Diamonds on large cardinals

Alex Hellsten November 13, 2003

Annales Academiæ Scientiarum Fennicæ Mathematica Dissertationes 134

Part of academic dissertation 2000 Mathematics Subject Classification: Primary: 03E05; Secondary 03E35, 03E55.

Academia Scientiarum Fennica

ISSN 1239-6303 ISBN 951-41-0929-5

Acknowledgements

I am grateful for having had the opportunity to work under the supervision of Professor Jouko Väänänen. He suggested to me the subject that was to be the starting point of this work that constitutes a part of my doctoral dissertation.

I would also like to thank all the other members of the Helsinki Logic Group, who have contributed to a pleasant and inspiring working environment. Especially I wish to thank Docent Tapani Hyttinen for always taking the time to discuss and giving his guidance.

Finally I wish to thank Professor Boban Veličković and Docent Kerkko Luosto for carefully reading the manuscript and suggesting many improvements.

Helsinki, November 2003 Alex Hellsten

Contents

1. Introduction 5

2. Ideals over regular cardinals 6 2.1. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2. Saturated ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3. Indescribability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4. n-closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3. Operations on ideals 15 3.1. Diamond principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2. Subtlety and ineffability . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3. Subtlety and diamonds . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4. Weak compactness 22 4.1. Some notes on ultraproducts . . . . . . . . . . . . . . . . . . . . . . . 22 4.2. Weakly compact diamond . . . . . . . . . . . . . . . . . . . . . . . . 25

5. Weak Compactness in generic extensions 26 5.1. Forcing preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2. Weakly compact diamond by forcing . . . . . . . . . . . . . . . . . . 30 5.3. Killing a weakly compact set . . . . . . . . . . . . . . . . . . . . . . . 34 5.4. Killing weakly compact diamond . . . . . . . . . . . . . . . . . . . . 36

References 47

1. Introduction

Let κ be a weakly compact cardinal. Consider the sets of the form {α < κ : ⟨V_{α}, ϵ, U ∩V_{α}⟩ ⊧ ¬ϕ}, where U ⊆ V_{κ} and ϕ is a Π_{1}^{1}-sentence such that ⟨V_{κ}, ϵ, U⟩ ⊧ ϕ. This collection of sets generates a normal ideal over κ, a proper extension of the nonstationary ideal over κ. Sets of positive measure with respect to this ideal are called weakly compact. Thus every weakly compact subset of κ is stationary but not vice versa.

The following combinatorial principle was defined by Sun in [18] and independently by Shelah in [16]: There exists a sequence (A_{α} : α < κ) such that

{α < κ : A ∩ α = A_{α}}

is weakly compact for every set A. This strengthening of the classical diamond principle can be referred to as the weakly compact diamond. It is applied in [16] where Shelah and Väänänen show that weakly compact diamond together with the assumption 2^{κ} = κ^{+} implies that the logic ℒ_{κκ} has no strongest extension with certain Löwenheim-Skolem and compactness properties.

In [16] it was stated without proof that weakly compact diamond holds for every measurable cardinal, holds in the constructible universe for every weakly compact cardinal, and can be obtained through forcing. After providing proofs of these three claims we discovered that Sun had independently proved the first two claims in [18]. Indeed in both proofs of the first claim, measurability is replaced with ineffability, a considerably weaker assumption. However this work initiated a broader study of normal ideals over regular cardinals and related diamond principles with the main focus on weak compactness.

In Section 2.4 the concept of an n-club is defined and it is proved that the n-club sets generate the Π_{n}^{1}-filter, provided that it is proper. The Π_{1}^{1}-filter is the dual of the weakly compact ideal. In the light of the results presented in this paper it can be argued that the n-clubs are a rather natural and canonical generalisation of closed unbounded sets. In Section 5.3 it is shown that a weakly compact set may be killed using forcing, by shooting a 1-club through its complement. It is interesting to note that the weak compactness of the complement of the set to be killed is the only assumption necessary for this argument. In this respect the weakly compact ideal behaves like the non-stationary ideal on ℵ_{1}.

In Section 3.1 a general definition for ideals like ND_{ℵ_{1}}, the “no diamond” ideal, is provided and some basic facts about these ideals are presented. Then the facts about ineffability and the axiom of constructibility implying weakly compact diamond are proved in a slightly more general form in sections 3.3 and 4.2.

Forcing arguments are developed in Section 5. Weakly compact diamond holds in many forcing extensions, but it can also be killed using a forcing notion that preserves weak compactness. The consistency of the failure of weakly compact diamond on a weakly compact cardinal follows from a result by Hauser [8].

This monograph is part of the doctoral dissertation of the author, which also includes the paper [9].

2. Ideals over regular cardinals

An ideal over a regular cardinal κ is a nonempty collection I ⊆ 𝒫(κ) which is closed under finite unions and subsets. The trivial ideal over κ is {∅} and an ideal is called proper if it is not 𝒫(κ). Thus an ideal I is proper if and only if κ ∉ I. The trivial ideal is a special case of a principal ideal; an ideal of the form {X ⊆ κ : X ∩ A = ∅} where A is some fixed subset of κ. The collection [κ]^{<κ} is sometimes called the ideal of small sets and it is proper and non-principal.

For W ⊆ 𝒫(κ) we let W^{∗} denote the collection {X ⊆ κ : κ \ X ∈ W }. A collection F ⊆ 𝒫(κ) is a filter over the regular cardinal κ if F^{∗} is an ideal over κ. If this is the case, I = F^{∗} will be called the dual ideal of F and F = I^{∗} is called the dual filter of I. All properties of ideals have their analogues for filters, which can be thought of to be defined via the dual. Thus a principal filter is the dual of a principal ideal and so forth. Also, if we find it more convenient to define some concept for filters, then it is tacitly meant to be defined on ideals too, via the dual.

A subset of κ which is not in the ideal I is said to have positive measure with respect to I. Thus the complement 𝒫(κ) \ I is the collection of sets of positive measure, and it is denoted I^{+}. If F is a filter, F^{+} means (F^{∗})^{+}. Likewise we can say that sets in the ideal have measure 0 and sets in the dual filter have measure 1. But note that this makes sense only for proper ideals, since an ideal I is proper if and only if the dual filter is contained in I^{+}, or equivalently, I is disjoint from its dual.

The following observations are trivial, but we want to state them explicitly since they are used tacitly, it seems, in virtually every proof that concerns itself with ideals. They can also be seen as a motivation for the measure theoretic terminology.

(a) A set has positive measure if and only if it intersects every set in the filter.

(b) A set is in the filter if and only if it intersects every set of positive measure.

(c) If E has positive measure and X is in the filter, then E ∩ X has positive measure.

Note that this holds regardless of whether the ideal in question is proper, which highlights our convention that there are no sets of positive measure with respect to the non-proper ideal over κ.

We shall only be interested in proper ideals that extend the ideal of small sets. These are necessarily non-principal. From now on we shall include the condition [κ]^{<κ} ⊆ I in the definition of I being an ideal over a regular cardinal κ. We would also like to include κ ∉ I in the conditions, but we shall frequently encounter explicit definitions of ideals which may not be proper in all situations. And some simple facts can be more conveniently stated if 𝒫(κ) is considered to be an ideal. Thus we shall take the following standpoint. In some contexts ideals are taken to be proper by definition, and in some contexts not. Usually it is evident which of the two “definitions” is meant.

2.1. Completeness

Let (X_{α} : α < κ) be a sequence of subsets of κ. The diagonal intersection △_{α<κ}X_{α} is the set

⋂_{α<κ}(X_{α} ∪ [α, κ)) = {β < κ : β ∈ ⋂_{α<β} X_{α}}

and the diagonal union ▽_{α<κ}X_{α} is the set

⋃_{α<κ}(X_{α} ∩ [0, α)) = {β < κ : β ∈ ⋃_{α<β} X_{α}}.

Since ▽_{α<κ}X_{α} = κ\(△_{α<κ}(κ\X_{α})), a collection W ⊆ 𝒫(κ) is closed under diagonal intersections if and only if W^{∗} is closed under diagonal unions.

Let I be an ideal over κ and let μ be another regular cardinal. The ideal I is said to be μ-complete if it is closed under unions of cardinality less than μ. As is customary, ℵ_{1}-completeness is referred to as σ-completeness or countable completeness. An ideal is normal if it is closed under diagonal unions.

2.1.1. Lemma. If a collection W ⊆ 𝒫(κ) is closed under subsets, diagonal unions, and the operations X ↦ X ∪ α for α < κ, then W is a normal κ-complete ideal.

Proof. We only need to verify that W is closed under unions of cardinality less than κ. But this is clear since ⋃_{ξ<α} X_{ξ} ⊆ (▽_{ξ<κ}X_{ξ}) ∪ α for any ordinal α < κ and sequence (X_{ξ} : ξ < κ) of subsets of κ. □

The canonical example of a normal ideal is the ideal of non-stationary sets, i.e. the collection of subsets of κ that are disjoint from some closed unbounded subset of κ. We shall denote this ideal by NS_{κ}. In fact NS_{κ} is the least normal ideal over κ. This is because any closed unbounded set C may be written as △_{α<κ}[min(C ∩ (α, κ)), κ), a diagonal intersection of final segments.

It is clear that an ideal over κ can not be μ-complete for any μ > κ. By Lemma 2.1.1 it follows that every normal ideal over κ is κ-complete. Thus one can say that normality is a stronger requirement than μ-completeness for any relevant μ. The facts discussed above depend on our convention that all ideals extend the ideal of small sets by definition. This is the main motivation for having this convention.

A function f from a set of ordinals to the ordinals is regressive if f(α) < α for every α ∈ dom(f) \ {0}. The connection between normal ideals and regressive functions is the following:

2.1.2. Lemma. Suppose that W ⊆ 𝒫(κ) is closed under subsets. The following conditions are equivalent:

(i) W is closed under diagonal unions

(ii) For every X ⊆ κ and f : X → κ, if f is a regressive function such that f^{−1}{α} ∈ W for every α < κ then X ∈ W.

Proof. Suppose that W is closed under diagonal unions and f : X → κ is regressive. If β ∈ X then β ∈ f^{−1}{f(β)} and thus X ⊆ ▽_{α<κ}f^{−1}{α}. It follows that X must be in W if f^{−1}{α} ∈ W for every α < κ.

For the other direction fix a sequence (X_{α} : α < κ) of sets in W and let X = ▽_{α<κ}X_{α}. It is straightforward to define a regressive function f : X → κ such that β ∈ X_{f(β)} for every β ∈ X. Now f^{−1}{α} ⊆ X_{α} ∈ W for every α < κ. □

Condition (ii) for NS_{ℵ_{1}} in the previous lemma is the classical Fodor’s lemma.

2.2. Saturated ideals

Let I be an ideal. Two sets X and Y in I^{+} are said to be almost disjoint with respect to I if X ∩ Y ∈ I. Let μ be a regular or finite cardinal. The ideal I is μsaturated if every subcollection of I^{+} of pairwise almost disjoint sets has cardinality less than μ. The least μ such that I is μ-saturated is denoted sat I.

Consider I^{+} to be ordered by inclusion. Then I is μ-saturated if and only if I^{+} satisfies the μ-chain condition in the sense of the standard definition for forcing notions.

A 2-saturated ideal is a prime ideal. As an exception to the convention that the same terminology is used for analogous properties of filters, the dual filter of a prime ideal is an ultrafilter. For a prime ideal I over κ it holds that I ∪ I^{∗} = 𝒫(κ) and thus a prime ideal is a maximal ideal.

For an ideal I over κ and a set E ⊆ κ we let I|E denote the collection {X ⊆ κ : X ∩ E ∈ I}. It is straightforward to see that I|E is an ideal with the same closure properties that I satisfies, i.e. if I is μ-complete then I|E is μ-complete and if I is normal then I|E is normal. Note that I|E is the ideal generated by the set I ∪ {κ \ E}. A basic observation is that

(a) I ⊆ I|E

(b) I|E = I if and only if E ∈ I^{∗}

(c) I|E is proper if and only if E ∉ I.

For the sake of completeness we wish to present the following two lemmas that are due to Baumgartner, Taylor, and Wagon [4].

2.2.1. Lemma. Let I and J be ideals over a regular cardinal κ such that I ⊆ J, and let μ be a regular cardinal. If either one of the conditions

(i) I is μ-saturated and J is μ-complete

(ii) I is κ^{+}-saturated and J is normal

hold, then J = I|E for some E ⊆ κ.

Proof. Assume that I is μ-saturated and suppose that J is μ-complete unless μ = κ^{+} in which case we assume that J is normal. Let {X_{i} : i < γ} be a maximal collection of sets in J \ I that are pairwise almost disjoint with respect to I. Since I is μ-saturated, γ < μ. Let S = ▽_{i<κ}X_{i} if γ = κ or else let S = ⋃_{i<γ} X_{i}. Let E = κ \ S. We shall show that J = I|E.

By our assumptions S ∈ J and thus it is clear that I|E ⊆ J. It is straightforward to see that E∩X_{i} ⊆ i+1 for every i < γ (in fact E∩X_{i} = ∅ in the case S = ⋃_{i<γ} X_{i}.) Let X ∈ J \ I be arbitrary. Since X ∩ E ∩ X_{i} ⊆ E ∩ X_{i} ∈ I, the set X ∩ E is almost disjoint from X_{i} for every i < γ. By the maximality of {X_{i} : i < γ} we must have X ∩ E ∈ I. □

Note that the conclusion J = I|E of the lemma above can be thought of as meaning that J has a maximal element with respect to I. Namely S ∈ J is maximal in the sense that X \ S ∈ I for every X ∈ J if and only if J = I|(κ \ S).

2.2.2. Lemma. Let I be an ideal over a regular cardinal κ and let μ be another regular cardinal.

(a) If I is μ-complete but not μ-saturated then I can be extended to a μ-complete ideal which is not of the form I|E.

(b) If I is normal but not κ^{+}-saturated then I can be extended to a normal ideal which is not of the form I|E.

Proof. Let {X_{i} : i < μ} be a subcollection of 𝒫(κ) and suppose that X_{i} ∩X_{j} ∈ I whenever i ≠ j. Let K be an ideal over μ and consider the collection J of all X ⊆ κ such that

{i < μ : X ∩ X_{i} ∉ I} ∈ K.

It is straightforward to check that J is an ideal and that I ∪ {X_{i} : i < μ} ⊆ J. Clearly J is a proper ideal if and only if {i < μ : X_{i} ∉ I} ∉ K. If J ⊆ I|E for some E ⊆ κ then E ∈ J directly by the definition and the fact that {X_{i} : i < μ} ⊆ I|E. Thus if J is a proper ideal it can not be of the form I|E.

For (a) let K = [μ]^{<μ}. Then since both I and K are μ-complete, also J must be μ-complete. Using the fact that I is not μ-saturated we can assume that X_{i} ∉ I for every i < μ. Collecting the facts stated above, we have a proof of (a). For (b) we still use K = [μ]^{<μ} but with μ = κ^{+}. Again it is not hard to verify that J is normal because I is normal and K is κ^{+}-complete. □

2.3. Indescribability

We shall be dealing with higher order formulae in an extended language of set theory that in addition to the relation symbol ∈ may include a finite number of unary predicate symbols and binary relation symbols. The higher order variables are always unary.

We shall often neglect to explicitly state the language we are using, but instead a statement like ⟨V_{κ}, ∈, U⟩ ⊧ ϕ is tacitly expressing that ϕ is in the language consisting of ∈ and one unary predicate symbol U. In fact any finite number of predicates and relations may be coded into one unary predicate using a first order definable coding. Thus we shall always use only one unary predicate unless some other language is motivated by notational convenience.

A formula is always equivalent with a formula in prenex normal form in which the quantifiers of the highest order are all collected in the beginning of the formula. Furthermore adjacent quantifiers of the same kind and order may be contracted into one, by coding the two variables into one by a first order definable coding.

Suppose that quantifiers of the highest order appearing in a formula ϕ have order p + 1. Let us assume that the quantifiers of order p + 1 are all collected in the beginning of ϕ and that they alternate so that no two existential nor two universal quantifiers are next to each other. Let n be the number of quantifiers of order p + 1 in ϕ. Then if the first quantifier is existential ϕ is said to be a Σ_{n}^{p}-formula and if the first quantifier is universal then ϕ is a Π_{n}^{p}-formula.

Also formulae that are obviously equivalent to a Π_{n}^{p}-formula or a Σ_{n}^{p}-formula are said to be Π_{n}^{p} or Σ_{n}^{p} respectively. This hierarchy of formulae and definable concepts is often referred to as the Levy hierarchy because a study of it was initiated by Levy in [13]. We shall mainly be interested in Π_{n}^{1}-formulae since they provide a generalisation of the non-stationary ideal that seems to be fruitful in many ways. By Π_{0}^{1}-formulae we mean first order formulae in a language including at least one unary predicate.

Let ϕ(m) be a Π_{n}^{1}-formula where the free variable m is of first order. We say that ϕ(m) is universal if the following holds: For every Π_{n}^{1}-sentence σ there exists a number m < ω such that ⟨V_{κ}, ∈, U⟩ ⊧ σ if and only if ⟨V_{κ}, ∈, U⟩ ⊧ ϕ[m] for any regular uncountable cardinal κ and predicate U ⊆ V_{κ}. We extend the definition to other formulae in the Levy hierarchy in the obvious way.

We shall now define an universal Π_{1}^{1}-formula ϕ(m). Fix some Gödel numbering of formulae with one free second order variable X and with all other variables bound and of first order. Let ϕ_{m}(X) denote the formula numbered with m. Simply by formalising the truth definition we can find a formula θ(T, X) with the following properties: Apart from the free second order variables T and X displayed, θ(T, X) contains only bounded first order variables. Furthermore for every X ⊆ V_{κ},

⟨V_{κ}, ∈, U⟩ ⊧ θ[T, X]

if and only if T is the set of pairs (m, a) such that m < ω, a is an assignment function ω → V_{κ} for the first order variables, and ⟨V_{κ}, ∈, U⟩ ⊧_{a} ϕ_{m}[X]. Since every Π_{1}^{1}-sentence σ is equivalent to the sentence ∀Xϕ_{m}(X) for some m < ω,

∀X∀T (θ(T, X) → ∀a ∈ ^{ω}V_{κ}((m, a) ∈ T))

is an universal formula. Note that the building blocks defined above can also be put together as

∃X∃T (θ(T, X) ∧ ∀a ∈ ^{ω}V_{κ}((m, a) ∈ T))

which is an universal Σ_{1}^{1}-formula.

The following results about universal formulae have been well known since early developments of the subject.

2.3.1. Lemma. There exists an universal Π_{n}^{1}-formula for every positive integer n.

Proof. We argued above that universal Π_{1}^{1}-formulae exist. The only change required to that argument is that the second order variable X in ϕ_{m}(X) and θ(T, X) must be replaced by a string of n second order variables. Then

∀X∃Y ∃T (θ(T, X, Y) ∧ ∀a ∈ ^{ω}V_{κ}((m, a) ∈ T))

is an universal Π_{2}^{1}-formula and

∀X∃Y ∀Z∀T (θ(T, X, Y, Z) → ∀a ∈ ^{ω}V_{κ}((m, a) ∈ T))

is an universal Π_{3}^{1}-formula and so forth, where the two forms alternate depending on whether n is even or odd. Note that θ is a different formula for each n. □

Of course universal Σ_{n}^{1}-formulae exist too, and in fact the result generalises to all higher order formulae in the Levy hierarchy, but we shall only be needing the result of Lemma 2.3.1.

In the case of order p + 1 the variables X and T are of order p + 1 and all other variables in ϕ_{m}(X) and θ(T, X) are bounded and of order at most p. The assignment functions a have to assign values to all variables of order at most p. Thus the pairs (m, a) ∈ T and the functions a must be coded using a flat pairing function in order to have T ∈ V_{κ+p}. After these changes the proof is the same as in the second order case.

A subset X of a regular cardinal κ is Π_{n}^{1}-indescribable if for every Π_{n}^{1}-sentence ϕ and every unary predicate U ⊆ V_{κ} such that ⟨V_{κ}, ∈, U⟩ ⊧ ϕ, there exists an ordinal α ∈ X such that ⟨V_{α}, ∈, U ∩ V_{α}⟩ ⊧ ϕ. A subset of κ which is not Π_{n}^{1}-indescribable is said to be Π_{n}^{1}-describable.

The following two lemmas also constitute well known observations (see e.g. Jech [10, Lemma 32.3]).

2.3.2. Lemma. A set X ⊆ α is first order indescribable if and only if α is inaccessible and X is stationary.

2.3.3. Lemma. A set is Σ_{n+1}^{1}-indescribable if and only it is Π_{n}^{1}-indescribable.

The following result is due to Levy [13].

2.3.4. Theorem. For every natural number n and cardinal κ the collection of Π_{n}^{1}-describable subsets of κ is a normal ideal.

Proof. Let W be the collection of Π_{n}^{1}-describable subsets of κ. It is evident that W is closed under subsets and rather easy to see that X ∪ α ∈ W for every X ∈ W and α < κ. By Lemma 2.1.1 it then suffices to check that W is closed under diagonal unions.

Let f : X → κ be regressive where X ⊆ κ and suppose that f^{−1}{i} ∈ W for every i < κ. Thus for each i < κ there is a Π_{n}^{1}-sentence ϕ_{i} and a predicate U_{i} ⊆ V_{κ} such that ⟨V_{κ}, ∈, U_{i}⟩ ⊧ ϕ_{i} and

⟨V_{α}, ∈, U_{i} ∩ V_{α}⟩ ⊧ ¬ϕ_{i}  (1)

whenever f(α) = i.

By Lemma 2.3.1 there exists an universal Π_{n}^{1}-formula. Let ϕ(U_{i}, m) be the formula obtained from the universal formula by replacing occurrences of the unary predicate symbol with the second order variable U_{i}. Let g be a function κ → ω such that ϕ_{i} is equivalent to ϕ(U_{i}, g(i)). Put U = {(ξ, i) : ξ ∈ U_{i}}. Now

⟨V_{κ}, ∈, U, g⟩ ⊧ ∀i(ϕ(U_{i}, g(i)))  (2)

and the right hand side is a Π_{n}^{1}-sentence if formalised properly. Of course U_{i} and g(i) are expressed in the formalisation using the predicate symbols for U and g.

Now suppose that α ∈ X and fix i = f(α). Since (1) holds and i < α,

⟨V_{α}, ∈, U ∩ V_{α}, g|α⟩ |̸= ∀i(ϕ(U_{i} ∩ V_{α}, g(i)))

where the right hand side is the same as in (2) when rendered formally. Thus X ∈ W and by Lemma 2.1.2 it follows that W is closed under diagonal unions. □

We shall simply talk about the Π_{n}^{1}-ideal over κ when we mean the ideal of Π_{n}^{1}- describable subsets of κ. In some connections where the cardinal κ is clear from the context we let the symbol Π_{n}^{1} denote this ideal. By Lemma 2.3.2 the Π_{0}^{1}-ideal over κ is proper if and only if κ is inaccessible, and then it coincides with NS_{κ}. The Π_{1}^{1}-indescribable sets are also called weakly compact and the Π_{1}^{1}-ideal over κ will be denoted WC_{κ}. Sometimes we refer to WC_{κ} as the weakly compact ideal.

2.4. n-closed sets

Lemma 2.3.1 has the following well known application.

2.4.1. Lemma. For every n < ω there exists a Π_{n+1}^{1}-sentence ϕ such that a set X ⊆ κ is Π_{n}^{1}-indescribable if and only if ⟨V_{κ}, ∈, X⟩ ⊧ ϕ.

Proof. We can find a Π_{1}^{1}-sentence that in ⟨V_{κ}, ∈, X⟩ expresses that κ is inaccessible and X is stationary. Thus the case n = 0 is handled by Lemma 2.3.2. Assume that n is positive for the rest of the proof.

Let ϕ(U, m) be obtained from an universal Π_{n}^{1}-formula in the same way as in the proof of Theorem 2.3.4. A proper formalisation of

∀U∀m(ϕ(U, m) → ∃α ∈ X(⟨V_{α}, ∈⟩ ⊧ ϕ(m, U ∩ V_{α})))

is the required Π_{n+1}^{1}-sentence involving the unary predicate X. The formalisation of ⟨V_{α}, ∈⟩ ⊧ ϕ(m, U ∩ V_{α}) can be assumed to imply that α is a regular uncountable cardinal. □

A subset X of κ is (n + 1)-closed if α ∈ X whenever α < κ and X ∩ α is Π_{n}^{1}- indescribable as a subset of α. If X is both (n+1)-closed and Π_{n}^{1}-indescribable then X is said to be a (n + 1)-club subset of κ. We let 0-club stand for closed unbounded.

In [18] Sun introduced a notion of 1-club as follows. A subset X of κ is 1-club if it is stationary and closed in the sense that for every regular α < κ, if X ∩ α is stationary then α ∈ X. For this notion Sun also proved Theorem 2.4.2 in the case n = 1. For our purposes the difference between our definition of 1-club using Π_{0}^{1}-indescribability from Sun’s notion is merely technical. In fact Sun’s notion can be seen as the analogue using weak indescribability.

2.4.2. Theorem. If a cardinal κ is Π_{n}^{1}-indescribable then a set X ⊆ κ is in the Π_{n}^{1}-filter if and only it contains an n-club.

Proof. For n = 0 the theorem follows from Lemma 2.3.2. Suppose now that n ≥ 1 and X is n-club. We shall show that X is in the Π_{n}^{1}-filter. Let E be Π_{n}^{1}-indescribable. By Lemma 2.4.1 there is a Π_{n}^{1}-sentence ϕ that expresses Π_{n−1}^{1}- indescribability. Thus ⟨V_{κ}, ∈, X⟩ ⊧ ϕ and therefore there exists an ordinal α ∈ E such that ⟨V_{α}, ∈, X ∩ α⟩ ⊧ ϕ. It follows that α ∈ X ∩ E.

For the other direction suppose that X = {α < κ : ⟨V_{α}, ∈, U ∩ V_{α}⟩ ⊧ ϕ} where U ⊆ V_{κ} and ϕ is a Π_{n}^{1}-sentence such that ⟨V_{κ}, ∈, U⟩ ⊧ ϕ. We shall show that X is n-club. Clearly X is Π_{n−1}^{1}-indescribable. Towards contradiction assume that α ∉ X although α < κ and X ∩ α is Π_{n−1}^{1}-indescribable. By Lemma 2.3.3 the set X ∩ α is also Σ_{n}^{1}-indescribable and since ⟨V_{α}, ∈, U ∩ V_{α}⟩ ⊧ ¬ϕ there exists an ordinal β ∈ X ∩ α such that ⟨V_{β}, ∈, U ∩ V_{β}⟩ ⊧ ¬ϕ. But this is a contradiction by the definition of X. □

The Mahlo operation M(X) on subsets of κ has traditionally been defined by putting

M(X) = {α < κ : cf α > ω and X ∩ α is stationary}.

We define the operations M^{n} : [κ]^{κ} → 𝒫(κ) for n < ω as follows:

M^{n}(X) = {α < κ : X ∩ α is Π_{n}^{1}-indescribable}.

We conclude this section by a simple fact that for n = 0 is analogous to a result proved by Baumgartner, Taylor, and Wagon [4] for the Mahlo operation. Their formulation is slightly different and related to the concept of an M-ideal defined in [4]. Note that the conclusion M^{n}(E) ∈ (Π_{n+1}^{1})^{∗} can be seen a straightforward generalisation of the fact that the set of limit points of an unbounded set is closed unbounded.

2.4.3. Lemma. If E is a Π_{n}^{1}-indescribable subset of κ then the set M^{n}(E) is in the Π_{n+1}^{1}-filter but not in the filter (Π_{n}^{1}|E)^{∗}, whereby Π_{n}^{1}|E ≠ Π_{n+1}^{1}.

Proof. That M^{n}(E) is in the Π_{n+1}^{1}-filter is an almost immediate corollary of Lemma 2.4.1. Suppose towards contradiction that M^{n}(E) ∈ (Π_{n}^{1}|E)^{∗}. It means that there exists a set X in the Π_{n}^{1}-filter such that X ∩ E ⊆ M^{n}(E). We can even assume that X = {α < κ : ⟨V_{α}, ∈, U ∩ V_{α}⟩ ⊧ ϕ} for some U ⊆ V_{κ} and Π_{n}^{1}-sentence ϕ. Let α = min(X ∩ E). Since α ∈ M^{n}(E) there exists an ordinal β ∈ X ∩ E ∩ α, a contradiction. □

An immediate corollary of Theorem 2.3.4 and Lemmas 2.2.1 and 2.4.3 is that the Π_{n}^{1}-ideal over κ is not κ^{+}-saturated whenever the Π_{n+1}^{1}-ideal is proper. (If Π_{n+1}^{1} is not proper then trivially Π_{n}^{1}|E = Π_{n+1}^{1} for any E ∈ Π_{n}^{1}.) In [4] it is proved that it suffices that κ is greatly Mahlo for NS_{κ} not to be κ^{+}-saturated. The version of Lemma 2.4.3 for the Mahlo operation is a central ingredient in Solovay’s [17] classical result that any stationary subset of a regular cardinal κ can be split into κ pairwise disjoint stationary sets.

3. Operations on ideals

3.1. Diamond principles

Let I be a normal ideal over a regular cardinal κ. By ◇(I) we shall denote the following statement: There exists a sequence (A_{α} : α < κ) such that

{α < κ : A ∩ α = A_{α}} ∉ I

for every set A. The classical diamond principle ◇_{κ} is ◇(NS_{κ}). A sequence witnessing that ◇(I) holds, as (A_{α} : α < κ) above, is said to be a ◇(I)-sequence.

The next result is well known in many settings (see e.g. [11] or [6]) but it is seldom, if ever, noted that it applies to any normal ideal whatsoever. This is why we give a complete proof. However it seems that the case I = NS_{κ} is in some sense the most interesting since the proof relies on closed unbounded sets.

3.1.1. Lemma. For any normal ideal I over κ, ◇(I) holds if and only if there exists a sequence (W_{α} : α < κ) such that |W_{α}| = |α| for every α < κ and

{α < κ : A ∩ α ∈ W_{α}} ∉ I

for every set A.

Proof. Clearly only one direction of the equivalence requires an argument. Assume that (W_{α} : α < κ) is as above. Let f : κ × κ → κ be a bijection. There is a closed unbounded set C such that f[α × α] = α for every α ∈ C. Because κ \ C ∈ NS_{κ} and NS_{κ} ⊆ I it is straightforward to use f to construct an indexed family (B_{α}^{i} : i < α < κ) such that

{α < κ : B ∩ (α × α) = B_{α}^{i} for some i < α} ∉ I  (3)

for every set B. To be more precise this is achieved by picking the sets B_{α}^{i} in such a way that {f[B_{α}^{i}] : i < α} = W_{α} ∩ 𝒫(α) whenever α ∈ C.

Let A_{α}^{i} = {ξ < α : (ξ, i) ∈ B_{α}^{i}} when i < α < κ and let A_{α}^{i} be arbitrary when α ≤ i < κ. Consider the sequences (A_{α}^{i} : α < κ) for i < κ. We shall derive a contradiction from the assumption that none of these sequences is a ◇(I)-sequence.

So we assume that for every i < κ there exists a set A_{i} such that X_{i} = {α < κ : A_{i} ∩ α ≠ A_{α}^{i}} is in the dual filter F of I. By normality X = Δ_{i<κ}X_{i} ∈ F. Let B = ⋃_{i<κ}(A_{i} × {i}). If α ∈ X then for all i < α we have α ∈ X_{i} and thus A_{i} ∩ α ≠ A_{α}^{i}. But if α is in the left hand side of (3) then A_{i} ∩ α = A_{α}^{i} for some i < α because for ξ, i < α we have ξ ∈ A_{i} ∩ α iff (ξ, i) ∈ B and (ξ, i) ∈ B_{α}^{i} iff ξ ∈ A_{α}^{i}. By (3) we have arrived at a contradiction. □

There is an immediate simple connection between diamond principles and saturation of ideals. Let A and B be subsets of κ and let γ < κ be the least ordinal such that A ∩ {γ} ≠ B ∩ {γ} i.e. γ is the least ordinal where the sets A and B differ. Then for any sequence (A_{α} : α < κ) we have {α < κ : A ∩ α = A_{α}} ∩ {α < κ : B ∩ α = A_{α}} ⊆ γ + 1. Thus ◇(I) implies that I is not 2^{κ}-saturated.

For a normal ideal I over κ, let ◇^{∗}(I) denote the following statement: There exists a sequence (W_{α} : α < κ) such that |W_{α}| = |α| for every α < κ and

{α < κ : A ∩ α ∈ W_{α}} ∈ I^{∗}

for every set A. Of course an analogous change to the initial formulation of ◇(I) would result in a provably false statement. We shall also talk about ◇^{∗}(I)-sequences in a similar fashion as with ◇(I).

By Lemma 3.1.1 it is clear that ◇^{∗}(I) → ◇(I) for any normal ideal I. Let I and J be normal ideals such that I ⊆ J. Then ◇(J) implies ◇(I) but ◇^{∗}(I) implies ◇^{∗}(J). Thus the principle ◇^{∗}(NS_{κ}), traditionally denoted by ◇_{κ}^{∗}, is the strongest of these statements and implies both ◇(I) and ◇^{∗}(I) for every normal ideal I over κ.

Let I be a normal ideal, let E ∈ I^{+}, and consider the following statements:

(a) There exists a sequence (A_{α} : α ∈ E) such that {α ∈ E : A ∩ α = A_{α}} ∈ I^{+} for every set A.

(b) There exists a sequence (W_{α} : α ∈ E) such that |W_{α}| = |α| for every α ∈ E and for every set A there exists a set X ∈ I^{∗} such that A∩α ∈ W_{α} for every α ∈ X ∩ E.

Traditionally (a) for I = NS_{κ} has been denoted ◇_{E} or ◇_{κ}(E) and (b) similarly ◇_{E}^{∗} or ◇_{κ}^{∗}(E). But it is not hard to see that in fact (a) and (b) are equivalent to ◇(I|E) and ◇^{∗}(I|E) respectively. Therefore there is no need for us to introduce special notation for this kind of principles. But we may still use e.g. ◇_{κ}(E) as a shorthand for ◇(NS_{κ}|E).

For a normal ideal I over κ we define the collections ND(I) ⊆ 𝒫(κ) and SD(I) ⊆ 𝒫(κ) as follows:

ND(I) = {X ⊆ κ : ◇(I|X) fails}
SD(I) = {X ⊆ κ : ◇^{∗}(I|X) holds}.

The letters ND and SD refer to “no diamond” and “strong diamond” respectively.

It turns out that these collections are ideals. By ND_{κ} and SD_{κ} we shall denote the ideals ND(NS_{κ}) and SD(NS_{κ}) respectively. ND I is proper iff ◇(I) holds, and SD I is proper iff ◇^{∗}(I) fails.

The ideal ND_{ℵ_{1}} has been studied in literature. The result that ND_{ℵ_{1}} is normal is due to Saharon Shelah and was announced in [5] where Devlin proves that ND_{ℵ_{1}} is countably complete.

3.1.2. Theorem. If I is a normal ideal over κ then ND(I) and SD(I) are both normal ideals over κ and ND(I) ∩ SD(I) = I.

Proof. It is easy to see that ND(I) is closed under subsets and that X ∈ ND(I) and α < κ implies X ∪ α ∈ ND(I). The same holds for SD(I). Thus, to see that ND(I) and SD(I) are normal ideals it suffices, by Lemma 2.1.1, to check that the collections in question are closed under diagonal unions.

Let X ⊆ κ and let f : X → κ be a regressive function such that f^{−1}{i} ∈ ND(I) for every i < κ. We shall show that X ∈ ND(I) by proving that an arbitrary sequence (A_{α} : α < κ) can not be a ◇(I|X)-sequence. It follows by Lemma 2.1.2 that ND(I) is closed under diagonal unions.

Let g : κ × κ → κ be a bijection and let C be a closed unbounded set such that g[α × α] = α for every α ∈ C. For ordinals i and α in κ we put

A_{α}^{i} = {ξ : (ξ, i) ∈ g^{−1}[A_{α}]}.  (4)

Since (A_{α}^{i} : α < κ) can not be a ◇(I|f^{−1}{i})-sequence there exists a set A^{i} ⊆ κ and a set Y_{i} ∈ I^{∗} such that

{α ∈ f^{−1}{i} : A^{i} ∩ α = A_{α}^{i}} ∩ Y_{i} = ∅.  (5)

Let A = g[⋃_{i<κ}(A^{i} × {i})]. We shall conclude the proof of X ∈ ND(I) by showing that

{α ∈ X : A ∩ α = A_{α}} ∩ △_{i<κ}Y_{i} ∩ C = ∅.  (6)

For α ∈ C we have A ∩ α = g[⋃_{i<α}((A^{i} ∩ α) × {i})]. So if A ∩ α = A_{α} we must have A_{α}^{i} = A^{i} ∩ α for every i < α by (4). But if α ∈ X and we fix i = f(α) it then follows from (5) that α ∉ Y_{i} and thus (6) holds.

For SD(I) the proof is somewhat easier. Suppose again that X ⊆ κ and f : X → κ is a regressive function such that f^{−1}{i} ∈ SD(I) for every i < κ. So for each i < κ there exists a ◇^{∗}(I|f^{−1}{i})-sequence (W_{α}^{i} : α < κ).

Let A be an arbitrary set. Now X_{i} = {α ∈ f^{−1}{i} : A ∩ α ∉ W_{α}^{i}} ∈ I for every i < κ and thus

{α ∈ X : A ∩ α ∉ W_{α}^{f(α)}} = ▽_{i<κ}X_{i} ∈ I

whereby (W_{α}^{f(α)} : α < κ) is a ◇^{∗}(I|X)-sequence.

Finally it is a triviality that I ⊆ ND(I) and I ⊆ SD(I) and by Lemma 3.1.1 we have that E ∈ SD(I) \ I implies E ∉ ND(I). □

3.2. Subtlety and ineffability

To make some phrasings more fluent we shall talk about a subset sequence when we mean a sequence (A_{i} : i ∈ X) such that A_{i} ⊆ i for every i ∈ X.

Let X be a set of ordinals. A set H ⊆ X is homogeneous for a sequence (A_{α} : α ∈ X) if A_{α} = A_{β} ∩ α for every pair of ordinals α < β in H. A simple observation sometimes used in proofs is that if H is a set of ordinals, then H is homogeneous for the sequence (A_{i} : i ∈ X) if and only if there exists a set A such that H ⊆ {i ∈ X : A ∩ i = A_{i}}.

Let I be an ideal over κ. By Sb I we denote the collection of sets X ⊆ κ with the property that there exists a set Y ∈ I^{∗} and a subset sequence (A_{α} : α ∈ X ∩ Y) for which there is no homogeneous set of cardinality 2. By In I we mean the collection of sets X ⊆ κ such that there exists a subset sequence (A_{α} : α ∈ X) for which every homogeneous set belongs to I.

To a large extent the following is due to Baumgartner [2].

3.2.1. Lemma. If I is an ideal over κ and NS_{κ} ⊆ In I then In I is a normal ideal extending I.

Proof. To begin with it is not difficult to see that In I is an ideal whenever I is an ideal and that I ⊆ In I. We shall now use Lemma 2.1.2 to check the normality.

Let X ⊆ κ, let f : X → κ be regressive and for each β < κ, suppose that the subset sequence (A_{α}^{β} : α ∈ f^{−1}{β}) witnesses that f^{−1}{β} ∈ In I. Let g : κ × κ → 2 be a bijection. Because we already know that In I is an ideal such that NS_{κ} ⊆ In I we may assume that g[α × 2] = α for every α ∈ X. Put

A_{α} = g[(A_{α}^{f(α)} × {0}) ∪ (f(α) × {1})]

for each α ∈ X. Let A be arbitrary and put H = {α ∈ X : A ∩ α = A_{α}}. For ξ < α both in H we have g^{−1}[A_{ξ}] = g^{−1}[A_{α}] ∩ (ξ × ξ) and thus f must be constant on H. Let β be the constant value of f in H. If H is unbounded in κ then

g^{−1}[A] = ⋃_{α∈H} g^{−1}[A_{α}] = ⋃_{α∈H}(A_{α}^{β} × {0}) ∪ (β × {1})

whereby H is homogeneous for (A_{α}^{β} : α ∈ f^{−1}{β}). It follows that H ∈ I. □

Note that we did not require in the previous lemma that I must be normal, but only that NS_{κ} ⊆ In I. This is utilised through the following fact.

3.2.2. Lemma. NS_{κ} ⊆ In([κ]^{<κ}).

Proof. Let C be a closed unbounded subset of κ. Let H ⊆ κ\C be homogeneous for the subset sequence (max(C ∩ α) : α ∈ κ \ C). Now suppose that α ∈ H and β = min(C \ α). Then we must have H ⊆ β and it follows that κ \ C ∈ In([κ]^{<κ}). □

Since [κ]^{<κ} is the smallest possible ideal over κ by our strict definition, it follows that In I is always a normal ideal. The operation Sb is not as well behaved as In.

3.2.3. Lemma. Let I be an ideal over κ and let μ be another regular cardinal. Then Sb I is an ideal extending I and if I is normal then Sb I is normal. If NS_{κ} ⊆ Sb I and I is μ-complete then Sb I is μ-complete.

Proof. As in the proof of Lemma 3.2.1 it is easy to see that Sb I is an ideal such that I ⊆ Sb I. We shall deal with normality through Lemma 2.1.1.

Let f : X → κ be regressive and for each β < κ, suppose that Y_{β} ∈ I^{∗} and (A_{α}^{β} : α ∈ f^{−1}{β} ∩ Y_{β}) witness that f^{−1}{β} ∈ Sb I. Let g : κ × κ → κ be a bijection. As in the proof of the normality of In I we can assume that g[α × α] = α for every α ∈ X. Still as in the proof of Lemma 3.2.1, put

A_{α} = g[(A_{α}^{f(α)} × {0}) ∪ (f(α) × {1})]

for α ∈ X. Let Y = Δ_{β<κ}Y_{β}. If {ξ, α} ⊆ X ∩ Y and A_{ξ} = A_{α} ∩ ξ then f(ξ) = f(α) and A_{ξ}^{f(ξ)} = A_{α}^{f(α)} ∩ ξ. Since {ξ, α} ⊆ Y it then follows that {ξ, α} ⊆ Y_{f(ξ)} and therefore we must have ξ = α. We conclude that X ∈ Sb I.

For the μ-completeness we can use the same argument as above, if we assume that ran(f) ⊆ μ and put Y = ⋂_{β<μ} Y_{β}. But in this case I does not necessary extend NS_{κ} so we need the assumption NS_{κ} ⊆ Sb I to be able to assume that g[α × α] = α for all α ∈ X without loosing generality. □

3.2.4. Lemma. Let I and J be ideals over κ. If J ⊆ In I then Sb J ⊆ In I.

Proof. Suppose that X ⊆ κ, Y ∈ J^{∗}, and the subset sequence (A_{α} : α ∈ X ∩ Y) has no homogeneous sets of cardinality 2. We wish to prove that X ∈ In I but since Y ∈ (In I)^{∗} it suffices to prove that X ∩ Y ∈ In I, but this of course is immediate.

□

By lemmas 3.2.2 and 3.2.4 and the obvious fact that the operation In is monotone, we have

NS_{κ} ⊆ Sb NS_{κ} ⊆ In([κ]^{<κ}) ⊆ In NS_{κ}

where each of the four collections involved is a normal ideal over κ by lemmas 3.2.1 and 3.2.3. The cardinal κ is said to be subtle if Sb NS_{κ} is proper, almost ineffable if In([κ]^{<κ}) is proper, and ineffable if In NS_{κ} is proper. These notions were introduced by Jensen and Kunen [11]. The letter combinations Sb and In used for the operations involved refer to these concepts. The sets that have positive measure with respect to the ideals above are also called subtle, almost ineffable, and ineffable respectively.

One can consider applying the operations defined in sections 3.1 and 3.2 repeatedly. Furthermore the union of a collection of ideals over a regular cardinal κ is itself an ideal over κ. Therefore let us put In^{0} I = I, In^{α+1} = In(In^{α} I), and In^{α} I = ⋃_{β<α} In^{β} I for limit ordinals α. We shall use analogous notation for the other operations defined.

If an operation is repeated as above, then sooner or later a fixpoint In^{α} I must be reached for which In^{α+1} I = In^{α} I. Let In^{∞} I denote this fixpoint. If In^{∞} NS_{κ} is proper we say that the cardinal κ is totally ineffable and if Sb^{∞} NS_{κ} is proper then κ is totally subtle. By a simple induction argument Lemma 3.2.4 also holds with Sb replaced by Sb^{∞}. Thus almost ineffable cardinals are totally subtle.

By virtue of Lemma 3.2.1 the ideal In^{α} I is normal for any successor ordinal α regardless of whether I is normal or not. In [2] Baumgartner studied the ideals In^{α} NS_{κ} and proved that if κ is totally subtle and In^{α} NS_{κ} = In^{∞} NS_{κ} then α ≥ κ^{+}.

Let f be a function on [κ]^{2}. We call f a partition as we think of the set [κ]^{2} being partitioned into parts labeled by the elements in ran f. A subset H of κ is homogeneous for the partition f if f is constant on the set [H]^{2}. For X ⊆ κ and an ideal I over κ we write X → (I^{+})^{2} if we want to say that X has the following partition property: For every function f : [κ]^{2} → 2 there exists a homogeneous set H ⊆ X such that H ∈ I^{+}. If S is not a subset of κ but S ⊆ 𝒫(κ) then S → (I^{+})^{2} is taken to mean that every X ∈ S has the partition property X → (I^{+})^{2}.

3.2.5. Lemma. Let I be an ideal over a regular cardinal κ and let X ⊆ κ. If NS_{κ} ⊆ I then X ∉ In I if and only if X → (I^{+})^{2}.

Proof. Suppose that X ∉ In I and f : [X]^{2} → 2 is arbitrary. Let A_{β} = {α ∈ X ∩β : f(α, β) = 1} for every β ∈ X. Fix A so that H = {β ∈ X : A∩β = A_{β}} ∉ I. Now either H ∩ A or H \ A is the homogeneous set we are looking for.

Now suppose that X → (I^{+})^{2} and the subset sequence (A_{α} : α ∈ X) is arbitrary. Define f : [X]^{2} → 2 by letting f(α, β) = 1 iff A_{α} lexicographically precedes A_{β}. (We consider the lexicographic ordering of the characteristic functions with domain κ.)

Pick a set H ∉ I that is homogeneous for f. We define A ⊆ κ by defining A ∩ ξ by induction on ξ < κ using the following requirement: For each ξ there exists an ordinal η(ξ) < κ such that A ∩ ξ = A_{α} ∩ ξ for every α ∈ H \ η(ξ). The limit steps are trivial and the successor steps are easily handled by the properties of the lexicographic ordering. Now H′ = {α ∈ H ∩ acc κ : η[α] ⊆ α} ∉ I (acc κ is the set of limit ordinals below κ) and A_{α} = A ∩ α whenever α ∈ H′. □

If D is a normal measure on κ then D → (D)^{2} whereby In^{∞} I ⊆ D^{∗} for any normal ideal I ⊆ D^{∗}. Since we know that NS_{κ} ⊆ D^{∗}, Lemma 3.2.5 implies that measurable cardinals are totally ineffable.

3.3. Subtlety and diamonds

If I and J are ideals over a regular cardinal κ then the collection

{X ∪ Y : X ∈ I, Y ∈ J}

is an ideal over κ. Clearly this is the ideal generated by I and J, i.e. the smallest ideal containing both I and J. We shall denote it by ⟨I ∪ J⟩. The ideals I and J are said to be coherent if ⟨I ∪ J⟩ is proper. In [1] Baumgartner proves the following result:

3.3.1. Lemma. Let κ be a regular cardinal.

(a) ⟨Π_{1}^{1} ∪ Sb NS_{κ}⟩ = In([κ]^{<κ}).

(b) ⟨Π_{n+2}^{1} ∪ Sb NS_{κ}⟩ = In Π_{n}^{1} for all n < ω.

Especially it follows that almost ineffable sets are weakly compact and ineffable sets are Π_{2}^{1}-indescribable. By Lemma 3.2.4 and monotonicity of the operation Sb it follows that Sb Π_{1}^{1} = In([κ]^{<κ}) and Sb Π_{n+2}^{1} = In Π_{n}^{1}.

3.3.2. Lemma. Let I be an ideal over a regular cardinal κ.

(a) κ ∈ In(I|E) if and only if E ∈ In I.

(b) κ ∈ Sb(I|E) implies that E ∈ Sb I.

Proof. Any subset sequence (A_{α} : α < κ) witnesses that κ ∈ In(I|E) if and only if its restriction (A_{α} : α ∈ E) witnesses that E ∈ In I. Given Y ⊆ κ put Z = Y ∪ (κ \ E). Then Y ∈ (I|E)^{∗} iff Z ∈ I^{∗}. If the sequence (A_{α} : α ∈ Y) witnesses that κ ∈ Sb(I|E) then the sequence (A_{α} : α ∈ E ∩ Z) witnesses that E ∈ Sb I. □

As a consequence of the following lemma, ◇_{κ}^{∗} fails whenever κ is ineffable which is a result due to Jensen and Kunen [11]. They also proved that the converse holds if V = L. Thus in L we have that ◇_{κ}^{∗} holds if and only if κ is not ineffable.

3.3.3. Lemma. SD I ⊆ In I for every normal ideal I over a regular cardinal κ.

Proof. Suppose that κ ∈ SD I and (W_{α} : α < κ) is a ◇^{∗}(I)-sequence. Pick a subset sequence (A_{α} : α < κ) such that A_{α} ∉ W_{α} for every α < κ. There can not exist a set A such that {α < κ : A ∩ α = A_{α}} ∉ I. Therefore κ ∈ In I. Since I was arbitrary it follows that ◇^{∗}(I|E) implies that κ ∈ In(I|E) for any E ⊆ κ. By Lemma 3.3.2 we are done. □

In [11] it was also pointed out that ◇_{κ} holds at any subtle cardinal κ (see [12]). In fact ◇_{κ}(E) holds for any subtle set E. Thus ND_{κ} ⊆ Sb NS_{κ} in terms of the notations for the ideals. The proof of this fact generalises to other levels in the Levy hierarchy in a rather straightforward way. Sun [18] proved that ◇(WC_{κ}) holds for almost ineffable cardinals.

3.3.4. Theorem. Let κ be a regular cardinal.

(a) ND(Π_{1}^{1}) ⊆ In([κ]^{<κ}).

(b) ND(Π_{n+2}^{1}) ⊆ In Π_{n}^{1} for all n < ω.

Proof. We shall prove (a). The proof of (b) is nearly the same. Let E be an almost ineffable subset of κ. By induction on α ≤ κ we define sets A_{α} ⊆ α, U_{α} ⊆ V_{α}, and Π_{1}^{1}-formulae ϕ_{α} such that the conditions (i) ⟨V_{α}, ∈, U_{α}⟩ ⊧ ϕ_{α}

(ii) If ξ ∈ E ∩ α and ⟨V_{ξ}, ∈, U_{α} ∩ V_{ξ}⟩ ⊧ ϕ_{α} then A_{α} ∩ ξ ≠ A_{ξ}

hold whenever it is possible to make them hold at a step α ≤ κ in the construction. To be precise we do not choose actual Π_{1}^{1}-formulae (whatever that means) but rather natural numbers that code the appropriate formulae.

For ordinals α ≤ κ such that conditions (i) and (ii) are met, put

X_{α} = {ξ ∈ E ∩ α : ⟨V_{ξ}, ∈, U_{α} ∩ V_{ξ}⟩ ⊧ ϕ_{α}}  (7)

and let X_{α} = α otherwise. By the construction (A_{α} : α < κ) will be a ◇_{κ}(WC_{κ}|E)- sequence if and only if X_{κ} = κ. (Note that we intend condition (ii) to imply 0 ∉ X_{α}, although the expression in the condition may not be meaningful for ξ = 0). We shall complete the argument by deriving a contradiction from the assumption X_{κ} ≠ κ.

We first notice that X_{α} ≠ α for every α ∈ X_{κ}. This is because A_{κ} ∩ α, U_{κ} ∩ V_{α}, and ϕ_{κ} would satisfy the conditions (i) and (ii) if they were to be chosen as A_{α}, U_{α}, and ϕ_{α} respectively.

Let f be a bijection κ × V_{κ} × ω → κ. There is a closed unbounded set C such that f[α×V_{α} ×ω] = α for every α ∈ C. Let B_{α} = f[A_{α} ×U_{α} ×{ϕ_{α}}] for α ∈ X_{κ} ∩C.

As pointed out earlier, the monotonicity of the operation Sb together with Lemma 3.2.4 and Lemma 3.3.1 implies that Sb WC_{κ} is the almost ineffable ideal. Since X_{κ} ∩ C is the intersection of E ∉ Sb WC_{κ} and a set in the weakly compact filter, there exist ordinals ξ < α both in X_{κ} ∩ C such that B_{α} ∩ ξ = B_{ξ}. It follows that A_{α} ∩ ξ = A_{ξ}, U_{α} ∩ V_{ξ} = U_{ξ} and ϕ_{α} = ϕ_{ξ}. But this means that X_{α} = α which is a contradiction since α ∈ X_{κ}. □

4. Weak compactness

From now on we shall concentrate on the weakly compact ideal WC_{κ} and the principle ◇(WC_{κ}) which we may call weakly compact diamond. We shall also consider the principles ◇(WC_{κ}|E) where E is a weakly compact subset of κ.

4.1. Some notes on ultraproducts

Let κ be a regular cardinal. A κ-complete algebra of sets is a non-empty collection of sets which contains the union of all its members and is closed under set difference and unions and intersections of cardinality less than κ. Thus a σ-algebra could also be referred to as a ℵ_{1}-complete algebra of sets. Sometimes one talks about a field of sets instead of an algebra of sets. A subcollection F of a κ-complete algebra of sets S is a filter if it is closed under finite intersections and X ∈ F and X ⊆ Y ∈ S implies that Y ∈ F. The notions of ultrafilter and a μ-complete filter where μ ≤ κ is a regular cardinal are defined as expected.

Let M be a transitive set. Typically M would be V_{κ} or a model of a large enough finite fragment of ZFC such that κ ∈ M. By def M we shall denote the collection of subsets of M that are definable over M. We shall require that κ ∈ def M which clearly holds for the examples of M mentioned above. Let S ⊆ 𝒫(κ) be a κ-complete algebra of sets such that def(M) ∩ 𝒫(κ) ⊆ S.

For a first order formula ϕ(x_{1}, …, x_{n}) and for functions f_{1}, …, f_{n} ∈ ^{κ}M we put

X_{ϕ[f_{1},...,f_{n}]} = {α < κ : ϕ^{M}[f_{1}(α), …, f_{n}(α)]}.

If the functions f_{1}, …, f_{n} are in def M then X_{ϕ[f_{1},...,f_{n}]} ∈ S because of the requirement κ ∈ def M. Let U be an ultrafilter on S. The condition X_{f=g} ∈ U defines an equivalence relation =_{U} on the set def(M) ∩ ^{κ}M and the condition X_{f∈g} ∈ U defines a binary relation [f] ∈_{U} [g] on the set of equivalence classes. Consider the structure 𝔑 = ⟨(def(M) ∩ ^{κ}M)/ =_{U}, ∈_{U}⟩.

4.1.1. Lemma. Let ϕ(x_{1}, …, x_{n}) be a first order formula and let f_{1}, …, f_{n} ∈ def(M) ∩ ^{κ}M. If def M contains a well-ordering of M then

𝔑 ⊧ ϕ[[f_{1}], …, [f_{n}]] if and only if X_{ϕ[f_{1},...,f_{n}]} ∈ U.

Proof. By induction on the complexity of ϕ as in the standard Łoś’s Theorem. The well-ordering of M is used in the existential quantifier step where it is required to find a function f ∈ def(M) ∩ ^{κ}M such that

ϕ^{M}[f(α), f_{1}(α), …, f_{n}(α)]

holds for all α < κ with the property that there exists at least one x ∈ M for which ϕ^{M}[x, f_{1}(α), …, f_{n}(α)] holds. □

By the above lemma 𝔑 is extensional. Let us now assume that U is σ-complete. Then 𝔑 is also well-founded. Let N be the Mostowski collapse of 𝔑. N is called the definable ultrapower of M modulo U. The corresponding canonical embedding j : M → N is defined by j(x) = π([f_{x}]) where π is the Mostowski collapse and f_{x} is the constant function κ → M with value x. By Lemma 4.1.1 the canonical embedding j is an elementary embedding.

Hereafter we shall let [f] denote the element of N that is the image of the equivalence class of f : κ → M under the Mostowski collapse, rather than the equivalence class itself.

4.1.2. Lemma. If U is a κ-complete ultrafilter over κ and every regressive function in def(M) ∩ ^{κ}κ is constant on a set in U, then κ = [id_{κ}] and for any X ∈ M ∩ 𝒫(κ) it holds that X ∈ U if and only if κ ∈ j(X).

Proof. For the second claim, just compare id_{κ} and the constant function f_{X}. □

Through lemmas 4.1.4 and 4.1.6 below we shall make extensive use of the following characterisation of weak compactness due to Baumgartner [2].

4.1.3. Theorem. Let S ⊆ 𝒫(κ) be a κ-complete algebra of sets and F a collection of regressive functions on κ such that |F | = |S| = κ, κ ∈ S, and f^{−1}{α} ∈ S for all f ∈ F and α < κ. If X ∈ S is weakly compact then there exists a κ-complete ultrafilter U on S such that X ∈ U and each f ∈ F is constant on a set in U. Conversely if such an ultrafilter U exists for every S and F as above such that X ∈ S then X is weakly compact.

4.1.4. Lemma. If E is a weakly compact subset of κ and M is a transitive set such that E ∈ M and κ ∈ def M then there exists a transitive set N and an elementary embedding j : M → N such that κ ∈ j(E).

What is the role of def M in the arguments above? We shall now consider replacing def M by some other collection D such that M ⊆ D ⊆ 𝒫(M). To be able to use Theorem 4.1.3 we must have |D ∩ 𝒫(κ)| = κ. Apart from that there are only two points to be watched. One is that for any functions f_{1}, …, f_{n} ∈ D ∩ ^{κ}M and any first order formula ϕ(x_{1}, …, x_{n}) we must have X_{ϕ[f_{1},...,f_{n}]} ∈ D ∩ 𝒫(κ). The other is that the function f : κ → M constructed in the proof of Lemma 4.1.1 must be in D.

Given M and an ultrafilter U, let us call N the κω-definable ultrapower of M modulo U, if the transitive set N is defined exactly as the definable ultrapower, except that def M is replaced by the collection of all subsets of M that are ℒ_{κω}- definable over M.

4.1.5. Lemma. If ^{<κ}M ⊆ M and N is the κω-definable ultrapower of M modulo U, then ^{<κ}N ⊆ N.

Proof. Suppose that α < κ and (f_{i} : i < α) represents an α-sequence in N. So each f_{i} is a function κ → M which is ℒ_{κω}-definable over M. For ξ < κ let s_{ξ} be the sequence (f_{i}(ξ) : i < α). Since ^{<κ}M ⊆ M we have s_{ξ} ∈ M for every ξ < κ. Let f : κ → M be defined by f(ξ) = s_{ξ}. Clearly f is ℒ_{κω}-definable over M since we may use a disjunction involving the formulae defining the functions f_{i}. Thus [f] is an element in N and it is straightforward to see that [f] is the sequence ([f_{i}] : i < α).

□

4.1.6. Lemma. If E is a weakly compact subset of κ and M is a transitive set such that M^{<κ} ⊆ M, E ∈ M, and κ ∈ M then there exists a transitive set N and an elementary embedding j : M → N such that N^{<κ} ⊆ N and κ ∈ j(E).

4.2. Weakly compact diamond

The following result was proved by Sun [18] in the case E = κ. Note that by the discussion preceding Lemma 3.3.3 this result is of interest only for ineffable cardinals.

4.2.1. Theorem. If V = L then ◇(WC_{κ}|E) holds for every weakly compact subset E of a weakly compact cardinal κ.

Proof. We pick sets A_{α} ⊆ α and U_{α} ⊆ V_{α} and Π_{1}^{1}-sentences ϕ_{α} by induction on α ≤ κ in the same manner as in the proof of Theorem 3.3.4. Only now we require that (A_{α}, U_{α}) is the <_{L}-least pair for which the conditions

(i) ⟨V_{α}, ∈, U_{α}⟩ ⊧ ϕ

(ii) If ξ ∈ E ∩ α and ⟨V_{ξ}, ∈, U_{α} ∩ V_{ξ}⟩ ⊧ ϕ then A_{α} ∩ ξ ≠ A_{ξ}

can be satisfied if they can be satisfied at all for some Π_{1}^{1}-sentence ϕ. Here we have dropped the subscript on ϕ since we only need to refer to one particular sentence in the forthcoming argument. Also

X_{α} = {ξ ∈ E ∩ α : ⟨V_{ξ}, ∈, U_{α} ∩ V_{ξ}⟩ ⊧ ϕ}  (8)

is defined exactly as in the proof of Theorem 3.3.4 and the theorem is proved by deriving a contradiction from the antithesis X_{κ} = κ. So suppose that there exists a Π_{1}^{1}-sentence ϕ satisfying conditions (i) and (ii) for α = κ.

For each pair (A, U) of subsets of κ such that (A, U) <_{L} (A_{κ}, U_{κ}) there exists a countable collection W_{A,U} ⊆ V_{κ+1} which contains witnesses for that fact that the pair (A, U) was not chosen in place of (A_{κ}, U_{κ}) in the construction. More exactly if (ii) holds for (A, U) <_{L} (A_{κ}, U_{κ}) and some Π_{1}^{1}-sentence ϕ which evaluated in ⟨V_{κ}, ∈, U⟩ is equivalent to ∀X ⊆ V_{κ}(⟨V_{κ}, ∈, U⟩ ⊧ ψ[X]) then there exists a set X ∈ W_{A,U} such that ⟨V_{κ}, ∈, U⟩ |̸= ψ[X]. The set

W = ⋃{W_{A,U} : (A, U) ∈ 𝒫(κ) × V_{κ+1}, (A, U) <_{L} (A_{κ}, U_{κ})}

has cardinality κ since 𝒫(κ) × V_{κ+1} ⊆ L_{κ^{+}} and |L_{α}| = |α| for every infinite ordinal α. Let M be a transitive set such that |M| = κ, ((A_{α}, U_{α}) : α ≤ κ) ∈ M, E ∈ M, V_{κ} ∪ W ⊆ M, and M satisfies a large enough finite fragment of ZFC.

Because E is weakly compact it is immediate from (8) that X_{κ} is weakly compact. Clearly X_{κ} ∈ def M. By Lemma 4.1.4 there exists a transitive set N and an elementary embedding j : M → N such that κ ∈ j(X_{κ}). Because V_{κ+1}^{M} ⊆ V_{κ+1}^{N} the set W is also contained in N. Thus the construction is absolute up to the point that (j((A_{α} : α ≤ κ)))_{κ} = A_{κ}. But j(A_{κ}) ∩ κ = A_{κ} and κ ∈ j(X_{κ}) by our choice of the embedding j. This is a contradiction by (ii) and (8) and the elementarity of j. □

5. Weak Compactness in generic extensions

5.1. Forcing preliminaries

We use the Boolean algebraic convention regarding the ordering relation of forcing notions i.e. a condition p is stronger than a condition q if p ≤ q. Trees are tacitly reversed as soon as they are used as forcing notions.

If we say that τ is a name for some object x in the generic extension then we shall take it to mean that any condition forces τ to have whatever properties we have attributed to x.

When we say that P is an iteration of the sequence (Q̇_{α} : α < η) we mean that P is an iteration of length η and Q̇_{α} is the name of the forcing notion used to construct P_{α+1} from P_{α}. Since there for a condition p ∈ P_{α} usually is a proper class of P_{α}-names τ such that p ⊩ “τ ∈ Q̇_{α}”, one must somehow choose a representative set of such names that will then be the set

{q(α) : q ∈ P, q|α = p}.

Often one picks all possible names up to equivalence without further specifying the choice, but we need to be more restrictive in some cases. Therefore we take the following somewhat nonstandard approach. With each Q̇_{α} we associate a set of P_{α}-names that we denote by dom Q̇_{α}. Then we require that for every p ∈ P and α ∈ dom p, we have p(α) ∈ dom Q̇_{α} and p|α ⊩ “p(α) ∈ Q̇_{α}”.

One may think of dom Q̇_{α} as a property of the name Q̇_{α} and even as the actual domain of the set Q̇_{α} if names are formally defined Shoenfield style. But one may also, if one prefers, think of dom Q̇_{α} as merely a notational convention for specifying how the definition of an iteration is to be interpreted in detail. Anyhow we shall require that for any P_{α}-generic set G the interpretation of the name Q̇_{α} is ⋃_{p∈G}{τ_{G} : τ ∈ dom Q̇_{α}, p ⊩ “τ ∈ Q̇_{α}”}.

We consider the conditions of an iteration of length α to be partial functions on α. The support of a condition p is identified with the domain of p. This is convenient in in many situations, but as a technicality (see the proof of Lemma 5.1.2, where this approach is less convenient) we must require that if α = max dom p and p(α) is a name for a maximal element of Q̇_{α} then p ≤ p|α ≤ p.

As exemplified in the last paragraph, iterations are not partial orders in the strict sense, since they are not antisymmetric. But antisymmetry of forcing notions is not needed for the development of the theory of forcing. We consider a forcing notion to be a base set endowed with a transitive and reflexive binary relation ≤. Such objects are also called preorders. The inequality p ≤ q ≤ p defines an equivalence relation which furthermore is a congruence with respect to the relation ≤. Thus there is a canonical partial ordering of the equivalence classes.

We shall say that two forcing notions P and Q are isomorphic if there exists a surjective function h : P → Q such that h(p) ≤ h(q) if and only if p ≤ q for every p and q in P. If P is antisymmetric then such a function h is necessarily injective. If P is not antisymmetric h still induces an injective mapping when we pass to equivalence classes, making the use of the term “isomorphism” permissible.

The following fact is well known. Note that the premises of the lemma imply that P_{α} is a direct limit.

5.1.1. Lemma. Let α be a limit ordinal of cofinality κ and let P be an iteration of length α such that P_{β} satisfies the κ-cc for every β < α. If there exists a stationary set of ordinals β < α such that P_{β} is a direct limit then P_{α} satisfies the κ-cc.

For an iteration P of length α, an ordinal β < α, and a P_{β}-generic set G_{β} we put

P^{β} = {p|[β, α) : p ∈ P and p|β ∈ G_{β}}

and define the ordering of P^{β} as follows: f ≤ g if there exists a condition p ∈ G_{β} such that p ∪ f and p ∪ g are conditions in P and p ∪ f ≤ p ∪ g. We may write P_{γ}^{β} for β < γ < α without fearing ambiguity since

(P_{γ})^{β} = {f|γ : f ∈ P^{β}} = {f ∈ P^{β} : dom f ⊆ γ}

and the orderings of (P_{γ})^{β} and P^{β} agree. If H is P^{β}-generic over V [G_{β}] then the set

G = {p ∈ P : p|β ∈ G_{β} and p|[β, α) ∈ H}

is P -generic over V and G_{β} = G ∩ P_{β}. It is useful to note that when choosing a condition p ∈ P to represent a particular condition p|[β, α) ∈ H, we may assume that p ∈ G. In the situation described above we shall denote G by G_{β} ∗ H. If we identify a condition p ∈ P with the pair (p|β, p|[β, α)) then this is in conformance with standard notation.

Another well known fact is that P^{β} is isomorphic to an iteration R of a sequence (Ṡ_{γ} : γ < η) through an isomorphism h : P^{β} → R such that the conditions

(i) dom(h(f)) = {γ : β + γ ∈ dom f}

(ii) The interpretations of Q̇_{β+γ} and Ṡ_{γ} coincide for every γ < η

hold. We shall say that P^{β} is a direct limit whenever R is a direct limit and similarly for inverse limits. By our definition of isomorphism R is unique only when we pass to equivalence classes. Therefore we shall say that a forcing notion P of length α is an inverse limit if there for every partial function p on α such that p|β ∈ P_{β} for every β < α exists a condition q ∈ P such that q|β ≤ p|β ≤ q|β for every β < α. Then there will be no ambiguity in the statement that P^{β} is an inverse limit.

5.1.2. Lemma. Suppose that P is an iteration of (Q̇_{α} : α < η), β < η, and κ is a regular cardinal. If P satisfies the conditions

(i) For every P_{β}-name τ and condition p ∈ P_{α} where β ≤ α < η, if p ⊩ “τ ∈ Q̇_{α}” then there exists a name σ ∈ dom Q̇_{α} such that p ⊩ “σ = τ”

(ii) P_{α} is either a direct or an inverse limit for every limit ordinal α such that β < α ≤ η

(iii) P_{β} satisfies the κ-cc

(iv) P_{α} is an inverse limit for every limit ordinal α > β such that cf α ≤ κ

then for every limit ordinal α > β, P_{α}^{β} is a direct limit iff P_{α} is a direct limit and P_{α}^{β} is an inverse limit iff P_{α} is an inverse limit.

Proof. The only implication which is not straightforward is the fact that if P_{α} is an inverse limit then P_{α}^{β} is an inverse limit.

So let ḟ be a P_{β}-name for partial function on α such that ⊩ “ḟ|γ ∈ Ṗ^{β}” for all γ < α. By condition (i) we can define a partial function p on α \ β such that

(v) p(γ) ∈ dom Q̇_{γ} for every γ ∈ dom p

(vi) If ⊩ “ḟ(γ) is undefined” then p(γ) is undefined

(vii) ⊩ “γ ∈ dom ḟ → ḟ(γ) = p(γ)”

(viii) ⊩ “p(γ) is maximal in Q̇_{γ} if defined when ḟ(γ) is undefined”.

We claim that p is a condition in P_{α}. It is clear that ⊩ “p(γ) ∈ Q̇_{γ}” for every γ ∈ dom p. It remains to check that dom p conforms with the definition of P_{α}. Let γ be a limit ordinal such that β < γ ≤ α. If P_{γ} is not a direct limit then P_{γ} must be an inverse limit and everything is fine. Suppose now that P_{γ} is a direct limit. We must check that dom(p) ∩ γ is a bounded subset of γ. But

⊩ “dom(ḟ) ∩ γ is a bounded subset of γ”

and thus using conditions (iii) and (iv) we can find an ordinal δ < γ such that ⊩ “dom(ḟ) ∩ γ ⊆ δ”. By condition (vi) we have now seen that p ∈ P_{α}.

Fix a P_{β}-generic set G_{β} and let f be the interpretation of ḟ. Now p ∈ P_{α}^{β} since p|β is the empty function. Fix an arbitrary ordinal γ such that β < γ < α. Since f|γ ∈ P^{β} there exists a condition q ∈ P such that q|β ∈ G_{β} and f|γ = q|[β, α). Also (q|β)∪(p|γ) is a condition in P and q ≤ (q|β)∪(p|γ) ≤ q. Therefore f|γ ≤ p|γ ≤ f|γ.

□

Suppose now that M and N are transitive classes or sets and j : M → N is an elementary embedding. Also assume that P ∈ M where P is an iteration of length α and j(P)_{β} = P where β < j(α) and the models M and N satisfy a large enough finite fragment of ZFC to make the use of the forcing theorem possible. Let G be P -generic over both M and N and let H be j(P)^{β}-generic over N[G]. As noted earlier N[G][H] is a generic extension by the forcing notion j(P). If it in addition holds that j(p)|[β, α) ∈ H for every p ∈ G then the equation

j(τ_{G}) = j(τ)_{G∗H}

holds for every P -name τ such that τ_{G} ∈ M. Furthermore, if the equation is applied to the class of all P -names τ, taking the right hand side as a definition of the left hand side, it defines an elementary embedding j : M[G] → N[G][H]. It is natural and convenient to denote this extended embedding by j too, since its restriction to M is the original j. Obviously j(G) = G ∗ H.

Consider the following game between players I and II on a forcing notion P. The length of the game is an ordinal α. The players take turns picking conditions in P that will eventually form a decreasing sequence (p_{i} : i < α) if player II is to win the game. The conditions p_{i} are picked in increasing order of the index so that player I picks p_{i} for i even and player II for i odd.

If the game successfully goes on for α rounds then player II wins. The game ends in the victory of player I if the moves played at some point form a decreasing sequence that has length less than α but no lower bound in P, making the next move impossible. If κ is a regular cardinal and player II has a winning strategy in this game for every α < κ then the forcing notion P is said to be κ-game closed. Let 𝒢_{α}(P) denote the game described above.

5.1.3. Lemma. Let P be an iteration of the sequence (Q̇_{α} : α < η) and let κ be a regular cardinal. If P satisfies the conditions

(i) For every α < η and β < κ there exists a name τ_{β} for a winning strategy for player II in 𝒢_{β}(Q̇_{α}) such that given any sequence (ρ_{i} : i < ξ) in the ground model of P_{α}-names for elements of Q̇_{α}, there exists a name ρ ∈ dom Q̇_{α} such that ⊩ “ρ = τ_{β}(ρ_{i} : i < ξ)”

(ii) If γ ≤ η is a limit ordinal such that P_{γ} is a direct limit then cf γ ≥ κ

(iii) If γ ≤ η is a limit ordinal such that P_{γ} is an inverse limit then there exists a strictly increasing continuous sequence (γ_{i} : i < ζ) such that sup_{i<ζ} γ_{i} = γ and P_{γ_{i}} is an inverse limit for every limit ordinal i < ζ

then P is κ-game closed.

Proof. Fix β < κ. By induction on α ≤ η we shall construct winning strategies σ_{α} : [P_{α}]^{<κ} → P_{α} for player II in 𝒢_{β}(P_{α}) such that

(iv) (σ_{α}(p_{i} : i < ξ))|γ = σ_{γ}(p_{i}|γ : i < ξ) for all γ < α

(v) dom(σ_{α}(p_{i} : i < ξ)) = ⋃_{i<ξ} dom p_{i}.

Suppose P_{α} is a direct limit. Then cf α ≥ κ whereby [P_{α}]^{<κ} = ⋃_{γ<α}([P_{γ}]^{<κ}). By conditions (iv) and (v) the winning strategies constructed at earlier stages extend each other. Thus we can put σ_{α} = ⋃_{γ<α} σ_{γ}. This is also the only possible way to construct σ_{α} if conditions (iv) and (v) are to remain true.

Let us then deal with the case where P_{α} is an inverse limit. Now we must have ⋃_{γ<α} σ_{γ} ⊆ σ_{α} but the inclusion is proper. Suppose that (p_{i} : i < ξ) is a decreasing sequence such that ⋃_{i<ξ} dom p_{i} is an unbounded subset of α. It is straightforward to check that putting

σ_{α}(p_{i} : i < ξ) = ⋃_{γ<α} σ_{γ}(p_{i}|γ : i < ξ)

does the job.

Finally suppose that α = γ + 1. We shall define σ_{α}(p_{i} : i < ξ) by induction on ξ. Since player II moves on odd indexes ξ is a successor ordinal in all relevant cases. Condition (i) is tailored for this step. We consider (ρ_{i} : i < ξ) to be (p_{i}(γ) : i < ξ) and put

σ_{α}(p_{i} : i < ξ) = σ_{γ}(p_{i}|γ : i < ξ)^{⌢}(ρ)

where ρ is as in condition (i). □

It is evident that if P is κ-game closed then P is α-Baire for every α < κ. That P is α-Baire is equivalent to the fact that forcing with P does not add any new functions with domain α. Obviously κ-closed forcing notions are κ-game closed.

Note that from the last proof we can extract a somewhat stronger result than the one appearing in the formulation of Lemma 5.1.3. Namely conditions (iv) and (v) can be taken as properties of the final winning strategy σ_{η} that are often useful in applications of the lemma. Let us call a winning strategy with the above mentioned properties uniform. In Section 5.4 we shall need to refer to a uniform winning strategy.

For a regular cardinal κ and a cardinal λ ≥ κ, let C_{κ}(λ) denote the standard forcing notion for adding λ many Cohen subsets of κ. Formally we shall view C_{κ}(λ) as the set of all partial functions f : λ → 2 such that |f| < κ, ordered by reverse inclusion. Sometimes it is convenient, for technical reasons, to replace λ by a set S of cardinality λ. Of course C_{κ}(S) is then isomorphic to C_{κ}(λ).

An iterated forcing notion P is said to have Easton support if the only restriction placed on the supports is that |dom(p) ∩ μ| < μ for every p ∈ P and every regular cardinal μ. In terms of limits, Easton support means taking direct limits at regular cardinals and inverse limits otherwise.

5.2. Weakly compact diamond by forcing

Let κ be a weakly compact cardinal and let P = P_{κ+1} be the Easton support iteration of (Q̇_{α} : α ≤ κ) where Q̇_{α} is a name for C_{α}(α) whenever α is an Mahlo cardinal and a name for the trivial forcing notion {1} otherwise.

The forcing notion P defined above is due to Jack Silver (unpublished) who used ideas originating from unpublished work of Ronald Jensen (see [14]). This technique is known as reverse Easton forcing and was originally used to violate the GCH at various large cardinals. It is well known that κ remains weakly compact in the generic extension V^{P}. The proof of Lemma 5.2.3 is essentially a proof of the preservation of weak compactness augmented with the ideas from the standard proof of the fact that ◇_{κ} holds in an extension by C_{κ}(κ).

Let us assume that the sets dom Q̇_{α} contain all relevant names up to equivalence and have the lowest possible rank.

5.2.1. Lemma. If α ≤ κ is inaccessible then P_{α} ⊆ V_{α} and if α is a Mahlo cardinal then P_{α} is α-cc, dom Q̇_{α} ⊆ V_{α}, and |P_{α+1}| = α.

Proof. Let α ≤ κ be inaccessible. Using the assumption made about the sets Q̇_{β} it is straightforward to check by induction that P_{β} ∈ V_{α} for all β < α. Since P_{α} is a direct limit P_{α} ⊆ V_{α}.

Suppose now that α is a Mahlo cardinal. Then the set of ordinals β < α such that P_{β} is a direct limit is stationary. By Lemma 5.1.1 P_{α} satisfies the α-cc. Using this fact and the assumption about dom Q̇_{α} being of lowest possible rank, it is again straightforward to see that dom Q̇_{α} ⊆ V_{α}. It follows that |P_{α+1}| = α. □

It follows from the previous lemma that any Mahlo cardinal α ≤ κ remains a regular cardinal in an extension by P_{α}. In fact we were a bit sloppy with the definition of P since the use of the notation C_{α}(α) assumed α to be regular.

5.2.2. Lemma. If α < κ is a Mahlo cardinal then V^{P_{α+1}} and V^{P} have the same subsets of α.

Proof. By Lemma 5.2.1 P_{α+1} satisfies the α^{+}-cc and thus by Lemma 5.1.2 for limit ordinals γ such that α + 1 < γ ≤ κ it holds that P_{γ}^{α+1} is a direct limit when γ is regular and an inverse limit otherwise. Still by the fact that P_{α+1} satisfies the α^{+}-cc all regular cardinals above α remain regular in V^{P_{α+1}}. Since

⊩ “Q̇_{γ} is α^{+}-closed”

for all γ such that α+1 ≤ γ < κ we see that P^{α+1} is isomorphic to an iteration that is α^{+}-game closed by Lemma 5.1.3. Note that condition (i) of Lemma 5.1.3 is trivially fulfilled since the names dom Q̇_{γ} contains all relevant names up to equivalence. □

5.2.3. Lemma. If E is a weakly compact subset of κ in the ground model then ◇(WC_{κ}|E) holds in V^{P}.

Proof. Given a P -generic set G we obtain a function g : κ → 2 by putting g = ⋃_{p∈G} p(κ)_{G_{κ}}. Define a function F on κ by

F (α) = {β < α : g(∑_{i<α} i + β) = 1}

and let Ḟ be a P -name for F. (The sums appearing in the definition of F are ordinal sums.) We shall show that F is the required sequence.

Let ϕ be a Π_{1}^{1}-sentence, let U̇ be a P -name for a subset of V_{κ} of the generic extension, and suppose that

p ⊩ “⟨V_{κ}, ∈, U̇⟩ ⊧ ϕ”  (9)

for some p ∈ P. Let Ȧ be a P -name for an arbitrary subset of κ. The lemma will be proved if we can find a condition q ≤ p such that

q ⊩ “Ȧ ∩ α = Ḟ (α), α ∈ Ě, and ⟨V_{α}, ∈, U̇ ∩ V_{α}⟩ ⊧ ϕ”  (10)

for some ordinal α < κ. Let us assume that the names Ḟ, U̇, and Ȧ discussed above have the lowest possible rank.

Using the reflection principle we can pick a set M′ of cardinality κ that reflects the formulae (9) and (10). Let us also assume that

V_{κ} ∪ {P, E, Ḟ , U̇, Ȧ} ⊆ M′

and M′ satisfies a large enough finite fragment of ZFC. Let π : M′ → M be the Mostowski collapse of M′. By Lemma 5.2.1 and similar arguments the sets P, E, Ḟ, U̇, and Ȧ and their respective elements remain fixed under the isomorphism π.

Using Lemma 4.1.4 we pick a transitive set N and an elementary embedding j : M → N such that κ ∈ j(E). The property of being a Mahlo cardinal is obviously preserved when moving to submodels. Therefore κ is a Mahlo cardinal in N.

By Lemma 5.2.1 applied on j(P) in N we have j(P)_{κ} = j(P) ∩ V_{κ} = P_{κ}. We can assume that the sets dom Q̇_{α} used in the definition of P are picked in a canonical way using some well ordering in V_{κ+1}. Absoluteness then gives j(P)_{κ+1} = P_{κ+1} = P whereby P ∈ N. Let p ∈ P be arbitrary. Since P_{κ} is a direct limit there exists an ordinal α < κ such that p|κ = p|α. By the properties of j and the fact that p|α ∈ V_{κ} we then have j(p)|j(κ) = j(p|κ) = j(p|α) = p|α.

Fix some P -generic set G. From the above discussion we know that j(p)|(κ+1) ∈ G for every p ∈ G and therefore

D = {j(p)|[κ + 1, j(κ) + 1) : p ∈ G}

is a subset of j(P)^{κ+1}. In fact every nonempty function in D is defined only on the singleton {j(κ)}. Using the fact that G is a filter it is straightforward to see that D is a directed set.

We define a function d on {j(κ)} by letting d(j(κ)) be a P -name for a function h : κ + κ → 2 such that ⋃_{s∈G} s(κ)_{G_{κ}} ⊆ h and

Ȧ_{G} = {α < κ : h(κ + α) = 1}.

This is possible since {s(α)_{G_{κ}} : s ∈ G} is C_{κ}(κ)-generic over V [G_{κ}]. Because P - names are also j(P)_{j(κ)}-names, we can require that d(j(κ)) ∈ dom(j(Q̇_{κ})). So d is in fact a condition in j(P). Because d = d|[κ + 1, j(κ) + 1) we even have d ∈ j(P)^{κ+1}.

Let f = j(p)|[κ + 1, j(κ) + 1) be an arbitrary element of D. Thus we naturally assume that p ∈ G. As noted before dom f = {j(κ)} unless f is the empty function. By the definition of d we have p(κ)_{G_{κ}} ⊆ d(j(κ))_{G} and thus there is a condition q ∈ G such that q ⊩ “p(κ) ⊆ d(j(κ))”. We may assume that q ≤ p. Now both q ∪ d and q∪f are conditions in j(P) and q∪d ≤ q∪f. Thus d ≤ f in the ordering of j(P)^{κ+1}.

Let H be a j(P)^{κ+1}-generic set over V [G] such that d ∈ H. Since d extends every condition in D we have D ⊆ H and thus by the definition of D it is clear that j(p)|[κ + 1, j(κ) + 1) ∈ H for every p ∈ G. So the embedding j can be extended to an embedding j : M[G] → N[G][H] via the definition discussed in Section 5.1.

Let us now proceed with the argument outlined in the beginning of the proof. Suppose that G is generic over V and p ∈ G where p is a condition satisfying (9).

In V [G] it holds that ⟨V_{κ}, ∈, U̇_{G}⟩ ⊧ ϕ. By absoluteness of V_{κ} and downward absoluteness of universal statements this holds in N[G] too. Lemma 5.2.2 can be applied with M as the ground model and by elementarity it can therefore be applied on j(P) with N as the ground model. Since κ is a Mahlo cardinal N[G][H] contains no subsets of κ besides those in N[G]. Furthermore V_{κ} is absolute with respect to the generic extensions under discussion. We can conclude that the statement ⟨V_{κ}, ∈, U̇_{G}⟩ ⊧ ϕ holds in N[G][H].

In addition j(Ȧ)_{G∗H}∩κ = j(Ȧ_{G})∩κ = Ȧ_{G}. Since d ∈ G∗H we have j(Ȧ)_{G∗H}∩κ = j(Ḟ)_{G∗H}(κ). Similarly j(U̇_{G}) = j(U̇)_{G∗H} and we have j(U̇)_{G∗H} ∩ V_{κ} = U̇_{G}. Thus it holds in N[G][H] that there exists an ordinal α < j(κ) such that

j(Ȧ)_{G∗H} ∩ α = j(Ḟ)_{G∗H}(α), α ∈ j(E) and ⟨V_{α}, ∈, j(U̇)_{G∗H} ∩ V_{α}⟩ ⊧ ϕ.

By elementarity it is true in M[G] that there exists an ordinal α < κ such that

Ȧ_{G} ∩ α = Ḟ_{G}(α), α ∈ E and ⟨V_{α}, ∈, U̇_{G} ∩ V_{α}⟩ ⊧ ϕ

and therefore it is true in M that there exists an ordinal α < κ and a condition q ≤ p such that (10) holds.

Through the fact that the collapsing isomorphism π : M′ → M does not move the parameters involved, (10) holds in M′. As M′ was chosen to reflect (10) the same holds in V. □

Thus we have proved the following result that was announced without proof in [16] in the case E = κ. Hamkins [7] has independently obtained related results.

5.2.4. Theorem. If E is a weakly compact subset of κ then there exists a generic extension in which ◇(WC_{κ}|E) holds.

Note that in the above proof we needed to argue that (10) holds in V since we started out by assuming that (9) holds in V. Had we only assumed that (9) holds in M, we would have had trouble arguing that (9) holds in N, since for all we know N might contain subsets of V_{κ} that are not present in M.

There is a well known version of Lemma 5.1.3, where κ-game closed forcing notions are replaced by forcing notion that are α-directed closed for all α < κ. For Lemma 5.2.3 this result would have sufficed. Furthermore, if we were only interested in the preservation of weak compactness, we could have picked the condition d in the proof of Lemma 5.2.3 just by referring to the fact that D is a small enough directed set.

5.3. Killing a weakly compact set

It follows from Lemma 5.2.3 that the forcing notion of the previous section does not only preserve weak compactness of κ, but preserves every weakly compact set. Can one kill a weakly compact set by forcing while preserving the weak compactness of κ? We shall now give an affirmative answer to this question. It even turns out that there are no extraneous requirements on the set to be killed. In the spirit of [3] it follows that a set is not weakly compact because of some intrinsic property of the set itself.

For A ⊆ κ, we shall denote the tree consisting of bounded 1-closed subsets of A by T^{1}(A). (We consider T^{1}(A) to be ordered by end extension.) Let E be a weakly compact subset of κ. Let P = P_{κ+1} be the Easton support iteration of (Q̇_{α} : α ≤ κ) where Q̇_{α} is a name for T^{1}(E ∩ α) whenever α is Mahlo and E ∩ α is an unbounded subset of α. For other ordinals α ≤ κ we let Q̇_{α} be a name for the trivial forcing notion. Be aware that we are reusing the letter P and some other symbols, but there should be no confusion since we shall not make any explicit references to the earlier forcing notion. Rather it is convenient since we can refer to the fact that some of the lemmas in Section 5.2 hold verbatim for the forcing notion defined above.

We are interested in the case were both E and its complement are weakly compact. We claim that forcing with P kills the weak compactness of κ \ E while preserving the weak compactness of κ. We would like to emphasise that no other requirements are placed on E. Note also that the situation at hand is drastically different from killing stationary sets by forcing. Stationary sets can never, at a later stage, regain the property of being stationary once it is lost. This is not the case with weak compactness.

5.3.1. Lemma. If α is regular and A is an unbounded subset of α then the set D_{β} = {p ∈ T^{1}(A) : sup(p) ≥ β} is dense in T^{1}(A) for every β < α and T^{1}(A) is α-game closed.

Proof. It is immediately clear that the sets D_{β} are dense since it suffices to add one point to a condition in order to extend it to a condition in one of the sets D_{β}.

Player II can ensure a victory already in the first round of the game 𝒢_{β}(T^{1}(A)). If p_{0} is the first move made by player I and there exists an inaccessible cardinal γ such that β ≤ γ < α then player II chooses p_{1} ∈ D_{γ}. It is then impossible for player II to loose since any limit ordinal j ≤ β has small enough cofinality so that ⋃_{i<j} sup(p_{i}) can not be inaccessible unless the sequence (p_{i} : i < j) is eventually constant. In either case ⋃_{i<j} p_{i} is a condition in T^{1}(A) and a valid move for player I. □

As in Section 5.2 we shall assume that the sets dom Q̇_{α} used in the definition of P contain all relevant names up to equivalence but are of lowest possible rank. Both the formulation and the proof of Lemma 5.2.1 hold verbatim for the forcing notion P of this section. The same goes for Lemma 5.2.2 except that in the proof we must refer to Lemma 5.3.1 rather than the fact that C_{γ}(γ) is α-closed for every regular γ above α.

5.3.2. Lemma. Every set S ⊆ E that is weakly compact in the ground model is weakly compact in V^{P}.

Proof. The argument we give is very similar to the proof of Lemma 5.2.3 if we omit the references to the name Ȧ. Thus, given a Π_{1}^{1}-sentence ϕ and a P -name U̇ for a subset of V_{κ}, then for every condition p ∈ P such that

p ⊩ “⟨V_{κ}, ∈, U̇⟩ ⊧ ϕ”  (11)

we shall find a condition q ≤ p and an ordinal α < κ such that

q ⊩ “α ∈ Š, and ⟨V_{α}, ∈, U̇ ∩ V_{α}⟩ ⊧ ϕ”.  (12)

We choose the models M and N and the elementary embedding j : M → N exactly as in the proof of Lemma 5.2.3 except that let S take the role of E so that κ ∈ j(S).

Again it is clear that j(P)_{κ} = P_{κ} and we use absoluteness to argue that j(P)_{κ+1} = P. This time we have the parameter E involved in the definition, but it causes no trouble since j(E) ∩ κ = E.

Now we define the function d on {j(κ)} by letting d(j(κ)) be a P -name for ⋃_{s∈G} s(κ)_{G_{κ}} ∪ {κ}. Fixing a P -generic set G we let C denote d(j(κ))_{G}. Working in V [G] we shall conclude the proof by checking that C ∈ T^{1}(j(E)). This suffices since one can then proceed as in the proof of Lemma 5.2.3, picking a j(P)^{κ+1}-generic set H over V [G] such that d ∈ H and defining the extended elementary embedding j : M[G] → N[G][H].

Directly by the definition it is clear that C ⊆ j(E). Since being 1-closed is upwards absolute C ∩ γ is 1-closed for every γ < κ. It follows that C is 1-closed as a subset of j(κ) since C ∩ κ is an unbounded subset of κ by Lemma 5.3.1. □

5.3.3. Lemma. The weakly compact filter of V^{P} contains E and every set that was in the weakly compact filter of the ground model.

Proof. Let G be a P -generic set. The 1-club subsets of κ in the ground model remain 1-closed in V^{P}. Likewise C = ⋃_{p∈G} p(κ)_{G_{κ}} is 1-closed as a subset of κ by similar arguments as in the proof of the previous lemma. It remains to prove that the above mentioned sets are stationary in V [G] which in fact implies that they are weakly compact.

If X is in the weakly compact filter in the ground model then X ∩ E is weakly compact in the ground model and remains weakly compact in V^{P} by Lemma 5.3.2. In the setting of the proof of Lemma 5.3.2 it holds that

j(C) = ⋃_{p∈G∗H} p(j(κ))_{(G∗H)_{j(κ)}}

and thus d(j(κ))_{G} ⊆ j(C). It follows that κ ∈ j(C). We can then redo the argument yielding Lemma 5.3.2 with C replacing S, and letting Ċ be a name for C we obtain

q ⊩ “α ∈ Ċ, and ⟨V_{α}, ∈, U̇ ∩ V_{α}⟩ ⊧ ϕ”

rather than (12). □

Summarising what was said in Lemmas 5.3.2 and 5.3.3 we have the following result:

5.3.4. Theorem. If both S and its complement are weakly compact subsets of κ then there exists a generic extension in which the weakly compact ideal is proper, extends the weakly compact ideal of the ground model, and contains the set S.

5.4. Killing weakly compact diamond

Let E denote the set of regular cardinals below κ and let m and n be positive natural numbers. Hauser [8] has proved that it is consistent that ◇_{κ}(E) fails although κ is Π_{n}^{m}-indescribable and GCH holds. Of course the consistency of the existence of a Π_{n}^{m}-indescribable cardinal must be assumed, and the result is an equiconsistency result, to be more accurate. When κ is weakly compact, E is in the weakly compact filter, so NS_{κ}|E ⊆ WC_{κ} whereby ◇_{κ}(WC) implies ◇_{κ}(E). Thus ◇_{κ}(WC) may fail at a weakly compact cardinal. In this section we shall prove the consistency of the failure of the weakly compact diamond in a more explicit way.

Suppose that α is a Mahlo cardinal, Ā is an α-sequence (A_{i} : i < α), and S is a set of ordinals. Let K_{S}^{1}(Ā) be the collection of pairs (C, A) such that

(i) C is a 1-closed and bounded subset of α

(ii) A ⊆ sup C

(iii) If i ∈ C ∩ S then A ∩ i ≠ A_{i}.

We order K_{S}^{1}(Ā) by letting (C′, A′) ≤ (C, A) if the conditions

(iv) C′ ∩ (sup(C) + 1) = C

(v) A′ ∩ sup C = A

hold.

Assume that V = L and S is a weakly compact subset of κ. We shall define an iteration P of (Q̇_{α} : α < κ^{+}). The limit P_{α} is direct if α is regular or if cf α is Mahlo and α < (cf α)^{+}. All other limits are inverse.

Simultaneously with the sequence (Q̇_{α} : α < κ^{+}) we shall define a sequence (τ_{α} : α < κ^{+}). If there exists a Mahlo cardinal β ≤ κ such that β ≤ α < β^{+} then (τ_{α}, Q̇_{α}) is the <_{L}-least pair such that

(vi) τ_{α} is a P_{α}-name for a subset sequence indexed by β

(vii) (τ_{α}, Q̇_{α}) ≠ (τ_{ξ}, Q̇_{ξ}) whenever β ≤ ξ < α

(viii) Q̇_{α} is a name for K_{S}^{1}(τ_{α})

(ix) dom Q̇_{α} is the set of all relevant P_{β}-names up to equivalence and is of lowest possible rank

if such a pair exists. Otherwise τ_{α} is arbitrary and Q̇_{α} is the canonical name for the trivial forcing notion. If the conditions above hold then for any s ∈ P_{α+1}, C_{s(α)} and A_{s(α)} denote P_{α}-names such that

s|α ⊩ “š(α) = (C_{s(α)}, A_{s(α)})”.

By conditions (viii) and (ix) C_{s(α)} and A_{s(α)} can actually be assumed to be P_{β}-names and the condition s|α could be replaced by s|β above.

We shall often encounter sets written as ⋃_{s∈G_{α+1}} C_{s(α)} or ⋃_{s∈G_{α+1}} A_{s(α)} where G is a fixed generic set. In this setting C_{s(α)} and A_{s(α)} do not denote the names but rather their interpretations by G_{α} where G is the same generic set as in the subscript of the union symbol.

Again both the statement and the proof of Lemma 5.2.1 holds verbatim for our latest forcing notion. But now Lemma 5.2.1 does not tell us anything about P itself since P has length κ^{+} and not κ + 1 as was the case with the forcing notions dealt with earlier. Nevertheless our forthcoming arguments rely on the fact that P to some extent resembles an iteration of length κ + 1. The idea is that for a Mahlo cardinal β ≤ κ one can think of the steps between β and β^{+} as a single step consisting of a product forcing.

5.4.1. Lemma. Let β ≤ κ be a Mahlo cardinal. In V^{P_{β}} there is an embedding P_{β^{+}}^{β} → C_{β}(β^{+}) that preserves incompatibility.

Proof. We shall find an embedding into C_{β}((β^{+} \β)×2×β) rather than C_{β}(β^{+}). Fix a P_{β}-generic set G_{β}. Given a condition p ∈ P_{β^{+}} such that p|β ∈ G_{β} we shall define a partial function f : (β^{+} \ β) × 2 × β → 2. For α ∈ dom p let C_{α} and A_{α} denote the interpretations of the P_{β}-names C_{p(α)} and A_{p(α)} respectively.

If α ∈ dom(p) \ β then f|({α} × {0} × β) is like the characteristic function of C_{α} with domain {α} × {0} × sup(C_{α} + 1). Furthermore f|({α} × {1} × β) has domain {α} × {1} × sup(C_{α}) and is like the characteristic function of A_{α}. If α ≥ β but α ∉ dom p then dom(f|({α} × 2 × β)) is empty. The embedding i is defined by letting p|[β, β^{+}) be mapped to f.

Suppose p and q are conditions in P_{β^{+}} such that p|β and q|β are in G_{β}. Let f = p|[β, β^{+}) and g = q|[β, β^{+}) and suppose that i(f) and i(g) are compatible in C_{β}((β^{+} \ β) × 2 × β). Then for α ∈ (dom(p) ∩ dom(q)) \ β we have either

i(f)|({α} × 2 × β) ⊆ i(g)|({α} × 2 × β)

or vice versa. We define a function h by letting h(α) = f(α) if the situation is as above and h(α) = g(α) if the subset relation is reversed. For ordinals α ≥ β such that only one of f(α) and g(α) is defined we let h(α) equal the defined one.

There exists is common extension r ∈ G_{β} of p|β and q|β such that r ∪ h is a common extension of r ∪ f ≤ r ∪ g and thereby a common extension of p and q. For P_{β^{+}}^{β} it follows that h is a common extension of f and g. □

5.4.2. Lemma. Let β ≤ κ be a Mahlo cardinal. P_{β^{+}} satisfies the β^{+}-cc.

Proof. Since C_{β}(β^{+}) satisfies the β^{+}-cc, Lemma 5.4.1 implies that P^{β} satisfies the β^{+}-cc in V^{P_{β}}. On the other hand P_{β} even satisfies the β-cc by the analogue of Lemma 5.2.1. □

5.4.3. Lemma. If α is regular and Ā is an α-sequence then the set D_{β} = {(C, A) ∈ K_{S}^{1}(Ā) : sup(C) ≥ β} is dense in K_{S}^{1}(Ā) for every β < α and K_{S}^{1}(Ā) is α-game closed.

5.4.4. Lemma. If β < κ is a Mahlo cardinal then V^{P_{β^{+}}} and V^{P} have the same subsets of β.

Proof. Because P_{β^{+}} is β^{+}-cc by Lemma 5.4.2, we can apply Lemma 5.1.2 to P^{β^{+}}. Thus for limit ordinals γ such that β^{+} < γ < κ^{+} the limit P_{γ}^{β^{+}} is direct if γ is regular or cf γ is Mahlo and γ < (cf γ)^{+}. We are going to finish the proof by showing that an application of Lemma 5.1.3 yields that P^{β^{+}} is b^{+}-game closed.

Clearly conditions (ii) and (iii) of Lemma 5.1.3 are fulfilled. For condition (i) we need to note that the winning strategy is definable using parameters only from the ground model, but the definition is indeed very simple, as was seen in the proof of Lemma 5.4.3. □

As an immediate corollary of the previous lemma is the following:

5.4.5. Lemma. The name Q̇_{α} represents a non-trivial forcing notion for every ordinal α < κ^{+} such that cf α is Mahlo and α < (cf α)^{+}.

We shall develop an argument similar to the proofs of Lemmas 5.2.3 and 5.3.2 to show that P preserves weakly compact sets. Since the details are slightly more involved we shall break it into a sequence of lemmas and fix some mathematical objects for the duration of this section as we go along.

Assume that E is weakly compact subset of κ and

p ⊩_{κ^{+}} “⟨V_{κ}, ∈, U̇⟩ ⊧ ϕ”  (13)

where p ∈ P, U̇ is a nice name for a subset of κ and ϕ is a Π_{1}^{1}-sentence. By Lemma 5.4.2 there exists an ordinal η < κ^{+} such that p ∈ P_{η} and U̇ is a P_{η}-name. Fix an arbitrary η < κ^{+} with these properties. For convenience we shall assume that η ≥ κ.

5.4.6. Lemma. p ⊩_{ζ} “⟨V_{κ}, ∈, U̇⟩ ⊧ ϕ” whenever η ≤ ζ ≤ κ^{+}.

Proof. This is because V_{κ} is absolute w.r.t. V^{P_{ζ}} and V^{P_{κ^{+}}}, p ∈ G_{ζ} implies p ∈ G_{κ^{+}}, and Π_{1}^{1}-truth is downward absolute. □

Pick a set M′ of cardinality κ that reflects the formulae (13) and

q ⊩ “α ∈ Š, and ⟨V_{α}, ∈, U̇ ∩ V_{α}⟩ ⊧ ϕ”  (14)

such that ^{<κ}M′ ⊆ M′ and M′ contains a bijection κ → η. Furthermore make sure that M′ satisfies a large enough finite fragment of ZFC, V_{κ} ⊆ M′, η + 1 ⊆ M′, and E, S, P_{η}, and U̇ are elements of M′. Let π : M′ → M be the Mostowski collapse.

5.4.7. Lemma. For every α ≤ η we have

(a) P_{α} ⊆ M′

(b) π|P_{α} = id_{P_{α}}

(c) π(P_{α}) = P_{α}

and furthermore E, S, and U̇ are mapped on themselves by π.

Proof. (c) always follows from the others, and all is clear for α ≤ κ. For α > κ we proceed by induction. Inverse limits work since ^{<κ}M′ ⊆ M′. For α = β + 1 we (just) need to show that dom Q̇_{β} ⊆ M′ and π| dom Q̇_{β} = id. But this is clear since M′ contains all antichains of P_{κ}. □

By Lemma 4.1.6 we can find a transitive set N and an elementary embedding j : M → N so that ^{<κ}N ⊆ N and κ ∈ j(E). Let ζ = (κ^{+})^{N}. By our choice of M′ we have η < ζ < j(κ).

5.4.8. Lemma. j(P_{η})_{α} = P_{α} for all α ≤ ζ.

Proof. The conclusion is immediate for α ≤ κ. We proceed by induction. Since <κN ⊆ N cofinality is absolute for ordinals α < ζ. For the same reason cf ζ = κ (|N| = κ by its definition). Thus the choices between direct and inverse limits are the same for j(P_{η})_{α} and P_{α}.

Suppose that P_{α} is an inverse limit and p ∈ P_{α}. Again ^{<κ}N ⊆ N together with the induction hypothesis implies that p ∈ j(P_{η})_{α}. We have now handeled the limit steps of the induction.

Now suppose that α = β +1 > κ. Here we just need to check that the conditions (vi)–(ix) in the definition of P are absolute for N. But these conditions can be rendered to refer to formulae in the forcing language only in quantifier free ways. □

We shall need to refer to the above lemma for α = η, but note that η can be arbitrarily large.

We define a function d on {j(α) : κ ≤ α < η} by letting d(j(α)) be a P_{α+1}-name for the pair (⋃_{s∈G_{α+1}} C_{s(α)} ∪ {κ}, ⋃_{s∈G_{α+1}} A_{s(α)}). Furthermore we choose d(j(α)) from dom(j(Q̇_{α})) whenever possible.

5.4.9. Lemma. Suppose that κ ≤ ν ≤ η. If there exists a condition p ∈ G_{j(ν)} such that dom d ⊆ dom p ⊆ [ζ, j(η)) and

p|j(α) ⊩ “d(j(α)) ∈ Q̇_{j(α)} and p(j(α)) ≤ d(j(α))”

for every α ∈ [κ, ν) then j extends to an elementary embedding j : M[G_{ν}] → N[G_{j(ν)}].

Proof. For notational simplicity we shall assume that ν = η. For other ordinals ν the proof is the same apart from taking the appropriate restrictions. We need to show that j(q) ∈ G_{j(η)} whenever q ∈ G_{η}. We shall first show that q ∪ p ≤ j(q) and then finish the proof by checking that q ∪ p ∈ G_{j(η)} (we know that q ∪ p ∈ j(P_{η}) since η < j(κ)).

First note that |dom q| < κ whereby dom(j(q)) = j[dom q]. We also have j(q)|j(κ) = j(q)|κ = q|κ. Thus q ≤ j(q)|j(κ) and dom(j(q)) ⊆ dom(q ∪ d). Since j(q(α)) = q(α) for all α ∈ dom q we have q ∪ p|j(α) ⊩ “d(j(α)) ≤ j(q(α))” for all α ∈ dom(q) ∩ [κ, η). Thus q ∪ p ≤ j(q). Finally there is a common extension of q and p in G_{j(η)} and since q ∪ p = inf{q, p} it is clear that q ∪ p ∈ G_{j(η)}. □

5.4.10. Lemma. Suppose that κ ∉ j(S) and κ ≤ ν ≤ η. Then d|j(ν) ∈ j(P_{ν}).

Proof. First note that {j(α) : κ ≤ α < η} = j(f)|κ where f is any function in M that maps κ onto [κ, η), and therefore dom d ∈ N. It is also easy to check that dom d is bounded below every j(κ)-cofinal cardinal.

By Lemma 5.4.8 the P_{α+1}-name d(j(α)) is also a j(P_{ν})_{j(κ)}-name for all α ∈ [κ, η). We shall now proceed by induction on ν. The limit steps are clear by the facts mentioned above. Suppose that ν = α + 1. By the induction hypothesis d|j(α) ∈ j(P_{α}) and by Lemma 5.4.9 it follows that

d|j(α) ⊩ “j(τ_{α})_{i} = (τ_{α})_{i}”  (15)

for all i < κ. Here the subscript i is used to indicate the ith element of the sequence that is the interpretation of the name in question. By the induction hypothesis we only need to worry about d(j(α)) to see that d|j(ν) ∈ j(P_{ν}). By (15) and the fact that κ ∉ j(S) we are basically done. We are left with checking that

d|j(α) ⊩ “⋃_{s∈G_{α+1}} C_{s(α)} ∪ {κ} is 1-closed”

but this is similar to the last paragraph of the proof of Lemma 5.3.2 and relies on Lemma 5.4.3. □

5.4.11. Lemma. If E ⊆ κ is disjoint from S and weakly compact in the ground model then ◇(WC_{κ}|E) holds in V^{P}.

Proof. Let Ȧ be an arbitrary P -name for a subset of κ. Since η was chosen to be arbitrary large we can assume that Ȧ is a P_{η}-name. Since we are now assuming that E is disjoint from S we have κ ∉ j(S) and therefore d is a condition in j(P_{η}) by Lemma 5.4.10.

Fix a P_{η}-generic set G and consider the set B ⊆ κ+κ such that B∩κ = (A_{d(j(κ))})_{G} and Ȧ_{G} = {α < κ : κ + α ∈ B}. Fix an ordinal γ < j(κ) such that κ + κ ≤ γ and γ ∉ j(S). Such ordinals certainly exists since j(E) and j(S) are disjoint. Let q be a condition that is exactly like d except that q(j(κ)) is a name for the pair (B, (C_{d(j(κ))})_{G} ∪ {γ}). Clearly q ≤ d.

Now the proof proceeds as the proof of Lemma 5.4.18 except that we assume that q ∈ G_{j(η)}. Using Lemma 5.4.10 the claim follows in similar fashion as we obtained the result of Lemma 5.2.3; the diamond sequence is essentially the set ⋃_{s∈G_{κ+1}} A_{s(κ)}.

□

One could hope to be able to prove the preservation of every weakly compact set using Lemma 5.4.9. The argument would be similar as the one in the proof of Lemma 5.4.10. But now we face the possibility that κ ∈ j(S), when trying to inductively extend the condition p of Lemma 5.4.9. So in addition we would have to find a condition q ∈ j(P_{ν}) such that q ≤ p, dom(q) ∩ ζ is empty, and

q ⊩ “⋃_{s∈G_{α+1}} A_{s(α)} ≠ j(τ_{α})_{κ}”

so that we could define an extended condition p′ by letting p′|j(ν) = q and p′(j(ν)) = d(j(ν)).

However, without any substantial restrictions on which names τ_{α} can be chosen in the definition of P, we will inevitably face a situation in which

p ⊩ “⋃_{s∈G_{α+1}} A_{s(α)} = j(τ_{α})_{κ}”

making the choice of q above impossible. On the other hand, any narrowed down set of possible names τ_{α} which enables us to put q = p is too restrictive. A simple argument as in the proof of Lemma 5.4.11 would then show that the forcing does not perform the intended task. Notice that a diamond sequence could equally well be “read” from some other coordinate besides κ, the coordinate that was used in the proof of Lemma 5.4.11.

Restricting the set of possible names τ_{α} does not seem to be fruitful. Instead we shall use ideas developed by Hauser [8] to find an extension of the embedding j by replacing a part of the generic set with an isomorphic copy.

We say that a subset S of κ^{+} is a complete set of coordinates (with respect to P) if p|S ∈ P for every condition p ∈ P. In this setting P |S denotes {p|S : p ∈ P }.

5.4.12. Lemma. If S and S′ are complete sets of coordinates such that κ ⊆ S ⊆ S′ then P |S is a complete suborder of P |S′.

Proof. Let p and q be conditions in P and let α ∈ dom(p) ∩ dom(q) \ κ. By condition (ix) in the definition of P we have p|α ⊩ “p(α) ≤ q(α)” if and only if p|κ ⊩ “p(α) ≤ q(α)”. Thus it is clear that p|S′ ≤ q|S′ implies p|S ≤ q|S. Using the observation above it is straightforward to check the claim of the lemma. □

Let π be a bijective partial function κ^{+} → κ^{+}. If

(i) π|κ = id_{κ}

(ii) Both the domain and range of π are complete sets of coordinates

(iii) p ↦ p ∘ π^{−1} defines an isomorphism between P | dom π and P | ran π

then we say that π is P -complete. Note that the definition is symmetric in the sense that π is P -complete if and only if π^{−1} is P -complete.

5.4.13. Lemma. Suppose that π is P -complete, α ⊆ dom π, α ∉ dom π, |π| < κ^{+} and β < κ^{+}. Then there exists an ordinal γ such that β ≤ γ < κ^{+} and the extended function π ∪ {(α, γ)} is a P -complete bijection.

Proof. Consider the P_{α}-name τ_{α}. Since α ⊆ dom π we can look upon τ_{α} as a (P | dom π)-name. Here we have to note that P_{α} is a complete suborder of P | dom π, but this is obvious since we are restricting to an initial segment α of dom π.

Let τ_{α}^{π} denote the (P | ran π)-name that corresponds to τ_{α} under the isomorphism p ↦ p ∘ π^{−1}. By Lemma 5.4.12 we can look upon τ_{α}^{π} as a P_{γ}-name for a subset sequence indexed by κ, where γ is any ordinal such that sup ran π ≤ γ < κ^{+}. Condition (vii) in the definition of P is taken to talk of the names literally as sets. Therefore there is unboundedly many ordinals γ below κ^{+} such that τ_{γ} is equivalent to τ_{α}^{π}. Let us fix such an ordinal γ ≥ max{β, sup ran π}. We shall prove that γ is as desired. Let us denote the extended mapping by π too, since there can be no confusion. Thus we have π(α) = γ.

Let p ∈ P | dom(π) be arbitrary. We shall first show that p ∘ π^{−1} ∈ P_{κ^{+}}. Now suppose that α ∈ dom p. (It should be clear from the argument that follows that other ordinals in dom p do not present any problems.) We must prove that

p(α) ∈ dom Q̇_{π(α)}  (16)

where Q̇_{π(α)} is a name for K_{S}^{1}(τ_{π(α)}) and that

(p ∘ π^{−1})|π(α) ⊩ “p(α) ∈ K1 ; Š(τ_{π(α)})”.  (17)

But by our choice of π(α) we have (p ∘ π^{−1})|π(α) = p|(κ^{+} \ {α}) ∘ π^{−1}. Furthermore τ_{π(α)} corresponds essentially to τ_{α} under the isomorphism between P |(dom(π)\{α}) and P |(ran(π) \ {π(α)}), and p(α), being a P_{κ}-name corresponds to itself. Thus

p|(κ^{+} \ {α}) ⊩ “p(α) ∈ K1 ; Š(τ_{α})”  (18)

is in fact equivalent to (17). But τ_{α} is a P_{α}-name and therefore the forcing language statement in (18) depends only on the P_{α}-generic part of a given generic set. So in fact

p|α ⊩ “p(α) ∈ K1 ; Š(τ_{α})”  (19)

is equivalent to (17), and (19) holds directly by the definition of P.

Since the equivalence between (17) and (19) holds not only for p(α) but also for other P_{κ}-names, (16) holds by absoluteness of the choice of the names Q̇_{α}. Using nearly the same argument we can see that the mapping p ↦ p ∘ π^{−1} from P | dom π into P | ran π is surjective. That the ordering is preserved by the mapping is obvious by arguments similar to those in the proof of Lemma 5.4.12.

The domain of the extended function π is clearly complete, since α ⊆ dom π. Now we are left with proving that ran π is complete, but this also becomes clear when one considers the equivalence of (17) and (19). □

5.4.14. Lemma. Suppose that S is a collection of P -complete partial functions κ^{+} → κ^{+} such that π ⊆ π′ or π′ ⊆ π for every pair of functions π and π′ from S. Then ⋃ S is a P -complete function.

Proof. Let A = ⋃_{π∈S} dom π and let p ∈ P. We shall show that p|A ∈ P. Let α ∈ dom(p) ∩ A be arbitrary. We are using induction on α so let us assume that p|(A ∩ α) ∈ P. There exists a function π ∈ S such that α ∈ dom π. Now p|(dom(π) ∩ α) ⊩ “p(α) ∈ Q̇_{α}”. Since p|(A ∩ α) ≤ p|(dom(π) ∩ α) we are done.

The above argument works for unions of P -complete sets of coordinates in full generality. Thus also B = ⋃_{π∈S} ran π is P -complete. Now let π = ⋃ S and suppose that p ∈ P | dom π. It is not difficult to show that p ∘ π^{−1} ∈ P. Note that just by considering properties of sets of ordinals, every strictly increasing sequence (γ_{i} : i < κ) of ordinals in ran π contains a subsequence (γ_{i} : i ∈ I) such that |I| = κ and (π^{−1}(γ_{i}) : i ∈ I) is strictly increasing. Thus dom(p ∘ π^{−1}) conforms to the definition of P. □

5.4.15. Lemma. Any P -complete partial function π : κ^{+} → κ^{+} such that |π| < κ^{+} can be extended to a P -complete total function κ^{+} → κ^{+}.

Proof. The before-mentioned symmetry allows us to apply Lemma 5.4.13 not only to a P -complete function π but also to the inverse π^{−1}. Thus using a standard back and forth type argument it is straightforward to use Lemmas 5.4.13 and 5.4.14 to complete a given partial function to a total one. □

In the argument that follows we shall cheat slightly. Actually Lemmas 5.4.16 and 5.4.17 are intertwined in an induction argument in the same way that Lemmas 5.4.9 and 5.4.10 are. But we shall actually refer to a modified version of Lemma 5.4.17 before proving it, in order not to break the natural flow of the presentation.

Fix a P_{ζ}-name σ for a uniform winning strategy for player II in 𝒢_{ζ}(j(P_{η})^{ζ}). We shall inductively define a P -complete function π : η → ζ simultaneously with a sequence (ḟ_{ν} : κ ≤ ν < η) of P_{ζ}-names for forcing conditions in j(P_{η})^{ζ} such that the conditions introduced below are upheld for as long as possible. As one might expect we shall then prove that the conditions indeed can be realised throughout the entire construction. The construction takes place in N.

The induction is on ν where κ ≤ ν < η. We shall try to make sure that

(iv) π|(ν + 1) is P -complete

and that the following holds in any generic extension by P_{ζ}:

(v) (f_{i} : i < ν) is a decreasing sequence of conditions in j(P_{η})^{ζ} that form the moves of player I in some instance of 𝒢_{ζ}(j(P_{η})^{ζ}) where player II plays according to the interpretation of σ

(vi) j(ν) = max dom f_{ν} and f_{ν}(j(ν)) is a P_{π(ν)+1}-name such that

⊩ “f_{ν}(j(ν)) = (⋃_{s∈Ġ_{π(ν)+1}} C_{s(π(ν))} ∪ {κ}, ⋃_{s∈Ġ_{π(ν)+1}} A_{s(π(ν))})”.

Note that condition (vi) closely resembles the definition of the function d apart from the coordinates being shifted by π.

5.4.16. Lemma. There exist a function π : η → ζ and a sequence (ḟ_{ν} : κ ≤ ν < η) satisfying the conditions in the definition for all ordinals ν such that κ ≤ ν < η.

Proof. Suppose that (ρ_{i} : κ ≤ i < ν) and π|ν are defined successfully maintaining conditions (iv)–(vi) for ordinals smaller than ν. We shall define π(ν) and ḟ_{ν} so that the conditions hold for ν.

Arguing in an arbitrary generic extension of N by P_{ζ}, we can use the induction hypothesis about condition (v) to conclude that there exists a lower bound f for {f_{i} : i < ν}. Fix such a condition f and if ν happens to be a successor ordinal, let f be the choice of player II by the winning strategy. (Recall that player I plays first at limit ordinals.) Since the winning strategy is uniform we can require that f ∈ j(P_{ν})^{ζ}.

Consider a further generic extension by j(P_{ν})^{ζ} such that f is in the generic set. Let j(τ_{ν})_{κ} denote the κth element in the sequence that is the interpretation of the j(P_{ν})-name j(τ_{ν}). By standard density arguments there exist at most one ordinal α < ζ for which ⋃_{s∈G_{α+1}} A_{s(α)} = j(τ_{ν})_{κ}.

Let us now go back to argue in the generic extension by P_{ζ} and let j(τ_{ν})_{κ} denote a j(P_{η})^{ζ}-name for the object under discussion. By the previous remarks we can find a condition f^{∗} ∈ j(P_{ν})^{ζ} and a fixed ordinal α_{ν} such that f^{∗} ≤ f and

f^{∗} ⊩ “⋃_{s∈G_{α+1}} A_{s(α)} ≠ j(τ_{ν})_{κ}”  (20)

for all ordinals α such that α_{ν} < α < ζ.

Continuing the argument in N we consider P_{ζ}-names for the objects α_{ν} and f^{∗}. Since P_{ζ} satisfies the ζ-cc by Lemma 5.4.2 applied inside N, the possible values of α_{ν} are bounded below ζ. Now we fix π(ν) above this upper bound so that

α_{ν} < π(ν)  (21)

holds in any generic extension by P_{ζ} where α_{ν} is the realisation of the name we chose and π(ν) is fixed in the ground model N.

At the same time we can make sure that π|(ν +1) is P -complete which is possible by Lemma 5.4.13. If ν is a limit ordinal we shall also need to refer to Lemma 5.4.14 to first conclude that π|ν is P -complete.

To deal with condition (vi) we first need to note that {j(ν) : κ ≤ ν < η} ∈ N, but this was already dealt with at the beginning of the proof of Lemma 5.4.10. Now pick ḟ_{ν} such that f^{∗} = f_{ν}|j(ν) in any generic extension by P_{ζ}, and condition (vi) is satisfied. The only thing left to verify is that f_{ν} actually is a condition in j(P_{η})^{ζ}.

At this point we are going refer to the fact that f^{∗} behaves for j(P_{ν})^{ζ} as f_{η} does for j(P_{η})^{ζ} in Lemma 5.4.17. The suspicious reader may read Lemma 5.4.17 and the definitions before and after it, replacing η by ν. So similarly as in the proof of Lemma 5.4.10 we may extend j to see that

f^{∗} ⊩ “j(τ_{ν})_{i} = (τ_{ν})_{i}”

for all i < κ. Furthermore

f^{∗} ⊩ “⋃_{s∈G_{π(ν)+1}} C_{s(π(ν))} ∪ {κ} is 1-closed”

but now also the potential problem that occurs if κ ∈ j(S) is handled by (20) and (21). □

Let p be the condition fixed by (13). Fix a P_{ζ}-generic set G_{ζ} such that p ∘ π^{−1} ∈ G_{ζ}. Since π is P -complete {q ∘ π^{−1} : q ∈ P_{η}} = P | ran π is a complete suborder of P_{ζ}. Thus by putting

G_{η}^{π} = {q ∈ P_{η} : q ∘ π^{−1} ∈ G_{ζ}}

we obtain a P_{η}-generic set G_{η}^{π} such that p ∈ G_{η}^{π}. Consider the forcing notion j(P_{η})^{ζ} in the extension N[G_{ζ}] and let f_{η} be a lower bound for the sequence (f_{ν} : ν < η). Fix a j(P_{η})^{ζ}-generic set H over N[G_{ζ}] such that f_{η} ∈ H.

5.4.17. Lemma. q ∘π^{−1} ∪f_{η} is a condition in j(P_{η}) such that q ∘π^{−1} ∪f_{η} ≤ j(q) for every q ∈ P_{η} and therefore j(q) ∈ G_{ζ} ∗ H for every condition q ∈ G_{η}^{π}.

Proof. Let q ∈ P_{η}. As before we know that dom(j(q)) = j[dom q]. Since j(q)|j(κ) = q|κ = (q ∘ π^{−1})|κ, we can conclude that q ∘ π^{−1} ≤ j(q)|j(κ) and dom(j(q)) ⊆ dom((q ∘ π^{−1}) ∪ f_{η}). Let α ∈ dom(q) ∩ [κ, η) be arbitrary. We first note that j(q)(j(α)) = q(α) = (q ∘ π^{−1})(π(α)). By condition (vi) in the definition of the sequence (f_{α} : α < η) we have

(q ∘ π^{−1})|(π(α) + 1) ⊩ “f_{α}(j(α)) ≤ (p ∘ π^{−1})(π(α))”

and since f_{η} ≤ f_{α} we can conclude that q∘π^{−1}∪f_{η} ≤ j(q). Since (q∘π^{−1})∪f_{η} ∈ G_{η}∗H the second claim is obviously true. □

By Lemma 5.4.17 the equation

j(τ_{G_{η}^{π}}) = j(τ)_{G_{ζ}∗H}

is well behaved and can be used to define an extended elementary embedding j : M[G_{η}^{π}] → N[G_{ζ}][H]. Let us fix such an embedding. Let us also extend π to a total P -complete bijection ζ → ζ. This is possible by applying Lemma 5.4.15 inside N. We shall denote the extended function by π too, so that our original function is π|η. Now we are set to prove the crucial lemma.

5.4.18. Lemma. Every set E ⊆ κ that is weakly compact in the ground model is weakly compact in V^{P}.

Proof. Recall that we had fixed a weakly compact set E such that κ ∈ j(E). Since this set E was completely arbitrary we can let it exemplify the E of the lemma. Let U̇^{π} denote the P_{ζ}-name that corresponds to the P_{η}-name U̇ under the automorphism p ↦ p ∘ π^{−1} of P_{ζ}. Since

p ⊩_{ζ} “⟨V_{κ}, ∈, U̇⟩ ⊧ ϕ”

by (13) and Lemma 5.4.6, it then clearly follows that

p ∘ π^{−1} ⊩_{ζ} “⟨V_{κ}, ∈, U̇^{π}⟩ ⊧ ϕ”.

Now U̇_{G_{ζ}}^{π} = U̇_{G_{η}^{π}} and since the condition p ∘ π^{−1} ∈ G_{ζ} we have that ⟨V_{κ}, ∈, U̇_{G_{η}^{π}}⟩ ⊧ ϕ holds in V [G_{ζ}] and thus in N[G_{ζ}].

Since it is true (Lemma 5.4.4) in N[G_{ζ}] that j(P_{η})^{ζ} does not add subsets of κ the statement holds in N[G_{ζ}][H]. By definition j(U̇_{G_{η}^{π}}) = j(U̇)_{G_{ζ}∗H}. Thus by elementarity and the fact that U̇_{G_{η}^{π}} = j(U̇_{G_{η}^{π}}) ∩ V_{κ} it holds in M[G_{η}^{π}] that there exists an ordinal α ∈ E such that ⟨V_{α}, ∈, U̇_{G_{η}^{π}} ∩ V_{α}⟩ ⊧ ϕ.

Since p ∈ G_{η}^{π} it holds in M that there exists an ordinal α ∈ E and a condition q ≤ p in P_{η} such that

q ⊩_{η} “⟨V_{α}, ∈, U̇ ∩ V_{α}⟩ ⊧ ϕ”

and by the choice of M′ this holds in V too. Finally by absoluteness with respect to the generic extensions involved, the above holds with η replaced with κ^{+}. □

Having proved that weak compactness is preserved, we move on to checking that forcing with P really kills the desired version of weakly compact diamond.

5.4.19. Lemma. For every α < κ^{+} the interpretation of τ_{α} is not a ◇(WC_{κ}|S)- sequence in V^{P}.

Proof. In similar fashion as for Lemma 5.3.3 we need to show that

C = ⋃_{s∈G_{α+1}} C_{s(α)}

is stationary in V^{P}. Since η < κ^{+} may be arbitrary large, we may assume that α < η. By Lemma 5.4.2 it is enough to show that C is stationary in V^{P_{η}}. But C_{d(j(κ))} ⊆ j(C) so κ ∈ j(C) and the argument of Lemma 5.4.18 yields that C is weakly compact in V^{P_{η}}. □

5.4.20. Lemma. ◇(WC_{κ}|S) fails in V^{P}.

Proof. By the previous lemma we only have to check that every subset sequence indexed by κ in V^{P} is the realisation of τ_{α} for some α < κ^{+}. □

The result of this section is a combination of Lemmas 5.4.11, 5.4.18, and 5.4.20.

5.4.21. Theorem. If S is a weakly compact subset of κ then there exists a generic extension that preserves every weakly compact subset of κ and in which ◇(WC_{κ}|S) fails and ◇(WC_{κ}|E) holds for every weakly compact E in the ground model such that E is disjoint from S.

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