CANONICAL STRUCTURE IN THE UNIVERSE OF SET THEORY: PART TWO

JAMES CUMMINGS, MATTHEW FOREMAN, AND MENACHEM MAGIDOR

Abstract. We prove a number of consistency results complementary to the ZFC results from our paper [4]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness.

1. Introduction

In our paper [4] we prove a number of ZFC results concerning PCF theory, mutual stationarity, square principles and stationary reflection. In that paper we discussed the informal notion of canonical structure. This notion is supposed to capture the idea of structure that is not arbitrarily determined by non-constructive existence assumption. For example, structure that requires the axiom of choice to prove its existence may still be independent of any choices made in proving it exists. Cardinals of uncountable cofinality fall into this category. Other examples might include fine structure models of large cardinals. Large cardinal axioms are nonconstructive assumptions (as opposed to e.g. the pairing axiom, where we know exactly what the intended object is). However, as a consequence of their existence there is various canonical structure, such as U ∩ L[U] for U a normal ultrafilter on a measurable cardinal κ.

The notion of canonical structure is different from the notion of absoluteness. We illustrate this with an example. Assuming the Axiom of Choice, the collection of real numbers has some well-ordered cardinality 𝔠 and this cardinality is independent of the choices made to show it exists. Similarly, one needs the Axiom of Choice to prove that the least regular uncountable ordinal (ℵ_{1}) exists. Both of these objects are “canonical” in our sense, but it is independent of ZFC whether they are in fact identical. We would like to say that these distinct examples of structure that may or may not determine identical objects.

1991 Mathematics Subject Classification. Primary 03E35, 03E55; Secondary 03E05.

Key words and phrases. PCF theory, good ordinal, approachable ordinal, the ideal I[λ], internally approachable structure, tight structure, square sequence, covering properties, precipitous ideal, mutual stationarity, stationary reflection.

The first author was partially supported by NSF Grants DMS-9703945 and DMS-0070549.

The second author was partially supported by DMS-9803126 and DMS-0101155. The second and third authors were partially supported by the US-Israel Binational Science Foundation.

In this paper we continue to explore canonical structure by proving consistency results complementary to the ZFC results in [4].

After some preliminaries in Section 2, we show the following results.

• In Section 3, we give a forcing construction for a sequence of stationary sets which is mutually stationary but not tightly stationary. The proof involves a combinatorial principle which we dub Coherent Squares.

• In Section 4, we give another forcing construction for a sequence of stationary sets which is mutually stationary but not tightly stationary. The proof involves some lemmas about uniform structures and mutual stationarity which are of independent interest. We also show the consistency of a splitting property for mutually stationary sequences.

• In Section 5 we show that on an increasing ω-sequence of measurable cardinals, any sequence of stationary sets is mutually stationary. We also show that for any Prikry-generic sequence, a tail of the sequence has this property.

• In Section 6 we give an alternative proof of a theorem by Shelah, that there can exist sequences of stationary sets on the ℵ_{n} for n finite which are mutually stationary and do not concentrate on a fixed cofinality.

• In Section 7 we give an alternative construction for a model in which the set of non-good points of cofinality ℵ_{1} in ℵ_{ω+1} is non-stationary. We also show that if we are given an increasing ω-sequence of measurable cardinals such that the successor of their supremum exhibits a certain stationary reflection property, then the reflection property is preserved by diagonal Prikry forcing.

• In Section 8 we show that the principle saying that for all λ any family of fewer than η many stationary subsets of [λ]^{ℵ_{0}} reflect does not imply simultaneous reflection of η many sets of ω-cofinal ordinals. The proof uses Martin’s Maximum.

• In Section 9 we show that it is consistent that the least λ for which □_{λ} fails is inaccessible.

• In Section 10 we show that if □_{μ}^{∗} holds for a singular cardinal μ of cofinality ω, then a cardinal-preserving countably closed forcing poset can not create any instances of supercompactness below μ. This shows that there is an essential problem in a result by Ben-David and Shelah [2].

We would like to thank John Krueger for his careful reading of an earlier version of this paper.

2. Preliminaries

In this section we give some background material on mutual and tight stationarity and PCF theory. For more details we refer the reader to [4].

The idea of mutual stationarity was introduced by Foreman and Magidor [12] in their work on the non-saturation of the non-stationary ideal on P_{κ}λ.

Definition 2.1. Let ⟨S_{κ} : κ ∈ K⟩ be such that S_{κ} ⊆ κ for all κ ∈ K, where K is a set of regular uncountable cardinals.

(1) If N is a set, then N meets ⟨S_{κ} : κ ∈ K⟩ if and only if sup(N ∩ κ) ∈ S_{κ} for all κ ∈ N ∩ K.

(2) ⟨S_{κ} : κ ∈ K⟩ is mutually stationary if and only if for every algebra A on sup(K) there exists N ≺ A such that N meets ⟨S_{κ} : κ ∈ K⟩.

If S ⊆ P(X) then S is a stationary subset of P(X) if and only if for every algebra A on X there is B ∈ S such that B ≺ A. The sequence ⟨S_{κ} : κ ∈ K⟩ is mutually stationary if and only if the set of subsets of sup(K) which meet ⟨S_{κ} : κ ∈ K⟩ is a stationary subset of P(sup(K)). By standard facts [13, Lemma 0] about generalised stationarity, if X is any set with sup(K) ⊆ X then ⟨S_{κ} : κ ∈ K⟩ is mutually stationary if and only if the set of subsets of X which meet ⟨S_{κ} : κ ∈ K⟩ is a stationary subset of P(X).

It is easy to see that if ⟨S_{κ} : κ ∈ K⟩ is mutually stationary then S_{κ} is stationary for each κ. Foreman and Magidor showed that the converse is false in general, but is true if S_{κ} ⊆ κ ∩ cof(ω) for all κ. In order to get versions of Solovay’s splitting theorem and Fodor’s theorem Foreman and Magidor introduced the notion of tight structure and tightly stationary sequence.

Definition 2.2. Let K be a set of regular cardinals, let θ = cf(θ) > sup(K), and let A = (H_{θ}, ∈, <_{θ}). Let M ≺ A.

Then M is tight for K if and only if

(1) K ∈ M.

(2) For all g ∈ ∏_{κ∈M∩K}(M ∩κ) there exists h ∈ M ∩∏ K such that g(κ) < h(κ)

for all κ ∈ M ∩ K.

If |K| ⊆ M then K ⊆ M, and in this case tightness has a simpler formulation. When K ⊆ M, M is tight for K exactly when M ∩ ∏ K is cofinal in ∏_{κ∈K} M ∩ κ.

Definition 2.3. Let K be a set of regular cardinals and let M be a set. The characteristic function of M (on K) is the function χ_{M}^{K} with domain K given by χ_{M}^{K} : κ ⟼ sup(M ∩ κ).

If a structure M is such that K ⊆ M, then tightness of M amounts to saying that every function in ∏ K which is pointwise dominated by χ_{M} is pointwise dominated by some function in M ∩ ∏ K, that is to say M ∩ ∏ K is cofinal in ∏ K below χ_{M}^{K}.

Definition 2.4. Let K be a set of regular cardinals and let ⟨S_{κ} : κ ∈ K⟩ be such that S_{κ} ⊆ κ for all κ ∈ K. Let θ = sup(K)^{+}. The sequence ⟨S_{κ} : κ ∈ K⟩ is tightly stationary if and only if for every algebra A on H_{θ} there is N ≺ A such that N is tight for K and N meets ⟨S_{κ} : κ ∈ K⟩.

PCF theory gives a very general technique for analysing singular cardinals, but for our purposes in this paper we will restrict ourselves to the special case when the singular cardinal is ℵ_{ω}. Shelah has shown that

• There is an infinite set A ⊆ ω and a sequence of functions ⟨f_{α} : α < ℵ_{ω+1}⟩ which is a scale (that is to say an increasing and cofinal sequence) in ∏_{n∈A} ℵ_{n} under the eventual domination ordering.

• Modulo finite sets there is a unique maximal choice for the set A. For the rest of this discussion we fix A to be the maximal set as above, and also

fix ⟨f_{α} : α < ℵ_{ω+1}⟩ a scale in ∏_{n∈A} ℵ_{n}. A function g from A to the ordinals is said to be an exact upper bound for ⟨f_{α} : α < β⟩ iff f_{α} <^{∗} g for all α < β, and for every h < g there is α < β such that h <^{∗} f_{α}. For example the function n ↦ ℵ_{n} is an exact upper bound for ⟨f_{α} : α < ℵ_{ω+1}⟩.

Without loss of generality we assume that the scale ⟨f_{α} : α < ℵ_{ω+1}⟩ is continuous, which means that whenever an exact upper bound for ⟨f_{α} : α < β⟩ exists then f_{β} is such an upper bound. It is easy to see that modulo finite sets exact upper bounds are unique, so that if h is any exact upper bound for ⟨f_{α} : α < β⟩ then h =^{∗} f_{β}.

We will be especially interested in the good points of this kind of scale. An ordinal β < ℵ_{ω+1} of uncountable cofinality is good if there exists an exact upper bound h for ⟨f_{α} : α < β⟩ such that cf(h(n)) = cf(β) for all n with cf(β) < ω_{n}. The set of good points is stationary in every uncountable cofinality and is an important invariant of the universe of set theory; see for example [11], [6] and [18].

There is a useful alternative characterization of good points. The point β is good if and only if it has uncountable cofinality and for every A unbounded in β there exist B ⊆ A unbounded in β and k < ω such that ⟨f_{α}(n) : α ∈ B⟩ is strictly increasing for all n > k.

One reason for us to be interested in good ordinals is that they give a characterization of tight structures. We showed [12] that if M ≺ H_{θ}, 0 < m < ω and PCF is trivial (that is to say that A = ω, so there is a scale of length ℵ_{ω+1} in ∏_{n} ℵ_{n} modulo the ideal of finite sets.) then the following are equivalent:

(1) The structure M is tight for {ℵ_{n} : n < ω} and cf(M ∩ ℵ_{n}) = ℵ_{m} for all

large n < ω.

(2) If γ = sup(M ∩ ℵ_{ω+1}) then γ is a good point of cofinality ℵ_{m} and f_{γ}(n) =

χ_{M}(ℵ_{n}) for all large n < ω.

The kind of uniform cofinality assumption which appears in the result we just quoted is ubiquitous enough to deserve a name. We will say that a structure M is ℵ_{m}-uniform if cf(M ∩ ℵ_{n}) = ℵ_{m} for m < n < ω.

If 0 < m < ω then every internally approachable structure of length and cardinality ℵ_{m} is tight for {ℵ_{n} : n < ω}, so there are stationarily many ℵ_{m}-uniform tight structures. Zapletal [12] showed that there are stationarily many ℵ_{m}-uniform non-tight structures.

Without the assumption that PCF is trivial, we can give a more complicated description of the uniform tight structures. We refer the reader to [4, Theorem 5.6] for the missing details. Let K = {ℵ_{n} : n < ω}, let B⃗ = ⟨B_{λ} : λ ∈ pcf(K)⟩ be a sequence of PCF generators for K and let f⃗ = ⟨f_{α}^{λ} : α < λ, λ ∈ pcf(K)⟩ be such that ⟨f_{α}^{λ} : α < λ⟩ is an ω-club minimal scale in ∏ B_{λ}/J_{<λ} for each λ. We showed [4] that if 0 < m < ω and M ≺ (H_{θ}, B⃗, f⃗) is ℵ_{m}-uniform and tight for K then χ_{M}^{K} can be written as the pointwise supremum of finitely many functions of the form f_{sup(M∩λ)}^{λ}.

If M is an ℵ_{m}-uniform substructure of some expansion A of (H_{θ}, ∈, <_{θ}), and M^{∗} is the Skolem hull in A of M ∪ℵ_{m}, then as we see later in Lemma 6.3 sup(M ∩ℵ_{n}) = sup(M^{∗} ∩ ℵ_{n}) for m < n < ω. So M^{∗} is ℵ_{m}-uniform and contains ℵ_{m}. We showed [4] that for 0 < m < ω, if N is ℵ_{m}-uniform and contains ℵ_{m} then the set N ∩ ℵ_{ω} is closed under bounded suprema of length less than ℵ_{m}; in particular for m < n < ω there is a club subset of N ∩ℵ_{n} which has order type ℵ_{n} and is contained in N ∩ℵ_{n}.

3. A non-tight mutually stationary sequence

Foreman and Magidor [12] raised the question as to whether every mutually stationary sequence is tightly stationary. In this section we give a forcing construction showing that a negative answer is consistent; we do not know whether a negative answer follows from the axioms of ZFC. Given k > 0 we will construct by forcing a sequence ⟨T_{n} : k < n < ω⟩ with T_{n} ⊆ ℵ_{n} ∩ cof(ℵ_{k}), which is mutually stationary but not tightly stationary.

We start by defining a combinatorial principle Coherent Squares (CS). The principle asserts the existence of □_{ℵ_{n}}-sequences for 0 < n ≤ ω, together with a scale of length ℵ_{ω+1} in ∏_{n<ω} ℵ_{n} which relates the □_{ℵ_{n}}-sequences for n < ω to the □_{ℵ_{ω}}- sequence. We note that the scale involved in the principle CS is a “Very Good Scale” in the sense of our paper [6]. This principle is closely related to some combinatorial principles of Donder, Jensen and Stanley [8] and Donder, Jensen and Koppelberg [7].

Definition 3.1. For each n ≤ ω let I_{n} = { α : ℵ_{n} < α < ℵ_{n+1} }. The principle CS asserts that there exist sequences

⟨C_{α}^{n} : α ∈ I_{n} ∩ LIM, 0 < n ≤ ω⟩, ⟨f_{α} : α ∈ I_{ω} ∩ LIM⟩,

such that

(1) For all n and all α in S_{n} ∩ LIM

(a) The set C_{α}^{n} is club in α, and C_{α}^{n} ⊆ I_{n}.

If the cofinality of α is less than ℵ_{n} then the order type of C_{α}^{n} is less than ℵ_{n}.

(b) For every limit point β of C_{α}^{n}, C_{β}^{n} = C_{α}^{n} ∩ β.

(2) For all α in S_{ω} ∩ LIM, f_{α} is a function such that

(a) There exists k < ω such that dom(f_{α}) = { n : k < n ≤ ω } and ot(C_{α}^{ω}) ≤ ℵ_{k} .

(b) For all n in dom(f_{α}), f_{α}(n) ∈ I_{n} ∩ LIM.

(c) For every limit point β of C_{α}^{ω}, dom(f_{α}) ⊆ dom(f_{β}).

(d) For all n in dom(f_{α}),

lim(C_{f_{α}(n)}^{n}) = { f_{β}(n) : β ∈ lim(C_{α}^{ω}) }.

(3) The sequence ⟨f_{α} : α ∈ I_{ω} ∩ LIM⟩ forms a scale in ∏_{n} ℵ_{n+1}, that is to say it is increasing and cofinal in the eventual domination ordering.

Remark 3.2. Notice that ⟨C_{α}^{n} : α ∈ I_{n} ∩ LIM⟩ is essentially a □_{ℵ_{n}}-sequence, with the (purely cosmetic) difference that the underlying set is I_{n} rather than ℵ_{n+1}.

Remark 3.3. If ⟨f_{α} : α ∈ I_{ω}⟩ is the scale in ∏ ℵ_{n} modulo the ideal of finite sets given by the principle CS, then ⟨f_{α}⟩ is continuous and ω-club minimal. Moreover, it is a very good scale in the sense of [6].

All this follows from the observation that if α has uncountable cofinality then for large n the sequence ⟨f_{γ}(n) : γ ∈ lim(C_{α}^{ω})⟩ is continuous and increasing with supremum f_{α}(n).

To motivate the principle CS we show that it can be used to generate a sequence of stationary sets which is not tightly stationary. We suppose that ⟨f_{α}⟩ and ⟨C_{α}^{n}⟩ are as in Definition 3.1. Given k < ω and a sequence of limit ordinals ⟨γ_{n} : k ≤ n < ω⟩ such that γ_{n} < ℵ_{n} for all n, we define a sequence of sets by

T_{n} = { α ∈ I_{n} ∩ LIM : cf(α) = ℵ_{k}, ot(C_{α}^{n}) ≥ γ_{n} }.

The stationarity of the sets T_{n} follows from a general fact about □_{κ}-sequences for κ regular.

Lemma 3.4. If κ is regular and ⟨C_{α} : α < κ^{+}⟩ is a □_{κ}-sequence, then for every η < κ the set {δ < κ^{+} : ot(C_{δ}) > η} is stationary.

Proof. Let C be club in κ^{+} and let ζ be a limit point of C with cofinality κ. C_{ζ} ∩ C is club in ζ with order type κ, and so we may find δ a limit point of C_{ζ} ∩ C such that ot(C_{ζ} ∩ δ) > η. Clearly δ ∈ C, and by the coherence property of the square sequence C_{ζ} ∩ δ = C_{δ} and so ot(C_{δ}) > η. We showed {δ < κ^{+} : ot(C_{δ}) > η} meets every club subset of κ^{+} and so it is stationary. □

Lemma 3.5. If ⟨γ_{n} : k ≤ n < ω⟩ is unbounded in ℵ_{ω} γ_{n} < ℵ_{n} for every n, then ⟨T_{n} : k ≤ n < ω⟩ is not tightly stationary.

Proof. Let N be a tight structure and for each i ≤ ω let α_{i+1} = sup(N ∩ ℵ_{i+1}). Suppose for a contradiction that α_{i} ∈ T_{i} for all i.

The sequence ⟨f_{α} : α ∈ I_{ω} ∩ LIM⟩ forms a continuous scale, so by the characterization of uniform tight structures in terms of PCF theory which we discussed in Section 2 there exists m < ω such that f_{α_{ω}}(n) = α_{n} for all n ≥ m. If n ≥ m then

lim(C_{α_{n}}^{n}) = lim(C_{f_{αω} (n)}^{n}) = { f_{β}(n) : β ∈ lim(C_{α_{ω}}^{ω}) }.

Notice also that if β, γ ∈ lim(C_{α_{ω}}^{ω}) and β < γ then β ∈ lim(C_{γ}^{ω}), and so f_{β}(n) ∈ lim(C_{f_{γ}(n)}^{n}) and in particular f_{β}(n) < f_{γ}(n). It follows that ot(C_{α_{n}}^{n}) = ot(C_{α_{ω}}^{ω}) for all n ≥ m. Since ⟨γ_{n} : k ≤ n < ω⟩ is unbounded in ℵ_{ω}, we may find n ≥ m such that γ_{n} > ot(C_{α_{ω}}^{ω}). It follows that α_{n} ∉ T_{n}, which is a contradiction. □

We define a forcing iteration of length ω + 1 which forces CS to hold. At stage n for n < ω we force with a version of Jensen’s poset for adding a square sequence C⃗^{n}, where conditions prescribe an initial segment of C⃗^{n}. At stage ω we force with conditions which prescribe initial segments of C⃗^{ω} and f⃗.

Definition 3.6. For n < ω, ℚ_{n} is the set of sequences

q = ⟨C_{α}^{q,n} : α ∈ I_{n} ∩ LIM ∩ (β + 1)⟩

where

(1) The ordinal β is a limit ordinal in S_{n}. We refer to β as the length of q and write β = lh(q).

(2) For all α in S_{n} ∩ LIM ∩ (β + 1)

(a) The set C_{α}^{q,n} is club in α and C_{α}^{q,n} ⊆ I_{n}.

(b) The order type of C_{α}^{q,n} is less than ℵ_{n} if the cofinality of α is less than ℵ_{n}.

(c) For every limit point γ of C_{α}^{q,n}, C_{γ}^{q,n} = C_{α}^{q,n} ∩ γ. If q, r ∈ ℚ_{n} then q ≤ r if and only if

(1) The length of q is greater than or equal to the length of r.

(2) For all α ∈ I_{n} ∩ LIM ∩ (lh(r) + 1), C_{α}^{q,n} = C_{α}^{r,n}.

Before stating the main facts about ℚ_{n} we recall the concept of strategic closure. Let λ be an infinite cardinal. A poset ℚ is (λ + 1)-strategically closed if and only if player Even has a winning strategy for the game in which two players (Even and Odd) build a decreasing λ + 1-sequence in ℚ, where Odd plays at all odd stages and Even plays at all non-zero even stages including limit stages, and Even wins if she can move at stage λ. We refer the reader to Foreman’s paper on games [9] for more about strategic closure, noting here only that a (λ + 1)-strategically closed poset adds no λ-sequences and hence preserves all cardinals less than or equal to λ^{+}.

The following facts are standard, see for example [6].

Fact 3.7. Let n < ω.

(1) The poset ℚ_{n} is countably closed.

(2) The poset ℚ_{n} is (ℵ_{n} + 1)-strategically closed.

(3) If 2^{ℵ_{n}} = ℵ_{n+1} then |ℚ_{n}| = ℵ_{n+1}, so in particular ℚ_{n} has the ℵ_{n+2}-c.c.

(4) If p ∈ ℚ_{n}, γ is a limit ordinal in S_{n} and lh(p) < γ, then there is q ≤ p such that lh(q) = γ.

(5) If C⃗^{n} = ⟨C_{α}^{n} : α ∈ I_{n} ∩ LIM⟩ is ℚ_{n}-generic and δ ∈ (ℵ_{n} + 1) ∩ LIM, then

V [C⃗^{n}] ⊧ “ { α ∈ I_{n} ∩ LIM : ot(C_{α}^{n}) = δ } is stationary in ℵ_{n+1}”.

We now assume that V satisfies GCH. We define ℙ_{ω} as an iteration with full support, where at stage n we force with ℚ_{n} as defined in V^{ℙ_{n}}. As usual we let ℚ̇_{n} be a ℙ_{n}-name for ℚ_{n}. In V^{ℙ_{ω}} let ⟨C_{α}^{n} : α ∈ I_{n} ∩ LIM⟩ be the sequence added by ℚ_{n}, and define ℚ_{ω} as follows.

Definition 3.8. The poset ℚ_{ω} is the set of pairs of sequences

q = (⟨C_{α}^{q,ω} : α ∈ I_{ω} ∩ LIM ∩ (β + 1)⟩, ⟨f_{α}^{q} : α ∈ I_{ω} ∩ LIM ∩ (β + 1)⟩)

where

(1) The ordinal β is a limit ordinal in I_{ω}. We call β the length of q and write β = lh(q).

(2) For all α ∈ I_{ω} ∩ LIM ∩ (β + 1)

(a) The set C_{α}^{q,ω} is club in α and C_{α}^{q,ω} ⊆ I_{ω}.

(b) The order type of C_{α}^{q,ω} is less than ℵ_{ω}.

(c) For every limit point γ of C_{α}^{q,ω}, C_{γ}^{q,ω} = C_{α}^{q,ω} ∩ γ.

(3) For all α in I_{ω} ∩ LIM ∩ (β + 1), f_{α} is a function such that

(a) There exists k < ω such that dom(f_{α}) = { n : k < n ≤ ω } and ot(C_{α}^{q,ω}) ≤ ℵ_{k}.

(b) For all n ∈ dom(f_{α}), f_{α}(n) ∈ I_{n} ∩ LIM.

(c) For all limit points β of C_{α}^{q,ω}, dom(f_{α}) ⊆ dom(f_{β}).

(d) For all n ∈ dom(f_{α}),

lim(C_{f_{α}(n)}^{n}) = { f_{β}(n) : β ∈ lim(C_{α}^{q,ω}) }.

(4) If α_{1} < α_{2} ≤ β, then f_{α_{1}} <^{∗} f_{α_{2}}. If q, r ∈ ℚ_{ω} then q ≤ r if and only if

(1) The length of q is greater than or equal to the length of r.

(2) For all α in S_{n} ∩ LIM ∩ (lh(r) + 1), C_{α}^{q,ω} = C_{α}^{r,ω} and f_{α}^{q} = f_{α}^{r}.

When we construct members of ℚ_{ω} we will generally only verify that the “coherence” clause 3d holds.

Lemma 3.9. The forcing poset ℚ_{ω} is countably closed in V^{ℙ_{ω}}.

Proof. Let ⟨q_{i} : i < ω⟩ be a strictly decreasing ω-sequence of conditions, and define β_{i} = lh(q_{i}) and β = sup_{i<ω} β_{i}. We define a condition q as follows.

(1) The length of q is β.

(2) For all α ∈ I_{n} ∩ LIM ∩ (lh(q_{i}) + 1), C_{α}^{q,ω} = C_{α}^{q_{i},ω} and f_{α}^{q} = f_{α}^{q_{i}}.

(3) The set C_{β}^{q,ω} is cofinal in β with order type ω.

(4) For all n < ω, f_{β}^{q}(n) ≥ sup { f_{β_{i}}^{q}(n) : i ∈ ω } and ot(C_{f_{β}^{q}(n)}^{n}) = ω.

The choice of f_{β}^{q}(n) is possible because ot(C_{δ}^{n}) = ω for a cofinal set of δ < ℵ_{n+1}. The clause 3d of the definition of ℚ_{n} is satisfied trivially because neither C_{β}^{q,ω} nor any of the C_{f_{β}^{q}(n)}^{n} has any limit points. □

We now define ℙ_{ω+1} = ℙ_{ω} ∗ ℚ_{ω}. A standard argument shows that

ℙ_{ω+1} ≃ ℙ_{n} ∗ ℝ_{n},

where ℝ_{n} is the full support iteration of length ω + 1 with factors ⟨ℚ_{i} : n ≤ i ≤ ω⟩. For notational convenience we will index the steps of ℝ_{n} by the set { i : n ≤ i ≤ ω } rather than ω + 1.

For the next few lemmas we work in V^{ℙ_{n}}.

Lemma 3.10. The set of p in ℝ_{n} such that p ↾ i \ n decides lh(p(i)) for all i with n ≤ i ≤ ω is dense.

Proof. This is easy, as each of the ℚ_{i} is countably closed. □

From now on we will assume that all p ∈ ℝ_{n} have this property. Accordingly we will write lh_{i}(p) for the unique ordinal γ such that p ↾ i ⊩ γ̌ = lh(p(i)).

Notation: If p ∈ ℝ_{n} then we write

p(i) = ⟨Ċ_{α}^{p,i} : α ∈ S_{i} ∩ LIM ∩ (lh_{i}(p) + 1)⟩

for i < ω, and let ⟨Ċ_{α}^{p,ω} : α ∈ I_{ω} ∩ LIM ∩ (lh_{ω}(p) + 1)⟩ be the first component of p(ω) and ⟨ḟ_{α}^{p} : α ∈ I_{ω} ∩ LIM ∩ (lh_{ω}(p) + 1)⟩ the second component.

Definition 3.11. A condition p ∈ ℝ_{n} is flat if and only if p ↾ ω \ n forces that (1) dom(ḟ_{lh_{ω}(p)}^{p}) = { i : n ≤ i < ω }.

(2) For all i ∈ dom(ḟ_{lh_{ω}(p)}^{p}), ḟ_{lh_{ω}(p)}^{p}(i) = lh_{i}(p).

Lemma 3.12. The set of flat conditions in ℝ_{n} is dense.

Proof. Given p we first find q ≤ p such that • lh_{ω}(q) = lh_{ω}(p).

• q ↾ ω \ n decides f_{lh_{ω}(p)}^{p}.

• lh_{i}(q) > f_{lh_{ω}(p)}^{p}(i) and ot(C_{lh_{i}(q)}^{q,i}) = ω for all i with n ≤ i < ω. Then we find r ≤ q such that

(1) lh_{i}(r) = lh_{i}(q) for all i < ω, and lh_{ω}(r) = lh_{ω}(q) + ω.

(2) ot(C_{lh_{ω}(r)}^{r,ω}) = ω.

(3) dom(f_{lh_{ω}(r)}^{r}) = { i : n ≤ i < ω } and f_{lh_{ω}(r)}^{r}(i) = lh_{i}(r) for all i.

Clearly r is a condition, r ≤ p and r is flat. □

Lemma 3.13. For all n < ω, ℝ_{n} is (ℵ_{n} + 1)-strategically closed.

Proof. We describe a strategy for player Even in the game, where, without loss of generality, we may assume that Odd plays a flat condition at each odd stage. Even’s moves will also be flat conditions. Suppose that γ is even and that so far the sequence ⟨p_{i} : i < γ⟩ has been played.

γ successor: If γ = δ + 1, then Even defines p_{γ} as follows.

(1) lh_{i}(p_{γ}) = lh_{i}(p_{δ}) + ω and C_{lh_{i}(p_{γ})}^{p_{γ},i} = {lh_{i}(p_{δ}) + j : j < ω} for n ≤ i ≤ ω.

(2) dom(f_{lh_{ω}(p_{γ})}^{p_{γ}}) = { i : n ≤ i < ω }.
(3) f_{lh_{ω}(p_{γ})}^{p_{γ}}(i) = lh_{i}(p_{γ}) for n ≤ i < ω.

γ limit: Even defines p_{γ} as follows.

(1) lh_{i}(p_{γ}) = ⋃_{β<γ} lh_{i}(p_{β}) and C_{lh_{i}(p_{γ})}^{p_{γ},i} = { lh_{i}(p_{β}) : β < γ } for n ≤ i ≤ ω.

(2) dom(f_{lh_{ω}(p_{γ})}^{p_{γ}}) = { i : n ≤ i < ω }.

(3) f_{lh_{ω}(p_{γ})}^{p_{γ}}(i) = lh_{i}(p_{γ}) for n ≤ i < ω.

As usual we only check that p_{γ}(ω) satisfies clause 3d from the definition of ℚ_{ω}. We observe first that if δ ∈ lim(C_{lh_{i}(p_{γ})}^{p_{γ},i}) then δ = lh_{i}(p_{β}) for some limit β < γ.

At stage β player Even defined C_{lh_{i}(p_{β})}^{p_{β},i} = { lh_{i}(p_{α}) : α < β } for n ≤ i ≤ ω, and f_{lh_{ω}(p_{β})}^{p_{β}}(i) = lh_{i}(p_{β}) for n ≤ i < ω.

It follows that for n ≤ i < ω

lim(C_{lh_{i}(p_{γ})}^{p_{γ},i}) = { lh_{i}(p_{β}) : β ∈ γ ∩ LIM }
= { f_{lh_{ω}(p_{β})}^{p_{β}}(i) : β ∈ γ ∩ LIM }
= { f_{δ}^{p_{γ}} (i) : δ ∈ lim(C_{lh_{ω}(p_{γ})}^{p_{γ},ω}) }
□

Lemma 3.14. Let G_{ω+1} be ℙ_{ω+1}-generic and let G_{ω} be the induced ℙ_{ω}-generic filter. Then

(1) The models V and V [G_{ω+1}] have the same cardinals and cofinalities up to ℵ_{ω+1}.

(2) The principle CS holds in V [G_{ω+1}].

(3) Every ℵ_{ω}-sequence of ordinals from V [G_{ω+1}] is in V [G_{ω}].

Proof. This is fairly routine. We only check that ⟨f_{α} : α ∈ I_{ω} ∩ LIM⟩ forms a scale. To see this let g ∈ (∏_{n} ℵ_{n+1})^{V [G_{ω+1}]}, and observe that g ∈ V because ℙ_{ω+1} is countably closed. Now let p be an arbitrary condition. Find q ≤ p such that lh_{ω}(q) = lh_{ω}(p) and lh_{i}(q) ≥ g(i) for all i, and then find r ≤ q such that r is flat. By construction f_{lh_{ω}(r)}^{r}(i) = lh_{i}(r) ≥ lh_{i}(q) ≥ g(i) for all i < ω, and we are done. □

We now work in V [G_{ω+1}]. We recall that given k < ω and a sequence of limit ordinals ⟨γ_{n} : k ≤ n < ω⟩ such that γ_{n} < ℵ_{n} for all n, we defined a sequence of sets by

T_{n} = { α ∈ I_{n} ∩ LIM : cf(α) = ℵ_{k}, ot(C_{α}^{n}) ≥ γ_{n} }.

We showed in Lemma 3.5 that a suitable choice of ⟨γ_{n} : k ≤ n < ω⟩ will generate a sequence which is not tightly stationary.

Lemma 3.15. Let G_{ω+1} ⊂ ℚ_{ω} be generic. Then in V [G_{ω+1}], for all sequences ⟨γ_{n} : k ≤ n < ω⟩ the sequence ⟨T_{n} : k ≤ n < ω⟩ is mutually stationary (where T_{n} is defined as above.)

Proof. Let ⟨γ_{n} : k ≤ n < ω⟩ be a sequence of ordinals in V [G_{ω+1}]. Then, by the closure of ℚ_{ω}, ⟨γ_{n} : k ≤ n < ω⟩ lies in V. We showed in Lemma 3.14 that the models V [G_{ω}] and V [G_{ω+1}] have the same ℵ_{ω}-sequences of ordinals, so it is enough to check that this is so in V [G_{ω}]. We use the fact that

ℙ_{ω} ≃ ℙ_{k} ∗ 𝕊_{k}

where 𝕊_{k} is the iteration of length ω with full support and factors ℚ_{n} for k ≤ n < ω. We will do a density argument in 𝕊_{k} similar to the proof given above that ℝ_{k} is strategically closed.

Let Ḣ be a name for a function from ^{<ω}ℵ_{ω} to ℵ_{ω}. Let p_{0} ∈ 𝕊_{n} be arbitrary. Extending p_{0} if necessary, we may assume without loss of generality that ot(C_{lh_{i}(p_{0})}^{p_{0},i}) = γ_{i}.

We will argue in the model V^{ℙ_{k}}, using the closure of 𝕊_{k}. In particular we will now understand Ḣ as an 𝕊_{k}-name appropriate for forcing over V^{ℙ_{k}}. We describe an inductive construction of a decreasing chain of conditions ⟨p_{j} : j ≤ ℵ_{k}⟩ in 𝕊_{k}, and an increasing chain of sets ⟨A_{j} : j ≤ ℵ_{k}⟩ such that |A_{j}| ≤ ℵ_{k}.

Stage zero: p_{0} has already been determined, and we set A_{0} = ∅. Successor stages: Suppose that p_{j}, A_{j} have been defined. We start by choosing B_{j+1} ⊆ ℵ_{ω} such that |B_{j+1}| ≤ ℵ_{k}, A_{j} ⊆ B_{j+1}, and lh_{i}(p_{j}) ≤ sup(B_{j+1} ∩ ℵ_{i+1}) for i ≥ k. We then choose q_{j} ≤ p_{j} and A_{j+1} so that q_{j} forces the Ḣ-closure of B_{j+1} to be A_{j+1}, and lh_{i}(q_{j}) ≥ sup(A_{j+1} ∩ ℵ_{i+1}) for i ≥ k. We define p_{j+1} as follows:

(1) p_{j+1} ≤ q_{j}.

(2) lh_{i}(p_{j+1}) = lh_{i}(q_{j}) + ω.
(3) C_{lh_{i}(p_{j+1})}^{p_{j+1},i} = { lh_{i}(q_{j}) + l : l < ω }.

Limit stages: Suppose that j is limit and we have defined ⟨p_{k} : k < j⟩ and ⟨A_{k} : k < j⟩. Define p_{j} by

(1) p_{j} ≤ p_{k} for all k < j.

(2) lh_{i}(p_{j}) = sup_{k<j} lh_{i}(p_{k}).
(3) C_{lh_{i}(p_{j})}^{p_{j},i} = C_{lh_{i}(p_{0})}^{p_{0},i} ∪ { lh_{i}(p_{k}) : k < j }.

Let A_{j} = ⋃_{k<j} A_{k}.

If j is limit it is routine to check that p_{j} is a condition, sup(A_{j} ∩ ℵ_{i+1}) = lh_{i}(p_{j}), and that p_{j} forces that A_{j} is closed under Ḣ.

Let A = A_{ℵ_{k}}, p = p_{ℵ_{k}}. Then p forces that sup(A ∩ ℵ_{i+1}) = lh_{i}(p) for all i, and p also forces that ot(C_{lh_{i}(p)}^{i}) = γ_{i} +ℵ_{k} for all i. It follows that p ⊩ ∀i sup(A∩ℵ_{i+1}) ∈ T_{i}. □

We summarise the main result of this section in a theorem.

Theorem 3.16. It is consistent that for every integer k > 0 there exists a sequence ⟨T_{n} : k < n < ω⟩ such that T_{n} ⊆ ℵ_{n} ∩ cof(ℵ_{k}), and the sequence is mutually stationary but not tightly stationary.

4. Another non-tight mutually stationary sequence

Steprans and Foreman found another consistency proof for the existence of a sequence ⟨S_{n} : k < n < ω⟩ such that S_{n} ⊆ ℵ_{n} ∩ cof(ℵ_{k}), and the sequence is mutually stationary but not tightly stationary. The model is easily described: fixing an integer k > 0, we force with the Cohen poset Add(ℵ_{0}, ℵ_{ω}) for adding a subset S of ℵ_{ω} with finite conditions and define S_{n} = ℵ_{n} ∩ cof(ℵ_{k}) ∩ S for each n > k. We claim that in V [S] the sequence ⟨S_{n} : k < n < ω⟩ is as required.

We start by showing that ⟨S_{n} : k < n < ω⟩ is not tightly stationary. This part of the argument is due to Steprans (under the assumption that 2^{ℵ_{ω}} = ℵ_{ω+1}).

We observe that the poset Add(ℵ_{0}, ℵ_{ω}) has the countable chain condition. Working in V we fix a sequence of PCF generators ⟨B_{λ} : λ ∈ pcf(K)⟩ and a family ⟨f_{α}^{λ} : λ ∈ pcf(K), α < λ⟩ of ω-club minimal scales, where K = {ℵ_{n} : n < ω}. By the countable chain condition it is still the case in V [S] that ⟨B_{λ}⟩ is a sequence of generators and ⟨f_{α}^{λ}⟩ is a matrix of ω-club minimal scales.

If N is a tight ℵ_{k}-uniform structure in V [S] then as we discussed in Section 2, it follows from [4, Theorem 5.6] that χ_{N} can be computed in an absolute way from finitely many of the functions f_{α}^{λ}, and so χ_{N} ∈ V. An easy density argument shows that V ∩∏_{n} S_{n} = ∅, so that the tight structure N can not meet the sequence ⟨S_{n} : n < ω⟩.

It remains to be seen that the sequence ⟨S_{n} : n < ω⟩ is mutually stationary in V [S]. Let F ∈ V [S] be a function from ^{<ω}ℵ_{ω} to ℵ_{ω}. We start by showing that it is enough to consider structures which lie in the ground model.

Lemma 4.1. Let ℙ be a c.c.c. forcing poset, let λ be a cardinal and let Ḟ be a ℙ-name for a function from ^{<ω}λ to λ. There is a function f ∈ V from ^{<ω}λ to λ such that if G is ℙ-generic and X ∈ V [G] is a subset of λ closed under f, then X is closed under Ḟ^{G}.

Proof. It follows from the c.c.c. that if x ∈ ^{<ω}λ then there are only countably many possibilities for Ḟ (x). Fix an enumeration of these possibilities as ⟨g(x, n) : n < ω⟩ and then define f as follows: if y ∈ ^{<ω}λ and lh(y) = 2^{m}(2n + 1) then f(y) = g(y ↾ m, n). □

For the rest of this section we mean by “structure” an elementary substructure of (H_{ℵ_{ω+1}}, ∈, <, F). Let F be the set of characteristic functions of ℵ_{k}-uniform structures with respect to the set {ℵ_{n} : k < n < ω}; for notational simplicity we consider the domain of an element of F to be {n : k < n < ω}. Let T be the tree consisting of all proper initial segments of all elements of F. We prove two lemmas about T, which may have some independent interest.

Lemma 4.2. Every infinite branch of T is a member of F.

Proof. Let χ be a branch of T, and find structures ⟨M_{j} : k < j < ω⟩ such that sup(M_{j} ∩ ℵ_{n}) = χ(n) for all n and j with k < n ≤ j. As we noted in Section 2, we may as well assume that ℵ_{k} ⊆ M_{j} and then may find C_{n}^{j} ⊆ M_{j} ∩ χ(n) which is club in χ(n), for all n and j with k < n ≤ j. For all n > k let D_{n} = ⋂_{j≥n} C_{n}^{j}, so that D_{n} is club in χ(n). We note that if γ ∈ D_{n} then γ ∈ M_{j} for all large j.

Let M be the Skolem hull of ⋃_{n>k} D_{n}. We claim that M is a structure with characteristic function χ. It is clear that χ_{M}(n) ≥ χ(n) for all n > k. To see that the reverse inequality holds, let α ∈ M ∩ ℵ_{n} and fix s a finite subset of ⋃_{n>k} D_{n} such that α is in the hull of s. Since s is finite we may find j so large that j ≥ n and s ⊆ M_{j}, so that α ∈ M_{j} ∩ ℵ_{n} and therefore α < χ_{M_{j}}(n) Since χ_{M_{j}}(n) = χ(n), we are done. □

Remark 4.3. Notice that in any cardinal-preserving extension of V, the argument works to show that every infinite branch of T is the characteristic function of some ℵ_{k}-uniform structure. In particular this is true in V [S].

We now show the tree T has a stationary branching subtree U.

Lemma 4.4. There is a tree U ⊆ T such that for all j > k and t ∈ U with dom(t) = {n : k < n < j}, {α < ℵ_{j} : t^{⌢}α ∈ U} is stationary in ℵ_{j}.

Proof. We will use the Gale-Stewart theorem [19] on the determinacy of games with open payoff sets. We denote by ⟨⟨β_{0}, … β_{j−1}⟩⟩ the function f with domain {n : k < n < k + j + 1} given by f(n) = β_{n−k−1}.

Consider the following two-player game of perfect information between two players I and II. Player I’s i^{th} move is a set of ordinals A_{i}, player II’s i^{th} move is an ordinal α_{i}. We suppose that

(1) ⟨⟨α_{i} : i < j⟩⟩ ∈ T for all j.

(2) A_{j} is a subset of {β < ℵ_{k+j+1} : ⟨⟨α_{0}, … α_{j−1}, β⟩⟩ ∈ T }, and A_{j} is non-

stationary in ℵ_{k+j+1}.
(3) α_{j} ∉ A_{j}.

The first player to violate these conditions loses, and if play continues for ω moves then II wins.

Intuitively the idea is that II is trying to build an infinite branch of T, and that player I is allowed to block a non-stationary set of potential successors at each stage. Similar games appear in the game analysis of Namba forcing by Shelah [21].

We claim that II has a winning strategy. Since the game is open for player I, it follows from the Gale-Stewart theorem that it suffices to show I has no winning strategy. Suppose for a contradiction that I has a winning strategy τ, and find an ℵ_{k}-uniform structure M with τ ∈ M.

We will construct a run of the game where I plays according to τ but the wrong player (player II) wins. At her j^{th} move player II will play α_{j} = sup(M ∩ ℵ_{k+j+1}). We check that this gives a win for player II.

Suppose that I has played ⟨A_{i} : i ≤ j⟩, II has played ⟨α_{i} : i < j⟩, and α_{i} ∉ A_{i} for i < j. In general A_{j} will not be in M. However if we define B to be the union of the set of all A such that I plays A at stage j in some run of the game where I plays according to τ, then B ∈ M because τ ∈ M. Since B is the union of at most ℵ_{k+j} non-stationary subsets of ℵ_{k+j+1}, B is non-stationary.

Let C ∈ M be a club subset of ℵ_{k+j+1} which is disjoint from B. Since C is unbounded in α_{j} by elementarity, α_{j} ∈ C and thus α_{j} ∉ B. By construction A_{j} ⊆ B, thus α_{j} ∉ A_{j}. It follows that II wins the game, contradiction!

We now fix a winning strategy σ for player II. We define U to be the set of all ⟨⟨α_{0}, … α_{j−1}⟩⟩ such that α_{0}, … α_{j−1} is an initial segment of II’s sequence of plays in some run of the game where II plays according to σ. To finish the proof, we show that U has stationary branching.

Let ⟨⟨α_{0}, … α_{j−1}⟩⟩ ∈ U and suppose that it represents II’s response to I’s playing A_{0}, A_{1}, … A_{j−1}. Let

B = {β < ℵ_{k+j+1} : ⟨⟨α_{0}, … α_{j−1}, β⟩⟩ ∈ U},

and suppose for a contradiction that B is non-stationary. Let I play B as his j^{th} move and let β be the response dictated by σ. Then by the definition of U, ⟨⟨α_{0}, … α_{j−1}, β⟩⟩ ∈ U and so β ∈ B. This means that player II loses immediately, contradicting the assumption that σ was a winning strategy.

□

It is easy to check that for every n > k, S_{n} meets every stationary subset of ℵ_{n} ∩ cof(ℵ_{k}) from the ground model. Since U has stationary branching, we may build by induction a branch b of U which is in ∏_{n>k} S_{n}. By Lemma 4.2 we may construct a structure M such that χ_{M} = b. This shows that the sequence ⟨S_{n}⟩ is mutually stationary.

We summarise the main result of this section in a theorem.

Theorem 4.5. Let S ⊆ ℵ_{ω} be V -generic for Add(ℵ_{0}, ℵ_{ω}), and define S_{n} = ℵ_{n} ∩ cof(ℵ_{k}) ∩ S for each n > k. In V [S] the sequence ⟨S_{n} : k < n < ω⟩ is mutually stationary but not tightly stationary.

Remark 4.6. It is clear from the proof that a large class of forcing posets could be used in place of Add(ℵ_{0}, ℵ_{ω}). To be more precise, essentially the same proof will work for any forcing poset ℙ such that

(1) The poset ℙ is ℵ_{ω}-c.c. and cardinal-preserving.

(2) Forcing with ℙ adds a sequence ⟨S_{n} : k < n < ω⟩ with S_{n} ⊆ ℵ_{n} ∩ cof(ℵ_{k}) such that

(a) In the extension by ℙ, V ∩ ∏_{n} S_{n} = ∅.

(b) For all S ∈ V such that S ⊆ ℵ_{n} ∩ cof(ℵ_{k}) and V ⊧ “S is stationary”,

S_{n} ∩ S ≠ ∅.

Similar ideas can be used to show that adding enough Cohen reals gives a model in which every mutually stationary sequence can be split.

Lemma 4.7. Let 0 < k < ω and let ⟨U_{n} : k < n < ω⟩ be a mutually stationary sequence of sets with U_{n} ⊆ ℵ_{n} ∩ cof(ℵ_{k}). Let T^{∗} be the tree of functions f such that

• dom(f) = {n : k < n ≤ j} for some j > k.

• There is a structure M such that M meets ⟨U_{n} : k < n < ω⟩ and f(n) =

sup(M ∩ ℵ_{n}) for k < n ≤ j.

Then there is a subtree U^{∗} ⊆ T^{∗} such that for all j > k and t ∈ U^{∗} with dom(t) = {n : k < n < j}, {α < ℵ_{j} : t^{⌢}α ∈ U^{∗}} is stationary in ℵ_{j}.

Proof. The proof is very similar to that of Lemma 4.4. Two players I and II collaborate to build a branch of T^{∗}, with player I blocking out a non-stationary set of possible successors of the current position and player II choosing a successor which was not blocked by player I.

We need to check that I does not win, and so we suppose that τ is a strategy for player II. Since ⟨U_{n} : k < n < ω⟩ is mutually stationary, we may find M such that M meets ⟨U_{n} : k < n < ω⟩ and τ ∈ M. As before, we may check that II can win against τ by playing sup(M ∩ ℵ_{k+j+1}) at move j of the game.

By the Gale-Stewart theorem there is a winning strategy σ for player II. As in Lemma 4.4 we may use σ to construct a suitable tree U^{∗}, consisting of finite initial segments of runs of the game in which II plays according to σ. □

Lemma 4.8. Let 0 < k < ω and let ⟨U_{n} : k < n < ω⟩ be a mutually stationary sequence of sets with U_{n} ⊆ ℵ_{n} ∩ cof(ℵ_{k}). Let G be Add(ℵ_{0}, ℵ_{ω})-generic over V. Then in V [G] the following is true: there are partitions ⟨U_{n}^{i} : i < ω⟩ of each U_{n} into ω disjoint stationary pieces, such that for all f : ω ⟶ ω the sequence ⟨U_{n}^{f(n)} : k < n < ω⟩ is mutually stationary.

Proof. We can regard Add(ℵ_{0}, ℵ_{ω}) as the finite support product of posets ℙ_{n} for k < n < ω, where ℙ_{n} is the poset of finite partial functions from ℵ_{n} to ℵ_{0}. We may then identify G with ⟨g_{n} : k < n < ω⟩ where g_{n} is a map from ℵ_{n} to ℵ_{0}, and we set U_{n}^{i} = {α ∈ U_{n} : g_{n}(α) = i}.

It is routine to check that each U_{n}^{i} is stationary. We now use Lemma 4.7 to construct a suitable tree U^{∗}, and then given f we build a branch of U^{∗} which lies in ∏_{n>k} U_{n}^{f(n)}. We may now finish the argument exactly as in the proof of Theorem 4.5. □

Theorem 4.9. If G is generic for Add(ℵ_{0}, ℵ_{ω+1}) then the following statement holds in V [G]: for all k > 0 and all mutually stationary ⟨U_{n} : k < n < ω⟩ with U_{n} ⊆ ℵ_{n} ∩ cof(ℵ_{k}), there are partitions ⟨U_{n}^{i} : i < ω⟩ of each U_{n} into ω disjoint stationary pieces, such that for all f : ω ⟶ ω the sequence ⟨U_{n}^{f(n)} : k < n < ω⟩ is mutually stationary.

Proof. By a chain condition argument the sequence ⟨U_{n} : k < n < ω⟩ lies in the generic extension of V by some proper initial segment Add(ℵ_{0}, λ) of Add(ℵ_{0}, ℵ_{ω+1}), where λ < ℵ_{ω+1}. Since Add(ℵ_{0}, ℵ_{ω+1}) ≃ Add(ℵ_{0}, λ) × Add(ℵ_{0}, ℵ_{ω+1}) we may as well assume that ⟨U_{n} : k < n < ω⟩ ∈ V. The theorem is now immediate from Lemma 4.8. □

5. Models in which every sequence is mutually stationary

Foreman and Magidor [12] pointed out that in general the question of which sequences ⟨S_{n} : n < ω⟩ with S_{n} ⊆ ℵ_{n} are mutually stationary is connected with the open question whether ℵ_{ω} can be a Jonsson cardinal. It is known that rather large singular cardinals of cofinality ω can be Jonsson: in particular Prikry proved that a singular limit of measurable cardinals is Jonsson and that a measurable cardinal remains Jonsson after doing Prikry forcing [20].

In this section we mildly strengthen these classical results by relating them to mutual stationarity. See the introduction to our previous paper [4] for more on the connection between mutual stationarity, Jonsson cardinals and Chang’s conjecture.

Remark 5.1. Baumgartner [1] proved Theorem 5.2 in the special case where S_{n} = κ_{n} ∩ cof(ℵ_{f(n)}) for f : ω ⟶ 2.

Theorem 5.2. Let ⟨κ_{i} : i < λ⟩ be an increasing sequence of measurable cardinals where λ = cf(λ) < κ_{0}. Let S_{i} ⊆ κ_{i} be stationary for each i < λ, then ⟨S_{i} : i < λ⟩ is mutually stationary.

Note that an immediate corollary is the well-known fact that sup ⟨κ_{i} : i < λ⟩ is Jonsson.

Proof. Note that the hypothesis imply that for all i < λ, κ_{i} > sup ⟨κ_{j} : j < i⟩. To simplify the bookkeeping we assume λ = ω. Let θ = sup_{i} κ_{i}, and fix M a structure on H_{θ}. For each i let U_{i} be a normal measure on κ_{i}.

We will construct sets J_{i} ∈ U_{i} such that J_{i+1} ⊆ (κ_{i}, κ_{i+1}) and the following indiscernibility property holds: for any positive integer n and any sequence ⟨k_{j} : j < n⟩ of positive integers, if t_{i}, u_{i} ∈ [J_{i}]^{k_{i}} for i < n and ϕ is any formula in the language of M then

M ⊧ ϕ(t_{0}, … t_{n−1}) ⟺ M ⊧ ϕ(u_{0}, … u_{n−1}).

To build the J_{i}, we define for each j < ω a sequence ⟨I_{n}^{j} : n < ω⟩ such that

(1) I_{n}^{j} ∈ U_{n}.

(2) I_{n+1}^{j} ⊆ (κ_{n}, κ_{n+1}).

(3) I_{n}^{j+1} ⊆ I_{n}^{j}.

(4) For all s ∈ [κ_{n−1}]^{<ω} and all ⟨t_{i} : i ≤ j⟩ with t_{i} ∈ [I_{n+i}^{j}]^{<ω}, the M-type of

(s, t_{0}, … t_{j}) is determined by (s, |t_{0}|, … |t_{j}|).

Base case j = 0: We choose I_{n}^{0} ∈ U_{n} as a set of order-indiscernibles for the structure obtained from M by adding a constant symbol for each element of κ_{n−1}. This is possible by Rowbottom’s theorem.

Successor step: suppose we have constructed I_{n}^{j}. Let s ∈ [κ_{n−1}]^{<ω}, t ∈ [I_{n}^{j}]^{<ω} and u_{i} ∈ I_{n+i}^{j} for 1 ≤ i ≤ j + 1. By induction the M-type of (s, t, u_{1}, … u_{j+1}) is determined by (s, t, |u_{1}|, … |u_{j+1}|). Using Rowbottom’s theorem and the completeness of U_{n} we may find I_{n}^{j+1} ⊆ I_{n}^{j} such that if t ∈ [I_{n}^{j+1}]^{<ω} then for all s, u_{1}, … u_{j+1} the M-type of (s, t, u_{1}, … u_{j+1}) is determined by (s, |t|, |u_{1}|, … |u_{j+1}|).

We now set J_{n} = ⋂_{j} I_{n}^{j}. To finish the proof of the theorem, we choose for each n a set z_{n} ⊆ J_{n} with limit order-type such that sup(z_{n}) ∈ S_{n}. Let N be the Skolem hull in M of the union of the sets z_{n}. We claim that sup(N ∩ κ_{n}) = sup(z_{n}) for each n.

Suppose that t is a Skolem term and that t(a_{0}, … a_{j}) < κ_{i} where a_{n} ∈ [z_{n}]^{<ω} and without loss of generality j ≥ i. Let β be the least element of z_{i} with β > max(a_{i}). It must be that t(a_{0}, … a_{j}) < β, for if not an application of indiscernibility shows that every element of J_{i} which is greater than max(a_{i}) is bounded by t(a_{0}, … a_{j}), and this is impossible since J_{i} is unbounded in κ_{i}. This shows that t(a_{0}, … a_{j}) < sup(z_{i}), so sup(N ∩ κ_{i}) = sup(z_{i}) and we are done. □

We now turn to the situation in which ⟨κ_{n} : n < ω⟩ is a Prikry-generic sequence in a measurable cardinal κ.

Remark 5.3. It is too much to ask that every Prikry-generic ω-sequence should have the property that every sequence of stationary sets is mutually stationary. For example if the sequence begins with ℵ_{1} and ℵ_{2} and Chang’s conjecture is false then we can not meet the sets S_{0} = ℵ_{1}, S_{2} = ℵ_{2} ∩ cof(ℵ_{1}).

Theorem 5.4. Let κ be measurable and let U be a normal measure on κ. Let ℙ be the Prikry forcing defined from U. Then there is a condition (∅, A) ∈ ℙ which forces that if ⟨κ_{n} : n < ω⟩ is the generic cofinal ω-sequence added by ℙ, then every sequence ⟨S_{n} : n < ω⟩ with S_{n} stationary in κ_{n} for all n is mutually stationary.

Proof. Suppose not. By the direct extension property for Prikry forcing, there is a condition (∅, A) and names Ṡ_{n} and Ȧ such that (∅, A) forces that

• Ȧ is an algebra on κ.

• Ṡ_{n} is stationary in κ_{n}.

• No substructure of Ȧ meets ⟨Ṡ_{n} : n < ω⟩.

Let Ḟ be a name for a function F : [κ]^{<ω} ⟶ κ which is a Skolem function for A. That is to say, X ⊆ F “[X]^{<ω} and F “[X]^{<ω} ≺ A for all infinite X ⊆ κ. Define a function F^{∗} : [κ]^{<ω} × [κ]^{<ω} ⟶ κ as follows: F^{∗}(s, t) is equal to the unique β such that there is E ∈ U with the property (s, E) ⊩ Ḟ (t) = β if such an E exists, and 0 otherwise.

By a standard application of Rowbottom’s theorem and a diagonal intersection argument, we may find B ∈ U such that for all δ < κ the set B \ (δ + 1) is a set of order-indiscernibles for (κ, <, F^{∗}, {γ : γ ≤ δ}). Now let D = {γ ∈ A ∩ B : sup(B ∩ γ) = γ}. It is easy to check that D ∈ U, since U is normal.

We now force below the condition (∅, D) to get a generic increasing ω-sequence G = ⟨κ_{n} : n < ω⟩. We use this to realise the names Ṡ_{n}, Ȧ and Ḟ to get stationary sets S_{n} , an algebra A on κ and a Skolem function F for A. Since (∅, D) refines (∅, A) there can be no substructure of A which meets ⟨S_{n} : n < ω⟩.

Working in V [G] we choose for each n a point γ_{n} ∈ S_{n} with sup(B ∩ γ_{n}) = γ_{n}. Let P = ⋃_{n} B ∩ [κ_{n−1}, γ_{n}) and let N be the closure of P under F. We claim that sup(N ∩ κ_{n}) = γ_{n} for all n.

To see this, suppose for a contradiction that F (w) = β for some w ∈ [P]^{<ω} and β such that γ_{n} ≤ β < κ_{n}.

Find a condition (s, E) ∈ G such that (s, E) ⊩ Ḟ (w) = β, and notice that s must be a finite initial segment of ⟨κ_{n} : n < ω⟩. Extending if necessary we may assume that lh(s) = m > n. It is convenient to break up w and s as follows;

• t = w ∩ κ_{n−1}, u = w ∩ [κ_{n−1}, κ_{n}), v = w ∩ (κ \ κ_{n}).

• s_{L} = (κ_{0}, … κ_{n−1}), s_{H} = (κ_{n}, … κ_{m−1}). By the definition of the function F^{∗}, we have

F^{∗}(s_{L}^{⌢}s_{H}, t^{⌢}u^{⌢}v) = β < κ_{n}.

All points of B above sup(u) are chosen from a set of indiscernibles for a structure which has symbols for F^{∗} and all ordinals below sup(u). Fix δ, ζ ∈ B ∩ κ_{n} with β < δ and sup(w ∩ κ_{n}) < ζ < γ_{n}. We may choose a suitable δ because κ_{n} ∈ D and so B ∩ κ_{n} is unbounded in κ_{n}.

The key points are that • The sequences s_{L}, t and u consist of ordinals below sup(u).

• The sequences s_{H} and v consist of ordinals in B above κ_{n}.

• The ordinals δ and ζ lie in B and are between sup(u) and κ_{n}. Since F^{∗}(s, w) < δ, it follows by indiscernibility that F^{∗}(s, w) < ζ. This is

a contradiction, so sup(N ∩ κ_{n}) = γ_{n} as required. It follows that N meets the sequence ⟨S_{n} : n < ω⟩, contradiction! □

Corollary 5.5. If ⟨κ_{n} : n < ω⟩ is any Prikry generic sequence, then there exists m such that all sequences ⟨S_{m} : m ≤ n < ω⟩ with S_{m} stationary in κ_{m} for all m ≥ n are mutually stationary.

Proof. Let A be as in the conclusion of Theorem 5.4, and find m such that κ_{n} ∈ A for all n ≥ m. □

6. Mutually stationary sequences not concentrating on a fixed cofinality

Theorems 5.2 and 5.4 show that if ⟨κ_{n} : n < ω⟩ is an increasing sequence of reasonably large cardinals then every sequence of stationary sets can be mutually stationary. We now return to the problem of mutual stationarity for small cardinals.

Let 0 < l < ω, let f : ω ⟶ {0, l} be any function, and define S_{n}^{f} = {α < ℵ_{n} : cf(α) = ℵ_{f(n)}} for n > l. We will construct a model in which for every function f the sequence ⟨S_{n}^{f} : l < n < ω⟩ is mutually stationary, starting from the assumption that there are infinitely many supercompact cardinals. This was originally done by Shelah, the simpler proof given here is due to Foreman and Magidor.

We will use some facts about IA structures. The first fact appears in in section 2 of [10].

Lemma 6.1. Let N ≺ A be a structure of some regular uncountable cardinality μ. Then the following are equivalent:

(1) N is IA of length and cardinality μ.

(2) For every μ-closed poset ℙ ∈ N there is a sequence of elements ⟨p_{α} : α < μ⟩ of N ∩ℙ, such that for every D ∈ N a dense open subset of ℙ there is α < μ with p_{α} ∈ D.

The next fact is implicit in Foreman, Magidor and Shelah’s paper [13] on Martin’s Maximum.

Lemma 6.2. Let N ≺ A be internally approachable of length and cardinality μ, where μ is an uncountable regular cardinal. Let β be an ordinal such that β < sup(N ∩ ON) and let M = Sk^{A}(N ∪ {β}). Then M is internally approachable of length and cardinality μ.

Proof. Let ⟨N_{α} : α < μ⟩ be an internally approaching chain of models with union M. We may assume without loss of generality that β < sup(N_{0} ∩ ON). Define M_{α} = Sk^{A}(N_{α} ∪ {β}), so that clearly the M_{α} form an increasing continuous chain of models of size less than μ whose union is M.

We claim that M_{α} = {f(β) : f ∈ N_{α}, β ∈ dom(f)}. Clearly if f ∈ N_{α} then f(β) ∈ M_{α}. Conversely if y ∈ M_{α} then y = t(x, β) for some Skolem term t and parameter x ∈ N_{α}. If γ ∈ N_{0} ∩ ON with β < γ then the (partial) function f with domain γ which maps α to t(x, α) is definable in H_{θ} from the parameters y, γ so f ∈ N_{α}.

Fix an ordinal ζ < μ. β and ⟨N_{α} : α ≤ ζ⟩ are members of M_{ζ+1}, so by the work of the last paragraph ⟨M_{α} : α ≤ ζ⟩ ∈ M_{ζ+1}. So M is internally approachable of length and cardinality μ, as claimed. □

The construction will proceed by starting with a structure which meets each ℵ_{n} for n > l in a set of cofinality ℵ_{l}, and judiciously adding ω many ordinals. The following well-known lemma [1] shows that adding an ordinal below ℵ_{m} does no damage above ℵ_{m}.

Lemma 6.3. Let A = (H_{θ}, ∈, <_{θ}) for some large regular θ. Let N ≺ A, where |N| = ℵ_{n} ⊆ N for some n < ω. Let n < m < ω, let β be an ordinal with sup(N ∩ ℵ_{m}) < β < ℵ_{m}, and let N^{∗} = Sk^{A}(N ∩ {β}). Then sup(N^{∗} ∩ ℵ_{j}) = sup(N ∩ ℵ_{j}) for m < j < ω.

Proof. Let t be a Skolem term. For each x ∈ N, N can compute the supremum of the set {t(x, δ) : δ < ℵ_{m}} ∩ ℵ_{j}. □

For the rest of this section we will make the following assumption: Assumption: there exists a sequence of ideals ⟨I_{n} : l + 2 ≤ n < ω⟩ such that

(1) I_{n} is a uniform, ℵ_{n}-complete, normal ideal on ℵ_{n}.

(2) P(ℵ_{n})/I_{n} has an ℵ_{l+1}-closed dense subset.

This assumption is known to be consistent relative to the existence of infinitely many supercompact cardinals.

We now fix some large regular cardinal θ and a structure A which is an expansion of (H_{θ}, ∈, <_{θ}, ⟨I_{n} : l + 2 ≤ n < ω⟩). If N ≺ A has cardinality ℵ_{l}, and sup(N ∩ℵ_{n}) < β < ℵ_{n} for some n, we will say that β is I_{n}-generic for N if and only if the following two conditions are satisfied

(1) For every C ∈ N ∩ I_{n}, β ∉ C.

(2) The set {A ∈ N ∩ P(ℵ_{n}) : β ∈ A} induces an N-generic filter on N ∩

P(ℵ_{n})/I_{n}.

Notice that by the first of these two conditions, if A and A′ are subsets of ℵ_{n} which both lie in N and are equivalent modulo I_{n}, then β ∈ A ⟺ β ∈ A′.

Lemma 6.4. If N is internally approachable of length and cofinality ℵ_{l}, then the set of β < ℵ_{n} which are I_{n}-generic for N is I_{n}-positive.

Proof. Let D be a dense ℵ_{l+1}-closed subset of P(ℵ_{n})/I_{n}. By Lemma 6.1 we may find a decreasing sequence ⟨[A_{α}] : α < ℵ_{l}⟩ of elements of N ∩ D, which meets every dense open subset of P(ℵ_{n})/I_{n} lying in N.

Let [B] ∈ D be a lower bound for the sequence ⟨[A_{α}] : α < ℵ_{l}⟩. Since the ideal I_{n} is ℵ_{l+1}-complete, the set C = ⋂ { ℵ_{n}\X : X ∈ N ∩ I_{n} } is in the dual of I_{n}. For all A ∈ N ∩ P (ℵ_{n}), α ∈ B ∩ C we have α ∈ A iff A is in the filter generated by the sequence ⟨[A_{α}] : α < ℵ_{l}⟩. In particular, all α ∈ B ∩ C are generic over N. □

The following lemma is the crucial one motivating our use of I_{n}-generic ordinals. It indicates that when we add a suitable I_{n}-generic ordinal we do not undo our work at cardinals below ℵ_{n}.

Lemma 6.5. Let β be such that sup(N ∩ ℵ_{n}) < β < ℵ_{n} and β is I_{n}-generic for N. Let N^{∗} = Sk^{A}(N ∪ {β}). Then N^{∗} ∩ ℵ_{n−1} = N ∩ ℵ_{n−1}.

Proof. By the same argument that we used in Lemma 6.2, N^{∗} = {f(β) : f ∈ N}. Let γ ∈ N^{∗} ∩ ℵ_{n−1} and fix f ∈ N such that f : ℵ_{n} ⟶ ℵ_{n−1} and γ = f(β).

The set of equivalence classes [A] such that f is constant on A lies in N, and by normality it is dense in P(ℵ_{n})/I_{n}. Since β is a generic ordinal there is A ∈ N such that β ∈ A and f is constant on A. It follows that γ ∈ N. □

Remark 6.6. We may also give an essentially equivalent proof of Lemma 6.5 phrased in the language of ultrafilters and elementary embeddings. Let M be the transitive collapse of N, let M^{∗} be the collapse of N^{∗}, and let j : M ⟶ M^{∗} be the elementary embedding from M to M^{∗} corresponding to the inclusion map from N to N^{∗}. Let U be the M-ultrafilter on the collapse of κ_{n} which is induced by β.

It is routine to check that M^{∗} = Ult(M, U) and j is the associated elementary embedding j_{U}^{M}. j has critical point equal to the collapse of κ_{n}, so in particular j fixes the collapse of κ_{n−1}. It follows that N ∩ κ_{n−1} = N^{∗} ∩ κ_{n−1}.

Theorem 6.7. Let f : ω ⟶ {0, l} be any function and let T_{n} = {α < ℵ_{n} : cf(α) = ℵ_{f(n)}} for n > l. The sequence ⟨T_{n} : l < n < ω⟩ is mutually stationary.

Proof. It will suffice to build a structure M ≺ A such that cf(M ∩ ℵ_{l+1}) = ℵ_{l} and cf(M ∩ ℵ_{n}) = f(n) for n > l + 1. If necessary we may then use Lemma 6.3 to add in ω ordinals below ℵ_{l+1} and adjust cf(M ∩ ℵ_{l+1}).

Let A be some expansion of (H_{θ}, ∈, <_{θ}). Let N ≺ A be an internally approachable structure of length and cardinality ℵ_{l}. In particular, sup(N ∩ℵ_{n}) has cofinality ℵ_{l} for every n > l.

If f is constant with value l there is nothing to do, so we assume that f takes the value 0 at least once. Let ⟨n_{k} : k < ω⟩ be a sequence of integers such that n_{k} > 2, f(n_{k}) = 0 for all k, and for all n > l + 1 such that f(n) = 0 there are infinitely many k such that n_{k} = n.

We construct sequences ⟨N_{k} : k < ω⟩ of structures and ⟨β_{k} : k < ω⟩ of ordinals by recursion on k.

• N_{0} = N.

• If n_{k} = n then β_{k} is some ordinal such that sup(N_{k} ∩ ℵ_{n}) < β_{k} < ℵ_{n} and

β_{k} is I_{n}-generic for N_{k}.

• N_{k+1} = Sk^{A}(N_{k} ∪ {β_{k}}). The construction can proceed, because by Lemma 6.2 the structure N_{k} is inter-

nally approachable of length and cofinality ℵ_{l} for every k < ω. Lemmas 6.3 and 6.5 imply that sup(N_{k+1} ∩ ℵ_{j}) = sup(N_{k} ∩ ℵ_{j}) for j ≠ k, so if we set N_{ω} = ⋃_{k} N_{k} then we see that

• N_{ω} ≺ A.

• cf(N_{ω} ∩ ℵ_{l+1}) = ℵ_{l}.
• cf(N_{ω} ∩ ℵ_{j}) = ℵ_{l} if f(j) = l, j > l + 1.
• cf(N_{ω} ∩ ℵ_{j}) = ℵ_{0} if f(l) = 0, j > l + 1.

This shows that ⟨T_{n} : l < n < ω⟩ is mutually stationary. □

If we are willing to leave gaps between the cardinals where we want cofinality ω, then we can reduce the hypothesis of Theorem 6.7 to infinitely many measurable cardinals. Explicitly: If there are infinitely many measurable cardinals and A ⊂ ω\2 is such that for all n ∈ A, n + 1 ∉ A, then there is a forcing extension where ℵ_{n} carries a normal ℵ_{n}-complete ideal on ℵ_{n} with a dense set that is closed under decreasing sequences of length ℵ_{n−2}. In the resulting model, it can be shown that if f : ω → {0, l} is such that f^{−1}(0) ⊂ A, then the sequence of sets ⟨T_{n} : l < n < ω⟩ is mutually stationary. The proof is exactly as above.

7. Good points and diagonal Prikry forcing

In this section we record two forcing constructions involving large cardinals, PCF and reflection. The first construction gives a simple proof that it is consistent for there to be stationarily many non-good points in ℵ_{ω+1}.

7.1. Good points. As we mentioned in the introduction to this paper, various models are known in which the set of non-good points of cofinality ℵ_{1} is stationary in ℵ_{ω+1}.

• Levinski, Magidor and Shelah [16] have shown that the Chang’s conjecture (ℵ_{ω+1}, ℵ_{ω}) ↠ (ℵ_{1}, ℵ_{0}) is consistent, and Foreman and Magidor [11] have shown that if (ℵ_{ω+1}, ℵ_{ω}) ↠ (ℵ_{1}, ℵ_{0}) then the set of non-good points of cofinality ℵ_{1} is stationary.

• In unpublished work Magidor [17] has shown that the same conclusion

follows from Martin’s Maximum. In this section we record the remark that Shelah’s construction [14] for making

the set of non-approachable points of cofinality ℵ_{1} stationary also makes the set of non-good points stationary.

We start by assuming that κ is supercompact and that the GCH holds. It follows from GCH that there exists a scale ⟨f_{α} : α < κ^{+ω+1}⟩ in ∏_{n<ω} κ^{+n} under the eventual domination ordering; to see this enumerate ∏_{n<ω} κ^{+n} as ⟨g_{η} : η < κ^{+ω+1}⟩, write each α < ℵ_{ω+1} as an increasing union ⋃_{n} X_{n}^{α} with |X_{n}^{α}| < κ^{+n}, and inductively choose f_{α} so that f_{α}(n) is greater than f_{η}(n) and g_{η}(n) for all η ∈ X_{n}^{α}.

The basic idea is that this scale contains many non-good points of cofinality less than κ, and that we will “miniaturise” this situation by some judicious cardinal collapsing. Fix j : V ⟶ M witnessing that κ is κ^{+ω+1}-supercompact, and note that j is discontinuous at κ^{+n} for n < ω and also at κ^{+ω+1}. Let γ = sup(j“κ^{+ω+1}) and let H ∈ ∏ j(κ^{+n}) be given by H(n) = sup(j“κ^{+n}). Let j(⟨f_{α} : α < κ^{+ω+1}⟩) = ⟨g_{β} : β < j(κ^{+ω+1})⟩, and observe that by the elementarity of j and the closure of M the sequence ⟨g_{β} : β < j(κ^{+ω+1})⟩ is a scale in ∏_{n} j(κ^{+n}) under eventual domination. It is easy to see that H is an exact upper bound for ⟨g_{β} : β < γ⟩.

We claim that there is an inaccessible δ < κ such that for stationarily many η ∈ κ^{+ω+1} ∩cof(δ^{+ω+1}) there an exact upper bound h for ⟨f_{α} : α < η⟩, with cf(h(n)) = δ^{+n} for all n. If the claim fails then fix for each δ a club C_{δ} witnessing the nonstationarity of the relevant set, and let C = ⋂_{δ} C_{δ}. Since C is club we see that γ ∈ j(C), and since cf(γ) = κ^{+ω+1} and ⟨g_{β} : β < γ⟩ has an exact upper bound H with cf(H(n)) = κ^{+n} for all n we get a contradiction by elementarity.

We now fix a suitable inaccessible δ < κ and let S be the stationary set of η ∈ κ^{+ω+1} ∩ cof(δ^{+ω+1}) such that there is an exact upper bound h for ⟨f_{α} : α < η⟩, with cf(h(n)) = δ^{+n} for all n. We force with ℙ × ℚ where ℙ = Col(ω, δ^{+ω}) and ℚ = Col(δ^{+ω+2}, < κ). Let G × H be ℙ × ℚ-generic.

The usual chain condition and closure arguments tell us that δ_{V}^{+ω+1} is the new ℵ_{1}, κ_{V}^{+n} = ℵ_{n+3} and κ_{V}^{+ω+1} = ℵ_{ω+1}. By Easton’s lemma all δ_{V}^{+ω+1}-sequences of ordinals from V [G][H] lie in V [G]. Since ℙ × ℚ is κ-c.c. it is also routine to check that S is still stationary in V [G][H] and that ⟨f_{α} : α < κ^{+ω+1}⟩ is a scale in ∏_{n<ω} κ_{V}^{+n}.

To finish we claim that if η ∈ S then η is not a good point in V [G][H]. Suppose for a contradiction that such an η is good, and fix an unbounded set A ⊆ η of order type δ_{V}^{+ω+1} and k < ω such that ⟨f_{α}(n) : α ∈ A⟩ is strictly increasing for n > k. As we pointed out above, A ∈ V [G]. Since ℙ has cardinality δ^{+ω} it follows that there is B ⊆ A with B ∈ V and B unbounded in η.

The set B will serve as a witness that in V the point η is good of cofinality δ^{+ω+1}. This implies that an exact upper bound g for ⟨f_{α} : α < η⟩ exists with cf(g(n)) = δ^{+ω+1} for all n, contradicting the fact that η ∈ S and that exact upper bounds are unique modulo finite alteration.

To summarise we have proved the following result.

Theorem 7.1. If κ is κ^{+ω+1}-supercompact then in some generic extension the set of non-good points of cofinality ℵ_{1} in ℵ_{ω+1} is stationary.

If we could make δ^{+ω+1} into ℵ_{2} by some small forcing we could get the consistency of the set of non-good points of cofinality ℵ_{2} being stationary. Unfortunately this kind of cardinal collapse is provably very difficult and conjectured to be impossible [3].

7.2. Diagonal Prikry forcing. We showed in a previous paper [6] that Prikry forcing at a measurable cardinal κ preserves some of the stationary reflection properties of κ^{+}. Here we prove a similar result for diagonal Prikry forcing, using a rather similar argument.

We start by fixing some notation that we will use through this section. Suppose that we are given an increasing ω-sequence of measurable cardinals κ_{n} together with a normal measure U_{n} on each κ_{n}. A condition in the diagonal Prikry forcing determined by these data is a sequence (α_{0}, … α_{m−1}, B_{m}, B_{m+1}, …) where κ_{i−1} < α_{i} < κ_{i} and B_{i} ∈ U_{i}. Given conditions p = (α_{0}, … α_{m−1}, B_{m}, B_{m+1}, …) and q = (β_{0}, … β_{n−1}, C_{n}, C_{n+1}, …), q extends p when n ≥ m, β_{i} = α_{i} for i < m and β_{i} ∈ B_{i} for m ≤ i < n.

We refer to the finite sequence (α_{0}, … α_{m−1}) as the lower part of the condition (α_{0}, … α_{m−1}, B_{m}, B_{m+1}, …). It is well-known that diagonal Prikry forcing has the Prikry property, in the sense that any question about the forcing extension can be decided by shrinking the measure one sets in a condition, or to put it another way without changing the lower part.

We now let κ = ⋃_{n} κ_{n} and suppose that κ^{+} has the following reflection property: for all n, any stationary susbet of κ^{+}∩cof(< κ_{n}) reflects at some point in κ^{+}∩cof(< κ_{n}). This will be the case for example if all of the κ_{n} are strongly compact. We claim that this reflection property is preserved by the diagonal Prikry forcing.

To see this fix n, a condition p and a name Ṫ for a stationary subset of κ^{+}∩cof(< κ_{n}). By extending p if necessary we may assume that the lower part of p has length at least n. For each lower part x which extends the lower part of p we let T_{x} be the set of α such that some extension of p with lower part x forces α into Ṫ; since there are only κ possibilities for x, we may find x such that T_{x} is stationary.

By hypothesis there is γ < κ^{+} with cf(γ) < κ_{n} such that T_{x} ∩ γ is stationary. We now fix C ⊆ γ with order type cf(γ), and then use the completeness of the measures U_{j} for j ≥ n to find a single condition q with lower part x such that q forces that C ∩ T_{x} ⊆ Ṫ. Then q forces that Ṫ reflects at γ and we are done.

We summarise the results of this discussion in a theorem.

Theorem 7.2. Let ⟨κ_{n} : n < ω⟩ be an increasing sequence of measurable cardinals with supremum κ, and suppose that for every n every stationary susbet of κ^{+}∩cof(< κ_{n}) reflects at some point in κ^{+}∩cof(< κ_{n}). Then this reflection property still holds in the generic extension by any diagonal Prikry forcing defined from some sequence of normal measures on the κ_{n}.

Gitik and Magidor have devised several forcing posets for adding many diagonal Prikry sequences simultaneously. It would be interesting to combine their methods with those of Theorem 7.2.

8. Reflection and Martin’s maximum revisited

Foreman, Magidor and Shelah [13] showed that Martin’s Maximum implies that for all λ ≥ ℵ_{2}, every stationary subset of [H_{λ}]^{ℵ_{0}} reflects to a structure of size and uniform cofinality ℵ_{1}. We showed in the last section of [6] that forcing over a model of MM^{+} we can get the consistency of this kind of reflection with the existence of two stationary subsets of a regular cardinal κ ≥ ℵ_{2} which do not reflect simultaneously, and Larson [15] independently obtained similar results. The following result generalises and sharpens these theorems: note in particular that we are reflecting to an IA structure and that we are only using MM (rather than MM^{+}) in the ground model.

Theorem 8.1. Assume Martin’s Maximum. Let κ be a regular cardinal with κ > ℵ_{1} and let η be a (possibly finite) cardinal with η ≤ ℵ_{1}. Then there is a forcing poset ℙ which adds no bounded subsets of κ and such that in V^{ℙ}

(1) There are η stationary subsets of κ ∩ cof(ω) which do not reflect simultaneously.

(2) For every λ > ℵ_{1}, every set of fewer than η stationary subsets of [H_{λ}]^{ℵ_{0}} simultaneously reflects to an internally approachable set in [H_{λ}]^{ℵ_{1}}.

In particular, every collection of less than η stationary subsets of κ ∩ cof(ω) simultaneously reflects.

Proof. We let ℙ be the natural poset for partitioning κ ∩ cof(ω) into η many stationary sets which do not reflect simultaneously. Conditions in ℙ are functions p such that p : η×(α∩cof(ω)) ⟶ 2 for some α < κ, with the properties that for each i we have p(ν, i) = 1 for exactly one ν < η, and that for every β ∈ (α + 1) ∩ cof(ℵ_{1}) there are ν < η and C club in β such that f ↾ {ν} × C is identically zero.

It is easy to see that ℙ is countably closed and is κ-strategically closed (a winning strategy for player II is to pick a coordinate ν < η and to write zero at that coordinate whenever it is her turn to play). It is also easy to see that ℙ adds a partition of κ ∩ cof(ω) into η stationary sets which do not reflect simultaneously.

By Martin’s Maximum the non-stationary ideal on ℵ_{1} is ℵ_{2}-saturated in V. We claim that this is also the case in V^{ℙ}. To see this let ⟨Ȧ_{i} : i < ℵ_{2}⟩ be a ℙ-name for a counterexample to saturation, and note that V and V^{ℙ} have the same subsets of ℵ_{1}; in particular they agree on the question of whether a subset of ℵ_{1} is club, stationary or non-stationary. Since κ ≥ ℵ_{2} we may use the strategic closure of ℙ to build a decreasing chain ⟨p_{i} : i < ℵ_{2}⟩ of conditions in ℙ such that p_{i} ⊩ Ȧ_{i} = B̌_{i} for some B_{i} ∈ V. Then ⟨B_{i} : i < ℵ_{2}⟩ is a counterexample to saturation in V, which is a contradiction.

We let ⟨T_{j} : j < η⟩ be the sequence of stationary subsets of κ added by ℙ. Let ζ < η be a cardinal, let λ be a cardinal in V^{ℙ} and let ⟨Ṡ_{i} : i < ζ⟩ be a sequence of ℙ-names for stationary subsets of [H_{λ}]^{ℵ_{0}}. (So there are at most countably many sets S_{i}.) Let μ be the maximum of λ and κ. We work towards showing that ℙ forces that the sets S_{i} reflect simultaneously.

We now work in V^{ℙ}. For each i < ζ we will say that S_{i} is social if there exists j < η such that for stationarily many N ∈ [H_{μ}]^{ℵ_{0}}, N ∩H_{λ} ∈ S_{i} and sup(N ∩κ) ∈ T_{j}. In this case we let j(i) be the least j with this property. If S_{i} is not social we say that S_{i} is antisocial. If η < ℵ_{1} then all S_{i} are social, but if η = ℵ_{1} this is not necessarily the case.

Let j^{∗} < η be least such that j^{∗} ≠ j(i) for any social S_{i}. Since there are only countably many sets S_{i} we may fix a club set C_{bad} in [H_{μ}]^{ℵ_{0}} such that if S_{i} is antisocial, then sup(N ∩ κ) ∉ T_{j^{∗}} for every N ∈ C_{bad} such that N ∩ H_{λ} ∈ S_{i}.

We now use a fact from [13]:

Claim. Suppose that the non-stationary ideal on ω_{1} is ℵ_{2}-saturated and S ⊂ [H_{μ}]^{ℵ_{0}} is stationary. Then there is a closed unbounded set C ⊂ [H_{μ}]^{ℵ_{0}} such that for all stationary T ⊂ { N ∩ ω_{1} : N ∈ C ∩ S } there are stationarily many N ∈ C ∩ S such that N ∩ ω_{1} ∈ T.

Proof. (Sketch) First note that if C, D are club sets in [H_{μ}]^{ℵ_{0}} and there is a ξ ∈ ω_{1} such that for all N ∈ C if ξ ∈ N then N ∈ D, then {N ∩ ω_{1} : N ∈ C} ⊂ {N ∩ ω_{1} : N ∈ D} modulo the non-stationary ideal on ω_{1}.

Now build a sequence of closed unbounded sets ⟨C_{ξ} : ξ < ξ^{∗}⟩ for some ξ^{∗} ≤ ω_{2} by induction. Let C_{0} = [H_{μ}]^{ℵ_{0}} and given C_{ξ} choose C_{ξ+1} ⊂ C_{ξ} if possible so that { N ∩ ω_{1} : N ∈ C_{ξ+1} } ⊊ { N ∩ ω_{1} : N ∈ C_{ξ} } modulo the non-stationary ideal. If this is not possible, then we set ξ^{∗} = ξ + 1. At limit stages we take diagonal intersections.

Since the non-stationary ideal on ω_{1} is ℵ_{2}-saturated, there is a ξ^{∗} < ω_{2} where this sequence stops. If ξ^{∗} = ξ +1, then C = C_{ξ} satisfies the conclusion of the claim.

□

By the claim we can fix for each i < ζ a stationary set U_{i} ⊆ ℵ_{1} such that (1) If S_{i} is social then for every stationary T ⊆ U_{i} there are stationarily many N ∈ [H_{μ}]^{ℵ_{0}} such that N ∩ H_{λ} ∈ S_{i}, sup(N ∩ κ) ∈ T_{j(i)} and N ∩ ℵ_{1} ∈ T.

(2) If S_{i} is antisocial then for every stationary T ⊆ U_{i} there are stationarily many N ∈ [H_{μ}]^{ℵ_{0}} such that N ∩ H_{λ} ∈ S_{i} and N ∩ ℵ_{1} ∈ T.

Thinning out if necessary we arrange that the U_{i} are pairwise disjoint. By the closure of ℙ we see that ⟨U_{i} : i < ζ⟩ ∈ V, and so working below a suitable condition in ℙ we may assume that we have a fixed sequence ⟨U_{i} : i < ζ⟩ which is in V.

Still working in V^{ℙ} we define ℚ to be Col(ℵ_{1}, H_{μ}^{V}). If F : ℵ_{1} ≃ H_{μ}^{V} is the bijection added by ℚ then in V^{ℙ∗ℚ} we let s_{i} = {δ < ℵ_{1} : F “δ ∩ H_{λ} ∈ S_{i}} and t_{j} = {δ < ℵ_{1} : sup(F “δ ∩ κ) ∈ T_{j}}. Working in V^{ℙ∗ℚ} we define a poset ℝ; conditions in ℝ are closed bounded subsets of ℵ_{1} consisting of ordinals δ such that δ ∉ t_{j^{∗}}, and such that δ ∈ U_{i} implies F “δ ∩ H_{λ} ∈ S_{i} for each i. The ordering on ℝ is end-extension.

With a view to applying Martin’s Maximum, we claim that ℙ∗ℚ∗ℝ is stationary set preserving. Let S be a stationary subset of ℵ_{1}. It is clear by the strategic closure of ℙ that S is still stationary in V^{ℙ}, and we will work in V^{ℙ} to argue that ℚ ∗ ℝ preserves the stationarity of S. Let Ċ be a ℚ ∗ ℝ-name for a club subset of ℵ_{1} and let (q_{0}, c_{0}) ∈ ℚ ∗ ℝ. As usual when we are proving the preservation of stationarity, our goal is to find (q, c) ≤ (q_{0}, c_{0}) forcing that Ċ meets S.

Shrinking S if necessary, and using the fact that there are only countably many sets U_{i}, we may assume that either S is disjoint from every U_{i} or S ⊆ U_{i} for some i. We will treat these cases separately, and will also break up the second case according to the sociality or otherwise of S_{i}. We start by fixing some large regular cardinal θ.

Case 1: S is disjoint from every U_{i}. In this case we will choose a countable M ≺ H_{θ} containing everything relevant such that δ =_{def} M ∩ ℵ_{1} ∈ S and sup(M ∩ κ) ∉ T_{j^{∗}}. We then build a chain ⟨(q_{n}, c_{n}) : n < ω⟩ of conditions in M ∩ ℚ ∗ ℝ which meets every dense subset of ℚ ∗ ℝ lying in M, and let q = ⋃ q_{n} and c = ⋃ c_{n}.

It is clear that q ∈ ℚ and q forces that F “δ = M ∩ H_{μ}^{V}. Since sup(M ∩ κ) ∉ T_{j^{∗}}, q forces that δ ∉ t_{j^{∗}} and so (q, c∪{δ}) is a condition in ℚ∗ℝ. This condition forces that δ ∈ Ċ and we are done.

Case 2a: S ⊆ U_{i} for a social S_{i}. In this case we choose a countable M ≺ H_{θ} such that δ =_{def} M ∩ ℵ_{1} ∈ S, M ∩ H_{λ} ∈ S_{i} and sup(M ∩ κ) ∈ T_{j(i)}; this is possible by the choice of j(i) and U_{i}. We define q and c as in case 1, and again q forces that F “δ = M ∩ H_{μ}^{V}.

Since j(i) ≠ j^{∗}, q forces that δ ∉ t_{j^{∗}}. By the choice of M we also see that q forces δ ∈ s_{i}. Thus (q, c ∪ {δ}) is a condition in ℚ ∗ ℝ and we are done.

Case 2b: S ⊆ U_{i} for an antisocial S_{i}. In this case we choose M ≺ H_{θ} such that δ =_{def} M ∩ ℵ_{1} ∈ S, M ∩ H_{λ} ∈ S_{i} and M ∩ H_{μ} ∈ C_{bad}. It follows from the choice of C_{bad} that sup(M ∩ κ) ∉ T_{j^{∗}}, and we may now proceed as in Case 2a.

We note that in the course of proving the claim, we also showed that if C is the club set added by ℝ then C ∩ U_{i} is stationary for every i.

To finish the argument we will now apply Martin’s Maximum to ℙ ∗ ℚ ∗ ℝ as in the last section of [6]. Meeting suitable dense sets we produce p, F and C together with j^{∗} < η and disjoint stationary U_{i} ⊆ ℵ_{1} such that

(1) dom(p) = η × α for some α < κ of cofinality ℵ_{1}, with p : η × α ⟶ 2.

(2) F : ℵ_{1} ⟶ H_{μ} and F “ℵ_{1} is internally approachable (this approachability is easy to arrange, observing that for each γ < ℵ_{1} the set of (p, q, r) such that q ↾ γ ∈ range(q) is dense)

(3) sup F “ℵ_{1} ∩ κ = α.

(4) C is club in ℵ_{1} and C ∩ U_{i} is stationary for every i.

(5) p(j^{∗}, sup(F “δ ∩ κ)) = 0 for every δ ∈ C.

(6) For every ᾱ < α, p ↾ η × ᾱ ∈ ℙ.

(7) For every δ ∈ C ∩ U_{i}, there is some ᾱ < α such that p ↾ η × ᾱ forces that F “δ ∩ H_{λ} ∈ S_{i}.

Since we have arranged that p(j^{∗}, ν) = 0 for a club set of ν < α, p is itself a condition in ℙ. For each i we have arranged that C ∩ U_{i} is stationary and that p forces {F “δ ∩ H_{λ} : δ ∈ C ∩ U_{i}} ⊆ S_{i}, so that p forces S_{i} to reflect to F “ℵ_{1}.

□

9. The least cardinal where square fails

We showed [4] that if □_{ℵ_{n}} holds for every n < ω and CH holds then a certain weakening of □_{ℵ_{ω}} holds. We then showed [5] that it is consistent for the full □_{ℵ_{ω}} to fail under these circumstances. In this section we show that the least cardinal where square fails can be the least inaccessible.

Theorem 9.1. It is consistent from large cardinals that the least λ where □_{λ} fails is the least inaccessible cardinal.

Zeman pointed out that consistency that square first fails at the first inaccessible is a Mahlo cardinal. The model he constructs is the “usual” model V^{Col(κ,λ)}, where κ is the first inaccesible cardinal. The “usual” arguments show that square fails at κ in this model, and moreover, that if square held below κ in the ground model, it holds below κ in this model. Nonetheless we give the proof below as it seems that it may be useful in some other context.

Proof. (Sketch) Let GCH hold, let κ be supercompact and let λ be the least inaccessible cardinal greater than κ. Force that □_{η} holds for every η with η < λ by a Reverse Easton iteration ℙ of length λ. Note that ℙ preserves cardinals, preserves the inaccessibility of λ and has cardinality λ. Now let ℚ be the Cohen forcing Add(ℵ_{0}, κ), so that in V^{ℙ∗ℚ} the cardinal λ is the least inaccessible cardinal.

We show that □_{λ} fails in V^{ℙ∗ℚ} by showing that every stationary subset of λ^{+} reflects. Let T be a stationary subset of λ^{+} in V^{ℙ∗ℚ}, and use the fact that ℙ ∗ ℚ has size λ to find a set U ⊆ T such that U ∈ V and U is stationary in V. Since κ is supercompact U reflects to some point η of cofinality δ^{+}, for some inaccessible δ with δ < κ. We finish by showing that ℙ∗ℚ preserves the stationarity of stationary subsets of δ^{+}, from which it follows that T ∩ η is stationary in V^{ℙ∗ℚ}.

We factor ℙ as ℙ_{δ} ∗ ℙ^{δ}, where ℙ_{δ} adds the □_{ζ} sequences for ζ < δ and ℙ^{δ} adds them for ζ ≥ δ. Since ℙ_{δ} is δ^{+}-c.c. and ℙ^{δ} is < δ^{+}-strategically closed in V^{ℙ_{δ}}, forcing with ℙ preserves stationary subsets of δ^{+}; since ℚ is is c.c.c the same is true of ℙ ∗ ℚ. □

10. A limiting result

In this last section of the paper we prove a result which limits the possibilities for creating a supercompact cardinal by forcing in the presence of weak squares. This result was motivated by the question “to what extent are weak squares compatible with stationary reflection?” A natural scenario for making a model with weak square at μ and some reflection is to make a model of □_{μ}^{∗} where some λ with λ < μ can be made supercompact by “mild” forcing. Ben-David and Shelah [2] attempted to give a proof of the consistency of weak square with reflection in which a generic supercompact embedding is resurrected by countably closed forcing, but the theorem that follows shows that their approach to the problem cannot work. See our paper on squares and reflection [6] for a consistency proof that uses the technique of resurrecting supercompact cardinals, but where the forcing which resurrects supercompactness is stationary set preserving for more delicate reasons.

A cardinal λ is generically η- supercompact by countably closed forcing iff there is a countable closed forcing ℙ such that in V^{ℙ}, there is an elementary embedding j : V → M with M a transitive class and j“η ∈ M.

Theorem 10.1. Let λ < μ be cardinals with λ regular, μ strong limit and cf(μ) = ω. If □_{μ}^{∗} holds then λ is not generically μ^{+}-supercompact by a countably closed forcing which preserves μ and μ^{+}.

Proof. We wish to fix a sequence which witnesses □_{μ}^{∗} and has some additional properties. Starting with an arbitrary □_{μ}^{∗}-sequence, we first replace each set C_{α} by its closure under the power set operation; since μ is strong limit and the elements of C have order type less than μ, the resulting set still has size at most μ. We have produced a sequence ⟨C_{α} : α < μ^{+}⟩ such that for all α

(1) C_{α} is a set of subsets of α and |C_{α}| ≤ μ.

(2) If C is in C_{α} then (a) P(C) ⊆ C_{α}.

(b) For every β < α with sup(C ∩ β) = β, C ∩ β ∈ C_{β}.

(3) C_{α} contains at least one set which is club in α and has order-type cf(α). Now let ℙ be some countably closed forcing poset which preserves μ and μ^{+},

let V_{1} be some generic extension by ℙ. Suppose that in V_{1}, the generic μ^{+}- supercompactness of λ is witnessed by j : V ⟶ M.

Let γ = sup j“μ^{+}, so that cf(γ) = cf_{M}(γ) = μ^{+} and γ < j(μ^{+}). Let the image of our original □_{μ}^{∗}-sequence under j be ⟨C_{α}^{j} : α < j(μ^{+})⟩, and fix C ∈ C_{γ}^{j} which is club in γ and has order-type μ^{+}. The embedding j is continuous at points of cofinality ω, so that j“μ^{+} ∩ C is ω-club in γ. Let Ċ name C.

Claim. There do not exist p ∈ ℙ and an unbounded subset D of μ^{+} such that p ⊩ j“D ⊆ Ċ.

Proof. If such p and D exist, let α be an accumulation point of D such that cf(α) = ω and ot(D ∩ α) ≥ μ. The embedding j must be continuous at α, so p forces that j(α) is an accumulation point of j(Ċ), and so by coherence that j(Ċ)∩j(α) ∈ C_{j(α)}^{j}. If x is any countable subset of D ∩ α then p forces that j(x) = j“x and that j“x ⊆ j(Ċ) ∩ j(α), so p forces that j(x) ∈ C_{j(α)}^{j}. By elementarity x ∈ C_{α}. This is impossible because there are μ^{ω} possibilities for x and μ^{ω} ≥ μ^{+} > |C_{α}|. □

Given α < μ^{+}, let α^{∗} be a term for the least β > α with j(β) ∈ Ċ; we say that p bounds α^{∗} if and only if there is γ < μ^{+} such that p ⊩ α^{∗} ≤ γ. Not that if p does not bound α^{∗} and β > α then p does not bound β^{∗}.

Claim. For every p there is an α < μ^{+} such that p does not bound α^{∗}.

Proof. Suppose that p bounds α^{∗} for every α, and define D to be the ω-club set of points β such that cf(β) = ω and p ⊩ α^{∗} < β for every α < β. If β ∈ D then it is forced that j is continuous at β, so that p forces j(β) to be a limit point of Ċ and hence p ⊩ j(β) ∈ Ċ. This contradicts the previous claim. □

Claim. There exist a tree of conditions ⟨p_{s} : s ∈ ^{<ω}μ⟩ and an increasing sequence ⟨α_{i} : i < ω⟩ of ordinals from μ^{+} such that

(1) If t extends s then p_{t} ≤ p_{s}.

(2) The condition p_{s} does not bound α_{lh(s)}^{∗}.

(3) For each i < μ, p_{s^{⌢}i} decides α_{lh(s)}^{∗} as some ordinal β(s^{⌢}i) with β(s^{⌢}i) < α_{lh(s)+1}.

(4) If i ≠ j then β(s^{⌢}i) ≠ β(s^{⌢}j).

Proof. We observe that if p does not bound α^{∗} and β > α then p does not bound β^{∗}. We start by setting p_{0} = 1_{ℙ} and choosing α_{0} such that p_{0} does not bound α_{0}^{∗}. Having defined α_{n} and p_{s} for lh(s) = n, we use the fact that no p_{s} bounds α_{n}^{∗} to choose the p_{s^{⌢}i} and β(s^{⌢}i) appropriately; we then choose α_{n+1} above all the β(t) for lh(t) = n + 1, with the property that α_{n+1}^{∗} is not bounded by any p_{t} with lh(t) = n + 1. □

Let α_{ω} = sup_{i<ω} α_{i}. For each f ∈ ^{ω}μ let p_{f} be a lower bound for ⟨p_{f↾n} : n < ω⟩ and let x_{f} = { β(f ↾ n) : n < ω }. By construction each p_{f} forces that j(α_{ω}) is a limit point of C, so that arguing as in the proof of our first claim p_{f} ⊩ j(x_{f}) ∈ C_{j(α_{ω})}^{j} and hence x_{f} ∈ C_{α_{ω}}.

By construction the x_{f} are all distinct, and there are μ^{ω} possibilities for f. Therefore |C_{α_{ω}}| > μ, contradiction! It follows that j can not be a generic μ^{+}- supercompact embedding in V^{ℙ}. □

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