}[Ṡ]) is equivalent to Add(λ, 1) ∗ Ȧdd(δ, λ). The proofs of Lemma 5 of [AS97a] and Lemma 6 of [AS97b] will then show P_{δ,λ}^{0} ∗ P_{δ,λ}^{1}[Ṡ] preserves cardinals and cofinalities, is λ^{+}-c.c., is < δ-strategically closed, and is so that V^{P_{δ,λ}^{0}∗P_{δ,λ}^{1}[Ṡ]} ⊧ “2^{κ} = λ for every cardinal κ ∈ [δ, λ)”.

Although the above definition of P_{δ,λ}^{1}[S] (henceforth to be referred to as the “simpler form”) is perfectly adequate for our purposes, as mentioned at the end of [AS97b], it will not suffice to prove Theorem 3 of [AS97b]. In order to do this, a more complicated form of P_{δ,λ}^{1}[S] is required. Since this version of P_{δ,λ}^{1}[S] will also work for Theorem 1, for completeness, we recall its definitions and properties here. First, we fix δ < λ as before. We then as was done in [AS97b] redefine P_{δ,λ}^{0} as the partial ordering which adds a nonreflecting stationary set of ordinals S of cofinality δ to λ. The definition of P_{δ,λ}^{2}[S] remains the same. Having fixed a ♣(S) sequence as before, P_{δ,λ}^{1}[S] is then the set of all 5-tuples ⟨w, α, r̄, Z, Γ⟩ satisfying the following properties.

1. w ⊆ λ is so that |w| = δ.

2. α < δ.

3. r̄ = ⟨r_{i} : i ∈ w⟩ is a sequence of functions from α to {0, 1}, i.e., a sequence of subsets of α.

4. Z is a function so that:

a) dom(Z) ⊆ {x_{β} : β ∈ S} and range(Z) ⊆ {0, 1}.

b) If z ∈ dom(Z), then for some y ∈ [w]^{δ}, y ⊆ z and z − y is bounded in the β so that z = x_{β}.

5. Γ is a function so that:

a) dom(Γ) = dom(Z).

b) If z ∈ dom(Γ), then Γ(z) is a closed, bounded subset of α such that if γ is inaccessible, γ ∈ Γ(z), and β is the γth element of z, then β ∈ w, and for some β′ ∈ β ∩ w ∩ z, r_{β′}(γ) = Z(z).

Note that the definitions of Z and Γ imply |dom(Z)| = |dom(Γ)| ≤ δ.

The ordering on P_{δ,λ}^{1}[S] is given by ⟨w^{1}, α^{1}, r̄^{1}, Z^{1}, Γ^{1}⟩ ≤ ⟨w^{2}, α^{2}, r̄^{2}, Z^{2}, Γ^{2}⟩ iff the following hold.

1. w^{1} ⊆ w^{2}.

2. α^{1} ≤ α^{2}.

3. If i ∈ w^{1}, then r_{i}^{1} ⊆ r_{i}^{2} and |{i ∈ w^{1} : r_{i}^{2}`|(α_{2} − α_{1}) is not constantly 0}| < δ.

4. Z^{1} ⊆ Z^{2}.

5. dom(Γ^{1}) ⊆ dom(Γ^{2}).

6. If z ∈ dom(Γ^{1}), then Γ^{1}(z) is an initial segment of Γ^{2}(z) and |{z ∈ dom(Γ^{1}) : Γ^{1}(z) ≠ Γ^{2}(z)}| < δ. The intuition behind the above definition of P_{δ,λ}^{1}[S] is the same as in [AS97b]. If δ is

measurable, then δ must carry a normal measure. The forcing P_{δ,λ}^{1}[S] has specifically been designed to destroy this fact. (See Lemma 3 of [AS97b] for a proof.) It has been designed, however, to destroy the measurability of δ “as lightly as possible”, making little damage, assuming δ is < λ supercompact. Specifically, if δ is < λ supercompact, then the nonreflecting stationary set S, having been added to λ, does not kill the < λ supercompactness of δ by itself. The additional forcing P_{δ,λ}^{1}[S] is necessary to do the job and has been designed so as not only to destroy the < λ supercompactness of δ but to destroy the measurability of δ as well. The forcing P_{δ,λ}^{1}[S], however, has been designed so that if necessary, we can resurrect the < λ supercompactness of δ by forcing further with P_{δ,λ}^{2}[S].

We conclude this section by noting that there are additional properties of the more complicated version of P_{δ,λ}^{1}[S] that will be relevant to our work. These will be discussed in more detail in later sections. §2 A Supercompact Cardinal with a Special Kind of Embedding

In this section, we force and construct a supercompact cardinal possessing a special sort of supercompactness embedding. Such cardinals will be critical in the proof of Theorem 1. Specifically, we prove the following.

Lemma 2. Suppose V ⊧ “ZFC + GCH + κ is supercompact”. There is then a partial ordering P^{κ,0} ∈ V so that V^{P^{κ,0}} ⊧ “κ is supercompact + 2^{κ} = κ^{++}”. In addition, there is an elementary embedding j^{∗} : V^{P^{κ,0}} → M^{j^{∗}(P^{κ,0})} definable in V^{P^{κ,0}} witnessing the κ^{+} supercompactness of κ so that M^{j^{∗}(P^{κ,0})} ⊧ “κ isn’t measurable”. Proof of Lemma 2: Fix f : κ → V_{κ} a Laver function [L], i.e., f is so that for every x and every λ ≥ |TC(x)|, there is a λ supercompact ultrafilter U_{λ,x} with associated embedding j_{U_{λ,x}} : V → M so that j_{U_{λ,x}}(f)(κ) = x. Also, let ⟨δ_{α} : α ≤ κ⟩ enumerate the inaccessibles ≤ κ, and let γ < δ_{0} be a fixed but arbitrary regular cardinal.

As Laver does in [L], we define now simultaneously an Easton support iteration P^{κ,0} = ⟨⟨P_{α}^{∗}, Q̇_{α}^{∗}⟩ : α ≤ κ⟩ and a sequence of ordinals ⟨ρ_{α} : α < κ⟩, where ρ_{λ} = ∪_{α<λ}ρ_{α} if λ is a limit ordinal. We use here in our definition the simpler form of P_{δ,λ}^{1}[S] of Section 1 defined using γ and the associated P_{δ,λ}^{0} and P_{δ,λ}^{2}[S] and indicate at the end of the section the modifications needed when the more complicated form of P_{δ,λ}^{1}[S] is used. Specifically, the definition has P_{0} being trivial with ρ_{0} = 0, and P_{α+1}^{∗} = P_{α}^{∗} ∗ Q̇_{α}^{∗}, where ∥–_{P_{α}}“Q̇_{α}^{∗} is trivial” and ρ_{α+1} = ρ_{α} unless one of the following holds:

1. If for all β < α, ρ_{β} < α and δ_{α} < κ is so that V ⊧ “δ_{α} isn’t δ_{α}^{+} supercompact”, then P_{α+1}^{∗} = P_{α}^{∗} ∗ Q̇_{α}^{∗}, where Q̇_{α}^{∗} is a term for P_{δ_{α},δ_{α}^{++}}^{0} ∗ P_{δ_{α},δ_{α}^{++}}^{1}[Ṡ_{δ_{α}^{++}}], and Ṡ_{δ_{α}^{++}} is a term for the non-reflecting stationary subset of δ_{α}^{++} introduced by P_{δ_{a},δ_{α}^{++}}^{0}. If f(α) is an ordinal and f(α) > ρ_{α}, then ρ_{α+1} = f(α). If this condition on f(α) doesn’t hold, then ρ_{α+1} = ρ_{α}.

2. If for β < α, ρ_{β} < α and δ_{α} ≤ κ is so that V ⊧ “δ_{α} is δ_{α}^{+} supercompact”, then P_{α+1}^{∗} = P_{α}^{∗} ∗ Q̇_{α}^{∗}, where Q̇_{α}^{∗} is a term for P_{δ_{α},δ_{α}^{++}}^{0} ∗ (P_{δ_{α},δ_{α}^{++}}^{1}[Ṡ_{δ_{α}^{++}}] × P_{δ_{α},δ_{α}^{++}}^{2}[Ṡ_{δ_{α}^{++}}]). If f(α) is an ordinal and f(α) > ρ_{α}, then ρ_{α+1} = f(α). If this condition on f(α) doesn’t hold, then ρ_{α+1} = ρ_{α}. Suppose now j : V → M is an embedding witnessing the κ^{+} supercompactness of κ so

that M ⊧ “κ isn’t κ^{+} supercompact”. Lemma 9 of [AS97b] shows that if P^{κ,0} = P_{κ}^{∗} ∗Q̇_{κ}^{∗} = P_{κ}^{∗} ∗ (Ṗ_{κ,κ^{++}}^{0} ∗ (P_{κ,κ^{++}}^{1}[Ṡ_{κ^{++}}] × P_{κ,κ^{++}}^{2}[Ṡ_{κ^{++}}])) were an iteration as defined in the proofs of Theorem 1 or Theorem 3 of [AS97b], then j : V → M extends to j^{∗} : V^{P^{κ,0}} → M^{j^{∗}(P^{κ,0})} witnessing the κ^{+} supercompactness of κ in a manner definable in V^{P^{κ,0}}. The type of iteration used in the proofs of Theorem 1 or Theorem 3 of [AS97b], however, is essentially the one just described here. The only real difference is that here, we use a Laver function to “space out” the iteration at successor stages below κ. At stage κ + 1 in V, however, the partial ordering used in the iteration is P_{κ,κ^{++}}^{0} ∗ (P_{κ,κ^{++}}^{1}[Ṡ_{κ^{++}}] × P_{κ,κ^{++}}^{2}[Ṡ_{κ^{++}}]), and at stage κ + 1 in M, the partial ordering used in the iteration is P_{κ,κ^{++}}^{0} ∗ P_{κ,κ^{++}}^{1}[Ṡ_{κ^{++}}]. These occurrences at stage κ+1 in V and M in conjunction with the definition of P^{κ,0} will then allow the arguments of Lemma 9 of [AS97b] to go through to yield that j extends to j^{∗} : V^{P^{κ,0}} → M^{j^{∗}(P^{κ,0})}. Note that since P_{κ,κ^{++}}^{0} ∗ P_{κ,κ^{++}}^{1}[Ṡ_{κ^{++}}] is used at stage κ + 1 in M, Lemma 3 of [AS97a] and Lemma 8 of [AS97b] show that M^{j^{∗}(P^{κ,0})} ⊧ “κ isn’t measurable”.

We show now that V^{P^{κ,0}} ⊧ “κ is supercompact”. To do this, we give an argument similar to the one given in the proof of Lemma 2 of [A∞]. Specifically, let γ > κ^{++} be an arbitrary cardinal, and let λ > 2^{[γ]^{<κ}} be a cardinal so that for some embedding k : V → M witnessing the λ supercompactness of κ, k(f)(κ) = λ. By the definition of P^{κ,0} and the properties of k, k(P^{κ,0}) = (P_{κ}^{∗} ∗ (P_{κ,κ^{++}}^{0} ∗ (P_{κ,κ^{++}}^{1}[Ṡ_{κ^{++}}] × P_{κ,κ^{++}}^{2}[Ṡ_{κ^{++}}]))) ∗ Q̇^{∗} = (P_{κ}^{∗} ∗ Q̇_{κ}^{∗}) ∗ Q̇^{∗} = P^{κ,0} ∗ Q̇^{∗} = P^{κ,0} ∗ Ṙ^{∗} ∗ Q̇_{k(κ)}^{∗}, where Ṙ^{∗} is a term for the M^{P^{κ,0}} partial ordering P_{k(κ)}^{∗}/P^{κ,0}. By the definition of P^{κ,0}, in M, ∥–_{P^{κ,0}}“The field of Q̇ is composed of cardinals > λ”. Further, by the definition of P^{κ,0} and the fact M^{λ} ⊆ M, it is true that in V and M, ∥–_{P^{κ,0}}“Both Ṙ^{∗} and Ṙ^{∗} ∗ Q̇_{k(κ)}^{∗} are λ-strategically closed and λ > 2^{[γ]^{<κ}}”. And, by our earlier remarks, in both V and M, ∥–_{P_{κ}^{∗}}“Q̇_{κ}^{∗} is forcing equivalent to Ȧdd(κ^{++}, 1) ∗ Ȧdd(κ, κ^{++}), a κ-directed closed partial ordering having size κ^{++}”. Therefore, V^{P_{κ}^{∗}∗Q̇_{κ}^{∗}} = V^{P^{κ,0}} ⊧ “2^{κ} = κ^{++}”, and the standard arguments (see, e.g., Lemma 2 of [A∞]) in turn show that M^{P^{κ,0}∗Ṙ^{∗}} remains λ-closed with respect to V^{P^{κ,0}∗Ṙ^{∗}} and that if G_{0} ∗G_{1} is V -generic over P_{κ}^{∗} ∗Q̇_{κ}^{∗} = P^{κ,0} and G_{2} is V [G_{0}][G_{1}]-generic over R^{∗}, in V [G_{0}][G_{1}][G_{2}], we can find a master condition q extending each p ∈ k′′G_{1}. If G_{3} is V [G_{0}][G_{1}][G_{2}]-generic over Q_{k(κ)} so that q ∈ G_{3}, in V [G_{0}][G_{1}][G_{2}][G_{3}], there is an elementary embedding k^{∗} : V [G_{0}][G_{1}] → M[G_{0}][G_{1}][G_{2}][G_{3}] extending k. Since in V, ∥–_{P^{κ,0}}“Ṙ^{∗} ∗ Q̇_{k(κ)}^{∗} is λ-strategically closed”, V [G_{0}][G_{1}] ⊧ “κ is γ supercompact”. This proves Lemma 2.

Lemma 2

When the more complicated version of P_{δ,λ}^{1}[S] and the associated versions of P_{δ,λ}^{0} and P_{δ,λ}^{2}[S] are employed as the building blocks of P^{∗}, instead of working with δ_{α}^{++}, we use δ_{α}^{+++}, i.e., at each non-trivial stage in our iteration, we force with either P_{δ_{α},δ_{α}^{+++}}^{0} ∗ P_{δ_{α},δ_{α}^{+++}}^{1}[Ṡ_{δ_{α}^{+++}}] or P_{δ_{α},δ_{α}^{+++}}^{0} ∗(P_{δ_{α},δ_{α}^{+++}}^{1}[Ṡ_{δ_{α}^{+++}}]×P_{δ_{α},δ_{α}^{+++}}^{2}[Ṡ_{δ_{α}^{+++}}]). This is since by Lemma 6 of [AS97b], both of the just mentioned partial orderings will collapse δ_{α}^{+}. Except for this difference, however, the proof of Lemma 2 is the same as before, making the appropriate references to Lemmas 6 and 8 of [AS97b] as necessary.

In conclusion to this section, we note that if we assume that κ has no inaccessible cardinals above it, no use of the Laver function f is needed in the definition of P^{κ,0}. At each δ_{α} < κ which isn’t δ_{α}^{+} supercompact, we can force as in Case 1 of the definition of P^{κ,0}, and at each δ_{α} ≤ κ which is δ_{α}^{+} supercompact, we can force as in Case 2 of the definition of P^{κ,0}. We leave it to any interested readers to verify that the proof of Lemma 2 becomes simpler under these circumstances. It is only when there are large enough cardinals above κ that the use of the Laver function f is required in the definition of P^{κ,0}. §3 The Case n = 1

We present in this section a proof of Theorem 1 when n = 1. We assume that V ⊧ “ZFC + κ is supercompact”. By Lemma 2, we also assume that V ⊧ “2^{κ} = κ^{++}” and that there is a κ^{+} supercompactness embedding k_{0}^{∗} : V → M^{∗} generated by a κ^{+} supercompact ultrafilter over P_{κ}(κ^{+}) so that M^{∗} ⊧ “κ isn’t measurable”. Further, we assume for the time being that there are no measurable cardinals in V above κ.

Fix now an arbitrary regular cardinal γ < κ. Let ⟨δ_{α} : α < κ⟩ this time enumerate the measurables < κ. The partial ordering P^{κ,1} we use in the proof of Theorem 1 when n = 1 is the Easton support iteration ⟨⟨P_{α}^{κ}, Q̇_{α}^{κ}⟩ : α < κ⟩, where P_{0}^{κ} is trivial and ∥–_{P_{α}^{κ}}“Q̇_{α}^{κ} adds a non-reflecting stationary set of ordinals of cofinality γ to δ_{α}”. Lemma 3. V^{P^{κ,1}} ⊧ “No cardinal δ < κ is measurable”. Proof of Lemma 3: Let δ < κ be so that V ⊧ “δ is measurable”. It must therefore be the case that δ = δ_{α} for some α < κ. This allows us to write P^{κ,1} = P_{α}^{κ}∗Q̇_{α}^{κ}∗Ṙ = P_{α+1}^{κ}∗Ṙ.

By the definition of P^{κ,1} and the fact that any stationary subset of a measurable (or weakly compact) cardinal must reflect, V^{P_{α+1}^{κ}} ⊧ “δ isn’t measurable since there is S ⊆ δ which is a non-reflecting stationary set of ordinals of cofinality γ”. Since by the definition of P^{κ,1}, ∥–_{P_{α+1}^{κ}}“Ṙ is δ′-strategically closed for δ′ the least inaccessible above δ”, V^{P_{α+1}^{κ}∗Ṙ} = V^{P^{κ,1}} ⊧ “S ⊆ δ is a non-reflecting stationary set of ordinals of cofinality γ, so δ isn’t measurable”. Thus, V^{P^{κ,1}} ⊧ “No V -measurable cardinal δ < κ is measurable”.

The proof of Lemma 3 will therefore be complete once we have shown there is no cardinal δ < κ so that ∥–_{P^{κ,1}}“δ is measurable”.

To do this, we give an argument similar to the one found in the last part of Lemma 8 of [A97b], which in turn is essentially the same as the arguments given in Theorem 2.1.5 of [H] and Theorem 2.5 of [KiM]. Assume that V^{P^{κ,1}} ⊧ “δ is measurable”. Since we have just shown that no V -cardinal is measurable in V^{P^{κ,1}}, we can write P^{κ,1} = P_{ζ}^{κ} ∗ Ṙ, where δ ∉ field(P_{ζ}^{κ}) and ∥–_{P_{ζ}^{κ}}“Ṙ is δ′-strategically closed for δ′ the least inaccessible above δ”. Thus, ∥–_{P_{ζ}^{κ}}“δ is measurable” iff ∥–_{P^{κ,1}}“δ is measurable”, so we show without loss of generality that ∥–_{P_{ζ}^{κ}}“δ isn’t measurable”.

Note now that since V^{P_{ζ}^{κ}} ⊧ “δ is Mahlo”, V ⊧ “δ is Mahlo”. Next, let p ∈ P_{ζ}^{κ} be so that p∥– “μ̇ is a measure over δ”. We show there is some q ≥ p, q ∈ P_{ζ}^{κ} so that for every X ∈ (℘(δ))^{V}, q∥“X ∈ μ̇”. To do this, we build in V a binary tree T of height δ, assuming no such q exists. The root of our tree is ⟨p, δ⟩. At successor stages β +1, assuming ⟨r, X⟩ is on the βth level of T , r ≥ p, and X ⊆ δ, X ∈ V is so that r∥– “X ∈ μ̇”, we let X = X_{0} ∪X_{1} be such that X_{0}, X_{1} ∈ V, X_{0} ∩ X_{1} = ∅, and for r_{0} ≥ r, r_{1} ≥ r incompatible, r_{0}∥– “X_{0} ∈ μ̇” and r_{1}∥– “X_{1} ∈ μ̇”. We can do this by our hypothesis of the non-existence of a q ∈ P_{ζ}^{κ} as mentioned earlier. We place both ⟨r_{0}, X_{0}⟩ and ⟨r_{1}, X_{1}⟩ in T at height β + 1 as the successors of ⟨r, X⟩. At limit stages λ < δ, for each branch B in T of height ≤ λ, we take the intersection of all second coordinates of elements along B. The result is a partition of δ into ≤ 2^{λ} many sets, so since δ is Mahlo in V, 2^{λ} < δ, i.e., the partition is into < δ many sets. Since V^{P_{ζ}^{κ}} ⊧ “δ is measurable”, there is at least one element Y of this partition resulting from a branch of height λ and a condition s ≥ p so that s∥– “Y ∈ μ̇”. For all such Y, we place a pair of the form ⟨s, Y ⟩ into T at level λ as the successor of each element of the branch generating Y.

Work now in V^{P_{ζ}^{κ}}. Since δ is measurable in V^{P_{ζ}^{κ}}, V^{P_{ζ}^{κ}} ⊧ “δ is weakly compact”. By construction, T is a tree having δ levels so that each level has size < δ. Thus, by the weak compactness of δ in V^{P_{ζ}^{κ}}, we can let B = ⟨⟨r_{β}, X_{β}⟩ : β < δ⟩ be a branch of height δ through T . If we define for β < δ Y_{β} = X_{β} − X_{β+1}, then since ⟨X_{β} : β < δ⟩ is so that 0 ≤ β < ρ < δ implies X_{β} ⊇ X_{ρ}, for 0 ≤ β < ρ < δ, Y_{β} ∩ Y_{ρ} = ∅. Since by the construction of T , at level β + 1, the two second coordinate portions of the successor of ⟨r_{β}, X_{β}⟩ are X_{β+1} and Y_{β}, for the s_{β} so that ⟨s_{β}, Y_{β}⟩ is at level β + 1 of T , ⟨s_{β} : β < δ⟩ must form in V^{P_{ζ}^{κ}} an antichain of size δ in P_{ζ}^{κ}.

In V^{P_{ζ}^{κ}}, P_{ζ}^{κ} is embeddable as a subordering of the Easton support product ∏_{α<ζ}Q_{α}^{κ} as calculated in V^{P_{ζ}^{κ}}. As V^{P_{ζ}^{κ}} ⊧ “δ is Mahlo”, this immediately implies that V^{P_{ζ}^{κ}} ⊧ “P_{ζ}^{κ} is δ-c.c.”, contradicting that ⟨s_{β} : β < δ⟩ is in V^{P_{ζ}^{κ}} an antichain of size δ. Thus, there is some q ≥ p so that for every X ∈ (℘(δ))^{V}, q∥“X ∈ μ̇”, i.e., δ is measurable in V. This contradiction proves Lemma 3.

Lemma 3

Lemma 4. V^{P^{κ,1}} ⊧ “κ is both strongly compact and κ^{+} supercompact”. Proof of Lemma 4: The proof of Lemma 4 heavily uses unpublished ideas of Magidor (which don’t even appear in the circulated manuscript of [KiM]). Let λ > 2^{[κ^{+}]^{<κ}} = 2^{κ^{+}} = 2^{κ} = κ^{++} be an arbitrary cardinal, and let k_{1} : V → M be an embedding witnessing the λ supercompactness of κ. λ has been chosen large enough so that any ultrafilter over P_{κ}(κ^{+}) present in V is an element of M, so we may assume by the remarks in the first paragraph of this section that k_{2} : M → N is an embedding witnessing the κ^{+} supercompactness of κ definable in M so that N ⊧ “κ isn’t measurable”. It is easily verifiable using the embedding definition of λ strong compactness given in Section 0 that j = k_{2} ∘ k_{1} is so that j : V → N is a λ strongly compact embedding that also witnesses the κ^{+} supercompactness of κ. We show that j extends to j^{∗} : V^{P^{κ,1}} → N^{j^{∗}(P^{κ,1})}, thus proving Lemma 4.

To do this, write j(P^{κ,1}) as P^{κ,1}∗Q̇^{κ}∗Ṙ^{κ}, where Q̇^{κ} is a term for the portion of j(P^{κ,1}) between κ and k_{2}(κ) and Ṙ^{κ} is a term for the rest of j(P^{κ,1}), i.e., the part above k_{2}(κ). Note that since N ⊧ “κ isn’t measurable”, κ ∉ field(Q̇^{κ}). Also, since M ⊧ “κ is measurable”, by elementarity, N ⊧ “k_{2}(κ) is measurable”. Thus, the field of Q̇^{κ} is composed of all N-measurable cardinals in the interval (κ, k_{2}(κ)] (so k_{2}(κ) ∈ field(Q̇^{κ})), and the field of Ṙ^{κ} is composed of all N-measurable cardinals in the interval (k_{2}(κ), k_{2}(k_{1}(κ))).

Let G_{0} be V -generic over P^{κ,1}. We construct in V [G_{0}] an N[G_{0}]-generic object G_{1} over Q^{κ} and an N[G_{0}][G_{1}]-generic object G_{2} over R^{κ}. Since P^{κ,1} is an Easton support iteration of length κ with no forcing done at stage κ, the construction of G_{1} and G_{2} automatically guarantees that j′′G_{0} ⊆ G_{0} ∗ G_{1} ∗ G_{2}, meaning that j : V → N extends to j^{∗} : V [G_{0}] → N[G_{0}][G_{1}][G_{2}].

To build G_{1}, note that since k_{2} can be assumed to be generated by an ultrafiler U over (P_{κ}(κ^{+}))^{M} = (P_{κ}(κ^{+}))^{V}, and since in both V and M, 2^{κ^{+}} = 2^{κ} = κ^{++}, |k_{2}(κ^{++})| = |k_{2}(2^{κ})| = |{f : f : P_{κ}(κ^{+}) → κ^{++} is a function}| = |[κ^{++}]^{κ^{+}}| = κ^{++}. Thus, as N[G_{0}] ⊧ “|Q^{κ}| = k_{2}(2^{κ})”, we can let ⟨D_{α} : α < κ^{++}⟩ enumerate in V [G_{0}] the dense open subsets of Q^{κ} present in N[G_{0}]. Since the κ^{+} closure of N with respect to either M or V implies the least element of the field of Q^{κ} is > κ^{++}, the definition of Q^{κ} as the Easton support iteration which adds a non-reflecting stationary set of ordinals of cofinality γ to each N[G_{0}]-measurable cardinal in the interval (κ, k_{2}(κ)] implies that N[G_{0}] ⊧ “Q^{κ} is ≺ κ^{++}- strategically closed”. By the fact the standard arguments show that forcing with the κ-c.c. partial ordering P^{κ,1} preserves that N[G_{0}] remains κ^{+} closed with respect to either M[G_{0}] or V [G_{0}], Q^{κ} is ≺ κ^{++}-strategically closed in both M[G_{0}] and V [G_{0}].

We can now construct G_{1} in either M[G_{0}] or V [G_{0}] as follows. Player I picks p_{α} ∈ D_{α} extending sup(⟨q_{β} : β < α⟩) (initially, q_{−1} is the empty condition) and player II responds by picking q_{α} ≥ p_{α} (so q_{α} ∈ D_{α}). By the ≺ κ^{++}-strategic closure of Q^{κ} in both M[G_{0}] and V [G_{0}], player II has a winning strategy for this game, so ⟨q_{α} : α < κ^{++}⟩ can be taken as an increasing sequence of conditions with q_{α} ∈ D_{α} for α < κ^{++}. Clearly, G_{1} = {p ∈ Q^{κ} : ∃α < κ^{++}[q_{α} ≥ p]} is our N[G_{0}]-generic object over Q^{κ}.

It remains to construct in V [G_{0}] the desired N[G_{0}][G_{1}]-generic object G_{2} over R^{κ}. To do this, we first note that as λ > 2^{κ}, M ⊧ “κ is measurable”. This means we can write k_{1}(P^{κ,1}) as P^{κ,1} ∗Ṡ^{κ} ∗Ṫ^{κ}, where ∥–_{P^{κ,1}}“Ṡ^{κ} adds a non-reflecting stationary set of ordinals of cofinality γ to κ”, and Ṫ^{κ} is a term for the rest of k_{1}(P^{κ,1}). Since we have assumed V ⊧ “No cardinal δ > κ is measurable”, the λ closure of M with respect to V implies M ⊧ “No cardinal δ ∈ (κ, λ] is measurable”. Thus, the field of Ṫ^{κ} is composed of all M-measurable cardinals in the interval (λ, k_{1}(κ)), which implies that in M, ∥–_{P^{κ,1}∗Ṡ^{κ}}“Ṫ^{κ} is ≺ λ^{+}-strategically closed”. Further, since we can assume λ is regular, |[λ]^{<κ}| = λ, and 2^{λ} = λ^{+} (our ground model V is constructed by forcing over a model of GCH using a set partial ordering), and since, as before, k_{1} can be assumed to be generated by an ultrafilter U over P_{κ}(λ), |k_{1}(λ^{+})| = |k_{1}(2^{λ})| = |2^{k_{1}(λ)}| = |{f : f : P_{κ}(λ) → λ^{+} is a function} = |[λ^{+}]^{λ}| = λ^{+}.

Work until otherwise specified in M. Consider the “term forcing” partial ordering T^{∗} (see [C], Section 1.5, p. 8) associated with Ṫ^{κ}, i.e., τ ∈ T^{∗} iff τ is a term in the forcing language with respect to P^{κ,1} ∗ Ṡ^{κ} and ∥–_{P^{κ,1}∗Ṡ^{κ}}“τ ∈ Ṫ^{κ}”, ordered by τ ≥ σ iff ∥–_{P^{κ,1}∗Ṡ^{κ}}“τ ≥ σ”. Clearly, T^{∗} ∈ M. Also, since ∥–_{P^{κ,1}∗Ṡ^{κ}}“Ṫ^{κ} is ≺ λ^{+}-strategically closed”, it can easily be verified that T^{∗} itself is ≺ λ^{+}-strategically closed in M and, since M^{λ} ⊆ M, in V as well. Therefore, as ∥–_{P^{κ,1}∗Ṡ^{κ}}“|Ṫ^{κ}| = k_{1}(λ) and 2^{k_{1}(λ)} = (k_{1}(λ))^{+} = k_{1}(λ^{+})”, we can assume without loss of generality that in M, |T^{∗}| = k_{1}(λ). This means we can let ⟨D_{α} : α < λ^{+}⟩ enumerate in V the dense open subsets of T^{∗} present in M and argue as before to construct in V an M-generic object H_{2} over T^{∗}.

Note now that since N can be assumed to be given by an ultrapower of M via a normal measure U ∈ M over (P_{κ}(κ^{+}))^{M}, Fact 2 of Section 1.2 of [C] tells us that k′′_{2}H_{2} generates an N-generic object G_{2}^{∗} over k_{2}(T^{∗}). By elementariness, k_{2}(T^{∗}) is the term forcing whose elements are names for elements of k_{2}(Ṫ^{κ}) = Ṙ^{κ} in N^{k_{2}(P^{κ,1}∗Ṡ^{κ})}. Therefore, since G_{2}^{∗} is N-generic over k_{2}(T^{∗}), and since G_{0} ∗ G_{1} is k_{2}(P^{κ,1} ∗ Ṡ^{κ})-generic over N, Fact 1 of Section 1.5 of [C] tells us that for G_{2} = {i_{G_{0}∗G_{1}}(τ) : τ ∈ G_{2}^{∗}}, G_{2} is N[G_{0}][G_{1}]-generic over R^{κ}. As G_{0} is a set of conditions in an Easton support iteration of length κ in which a direct limit was taken at κ, each condition in G_{0} has a support which is bounded in κ. It follows that in V [G_{0}], j : V → N extends to j^{∗} : V [G_{0}] → N[G_{0}][G_{1}][G_{2}]. This proves Lemma 4.

Lemma 4

Since V ⊧ “|P^{κ,1}| = κ”, V^{P^{κ,1}} ⊧ “2^{κ} = κ^{++}”. Thus, Lemmas 3 and 4 complete the proof of Theorem 1 when n = 1.

Theorem 1 (n = 1)

We remark that Magidor’s unpublished ideas mentioned at the beginning of the proof of Lemma 4 were used in the context of a ground model V so that V ⊧ “GCH + κ is supercompact”. The partial ordering used in this situation was, as now, the P^{κ,1} of Lemma 4. The embedding k_{1} in this circumstance was as described above, but the embedding k_{2} was generated by a normal measure U ∈ M concentrating on non-measurable cardinals. The proof given in Lemma 4 then went through in this situation as well to show j = k_{2} ∘k_{1} extends, using that although N is only κ closed with respect to V, GCH gives fewer dense open sets to meet, i.e., in the construction of G_{1}, only κ^{+} instead of κ^{++} dense open subsets of Q^{κ} have to be met.

In conclusion to this section, we note that the first author’s non-iterative proof of Theorem 1 for the case n = 1 [A97a] used Magidor’s notion of iterated Prikry forcing [Ma] to destroy all measurable cardinals found below the cardinal κ produced in Theorem 3 of [AS97b], thereby requiring an initial assumption of a supercompact limit of supercompact cardinals. At the Oberwolfach meeting at which this proof was discovered, Magidor and Woodin independently of one another told both authors a (non-iterative) proof of Theorem 1 for the case n = 1 could be given using Radin forcing, starting from only one supercompact cardinal. Neither Woodin’s nor Magidor’s proof has been published (and both proofs seem unlikely to be published anywhere in the foreseeable future), but our methods here and the methods of [A97a] provide another non-iterative proof starting from only one supercompact cardinal for the case n = 1.

An outline of this proof is as follows: First, start with a ground model V so that V ⊧ “GCH + κ is supercompact”. Next, force using the partial ordering P^{κ,0} of Lemma 2 to preserve the supercompactness of κ, make 2^{κ} = κ^{++}, and construct a κ^{+} supercompact embedding j^{∗} : V^{P^{κ,0}} → M^{j^{∗}(P^{κ,0})} so that M^{j^{∗}(P^{κ,0})} ⊧ “κ isn’t measurable”. Finally, force over V^{P^{κ,0}} using Q^{∗}, Magidor’s notion of iterated Prikry forcing of [Ma], to destroy all measurable cardinals below κ. Magidor’s arguments of [Ma] yield V^{P^{κ,0}∗Q̇^{∗}} ⊧ “κ is both strongly compact and the least measurable cardinal”, and the exact same argument as given in the Lemma of [A97a] shows V^{P^{κ,0}∗Q̇^{∗}} ⊧ “κ is κ^{+} supercompact”. Also, since ∥–_{P^{κ,0}}“|Q̇^{∗}| = κ”, V^{P^{κ,0}∗Q̇^{∗}} ⊧ “2^{κ} = κ^{++}”.

The advantage of the non-iterative proof just given and the earlier non-iterative proofs previously mentioned is that the large cardinal structure above κ can be arbitrary in any of these proofs. The proof of Lemma 4 and the proofs to be given in the next section require severe restrictions on the large cardinal structure of the universe. We will comment more upon this in the concluding remarks of the paper. §4 The Case of Arbitrary Finite n

In this section, we give a proof of Theorem 1 for arbitrary finite n. Proof of Theorem 1: Let V_{0} ⊧ “ZFC + GCH + κ_{1} < κ_{2} < ⋯ < κ_{n} are the first n (for n ∈ ω) supercompact cardinals + No cardinal λ > κ_{n} is measurable”. Let P^{∗} ∈ V_{0} be a partial ordering so that V = V_{0}^{P^{∗}} ⊧ “Each κ_{i} for i = 1, …, n is indestructible under κ_{i}-directed closed forcing”. The existence of this sort of generalized version of Laver’s partial ordering of [L] is easy to show and is found in many places, e.g., [A83], [A98], or [CFM]. Note that since as in [L], P^{∗} can be defined as an iteration so that for P_{κ_{i}}^{∗} the portion of P^{∗} up through stage κ_{i}, |P_{κ_{i}}^{∗}| = κ_{i}, and since also we can assume that the portion of P^{∗} defined beyond stage κ_{i} is at least λ_{i}-directed closed, where for the rest of this section, λ_{i} is the least inaccessible above κ_{i}, V ⊧ “2^{κ_{i}} = κ_{i}^{+} for i = 1, …, n”. Also, by the Lévy-Solovay results [LS], since V_{0} ⊧ “No cardinal λ > κ_{n} is measurable”, V ⊧ “No cardinal λ > κ_{n} is measurable” as well.

We take now V as our ground model and let κ_{0} = ω. P will be defined as the cartesian product ∏_{1≤i≤n}P^{κ_{i}} where P^{κ_{i}} for i = 1, …, n can be defined in two ways, depending upon whether the simpler or more complicated version of the partial ordering P_{δ,λ}^{1}[S] is used. If the simpler version of P_{δ,λ}^{1}[S] is used, then P^{κ_{i}} = P^{κ_{i},0} ∗ Ṗ^{κ_{i},1}, where P^{κ_{i},0} is defined as in Section 2, using only those inaccessibles in the interval (κ_{i−1}, κ_{i}] satisfying GCH in its field and fixing κ_{i−1} as the cofinality of the non-reflecting stationary sets added by each P_{δ,λ}^{0} (since V ⊧ “2^{κ_{i}} = κ_{i}^{+}”, reflection shows that unboundedly many cardinals in (κ_{i−1}, κ_{i}) will be in the field of P^{κ_{i},0}), and P^{κ_{i},1} also adds non-reflecting stationary sets of ordinals of cofinality κ_{i−1} to every V^{P^{κ_{i},0}}-measurable cardinal in the interval (κ_{i−1}, κ_{i}). If the more complicated version of P_{δ,λ}^{1}[S] is used, then P^{κ_{i},0} is defined as in Section 2, using only those inaccessibles in the interval (κ_{i−1}, κ_{i}] in its field satisfying GCH, and P^{κ_{i},1} is as just defined when the simpler version of P_{δ,λ}^{1}[S] is used.

For i = 1, …, n, write P = P_{i} × P^{κ_{i}} × P^{i}, where P_{i} = ∏_{1≤j≤i−1}P^{κ_{j}} and P^{i} =

∏_{i+1≤j≤n}P^{κ_{j}}. When the simpler version of P_{δ,λ}^{1}[S] is used, it easily follows that P^{i} is κ_{i}-directed closed. This is since by the remarks in the middle of p. 108 of [AS97a], each P_{δ,λ}^{1}[S] used in the definition of each P^{κ_{j},0} for j = i+1, …, n is at least κ_{i}-directed closed, so as the cofinalities of the ordinals present in the non-reflecting stationary sets added by

P^{κ_{j},1} for j = i + 1, …, n are at least κ_{i}, P^{κ_{j}} for j = i + 1, …, n and P^{i} = ∏_{i+1≤j≤n}P^{κ_{j}} are

all κ_{i}-directed closed. Also, when the more complicated version of P_{δ,λ}^{1}[S] is used, it is the case that a V -generic object for P^{i} is V -generic over a κ_{i}-directed closed partial ordering. This follows by the argument of Lemma 14 of [AS97b] combined with the fact that any partial ordering of the form P_{δ,λ}^{0} ∗ (P_{δ,λ}^{1}[Ṡ] × P_{δ,λ}^{2}[Ṡ]) used in the definition of P^{κ_{j},0} for j = i + 1, …, n is κ_{i}-directed closed. (See Lemma 4 of [AS97b] for a proof.) Thus, by the indestructibility properties of κ_{i} and the fact P^{i} is λ_{i}-strategically closed, V^{P^{i}} ⊧ “κ_{i} is supercompact and 2^{κ_{i}} = κ_{i}^{+}”.

By the definition of P^{κ_{i+1}}, V^{P^{i}} ⊧ “No cardinal δ ∈ (κ_{i}, κ_{i+1}) is measurable”, for i = 1, …, n−1. When i = n, we take (κ_{i}, κ_{i+1}) as all ordinals > κ_{i} and P^{i} as being trivial, so our initial assumptions on V once more give us V^{P^{i}} ⊧ “No cardinal δ ∈ (κ_{i}, κ_{i+1}) is measurable”. Therefore, the arguments of Lemmas 2 - 4 apply to show V^{P^{i}×P^{κ_{i}}} ⊧ “κ_{i} is < κ_{i+1} strongly compact, κ_{i} is κ_{i}^{+} supercompact, 2^{κ_{i}} = κ_{i}^{++}, and no cardinal δ ∈ (κ_{i−1}, κ_{i}) is measurable”, taking when i = n “< κ_{i} strongly compact” as meaning “δ strongly compact for all cardinals δ > κ_{n}”, i.e., as meaning fully strongly compact. Since by the definition of P_{i}, |P_{i}| < λ_{i−1}, the results of [LS] tell us V^{P^{i}×P^{κ_{i}} ×P_{i}} = V^{P} ⊧ “κ_{i} is < κ_{i+1} strongly compact, κ_{i} is κ_{i}^{+} supercompact, 2^{κ_{i}} = κ_{i}^{++}, and no cardinal δ ∈ (κ_{i−1}, κ_{i}) is measurable”. Therefore, since a result of DiPrisco [DH] tells us that if δ is < γ strongly compact and γ is ρ strongly compact, then δ is ρ strongly compact, and since we are working with finitely many cardinals κ_{1}, …, κ_{n}, we can apply finitely often the result of [DH] to infer V^{P} ⊧ “κ_{i} is strongly compact, κ_{i} is κ_{i}^{+} supercompact, 2^{κ_{i}} = κ_{i}^{++}, and no cardinal δ ∈ (κ_{i−1}, κ_{i}) is measurable”. This completes the proof of Theorem 1 for arbitrary finite n.

Theorem 1

§5 Concluding Remarks

In conclusion to this paper, we note that for the moment, we are restricted to proving Theorem 1, as was Magidor, only to finite values of n. The proof of Theorem 1, and Magidor’s original proof of the consistency of the first n ∈ ω strongly compact cardinals being the first n measurable cardinals, both heavily use that the ground model contains only finitely many supercompact cardinals and no measurable cardinals beyond their supremum. This is evident in the proof of Lemma 4, its equivalent form in Magidor’s original proof, and in the proof given in Section 4. Although it is possible to give alternate proofs, in both our situation and Magidor’s, by composing supercompact embeddings either with embeddings generated by normal measures over measurable cardinals of Mitchell order 0 in Magidor’s case or with a κ^{+} supercompact embedding as constructed in Lemma 2 in our case, the proofs still require the initial assumption of only finitely many supercompact cardinals with no measurable cardinals above their supremum. (See the end of [A95] for a further discussion of this problem.) Thus, the prime question of the relative consistency of the first ω measurable and strongly compact cardinals coinciding, with or without any additional degrees of supercompactness, remains open.

References

[A83] A. Apter, “Some Results on Consecutive Large Cardinals”, Annals of Pure and Applied Logic 25, 1983, 1-17.

[A95] A. Apter, “On the First n Strongly Compact Cardinals”, Proc. Amer. Math. Soc. 123, 1995, 2229-2235.

[A97a] A. Apter, “More on the Least Strongly Compact Cardinal”, Math. Logic Quarterly 43, 1997, 427-430.

[A97b] A. Apter, “Patterns of Compact Cardinals”, Annals of Pure and Applied Logic 89, 1997, 101-115.

[A98] A. Apter, “Laver Indestructibility and the Class of Compact Cardinals”, J. Symbolic Logic 63, 1998, 149-157.

[A∞] A. Apter, “On Measurable Limits of Compact Cardinals”, to appear in the J. Symbolic

Logic.

[AS97a] A. Apter, S. Shelah, “On the Strong Equality between Supercompactness and Strong Compactness”, Trans. Amer. Math. Soc. 349, 1997, 103-128.

[AS97b] A. Apter, S. Shelah, “Menas’ Result is Best Possible”, Trans. Amer. Math. Soc. 349,

1997, 2007-2034.

[B] J. Burgess, “Forcing”, in: J. Barwise, ed., Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, 403-452.

[C] J. Cummings, “A Model in which GCH Holds at Successors but Fails at Singulars”, Trans. Amer. Math. Soc. 329, 1992, 1-39.

[CFM] J. Cummings, M. Foreman, M. Magidor, “Squares, Scales, and Stationary Reflection”, in preparation.

[CW] J. Cummings, W.H. Woodin, Generalised Prikry Forcings, circulated manuscript of a forthcoming book.

[DH] C. DiPrisco, J. Henle, “On the Compactness of ℵ_{1} and ℵ_{2}”, J. Symbolic Logic 43, 1978, 394-401.

[H] J. Hamkins, Lifting and Extending Measures; Fragile Measurability, Doctoral Dissertation, University of California, Berkeley, 1994.

[J] T. Jech, Set Theory, Academic Press, New York, 1978.

[Ka] A. Kanamori, The Higher Infinite, Springer-Verlag, Berlin, 1994.

[KaM] A. Kanamori, M. Magidor, “The Evolution of Large Cardinal Axioms in Set Theory”, in: Lecture Notes in Mathematics 669, Springer-Verlag, Berlin, 1978, 99-275.

[KiM] Y. Kimchi, M. Magidor, “The Independence between the Concepts of Compactness

and Supercompactness”, circulated manuscript.

[L] R. Laver, “Making the Supercompactness of κ Indestructible under κ-Directed Closed Forcing”, Israel J. Math. 29, 1978, 385-388.

[LS] A. Lévy, R. Solovay, “Measurable Cardinals and the Continuum Hypothesis”, Israel J. Math. 5, 1967, 234-248.

[Ma] M. Magidor, “How Large is the First Strongly Compact Cardinal?”, Annals of Math. Logic 10, 1976, 33-57.

[Me] T. Menas, “On Strong Compactness and Supercompactness”, Annals of Math. Logic 7, 1974, 327-359.

[MS] A. Mekler, S. Shelah, “When does κ-Free Imply Strongly κ-Free?”, in: Proceedings of the Third Conference on Abelian Group Theory, Gordon and Breach, Salzburg, 1987, 137-148.

[SRK] R. Solovay, W. Reinhardt, A. Kanamori, “Strong Axioms of Infinity and Elementary Embeddings”, Annals of Math. Logic 13, 1978, 73-116.