DESTROYING STATIONARY SETS

JOHN KRUEGER

Abstract. We present a forcing poset for destroying the stationarity of certain subsets of P_{κ}κ^{+}. Using this poset along with Prikry forcing techniques we establish some consistency results concerning saturated ideals and S(κ, κ^{+}).

In this paper we present a forcing poset for destroying stationary subsets of P_{κ}κ^{+} where κ is strongly inaccessible. We use this forcing poset, along with Prikry forcing techniques, to prove a variety of consistency results concerning saturated ideals and the set S(κ, κ^{+}) = {a ∈ P_{κ}κ^{+}: o.t.(a) = (a ∩ κ)^{+}}.

Previously Gitik [6] introduced a forcing poset for destroying stationary subsets of P_{κ}κ^{+}. He used this poset to construct a model with a non-splitting stationary subset of P_{κ}κ^{+}, that is, a stationary set S such that NS ↾ S is κ^{+}-saturated. However, since Gitik’s poset is designed only to destroy stationary sets contained in a fixed inner model of the universe, its applications are limited. Our forcing poset is a modification of Gitik’s poset which can destroy stationary sets not necessarily contained in the inner model.

The two main consistency results of the paper can be summarized as follows. Assume the existence of a supercompact cardinal. Then the following two statements are consistent:

(1) The set S(κ, κ^{+}) is stationary and for almost all a in S(κ, κ^{+}), cf(a ∩ κ) = ω.

(2) The ideal NS ↾ S(κ, κ^{+}) is κ^{+}-saturated. We also prove that (2) is optimal in the sense that whenever NS ↾ S is κ^{+}-saturated for some set S ⊆ P_{κ}κ^{+}, then S ⊆ S(κ, κ^{+}) modulo clubs.

The contents of the paper are as follows. Section 1 outlines notation and preliminaries. Section 2 describes some of the forcing posets we use in the paper. Section 3 introduces the set S(κ, κ^{+}). Section 4 presents the forcing poset for destroying stationary sets. Section 5 describes how to iterate such posets over different cardinals using Easton support Prikry iterations. Section 6 constructs a model using a Prikry iteration in which almost all a in S(κ, κ^{+}) have cf(a ∩ κ) = ω. Section 7 constructs a model in which NS ↾ S(κ, κ^{+}) is κ^{+}-saturated.

I would like to thank my thesis advisor James Cummings for many helpful discussions regarding this material. I would also like to thank Moti Gitik for explaining to me certain details from his paper [6].

1. Preliminaries

We assume that the reader is familiar with iterated forcing, supercompact cardinals, and Prikry forcing; see [2] and [8].

This paper forms a part of my Ph.D. dissertation written under the supervision of Professor Cummings at Carnegie Mellon University.

If κ is regular and κ ⊆ X, we define P_{κ}X = {a ⊆ X: |a| < κ, a ∩ κ ∈ κ}. A subset of P_{κ}X is club if it is closed under unions of increasing sequences of length less than κ and is cofinal in P_{κ}X. A set is stationary if it intersects every club.

If C is a club subset of P_{κ}X then there is a function f: X^{<ω} → X such that whenever a is in P_{κ}X and is closed under f then a is in C. If A is a directed subset of C with size less than κ then ⋃ A is in C.

When we say that a statement is true for almost all a in P_{κ}X, we mean there is a club set whose elements each satisfy the statement. We write A = B modulo clubs if there is a club set C such that A ∩ C = B ∩ C. A function f: P_{κ}X → X is regressive if f(a) is in a for all a.

The ideal of non-stationary subsets of P_{κ}λ for λ ≥ κ is denoted by NS_{κ,λ} or NS. If S is stationary then NS_{κ,λ} ↾ S denotes the ideal generated by the elements of NS_{κ,λ} along with the complement of S. An ideal on P_{κ}λ is fine if it contains the set {a ∈ P_{κ}λ: ξ ∉ a} for every ξ < λ. An ideal I is normal if for every set S ⊆ P_{κ}λ not in I and for every regressive function f: S → λ, there is an i < λ such that the set {a ∈ S: f(a) = i} is not in I.

By an ultrafilter on P_{κ}λ, we always mean a non-principal, fine ultrafilter.

If I is an ideal on P_{κ}λ the set I^{∗} = {P_{κ}λ\A: A ∈ I} is the dual filter of I. The collection of I-positive sets I^{+} = {A ⊆ P_{κ}λ: A ∉ I} is a forcing poset ordered by A ≤ B if A \ B is in I.

An ideal I is μ-saturated if I^{+} is μ-c.c. If I = NS ↾ S for some S then I is μ-saturated iff there is no family {S_{i}: i < μ} of stationary subsets of S such that S_{i} ∩ S_{j} is non-stationary for i < j. If κ is weakly inaccessible, a set S ⊆ P_{κ}κ^{+} is non-splitting if NS ↾ S is κ^{+}-saturated. A set S ⊆ P_{κ}κ^{+} is non-splitting iff there does not exist a partition of S into κ^{+}-many disjoint stationary subsets.

If I is an ideal on P_{κ}λ, the poset I^{+} adds a set U which is an ultrafilter on the Boolean algebra P(P_{κ}λ)^{V}. If M is the ultrapower of V by U in the generic extension by I^{+}, M is called the generic ultrapower and the ultrapower map j: V → M is the generic embedding. If I is λ^{+}-saturated then the generic ultrapower M is well-founded and ^{λ}M ⊆ M.

A cardinal κ is λ-supercompact if there is a normal ultrafilter on P_{κ}λ, or equivalently, there is an elementary embedding j: V → M, where M is a transitive inner model, such that crit j = κ, j(κ) > λ, and ^{λ}M ⊆ M. If U_{0} and U_{1} are normal fine ultrafilters on P_{κ}λ we write U_{0} ◁ U_{1} if U_{0} is in the ultrapower of V by U_{1}. This ordering is called the Mitchell ordering, and is well-founded. A cardinal κ is λ-strongly compact if there is a fine ultrafilter on P_{κ}λ.

The Magidor ordering on P_{κ}λ is defined by letting a ⊂ ∼ b if a ⊆ b and |a| < b∩κ.

We say that □_{κ} holds if there is a sequence ⟨c_{α}: α < κ^{+}, α limit⟩ such that for all limit ordinals α < κ^{+}:

(1) c_{α} club in α,

(2) if cf(α) < κ then o.t.(c_{α}) < κ,

(3) for all β in lim(c_{α}), c_{α} ∩ β = c_{β}. Similarly, we say that □_{κ,<δ} holds if there is a sequence ⟨C_{α}: α < κ^{+}, α limit⟩ such that for all limit ordinals α < κ^{+},

(1) C_{α} is a non-empty collection of less than δ many club subsets of α,

(2) if cf(α) < κ then every c in C_{α} has order type less than κ,

(3) if c is in C_{α} then for all β in lim(c), c ∩ β is in C_{β}.

If a is a set of ordinals then o.t.(a) is the order type of a, and if o.t.(a) is a limit ordinal then cf(a) is cf(o.t.(a)). Note that cf(a) = cf(sup a).

The expression θ ≫ κ means that θ is larger than 2^{2^{|H(κ)|}}. Suppose that N is in P_{κ}H(θ) and N is an elementary substructure of ⟨H(θ), ∈⟩ for some regular θ ≫ κ. If x is a set in N with size less than κ, then x ⊆ N.

We denote the class of singular limit ordinals by Sing and the class of regular cardinals by Reg.

If F: λ^{<ω} → λ is a partial function and A ⊆ λ, we say that F is Jonsson for A if A is closed under F, and whenever B ⊊ A is closed under F, it follows that |B| < |A|.

A standard presentation of the theory of forcing uses partially ordered sets, that is, orderings which are reflexive, antisymmetric, and transitive. It is well known however that this theory does not actually require antisymmetry. For example, forcing iterations as presented in [2] are not antisymmetric (i.e. it is possible that q ≤ p and p ≤ q, but p ≠ q).

Due to the author’s treatment of iterated forcing, we use this more general theory of forcing. So we define a forcing poset to be an ordering ⟨ℙ, ≤⟩ which is reflexive and transitive.

A forcing poset ⟨ℙ, ≤⟩ is separative if whenever q ≰ p, there is r ≤ q such that r and p are incompatible; equivalently, q ⊩ p ∈ Ġ iff q ≤ p.

We say that forcing posets ℙ and ℚ are equivalent if they have the same generic extensions. This is only an intuitive definition, not a formal one, since generic filters do not exist in the universe. We prove that two forcing posets are equivalent in this informal sense by showing how to construct a generic filter for one, given a generic filter for the other. For example, if D is a dense subset of ℙ, then D and ℙ are equivalent.

For any forcing poset ℙ there exists a separative poset ℙ^{∗} which is equivalent to ℙ. Define ℙ^{∗} as follows. The underlying set of ℙ^{∗} consists of equivalent classes of ℙ under the following equivalence relation: p is equivalent to q if for all r, r is compatible with p iff r is compatible with q. Define [q] ≤ [p] if for all r ≤ q, r is compatible with p. Then ℙ^{∗} is a separative forcing poset. Moreover, the surjective mapping i: ℙ → ℙ^{∗} defined by i(p) = [p] satisfies:

(1) q ≤ p implies i(q) ≤ i(p),

(2) If p and q are incompatible in ℙ, then i(p) and i(q) are incompatible in ℚ. The equivalence of ℙ and ℙ^{∗} will follow from the next lemma.

Lemma 1.1. Suppose that ℙ and ℚ are forcing posets, D ⊆ ℙ and E ⊆ ℚ are dense sets, and i: D → E is a surjective mapping satisfying:

(1) q ≤ p implies i(q) ≤ i(p),

(2) If p and q are incompatible in ℙ, then i(p) and i(q) are incompatible in ℚ. Then ℙ and ℚ are equivalent.

If ℙ is a separative forcing poset, then there exists a unique complete Boolean algebra r.o.(ℙ) and a mapping j: ℙ → r.o.(ℙ) satisfying:

(1) q < p iff j(q) < j(p),

(2) j“ℙ is a dense subset of r.o.(ℙ). The algebra r.o.(ℙ) is defined by taking regular cuts of ℙ ordered by inclusion.

If ℙ is a forcing poset, define r.o.(ℙ) to be r.o.(ℙ^{∗}), where ℙ^{∗} is the separative poset equivalent to ℙ as defined above. Define B(ℙ) to be the forcing poset consisting of the Boolean algebra r.o.(ℙ) minus the 0 element. Composing the maps ℙ → ℙ^{∗} and ℙ^{∗} → r.o.(ℙ^{∗}) described above, there exists a mapping k: ℙ → B(ℙ) satisfying:

(1) q ≤ p implies k(q) ≤ k(p),

(2) If p and q are incompatible in ℙ, then k(p) and k(q) are incompatible in B(ℙ),

(3) k“ℙ is a dense subset of B(ℙ). So by Lemma 1.1, ℙ and B(ℙ) are equivalent forcing posets.

When we discuss B(ℙ), we consider it as a forcing poset in the general sense, and avoid using Boolean valued models.

The following genericity criterion for subsets of B(ℙ) is straightforward: G is a generic filter for B(ℙ) over V iff G is an ultrafilter on r.o.(ℙ) which contains 1, such that whenever A ⊆ G is in V, ⋀ A is in G.

Now we discuss projection mappings. A mapping π: ℚ → ℙ is a projection mapping if it satisfies:

(1) q ≤ p implies π(q) ≤ π(p),

(2) π(1) = 1,

(3) if p ≤ π(q), then there is r ≤ q such that π(r) ≤ p.

Conditions (2) and (3) imply that π“ℚ is dense in ℙ. If π: ℚ → ℙ is a projection mapping and G is a generic filter for ℚ then π“G generates a generic filter for ℙ.

Suppose that π: ℚ → ℙ is a projection mapping and G is generic for ℙ. In V [G] define a forcing poset ℚ/ℙ as follows. The underlying set of this poset is {q ∈ ℚ: π(q) ∈ G}. We let q ≤ p in ℚ/ℙ iff q ≤ p in ℚ. Now in V let ℚ/ℙ be a ℙ-name for the forcing poset just described.

Define k: ℚ → ℙ ∗ (ℚ/ℙ) by letting k(q) = π(q) ∗ q̌. Note that k(q) really is in ℙ ∗ (ℚ/ℙ), since π(q) forces that π(q̌) is in Ġ. Then the mapping k satisfies:

(1) q ≤ p implies k(q) ≤ k(p),

(2) If p and q are incompatible in ℚ, then k(p) and k(q) are incompatible in ℙ ∗ (ℚ/ℙ),

(3) k“ℚ is dense in ℙ ∗ (ℚ/ℙ). So ℚ and ℙ ∗ (ℚ/ℙ) are equivalent.

Let ℙ and ℚ be forcing posets, and suppose that there is a ℚ-name Ġ_{ℙ} satisfying:

(1) ℚ forces that Ġ_{ℙ} is a generic filter for ℙ over V,

(2) for any p in ℙ, there is q in ℚ such that q forces that p is in Ġ_{ℙ}. Then since ℙ and B(ℙ) are equivalent, there exists a ℚ-name Ġ such that ℚ forces that Ġ is generic over V for B(ℙ), and for any a in B(ℙ) there is q in ℚ which forces that a is in Ġ.

Define π: ℚ → r.o.(ℙ) by

π(q) = ⋀{a ∈ B(ℙ): q ⊩ a ∈ Ġ}.

The set {a ∈ B(ℙ): q ⊩ a ∈ Ġ} is in the ground model, and q forces that it is contained in Ġ. By the criterion for genericity of filters on B(ℙ), q forces that π(q) is in Ġ, and hence is non-zero. Therefore π is a map ℚ → B(ℙ). It is also a projection mapping.

Suppose that κ is regular and ℙ is a forcing poset. We say that ℙ is < κdistributive if whenever {D_{i}: i < β} is a family of dense open subsets of ℙ and β < κ, then ⋂ D_{i} is dense open. Equivalently, ℙ is < κ-distributive if forcing with ℙ does not add any new sequences of ordinals with order type less than κ.

Suppose that ℙ is a forcing poset and λ is a cardinal. A canonical name for a subset of λ is a ℙ-name of the form

{⟨p, α̌⟩: p ∈ A_{α}, α < λ}

where A_{α} is an antichain for each α < λ. If p forces that Ȧ is a subset of λ, then there is a canonical name Ḃ for a subset of λ such that p forces that Ȧ = Ḃ. If Ẋ is a ℙ-name and ℙ forces that ḟ: λ → Ẋ is a bijection, then a canonical name for a subset of Ẋ is a name of the form

{⟨p, ḟ(α)⟩: p ∈ A_{α}, α < λ}

where each A_{α} is an antichain. If p forces that Ȧ is a subset of Ẋ, then there is a canonical name Ḃ for a subset of Ẋ such that p forces that Ȧ = Ḃ.

Suppose that ℙ is a forcing poset and p forces ϕ(Ẋ) for some ℙ-name Ẋ. Also assume that ℙ forces that there exists x such that ϕ(x) holds. Then there exists a ℙ-name Ẏ such that ℙ forces ϕ(Ẏ) and p forces Ẋ = Ẏ.

If H is a subset of a forcing poset ℙ, we say that H generates J if J is the set of q in ℙ such that there is p in H with p ≤ q.

Suppose that j: M → N is an elementary embedding between transitive models of set theory, ℙ is a forcing poset in M, G is generic for ℙ over M, and H is generic for j(ℙ) over N. Then j can be lifted to j: M[G] → N[H] iff j“G ⊆ H. In this case, j(G) = H. In particular, j can be lifted if there exists a condition s in H such that s ≤ j(p) for all p in G. We will use Silver’s notation and refer to such a condition s as a master condition.

2. Forcing Posets

In this section we describe some of the forcing posets we use in the paper.

Suppose that κ is strongly inaccessible and S is a stationary subset of κ. Define a forcing poset ℙ_{S} as follows. A condition in ℙ_{S} is a closed bounded subset of κ\S. We let q ≤ p if q end-extends p. Clearly ℙ_{S} has size κ.

The poset ℙ_{S} is < κ-distributive iff κ \ S is fat, i.e. for any club C and any ordinal β < κ, C \ S contains a closed subset with order type β. If G is a generic filter for ℙ_{S} then ⋃ G is a club subset of κ which is disjoint from S. So ℙ_{S} destroys the stationarity of S.

The set Sing ∩ κ is a fat set. So ℙ_{S} is < κ-distributive whenever S ⊆ Reg ∩ κ. For more on the poset ℙ_{S} see [1].

Gitik [6] generalized the forcing poset ℙ_{S} to destroy certain stationary subsets of P_{κ}κ^{+}, where κ is strongly inaccessible. Suppose that S is a subset of P_{κ}κ^{+}. Define ℙ_{S} as follows. A condition in ℙ_{S} is a set x ⊆ P_{κ}κ^{+} disjoint from S with size less than κ, and whenever ⟨a_{i}: i < β⟩ is ⊆-increasing in x, then ⋃ a_{i} is in x. Let y ≤ x if y end-extends x in the following sense: x ⊆ y, and whenever a is in y \ x and b is in x, a is not a subset of b.

The poset ℙ_{S} is always κ^{+}-c.c. The conditions under which ℙ_{S} is < κ-distributive are somewhat complex. We will consider this situation in detail in Section 4.

Suppose that κ is strongly inaccessible in V. In what follows we will need a generic extension V ⊆ W satisfying the following properties:

(1) κ is strongly inaccessible in W,

(2) V and W have the same limit cardinals,

(3) κ^{+V} = κ^{+W},

(4) there exists a club of former regulars in κ, that is, a club C ⊆ κ in W such that every α in C is a regular cardinal in V. There are a variety of ways to construct such a generic extension. We will use Radin forcing. The following lemma contains all the information about Radin forcing which we will need.

Lemma 2.1. Suppose that κ is λ-supercompact for some λ ≥ 2^{κ}. Then there exists a forcing poset ℝ, called Radin forcing, which adds a generic club set C to κ consisting of former regulars, and satisfies the following properties:

(1) ℝ is κ^{+}-c.c.,

(2) ℝ preserves all cardinals and the function α ↦ 2^{α},

(3) ℝ preserves the λ-supercompactness of κ.

See [5] for an exposition of Radin forcing.

A triple ⟨ℚ, ≤, ≤^{∗}⟩ is called a Prikry type forcing poset if ⟨ℚ, ≤⟩ and ⟨ℚ, ≤^{∗}⟩ are forcing posets, q ≤^{∗} p implies q ≤ p, and ℚ satisfies the Prikry property: for any ϕ in the forcing language for ⟨ℚ, ≤⟩ and for any p in ℚ, there exists q ≤^{∗} p such that q decides ϕ. If α is a cardinal we say that ℚ is α-weakly closed if ⟨ℚ, ≤^{∗}⟩ is α-closed. We say that ℚ satisfies the direct extension property if whenever q, r ≤^{∗} p, there is s ≤^{∗} q, r. It should be made clear that when we say that we force with a Prikry type forcing poset ℚ, we always mean that we force with ⟨ℚ, ≤⟩.

Suppose that U is a κ-complete fine ultrafilter on P_{κ}λ. We define a forcing poset ℙℝ(U) as follows. If U is a normal ultrafilter, ℙℝ(U) will denote the supercompact Prikry forcing. A condition in ℙℝ(U) is a pair ⟨a⃗, A⟩, where A is in U and a⃗ is a finite Magidor increasing sequence from P_{κ}λ. If p = ⟨a⃗, A⟩ and q = ⟨b⃗, B⟩ are conditions, let q ≤ p if B ⊆ A, a⃗ is an initial segment of b⃗, and any element of b⃗ which is not in a⃗ is in A. If q ≤ p and a⃗ = b⃗, we let q ≤^{∗} p.

Suppose that U is a non-normal ultrafilter. Then we let ℙℝ(U) denote the tree Prikry forcing. A condition in ℙℝ(U) is a pair ⟨t, T ⟩ satisfying:

(1) T is a collection of finite Magidor-increasing sequences from P_{κ}λ, ordered by u ⊴ v if u is an initial segment of v.

(2) ⟨T, ⊴⟩ is a tree ordering, that is, if u ⊴ v and v is in T then u is in T.

(3) t is the trunk of T, that is, t is in T and whenever u is in T, either u ⊴ t or t ⊴ u.

(4) For every u in T with t ⊴ u, the set {a ∈ P_{κ}λ: u ⟨a⟩ ∈ T } is in U. The ordering on ℙℝ(U) is ⟨s, S⟩ ≤ ⟨t, T ⟩ if S ⊆ T. We let ⟨s, S⟩ ≤^{∗} ⟨t, T ⟩ if ⟨s, S⟩ ≤ ⟨t, T ⟩ and s = t.

In either case, ℙℝ(U) satisfies the following properties. The poset ℙℝ(U) is a Prikry type forcing poset. If {⟨s, S_{i}⟩: i < β} is a family of fewer than κ many conditions with the same first coordinate, then ⟨s, ⋂ S_{i}⟩ is a condition which directly extends each ⟨s, S_{i}⟩. So ℙℝ(U) is κ-weakly closed and satisfies the direct extension property. Since there are only λ^{<κ} many possible first coordinates for a condition, ℙℝ(U) is (λ^{<κ})^{+}-c.c.

Suppose that G is a generic filter for ℙℝ(U). Then the set ⋃{a: ∃A ⟨a, A⟩ ∈ G} is a Magidor-increasing sequence ⟨a_{n}: n < ω⟩ cofinal in P_{κ}λ, and for any A in U there is n < ω such that a_{m} is in A for all m ≥ n. For each κ ≤ β ≤ λ with cf(β) ≥ κ, β = ⋃{a_{n} ∩ β: n < ω}, and therefore β has cofinality ω in the generic extension.

If the ultrafilter U is understood, we sometimes write ℙℝ(κ, λ) or ℙℝ for ℙℝ(U). For more information about ℙℝ(U) see [5].

Suppose that U is a κ-complete fine ultrafilter on P_{κ}λ. Assume that W is a generic extension of V and ⟨a_{n}: n < ω⟩ is an increasing sequence in W which is cofinal in P_{κ}λ. Assume that the sequence satisfies the Prikry genericity criterion: for every function h: (P_{κ}λ)^{<ω} → U in the ground model there is n such that for all m ≥ n, a_{m} is in h(⟨a_{0}, …, a_{m−1}⟩). Then the set {a_{n}: n < ω} generates a generic filter for ℙℝ(U) over V. In case U is non-normal, this generic filter is the set

{⟨t, T ⟩ ∈ ℙℝ(U): ⟨a_{0}, …, a_{n}⟩ ∈ T for all n < ω}

If U is normal, the generic filter is

{⟨a⃗, A⟩ ∈ ℙℝ(U): a⃗ is an initial segment of ⟨a_{n}: n < ω⟩, ∀n ≥ |a⃗| a_{n} ∈ A}.

If U is normal, a weaker genericity criterion is sufficient: for all A in U, there is n such that a_{m} is in A for all m ≥ n. For a proof see [8].

The following two results summarize a technique due to Gitik which is implicit in [6].

Lemma 2.2. Suppose that U is a κ-complete fine ultrafilter on P_{κ}λ and π: ℙℝ(U) → ℙ is a projection mapping. Then there exists a Prikry type forcing poset ⟨ℚ, ≤, ≤^{∗}⟩ satisfying:

(1) ⟨ℚ, ≤⟩ and ⟨ℙ, ≤⟩ are equivalent posets,

(2) ℚ is κ-weakly closed and satisfies the direct extension property.

Proof. Define ⟨ℚ, ≤, ≤^{∗}⟩ as follows. The underlying set of ℚ is exactly the underlying set of ℙℝ(U). Define ≤^{∗} on ℚ to be exactly ≤^{∗} on ℙℝ(U). It is obvious then that ⟨ℚ, ≤^{∗}⟩ is a forcing poset which is κ-closed, and whenever q, r ≤^{∗} p in ℚ, there is s ≤^{∗} q, r. Define ≤ on ℚ by letting q ≤ p iff π(q) ≤ π(p) holds in ℙ.

Define i: π“ℚ → ℚ by letting i(a) be some q in ℚ such that π(q) = a. Then i is an order preserving and incompatibility preserving surjective mapping between a dense subset of ℙ and a dense subset of ℚ. So ℚ and ℙ are equivalent posets.

We show that ℚ satisfies the Prikry property. First note that the identity mapping id: ℙℝ(U) → ℚ is a projection mapping. Therefore if H is generic for ℙℝ(U) over V, then H generates a generic filter J for ⟨ℚ, ≤⟩ over V.

Now let p be in ℚ and suppose that ϕ is a statement in the forcing language for ⟨ℚ, ≤⟩. Then p is also a condition in ℙℝ(U). Apply the Prikry property in ℙℝ(U) to find q ≤^{∗} p such that q decides over V in ℙℝ(U) the statement ϕ^{V [J]}. Without loss of generality suppose that q forces ϕ^{V [J]}. By the definition of ≤^{∗} in ℚ, we have q ≤^{∗} p in ℚ.

We claim that q forces ϕ. If not, there is r ≤ q in ℚ which forces the negation of ϕ. Since r ≤ id(q), apply the fact that id is a projection mapping to get s ≤ q in ℙℝ(U) such that s = id(s) ≤ r in ℚ. Now force with s in ℙℝ(U) to get a generic filter H and generic extensions V ⊆ V [J] ⊆ V [H].

Since s ≤ q in ℙℝ(U), ϕ^{V [J]} holds in V [H]. Therefore ϕ holds in V [J]. But also s is in J, and s ≤ r in ℚ and r forces the negation of ϕ. Therefore ϕ fails in V [J], and we have a contradicton.

□

Lemma 2.3. Suppose that U is a κ-complete fine ultrafilter on P_{κ}λ. Let ℙ be a < κ-distributive forcing poset such that |ℙ| ≤ λ, and assume that ℙ has no more than λ-many maximal antichains. Then there exists a projection mapping π: ℙℝ(U) → B(ℙ).

Therefore there exists a Prikry type forcing poset ⟨ℚ, ≤, ≤^{∗}⟩ which is κ-weakly closed, satisfies the direct extension property, and ⟨ℚ, ≤⟩ is equivalent to B(ℙ), and hence to ℙ.

Proof. It suffices to prove that there exists a ℙℝ(U)-name Ġ such that ℙℝ(U) forces that Ġ is a generic filter for ℙ over V, and for any p in ℙ, there is q in ℙℝ(U) which forces that p is in Ġ.

Let ⟨p_{i}: i < λ⟩ and ⟨A_{i}: i < λ⟩ enumerate ℙ and the maximal antichains of ℙ, possibly with repetitions.

Suppose that H is generic for ℙℝ(U) over V. Let ⟨a_{n}: n < ω⟩ be the set ⋃{s: ∃S ⟨s, S⟩ ∈ H}. Then each a_{n} is in V and ⋃{a_{n}: n < ω} = λ.

Define a decreasing sequence ⟨q_{n}: n < ω⟩ in ℙ inductively as follows. Let q_{0} be p_{sup a_{0}}. Now suppose that q_{n} is defined. Since a_{n} has size less than κ, the set D = ⋂{D_{i}: i ∈ a_{n}} is dense open in ℙ. So pick q_{n+1} in D below q_{n}. Clearly the set {q_{n}: n < ω} generates a generic filter for ℙ over V. Let Ġ be a name for G. The definition of q_{0} ensures that it is possible for any particular condition to be in Ġ. □

We will need a generalization of the last lemma.

Lemma 2.4. Suppose that U is a κ-complete fine ultrafilter on P_{κ}λ. Let ℙ be a < κ-distributive forcing poset with |ℙ| ≤ λ and such that that ℙ has no more than λ many antichains.

Assume that ℙ forces that U′ is a κ-complete fine ultrafilter on P_{κ}μ, where 2^{λ^{<μ}} ≤ λ and |ℙ ∗ ℙℝ(U′)| ≤ λ. Then there is a projection mapping

π: ℙℝ(U) → B(ℙ ∗ ℙℝ(U′)).

So there exists a Prikry type forcing poset ⟨ℚ, ≤, ≤^{∗}⟩ which is κ-weakly closed, satisfies the direct extension property, and ⟨ℚ, ≤⟩ is equivalent to ℙ ∗ ℙℝ(U′).

Proof. Enumerate ℙ ∗ ℙℝ(U′) as ⟨r_{i}: i < λ⟩, possibly with repetitions. Let H be generic for ℙℝ(U) over V and let ⟨a_{n}: n < ω⟩ be generic Prikry sequence given by H.

Fix q and ⟨ṫ, Ṫ ⟩ such that r_{sup a_{0}} = q∗⟨ṫ, Ṫ ⟩. As in Lemma 2.3, contruct a generic filter G_{0} for ℙ over V containing q. Let ⟨h_{i}: i < λ⟩ enumerate all the functions h: (P_{κ}μ)^{<ω} → U′ in V [G_{0}], possibly with repetitions. Now inductively define a Prikry sequence for ℙℝ(U′) as follows. Write t = ⟨b_{i}: i ≤ k⟩. Given b_{n} for n ≥ k, choose b_{n+1} which is a Magidor extension of b_{n} and is in the measure one set

⋂{h_{i}(⟨b_{0}, …, b_{n}⟩): i ∈ a_{n}} ∩ {a: ⟨b_{0}, …, b_{n}⟩ a ∈ T }.

By the Prikry genericity criterion, {b_{n}: n < ω} generates a generic filter G_{1} for ℙℝ(U′) over V [G_{0}] which contains ⟨t, T ⟩. □

3. The Stationarity of S(κ, κ^{+})

Now we begin our study by introducing the set S(κ, κ^{+}).

Lemma 3.1. There is a club set of a in P_{κ}κ^{+} such that o.t.(a) ≤ (a ∩ κ)^{+}.

Proof. Fix θ ≫ κ regular, and let N be an elementary substructure of ⟨H(θ), ∈⟩ in P_{κ}H(θ). Let a = N ∩ κ^{+}. We claim that o.t.(a) ≤ (a ∩ κ)^{+}.

We show that for all β in a, |a ∩ β| ≤ |a ∩ κ|. This is clear if β < κ. Suppose that β ≥ κ is in a. Fix a bijection f_{β}: κ → β in N. By elementarity, for all j in a ∩ β, f_{β}^{−1}(j) is in a ∩ κ. Therefore f_{β} ↾ a ∩ κ is a bijection of a ∩ κ onto a ∩ β, and so |a ∩ β| = |a ∩ κ|. Since each initial segment of a has size at most |a ∩ κ|, clearly o.t.(a) ≤ (a ∩ κ)^{+}. □

Now define S(κ, κ^{+}) as the set of a in P_{κ}κ^{+} such that o.t.(a) = (a ∩ κ)^{+}.

In [4] it was established that the stationarity of S(κ, κ^{+}) implies an inner model with a measurable. The exact consistency strength of the stationarity of S(κ, κ^{+}) is currently an open problem.

Lemma 3.2. For almost all a in P_{κ}κ^{+}, a contains all its limit points below sup a which have cofinality different from cf(a ∩ κ).

Proof. Fix θ ≫ κ regular and suppose that N is an elementary substructure of H(θ) in P_{κ}H(θ). Let a = N ∩ κ^{+}. Suppose for a contradiction that α is a limit ordinal in a below sup a, cf(α) ≠ cf(a ∩ κ), and α is not in a.

Let β be the least element of a greater than α. We claim that cf(β) = κ. Fix g: cf(β) → β in N increasing and cofinal in β. If cf(β) < κ, then cf(β) ⊆ β so g“cf(β) is a subset of a, which contradicts that a is bounded below β.

For all γ in a ∩ β there is i in a ∩ κ such that g(i) > γ. So g“(a ∩ κ) is increasing and unbounded below α. Therefore a ∩ κ and α have the same cofinality, which is a contradiction. □

Theorem 3.3. If □_{κ,<κ} holds then S(κ, κ^{+}) is non-stationary.

Proof. Let ⟨C_{α}: α < κ^{+}⟩ be a □_{κ,<κ}-sequence. Fix θ ≫ κ^{+} regular. Suppose for a contradiction that S(κ, κ^{+}) is stationary. Then there exists N in P_{κ}H(θ) which is an elementary substructure of the model

⟨H(θ), ∈, ⟨C_{α}: α < κ^{+}⟩⟩.

such that N ∩ κ^{+} is in S(κ, κ^{+}). Let a = N ∩ κ^{+}, β = sup a, and κ_{a} = N ∩ κ. Note that cf(β) = κ_{a}^{+}.

Let c be some member of C_{β}. Fix a regular cardinal δ < κ_{a} different from cf(κ_{a}). By Lemma 3.2, a is closed under suprema of subsets with order type δ. In particular, a is stationary in sup a, and therefore a ∩ lim(c) is unbounded in β. Now c has order type at least cf(β) = κ_{a}^{+}. So there is γ in a ∩ lim(c) such that o.t.(c ∩ γ) ≥ κ_{a}. But c ∩ γ is in C_{γ}. Since γ is in a, C_{γ} is a member of N. But |C_{γ}| < κ, so C_{γ} is a subset of N. Therefore c ∩ γ is in N, and hence o.t.(c ∩ γ) is in N ∩ κ = κ_{a}. This contradicts that o.t.(c ∩ γ) ≥ κ_{a}. □

A different proof of this theorem using ill-founded generic ultrapowers appears in [3].

The set S(κ, κ^{+}) has some importance in inner model theory. A regular cardinal κ is subcompact if for every set X ⊆ H(κ^{+}) there is a cardinal δ < κ, a set a ⊆ H(δ^{+}), and an elementary embedding

i: ⟨H(δ^{+}), ∈, a⟩ → ⟨H(κ^{+}), ∈, X⟩

such that crit(i) = δ and i(δ) = κ. Jensen defined subcompactness and proved that if κ is subcompact then □_{κ} fails. The following result was pointed out to the author by Ernest Schimmerling.

Theorem 3.4. Suppose that V = L[E⃗] where E⃗ is coherent sequence of extenders. Then the following are equivalent for an inaccessible cardinal κ.

(1) κ is subcompact,

(2) S(κ, κ^{+}) is stationary,

(3) □_{κ,<κ} fails.

Proof. The proof of (3) implies (1) appears in [12], and (2) implies (3) was proved in Theorem 3.3. For (1) implies (2), let C be a club subset of P_{κ}κ^{+}. Then C ⊆ H(κ^{+}), so there is δ < κ, c ⊆ H(δ^{+}), and an elementary embedding

i: ⟨H(δ^{+}), ∈, c⟩ → ⟨H(κ^{+}), ∈, C⟩

with critical point δ such that i(δ) = κ. Applying elementarity it is straightforward to check that i“c is a directed subset of C with size less than κ and with union i“δ^{+}. Therefore i“δ^{+} is in C ∩ S(κ, κ^{+}). □

In general the stationarity of S(κ, κ^{+}) does not imply that κ has any large cardinal property. For example, in the model which we construct in Section 6, S(κ, κ^{+}) is stationary, GCH holds, and κ is a non-Mahlo cardinal.

Note: Most of the results presented in this section are already known. The proof of Theorem 3.3 is due to the author. This proof allows for a different approach to destroying stationary subsets of P_{κ}κ^{+} than the one which we use in this paper. For more on this topic see [9] and [10].

4. Destroying Stationary Sets

In this section we present a forcing poset for destroying the stationarity of certain subsets of P_{κ}κ^{+}. Assume for the remainder of this section that V ⊆ W are transitive models of set theory with the same ordinals satisfying the properties:

(1) V and W have the same limit cardinals,

(2) κ is strongly inaccessible in W,

(3) κ^{+V} = κ^{+W},

(4) in W there is a club set C ⊆ κ such that every α in C is regular in V.

Now work in W. Consider the following two properties of a set S ⊆ P_{κ}κ^{+}:

Property 1: There exists a set A ⊆ κ^{+} such that

S = {a ∈ S(κ, κ^{+}): sup a ∈ A}.

Property 2: There exists a partial function F: κ^{+<ω} → κ^{+} in V such that in W,

S = {a ∈ S(κ, κ^{+}): F is not Jonsson for a}.

We will prove that if S is a set which satisfies either one of these properties, there exists a κ^{+}-c.c., < κ-distributive forcing poset which destroys the stationarity of S. (Note that if a forcing poset is κ^{+}-c.c. and < κ-distributive, the sets P_{κ}κ^{+} and S(κ, κ^{+}) are the same in the ground model and in the generic extension). If S has Property 1, we will denote this forcing poset by ℙ_{S}. If S has Property 2, we will denote this poset by ℙ_{F}.

The two conditions on S work together quite nicely. Namely, after forcing with ℙ_{F}, every stationary subset of S(κ, κ^{+}) has Property 1. For a proof see Proposition 4.9 below.

Before we can define ℙ_{S} and ℙ_{F}, we need the following lemma.

Lemma 4.1. There exists a function H: κ^{+<ω} → κ^{+} in V such that the following holds in W. For any a in P_{κ}κ^{+} which is closed under H:

(1) a ∩ κ is a limit cardinal,

(2) If |a| > a ∩ κ then o.t.(a) = (a ∩ κ)^{+}.

(3) If x is a bounded subset of a and cf(x) ≠ cf(a ∩ κ), then sup x ∈ a.

(4) If a is in S(κ, κ^{+}) then cf^{V} (sup a) is not a limit point of C.

Proof. In V fix θ ≫ κ^{+} regular. Let H^{∗}: H(θ)^{<ω} → H(θ) be a Skolem function for the structure ⟨H(θ), ∈⟩. Let H: κ^{+<ω} → κ^{+} be a function in V such that whenever b ⊆ κ^{+} in V is closed under H, cl_{H^{∗}}(b) ∩ κ^{+} = b.

Now work in W. Suppose that a is in P_{κ}κ^{+} and is closed under H. Since a is closed under V -cardinal successors and V and W have the same limit cardinals, a ∩ κ is a limit cardinal. It is easy to check (2) using the proof of Lemma 3.1.

For (3), let x be a bounded subset of a with order type a regular cardinal different from cf(a ∩ κ). Let β = sup x. Suppose for a contradiction that β is not in a. Let α be the least element of a greater than β. Let N be the H^{∗}-closure of (a ∩ κ) ∪ {α} in V. Clearly N ∩ κ = a ∩ κ. In N fix f_{α}: cf^{V} (α) → α increasing and unbounded in α. If cf^{V} (α) < κ then cf^{V} (α) ⊆ N, so f_{α}“cf^{V} (α) is an unbounded subset of α contained in a, which is impossible. So cf^{V} (α) = κ. But then f_{β}“a ∩ κ is increasing and unbounded in a ∩ α, so cf(a ∩ κ) = cf(β), which is a contradiction.

For (4), let a be in S(κ, κ^{+}) and suppose for a contradiction that cf^{V} (sup a) is a limit point of C. Let κ_{a} = a ∩ κ, β = sup a, and δ = cf^{V} (β). By (2) and (3), o.t.(a) = κ_{a}^{+} and a is stationary in β. Since κ_{a}^{+} = cf^{W} (β), κ_{a}^{+} ≤ δ. In V let π: δ → β be an increasing and continuous map with range unbounded in β. Since δ is a limit point of C, C ∩ δ is club in δ, and therefore π“(C ∩ δ) is club in β. Since a is stationary in β, there is γ in C ∩ δ with γ > κ_{a} such that π(γ) is in a. Note that cf^{V} (π(γ)) = γ, since γ is regular in V and sup(π“γ) = π(γ). But π(γ) is in a, and therefore cf^{V} (π(γ)) = γ is in a ∩ κ. This is a contradiction since γ > a ∩ κ. □

For the remainder of the section fix a function H: κ^{+<ω} → κ^{+} as in Lemma 4.1. We will now define our forcing poset ℙ.

If x and y are subsets of P_{κ}κ^{+}, we say x end-extends y if y ⊆ x and for every a in x \ y and b in y, a is not a subset of b.

Now suppose that S ⊆ P_{κ}κ^{+} is a set which satisfies either Property 1 or Property 2 above. If S has Property I then let F: κ^{+} → κ^{+} be the identity map. In this case we will write ℙ = ℙ_{S}. If S has Property II we will write ℙ = ℙ_{F}. Let E be the club set of a in P_{κ}κ^{+} such that a is closed under H and F.

Definition 4.2. A condition in ℙ is a set x which satisfies the following properties: (1) x ⊆ E \ S,

(2) x has size less than κ,

(3) x is closed under unions of ⊆-increasing sequences,

(4) if a is in x and u is a set in E \ S such that a ∩ κ ⊆ u ⊆ a, then u is in x,

(5) x contains an element max(x) such that b ⊆ max(x) for every b in x. The ordering on ℙ is x ≤ y if x end-extends y. □

The only significant differences between this definition and the definition of Gitik’s poset ℙ_{S} given in Section 2 is the addition of clause (4) and the fact that we require the elements of our conditions to be inside the club E. These changes are technical conditions which enable us to prove the < κ-distributivity of ℙ without assuming Gitik’s requirement that S is a subset of V.

We make some observations. In (4) the set u satisfies that u ∩ κ = a ∩ κ. If a and b are in E then a ∩ b is in E. If a is in E then o.t.(a) ≤ (a ∩ κ)^{+}. If x and y are in ℙ then x ≤ y iff y ⊆ x and for all a in x \ y, a ⊈ max(y). We can strengthen condition (4) as follows.

Lemma 4.3. Suppose that x is a condition in ℙ, a is in x, and u is a set in E such that a ∩ κ ⊆ u ⊆ a. Then u is not in S, therefore u is in x.

Proof. Recall that x ∩ S is empty so a is not in S. Suppose for a contradiction that u is in S. Then u is in S(κ, κ^{+}) and is closed under H. We know that o.t.(u) = (u∩κ)^{+} = (a∩κ)^{+}. Since o.t.(a) ≤ (a∩κ)^{+} and u ⊆ a, o.t.(a) = (a∩κ)^{+}. Clearly then sup a = sup u, for otherwise o.t.(a) would be larger than o.t.(u). If ℙ = ℙ_{S}, then sup u is in A so a is in S, and we have a contradiction. If ℙ = ℙ_{F}, then u is a subset of a closed under F and |u| = |a|. But a is not in S, so F is Jonsson for a, therefore u = a. But this is impossible since u is in S and a is not in S. □

Clearly ℙ has size κ^{+}. We will need the next several lemmas in order to prove that ℙ is < κ-distributive. The following lemma will also be useful for constructing master conditions in later sections.

Lemma 4.4. Suppose that X is a directed subset of ℙ with size less than κ. Let z be the closure of ⋃ X under unions of increasing sequences. Then the following statements hold.

(1) z is a subset of E with size less than κ which is closed under unions of increasing sequences,

(2) every a in z is a subset of ⋃{max(x): x ∈ X},

(3) if a is in z and u is a set in E such that a ∩ κ ⊆ u ⊆ a then u is in z,

(4) if a is in z \ ⋃ X then for all x in X, a ⊈ max(x).

(5) z end-extends x for all x in X.

Proof. Let ζ be the least ordinal such that z = ⋃{z_{i}: i < ζ} where z_{0} = ⋃ X, z_{δ} = ⋃{z_{i}: i < δ} for δ limit, and for each i, z_{i+1} is the collection of sets a such that there is an increasing sequence ⟨a_{i}: i < ξ⟩ of elements of z_{i} whose union is a. Statements (1) and (2) are straightforward.

(3) We prove by induction on i < ζ that if a is in z_{i} and u is a set in E such that a∩κ ⊆ u ⊆ a then u is in z_{i}. This is true for z_{0} since X consists of conditions. The limit step is clear. Suppose that the claim is true for z_{i} for some i < ζ. Let a be a set in z_{i+1} and u a set in E such that a ∩ κ ⊆ u ⊆ a. Fix an increasing sequence ⟨a_{j}: j < ξ⟩ from z_{i} whose union is a. For each j < ξ the set u ∩ a_{j} is in E and a_{j} ∩ κ ⊆ a_{j} ∩ u ⊆ a_{j}, so by induction a_{j} ∩ u is in z_{i}. Therefore u = ⋃{a_{j} ∩ u: j < ξ} is in z_{i+1}.

(4) We prove by induction on 0 < i < ζ that for all x in X, if a is in z_{i} \⋃ X then a ⊈ max(x). Suppose that a is in z_{1} \ ⋃ X. Write a = ⋃{a_{j}: j < ξ} where each a_{j} is in ⋃ X. Since a is not in ⋃ X, not all a_{j}’s are in x since x is closed. So fix l with a_{l} not in x. Then there is some y in X with a_{l} in y. Since X is directed, fix x^{∗} in X which end-extends both x and y. Then a_{l} is in x^{∗} \ x, so by the definition of the ordering, a_{l} ⊈ max(x). Therefore a ⊈ max(x). Now suppose that 0 < i < ζ and the claim holds for z_{i}. Let a be in z_{i+1} \ ⋃ X, and write a = ⋃{a_{j}: j < ξ} where each a_{j} is in z_{i}. If all the a_{j}’s are in ⋃ X then a is in z_{1} and we are done by what we just proved. Otherwise some a_{l} is in z_{i} \ ⋃ X so by induction a_{l} ⊈ max(x), so a ⊈ max(x).

(5) Fix x in X. Then x ⊆ ⋃ X ⊆ z. Suppose that a is in z \ x. If a is in ⋃ X, then since X is directed there is y ≤ x in X with a in y. By the definition of the ordering, a ⊈ max(x). Otherwise a is in z \ ⋃ X, so by (4) a ⊈ max(x). □

The following lemma is the core technical fact for the proof of < κ-distributivity. We state it abstractly since we will need to use it later in the paper.

Lemma 4.5. Suppose that γ < κ is a limit ordinal, f: κ → κ^{+} is in V, and ⟨N_{i}: i ≤ γ⟩ is an increasing and continuous sequence of sets with size less than κ which are closed under H and F such that for all i ≤ γ:

(1) N_{i} ∩ κ = δ_{i} is in lim(C),

(2) f“δ_{i} = N_{i} ∩ κ^{+}. Suppose that ⟨x_{i}: i < γ⟩ is a decreasing sequence in ℙ such that for each i < γ:

(3) x_{i} is in N_{i+1},

(4) max(x_{i}) = N_{i} ∩ κ^{+}. Let x_{γ} be the closure of ⋃{x_{i}: i < γ} under unions of increasing sequences. Then the following statements are true:

(A) x_{γ} is in ℙ,

(B) x_{γ} ≤ x_{i} for all i,

(C) max(x_{γ}) = N_{γ} ∩ κ^{+}.

Proof. Apply Lemma 4.4 with X = {x_{i}: i < γ} and z = x_{γ}. Then the only way that x_{γ} would not be a condition is if x∩S ≠ ∅. We also know that x_{γ} end-extends x_{i} for all i. Since N_{γ} ∩ κ^{+} = ⋃{N_{i} ∩ κ^{+}: i < γ}, N_{γ} ∩ κ^{+} is in x_{γ} and by Lemma 4.4 this set is the maximum element of x_{γ}. The only thing left to check is that x_{γ} is disjoint from S.

Suppose for a contradiction that x_{γ} ∩S contains an element a. Let κ_{a} = a∩κ and β = sup a. Since a is in E ∩ S(κ, κ^{+}), by Lemma 4.1 we know that o.t.(a) = κ_{a}^{+}, a is closed under suprema of bounded subsets with cofinality different from cf^{W} (κ_{a}), and cf^{V} (β) is not a limit point of C. Since N_{γ} ∩ κ^{+} = f“δ_{γ}, the set N_{γ} ∩ β is in V and is unbounded in β since it contains a. Therefore since |N_{γ} ∩ β|^{V} ≤ δ_{γ}, cf^{V} (β) ≤ δ_{γ}. But δ_{γ} is a limit point of C, therefore cf^{V} (β) < δ_{γ}. Since cf^{W} (β) = κ_{a}^{+W}, κ_{a}^{+W} ≤ cf^{V} (β) < δ_{γ}.

The set a is difficult to work with because it might not be in V. So what we do is find a set u such that a ∩ κ ⊆ u ⊆ a and u is covered by a small set d from V. Define an increasing sequence ⟨d_{i}: i < ω_{2}^{W} ⟩ in V as follows. Let d_{0} be any club subset of β in V with order type cf^{V} (β) and κ_{a} ⊆ d_{0}. Take unions at limits. Suppose that d_{i} is defined for a fixed i < ω_{2}^{W}. Define

d_{i}^{∗} = d_{i} ∪ H“d_{i}^{<ω} ∪ F “d_{i}^{<ω}

and let

d_{i+1} = d_{i}^{∗} ∪ lim(d_{i}^{∗}).

(This is where we use the fact that H and F are in V). Let d = ⋃{d_{i}: i < ω_{2}^{W} }. Then d is closed under H and F and |d| = cf^{V} (β) < δ_{γ}. We claim that in W the set d is closed under suprema of any bounded subset x with order type ω or ω_{1}. For if x is such a set, then there is i < ω_{2}^{W} such that x ⊆ d_{i}. Then sup x is a limit point of d_{i}, so sup x is in d_{i+1} ⊆ d.

Now let u = a∩d. Let μ be one of ω or ω_{1} which is different from cf(κ_{a})^{W}. Then both a and d are closed under suprema of subsets with order type μ. It follows that u = a ∩ d is also closed under such suprema, and therefore is an unbounded subset of a. Since u is unbounded in a, it must have order type κ_{a}^{+W}. Also κ_{a} = u ∩ κ. So u is in S(κ, κ^{+}). The set u is in E since it is closed under H and F. But a ∩ κ ⊆ u ⊆ a, so by (3) of Lemma 4.4, u is in x_{γ}.

The rest of the proof splits into two cases. Suppose first that ℙ = ℙ_{S} where S = {a ∈ S(κ, κ^{+}): sup a ∈ A} for some A ⊆ κ^{+}. Since a is in S, sup a is in A, therefore sup u is in A so u is in S. Now we get a contradiction as follows. Since ⋃ X ∩ S is empty, u is in x_{γ} \ ⋃ X. By (4) of Lemma 4.4, for all l < γ, u ⊈ N_{l} ∩ κ^{+}. Now the set N_{γ} ∩ κ^{+} = f“δ_{γ} is in V, therefore d ∩ N_{γ} is in V. Also |d ∩ N_{γ}|^{V} ≤ |d|^{V} < δ_{γ}. Since δ_{γ} is regular in V and d ∩ N_{γ} ⊆ N_{γ} ∩ κ^{+} = ⋃{f“i: i < δ_{γ}}, there is i < δ_{γ} such that d ∩ N_{γ} ⊆ f“i. Fix l < γ with i < δ_{l}. Then u = a ∩ d ⊆ d ∩ N_{γ} ⊆ f“i ⊆ f“δ_{l} = N_{l} ∩ κ^{+}, so u ⊆ N_{l} ∩ κ^{+} and we have a contradiction.

Suppose now that ℙ = ℙ_{F}. If u is in S then u is in x_{γ} \ ⋃ X, so we can get a contradiction as in the previous paragraph. Suppose that u is not in S. Since a is in S, a ≠ u, therefore u ⊊ a. Fix an ordinal δ^{∗} in a \ u. Let v be the closure of u ∪ {δ^{∗}} under H and F. Since a is closed under H and F, v ⊆ a. Therefore v ∩ κ = κ_{a}. So v is in P_{κ}κ^{+}. Since u ⊆ v, v is in S(κ, κ^{+}), and also v is in E. Since a ∩ κ ⊆ v ⊆ a, by (3) of Lemma 4.4, v is in x_{γ}. Now u is a proper subset of v with the same size as v and u is closed under F. Therefore F is not Jonsson for v, so v is in S. Since ⋃ X ∩ S is empty, v is in x_{γ} \ ⋃ X. By (4) of Lemma 4.4, for all l < γ, v ⊈ N_{l} ∩ κ^{+}.

Let Y be the closure of (N_{γ} ∩ d) ∪ {δ^{∗}} under H and F. Since N_{γ} ∩ d is in V, Y is in V, and |Y |^{V} = |N_{γ} ∩ d|^{V} ≤ |d|^{V} < δ_{γ}. Since δ^{∗} is in a ⊆ N_{γ}, Y ⊆ N_{γ} ∩κ^{+} = ⋃{f“i: i < δ_{γ}}. As above we can find l < γ such that Y ⊆ N_{l}∩κ^{+}. Since u ∪ {δ^{∗}} ⊆ (N_{γ} ∩ d) ∪ {δ^{∗}}, we have v ⊆ Y ⊆ N_{l} ∩ κ^{+}, which is a contradiction. □

Proposition 4.6. The forcing poset ℙ is < κ-distributive.

Proof. Suppose that {D_{i}: i < δ} is a family of dense open subsets of ℙ with δ < κ. Let x be a condition in ℙ. Fix θ ≫ κ^{+} regular, and let <_{θ} be a well-ordering of H(θ). Since κ is strongly inaccessible, there is M ≺ ⟨H(θ), ∈⟩ such that |M| = κ, κ ⊆ M, M ∩ κ^{+} is in κ^{+}, and ^{<κ}M ⊆ M. Choose M to contain the sets ℙ, x, C, H, F, E, and ⟨D_{i}: i < δ⟩. Let β = M ∩ κ^{+}. Fix a bijection f: κ → β in V. Let ⟨z_{i}: i < κ⟩ enumerate all the elements of M.

Define a chain ⟨M_{i}: i < κ⟩ of elementary substructures of

M = ⟨M, ∈, ℙ, x, C, H, F, E, ⟨D_{i}: i < δ⟩, f, <_{θ}⟩

as follows. Choose M_{0} in P_{κ}M with M_{0} ≺ M. Take unions at limits. Given M_{i}, let M_{i+1} be some elementary substructure of M in P_{κ}M which contains the sequence ⟨M_{j}: j ≤ i⟩ and also the set z_{i}. Note that M_{i} ∩ κ is in lim(C) and M_{i} ∩ κ^{+} is in E.

Since z_{i} is in M_{i+1} for each i, the collection {M_{i}: i < κ} is a club subset of P_{κ}M. Also there is a club set of i < κ such that M_{i} ∩ κ = i. Since f“κ = β, the collection {f“α: α < κ} is club in P_{κ}β. So it is possible to choose a subsequence ⟨N_{i}: i < κ⟩ of ⟨M_{i}: i < κ⟩ such that:

(1) N_{i} = M_{δ_{i}} for some δ_{i} in lim(C),

(2) N_{i} ∩ κ = δ_{i},

(3) ⟨N_{j}: j ≤ i⟩ is in N_{i+1}.

Since N_{i} is closed under f and f^{−1},

(4) f“δ_{i} = N_{i} ∩ κ^{+}. Since f is in V, N_{i} ∩ κ^{+} is in V. Also |N_{i} ∩ κ^{+}| = δ_{i} = N_{i} ∩ κ, so N_{i} ∩ κ^{+} is not in S.

We define a decreasing sequence of conditions ⟨x_{i}: i ≤ δ⟩ satisfying the following conditions for all i < δ:

(5) ⟨x_{j}: j ≤ i⟩ is in N_{i+1},

(6) max(x_{i}) = N_{i} ∩ κ^{+}. Define

x_{0} = x ∪ {u ∈ E: N_{0} ∩ κ ⊆ u ⊆ N_{0} ∩ κ^{+}}.

Since x is in N_{0} and |x| < κ, x ⊆ N_{0}. So if a is in x then since |a| < κ, a ⊆ N_{0}, and |a| < N_{0} ∩ κ. Using this fact it is easy to check that x_{0} is a condition below x satisfying the requirements. Suppose that x_{i} is defined for a fixed i < δ. Let y_{i} be the <_{θ}-least condition such that y_{i} ≤ x_{i} and y_{i} is in D_{i}. Since D_{i} is in N_{i+1}, y_{i} is in N_{i+1}. Define

x_{i+1} = y_{i} ∪ {u ∈ E: N_{i+1} ∩ κ ⊆ u ⊆ N_{i+1} ∩ κ^{+}}.

It is easy to check that x_{i+1} is a condition which is below x_{i} and satisfies the requirements.

Now suppose that γ ≤ δ is a limit ordinal and ⟨x_{i}: i < γ⟩ is defined. Since ⟨N_{i}: i < γ⟩ and ⟨D_{i}: i < γ⟩ are in N_{γ+1}, the sequence ⟨x_{i}: i < γ⟩ can be defined in N_{γ+1}. Let x_{γ} be the closure of ⋃{x_{i}: i < γ} under unions of increasing sequences. By Lemma 4.5, x_{γ} is a condition in N_{γ+1} below each x_{i} and max(x_{γ}) = N_{γ} ∩ κ^{+}.

This completes the construction. The condition x_{δ} is in ⋂{D_{i}: i < δ} and is below x. So ℙ is < κ-distributive. □

The proof of the κ^{+}-chain condition is basically the same as with Gitik’s poset.

Proposition 4.7. The forcing poset ℙ is κ^{+}-c.c.

Proof. Suppose for a contradiction that ⟨x_{i}: i < κ^{+}⟩ is an antichain in ℙ. By the Δ-system lemma, we can assume without loss of generality that there is a set d such that max(x_{i}) ∩ max(x_{j}) = d for i < j. Since κ is strongly inaccessible, there exist i < j such that P(d) ∩ x_{i} = P(d) ∩ x_{j}.

Choose a set a in E ∩ (P_{κ}κ^{+} \ S(κ, κ^{+})) which contains max(x_{i}) ∪ max(x_{j}), and |a| > | max(x_{i}) ∪ max(x_{j})|. Define X = {u ∈ E \ S: a ∩ κ ⊆ u ⊆ a}, and let y = x_{i} ∪ x_{j} ∪ X. Note that if u is in X then |u| = |a|, since |a| = a ∩ κ. Clearly y is a condition. But since x_{i} and x_{j} are incompatible, y is not below one of them.

Without loss of generality assume that y is not below x_{i}, i.e. y does not endextend x_{i}. Fix b in y \ x_{i} and c in x_{i} such that b ⊆ c. If b is in X, then |b| = |a| > | max(x_{i})| ≥ |c|, which is impossible. So b is in x_{j} \ x_{i}. Hence b ⊆ max(x_{j}) ∩ c ⊆ max(x_{j}) ∩ max(x_{i}) = d. Therefore b is in P(d) ∩ x_{j} = P(d) ∩ x_{i}. So b is in x_{i}, which is a contradiction. □

Proposition 4.8. The forcing poset ℙ forces that S is non-stationary.

Proof. Fix a name Ḋ such that ℙ forces that ⋃ Ġ_{ℙ} = Ḋ. We claim that ℙ forces that Ḋ is a club subset of P_{κ}κ^{+} disjoint from S. Clearly Ḋ is disjoint from S. To show that Ḋ is cofinal, suppose that x ⊩ ȧ ∈ P_{κ}κ^{+}. Since ℙ is < κ-distributive, there is x_{0} ≤ x and a such that x_{0} ⊩ ǎ = ȧ. Let b be any set in P_{κ}κ^{+} \ S(κ, κ^{+}) which is closed under H and F such that

| ⋃ x_{0}|^{+} ∪ max(x_{0}) ∪ a ⊆ b.

Define

y = x_{0} ∪ {u ∈ E: b ∩ κ ⊆ u ⊆ b}.

Then y is a condition refining x_{0} and y ⊩ a ⊆ b ∈ Ḋ. So Ḋ is cofinal.

Suppose that x ⊩ ⟨ȧ_{i}: i < β⟩ is an increasing sequence from Ḋ with size less than κ. Since ℙ is < κ-distributive there is y ≤ x and a family {a_{i}: i < β} such that for each i < β, y ⊩ ǎ_{i} = ȧ_{i}. Note that since y forces that a_{i} is in Ḋ, a_{i} must be in y. For otherwise we can extend y to a condition z for which it is impossible to add a_{i}. Since y is closed, ⋃{a_{i}: i < β} is in y. So y forces that ⋃{ȧ_{i}: i < β} is in Ḋ. □

We now verify the comments made prior to Lemma 4.1 concerning the relationship between ℙ_{F} and ℙ_{S}.

Proposition 4.9. Suppose that F: κ^{+<ω} → κ^{+} is a partial function in V, and in W, ℙ is a ℙ_{F} -name such that ℙ_{F} ∗ ℙ is < κ-distributive and κ^{+}-c.c. Then ℙ_{F} ∗ ℙ forces that for every stationary set T ⊆ S(κ, κ^{+}), there is A ⊆ κ^{+} such that

T = {a ∈ S(κ, κ^{+}): sup a ∈ A}

modulo clubs.

Proof. Let G_{F} ∗ G_{ℙ} be generic for ℙ_{F} ∗ ℙ over W. Suppose that T is a stationary subset of S(κ, κ^{+}) in W [G_{F} ∗ G_{ℙ}]. Let D be the club set of a in P_{κ}κ^{+} such that a is in ⋃ G_{F} and is closed under F and H. Let

A = {α < κ^{+}: ∃a ∈ T ∩ D sup a = α}

We show that a is in T ∩ D iff a is in S(κ, κ^{+}) ∩ D and sup a is in A. Note that if a is in T ∩ D then sup a is in A by definition of A.

Suppose that a is in S(κ, κ^{+}) ∩ D and sup a is in A. Fix b in T ∩ D such that sup a = sup b = β. Let κ_{a} = a ∩ κ. Now go back to W. By Lemma 4.1, o.t.(a) = κ_{a}^{+} = cf(β) = o.t.(b). So b ∩ κ = κ_{a}. Since a and b are closed under suprema of bounded subsets with cofinality different from cf(κ_{a}), a ∩ b is an unbounded subset of β and so |a ∩ b| = |a| = |b| = κ_{a}^{+}. But a ∩ b is closed under H and F. Since a and b are in ⋃ G_{F} ∩ S(κ, κ^{+}), F is Jonsson for a and b. Since a ∩ b ⊆ a, a ∩ b = a, and similarly a ∩ b = b. So a = b, and therefore a is in T. □

5. Iterated Forcing

In the sections which follow we will need to iterate forcing posets for destroying stationary sets over different cardinals. These posets are distributive, but not strategically closed. The usual methods of Easton support iterations require that the posets be strategically closed. To overcome this problem we use Prikry forcing techniques.

In Section 2 we described how to turn a distributive forcing poset into a Prikry type forcing poset. So we can iterate the poset for destroying stationary sets by using iterations of Prikry type forcing posets.

Magidor [11] first showed how to iterate Prikry forcing using conditions with full support. Gitik [6] used this style of iteration to construct a model with a nonsplitting subset of P_{κ}κ^{+}. However, lifting an elementary embedding with such an iteration is problematic. To overcome this difficulty, Gitik [7] invented a method for iterating Prikry type forcing posets using conditions with Easton support. We will use Easton support Prikry iterations in this paper.

In this section we describe a general schema for iterating Prikry type forcing posets with Easton support, and give a sample application.

An Easton support Prikry iteration is an iterated forcing

⟨ℙ_{α}, ℚ_{α}: α < κ⟩,

for some ordinal κ, satisfying the following properties:

(1) There exists a set A ⊆ κ consisting of strongly inaccessible cardinals such that if ℚ_{α} is non-trivial, then α is in A,

(2) ℙ_{α} forces that |ℚ_{α}| < min(A \ (α + 1)),

(3) ℙ_{α} forces that ⟨ℚ_{α}, ≤, ≤^{∗}⟩ is a Prikry type forcing poset,

(4) ℙ_{α+1} = ℙ_{α} ∗ ℚ_{α},

(5) If α is a limit ordinal, then ℙ_{α} consists of functions p with domain an Easton subset of α ∩ A such that p ↾ β is in ℙ_{β} for all β less than α,

(6) If α is a limit ordinal and p, q are in ℙ_{α}, then q ≤ p if q ↾ β ≤ p ↾ β for β less than α, and there is a finite set a such that for all β in dom(p) \ a, q ↾ β ⊩ q(β) ≤^{∗} p(β).

(7) If α is a limit ordinal, q ≤^{∗} p in ℙ_{α} if q ≤ p and the finite set a in (6) is empty; i.e. for all β in supp(p), q ↾ β ⊩ q(β) ≤^{∗} p(β),

(8) q ḃ ≤^{∗} p ȧ in ℙ_{β+1} if q ≤^{∗} p in ℙ_{β} and q ⊩ ḃ ≤^{∗} ȧ.

Such an iteration satisfies the Prikry property; see [7] for a proof. We say that the iteration above is defined on A. If p is a condition in ℙ_{α}, the support of p, denoted by supp(p), is the domain of p as a function.

If β is less than α, then ℙ_{α} factors into ℙ_{β} ∗ ℚ_{β} ∗ ℙ_{β,α}. Suppose that ℚ_{γ} is forced to be γ-weakly closed for all γ greater than β; then ⟨ℙ_{β,α}, ≤^{∗}⟩ is forced to be min(A \ (β + 1))-weakly closed. Therefore ℙ_{β,α} does not add any bounded subsets to min(A \ (β + 1)).

The following result follows from the usual proof of the corresponding fact for Easton support iterations.

Proposition 5.1. If |ℙ_{β}| < α for all β less than α and α is a Mahlo cardinal, then ℙ_{α} is α-c.c.

The following lemma describes in more detail the sort of iteration we will use later.

Lemma 5.2. Suppose that ⟨ℙ_{α}, ℚ̇_{α}: α < κ⟩ is an Easton support Prikry iteration defined on a set A. Assume that for all α in A:

(1) ℙ_{α} forces that ℚ_{α} is α-weakly closed and satisfies the direct extension property,

(2) there is μ_{α} and λ_{α} such that

α ≤ μ_{α} < 2^{(μ_{α}^{<α})} ≤ λ_{α}^{+} < min(A \ (α + 1)),

(3) there is a normal ultrafilter on P_{α}λ_{α} with j_{α}: V → M_{α} = Ult(V, U_{α}) such that min(j_{α}(A) \ (α + 1)) > λ_{α}.

Then for each α in A, ℙ_{α} forces that α is μ_{α}-strongly compact.

Proof. If α is not a limit point of A, then |ℙ_{α}| < α, so the result follows. Suppose that α is a limit point of A. Then

j_{α}(ℙ_{α}) = ℙ_{α} ∗ ℚ_{α} ∗ ℙ_{tail}

where ℙ_{tail} is forced to be min(j_{α}(A) \ (α + 1))-weakly closed. Let G_{α} be generic for ℙ_{α} over V.

Since ℙ_{α} is α-c.c., every member of (P_{α}μ_{α})^{V [G_{α}]} is covered by a set in (P_{α}μ_{α})^{V}. Since α is strongly inaccessible in V [G_{α}], |(P_{α}μ_{α})^{V [G_{α}]}| = (μ_{α}^{<α})^{V}. So we can enumerate all canonical ℙ_{α}-names for subsets of P_{α}μ_{α} as ⟨Ẋ_{i}: i < λ_{α}^{+}⟩, possibly with repetitions. By the closure of M_{α}, every initial segment of ⟨j_{α}(Ẋ_{i}): i < λ_{α}^{+}⟩ is in M_{α}.

Now j_{α}“G_{α} = G_{α}, so if H is generic for ℚ_{α} ∗ ℙ_{tail} over V [G_{α}], we can lift j_{α} to j_{α}: V [G_{α}] → M_{α}[G_{α} ∗ H] in V [G_{α} ∗ H]. Let G be generic for ℚ_{α} over V [G_{α}]. In V [G_{α}∗G] define a ≤^{∗}-decreasing sequence ⟨q_{i}: i < λ_{α}^{+}⟩ in ℙ_{tail} as follows. Let q_{0} = 1, and given q_{i}, let q_{i+1} ≤^{∗} q_{i} such that q_{i+1} decides the statement j_{α}“μ_{α} ∈ j_{α}(Ẋ_{i}). For limit δ < λ_{α}^{+} apply the weak closure of ℙ_{tail} to obtain q_{δ}. Note that each initial segment of the sequence is in M_{α}[G_{α} ∗ G].

Now ℚ_{α} is α-weakly closed and satisfies the direct extension property (if ℚ_{α} is the trivial poset, this is true vacuously). In V [G_{α}] define U^{∗} by letting X ∈ U^{∗} iff there is s ≤^{∗} 1 in ℚ_{α} and Ẋ_{i} such that Ẋ_{i}^{G_{α}} = X and

M_{α}[G_{α}] ⊧ s ⊩_{ℚ_{α}} q_{i+1} ⊩_{ℙ_{tail}} j_{α}“μ_{α} ∈ j_{α}(Ẋ_{i}).

We show that U^{∗} is a fine ultrafilter on P_{α}μ_{α}.

First we prove that the definition does not depend on i. Suppose X_{i}^{G_{α}} = X_{j}^{G_{α}} = X, where i < j. Fix p in G_{α} which forces that Ẋ_{i} = Ẋ_{j}. Now j_{α}(p) = p 1 in j_{α}(ℙ_{α}), and p is in G_{α}. Therefore

M_{α}[G_{α}] ⊧ ℚ_{α} ∗ ℙ_{tail} ⊩ j_{α}(Ẋ_{i}) = j_{α}(Ẋ_{j}).

Apply the Prikry property of ℚ_{α} to find s_{i}, s_{j} ≤^{∗} 1 such that s_{i} decides which way q_{i+1} decides the statement j_{α}“μ_{α} ∈ j_{α}(Ẋ_{i}), and similarly for s_{j} and Ẋ_{j}. By the direct extension property, s_{i} and s_{j} are compatible, and j_{α}(Ẋ_{i}) = j_{α}(Ẋ_{j}). So clearly s_{i} and s_{j} must decide their respective statements the same way.

Similar arguments show that U^{∗} is a fine ultrafilter. We show that U^{∗} is αcomplete. Suppose that p is in G, ȧ is a name for a subset of λ_{α}^{+} with size less than α, and p forces that ⟨Ẋ_{i}: i ∈ ȧ⟩ is a partition of P_{α}μ_{α}. Then j_{α}(p) = p 1 forces in j_{α}(ℙ_{α}) that j_{α}(⟨Ẋ_{i}: i ∈ ȧ⟩) = ⟨j_{α}(Ẋ_{i}): i ∈ a⟩ is a partition of j_{α}(P_{α}μ_{α}). So

M_{α}[G_{α}] ⊧ ℚ ∗ ℙ_{tail} ⊩ ⟨j_{α}(Ẋ_{i}): i ∈ a⟩ is a partition of j_{α}(P_{α}μ_{α}).

Apply the α-weak closure of ℚ_{α} to obtain s, a direct extension of 1 in ℚ_{α}, which decides which way that q_{i+1} decides the statement j_{α}“μ ∈ j_{α}(Ẋ_{i}), for all i in a. Clearly there must be some i in a such that

M_{α}[G_{α}] ⊧ s ⊩_{ℚ_{α}} q_{i+1} ⊩_{ℙ_{tail}} j_{α}“μ ∈ j_{α}(Ẋ_{i}).

So Ẋ_{i}^{G_{α}} is in U^{∗}. □

We now give a sample application of the forcing poset from Section 4 and the Prikry techniques above to prove the following theorem.

Theorem 5.3. Suppose that GCH holds and there exists a κ^{++}-supercompact cardinal κ. Then there exists a model in which κ is κ^{+}-supercompact and almost all a in S(κ, κ^{+}) satisfy that a ∩ κ is strongly inaccessible.

There are limits to this sort of result; see Proposition 5.7 below. In the next section we give a more elaborate construction of a model in which almost all a in S(κ, κ^{+}) satisfy that cf(a ∩ κ) = ω.

Start with a model V in which κ is κ^{++}-supercompact and GCH holds. Let ℝ be the Radin forcing for adding a club C of V -regulars to κ, while preserving the κ^{++}-supercompactness of κ. Let G_{ℝ} be generic for ℝ over V and let W = V [G_{ℝ}].

We would like to force with the poset ℙ_{S} which destroys the stationarity of the set of a in S(κ, κ^{+}) such that a ∩ κ is not strongly inaccessible. But in order to preserve the κ^{+}-supercompactness of κ, or even to preserve the stationarity of S(κ, κ^{+}), we need to iterate posets like ℙ_{S} below κ.

Let A be the set of β < κ in lim(C) such that β is β^{+}-supercompact. We define an Easton support Prikry iteration

⟨ℙ_{β}, ℚ̇_{β}: β < κ⟩.

Suppose that ℙ_{β} is defined for some fixed β in A ∪ {κ}. Let Ṡ_{β} be a ℙ_{β}-name for the set of a in S(β, β^{+}) such that a ∩ β is not strongly inaccessible. Let Ḃ_{β} be a ℙ_{β}-name for the set of α < β^{+} such that cf(α) = μ^{+} for some μ < κ which is not strongly inaccessible.

We claim that ℙ_{β} forces that

Ṡ_{β} = {a ∈ S(β, β^{+}): sup a ∈ Ḃ_{β}}.

For suppose that a is in S_{β}. Then a is in S(β, β^{+}) and a ∩ β is not inaccessible. Since cf(sup a) = (a ∩ β)^{+}, sup a is in B_{β}. On the other hand, suppose that a is in S(κ, κ^{+}) and sup a is in B_{β}. Then o.t.(a) = (a ∩ β)^{+}, so cf(sup a) = (a ∩ β)^{+}. Since sup a is in B_{β}, a ∩ β is not inaccessible.

So Ṡ_{β} is of the form required to define the < β-distributive, β^{+}-c.c. forcing poset ℙ_{S_{β}} from Definition 4.2.

By Lemma 5.2, ℙ_{β} forces that β is β^{+}-strongly compact. The forcing poset ℙ_{S_{β}} is < β-distributive, has size β^{+}, and has no more than β^{+}-many maximal antichains. By Lemma 2.3 there exist a Prikry type forcing poset ⟨ℚ_{β}, ≤, ≤^{∗}⟩ which is β-weakly closed, has the direct extension property, and ⟨ℚ, ≤⟩ is equivalent to ℙ_{S_{β}}. So at stage β we force with ⟨ℚ_{β}, ≤⟩.

This completes the definition of the iteration. Let Ṡ = Ṡ_{κ}. Clearly ℙ_{κ} ∗ℙ_{S} forces that for almost all a in S(κ, κ^{+}), a ∩ κ is strongly inaccessible. It remains to check that ℙ_{κ} ∗ ℙ_{S} preserves the κ^{+}-supercompactness of κ.

Fix an elementary embedding j: W → M such that crit(j) = κ, j(κ) > κ^{++}, and ^{κ^{++}}M ⊆ M. By GCH, any ultrafilter on P_{κ}κ^{+} has size κ^{++} and therefore is in M. So M models that κ is κ^{+}-supercompact and κ is in j(A). Enumerate all canonical ℙ_{κ} ∗ ℙ_{S}-names for subsets of P_{κ}κ^{+} as ⟨Ẋ_{i}: i < κ^{++}⟩. Write

j(ℙ_{κ}) = ℙ_{κ} ∗ ℚ_{κ} ∗ ℙ_{tail},

where ℚ_{κ} is a Prikry type forcing poset which is equivalent to ℙ_{S} and ℙ_{tail} is forced to be κ^{+3}-weakly closed. Let G_{κ} ∗ G_{S} ∗ G_{tail} be generic for ℙ_{κ} ∗ ℙ_{S} ∗ ℙ_{tail} over W. In W [G_{κ} ∗ G_{S} ∗ G_{tail}] the map j can be extended to

j: W [G_{κ}] → M[G_{κ} ∗ G_{S} ∗ G_{tail}].

Let M_{1} = M[G_{κ} ∗ G_{S} ∗ G_{tail}].

Now we construct a master condition. In M_{1} define

t = ⋃ j“G_{S} ∪ {u ∈ j(E): κ ⊆ u ⊆ j“κ^{+}}

Lemma 5.4. The set t is a condition in j(ℙ_{S}) and t ≤ j(x) for all x in G_{S}.

Proof. It suffices to prove that t is the closure of ⋃ j“G in M_{1} under unions of ⊆-increasing sequences. For if this is true, then Lemma 4.4 and the fact that κ is strongly inaccessible in M_{1} imply that t ∩ j(S) = ∅ and t is a condition in j(ℙ_{S}) which is below j(x) for all x in G_{S}.

Clearly the set {u ∈ j(E): κ ⊆ u ⊆ j“κ^{+}} is itself closed. Since ℙ_{tail} is κ^{+3}- weakly closed, ⋃ G_{S} remains a club subset of P_{κ}κ^{+} in M_{1}. Therefore ⋃ j“G_{S} is closed under unions of ⊆-increasing sequences with length less than κ. On the other hand, suppose that ⟨j(a_{i}): i < κ⟩ is ⊂-increasing. Then the sequence of ordinals

⟨j(a_{i}) ∩ j(κ): i < κ⟩ = ⟨a_{i} ∩ κ: i < κ⟩

must be unbounded in κ, since o.t.(a_{i}) = (a_{i} ∩ κ)^{+} and the a_{i}’s are getting larger. Therefore

u = ⋃{j(a_{i}): i < κ}

must satisfy that κ ⊆ u ⊆ j“κ^{+}. So the closure of ⋃ j“G is contained in t.

To show that the closure is equal to t, by Lemma 4.4 (3) it suffices to show that for any u in E with κ ⊆ u ⊆ j“κ^{+}, there is v in the closure with u ⊆ v. Suppose first that |u| = κ. Let ⟨α_{i}: i < κ⟩ enumerate u. Inductively define a ⊆-increasing sequence ⟨a_{i}: i < κ⟩ from ⋃ G_{S} by choosing a_{i+1} so that j(a_{i+1}) contains α_{i} and i as members. Then u ⊆ ⋃ j(a_{i}) and ⋃ j(a_{i}) is in the closure of ⋃ j“G_{S}. In particular this proves that all initial segments of j“κ^{+} which are in E are in the closure of ⋃ j“G_{S}, and we can write j“κ^{+} as the union of such initial segments. Hence j“κ^{+} is in the closure of ⋃ j“G_{S} and t is equal to this closure.

□

Lemma 5.5. In W [G_{κ} ∗ G_{S}], κ is κ^{+}-supercompact.

Proof. For each i < κ^{++}, let D_{i} be the dense set of conditions in j(ℙ_{S}) which decide the statement j“κ^{+} ∈ j(Ẋ_{i}). Since j(ℙ_{S_{κ}}) is < j(κ)-distributive and κ^{++} < j(κ), D = ⋂ D_{i} is a dense set. So fix s ≤ t in D. Let ṡ be a ℙ_{tail}-name for s. That is, ṡ is a name such that ℙ_{tail} forces that ṡ decides the statements j“κ^{+} ∈ j(Ẋ_{i}), and s ≤ j(x) for all x in G_{S}.

Apply the weak-closure of ℙ_{tail} to get a condition q in ℙ_{tail} such that for all i < κ^{++}, q decides the statement

ṡ ⊩_{j(ℙ_{S})} j“κ^{+} ∈ j(Ẋ_{i}).

Define U in W [G_{κ} ∗ G_{S}] by letting X_{i} be in U if

q ⊩_{ℙ_{tail}} ṡ ⊩_{j(P_{S})} j“κ^{+} ∈ j(Ẋ_{i}).

We claim that U is a normal ultrafilter on P_{κ}κ^{+}. The proof that U is well-defined and is a fine ultrafilter on P_{κ}κ^{+} is essentially the same as the proof of Lemma 5.2.

For normality, suppose that p x is in G_{κ} ∗ G_{S} and forces that ḟ: P_{κ}κ^{+} → κ^{+} is a regressive function. Then j(p x) = p 1 1 j(x), which is above the condition p 1 q s. So

M[G_{κ} ∗ G_{S}] ⊧ q ⊩ s ⊩ j(ḟ): P_{j(κ)}j(κ^{+}) → j(κ^{+}) is regressive.

Applying the weak closure of ℙ_{tail} and distributivity of j(ℙ_{S_{κ}}), we can choose q_{1} s_{1} ≤ q s and β < κ^{+} such that

q_{1} ⊩ s_{1} ⊩ j(ḟ)(j“κ^{+}) = j(β).

Let Ẋ_{i} be a name for the set {a ∈ P_{κ}κ^{+}: f(a) = β}. Then

q_{1} ⊩ s_{1} ⊩ j“κ^{+} ∈ j(Ẋ_{i}).

Since q ṡ decides the statement j“κ^{+} ∈ j(Ẋ_{i}), clearly

q ⊩ ṡ ⊩ j“κ^{+} ∈ j(Ẋ_{i})

and X_{i} is in U. Therefore U is normal. □

This completes the proof of Theorem 5.3.

We cannot replace “strongly inaccessible” in the last result with “supercompact”, as we see next.

Lemma 5.6. If β is β^{+}-supercompact, then there is a stationary set of a in S(β, β^{+}) such that a ∩ β is not (a ∩ β)^{+}-supercompact.

Proof. Let S be the set of a in S(β, β^{+}) such that a∩β is not (a∩β)^{+}-supercompact. Fix U a normal ultrafilter on P_{β}β^{+} which is minimal in the Mitchell ordering of normal ultrafilter on P_{β}β^{+}. Let j: V → M = Ult(V, U). Since M contains all subsets of P_{β}β^{+} and U is minimal, M models that β is not β^{+}-supercompact. So j“β^{+} is in j(S). Suppose that C is club subset of P_{β}β^{+}. Then j“C is a directed subset of j(C) with size less than j(β), so ⋃ j“C = j“β^{+} is in j(C) ∩ j(S). By elementarity, S ∩ C is non-empty. So S is stationary. □

Proposition 5.7. Suppose that S(κ, κ^{+}) is stationary. Then there is a stationary set of a in S(κ, κ^{+}) such that a ∩ κ is not (a ∩ κ)^{+}-supercompact.

Proof. Suppose that S(κ, κ^{+}) is stationary. Let C be a club subset of P_{κ}κ^{+}. Fix a function f: κ^{+<ω} → κ^{+} such that any set a in P_{κ}κ^{+} which is closed under f is in C. We will find a set a in S(κ, κ^{+}) which is closed under f such that a ∩ κ is not (a ∩ κ)^{+}-supercompact.

Fix b in S(κ, κ^{+}) which is closed under f. If b ∩ κ is not (b ∩ κ)^{+}-supercompact then we are done. Otherwise b ∩ κ = κ_{b} is κ_{b}^{+}-supercompact. Let π: κ_{b}^{+} → b be the unique order preserving bijection. Note that crit π = κ_{b}. Define f_{b}: κ_{b}^{+<ω} → κ_{b}^{+} by letting

f_{b}(α_{0}, …, α_{n}) = π^{−1}(f(π(α_{0}), …, π(α_{n}))).

Since κ_{b} is κ_{b}^{+}-supercompact, by Lemma 5.6 there is c in P_{κ_{b}}κ_{b}^{+} such that c is closed under f_{b}, o.t.(c) = (c ∩ κ_{b})^{+}, and c ∩ κ_{b} not (c ∩ κ_{b})^{+}-supercompact. Let a = π“c. It is straightforward to check that a is closed under f. Then a∩κ = π“c∩κ = c∩κ_{b} and o.t.(a) = o.t.(c) = (c ∩ κ_{b})^{+}. So a is in S(κ, κ^{+}) ∩ C and a ∩ κ is not (a ∩ κ)^{+}- supercompact. □

6. S(κ, κ^{+}) and Cofinalities

In this section we give another application of the forcing poset from Section 4 to prove the following theorem.

Theorem 6.1. Suppose that GCH holds and κ is κ^{+4}-supercompact. Then there exists a model in which κ is a non-Mahlo inaccessible cardinal, GCH holds, S(κ, κ^{+}) is stationary, and for almost all a in S(κ, κ^{+}), cf(a ∩ κ) = ω.

Start with a model V in which κ is κ^{+4}-supercompact and GCH holds. Let W be a generic extension of V by a Radin forcing ℝ which preserves the κ^{+4}- supercompactness of κ and adds a club set C ⊆ κ such that every α in C is regular in V.

Let A be the set of β < κ in lim(C) such that β is β^{+}-supercompact. We define an Easton support Prikry iteration ⟨ℙ_{i}, ℚ̇_{i}: i < κ⟩ such that ℚ_{α} is non-trivial iff α is in A. We will split the definition of ℚ_{α} into three cases. So partition A into disjoint sets as follows. Let A_{1} be the set of β in A such that β is β^{+}-supercompact but not β^{+2}-supercompact, and let A_{2} be the set of β in A such that β is β^{+2}- supercompact but not β^{+3}-supercompact. Let A_{3} be the set of β in A such that β is β^{+3}-supercompact.

Suppose that ℙ_{β} is defined for a fixed β in A. Let Ṡ_{β} be a ℙ_{β}-name for the set of a in S(β, β^{+}) such that a ∩ κ is a singular ordinal with uncountable cofinality. Let ℙ_{S_{β}} be a ℙ_{β}-name for the forcing poset for destroying the stationarity of Ṡ_{β}.

The definition of ℚ_{β} depends on which set A_{i} the ordinal β is in. Suppose that β is in A_{1}. By Lemma 5.2, ℙ_{β} forces that β is β^{+}-strongly compact. By Lemma 2.3 there is a Prikry type forcing poset ℚ̇_{β} which is β-weakly closed, has the direct extension property, and is equivalent to ℙ_{S_{β}}.

Suppose that β is in A_{2}. By the same argument as in Lemma 5.5, ℙ_{β} ∗ℙ_{S_{β}} forces that β is β^{++}-supercompact. Apply Lemma 2.4 to choose a Prikry type forcing poset ℚ̇_{β} which is β-weakly closed, satisfies the direct extension property, and is equivalent to ℙ_{S_{β}} ∗ ℙℝ(U), where U is a name for a normal ultrafilter on P_{β}β^{+}.

Suppose that β is in A_{3}. We will prove that ℙ_{β} ∗ ℙ_{S_{β}} forces that there is a β-complete ultrafilter U_{β} on β which contains the set A_{2} ∩ β. We then define ℚ_{β} as a Prikry type forcing poset which is equivalent to ℙ_{S_{β}} ∗ ℙℝ(U_{β}).

Lemma 6.2. The poset ℙ_{β} ∗ ℙ_{S_{β}} forces that there is a β-complete ultrafilter U_{β} on β which contains the set A_{2} ∩ β.

Proof. Let j: W → M = Ult(W, U) for some normal ultrafilter U on P_{β}β^{+3} which is minimal in the Mitchell ordering. Note that β is in j(A_{2}). Write

j(ℙ_{β}) = ℙ_{β} ∗ ℚ̇_{β} ∗ ℙ_{tail}

where ℚ̇_{β} is a Prikry type forcing poset equivalent to ℙ_{S_{β}} ∗ ℙℝ(β, β^{+}). Let G_{β} ∗ G_{S} be generic for ℙ_{β} ∗ ℙ_{S_{β}}.

Let ṡ be a ℙℝ(β, β^{+}) ∗ ℙ_{tail}-name for the closure under unions of ⊆-increasing sequences of the set

⋃ j“G_{S} ∪ {u ∈ j(E): β ⊆ u ⊆ j“β^{+}}

in the generic ultrapower, where G_{S} is the generic filter for ℙ_{S_{β}}. We will show that ṡ is forced to be a master condition.

Let G_{β} ∗ G_{S} ∗ H ∗ G_{tail} be generic for j(ℙ_{β}) over M. Let M_{1} = M[G_{β} ∗ G_{S} ∗ H ∗ G_{tail}]. Extend j to j: W [G_{β}] → M_{1}. The poset ℙ_{tail} is sufficiently weakly closed, so s is in M[G_{β} ∗ G_{S} ∗ H]. Since j“G_{S} is a directed subset of j(P_{S}), s satisfies all the conditions of Lemma 4.4, and in particular end-extends each member of j“G_{S}. So in order to check that s is a condition in j(ℙ_{S}), we need to show that s is disjoint from j(S).

Write

s^{∗} = {a ∈ s: a ∩ j(β) < β}.

Note that

s = s^{∗} ∪ {u ∈ j(E): β ⊆ u ⊆ j“β^{+}}.

Since cf(j“β^{+} ∩ j(β)) = cf(β) = ω in M_{1}, it suffices to show that s^{∗} is disjoint from j(S).

Define {t_{i}: i ∈ On} as follows. Let t_{0} = ⋃ j“G_{S}, and t_{δ} = ⋃{t_{i}: i < δ} for limit δ ≤ ζ. For each i, let t_{i+1} be the collection of sets a such that a is the union of an increasing sequence ⟨a_{j}: j < ξ⟩ of elements from t_{i} with length less than β. Let ζ be the first ordinal such that t_{ζ+1} = t_{ζ}. Then s^{∗} = ⋃{t_{i}: i < ζ}. We only need to take unions of sequences with length less than β in the successor case since any strictly increasing sequence with length at least β has a union which contains β.

Sublemma 6.3. If x is a set in M[G_{β}] with size less than β and y ⊆ x in M_{1}, then y is in M[G_{β}].

Proof. Since ℙ_{S_{β}} is < β-distributive and ℙℝ(β, β^{+}) ∗ ℙ_{tail} is β-weakly closed over M[G_{β} ∗G_{S}], ℙ_{S_{β}} ∗ℙℝ(β, β^{+})∗ℙ_{tail} does not add bounded subsets to β over M[G_{β}]. By considering a bijection between x and its cardinality in M[G_{β}], it is clear that the poset does not add subsets to x. □

Sublemma 6.4. s^{∗} ∩ W [G_{β}] = t_{0}.

Proof. Since G_{S} ⊆ W [G_{β}] and j ↾ β^{+} is in W [G_{β}], clearly t_{0} ⊆ s^{∗} ∩ W [G_{β}]. We prove by induction on i < ζ that t_{i} ∩ W [G_{β}] ⊆ t_{0}. The claim is obvious for t_{0} and for limit stages. Suppose that t_{i} ∩ W [G_{β}] ⊆ t_{0} for a fixed i < ξ, and let a be in t_{i+1} ∩ W [G_{β}]. Fix an increasing sequence ⟨a_{j}: j < ξ⟩ from t_{i} whose union is a. Since a is in W [G_{β}], it is in M[G_{β}], so each a_{j} is in M[G_{β}] ∩ t_{i} ⊆ W [G_{β}] ∩ t_{i} ⊆ t_{0}. Since P(a) is in M[G_{β}], so also is the set {a_{j}: j < ξ}. But t_{0} is a closed subset of P_{β}(j“β^{+}) in M[G_{β} ∗ G_{S}], so a = ⋃{a_{j}: j < ξ} is in t_{0}. □

Sublemma 6.5. The set s^{∗} is disjoint from j(S).

Proof. It suffices to prove that whenever a is in s^{∗} and o.t.(a) = (a ∩ j(β))^{+}, then either cf(a ∩ j(β)) = ω or a ∩ j(β) is strongly inaccessible. Suppose that a is in s^{∗} and o.t.(a) = (a ∩ j(β))^{+}. Let b = j^{−1}[a]. Note that b ∩ β = a ∩ j(β), so it suffices to show that b ∩ β is either strongly inaccessible or has cofinality ω. Consider the case when a is in W [G_{β}]. Then by the last sublemma, a is in t_{0}, so clearly a = j(b) and b is in ⋃ G_{S}. Then b is in S(β, β^{+}) ∩ ⋃ G_{S}, therefore b ∩ β = a ∩ j(β) either has cofinality ω or is strongly inaccessible.

Suppose that a is not in W [G_{β}]. Let μ = b ∩ β and let γ = sup b. Note that μ^{+} is the same in M[G_{β}] and M_{1}. We claim that cf(γ)^{M[G_{β}]} = μ^{+}. Since ℙ_{tail} is weakly closed enough that it does not add subsets to β^{+}, it does not change the cofinality of γ. But ℙ_{S_{β}} ∗ ℙℝ(β, β^{+}) only changes cofinalities to ω. This is true because ℙ_{S_{β}} preserves all cofinalities, since it is < β-distributive and β^{+}-c.c.; also ℙℝ(β, β^{+}) does not add bounded subsets to β, changes the cofinality of β and β^{+} to ω, and is β^{++}-c.c. So γ must have cofinality μ^{+} in M[G_{β}].

Define an increasing sequence ⟨d_{i}: i < μ^{+}⟩ in M[G_{β}] as follows. Let d_{0} be any club subset of γ with order type μ^{+} and μ ⊆ d_{0}. Take unions at limits. Given d_{i}, define

d_{i}^{∗} = d_{i} ∪ H“d_{i}^{<ω},

and define

d_{i+1} = d_{i}^{∗} ∪ lim(d_{i}^{∗}).

Now let d = ⋃{d_{i}: i < μ^{+}}. Then d is closed under H, has size μ^{+}, and d ∩ γ is a club subset of γ which contains μ. Note that b is closed under H. Let u = d ∩ b. Then u is closed under H. Since u is a subset of d, |d| < β, and d is in M[G_{β}], u is in M[G_{β}]. The set u = b ∩ d has order type μ^{+} and so is in S(β, β^{+}). Also

μ = a ∩ j(β) ⊆ j(u) ⊆ a

and j(u) is in j(E), so by Lemma 4.4 (3), j(u) is in s^{∗}. Since j ↾ β^{+} is in W [G_{β}], j(u) = j“u is in s^{∗} ∩ W [G_{β}]. So by the last sublemma, j(u) is in t_{0} = ⋃ j“G_{S}.Therefore u is in ⋃ G_{S} ∩ S(β, β^{+}), and so either μ = u ∩ β is strongly inaccessible or has cofinality ω. □

Enumerate all canonical ℙ_{β} ∗ ℙ_{S_{β}}-names for subsets of β as ⟨Ẋ_{i}: i < β^{+}⟩. Let G_{β} ∗ G_{S} be generic for ℙ_{β} ∗ ℙṠ_{β} over W. Let ṡ be a name for the master condition defined above.

Since j(ℙ_{S_{β}}) is j(β)-distributive, let ṫ be a name for a refinement of ṡ which decides the statement β ∈ j(Ẋ_{i}) for every i < β^{+}. Let q be a condition in ℙ_{tail} which decides each statement s ⊩ β ∈ j(Ẋ_{i}). Define U_{β} in W [G_{β} ∗ G_{S_{β}}] as the set of X_{i} such that there is a condition p in ℙℝ(β, β^{+}) which is a direct extension of 1 and

p ⊩ q ⊩ t ⊩ β ∈ j(Ẋ_{i}).

Recall that whenever A is a collection of less than β many direct extensions of 1 in ℙℝ(β, β^{+}), there is p such that p ≤^{∗} p_{0} for all p_{0} in A. Using this fact it is straightforward to check that U_{β} is a β-complete ultrafilter and contains A_{2} ∩ β. This completes the proof of the lemma. □

Let ℙ_{β}^{sing} denote the poset ℙ_{Sing∩β} from Section 2 for adding a club set to Sing∩β using closed, bounded subsets ordered by end-extension.

Lemma 6.6. The poset ℙ_{β} ∗ ℙ_{S_{β}} forces that there is a projection mapping

ℙℝ(U_{β}) → B(ℙ_{β}^{sing}),

where ℙ_{β}^{sing} is a name for the poset for adding a club to the set β ∩ Sing.

Proof. Let G_{β} ∗ G_{S} be generic for ℙ_{β} ∗ ℙ_{S_{β}} over W. We describe a canonical way to define a generic for ℙ_{β}^{sing} over W [G_{β} ∗ G_{S}] from a generic for ℙℝ(U_{β}).

Working in the model W [G_{β} ∗ G_{S}], enumerate ℙ_{β}^{sing} as ⟨c_{i}: i < β⟩. Let H be generic for ℙℝ(U_{β}) and let ⟨β_{n}: n < ω⟩ be the generic Prikry sequence given by H.

We define by induction a sequence ⟨d_{n}, α_{n}: n < ω⟩ satisfying the following induction hypotheses:

(1) each α_{n} is in A_{2},

(2) ⟨α_{n}: n < ω⟩ is a subsequence of ⟨β_{n}: n < ω⟩,

(3) d_{n} is in ℙ_{α_{n}}^{sing} in the model W [G_{α_{n}} ∗ G_{S_{αn}}], where G_{α_{n}} ∗ G_{S_{αn}} is the generic filter for ℙ_{α_{n}} ∗ ℙ_{S_{αn}} given by G_{β},

(4) d_{n+1} is an end-extension of d_{n}, Let d_{0} = c_{β_{0}}. Let α_{0} be the least element in the Prikry sequence such that α_{0} > max(d_{0}) and for all α ≥ α_{0} in the Prikry sequence, α is in A_{2}. Since d_{0} is a bounded subset of α_{0} and ℙ_{β} = ℙ_{α_{0}} ∗ ℙ_{S_{α_{0}}} ∗ ℙℝ(α_{0}, α_{0}^{+}) ∗ ℙ_{tail}, clearly d_{0} is in W [G_{α_{0}} ∗ G_{S_{α_{0}}} ].

Suppose that d_{n} and α_{n} are defined. Since α_{n} is in A_{2}, ℙ_{α_{n}+1} = ℙ_{α_{n}} ∗ ℙ_{S_{αn}} ∗ ℙℝ(α_{n}, α_{n}^{+}). Let G_{α_{n}+1} = G_{α_{n}} ∗ G_{S_{αn}} ∗ H_{α_{n}} be the generic for ℙ_{α_{n}+1} given by G_{β}.

Since ℙ_{α_{n}}^{sing} is < α_{n}-distributive and has size α_{n}, there is a generic filter G_{α_{n}}^{sing} for ℙ_{α_{n}}^{sing} over W [G_{α_{n}} ∗ G_{S_{αn}}] in the model W [G_{α_{n}+1}] which contains the condition d_{n}. Define

d_{n+1} = ⋃ G_{α_{n}}^{sing} ∪ {α_{n}}.

Let α_{n+1} be the least element of the Prikry sequence greater than α_{n}. Since α_{n} has cofinality ω in W [G_{α_{n+1}}], d_{n+1} is a condition in ℙ_{α_{n+1}}^{sing}. This completes the induction.

Let d = ⋃{d_{n}: n < ω} and let

G^{sing} = {d ∩ (γ + 1): γ < β}.

We prove that G^{sing} is generic for ℙ_{β}^{sing} over W [G_{β} ∗ G_{S}].

Let D be a dense subset of ℙ_{β}^{sing} in W [G_{β} ∗ G_{S}]. Define

B = {α ∈ A_{2}: D ∩ ℙ_{α}^{sing} is a dense subset of ℙ_{α}^{sing} in W [G_{α} ∗ G_{S_{α}}] }.

We claim that B is in U. It suffices to prove that in M[G_{β} ∗ G_{S}]:

ℙℝ(β, β^{+}) ⊩ q ⊩ s ⊩ β ∈ j(B),

where q and s are the names we constructed to define U_{β}. So let G ∗ G_{tail} ∗ g be generic for ℙℝ(β, β^{+}) ∗ ℙ_{tail} ∗ j(S_{β}) such that q is in G_{tail} and t is in g. Since t is a master condition, we can extend j to

j: W [G_{β} ∗ GṠ_{β}] → M[G_{β} ∗ G_{S} ∗ G ∗ G_{tail} ∗ g].

Since j ↾ ℙ_{β}^{sing} is the identity, j(D) ∩ ℙ_{β}^{sing} = D. But D is a dense subset of ℙ_{β}^{sing} in M[G_{β} ∗ GṠ_{β}], so by the definition of B we have β ∈ j(B), and B is in U_{β}.

Since a tail of the Prikry sequence is contained in B, fix α_{n} such that D ∩ℙ_{α_{n}}^{sing} is a dense subset of ℙ_{α_{n}}^{sing} in W [G_{α_{n}} ∗ G_{S_{αn}}]. Since G_{α_{n}}^{sing} is generic, fix a in D ∩ G_{α_{n}}^{sing}. Then clearly a is in D ∩ G^{sing}. So indeed G is generic. □

Note: The proof of Lemma 6.6 is similar to a method used by Gitik in [7].

This completes the definition of the iteration. Let Ṡ be a ℙ_{κ}-name for the set of a in S(κ, κ^{+}) such that a ∩ κ is a singular ordinal with uncountable cofinality. We force with the poset ℙ_{κ} ∗ ℙ_{S} ∗ ℙ^{sing}, where ℙ^{sing} adds a club through κ ∩ Sing. Let G_{κ} ∗ G_{S} ∗ G^{sing} be generic for ℙ_{κ} ∗ ℙ_{S} ∗ ℙ^{sing} over W. Let D = ⋃ G_{S}, which is a club subset of P_{κ}κ^{+}. By definition of S, if a is in S(κ, κ^{+}) ∩ D, then a ∩ κ is either inaccessible or has cofinality ω. But κ is non-Mahlo in W [G_{κ} ∗ G_{S} ∗ G^{sing}], so there is a club of singulars in κ. It follows that for almost all a in S(κ, κ^{+}) ∩ D, a ∩ κ is singular, and therefore has cofinality ω. The only thing left to prove is that S(κ, κ^{+}) remains stationary.

Fix an elementary embedding j: W → M in W with critj = κ, j(κ) > κ^{+4}, and κ^{+4}M ⊆ M. By GCH κ is a κ^{+3}-supercompact cardinal in M, so κ is in j(A_{3}). Write

j(ℙ_{κ}) = ℙ_{κ} ∗ ℚ_{κ} ∗ ℙ_{tail}

where ℙ_{tail} is forced to be κ^{+5}-weakly closed and ℚ̇_{κ} is equivalent to ℙ_{S} ∗ ℙℝ(U_{κ}), where U_{κ} is an ultrafilter on κ which contains A_{2}. Since κ is in j(A_{3}), by Lemma 6.6 ℙ_{κ} ∗ ℙ_{S} forces that there is a projection mapping

ℙℝ(U_{κ}) → B(ℙ^{sing}).

Let G_{κ} ∗ G_{S} ∗ G^{sing} ∗ G^{∗} ∗ G_{tail} be generic over W for the poset

ℙ_{κ} ∗ ℙ_{S} ∗ ℙ^{sing} ∗ (ℙℝ(U_{κ})/B(ℙ_{sing})) ∗ ℙ_{tail}.

Let M_{1} = M[G_{κ} ∗ G_{S} ∗ G^{sing} ∗ G^{∗} ∗ G_{tail}]. Extend j to j: W [G_{κ}] → M_{1}.

We would like to further extend j to W [G_{κ} ∗ G_{S} ∗ G^{sing}]. In order to do this we construct a master condition in M_{1} for j(ℙ_{S} ∗ ℙ^{sing}). Note that j(c) = c for all c in G^{sing}. Define s_{1} by

s_{1} = ⋃ G^{sing} ∪ {κ}.

Since κ has cofinality ω in M_{1}, it is straightforward to check that s_{1} is a closed bounded subset of j(κ) ∩ Sing which end-extends j(c) = c for all c in G^{sing}. Define s_{0} as the closure under unions of increasing sequences in M_{1} of the set

⋃ j“G_{S} ∪ {u ∈ j(E): κ ⊆ u ⊆ j“κ^{+}}

Exactly the same argument as in the proof of Lemma 6.2 shows that s_{0} is a master condition in j(ℙ_{S}).

So the condition ⟨s_{0}, s_{1}⟩ is a master condition. Let g be a generic for j(ℙ_{S} ∗ℙ^{sing}) over W [G_{κ} ∗ G_{S} ∗ G^{sing} ∗ G^{∗} ∗ G_{tail}] which contains ⟨s_{0}, s_{1}⟩. Then j extends to

j: W [G_{κ} ∗ G_{S} ∗ G^{sing}] → M_{1}[g].

We show that S(κ, κ^{+}) is stationary in W [G_{κ} ∗ G_{S} ∗ G^{sing}]. Suppose that C is a club subset of P_{κ}κ^{+}. Since κ^{+W} is a cardinal in M_{1}[g], j“κ^{+} is in j(S(κ, κ^{+})). Also j“C is a directed subset of j(C) with size less than j(κ), therefore

⋃ j“C = j“κ^{+} ∈ j(C) ∩ j(S(κ, κ^{+})).

By elementarity, S(κ, κ^{+}) ∩ C is non-empty. So S(κ, κ^{+}) is stationary. This completes the proof of Theorem 6.1.

7. S(κ, κ^{+}) can be Non-Splitting

In this section we prove the following theorem.

Theorem 7.1. Suppose GCH holds and there exists a κ^{+3}-supercompact cardinal κ. Then there exists a model in which κ is strongly inaccessible and NS ↾ S(κ, κ^{+}) is κ^{+}-saturated.

We will need the following result, which is a variation of the proof due to Tryba and Woodin [13] that successors of regular cardinals are not Jonsson cardinals.

Proposition 7.2. Suppose that β is a regular cardinal. Then there is a partial function F: β^{+<ω} → β^{+} such that whenever ℚ is a forcing poset which preserves stationary subsets of β^{+}, then ℚ forces that F is Jonsson for β^{+}.

Proof. Partition β^{+} ∩ cof(β) into disjoint stationary sets ⟨S_{i}: i < β^{+}⟩. For each limit ordinal α below β^{+}, fix a set d_{α} closed and unbounded in α with order type cf(α). Define a partial function F as follows:

(1) F (γ + 1) = γ,

(2) F (ξ, ξ) is the unique i such that ξ is in S_{i},

(3) F (α, γ) = min(d_{α} \ γ) when γ < α and α is a limit ordinal.

Let G be generic for ℚ over V. In V [G] suppose that A is a subset of β^{+} with size β^{+} and is closed under F. By the assumption on ℚ, each S_{i} is still stationary in V [G]. We claim that for all i < β^{+}, the set A ∩ S_{i} is non-empty. This is enough, since if ξ is in A ∩ S_{i} then F (ξ, ξ) = i is in A, so β^{+} ⊆ A.

Since S_{i} is stationary in V [G], fix ξ in S_{i} ∩ lim(A). We show that ξ is in A. Otherwise let α be the least element of A greater than ξ. By (1), α is a limit ordinal. By (3), d_{α} is unbounded in ξ. Now o.t.(d_{α}) ≤ β, and since d_{α} ∩ ξ is a proper initial segment of d_{α}, o.t.(d_{α} ∩ ξ) < β. But d_{α} ∩ ξ is in V, so cf^{V} (ξ) < β. This is a contradiction since ξ is in S_{i} ⊆ cof^{V} (β). □

Using this proposition we can obtain some information about non-splitting subsets of P_{κ}κ^{+}.

Corollary 7.3. Suppose that S is a stationary subset of P_{κ}κ^{+} such that NS ↾ S is κ^{+}-saturated. Then there is a club set C such that:

1. S ∩ C ⊆ S(κ, κ^{+}),

2. for all a ⊊ b in S ∩ C, |a| < b ∩ κ.

Proof. Let F: κ^{+<ω} → κ^{+} be a partial function as in Proposition 7.2. Let S^{∗} be the set of a in S such that a ∩ κ is a limit cardinal, o.t.(a) = (a ∩ κ)^{+}, and F is Jonsson for a. Suppose that a and b are in S^{∗} and a ⊊ b. Then since a is closed under F and F is Jonsson for b, |a| < |b| = (b ∩ κ)^{+}. But |a| is a successor cardinal and b ∩ κ is a limit cardinal, therefore |a| < b ∩ κ. So it suffices to show that T = S \ S^{∗} is non-stationary.

Suppose for a contradiction that T is stationary. Let U be a generic filter for (NS ↾ S)^{+} with T in U. Let j: V → M be the generic elementary embedding induced by U. Since (NS ↾ S)^{+} is κ^{+}-c.c., by Proposition 7.2 the function F is still Jonsson for κ^{+} in V [U]. It is straightforward to check that this implies that j(F) is Jonsson for j“κ^{+} in M, and therefore j“κ^{+} is in j(S^{∗}). So j“κ^{+} is not in j(T), which contradicts the fact that T is in U. □

The remainder of the section will be devoted to proving Theorem 7.1 by constructing a model in which NS ↾ S(κ, κ^{+}) is κ^{+}-saturated. Start with a model V in which GCH holds and κ is a κ^{+3}-supercompact cardinal. Let W be a generic extension of V by a Radin forcing ℝ which preserves the κ^{+3}-supercompactness of κ and adds a club set C to κ with order type κ such that every α in C is regular in V. For each strongly inaccessible β < κ choose partial functions H_{β} and F_{β} in V as in Lemma 4.1 and Proposition 7.2.

Let B be the set of β < κ in lim(C) such that β is β^{++}-supercompact. We define an Easton support Prikry iteration ⟨ℙ_{i}, ℚ̇_{i}: i < κ⟩ as follows. Suppose that ℙ_{β} is defined for some β < κ. If β is not in B, then let ℚ̇_{β} be trivial.

Suppose that β is in B ∪ {κ}. Let G_{β} be generic for ℙ_{β} over W. We describe a forcing poset

ℙ_{F_{β}} ∗ ℙ^{β}

in W [G_{β}] and then define a Prikry type forcing poset ℚ_{β} which is equivalent to it. This will complete the definition of ℙ_{κ} ∗ ℙ_{F_{κ}} ∗ ℙ^{κ}. Let ℙ_{F_{β}} be the < β-distributive, β^{+}-c.c. forcing poset from Definition 4.2 for destroying the stationarity of the set of a in S(β, β^{+}) such that F_{β} is not Jonsson for a. Let G_{F} be generic for ℙ_{F_{β}} over W [G_{β}].

Working in W [G_{β} ∗ G_{F}], fix a surjective function f: β^{++} → β^{++} × β^{++} such that

∀α, i, j < β^{++} ( f(α) = ⟨i, j⟩ ⟹ i ≤ α ).

Let E be the set of a in P_{β}β^{+} which are closed under H_{β} and F_{β}. We define a < β-support forcing iteration

⟨ℙ_{α}^{β}, ℚ̇_{α}^{β}: α < β^{++}⟩

and a family of names

{Ẋ_{i}^{α}: α, i < β^{++}}

by recursion on α < β^{++}. The following properties will hold for each α: (1) ℙ_{α}^{β} is < β-distributive,

(2) Let D_{α}^{β} denote the set of q in ℙ_{α}^{β} such that for all γ in supp(q) there is x in the ground model W [G_{β} ∗ G_{F}] such that q(γ) = x̌. Then D_{α}^{β} is dense.

(3) |D_{α}^{β}| < β^{++} if α < β^{++}.

Note that D_{i}^{β} ⊆ D_{j}^{β} for i < j.

First we consider the successor case. Suppose that ℙ_{α}^{β} is defined and satisfies the required properties for a fixed α < β^{++}. Since D_{α}^{β} has size less than β^{++}, we can enumerate all canonical D_{α}^{β}-names for stationary subsets of S(β, β^{+})∩⋃ G_{F} as ⟨Ẋ_{i}^{α}: i < β^{++}⟩. Write f(α) = ⟨i, j⟩. Then i ≤ α, so Ẋ_{j}^{i} is defined. By Proposition 4.9 and the recursion hypotheses, Ẋ_{j}^{i} is of the form required to define the < βdistributive, β^{+}-c.c. poset ℙẊ_{j}^{i} from Definition 4.2 which destroys the stationarity

of Ẋ_{j}^{i}. So let ℚ̇_{α}^{β} = ℙẊ_{j}^{i}. It is straightforward to check that (1), (2), and (3) hold

for ℙ_{α+1}^{β}.

Now we consider the limit case. Suppose that α ≤ β^{++} is a limit ordinal and ℙ_{γ}^{β} is defined as required for all γ < α. Define ℙ_{α}^{β} by taking conditions with < βsupport. That is, a condition in ℙ_{α}^{β} is a sequence p with domain a subset of α with size less than β such that p ↾ γ is in ℙ_{γ}^{β} for all γ < α.

Now we verify properties (1), (2), and (3) for ℙ_{α}^{β}. First we prove (2) by showing that D_{α}^{β} is dense.

Lemma 7.4. Property (2) holds for ℙ_{α}^{β}.

Proof. Let p be a condition. If cf(α) ≥ β then there is γ < α such that supp(p) ⊆ γ. Then p is in ℙ_{γ}^{β}, so by the recursion hypotheses there is q ≤ p in D_{γ}^{β}. But then q is in D_{α}^{β}.

Suppose on the other hand that cf(α) < β. Choose an increasing sequence ⟨α_{i}: i < cf(α)⟩ unbounded in α, with α_{0} = 0. Exactly as in Proposition 4.6 construct an increasing and continuous sequence of sets ⟨N_{i}: i ≤ cf(α)⟩ closed under H and F such that for some function f: β → β^{+} in V, for all i ≤ cf(α):

(1) N_{i} ∩ β = δ_{i} is in lim(C),

(2) f“δ_{i} = N_{i} ∩ β^{+}. Choose N_{0} to contain as elements the sets ℙ_{α}^{β}, p, C, H, E, ⟨α_{i}: i < cf(α)⟩, and ⟨D_{i}^{β}: i < α⟩. We construct a decreasing sequence of conditions ⟨p_{i}: i ≤ cf(α)⟩ satisfying:

(3) ⟨p_{j}: j ≤ i⟩ is in N_{i+1},

(4) p_{i} ↾ α_{i} is in D_{α_{i}}^{β},

(5) for all γ in supp(p_{i}) ∩ α_{i}, N_{i} ∩ β^{+} = max(p_{i}(γ)),

(6) p_{i}(ξ) = p(ξ) for all α_{i} ≤ ξ < α.

Suppose that p_{i} is given satisfying the requirements. Let p_{i}^{∗} be the <_{θ}-least extension of p_{i} ↾ α_{i+1} in D_{α_{i+1}}^{β}. Define p_{i+1} as follows. For each ξ in the support of p_{i}^{∗}, define

p_{i+1}(ξ) = p_{i}^{∗}(ξ) ∪ {u ∈ E: N_{i+1} ∩ β ⊆ u ⊆ N_{i+1} ∩ β^{+}}.

For each ξ in supp(p_{i}) \ α_{i+1}, let p_{i+1}(ξ) = p_{i}(ξ). It is straightforward to check that p_{i+1} satisfies the inductive requirements.

Suppose that γ ≤ cf(α) is a limit ordinal and p_{i} is defined for all i < γ. Define p_{γ} as follows. The support of p_{γ} is ⋃{supp(p_{i}): i < γ}. Let ξ be in this support. If ξ ≥ α_{γ} then let p_{γ}(ξ) = p(ξ). Otherwise there exists i < γ such that ⟨p_{j}(ξ): i ≤ j < γ⟩ is a sequence of canonical names for sets in the ground model. Apply Lemma 4.5 to this sequence to obtain p_{γ}(ξ).

This completes the construction. The condition p_{cf(α)} refines p and is in D_{α}^{β}. Therefore property (2) holds. □

Now we verify property (1) by showing that ℙ_{α}^{β} is β-distributive. This proof is basically the same as the proof of property (2).

Lemma 7.5. Property (1) holds for ℙ_{α}^{β}.

Proof. Suppose that ⟨D_{i}: i < δ⟩ is a sequence of less than β many dense subsets of ℙ_{α}^{β}. Suppose that p is a condition. Choose an increasing and continuous sequence ⟨N_{i}: i ≤ δ⟩ of sets closed under H and F such that for all i ≤ δ:

(1) N_{i} ∩ β = δ_{i} is in lim(C),

(2) f“δ_{i} = N_{i} ∩ β^{+}. Choose N_{0} to contain as elements the sets ℙ_{α}^{β}, p, C, H, E, ⟨α_{i}: i < cf(α)⟩, and ⟨D_{i}: i < δ⟩. We construct a decreasing sequence of conditions ⟨p_{i}: i ≤ δ⟩ satisfying:

(3) ⟨p_{j}: j ≤ i⟩ is in N_{i+1},

(4) p_{i+1} is in D_{α}^{β} ∩ D_{i},

(5) for all γ in supp(p_{i}), N_{i} ∩ β^{+} = max(p_{i}(γ)). The construction is straightforward and uses Lemma 4.5 for the limit stages. □

For property (3), first suppose that cf(α) ≥ β. In this case D_{α}^{β} = ⋃{D_{i}^{β}: i < α}, so clearly |D_{α}^{β}| < β^{++} if α < β^{++}. Otherwise fix an increasing sequence ⟨α_{i}: i < cf(α)⟩ unbounded in α. Then D_{α}^{β} is the set of p in ℙ_{α}^{β} such that p ↾ α_{i} is in D_{α_{i}}^{β} for each i. Since each |D_{α_{i}}^{β}| < β^{++}, |D_{α}^{β}| ≤ (β^{+})^{cf(α)} = β^{+}.

This completes the definition of the forcing iteration and the proof of properties (1), (2), and (3). Let ℙ^{β} = ℙ_{β^{++}}^{β} and D^{β} = ⋃{D_{i}^{β}: i < β^{++}}. Then ℙ_{F_{β}} ∗ ℙ^{β} is < β-distributive and has size β^{++}, and D^{β} is dense in ℙ^{β}.

Lemma 7.6. The poset ℙ^{β} is β^{+}-c.c.

Proof. Suppose for a contradiction that ℙ^{β} is not β^{+}-c.c. Then there is an antichain ⟨p_{i}: i < β^{+}⟩ in D^{β}. Without loss of generality, for each i and for each α in the support of p_{i}, p_{i}(α) = x̌_{α}^{i} for some x_{α}^{i} in W [G_{β} ∗ G_{F}]. Applying the Δ-system lemma we can assume that there is d ⊆ β^{++} with size less than β such that supp(p_{i}) ∩ supp(p_{j}) = d for i < j. Let a_{i} = ⋃{max(x_{α}^{i}): α ∈ supp(p_{i})}. Again by the Δ-system lemma we can assume there is e ⊆ β^{+} with size less than β such that a_{i} ∩ a_{j} = e for i < j. Now there are less than β many possibilities for a sequence ⟨P(e) ∩ x_{α}^{i}: α ∈ d⟩, so we can assume that all such sequences are equal for different values of i.

Consider any i < j. Since p_{i} and p_{j} are incompatible, then exactly as in the proof of Proposition 4.7 there is α in d and (without loss of generality) a in x_{α}^{i} \ x_{α}^{j} with a ⊆ max x_{α}^{j}. But then a ⊆ a_{i} ∩ a_{j} = e. So a is in P(e) ∩ x_{α}^{i}, therefore a is in P(e) ∩ x_{α}^{j} ⊆ x_{α}^{j}, which is a contradiction. □

Therefore ℙ^{β} has no more than β^{++}-many antichains. By Lemma 2.3 there exists a β-weakly closed Prikry type forcing poset ℚ_{β} which satisfies the direct extension property and is equivalent to ℙ_{F_{β}} ∗ ℙ^{β}.

This completes the definition of ℙ_{κ} ∗ ℙ_{F_{κ}} ∗ ℙ^{κ}. Let F = F_{κ}. We will now define a complete suborder ℙ_{F} ∗ ℙ_{A}^{κ} of ℙ_{F} ∗ ℙ^{κ} and prove that this suborder forces that NS ↾ S(κ, κ^{+}) is κ^{+}-saturated.

In order to define ℙ_{A}^{κ}, we need to consider a supercompact elementary embedding. Let j: W → M be an elementary embedding with crit(j) = κ, j(κ) > κ^{+3}, and κ^{+3}M ⊆ M. Then κ is in j(B), so write

j(ℙ_{κ}) = ℙ_{κ} ∗ ℚ̇_{κ} ∗ ℙ_{tail}

where ℚ̇_{κ} is equivalent to ℙ_{F} ∗ ℙ^{κ}. Let G_{κ} be generic for ℙ_{κ} over W. In the model W [G_{κ}], let ⟨Ẋ_{i}: i < κ^{++}⟩ be the sequence of names defined by Ẋ_{i} = Ẋ_{β}^{α}, where f(i) = ⟨α, β⟩ in the definition of ℙ^{κ}. So for all i < κ^{++}, ℙ_{F} ∗ ℙ_{i}^{κ} forces that ℚ̇_{i}^{κ} = ℙẊ_{i}.

First let us consider how to extend j by constructing a master condition for ℙ_{F}. Let G_{F} ∗ G^{κ} ∗ G_{tail} be generic for ℙ_{F} ∗ ℙ^{κ} ∗ ℙ_{tail} over W [G_{κ}]. Extend j to

j: W [G_{κ}] → M[G_{κ} ∗ G_{F} ∗ G^{κ} ∗ G_{tail}].

Write M_{1} = M[G_{κ} ∗ G_{F} ∗ G^{κ} ∗ G_{tail}]. We define a master condition t_{F} for G_{F} and j. Let

t_{F} = ⋃ j“G_{F} ∪ {u ∈ j(E): κ ⊆ u ⊆ j“κ^{+}}.

Since ⋃ G_{F} is a club subset of P_{κ}κ^{+} in M[G_{κ} ∗ G_{F}], by the distributivity of ℙ^{κ} and the weak closure of ℙ_{tail}, ⋃ j“G_{F} is a club subset of P_{κ}(j“κ^{+}) in M_{1}. So t_{F} is closed under unions of increasing sequences. Also ℝ ∗ j(ℙ_{κ}) does not destroy stationary subsets of κ^{+}, since ℝ is κ^{+}-c.c. and j(ℙ_{κ}) = ℙ_{κ+1} ∗ ℙ_{tail}, where ℙ_{κ+1} is κ^{+}-c.c. and ℙ_{tail} does not add subsets to κ^{+}. So by Proposition 7.2, j(F) is Jonsson for j“κ^{+} in M_{1}. But every member of t_{F} ∩ j(S(κ, κ^{+})) which is not in ⋃ j“G_{F} must be unbounded in j“κ^{+} and hence by the Jonsson property is equal to j“κ^{+}. Therefore j(F) is Jonsson for a for every a in t_{F} ∩ j(S(κ, κ^{+})). So t_{F} is a master condition.

Still working in M_{1}, fix a regular θ ≫ j(κ) and a well-ordering <_{θ} of H(θ). As in Proposition 4.6, construct an increasing sequence ⟨N_{i}: i < j(κ)⟩ of elementary substructures of H(θ) in P_{j(κ)}H(θ) such that for some function f: j(κ) → j(κ^{+}) in V:

(1) N_{i} ∩ j(κ) = δ_{i} is in j(lim(C)),

(2) N_{i} ∩ j(κ^{+}) = f“δ_{i},

(3) ⟨N_{j}: j ≤ i⟩ is in N_{i+1}. Choose N_{0} to contains the sets j(ℙ_{κ}), G_{F}, G^{κ}, j(F), j(H), j(E), t_{F}, and ⟨j(Ẋ_{i}): i < κ^{++}⟩. For each i < κ^{++} let G_{i}^{κ} be the generic filter for ℚ_{i}^{κ} given by G^{κ}. Now go back to W [G_{κ} ∗G_{F}] and identify the objects just defined with their ℙ^{κ} ∗ℙ_{tail}-names.

Working in W [G_{κ} ∗ G_{F}] we define a complete suborder ℙ_{A}^{κ} of ℙ^{κ} by inductively defining a < κ-support forcing iteration

⟨ℙ_{A_{i}}^{κ}, ℚ̇_{A_{i}}^{κ}: i < κ^{++}⟩.

For each i, A_{i+1} is a ℙ_{A_{i}}^{κ}-name for a subset of i + 1, and ℚ̇_{A_{i}}^{κ} is non-trivial iff i is in A_{i+1}. If ℚ̇_{A_{i}}^{κ} is non-trivial then ℚ̇_{A_{i}}^{κ} = ℙẊ_{i}. Let D_{A_{i}}^{κ} = ℙ_{A_{i}}^{κ} ∩ D_{i}^{κ}. Then D_{A_{i}}^{κ} is dense in ℙ_{A_{i}}^{κ} and there is a projection mapping

D_{i}^{κ} → D_{A_{i}}^{κ}.

Since the posets are < κ-distributive, the poset ℙẊ_{i} is the same in the generic extensions by ℙ_{i}^{κ} and ℙ_{A_{i}}^{κ}. At the same time we construct sequences

⟨ṡ_{i} ẋ_{i}: i ≤ κ^{++}⟩

and

⟨q̇_{i}: i ≤ κ^{++}⟩.

Let D^{κ} = D_{κ^{++}}^{κ} = ⋃{D_{i}^{κ}: i < κ^{++}}. Our induction hypothesis is that ℙ^{κ} forces: (1) q_{i} is in ℙ_{tail}, and q_{j} ≤^{∗} q_{i} for all i < j,

(2) ℙ_{tail} forces that ṡ_{i} ẋ_{i} is in j(ℙ_{F} ∗ ℙ_{A_{i}}^{κ}), ẋ_{i} is in j(D_{A_{i}}^{κ}), and ṡ_{j} ẋ_{j} ≤ ṡ_{i} ẋ_{i} for i < j,

(3) ℙ_{tail} forces that ⟨ṡ_{j} ẋ_{j}: j ≤ i⟩ is in N_{i+1},

(4) ℙ_{tail} forces that for all α in supp(ẋ_{i}), max(ẋ_{i}(α)) = N_{i} ∩ j(κ^{+}), and also max(ṡ_{i}) = N_{i} ∩ j(κ^{+}).

Let A_{0} = ∅ and let q̇_{0}, ẋ_{0} be names for the maximum conditions in ℙ_{tail} and j(ℙ^{κ}). Let ṡ_{0} = ṫ_{F}. Suppose that i < κ^{++} and A_{i}, ⟨ṡ_{0} ẋ_{j}: j ≤ i⟩, and ⟨q̇_{j}: j ≤ i⟩ are defined satisfying the induction hypotheses. Consider the name Ẋ_{i}. If this is not a ℙ_{A_{i}}^{κ}-name then let A_{i+1} = A_{i} and let q̇_{i+1} = q̇_{i}. Let ẋ_{i+1} be a name for the condition with the same support as ẋ_{i} such that for all α in supp(ẋ_{i}),

ẋ_{i+1}(α) = ẋ_{i}(α) ∪ {u ∈ j(E): N_{i+1} ∩ j(κ) ⊆ u ⊆ N_{i+1} ∩ j(κ^{+})},

and similarly define ṡ_{i+1}. Since |N_{i+1} ∩ j(κ^{+})| = N_{i+1} ∩ j(κ), ẋ_{i+1}(α) is disjoint from j(Ẋ_{α}). Using the induction hypotheses it is easy to check that ṡ_{i+1} ẋ_{i+1} is a condition in j(ℙ_{F} ∗ ℙ_{A_{i+1}}^{κ}) which refines ṡ_{i} ẋ_{i}.

Suppose that Ẋ_{i} is a ℙ_{A_{i}}^{κ}-name. By the induction hypothesis ṡ_{i} ẋ_{i} is in j(ℙ_{F} ∗ ℙ_{A_{i}}^{κ}), so let ṡ_{i}^{∗} ∗ ẋ_{i}^{∗} be a name for the <_{θ}-least refinement of ṡ_{i} ẋ_{i} in j(ℙ_{F} ∗ ℙ_{A_{i}}^{κ}) such that ẋ_{i} is in j(D_{A_{i}}^{κ}) and ṡ_{i}^{∗} ẋ_{i}^{∗} decides the statement j“κ^{+} ∈ j(Ẋ_{i}). Choose a name q̇_{i+1} for a direct extension of q̇_{i} such that q̇_{i+1} decides the statement

ṡ_{i}^{∗} ∗ ẋ_{i}^{∗} ⊩ j“κ^{+} ∈ j(Ẋ_{i}).

Let A_{i+1} be a ℙ_{A_{i}}^{κ}-name for the set which is equal to A_{i} unless

ℙ^{κ}/ℙ_{A_{i}}^{κ} ⊩ q̇_{i+1} ⊩ ṡ_{i}^{∗} ⊩ ẋ_{i}^{∗} ⊩ j“κ^{+} ∉ j(Ẋ_{i}),

in which case A_{i+1} = A_{i} ∪ {i}. Now define a ℙ^{κ} ∗ ℙ_{tail}-name ṡ_{i+1} ∗ ẋ_{i+1} as follows. If i is not in A_{i+1} then define ẋ_{i+1} as in the previous case, namely, as a name for the condition with the same support as ẋ_{i} and for all α in supp(ẋ_{i}),

ẋ_{i+1}(α) = ẋ_{i}(α) ∪ {u ∈ j(E): N_{i+1} ∩ j(κ) ⊆ u ⊆ N_{i+1} ∩ j(κ^{+})};

define ṡ_{i+1} similarly. If i is in A_{i+1}, then define

t_{i} = ⋃ j“G_{i}^{κ} ∪ {u ∈ j(E): κ ⊆ u ⊆ j“κ^{+}}.

Since s_{i}^{∗} ∗ x_{i}^{∗} forces that j“κ^{+} ∉ j(Ẋ_{i}) and j(F) is Jonsson for j“κ^{+}, s_{i}^{∗} ∗ x_{i}^{∗} forces that t_{i} is a condition in j(ℙẊ_{i}). Let ṡ_{i+1} ∗ x_{i+1} be obtained by taking the condition ṡ_{i}^{∗} ∗ x_{i}^{∗} t_{i} and adding to each coordinate the elements of

{u ∈ j(E): N_{i+1} ∩ j(κ) ⊆ u ⊆ N_{i+1} ∩ j(κ^{+})}.

Suppose that δ ≤ κ^{++} is a limit ordinal. Let A_{δ} denote ⋃{A_{i}: i < δ}. Since ⟨q_{i}: i < δ⟩ is forced to be ≤^{∗}-decreasing in the j(κ)-weakly closed poset ℙ_{tail}, we can choose a name q̇_{δ} for a condition in ℙ_{tail} which directly refines each q̇_{i}. By the inductive requirements of ṡ_{i} ẋ_{i} for i < δ, apply Lemma 4.5 to obtain ṡ_{δ} ∗ ẋ_{δ} refining each ṡ_{i} ∗ ẋ_{i}.

This completes the definition of ℙ_{A}^{κ}. Let ṡ_{F} = ṡ_{κ^{++}}, ẋ = ẋ_{κ^{++}} and q̇ = q̇_{κ^{++}}. It is straightforward to check that ṡ_{F} ∗ ẋ is forced to be a master condition for G_{F} ∗ G_{A}^{κ} and j. Let D_{A}^{κ} = D^{κ} ∩ ℙ_{A}^{κ}. Note that D_{A}^{κ} is a dense subset of ℙ_{A}^{κ}.

Suppose that G_{A}^{κ} is generic for ℙ_{A}^{κ} over W [G_{κ} ∗ G_{F}]. By the properties of ℙ_{A}^{κ}, any set X ⊆ P_{κ}κ^{+} in W [G_{κ} ∗ G_{F} ∗ G_{A}^{κ}] is in W [G_{κ} ∗ G_{F} ∗ G_{A_{i}}^{κ}] for some i < κ^{++}. Using this fact it is straightforward to check that if X is a subset of S(κ, κ^{+}) in W [G_{κ} ∗ G_{F} ∗ G_{A}^{κ}], then there is i such that X = X_{i}, Ẋ_{i} is a D_{i}^{κ}-name, and if X is non-stationary then we can choose such an i so that X is non-stationary in W [G_{κ} ∗ G_{F} ∗ G_{A_{i}}^{κ}].

Lemma 7.7. The poset ℙ_{F} ∗ ℙ_{A}^{κ} forces that NS ↾ S(κ, κ^{+}) is κ^{+}-saturated.

Proof. Let G_{F} ∗ G_{A} be generic for ℙ_{F} ∗ ℙ_{A}^{κ} over W [G_{κ}]. For any ℙ_{A}^{κ}-name Ẋ, let Φ(Ẋ) be the statement

q̇ ⊩ ṡ_{F} ⊩ ẋ ⊩ j“κ^{+} ∉ j(Ẋ),

First we show that for every set X ⊆ S(κ, κ^{+}) in W [G_{κ} ∗ G_{F} ∗ G_{A}], X is in NS ↾ S(κ, κ^{+}) iff there is a ℙ_{A}^{κ}-name Ẋ for X such that ℙ^{κ}/ℙ_{A}^{κ} forces Φ(Ẋ). Suppose that ℙ^{κ}/ℙ_{A}^{κ} forces Φ(Ẋ). Fix p in G_{A}^{κ} such that p ∗ 1 forces Φ(Ẋ) in ℙ^{κ}. Then we can choose Ẏ such that ℙ^{κ} forces Φ(Ẏ) and p ∗ 1 forces Ẋ = Ẏ. Therefore Ẏ is a name for X. Fix i such that Ẋ_{i} is a D_{A_{i}}^{κ}-name and ⊩ Ẏ = Ẋ_{i}. Then clearly i is in A_{i+1} and so ℚ̇_{A_{i}}^{κ} = ℙẎ_{i}. So ℙ_{A}^{κ} destroys X and X is non-stationary. The other direction is straightforward.

If X is in (NS ↾ S(κ, κ^{+}))^{+} then there is p_{X} in ℙ^{κ}/ℙ_{A}^{κ} such that

p_{X} ⊩ q̇ ⊩ ṡ_{F} ⊩ ẋ ⊩ j“κ^{+} ∈ j(Ẋ).

Suppose that X ∩ Y is disjoint from a club D. If r ≤ p_{X}, p_{Y} then

r ⊩ q̇ ⊩ ṡ_{F} ⊩ ẋ ⊩ j“κ^{+} ∈ j(Ẋ) ∩ j(Ẏ) ∩ j(D)

which is a contradiction. So p_{X} and p_{Y} are incompatible. Since ℙ^{κ}/ℙ_{A}^{κ} is κ^{+}-c.c., NS ↾ S(κ, κ^{+}) is κ^{+}-saturated. □

References

[1] U. Abraham and S. Shelah. Forcing closed unbounded sets. J. Symbolic Logic, 48(3):643–657, 1983.

[2] J. Baumgartner. Iterated forcing. In Surveys in Set Theory. Cambridge Univ. Press, 1983.

[3] D. Burke. Generic embeddings and the failure of box. Proc. Amer. Math. Soc., 123(9):2867– 2871, 1995.

[4] Donder, Koepke, and Levinski. Some stationary subsets of P_{κ}λ. Proc. Amer. Math. Soc., 102(4):1000–1004, 1988.

[5] M. Gitik. Introduction to prikry type forcing notions. To appear.

[6] M. Gitik. Nonsplitting subset of P_{κ}κ^{+}. J. Symbolic Logic, 50(4):881–894, 1985.

[7] M. Gitik. Changing cofinalities and the nonstationary ideal. Israel J. Math, 56(3):280–314, 1986.

[8] A. Kanamori. The Higher Infinite. Springer, 1994.

[9] J. Krueger. Adding clubs with square. To appear.

[10] J. Krueger. Strong compactness and stationary sets. To appear.

[11] M. Magidor. How large is the first strongly compact cardinal? Annals of Mathematical logic, 10:33–57, 1976.

[12] E. Schimmerling and M. Zeman. Square in core models. Bull. Symbolic Logic, 7(3):305–314, 2001.

[13] J. Tryba. On Jonsson cardinals with uncountable cofinality. Israel J. Math, 49(4):315–324, 1984.

Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währingerstrasse 25, 1090 Vienna, Austria

   E-mail address: jkrueger@logic.univie.ac.at
   URL: http://www.logic.univie.ac.at/∼jkrueger