INDEXED SQUARES

JAMES CUMMINGS AND ERNEST SCHIMMERLING

Abstract. We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1extendibility is incompatible with square; we prove this is close to optimal by showing that 1-extendibility is compatible with square.

1. Introduction

Several lines of research motivate the results in this paper. What the lines have in common is Jensen’s celebrated combinatorial principle □_{κ}, which is pronounced “square kappa”.

Definition 1.1 (Jensen [12]). Let κ be an infinite cardinal. A sequence ⟨C_{α}: α < κ^{+}⟩ is called a □_{κ}-sequence iff whenever β is a limit ordinal and κ < β < κ^{+},

(1) C_{β} is a closed unbounded subset of β,

(2) C_{β} has order type at most κ, and

(3) if α is a limit point of C_{β}, then C_{α} = C_{β} ∩ α. We say that □_{κ} holds iff there exists a □_{κ}-sequence.

If β is a limit ordinal between κ and κ^{+}, then clearly there exists a closed unbounded C ⊆ β with ot(C) ≤ κ. What gives □_{κ} strength is the last clause in the definition, which is commonly referred to as coherence. Although □_{κ} is a technical principle, it has turned out to be one of the most important links between diverse parts of set theory.

First author partially supported by NSF grants DMS-9703945 and DMS-0070549.

Second author partially supported by NSF Grants DMS-9305990, DMS-9712580, DMS-9996280 and DMS-0088948.

A major theme in set theory is the tension between “compactness” and “incompactness”. Examples of the kind of compactness phenomena we have in mind include stationary reflection (qv), the tree property, Shelah’s singular compactness theorem, or Silver’s theorem that GCH does not first fail at a singular cardinal of uncountable cofinality; some examples of incompactness are non-reflecting stationary sets, Aronszajn trees, or Magidor’s theorem that GCH can fail first at ℵ_{ω}. As we see shortly square and related principles are generally on the incompactness side, and can be used as a measure of the extent of compactness in the universe of set theory.

The following results are due to Jensen, and indicate some of the power of square principles. Let κ be a singular strong limit cardinal; then □_{κ} implies that

(1) there exists a special κ^{+}-Aronszajn tree,

(2) under GCH, there exists a κ^{+}-Souslin tree, and

(3) under GCH, for every structure 𝔄 of type (ℵ_{1}, ℵ_{0}) there exists a structure 𝔅 of type (κ^{+}, κ) such that 𝔄 ≡ 𝔅.

We recall that a stationary subset S of a regular cardinal λ is said to reflect if there is α < λ of uncountable cofinality such that S ∩ α is stationary. Solovay showed that if □_{κ} holds then every stationary subset of κ^{+} contains a non-reflecting stationary set. He also showed that if κ is κ^{+}-strongly compact then every stationary subset of the set κ^{+} ∩ cof(< κ) reflects, so that □_{κ} fails. This is a typical example of the kind of tension between compactness and incompactness which we discussed above.

Another point of departure for the work in this paper was the results of Cummings, Foreman and Magidor [5] on the relationship between square principles, stationary reflection and PCF theory. That paper uses ideas from PCF theory (in particular the concept of a “very good scale” as discussed in section 2) to clarify the relationship between squares and reflection. Several of our results are directly motivated by the results of [5].

It is natural to ask when □_{κ} holds. Jensen showed that in L the principle □_{κ} holds for all cardinals κ. On the other hand, the consistency strength of the failure of □_{κ} is strictly greater than that of ZFC. For example, the theory ZFC + □_{ℵ_{1}} fails is equiconsistent with ZFC + there exists a Mahlo cardinal; the lower bound is by Jensen and uses L, while the upper bound is a forcing argument by Solovay that uses the Levy algebra. More generally, the failure of □_{κ} with κ regular is well-understood. There is a serious gap, however, in our understanding the case in which κ is singular: roughly, there are upper bounds of about one supercompact cardinal, and lower bounds of many Woodin cardinals. This leads us to the second point of departure for [5] and us too, namely inner model theory.

The main technique for proving lower bounds on the failure of □_{κ} for κ singular involves generalizations of Jensen’s theorem that □_{κ} holds in L for all infinite cardinals κ, and Jensen’s Covering Theorem [6], which implies that if 0^{#} does not exist then (κ^{+})^{L} = κ^{+} for all V -singular cardinals κ. An immediate consequence of these results is that if 0^{#} does not exist and κ is a singular cardinal, then □_{κ} holds. We note that this helps to explain a well-known phenomenon in combinatorial set theory; if κ is singular then it is typically hard to build models without “incompact” objects of size κ^{+}, such as κ^{+}- Aronszajn trees [18] or non-reflecting stationary sets [16].

However, this sort of covering fails for L if 0^{#} exists, so other models must be used. The kind of transitive proper class models to which Jensen’s theorems have been generalized are known as core models. In his early attempts in this direction, Schimmerling introduced the following hierarchy of weakenings of □_{κ}.

Definition 1.2 (Schimmerling [20]). Let κ be an infinite cardinal and let λ be a cardinal such that 1 < λ < κ^{+}. A sequence of sets ⟨C_{α}: α < κ^{+}⟩ is called a □_{κ}^{<λ}-sequence iff whenever β is a limit ordinal and κ < β < κ^{+}, then

(1) 1 ≤ |C_{β}| < λ and

(2) for all C ∈ C_{β}, (a) C is a closed unbounded subset of β,

(b) C has order type at most κ, and

(c) if α is a limit point of C, then C ∩ α ∈ C_{α}. We say that □_{κ}^{<λ} holds iff there exists a □_{κ}^{<λ}-sequence.

We write □_{κ}^{λ} for □_{κ}^{<λ^{+}}. Clearly, □_{κ}^{1} is equivalent to □_{κ}. The principle □_{κ}^{κ} appears in the literature as □_{κ}^{∗} or “weak square kappa”; Jensen isolated □_{κ}^{∗} roughly at the same time as □_{κ}, and showed that it holds if and only if there is a special κ^{+}-Aronszajn tree. It is worth noticing that if κ^{<κ} = κ then by an easy argument □_{κ}^{∗} holds, so that □_{κ}^{∗} is of greatest interest when κ is singular.

As an example of how the □_{κ}^{λ} hierarchy has been used, we review some results about forcing axioms and square principles. The Proper Forcing Axiom (PFA) and Martin’s Maximum (MM) are strong versions of Martin’s Axiom (MA) which are consistent relative to a supercompact cardinal, and whose exact consistency strength is still unknown.

Todorcevic [25] proved that under PFA, □_{κ} fails for all κ ≥ ℵ_{1}. Hence, by the results of Jensen mentioned earlier, PFA implies that 0^{#} exists. Magidor observed that Todorcevic’s proof actually shows that PFA implies the failure of □_{κ}^{ℵ_{1}} for all κ ≥ ℵ_{1}. Schimmerling used Magidor’s observation together with the Mitchell-Schimmerling-Steel Covering Theorem to prove that one Woodin cardinal is a lower bound on the large cardinal consistency strength of PFA (the absolute square principles (qv) from Section 3 played a role in the original version of this argument).

We can use the □_{κ}^{λ} hierarchy to measure the combinatorial strength of a proposition P, by computing the least λ such that P is consistent with □_{κ}^{<λ}. Magidor showed that PFA + □_{κ}^{ℵ_{2}} for all κ ≥ ℵ_{2} is consistent relative to a supercompact cardinal, while by contrast MM is incompatible with □_{κ}^{∗} for any singular κ of cofinality ω; this is one measure of the gap in strength between these two axioms. In a similar vein, in [5] the □_{κ}^{λ} hierarchy is used to calibrate the strength of various stationary reflection principles (see the discussion in Section 2). An interesting open problem is to determine whether GCH + □_{ℵ_{ω}}^{ω} suffices to construct an ℵ_{ω+1}-Souslin tree; a straightforward adaptation of Jensen’s arguments shows that GCH + □_{ℵ_{ω}}^{<ω} is sufficient.

As the authors were writing this paper, the story of □_{κ} took an interesting turn. Schimmerling and Zeman proved that in all core models^{1} if κ is not a subcompact cardinal, then □_{κ} holds. Subcompactness is a new large cardinal property that was introduced by Jensen; subcompactness follows from supercompactness. Recall the fact due to Solovay that was mentioned earlier: if κ is κ^{+}-strongly compact, then □_{κ} fails. Jensen observed that subcompactness suffices in Solovay’s theorem and hence the converse to the Schimmerling-Zeman theorem holds. Therefore in the relevant core models □_{κ} holds iff κ is not subcompact. The results in the last section of our paper were inspired by these new developments.

1By core model we mean any proper class model of the form L[E⃗] where E⃗ is a coherent sequence of extenders, subject to certain fine structural conditions.

See [22] for a full explanation.

We end this introduction with a summary of our results. The sections can be read independently, though some of them use definitions from earlier sections.

• In [5] it is shown that certain forms of stationary reflection are consistent with forms of □_{κ}^{λ}. In Section 2 we show that these results are optimal in the sense that λ cannot be decreased. This is an example of the sort of calibration of combinatorial strength discussed above.

• In Section 3, we define and study “indexed square” principles stronger than □_{κ}^{λ}. The main new idea is that the members of C_{ν} are given indices up to λ and we explore various forms of coherence along the indices. These combinatorial principles were originally motivated by core model theory. The main result is Theorem 3.1 which identifies a square principle with some strong upwards absoluteness properties.

• In Section 4 we show that if ℙ is the Prikry forcing notion associated to a normal measure over κ, then □_{κ}^{ℵ_{0}} holds in V^{ℙ}. The proof uses ideas from the results about absolute squares in Section 3 but is self-contained; the main technical point is that given an inaccessible κ we can construct a “good matrix”, which is (roughly speaking) a form of □_{κ}^{∗}-sequence with additional coherence properties. We note that by results from [5], if κ is κ^{+}-supercompact then □_{κ}^{<ℵ_{0}} fails in V^{ℙ}.

• Jensen showed in unpublished work [11] that □_{ℵ_{1}} is strictly stronger than □_{ℵ_{1}}^{2}. In Section 5 we use similar methods to compare these principles with some of the simplest of their “indexed” counterparts from Section 3. The upshot is that the indexed □_{κ}^{λ}-hierarchy is interleaved with the original one.

• The results in Section 6 were obtained by the first author after he learned of the Schimmerling-Zeman result mentioned a few paragraphs above. Subcompactness is a natural strengthening of 1-extendibility, and so one would expect κ being 1-extendible to be consistent with □_{κ}. Actually we prove something stronger but more technical to state: there is a transitive set W and a predicate C⃗ on W such that (W, ∈, C⃗) is a model of ZFCC⃗ + there is a 1-extendible cardinal + C⃗ is a global square sequence. Here ZFCC⃗ is the version of ZFC written in the language of set theory expanded by a predicate symbol for C⃗, and “global square” is a class version of □ (introduced by Jensen) which implies that □_{λ} holds for all λ; this result leaves open whether we can have a 1-extendible cardinal and a definable global square sequence.

2. Stationary reflection

In this section we make two observations on the relationship between the □_{κ}^{λ} principles and stationary reflection. The following result was observed by Schimmerling and independently by Foreman and Magidor.

Theorem 2.1. Assume that κ^{<λ} = κ and □_{κ}^{<λ} holds. Let T ⊆ κ^{+} be stationary. Then there exists S ⊆ T such that S is stationary and S does not reflect at any ν < κ^{+} with cf(ν) ≥ λ.

Proof. Let C⃗ witness □_{κ}^{<λ}. Define F (ν) = {ot(C): C ∈ C_{ν}} for all ν ∈ T, and find S ⊆ T stationary such that F is constant on S with value A. Assume S reflects to ν, and choose C ∈ C_{ν}. Now the function which takes μ ∈ lim(C) ∩ S to ot(C ∩ μ) is an injection from lim(C) ∩ S to A. Since lim(C) ∩ S is unbounded (indeed stationary) in ν, we have

cf(ν) ≤ | lim(C) ∩ S| ≤ |A| < λ.
□

In particular □_{κ}^{<ω} implies that every stationary subset of κ^{+} has a non-reflecting stationary subset.

Cummings, Foreman and Magidor [5] have made a systematic study of the connection between □_{κ}^{λ} and other combinatorial principles. To set our second result in context we quote some theorems from [5].

Fact 2.2 ([5]). Let κ be singular, and let □_{κ}^{λ} hold for some λ < κ. Then for every stationary T ⊆ κ^{+} there exists ⟨S_{i}: i < cf(κ)⟩ such that each S_{i} is a stationary subset of T, and there is no ν < κ^{+} such that cf(ν) > cf(κ) and all the S_{i} reflect at ν.

The proof of Fact 2.2 falls into two parts; in the first part a principle VGS_{κ} (“Very Good Scale at κ”) is derived from □_{κ}^{λ}, and in the second part the conclusion of Fact 2.2 is derived from VGS_{κ}. One theme of [5] is that certain constructions of incompact objects from square principles can be done using very good scales: Fact 2.2 is an example of this. The next result shows that □_{κ}^{∗} is not powerful enough to imply the conclusion of Fact 2.2.

Fact 2.3 ([5]). If the existence of infinitely many supercompact cardinals is consistent, then it is consistent that

(1) □_{ℵ_{ω}}^{∗} holds.

(2) For all m, n with 1 ≤ m ≤ n < ω, if ⟨S_{i}: i < ℵ_{m}⟩ is a sequence of stationary subsets of {α < ℵ_{ω+1}: cf(α) < ℵ_{m}} then there exists ν < ℵ_{ω+1} such that cf(ν) = ℵ_{n} and all the S_{i} reflect at ν.

On the other hand, Fact 2.2 cannot in general be strengthened to rule out simultaneous reflection of fewer than cf(κ) many sets.

Fact 2.4 ([5]). If the existence of infinitely many supercompact cardinals is consistent, then it is consistent that

(1) □_{ℵ_{ω}}^{ω} holds.

(2) For every n < ω, if ⟨S_{i}: i < n⟩ is a sequence of stationary subsets of ℵ_{ω+1} then there exists M < ω such that for all m with M ≤ m < ω there exists ν < ℵ_{ω+1} such that cf(ν) = ℵ_{m} and all the S_{i} reflect at ν.

The second theorem of this section shows that Fact 2.4 is close to optimal, in that □_{ℵ_{ω}}^{<ℵ_{ω}} is incompatible with the conclusion. Before proving it we need a technical lemma (an easy generalisation of a well-known fact about □_{κ}).

Lemma 2.5. Let κ be singular. If □_{κ}^{<λ} holds then there exists a □_{κ}^{<λ}-sequence ⟨D_{α}: α < κ^{+}⟩, with the additional property that all the clubs in ⋃_{α} D_{α} are of order type less than κ.

Proof. Fix D ⊆ κ such that D is closed and unbounded and ot(D) = cf(κ). Given C a club subset of some α < κ^{+} with ot(C) ≤ κ we define C^{∗} ⊆ C:

• If ot(C) ∈ lim(D) or ot(C) = κ then

C^{∗} = {δ ∈ C: ot(C ∩ δ) ∈ D}.

• If ot(C) ∉ lim(D), then

C^{∗} = {δ ∈ C: ot(C ∩ δ) > max(ot(C) ∩ lim(D))}.

It is easy to check that C^{∗} is club in sup(C), and that if γ ∈ lim(C^{∗}) then C^{∗} ∩ γ = (C ∩ γ)^{∗}. Given a □_{κ}^{<λ}-sequence ⟨C_{α}: α < κ^{+}⟩, we set D_{α} = {C^{∗}: C ∈ C_{α}}. □

Theorem 2.6. Assume that κ is a singular strong limit cardinal, and let T be a stationary subset of κ^{+}. Suppose that □_{κ}^{<κ} holds. Then there is a sequence ⟨S_{i}: i < cf(κ)⟩ of stationary subsets of κ and a cardinal μ < κ such that

(1) S_{i} ⊆ T ∩ cof(< μ).

(2) If ν < κ^{+} is any point with cf(ν) ≥ μ, then ⟨S_{i}: i < cf(κ)⟩ does not reflect simultaneously to ν.

Proof. We fix a □_{κ}^{<κ} sequence C, and we assume (as we may by Lemma 2.5) that all the club sets appearing in C have order type less than κ. We also fix an increasing sequence ⟨κ_{i}: i < cf(κ)⟩ of regular cardinals cofinal in κ, and a stationary T ⊆ κ^{+}. Let T′ be a stationary subset of T on which the functions ν ↦ |C_{ν}| and ν ↦ cf(ν) are constant. Let i < cf(κ) be large enough so that both of these constant values are less than κ_{i}. For each ν < κ^{+} and j < cf(κ), the set

{ot(C): C ∈ C_{ν}} ∩ κ_{j}

is an element of V_{κ}. Since κ is a strong limit cardinal, by Fodor’s lemma there are sequences ⟨S_{j}: j < cf(κ)⟩ and ⟨A_{j}: j < cf(κ)⟩ such that for every j < cf(κ), S_{j} is a stationary subset of T′ and for every ν ∈ S_{j},

A_{j} = {ot(C): C ∈ C_{ν}} ∩ κ_{j} .

Suppose that ⟨S_{j}: j < cf(κ)⟩ reflects simultaneously to an ordinal ν < κ^{+}. Let C be any element of C_{ν}; by our assumptions on C we know that ot(C) < κ and so we may choose j < cf(κ) large enough that κ_{j} > ot(C). Then lim(C) ∩ S_{j} has fewer than κ_{i} many elements. This is because, if μ_{0} < μ_{1} are both elements of lim(C) ∩ S_{j}, then ot(C ∩ μ_{0}) and ot(C ∩ μ_{1}) are distinct elements of A_{j}, but |A_{j}| < κ_{i}. Therefore cf(ν) < κ_{i}. □

3. Indexed square principles

In this section, we introduce several weak square principles that were distilled from the core model combinatorics of [20] and [21]. Perhaps the most interesting of these principles is Slick-□_{κ}^{∗}. Rather than give the definition of Slick-□_{κ}^{∗} here, at the start of this section, we will lead up to it in steps. However, to give the reader an idea of the goal, let us go ahead and state a corollary to what we are about to do.

Theorem 3.1. Let V ⊆ W be transitive models of ZFC and κ be a limit cardinal in W. Suppose that (κ^{+})^{V} = (κ^{+})^{W} and Slick-□_{κ}^{∗} holds in V. Then both Slick-□_{κ}^{∗} and □_{κ}^{λ} hold in W where λ = cf^{W} (κ).

Combinatorics similar to the proof of this theorem were used to obtain lower bounds on the consistency strength of PFA in [20]. Theorem 3.1 was the inspiration for the result in the next section that Prikry forcing at a measurable cardinal κ adds a □_{κ}^{ℵ_{0}} sequence.

Whacky-□_{κ}^{∗} is another principle that we will define later in this section. On the surface, Whacky-□_{κ}^{∗} seems like a slight improvement of □_{κ}^{<κ}, but there are cases in which it does the work of □_{κ}^{<ℵ_{0}}. Again, we will state a result well in advance of giving the definitions. Combinatorics similar to the proof of Theorem 3.2 were used to obtain consistency strength lower bounds on stationary reflection in [20].

Theorem 3.2. Suppose that κ is a strong limit cardinal and that the principle Whacky-□_{κ}^{∗} holds. Then every stationary subset of κ^{+} has a non-reflecting stationary subset.

To orient the reader we will fill in the missing definitions to make sense of the series of implications

□_{κ}^{<κ} ⟸ Index-□_{κ}^{<κ} ⟸ Card-index-□_{κ}^{<κ} ⟸ Whacky-□_{κ}^{∗} from left to right, after which we will define Slick-□_{κ}^{∗}.

Our first definition is that of Index-□_{κ}^{<λ}, which strengthens □_{κ}^{<λ} by assigning ordinal indices from κ + 1 to each club in each C_{ν}, and demanding that initial segments of a club with a given index all get the same index.

Definition 3.3. The pair (A, C) is said to witness that Index-□_{κ}^{<λ} holds iff A and C are functions such that if ν is a limit ordinal and κ < ν < κ^{+}, then

(1) A(ν) is a non-empty subset of κ + 1 of cardinality < λ, and (2) if α ∈ A(ν), then

(a) C(ν, α) is a club subset of ν,

(b) ot(C(ν, α)) ≤ κ, and

(c) if μ ∈ lim(C(ν, α)), then α ∈ A(μ) and C(ν, α) ∩ μ = C(μ, α).

As one would expect, Index-□_{κ}^{λ} means Index-□_{κ}^{<λ^{+}}. It might seem more natural to require A(ν) ⊆ κ in Definition 3.3; the value of allowing κ as an index will be remarked on after Definition 3.4. In Section 5, we will prove that indexing gives something new, namely that Index-□_{ℵ_{1}}^{2} is strictly between □_{ℵ_{1}} and □_{ℵ_{1}}^{2} in its strength. However, for the rest of this section, most of our results will be about the case in which κ is a limit cardinal.

The point of our next principle, Card-index-□_{κ}^{<λ}, is to link information about the order type of a given club to its index. Of course, several ways of doing this are possible. Considerations tied to the core model combinatorics of [20] led to our choice of “α^{+}” here, in a way that we will not make precise.

Definition 3.4. We say that (A, C) witnesses that Card-index-□_{κ}^{<λ} holds iff (A, C) witnesses that Index-□_{κ}^{<λ} holds and for each limit ordinal ν between κ and κ^{+},

(1) either A(ν) ⊆ κ and |C(ν, α)| ≤ α^{+} for all α ∈ A(ν),

(2) or A(ν) = {κ}.

Here are some remarks on the definition.

• If κ is a successor cardinal and the principle Index-□_{κ}^{<λ} holds, then the principle Card-index-□_{κ}^{<λ+1} holds.

• The only point of allowing κ to be an index (rather than requiring A(ν) ⊆ κ in all cases) is to get a consistent principle when κ is an inaccessible cardinal. Note that if κ is an inaccessible cardinal and (A, C) is a witness that Card-index-□_{κ}^{λ} holds, then A(ν) = {κ} whenever cf(ν) = κ. Also note that if μ ∈ lim(C(ν, κ)), then A(μ) = {κ}.

• Let κ be a singular cardinal and λ = cf(κ). Suppose that there exists a witness that Card-index-□_{κ}^{λ} holds. Then there exists a witness (A, C) that Card-index-□_{κ}^{λ} holds such that A(ν) ⊆ κ for all ν. The construction is as follows. First arrange that ot(C(ν, κ)) < κ whenever A(ν) = {κ} as in the proof of Lemma 2.5. Then fix a sequence ⟨κ_{i}: i < λ⟩ that is increasing and unbounded in κ. If A(ν) ⊆ κ, then let B(ν) = A(ν) and D(ν, α) = C(ν, α). If A(ν) = {κ}, then let

B(ν) = {κ_{i} < κ: ot(C(ν, κ)) ≤ κ_{i}^{+}}

and D(ν, κ_{i}) = C(ν, κ) for κ_{i} ∈ B(ν). Then (B, D) is a witness that Card-index-□_{κ}^{λ} holds and B(ν) ⊆ κ for all ν.

We will prove Theorem 3.2 after defining the principle Whacky-□_{κ}^{∗}, which strengthens Card-index-□_{κ}^{<κ} by requiring, roughly, that the set of indices for a given ordinal ν be bounded in κ.

Definition 3.5. We say that (A, C) is a witness that Whacky-□_{κ}^{∗} holds iff (A, C) is a witness that Card-index-□_{κ}^{<κ} holds and for every limit ordinal ν between κ and κ^{+}, if A(ν) ⊆ κ, then sup(A(ν)) < κ.

We remark that if κ is a regular cardinal, then Whacky-□_{κ}^{∗} holds iff Card-index-□_{κ}^{<κ} holds.

Proof of Theorem 3.2. Say κ is a strong limit cardinal and (A, C) is a witness that Whacky-□_{κ}^{∗} holds. Let S ⊆ κ^{+} be a stationary set of limit ordinals.

First suppose that there exists a set S′ ⊆ S that is stationary in κ such that A(ν) = {κ} for all ν ∈ S′. By Fodor’s lemma, there exists an ordinal τ ≤ κ and a stationary set S′′ ⊆ S′ such that ot(C(ν, κ)) = τ for all ν ∈ S′′. Consider an arbitrary ν. Let α ∈ A(ν). Suppose that μ ∈ S′′ ∩ lim(C(ν, α)). Then α = κ, A(μ) = {κ} and

ot(C(ν, κ) ∩ μ) = ot(C(μ, κ)) = τ.

But this can hold for at most one μ, so S′′ does not reflect to ν.

Thus, without loss of generality A(ν) ⊆ κ for all ν ∈ S. Then

sup(A(ν)) < κ

and

sup ({ot(C(ν, α)): α ∈ A(ν)}) ≤ sup ({|C(ν, α)|^{+}: α ∈ A(ν)})
≤ sup ({α^{++}: α ∈ A(ν)})
< κ

for all ν ∈ S. Since κ is a strong limit cardinal,

⟨ot(C(ν, α)): α ∈ A(ν)⟩ ∈ H_{κ}

for all ν ∈ S. By Fodor’s lemma, there is a stationary S′ ⊆ S, a set B ∈ H_{κ} and a sequence of ordinals τ = ⟨τ_{α}: α ∈ B⟩ such that

A(ν) = B

and

⟨ot(C(ν, α)): α ∈ A(ν)⟩ = τ

for all ν ∈ S′. Consider an arbitrary ν. Let α ∈ A(ν). Suppose that μ ∈ S′ ∩ lim(C(ν, α)). Then α ∈ B = A(μ) and

ot(C(ν, α) ∩ μ) = ot(C(μ, α)) = τ_{α}.

This can hold for at most one μ, therefore S′ does not reflect to ν. □

Definition 3.6. We say that (A, C) is a witness that Slick-□_{κ}^{∗} holds iff A and C are functions such that for all limit ν with κ < ν < κ^{+},

(1) either (a) A(ν) is a non-empty closed subset of κ, and (b) if α ∈ A(ν), then

(i) C(ν, α) is a club subset of ν, (ii) ot(C(ν, α)) ≤ κ and |C(ν, α)| ≤ α^{+}, (iii) if μ ∈ lim(C(ν, α)), then α ∈ A(μ) and

C(ν, α) ∩ μ ⊆ C(μ, α),

and (iv) if α < β and β ∈ A(ν), then C(ν, α) ⊆ C(ν, β),

(2) or else (a) A(ν) = {κ},

(b) C(ν, κ) is a club subset of ν,

(c) ot(C(ν, κ)) ≤ κ, and

(d) if μ ∈ lim(C(ν, κ)), then A(μ) = {κ} and

C(ν, κ) ∩ μ = C(μ, κ).

A witness to Slick-□_{κ}^{∗} need not be a witness to any of the other indexed square principles that we have defined, not even □_{κ}^{∗}, because of the weaker coherence condition (“⊆” instead of “=”) in clause 1(b)(iii). In fact, the proof of Theorem 3.1 shows that if Slick-□_{κ}^{∗} holds, then there exists a witness with the stronger form of coherence. On the other hand, we have added clause 1(b)(iv), which is a different kind of coherence across indices. Another new feature is the requirement, clause 1(a), that A(ν) be closed; note that A(ν) is not required to be bounded or have small cardinality.

Let us also remark that if κ is a singular cardinal, then the second possibility in Definition 3.6 is not needed in the sense that Slick-□_{κ}^{∗} holds iff there is a witness (A, C) that Slick-□_{κ}^{∗} holds such that A(ν) ⊆ κ for all ν. The reason is just like that given in the third remark after Definition 3.4.

Theorem 3.1 follows immediately from the following two lemmas, the first of which is obvious.

Lemma 3.7. Let V ⊆ W be transitive models of ZFC and κ be a cardinal in W. Suppose that (κ^{+})^{V} = (κ^{+})^{W} and

V ⊧ (A, C) is a witness that Slick-□_{κ}^{∗} holds.

Then

W ⊧ (A, C) is a witness that Slick-□_{κ}^{∗} holds.

Lemma 3.8. Suppose that κ is a limit cardinal, λ = cf(κ), and there exists a witness that Slick-□_{κ}^{∗} holds. Then there exists a witness (A, C) that Slick-□_{κ}^{∗} holds such that if A(ν) ⊆ κ, then

(1) |A(ν)| ≤ λ, and

(2) if μ ∈ lim(C(ν, α)), then α ∈ A(μ) and C(ν, α)∩μ = C(μ, α). In particular, (A, C) is also a witness that Card-index-□_{κ}^{λ} holds.

Proof. Let ⟨κ_{i}: i < λ⟩ be a strictly increasing continuous sequence of cardinals that is unbounded in κ. Let (A, C) witness that Slick-□_{κ}^{∗} holds. From this data, we will define a pair (A^{fat}, C^{fat}) satisfying the requirements of the lemma. The intuitive idea is to recursively “fatten up” each C(ν, α) for α < κ.

If A(ν) = {κ}, then let A^{fat}(ν) = {κ} and C^{fat}(ν, κ) = C^{fat}(ν, κ). There is nothing to check in this case.

For the rest of the proof, we turn to the only other case, namely A(ν) ⊆ κ. Define

A^{fat}(ν) = {κ_{i}: A(ν) ∩ (κ_{i} + 1) ≠ ∅} .

Clearly A^{fat}(ν) is a club subset of κ of cardinality at most λ. If κ_{i} ∈ A^{fat}(ν), then define an ordinal

α(ν, i) = sup {A(ν) ∩ (κ_{i} + 1)} .

Since A(ν) is closed,

α(ν, i) ∈ A(ν) ∩ (κ_{i} + 1)

whenever κ_{i} ∈ A^{fat}(ν).

Claim 3.9. If κ_{i} ∈ A^{fat}(ν) and μ ∈ lim(C(ν, α(ν, i)), then

α(ν, i) ∈ A(μ) ∩ (κ_{i} + 1),
κ_{i} ∈ A^{fat}(μ),
α(ν, i) ≤ α(μ, i)

and

C(ν, α(ν, i)) ∩ μ ⊆ C(μ, α(ν, i)) ⊆ C(μ, α(μ, i)).

Proof. This is immediate from the definitions given before the statement of Claim 3.9 with clause 1(b)(iii) and clause 1(b)(iv) of Definition 3.6. □

By recursion on ν, define for κ_{i} ∈ A^{fat}(ν),

C^{fat}(ν, κ_{i}) = C(ν, α(ν, i)) ∪ ⋃ {C^{fat}(μ, κ_{i}): μ ∈ lim (C(ν, α(ν, i)))} .

This definition makes sense since if μ ∈ lim(C(ν, α(ν, i))), then μ < ν and, by Claim 3.9, κ_{i} ∈ A^{fat}(μ).

The next claim shows that (A^{fat}, C^{fat}) satisfies clause 1(b)(ii) of Definition 3.6.

Claim 3.10. If κ_{i} ∈ A^{fat}(ν), then |C^{fat}(ν, κ_{i})| ≤ κ_{i}^{+}.

Proof. By induction on ν, we see that C^{fat}(ν, κ_{i}) is the union of at most (α(ν, i))^{+}-many sets, each of cardinality at most κ_{i}^{+}. Since α(ν, i) ≤ κ_{i}, we are done. □

The following claim is another step towards seeing that (A^{fat}, C^{fat}) satisfies clause 2 of Lemma 3.8; the full verification of this will be given in Claim 3.14.

Claim 3.11. If κ_{i} ∈ A^{fat}(ν) and μ ∈ lim (C(ν, α(ν, i))), then

κ_{i} ∈ A^{fat}(μ)

and

C^{fat}(ν, κ_{i}) ∩ μ = C^{fat}(μ, κ_{i}).

Proof. The part about κ_{i} ∈ A^{fat}(μ) was already proved in Claim 3.9. We prove the other part by induction on ν. Assume that Claim 3.11 holds for all ν′ < ν and that μ ∈ lim (C(ν, α(ν, i))).

First suppose that μ is the largest limit point of C(ν, α(ν, i)). Consider an arbitrary μ′ < μ such that

μ′ ∈ lim(C(ν, α(ν, i))).

By Claim 3.9,

C(ν, α(ν, i)) ∩ μ ⊆ C(μ, α(μ, i)).

In particular,

μ′ ∈ lim(C(μ, α(μ, i))).

By the induction hypothesis

C^{fat}(μ, κ_{i}) ∩ μ′ = C^{fat}(μ′, κ_{i}).

By the arbitrariness of μ′ and the definition of C^{fat}(ν, κ_{i}),

C^{fat}(ν, κ_{i}) = C(ν, α(ν, i)) ∪ C^{fat}(μ, κ_{i}).

But

C(ν, α(ν, i)) ∩ μ ⊆ C(μ, α(μ, i)) ⊆ C^{fat}(μ, κ_{i})

by Claim 3.9 and the definition of C^{fat}(μ, κ_{i}). Thus

C^{fat}(ν, κ_{i}) = C^{fat}(μ, κ_{i}) ∪ (C(ν, α(ν, i)) − μ) .

It follows from the last equation that Claim 3.11 holds in the first case.

Second suppose that C(ν, α(ν, i)) has no largest limit point. Consider an arbitrary ν′ > μ such that ν′ ∈ lim (C(ν, α(ν, i))). By Claim 3.9 applied to ν′ and ν,

C(ν, α(ν, i)) ∩ ν′ ⊆ C(ν′, α(ν′, i)).

In particular,

μ ∈ lim (C(ν′, α(ν′, i))) .

By the induction hypothesis,

C^{fat}(ν′, κ_{i}) ∩ μ = C^{fat}(μ, κ_{i}).

By the arbitrariness of ν′ and the definition of C^{fat}(ν, κ_{i}),

C^{fat}(ν, κ_{i}) ∩ μ = (C(ν, α(ν, i)) ∩ μ) ∪ C^{fat}(μ, κ_{i}).

But

C(ν, α(ν, i)) ∩ μ ⊆ C(μ, α(μ, i)) ⊆ C^{fat}(μ, κ_{i})

by Claim 3.9 and the definition of C^{fat}(μ, κ_{i}). Thus Claim 3.11 follows in the second case too. □

Claim 3.12. Suppose that κ_{i} ∈ A^{fat}(ν) and

μ = sup {lim (C(ν, α(ν, i)))} .

Then

C^{fat}(ν, κ_{i}) = \begin{cases}C^{fat}(μ, κ_{i}) ∪ (C(ν, α(ν, i)) − μ) & if μ < ν \\ ⋃ \{C^{fat}(ν′, κ_{i}): ν′ ∈ lim (C(ν, α(ν, i)))\} & if μ = ν \end{cases}

Proof. The characterization follows by induction on ν from the definition of (A^{fat}, C^{fat}) and Claim 3.11. □

Claim 3.13. If κ_{i} ∈ A^{fat}(ν), then C^{fat}(ν, κ_{i}) is club in ν.

Proof. By induction on ν. Assume that Claim 3.13 holds for all ν′ < ν and that μ ∈ lim (C^{fat}(ν, κ_{i})). We will show that μ ∈ C^{fat}(ν, κ_{i}).

First suppose that there is no limit point of C(ν, α(ν, i)) strictly greater than μ. By Claim 3.12, μ must be the largest limit point of C(ν, α(ν, i)). So

μ ∈ C(ν, α(ν, i)) ⊆ C^{fat}(ν, κ_{i}).

On the other hand, if ν′ > μ and ν′ ∈ lim (C(ν, α(ν, i))), then by the induction hypothesis and Claim 3.11,

μ ∈ lim (C^{fat}(ν, κ_{i}) ∩ ν′) = lim (C^{fat}(ν′, κ_{i}))
⊆ C^{fat}(ν′, κ_{i}) = C^{fat}(ν, κ_{i}) ∩ ν′.
□

Claim 3.14. If κ_{i} ∈ A^{fat}(ν) and μ ∈ lim (C^{fat}(ν, κ_{i})), then

κ_{i} ∈ A^{fat}(μ)

and

C^{fat}(ν, κ_{i}) ∩ μ = C^{fat}(μ, κ_{i}).

Proof. By induction on ν. Assume that Claim 3.14 holds for all ν′ < ν and that μ ∈ lim (C^{fat}(ν, κ_{i})). If μ ∈ lim (C(ν, α(ν, i))), then we are done by Claim 3.11. So we may assume that there is a ν′ > μ such that ν′ ∈ lim (C(ν, α(ν, i))). Then

C^{fat}(ν, κ_{i}) ∩ ν′ = C^{fat}(ν′, κ_{i})

by Claim 3.11, so μ ∈ lim (C^{fat}(ν′, κ_{i})). By the induction hypothesis, C^{fat}(ν′, κ_{i}) ∩ μ = C^{fat}(μ, κ_{i}). Putting the equations together, we are done. □

The next result implies that (A^{fat}, C^{fat}) satisfies clause 1(b)(iv) of Definition 3.6.

Claim 3.15. If κ_{i} ∈ A^{fat}(ν) and i < j, then κ_{j} ∈ A^{fat}(ν) and

C^{fat}(ν, κ_{i}) ⊆ C^{fat}(ν, κ_{j}).

Proof. Obvious from the corresponding assumption on (A, C) and the definition of (A^{fat}, C^{fat}). □

From the claims above, it is immediate that (A^{fat}, C^{fat}) satisfies the requirements of Lemma 3.8.

□

4. Prikry forcing, good matrices and weak square

It is proved in [17] that after forcing with Prikry forcing at a measurable cardinal κ the weak square principle □_{κ}^{∗} holds. In this section we strengthen this result, showing that if κ is measurable in V and W is a Prikry extension of V then □_{κ}^{ω} holds in W. In general we can not hope to improve this; by Theorem 2.1 and the following result, doing Prikry forcing at a sufficiently large cardinal κ will make □_{κ}^{<ω} fail in the generic extension.

Fact 4.1 ([5]). If κ is κ^{+}-supercompact, ℙ is Prikry forcing defined from some normal measure on κ, and S = {α < κ^{+}: cf(α) < κ} then

V^{ℙ} ⊧ “finite sets of stationary subsets of S reflect simultaneously”.

A note on history: Originally we had a false proof of Theorem 4.2 based on Theorem 3.1 and an incorrect version of Lemma 4.4. Matt Foreman pointed out that we could get the conclusion more directly from the first version of Lemma 4.4. We then discovered and fixed the problem in Lemma 4.4, retaining Foreman’s direct way of drawing the desired conclusion.

Theorem 4.2. Let κ be measurable in V. Let U be a normal measure on κ and let ℙ_{U} be the Prikry forcing defined from U. If W is a generic extension of V by ℙ_{U} then □_{κ}^{ω} holds in W.

Proof. The key idea is to do most of the work in V. We will build in V an object called a “good matrix”, and then working in W we will read off the required □_{κ}^{ω}-sequence. It is helpful to think of the construction of a good matrix as a refinement of the (very easy) construction of a □_{κ}^{∗}-sequence for κ inaccessible.

The proof will be structured as follows: we will start by defining a good matrix, will show how to use one to build a □_{κ}^{ω}-sequence (hopefully motivating the definition) and will finish by constructing one.

Let λ be a regular cardinal with λ > ℵ_{1}. We will say that a set A ⊆ λ is a club^{∗} subset of λ if and only if there is C club in λ such that {α ∈ C: cf(α) > ω} ⊆ A. It is easy to see that the collection of club^{∗} subsets of λ is a normal filter on λ, and that any unbounded subset of λ which is closed under uncountable suprema is club^{∗}.

We claim that every club^{∗} subset A of κ has measure one for the normal measure U. To see this let C be club in κ such that {α ∈ C: cf(α) > ω} ⊆ A, and let j: V ⟶ M be the ultrapower map associated with U. Since j(C)∩κ = C and j(C) is closed we see that κ ∈ j(C), and since ^{κ}M ⊆ M we see that M ⊧ cf(κ) > ω; it follows by the elementarity of j that κ ∈ j(A), and so by the normality of U that A ∈ U.

It is a well-known fact about Prikry forcing that any ℙ_{U}-generic ω-sequence is eventually contained in any set in U. In particular we see that a ℙ_{U}-generic ω-sequence is eventually contained in any club^{∗} subset of κ from the ground model.

Definition 4.3. Let λ be an inaccessible cardinal, and let

S = {α < λ^{+}: cf(α) < λ}.

A good matrix for λ is an array of sets

⟨C(α, i): α ∈ S, i ∈ X_{α}⟩

such that (1) C(α, i) is club in α.

(2) X_{α} is a club^{∗} subset of λ.

(3) ot(C(α, i)) < λ.

(4) If i ∈ X_{α} and β ∈ lim(C(α, i)) then i ∈ X_{β} and C(α, i) ∩ β = C(β, i).

(5) If i, j ∈ X_{α} and i < j then C(α, i) ⊆ C(α, j).

(6) If α, β ∈ S with β < α then β ∈ lim(C(α, i)) for some i ∈ X_{α} (and thus for all larger i ∈ X_{α} by the preceding clause).

We now show how to finish the proof of Theorem 4.2, given the existence of a good matrix for κ. Let ⟨C(α, i): α ∈ S, i ∈ X_{α}⟩ be such a matrix. Let ⟨κ_{i}: i < ω⟩ be a Prikry sequence generic for the forcing ℙ_{U}. As we showed above, for every α the club^{∗} set X_{α} contains a final segment of ⟨κ_{i}: i < ω⟩.

We define our □_{κ}^{ω}-sequence ⟨D_{α}: α < κ^{+}, lim(α)⟩. Let α < κ^{+} be a limit ordinal. We distinguish two cases. Case I. α ∈ S. Let D_{α} = {C(α, κ_{j}): κ_{j} ∈ X_{α}}. Case II. α ∉ S, so that V ⊧ cf(α) = κ and W ⊧ cf(α) = cf(κ) = ω. Choose C_{α} to be any set which is club in α with ot(C_{α}) = ω, and then set D_{α} = {C_{α}}.

We need to verify that we have defined a □_{κ}^{ω}-sequence. It is clear that |D_{α}| ≤ ω and D_{α} is a family of clubs each with order type less than κ. To finish, suppose that C ∈ D_{α} and β ∈ lim(C). Clearly α ∈ S, because otherwise C = C_{α} and C_{α} has no limit points. So C = C(α, κ_{j}) for some j with κ_{j} ∈ X_{α}. By the properties of a good matrix κ_{j} ∈ X_{β} and C(β, κ_{j}) = C ∩ β, so that C ∩ β ∈ D_{β}.

This shows that ⟨D_{α}: α < κ^{+}⟩ is a □_{κ}^{ω}-sequence, so □_{κ}^{ω} holds in W and we are done once we have shown the following Lemma.

Lemma 4.4. If λ is inaccessible there is a good matrix.

Proof. We construct a good matrix by induction on α ∈ S.

Case 1: α = ω. We set X_{ω} = λ and C(ω, i) = ω for all i.

Case 2: α = β + ω for some limit ordinal β with cf(β) < λ (that is to say β ∈ S). We set X_{α} = X_{β} and C(α, i) = C(β, i) ∪ [β, α) for all i ∈ X_{α}.

Clearly C(α, i) is club in α. By definition X_{α} = X_{β}, and so X_{α} is club^{∗}. Since α = β + ω, ot(C(α, i)) = ot(C(β, i)) + ω and so ot(C(α, i)) < λ.

If i ∈ X_{α} and γ ∈ lim C(α, i) then either γ ∈ lim C(β, i) or γ = β. In the former case we have by induction that i ∈ X_{γ} and C(γ, i) = C(β, i) ∩ γ, in the latter that i ∈ X_{β} = X_{γ} and C(γ, i) = C(β, i): in either case C(α, i) ∩ γ = C(γ, i).

If i, j ∈ X_{α} with i < j then by induction C(β, i) ⊆ C(β, j), so that C(α, i) ⊆ C(α, j). Finally if γ ∈ S ∩α then either γ ∈ S ∩β or γ = β: if γ ∈ S ∩ β then by induction γ ∈ lim(C(β, i)) for some i and then γ ∈ lim(C(α, i)) for the same i, while if γ = β then γ ∈ lim(C(α, i)) for every i ∈ X_{α}.

Case 3: cf(α) = ω and α is a limit of limit ordinals. We choose ⟨α_{m}: m < ω⟩ an increasing sequence of ordinals in S which is cofinal in α. We set

X_{α} = {i < λ: ∀m < ω i ∈ X_{α_{m}} ∧ ∀m < n < ω α_{m} ∈ lim(C(α_{n}, i))}.

X_{α} is a club^{∗} set because it is a final segment of ⋂_{j} X_{α_{j}}.

We observe that if i ∈ X_{α} then C(α_{m}, i) = C(α_{n}, i) ∩ α_{m} for all m < n < ω. We now set C(α, i) = ⋃_{m} C(α_{m}, i) for all i ∈ X_{α}.

C(α, i) is club in α because every initial segment is an initial segment of C(α_{m}, i) for some m. A similar argument shows that ot(C(α, i)) < λ. If β ∈ lim(C(α, i)) then β ∈ lim(C(α_{m}, i)) for some m, and by induction i ∈ X_{β} and C(β, i) = C(α_{m}, i)∩β = C(α, i)∩β.

If i, j ∈ X_{α} with i < j then by induction C(α_{m}, i) ⊆ C(α_{m}, j) for all m < ω, so that C(α, i) ⊆ C(α, j). Finally if β ∈ S ∩ α then β ∈ S ∩α_{m} for some m, and so by induction β ∈ lim(C(α_{m}, i)) for all large i ∈ X_{α_{m}}; it follows that β ∈ lim(C(α, i)) for any large enough i ∈ X_{α}.

Case 4: ω < cf(α) < λ. Let cf(α) = ρ say. As in Case 3 we fix ⟨α_{m}: m < ρ⟩ an increasing and continuous sequence of members of S which is cofinal in α. We define

Y_{α} = {i < λ: ∀m < ρ i ∈ X_{α_{m}} and ∀m < n < ρ α_{m} ∈ lim C(α_{n}, i)}.

Note that Y_{α} depends on the choice of the sequence ⟨α_{m}: m < ρ⟩ used in its definition. Exactly as in Case 3 Y_{α} is a club^{∗} set, and if i ∈ Y_{α} then C(α_{m}, i) = C(α_{n}, i) ∩ α_{m} for all m < n < ρ.

Unfortunately Y_{α} will not quite do as a candidate for X_{α} because its dependence on the choice of ⟨α_{m}: m < ρ⟩ would cause a problem in Case 5. We choose X_{α} in a more canonical way and make it as large as possible. To be more precise we let

X_{α} = {i < λ: ∃E club in α ∀γ ∈ lim(E)(i ∈ X_{γ} ∧ E ∩ γ = C(γ, i))}.

If i ∈ Y_{α} and we let E = ⋃_{m} C(α_{m}, i) then it is easy to check that E witnesses i ∈ X_{α}, so that Y_{α} ⊆ X_{α}.

Suppose that i ∈ X_{α} and E, E′ are both clubs in α witnessing this. Then E ∩ E′ is club in α and

E = ⋃_{γ∈lim(E∩E′)} C(γ, i) = E′.

For each i ∈ X_{α}, we now define C(α, i) to be the unique E which is club in α and is such that ∀γ ∈ lim(E) E ∩ γ = C(γ, i). Notice that if i ∈ Y_{α} then automatically C(α, i) = ⋃_{m} C(α_{m}, i).

Since every initial segment of C(α, i) is an initial segment of C(γ, i) for some γ < α, ot(C(α, i)) < λ. If β ∈ lim(C(α, i)) then β ∈ lim(C(γ, i)) for some γ ∈ lim(C(α, i)), and we have by induction that i ∈ X_{β} and C(β, i) = C(γ, i) ∩ β = C(α, i) ∩ β.

Let i, j ∈ X_{α} with i < j. Let C(α, i) = E and C(α, j) = F. Then

E = ⋃_{γ∈lim(E∩F)} C(γ, i) ⊆ ⋃_{γ∈lim(E∩F)} C(γ, j) = F,

that is to say that C(α, i) ⊆ C(α, j). Finally we may argue as in Case 3 that S ∩ α ⊆ ⋃_{i∈Y_{α}} lim C(α, i), which suffices since Y_{α} ⊆ X_{α}.

Case 5: α = β +ω where cf(β) = λ. We fix ⟨β_{i}: i < λ⟩ an increasing and continuous sequence of members of S which is cofinal in β. Let

Z = {i < λ: ∀j < i i ∈ X_{βj} and ∀j < k < i β_{j} ∈ lim(C(β_{k}, i))}.

We claim that Z is club^{∗} in λ. To see this first observe that if D = {i < λ: ∀j < i i ∈ X_{β_{j}}} then D is a diagonal intersection of sets in the club^{∗} filter, and since that filter is normal D is a club^{∗} set. Define f: [λ]^{2} ⟶ λ by setting f(j, k) equal to the least i ∈ X_{β_{k}} with β_{j} ∈ lim C(β_{k}, i), and let C be the club set of i < λ which are closed under f. If i ∈ D ∩ C then

(1) Since i ∈ D, i ∈ X_{β_{j}} for all j < i.

(2) If j, k < i then since i ∈ C we have f(j, k) < i, and by definition f(j, k) ∈ X_{β_{k}} and β_{j} ∈ lim(C(β_{k}, f(j, k))). Since i ∈ D we also have i ∈ X_{β_{k}}, and so by the properties of a good matrix C(β_{k}, f(j, k)) ⊆ C(β_{k}, i) and so β_{j} ∈ lim(C(β_{k}, i)).

It follows that D ∩ C ⊆ Z, and so Z is a club^{∗} set.

We let X_{α} = {i ∈ D ∩ C: cf(i) > ω}. Let i ∈ X_{α} and consider the construction at level β_{i}; since cf(i) > ω and the sequence ⟨β_{j}: j < λ⟩ is continuous, cf(β_{i}) = cf(i) > ω and the relevant clause of the definition is Case 4.

If we let E = ⋃_{j<i} C(β_{j}, i) then the fact that i ∈ Z and the coherence properties of the good matrix imply that ∀γ ∈ lim(E) E ∩ γ = C(γ, i), so that by the definition of X_{β_{i}} and C(β_{i}, i) from Case 4 i ∈ X_{β_{i}} and C(β_{i}, i) = ⋃_{j<i} C(β_{j}, i).

We define

C(α, i) = C(β_{i}, i) ∪ {β_{i}} ∪ [β, α).

Clearly C(α, i) is club in α, and ot(C(α, i)) = ot(C(β_{i}, i))+ω < λ. If γ ∈ lim C(α, i) then either γ ∈ lim C(β_{i}, i) or γ = β_{i}, and in either case it is easy to see that i ∈ X_{γ} and C(γ, i) = C(β_{i}, i) ∩ γ = C(α, i) ∩ γ.

Let i, j ∈ X_{α} with i < j. By induction

C(β_{i}, i) = ⋃_{k<i} C(β_{k}, i) ⊆ ⋃_{k<i} C(β_{k}, j) ⊆ ⋃_{k<j} C(β_{k}, j) = C(β_{j}, j).

Since C(β_{j}, j) is club in β_{j} and C(β_{i}, i) is cofinal in β_{i}, it follows that β_{i} ∈ C(β_{j}, j). Therefore by definition C(α, i) ⊆ C(α, j).

Finally let γ ∈ S ∩ α, and observe that since β ∉ S we have S ∩ α = S ∩ β. Find i such that γ < β_{i}, and then j ∈ X_{α} such that i < j and γ ∈ lim C(β_{i}, j). Since C(β_{j}, j) = ⋃_{k<j} C(β_{k}, j), γ ∈ lim C(β_{j}, j).

This concludes the proof of Lemma 4.4. □

The construction of a good matrix for κ in Lemma 4.4 concludes the proof of Theorem 4.2. □

It is natural to ask what happens when the cofinality of κ is changed to some value other than ω, for example by Radin forcing. Apter and Cummings [1] studied this question and used the ideas of Theorem 4.2 and Fact 4.1 to show

Fact 4.5. Let GCH hold and let κ be a κ^{+5}-supercompact cardinal. Then there exists a forcing poset ℙ such that in V^{ℙ}

(1) κ is κ^{+5}-supercompact.

(2) For every singular cardinal λ < κ

(a) There exists S ⊆ λ^{+} stationary such that any family of size less than cf(λ) of stationary subsets of S reflects simultaneously to a point of cofinality μ for unboundedly many μ < λ.

(b) The combinatorial principle □_{λ}^{cf(λ)} holds.

We also note a connection with some work of Gitik, Dzamonja and Shelah. Strengthening a result of Gitik [10], Dzamonja and Shelah [7] showed some results on “outside guessing of clubs” which have the following corollary:

Fact 4.6. Let V ⊆ W be inner models of ZFC and let GCH hold in V. Suppose that κ is a W -cardinal such that κ_{V}^{+} = κ_{W}^{+} , W ⊧ cf(κ) = ω and V ⊧ κ is inaccessible. Then there is in W an ω-sequence which is cofinal in κ and is eventually contained in every club^{∗} subset of κ from V.

It follows that Theorem 4.2 can be generalised to a wider class of extensions.

5. Distinguishing squares

Jensen showed in unpublished work [11] that □_{ℵ_{1}} is strictly stronger than □_{ℵ_{1}}^{2}. His methods can be used to distinguish the principles □_{κ}^{λ} for a fixed regular κ, and similar results can be proved [5] for κ singular. In this section we use methods similar to those of [11] to show where the simplest indexed versions of weak square principles fit in.

Theorem 5.1. Let κ be Mahlo. Then (1) There is a forcing extension in which κ = ℵ_{2}, □_{ℵ_{1}}^{2} holds and

Index-□_{ℵ_{1}}^{2} fails.

(2) There is a forcing extension in which κ = ℵ_{2}, Index-□_{ℵ_{1}}^{2} holds and □_{ℵ_{1}} fails.

Proof. We will prove the first claim of the theorem in some detail, and then indicate how to modify the proof to give the second claim.

Let δ be inaccessible. We begin by describing a countably closed forcing ℙ_{δ} which will collapse δ to be ℵ_{2} and at the same time will add a □_{ℵ_{1}}^{2}-sequence. The sequence we add will have the special property that at points of uncountable cofinality it only gives a single club set.

p ∈ ℙ_{δ} iff p is a function such that

(1) dom(p) is a countable set of limit ordinals less than δ.

(2) If cf(α) = ω and α ∈ dom(p) then 1 ≤ |p(α)| ≤ 2 and each set in p(α) is a club subset of α with countable order type.

(3) If cf(α) > ω then p(α) = {C} where C is a closed bounded subset of α with countable order type, and the largest point of C is greater than sup(dom(p) ∩ α).

(4) If α ∈ dom(p), C ∈ p(α) and β ∈ lim(C), then β ∈ dom(p) and C ∩ β ∈ p(β).

If p, q ∈ ℙ_{δ} then p ≤ q iff

(1) dom(q) ⊆ dom(p).

(2) For all α ∈ dom(q)

(a) If cf(α) = ω then p(α) = q(α).

(b) If cf(α) > ω, p(α) = {C} and q(α) = {D} then D = C ∩ (max(D) + 1).

Lemma 5.2. Let δ be inaccessible. Then • ℙ_{δ} is δ-c.c. and countably closed.

• ℙ_{δ} collapses δ to ℵ_{2} and adds a □_{ℵ_{1}}^{2}-sequence.

Proof. This is routine. The only slightly delicate point comes in checking that ℙ_{δ} is countably closed. Let ⟨p_{n}: n < ω⟩ be a decreasing sequence of conditions, and let α ∈ ⋃_{n} dom(p_{n}) be an ordinal such that cf(α) > ω and the value of p_{n}(α) does not eventually stabilise for large n. The third clause in the definition of a condition implies that max p_{n}(α) > sup(dom(p_{n}) ∩ α), so that if β = sup_{n} max p_{n}(α) then β ∉ ⋃_{n} dom(p_{n}) and we are at liberty to define a lower bound p_{ω} for ⟨p_{n}: n < ω⟩ with p_{ω}(β) = {⋃_{n} p_{n}(α)}. □

Now we suppose that γ, δ are inaccessible with γ < δ. We will show that ℙ_{δ} can be viewed as a three step iteration ℙ_{γ} ∗ 𝕋 ∗ ℚ, where 𝕋 adds a suitable club at γ and ℚ adds suitable clubs in the interval (γ, δ). Conditions in 𝕋 and ℚ are countable sets of ordinals, and so since ℙ_{δ} is countably closed we will have 𝕋 ⊆ V and ℚ ⊆ V (though of course these posets will not be members of V).

Definition 5.3. Let γ, δ be inaccessible with γ < δ.

(1) If C⃗ = ⟨C_{α}: α < γ⟩ is the sequence added by ℙ_{γ}, then 𝕋 is the poset in V [C⃗] defined as follows.

(a) t ∈ T iff t is a countable, closed and bounded subset of γ such that ∀α ∈ lim(t) t ∩ α ∈ C_{α}.

(b) If t, t′ ∈ 𝕋 then t ≤ t′ iff t = t′ ∩ (max(t) + 1).

(2) If C⃗ = ⟨C_{α}: α < γ⟩ is the sequence added by ℙ_{γ} then ℚ is the poset in V [C⃗] defined as follows:

(a) q ∈ ℚ iff q is a function such that

(i) dom(q) is a countable set of limit ordinals in the interval (γ, δ).

(ii) If cf(α) = ω and α ∈ dom(q) then 1 ≤ |q(α)| ≤ 2 and each set in q(α) is a club subset of α with countable order type.

(iii) If cf(α) > ω then q(α) = {C} where C is a closed bounded subset of α, C has countable order type, and max(C) > sup(dom(q) ∩ α).

(iv) If α ∈ dom(q), C ∈ q(α) and β ∈ lim(C) then (A) If β > γ, then β ∈ dom(q) and C ∩ β ∈ q(β). (B) If β < γ, then C ∩ β ∈ C_{β}.

(b) If q, q′ ∈ ℚ then q′ ≤ q iff

(i) dom(q) ⊆ dom(q′). (ii) For all α ∈ dom(q)

(A) If cf(α) = ω then q′(α) = q(α).

(B) If cf(α) > ω, q(α) = {C} and q′(α) = {D} then D = C ∩ (max(D) + 1).

Remark: We can define ℚ in V^{ℙ_{γ}} because γ can not be a limit point of any club in C_{α} for γ < α < δ.

Lemma 5.4. Let γ, δ be inaccessible cardinals with γ < δ. Then there is an isomorphism between a dense subset of ℙ_{δ} and a dense subset of ℙ_{γ} ∗ 𝕋 ∗ ℚ.

Moreover, ℚ is countably closed in V^{ℙ_{γ}∗𝕋}.

Proof. Let D_{0} = {p ∈ ℙ_{δ}: γ ∈ dom(p)} and D_{1} = {q: ∃p ∈ D_{0} q = (p ↾ γ, p(γ), p ↾ (γ, δ))}.

It is easy to see that D_{0} is dense in ℙ_{δ}, D_{1} ⊆ ℙ_{γ} ∗ 𝕋 ∗ ℚ, and the map ϕ: p ⟼ (p ↾ γ, p(γ), p ↾ (γ, δ)) is an isomorphism between D_{0} and D_{1}. In fact we wrote the definitions of 𝕋 and ℚ to make this true.

It remains to be seen that D_{1} is dense in ℙ_{γ} ∗ 𝕋 ∗ ℚ. To see this let (p, ṫ, q̇) be an arbitrary condition in ℙ_{γ} ∗ 𝕋 ∗ ℚ. Since ℚ ⊆ V we may find (p_{1}, ṫ_{1}) ≤ (p, ṫ) and q such that (p_{1}, ṫ_{1}) ⊩ q̇ = q̌ and then p_{2} ≤ p_{1} and t_{1} such that p_{2} ⊩ ṫ_{1} = ť_{1}. By construction (p_{2}, t_{1}, q) ∈ P_{γ} ∗ 𝕋 ∗ ℚ and (p_{2}, t_{1}, q) ≤ (p, ṫ, q̇).

Now p_{2} ⊩ ť_{1} ∈ 𝕋 and (p_{2}, t_{1}) ⊩ q̌ ∈ ℚ. It is routine to check that if we define p^{∗} = p_{2} ∪ {(γ, t_{1})} ∪ q then p^{∗} ∈ ℙ_{δ} and ϕ(p^{∗}) = (p_{2}, t_{1}, q).

The proof that ℚ is countably closed in V^{ℙ_{γ}∗𝕋} is just like the proof that ℙ_{δ} is countably closed in V. □

We will be done once we have proved the following result.

Claim 5.5. If κ is Mahlo then Index-□_{ℵ_{1}}^{2} fails in V^{ℙ_{κ}}.

Proof. Suppose not. For simplicity we assume that the empty condition forces that the principle holds, say

⊩_{ℙ_{κ}} “(Ȧ, Ċ) witnesses Index-□_{ℵ_{1}}^{2}.”

By the κ-c.c. for ℙ_{κ} and the Mahloness of κ we may find δ < κ such that δ is inaccessible and (Ȧ ↾ δ, Ċ ↾ δ × ℵ_{1}) is a name in V^{ℙ_{δ}}. This implies that

⊩_{ℙ_{δ}} “(Ȧ ↾ δ, Ċ ↾ δ × ℵ_{1}) witnesses Index-□_{ℵ_{1}}^{2}.”

We now identify ℙ_{κ} with ℙ_{δ} ∗ 𝕋 ∗ ℚ where 𝕋, ℚ are defined as in Lemma 5.4. Fix a condition (p, t, q) which forces that α ∈ A(δ) for some α < ℵ_{1}. If Ḋ = Ċ(δ, α) then (p, t, q) forces that Ḋ is club in δ, ot(Ḋ) = ℵ_{1} and ∀γ ∈ lim(Ḋ) Ḋ ∩ γ = Ċ(γ, α).

The object Ḋ cannot exist in the generic extension by ℙ_{δ}, so we claim that we may find conditions (p′, t_{0}, q_{0}) and (p′, t_{1}, q_{1}) both extending (p, t, q) and an ordinal ζ < δ such that

(p′, t_{0}, q_{0}) ⊩ ζ̌ ∈ Ḋ
(p′, t_{1}, q_{1}) ⊩ ζ̌ ∉ Ḋ

If this were not so then we would have

∀p′ ≤ p ∀ζ, t_{0}, q_{0}, t_{1}, q_{1} (p′, t_{0}, q_{0}) ⊩ ζ̌ ∈ Ḋ ⟺ (p′, t_{1}, q_{1}) ⊩ ζ̌ ∈ Ḋ,

which would imply that below (p, t, q) the name Ḋ was equivalent to a ℙ_{δ}-name.

We build sequences ⟨p_{n}: 1 ≤ n < ω⟩, ⟨t_{0}^{2n+1}: n < ω⟩, ⟨t_{1}^{2n+2}: n < ω⟩, ⟨q_{0}^{2n+1}: n < ω⟩, ⟨q_{1}^{2n+2}: n < ω⟩, and ⟨ζ_{n}: 1 ≤ n < ω⟩ such that

(1) p_{n} ∈ ℙ_{δ}, p_{1} ≤ p′ and ⟨p_{n}: 1 ≤ n < ω⟩ is decreasing.

(2) (p_{1}, t_{0}^{1}, q_{0}^{1}) ≤ (p′, t_{0}, q_{0}) and ⟨(p_{2n+1}, t_{0}^{2n+1}, q_{0}^{2n+1}): n < ω⟩ is decreasing.

(3) (p_{2}, t_{1}^{2}, q_{1}^{2}) ≤ (p′, t_{1}, q_{1}) and ⟨(p_{2n+2}, t_{1}^{2n+2}, q_{1}^{2n+2}): n < ω⟩ is decreasing.

(4) ⟨ζ_{n}: 1 ≤ n < ω⟩ is an increasing sequence of ordinals such that (a) ζ_{1} > max{max(t_{0}), max(t_{1}), ζ}.

(b) ζ_{2n+1} < max{max(t_{0}^{2n+1}), sup dom(p_{2n+1})} < ζ_{2n+2}.

(c) ζ_{2n+2} < max{max(t_{1}^{2n+2}), sup dom(p_{2n+2})} < ζ_{2n+3}.

(d) (p_{2n+1}, t_{0}^{2n+1}, q_{0}^{2n+1}) ⊩ ζ̌_{2n+1} ∈ Ḋ.

(e) (p_{2n+2}, t_{1}^{2n+2}, q_{1}^{2n+2}) ⊩ ζ̌_{2n+2} ∈ Ḋ. Let p_{ω} ∈ ℙ_{δ} be a lower bound for the sequence ⟨p_{n}: 1 ≤ n < ω⟩.

Since ℚ is countably closed in V^{ℙ_{δ}} we may find q_{0}^{∗} and q_{1}^{∗} such that (p_{ω}, q_{0}^{∗}) is a lower bound for ⟨(p_{2n+1}, q_{0}^{2n+1}): n < ω⟩ and (p_{ω}, q_{1}^{∗}) is a lower bound for ⟨(p_{2n+2}, q_{1}^{2n+2}): n < ω⟩. Now define

ζ^{∗} = sup_{n} ζ_{n},
t_{0}^{∗} = ⋃_{n} t_{0}^{2n+1} ∪ {ζ^{∗}},
t_{1}^{∗} = ⋃_{n} t_{1}^{2n+2} ∪ {ζ^{∗}},
p^{∗} = p_{ω} ∪ {(ζ^{∗}, {t_{0}^{∗}, t_{1}^{∗}})}.

It is routine to check that (p^{∗}, t_{0}^{∗}, q_{0}^{∗}) and (p^{∗}, t_{1}^{∗}, q_{1}^{∗}) are both conditions in ℙ_{δ} ∗ 𝕋 ∗ ℚ.

The conditions (p^{∗}, t_{0}^{∗}, q_{0}^{∗}) and (p^{∗}, t_{1}^{∗}, q_{1}^{∗}) both force ζ^{∗} to be a limit point of Ḋ, so

(p^{∗}, t_{0}^{∗}, q_{0}^{∗}) ⊩ ζ̌ ∈ Ċ(ζ^{∗}, α)
(p^{∗}, t_{1}^{∗}, q_{1}^{∗}) ⊩ ζ̌ ∉ Ċ(ζ^{∗}, α)

This is absurd because Ċ(ζ^{∗}, α) is a name in V^{ℙ_{δ}}, so that the preceding equations imply p^{∗} ⊩ ζ̌ ∈ Ċ(ζ^{∗}, α) and p^{∗} ⊩ ζ̌ ∉ Ċ(ζ^{∗}, α).

□

This concludes the proof of the first claim of Theorem 5.1.

For the second claim, we start by defining a poset ℙ_{δ}^{∗} which is designed to add an Index-□_{ℵ_{1}}^{2}-sequence while collapsing an inaccessible δ to become ℵ_{2}. This sequence will have the special properties that it only gives one club set at limit ordinals of cofinality greater than ω, and that the only indices which are used are 0 and 1.

p ∈ ℙ_{δ}^{∗} iff p is a pair (a, c) where

(1) a is a function with dom(a) a countable set of limit ordinals less than δ.

(2) For every ν ∈ dom(a), a(ν) is a nonempty subset of {0, 1}. If cf(ν) > ω then |a(ν)| = 1.

(3) c is a function with domain {(ν, α): ν ∈ dom(a), α ∈ a(ν)}.

(4) If cf(ν) = ω and (ν, α) ∈ dom(c) then c(ν, α) is a club subset of ν with countable order type.

(5) If cf(ν) > ω and (ν, α) ∈ dom(c) then c(ν, α) is a closed bounded subset of ν with countable order type, with the additionla property that max(c(ν, α)) > sup(dom(a) ∩ ν).

(6) If (ν, α) ∈ dom(c) and β ∈ lim(c(ν, α)) then (ν, β) ∈ dom(c) and c(ν, β) = β ∩ c(ν, α).

Conditions in ℙ_{δ}^{∗} are ordered as follows: (a_{1}, c_{1}) ≤ (a_{0}, c_{0}) iff

(1) dom(a_{0}) ⊆ dom(a_{1}).

(2) For all ν ∈ dom(a_{0})

(a) a_{0}(ν) = a_{1}(ν).

(b) For all α ∈ a_{0}(ν), c_{0}(ν, α) = c_{1}(ν, α). As before it is easy to see that (1) ℙ_{δ}^{∗} is countably closed and δ-c.c.

(2) ℙ_{δ}^{∗} collapses δ to ℵ_{2} and adds (A, C) witnessing Index-□_{ℵ_{1}}^{2}. ℙ_{δ}^{∗} is susceptible to a factor analysis very similar to that which we

gave for ℙ_{δ} above. The main difference is that we need two versions of 𝕋 and ℚ, reflecting the fact that at γ we must decide whether to put a club set with index 0 or a club set with index 1.

Lemma 5.6. Let γ, δ be inaccessible with γ < δ. There exist posets 𝕋^{0}, 𝕋^{1}, ℚ^{0}, ℚ^{1} ∈ V^{ℙ_{γ}} such that

(1) If p = (a, c) ∈ ℙ_{δ}^{∗} and (γ, 0) ∈ dom(c) then ℙ_{δ}^{∗}/p is isomorphic to a dense subset of

ℙ_{γ}^{∗}/(a ↾ γ, c ↾ γ × 2) × 𝕋^{0}/c(γ, 0) × ℚ^{0}/(a ↾ (γ, δ), c ↾ (γ, δ) × 2).

(2) If p = (a, c) ∈ ℙ_{δ}^{∗} and (γ, 1) ∈ dom(c) then ℙ_{δ}^{∗}/p is isomorphic to a dense subset of

ℙ_{γ}^{∗}/(a ↾ γ, c ↾ γ × 2) × 𝕋^{1}/c(γ, 1) × ℚ^{1}/(a ↾ (γ, δ), c ↾ (γ, δ) × 2).

(3) ℚ^{j} is countably closed in V^{ℙ_{γ}^{∗}∗𝕋^{j}}.

Proof. The definitions and proofs are like those of Definition 5.3 and Lemma 5.4. □

Claim 5.7. If κ is Mahlo then □_{ℵ_{1}} fails in V^{ℙ_{κ}^{∗}}.

Proof. Suppose that

⊩_{ℙ_{κ}^{∗}} “⟨Ḋ_{α}: α < κ⟩ is a □_{ℵ_{1}}-sequence”

By the κ-c.c. for ℙ_{κ}^{∗} and the Mahloness of κ we may find δ < κ such that δ is inaccessible and ⟨Ḋ_{α}: α < δ⟩ is a name in V^{ℙ_{δ}^{∗}}. This implies that

⊩_{ℙ_{δ}^{∗}} “⟨Ḋ_{α}: α < δ⟩ is a □_{ℵ_{1}}-sequence”

We now consider the ℙ_{κ}^{∗}-name Ḋ = Ḋ_{δ}. We claim that we may find conditions (p, t^{0}, q^{0}) ∈ ℙ_{δ}^{∗} ∗ 𝕋^{0} ∗ ℚ^{0} and (p, t^{1}, q^{1}) ∈ ℙ_{δ}^{∗} ∗ 𝕋^{1} ∗ ℚ^{1} together with an ordinal ζ < δ such that either

(p, t^{0}, q^{0}) ⊩ ζ̌ ∈ Ḋ
(p, t^{1}, q^{1}) ⊩ ζ̌ ∉ Ḋ

or

(p, t^{0}, q^{0}) ⊩ ζ̌ ∉ Ḋ

(p, t^{1}, q^{1}) ⊩ ζ̌ ∈ Ḋ

To see this we first find conditions (p^{0}, u_{0}^{0}, r_{0}^{0}) and (p^{0}, u_{1}^{0}, r_{1}^{0}) from ℙ_{δ} ∗ 𝕋^{0} ∗ ℚ^{0} and an ordinal ζ such that

(p^{0}, u_{0}^{0}, r_{0}^{0}) ⊩ ζ̌ ∈ Ḋ,
(p^{0}, u_{1}^{0}, r_{1}^{0}) ⊩ ζ̌ ∉ Ḋ.

This is possible because Ḋ names a set which is not in V^{ℙ_{δ}}. We now find (p, u^{1}, r^{1}) in ℙ_{δ} ∗ 𝕋^{1} ∗ ℚ^{1} such that p ≤ p^{0} and (p, u^{1}, r^{1}) decides the statement “ζ ∈ Ḋ”, and then choose t^{i} and q^{i} accordingly.

We build sequences ⟨p_{n}: 1 ≤ n < ω⟩, ⟨t_{0}^{2n+1}: n < ω⟩, ⟨t_{1}^{2n+2}: n < ω⟩, ⟨q_{0}^{2n+1}: n < ω⟩, ⟨q_{1}^{2n+2}: n < ω⟩, and ⟨ζ_{n}: 1 ≤ n < ω⟩ such that

(1) p_{n} ∈ ℙ_{δ}, p_{1} ≤ p′ and ⟨p_{n}: 1 ≤ n < ω⟩ is decreasing.

(2) (p_{2n+1}, t_{0}^{2n+1}, q_{0}^{2n+1}) ∈ ℙ_{δ} ∗𝕋^{0} ∗ℚ^{0}, (p_{1}, t_{0}^{1}, q_{0}^{1}) ≤ (p, t^{0}, q^{0}), and ⟨(p_{2n+1}, t_{0}^{2n+1}, q_{0}^{2n+1}): n < ω⟩ is decreasing.

(3) (p_{2n+2}, t_{1}^{2n+2}, q_{1}^{2n+2}) ∈ ℙ_{δ} ∗𝕋^{1} ∗ℚ^{1}, (p_{2}, t_{1}^{2}, q_{1}^{2}) ≤ (p, t^{1}, q^{1}), and ⟨(p_{2n+2}, t_{1}^{2n+1}, q_{1}^{2n+2}): n < ω⟩ is decreasing.

(4) ⟨ζ_{n}: 1 ≤ n < ω⟩ is an increasing sequence of ordinals such that (a) ζ_{1} > max{max(t^{0}), max(t^{1}), ζ}.

(b) ζ_{2n+1} < max{max(t_{0}^{2n+1}), sup dom(p_{2n+1})} < ζ_{2n+2}.

(c) ζ_{2n+2} < max{max(t_{1}^{2n+2}), sup dom(p_{2n+2})} < ζ_{2n+3}.

(d) (p_{2n+1}, t_{0}^{2n+1}, q_{0}^{2n+1}) ⊩ ζ̌_{2n+1} ∈ Ḋ.

(e) (p_{2n+2}, t_{1}^{2n+2}, q_{1}^{2n+2}) ⊩ ζ̌_{2n+2} ∈ Ḋ.

Let ζ^{∗} = ⋃_{n} ζ_{n}, and let p_{ω} be a lower bound for ⟨p_{n}: 1 ≤ n < ω⟩. Define

u^{0} = ⋃_{n} t_{2n+1}^{0} ∪ {ζ^{∗}},
u^{1} = ⋃_{n} t_{2n+2}^{1} ∪ {ζ^{∗}},
p^{∗} = p_{ω} ∪ {((ζ^{∗}, 0), u^{0}), ((ζ^{∗}, 1), u^{1}).}

Using the countable closure of the ℚ^{j}, we find r^{0} and r^{1} such that (p^{∗}, u^{0}, r^{0}) is a lower bound for ⟨(p_{2n+1}, t_{0}^{2n+1}, q_{0}^{2n+1}): n < ω⟩, and (p^{∗}, u^{1}, r^{1}) is a lower bound for ⟨(p_{2n+2}, t_{1}^{2n+2}, q_{1}^{2n+2}): n < ω⟩.

The conditions (p^{∗}, u^{0}, r^{0}) and (p^{∗}, u^{1}, r^{1}) both force ζ^{∗} to be a limit point of Ḋ, so

(p^{∗}, u^{0}, r^{0}) ⊩ ζ̌ ∈ Ḋ_{ζ^{∗}}
(p^{∗}, u^{1}, r^{1}) ⊩ ζ̌ ∉ Ḋ_{ζ^{∗}}

But Ḋ_{ζ^{∗}} is a V^{ℙ_{δ}}-name so p^{∗} ⊩ ζ̌ ∈ Ḋ_{ζ^{∗}} and p^{∗} ⊩ ζ̌ ∉ Ḋ_{ζ^{∗}}. This is a contradiction. □

This concludes the proof of Theorem 5.1. □

6. Global square and 1-extendible cardinals

In this section we investigate the question of how strong a large cardinal axiom has to be before it becomes incompatible with the existence of square sequences. We start by recalling the definition of a 1-extendible cardinal.

Definition 6.1. κ is 1-extendible iff there exist a cardinal λ > κ and π: H_{κ^{+}} ⟶ H_{λ^{+}} an elementary embedding with crit(π) = κ and π(κ) = λ.

For more information about extendible cardinals see Kanamori’s book[14]. We note that if γ is a cardinal then γ is definable in H_{γ^{+}} as the largest cardinal, so that the demand that π(κ) = λ in the definition of 1-extendibility is superfluous; it follows from the elementarity of the map π.

Jensen [13] introduced a strengthening of 1-extendibility called quasicompactness. For expository purposes we will also define an intermediate notion 1-extendible in A.

Definition 6.2. Let κ be a cardinal.

(1) For A ⊆ H_{κ^{+}}, κ is 1-extendible in A iff there exist a cardinal λ > κ, a set B ⊆ H_{λ^{+}} and an elementary embedding π from (H_{κ^{+}}, ∈, A) to (H_{λ^{+}}, ∈, B), such that π(κ) = λ and the critical point of π is κ.

(2) κ is quasicompact iff κ is 1-extendible in A for all A ⊆ H_{κ^{+}}.

Jensen showed that if C⃗ = ⟨C_{α}: α < κ^{+}⟩ is such that C_{α} ⊆ α for all α, and κ is 1-extendible in C⃗ then C⃗ is not a □_{κ}-sequence. In particular if κ is quasicompact then □_{κ} fails. Reflecting on this proof Jensen introduced the notion of subcompactness.

Definition 6.3. Let κ be a cardinal. κ is subcompact iff for all A ⊆ H_{κ^{+}} there exist a cardinal α < κ, a set a ⊆ H_{α^{+}} and an elementary embedding π from (H_{α^{+}}, ∈, a) to (H_{κ^{+}}, ∈, A), such that crit(π) = α and π(α) = κ.

Jensen’s argument shows that if κ is subcompact then □_{κ} fails. We note that a subcompact cardinal need not be measurable. In fact if κ is measurable and subcompact and U is any normal measure on κ then it is routine to check that κ is subcompact in Ult(V, U), so that there are many subcompact cardinals below κ.

At this point a few words about the inner model program are in order. The goals of the program are to construct canonical “L-like” inner models for large cardinal axioms, and to analyse the internal structure of these models and their relation to V. This analysis can be used to obtain lower bounds on consistency strength for combinatorial statements. We refer the reader to the survey papers [23] and [15] and the books [19] and [24] for more information.

The inner models which are studied in the inner model program have the form L[E⃗], where E⃗ is a sequence of extenders which is subject to certain fine-structural conditions; we will refer to models of this standard type as “L[E⃗] models”. It is anticipated that all large cardinal axioms below the level of supercompactness can hold in L[E⃗]-models, but currently this has only been proved up to slightly beyond the level of a measurable limit of Woodin cardinals.

Schimmerling and Zeman have shown that in any L[E⃗]-model, if there are no subcompact cardinals then □_{λ} holds for all λ. From the discussion in the previous paragraph, this shows that □_{λ} holds for every λ is consistent with large cardinals up to slightly beyond a measurable limit of Woodin cardinals. It should eventually be possible to show that □_{λ} holds for every λ is consistent with the existence of a 1-extendible cardinal by constructing a suitable L[E⃗]-model; in this section we will use forcing to prove this consistency result. Actually we prove something slightly stronger but more technical to state, which needs a preliminary definition.

Definition 6.4. ⟨C_{α}: α ∈ ON, cf(α) < α⟩ is a global □-sequence iff (1) For every singular ordinal α, C_{α} is club in α with ot(C_{α}) < α.

(2) If cf(α) < α and β ∈ lim(C_{α}), then cf(β) < β and C_{β} = C_{α} ∩ β.

Jensen proved that if V = L there is a global square sequence, and that if a global square sequence exists then □_{κ} holds for all κ. We can now state the result of this section precisely.

Theorem 6.5. Let GCH hold, let κ be 1-extendible as witnessed by π: H_{κ^{+}} ⟶ H_{λ^{+}}, and let δ be inaccessible with δ > λ. Then in some generic extension there is a transitive set W and a predicate C⃗ on W such that (W, ∈, C⃗) is a model of ZFCC̃ + κ is 1-extendible + C⃗ is a global square sequence.

The rest of this section will be devoted to a proof of this theorem. Before starting the proof a few remarks are in order:

(1) Doug Burke [3] showed that the existence of a superstrong cardinal is consistent with □_{λ} holds for every λ.

(2) At first sight the most natural procedure for showing that a 1-extendible cardinal is consistent with global square would be to start with a model with some large cardinal κ, use class forcing to add a global square sequence and then argue that the resulting structure is a model of set theory in which κ is 1-extendible. We were unable to make this scenario work without assuming some additional reflection properties for the class of ordinals, which amounted to assuming that the universe has the form V_{δ} for δ inaccessible; we therefore decided to eliminate the complications of class forcing and build a transitive set model of our desired hypothesis by set forcing.

(3) It is easy to see that if κ is 1-extendible then κ is 1-extendible in A for every definable A, so that there can be no □_{κ}- sequence which is definable in H_{κ^{+}}. While we are on the subject of definability we note that in Theorem 6.5 the sequence C⃗ is not definable in W, so our theorem leaves open whether a definable global square sequence is consistent with the existence of a 1-extendible cardinal.

(4) Jensen showed that if □_{κ} holds for all κ and a weak form of global square holds on singular cardinals, then global square holds. Zeman showed that the weak form of global square holds in all L[E⃗] models. Combining these results with the Schimmerling-Zeman result, we see that global square holds in L[E⃗] if L[E⃗] has no subcompact cardinals. See [22].

The following definition is not standard usage but is convenient here.

Definition 6.6. Let η be an ordinal. A GS(η)-sequence is a sequence ⟨C_{α}: α < η, cf(α) < α⟩ where

(1) For every singular ordinal α < η, C_{α} is a club subset of α with ot(C_{α}) < α.

(2) If cf(α) < α and β ∈ lim(C_{α}), then cf(β) < β and C_{β} = C_{α} ∩ β.

Intuitively a GS(η)-sequence is a potential initial segment of a global square sequence.

We now state our large cardinal hypothesis, which will be in effect for the rest of this section:

Hypothesis: GCH holds and there are regular cardinals κ < λ < δ such that

(1) There exists j: H_{κ^{+}} ⟶ H_{λ^{+}} such that crit(j) = κ, j(κ) = λ and j is elementary (that is to say j witnesses that κ is 1extendible).

(2) δ is inaccessible.

Our plan for proving Theorem 6.5 is as follows: we will build a two-step generic extension V [G][g] such that

(1) δ is inaccessible in V [G][g].

(2) V_{δ}^{V [G]} = V_{δ}^{V [G][g]} (we denote this model by V_{δ}[G] below).

(3) V_{δ}[G] ⊧ “κ is 1-extendible”.

(4) In V [G][g] there is a sequence C⃗ = ⟨C_{α}: α < δ, cf(α) < α⟩ such that

(a) ot(C_{α}) < α, and ∀β ∈ lim(C_{α}) C_{β} = C_{α} ∩ β.

(b) (V_{δ}[G], ∈, C⃗) is a model of ZFCC⃗.

Before giving the details of the construction we discuss a couple of distinctive features. We note that very similar issues arise (and are discussed in more detail) in a paper by Cummings, Dzamonja and Shelah [4].

The construction is a “Reverse Easton” iteration of the same general type as those discussed in Baumgartner’s survey [2]. It is common in Reverse Easton iterations for the forcing being done at stage γ to be γ-closed, but in our situation we will only assume that it is < γ-strategically closed. We recall the definition of strategic closure.

Definition 6.7 (Foreman [9]). Let γ be a cardinal. A poset ℙ is < γ-strategically closed if and only if for every ordinal ζ < γ player II wins the following two-player game of perfect information. Players I and II collaborate to build a decreasing chain ⟨p_{α}: 0 < α⟩ in ℙ with player I playing at odd α and player II at even α (including all limit stages). Player II wins if play proceeds for ζ many moves, that is to say p_{α} is defined for all α < ζ.

Replacing closure by strategic closure necessitates a few changes in the standard Reverse Easton arguments. We outline these changes at the relevant points below.

In our iteration, at each regular γ we will force with a poset ℚ_{γ} which adds a GS(γ)-sequence by approximation via initial segments. A potential problem with this strategy is that a priori there may not be enough conditions in ℚ_{γ}, in fact what we need (see Claim 6.11 for the details) is that GS(α)-sequences already exist for all ordinals α < γ; we will arrange this using the fact that we already forced with ℚ_{μ} for all regular μ < γ and the following sequence of technical lemmas.

Lemma 6.8. Let ν be an infinite cardinal. If there exists a GS(ν)- sequence, then there exists a GS(η)-sequence for every η < ν^{+}.

Proof. Let ⟨C_{α}: α < ν, cf(α) < α⟩ be a GS(ν)-sequence. We prove the existence of a GS(η + 1)-sequence by induction on limit η in the interval [ν, ν^{+}). Case 1: η = ν. If ν is regular there is nothing to do, so we assume that ν is singular. Choose ⟨ν_{i}: i < cf(ν)⟩ increasing, continuous and cofinal in ν with ν_{0} = 0 and cf(ν) < ν_{1}. Define for singular ordinals α ≤ ν

D_{α} = \begin{cases}\{ν_{i}: i < j\} & α = ν_{j}, j limit \\ \{ν_{i}: i < cf(ν)\} & α = ν \\ C_{α} \ (ν_{i} + 1) & ν_{i} < α ≤ ν_{i+1} \end{cases}

Case 2: η = γ + ω, γ limit. Let ⟨D_{α}: α ≤ γ, cf(α) < α⟩ be a GS(γ + 1)-sequence. We may extend this to be a GS(η + 1)-sequence by defining D_{η} = {γ + n: n < ω}. Case 3: ν < η < ν^{+}, η a limit of limit ordinals. Let cf(η) = μ, where necessarily μ ≤ ν. Choose ⟨η_{i}: i < μ⟩ increasing, continuous and cofinal in η, in such a way that

(1) η_{0} = 0.

(2) η_{i+1} is a singular limit ordinal for all i.

(3) η_{1} > ν. Fix ⟨C_{α}^{i+1}: α ≤ η_{i+1}, cf(α) < α⟩ a GS(η_{i+1} + 1)-sequence for each i < μ.

Define for singular ordinals α ≤ η

D_{α} = \begin{cases}\{η_{i}: i < j\} & α = η_{j}, j limit \\ \{η_{i}: i < μ\} & α = η \\ C_{α}^{i+1} \ (η_{i} + 1) & η_{i} < α ≤ η_{i+1} \end{cases}

□

Lemma 6.9. If ν is a singular cardinal and there is a GS(μ)-sequence for every regular μ < ν, then there is a GS(ν)-sequence.

Proof. Like Case 3 in Lemma 6.8. □

Lemma 6.10. Let γ be a cardinal and suppose that for every regular cardinal μ < γ there is a GS(μ)-sequence. Then for every ordinal α < γ there is a GS(α)-sequence.

Proof. If γ is a limit cardinal then there are unboundedly many regular cardinals less than γ, and the result is clear. So suppose γ = μ^{+} for some cardinal μ. If μ is regular then there is a GS(μ)-sequence by assumption, if μ is singular then there is a GS(μ)-sequence by Lemma 6.9. In either case, by Lemma 6.8 there is a GS(α)-sequence for every α < μ^{+} = γ. □

We can now describe our iterated forcing construction. Given a regular cardinal γ we define a poset ℚ_{γ}. p ∈ ℚ_{γ} if and only if p = ⟨C_{α}: cf(α) < α, α ≤ β⟩ where

(1) β is a singular limit ordinal less than γ.

(2) p is a GS(β + 1)-sequence.

If p = ⟨C_{α}: cf(α) < α, α ≤ β⟩ and q = ⟨D_{α}: cf(α) < α, α ≤ β^{∗}⟩ are in ℚ_{γ} then p ≤ q iff β ≥ β^{∗} and C_{α} = D_{α} for all α ≤ β^{∗}. We note that by GCH ℚ_{γ} is a poset of size at most γ, and so trivially has the γ^{+}-chain condition.

Claim 6.11. If there is a GS(α)-sequence for every α < γ then forcing with ℚ_{γ} adds a GS(γ)-sequence.

Proof. We need to check that for every ζ < γ the set of GS(ζ + 1)- sequences is dense. Let p = ⟨C_{α}: cf(α) < α, α ≤ β⟩ be a GS(β + 1) sequence for some singular ordinal β, and let ζ be a singular ordinal with β < ζ < γ. Let q = ⟨D_{α}: cf(α) < α, α ≤ ζ⟩ be a GS(ζ + 1)- sequence.

We define E_{α} for singular α with α ≤ ζ by letting E_{α} = C_{α} for α ≤ β and E_{α} = D_{α} \ (β + 1) for β < ζ ≤ ζ. It is routine to check that if r = ⟨E_{α}: cf(α) < α, α ≤ ζ⟩ then r is a GS(ζ + 1)-sequence extending p. □

We define ℙ_{δ+1} to be the Reverse Easton iteration of ℚ_{γ} for regular γ ≤ δ. To be a little more explicit we define sequences ⟨ℙ_{α}: α ≤ δ + 1⟩ and ⟨ℚ̇_{α}: α ≤ δ⟩ inductively by

(1) ℚ̇_{α} is a ℙ_{α}-name for the version of ℚ_{α} computed by V^{ℙ_{α}}, if α is regular in V^{ℙ_{α}} (which will turn out to be the case for every regular α, see claim 6.12). Otherwise ℚ̇_{α} names the trivial forcing.

(2) ℙ_{α+1} = ℙ_{α} ∗ ℚ̇_{α}.

(3) For λ ≤ δ a limit ordinal, ℙ_{λ} is the direct limit of ⟨ℙ_{α}: α < λ⟩ for λ inaccessible, and the inverse limit otherwise.

Claim 6.12. Let γ be regular. Then in V^{ℙ_{γ}}

1_{γ}. For every condition p ∈ ℚ_{γ} and every ζ < γ there is a condi-

tion q ∈ ℚ_{γ} with q ≤ p and max(dom(q)) ≥ ζ. 2_{γ}. ℚ_{γ} is < γ-strategically closed. 3_{γ}. Cardinals and cofinalities are preserved.

Proof. We proceed by induction. Assume that we have 1_{δ}, 2_{δ} and 3_{δ} for regular δ < γ. We start by outlining the argument that ℙ_{γ} preserves all cardinals and cofinalities.

Given a cardinal μ < γ, we factor ℙ_{γ} in the standard way as ℙ_{μ} ∗ ℚ_{μ} ∗ ℝ and note that μ^{+} is the first point at which the iteration ℝ does non-trivial forcing. The arguments of [2], suitably adapted for strategically closed forcing, gives us that ℝ is < μ^{+}-strategically closed in V^{ℙ_{μ+1}}. The usual counting arguments give us that ℙ_{μ+1} is always μ^{+}-c.c. A suitable adaptation of Easton’s arguments from [8] shows that all μ-sequences from V^{ℙ_{γ}} must lie in V^{ℙ_{μ+1}}, and arguments exactly like those of [8] then show that all cardinals and cofinalities are preserved in V^{ℙ_{γ}}.

It follows by the induction hypothesis and Claim 6.11 that in V^{ℙ_{γ}} we will have a GS(δ)-sequence for every regular δ with δ < γ. By Lemma 6.10 there is a GS(α)-sequence for every ordinal α < γ. By Lemma 6.11 again 1_{γ} holds in V^{ℙ_{γ}}.

Recall the strategic closure game from Definition 6.7. We describe a winning strategy for player Even in the game of length α + 1 played on ℚ_{γ}, where α < γ is a limit ordinal. Let p_{β} = ⟨E_{ζ}: cf(ζ) < ζ, ζ ≤ γ_{β}⟩ be the condition which is played at stage β, where player Even’s strategy will guarantee that ⟨γ_{β}: β ≤ α⟩ is continuous.

Case 1: β = 2: Even plays a condition p_{2} ≤ p_{1} with γ_{2} > α. Notice that for all limit β ≤ α we will have cf(γ_{β}) = cf(β) ≤ α < γ_{2} < γ_{β}.

Case 2: β = β_{0} + 2, β_{0} > 0 even: Even sets γ_{β} = γ_{β_{0}+1} + ω and E_{γ_{β}} = {γ_{β_{0}+1} + n: n < ω}.

Case 3: lim(β): Even sets γ_{β} = sup_{β̄<β} γβ̄ and E_{γ_{β}} = {γβ̄: β̄ < β}. This is a legal move because

(1) If ζ ∈ lim(E_{γ_{β}}) then ζ = γβ̄ for β̄ limit, and so

E_{ζ} = {γ_{η}: η < β̄} = E_{γ_{β}} ∩ ζ.

(2) ot(E_{γ_{β}}) = β ≤ α < γ_{2} < γ_{β}.

It is routine to check that this is a winning strategy, concluding the proof of Claim 6.12. □

We will choose G to be some ℙ_{δ}-generic filter subject to a certain technical condition; if C⃗^{κ} and C⃗^{λ} are the sequences added at stages κ and λ, then we choose G so that C⃗^{λ} ↾ κ = C⃗^{κ}. This is possible because κ is regular in V [G_{λ}], so that C⃗^{κ} can be extended ^{2} to a condition M in ℚ_{λ} and we may force below M to get C⃗^{λ} as desired.

The reason for doing this is explained in detail in Claim 6.15 below; in the jargon of large cardinal theorists M is a “master condition” in ℚ_{λ}, which is to say that forcing below M at stage λ will guarantee that we can lift our original embedding j to an elementary embedding j: H_{κ^{+}}[G_{κ} ∗ g_{κ}] ⟶ H_{λ^{+}}[G_{λ} ∗ g_{λ}]. For more about master conditions see the section on Reverse Easton forcing in Baumgartner’s survey paper [2].

By Lemma 6.12, V [G][g] has the same cardinals and cofinalities as V and δ is inaccessible in V [G][g]. This implies that every set of rank less than δ is coded by a bounded subset of δ, so V_{δ}^{V [G]} = V_{δ}^{V [G][g]}; to save the reader from a plague of superscripts we denote this set by V_{δ}[G] in what follows.

Let C⃗ be the generic GS(δ)-sequence added by g. The structure (V_{δ}[G], ∈, C⃗) is a model of ZFCC⃗ because V [G][g] is a model of ZFC, δ is inaccessible in V [G][g] and C⃗ ∈ V [G][g].

To finish, we show that in V_{δ}[G] there is an elementary embedding from H_{κ^{+}}^{V_{δ}[G]} to H_{λ^{+}}^{V_{δ}[G]}. It will then follow that κ is 1-extendible in V_{δ}[G]. The elements of H_{κ^{+}} are coded by subsets of κ and no subsets of κ are added past stage κ of the iteration so H_{κ^{+}}^{V_{δ}[G]} = H_{κ^{+}}^{V [G_{κ+1}]} = H_{κ^{+}}^{V_{δ}[G_{κ+1}]} Similarly we see that H_{λ^{+}}^{V_{δ}[G]} = H_{λ^{+}}^{V [G_{λ+1}]} = H_{λ^{+}}^{V_{δ}[G_{λ+1}]}.

The intuition behind the rest of the proof is that we want to treat H_{κ^{+}}^{V_{δ}[G_{κ+1}]} as a generic extension of H_{κ^{+}} by ℙ_{κ+1}, and then apply the techniques of Reverse Easton forcing to lift the embedding j. The argument requires a little care because H_{κ^{+}} is not a model of ZFC.

Since |ℙ_{κ+1}| = κ, every element of H_{κ^{+}}^{V_{δ}[G_{κ+1}]} has the form τ̇^{G_{κ+1}} for some ℙ_{κ+1}-name τ̇ ∈ H_{κ^{+}}. What is more ℙ_{κ+1} ∈ H_{κ^{+}}. A tedious but routine argument now shows that for any formula ϕ there exists a formula ϕ^{∗} such that for any ℙ_{κ+1}-name τ̇ and condition p ∈ ℙ_{κ+1}

p ⊩_{ℙ_{κ+1}}^{V} “H_{κ^{+}}^{V_{δ}[Ġ_{κ+1}]} ⊧ ϕ(τ̇ Ġ_{κ+1})” ⟺ H_{κ^{+}} ⊧ ϕ^{∗}(p, ℙ_{κ+1}, τ̇).

2Define M to agree with C⃗^{κ} up to κ, to have κ + ω as the largest point in its domain and to associate {κ + n: n < ω} to κ + ω.

Abusing notation slightly we write p ⊩_{ℙ_{κ+1}}^{H_{κ}^{+}} ϕ(τ̇) for this relation, where the key point is that the relation is definable in H_{κ^{+}}. An exactly similar analysis works for H_{κ^{+}}^{V_{δ}[G_{κ}]} and we write p ⊩_{ℙ_{κ}}^{H_{κ}^{+}} ϕ(τ̇)

as an abbreviation for the indigestible p ⊩_{ℙ_{κ}}^{V} “H_{κ^{+}}^{V_{δ}[Ġ_{κ}]} ⊧ ϕ(τ̇Ġ_{κ})”. In line with our intuitive remarks above we further abuse notation and write H_{κ^{+}}[G_{κ}] for H_{κ^{+}}^{V_{δ}[G_{κ}]}, H_{κ^{+}}[G_{κ+1}] for H_{κ^{+}}^{V_{δ}[G_{κ+1}]}, H_{λ^{+}}[G_{λ}] for H_{λ^{+}}^{V_{δ}[G_{λ}]}, and H_{λ^{+}}[G_{λ+1}] for H_{λ^{+}}^{V_{δ}[G_{λ+1}]}.

As usual, the problem is to lift the embedding j. We break up G_{λ+1} as G_{κ} ∗ g_{κ} ∗ H ∗ g_{λ}, where g_{κ} is the generic object added at κ, H the generic object added between κ and λ, and g_{λ} the generic object added at λ. Here G_{λ} = G_{κ} ∗ g_{κ} ∗ H will be the generic object for ℙ_{λ}.

Claim 6.13. j“G_{κ} ⊆ G_{λ}.

Proof. Let p ∈ G_{κ}, then since we did a Reverse Easton iteration we know that the support of p is some ordinal α with α < κ. Now crit(j) = κ, so the support of j(p) is also α and p ↾ α = j(p) ↾ α. So clearly we have j(p) ∈ G_{κ} ∗ g_{κ} ∗ H = G_{λ}, as desired. □

We now attempt to extend the embedding j to the larger domain H_{κ^{+}}[G_{κ}] by defining j(τ̇^{G_{κ}}) = j(τ̇)^{G_{λ}} for all τ̇ ∈ H_{κ^{+}}.

Claim 6.14. This definition gives a well-defined elementary embedding j: H_{κ^{+}}[G_{κ}] ⟶ H_{λ^{+}}[G_{λ}] which extends our original map j: H_{κ^{+}} ⟶ H_{λ^{+}}.

Proof. Suppose that τ̇^{G_{κ}} = σ̇^{G_{κ}}. By the truth lemma there is p ∈ G_{κ} such that p ⊩_{ℙ_{κ}}^{H_{κ^{+}}} τ̇ = σ̇. This is a first-order statement in H_{κ^{+}} and so since j is elementary j(p) ⊩_{ℙ_{λ}}^{H_{λ^{+}}} j(τ̇) = j(σ̇). j(p) ∈ G_{λ} and so j(τ̇)^{G_{λ}} = j(σ̇)^{G_{λ}}.

The proofs that the map we have defined is elementary and extends the original map are very similar. □

Claim 6.15. j“g_{κ} ⊆ g_{λ}.

Proof. Let p ∈ g_{κ}. Then p is in an initial segment of C⃗^{κ}, and j(p) = p. Since we chose C⃗^{λ} to extend C⃗^{κ}, j(p) ∈ g_{λ}. □

By the same method as in Claim 6.14 we may further extend j to get an elementary embedding j: H_{κ^{+}}[G_{κ} ∗g_{κ}] ⟶ H_{λ^{+}}[G_{λ} ∗g_{λ}]. Since H_{κ^{+}}[G_{κ} ∗g_{κ}] = H_{κ^{+}}^{V_{δ}[G]} and H_{λ^{+}}[G_{λ} ∗g_{λ}] = H_{λ^{+}}^{V_{δ}[G]}, we have shown that κ is 1-extendible in V_{δ}[G]. This concludes the proof of Theorem 6.5.

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Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA

  E-mail address: jcumming@andrew.cmu.edu

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA

  E-mail address: eschimme@andrew.cmu.edu