Some Consistency Strength Analyses using Higher Core Models

Dissertation

zur

Erlangung des Doktorgrades (Dr. rer. nat.)

der

Mathematisch-Naturwissenschaftlichen Fakultät

der

Rheinischen Friedrich-Wilhelms-Universität Bonn

vorgelegt von Florian Rudolph

aus Bonn

Bonn 2000 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn

1. Referent: Prof. Dr. Peter Koepke

2. Referent: Prof. Dr. Hans-Dieter Donder Tag der Promotion: 14. April 2000

Druck: Rheinische Friedrich-Wilhelms-Universität Bonn ISSN: 0524 – 045X

Contents

1 Introduction 1 2 Short Core Models 9 3 Chang’s Conjecture at club-many Points 13 4 The Transversal Hypothesis 25 5 Core Models up to a Strong Cardinal 37 6 Short Iterations 45 7 Irregular Ultrafilters 67 Bibliography 99 Index 103

Acknowledgements

I would like to thank Prof. Peter Koepke, who first got me hooked on the subject of set theory and later on that of inner models and large cardinals. Many fruitful discussions for which he made himself readily available have been helpful for all aspects of my research. I would also like to thank Prof. Hans-Dieter Donder for taking a sincere interest in this work. I am indebted to Prof. Sy Friedman, whom I owe the privilege of studying at MIT for one year. During this time and with his thoughtful guidance I was able to make significant progress in understanding the complexities of core model theory. Discussions with members of the Boston logic group, especially Zoran Spasojević, were just as profitable (and enjoyable) as those with the logicians in Bonn, where Dr. Ralf Schindler and Dr. Heike Mildenberger deserve special mention for their unfailing willingness to bear with my questions.

I thankfully acknowledge a scholarship of the Studienstiftung des Deutschen Volkes, which not only financially supported the major part of my research but also provided a number of opportunities for straying aside and meeting interesting people in interesting places. I would also like to thank the Graduiertenkolleg at the Mathematical Institute of the University of Bonn for a scholarship during the last months of the preparation of this thesis. Finally, I should not fail to record my gratitude to my parents, who unconditionally supported a project which to them, I fear, must still seem a large mystery.

Chapter 1

Introduction

’Twas brillig, and the slythy toves did gyre and gimble in the wabe. All mimsy were the borogoves and the mome rath outgrabe.

(Lewis Carroll, Jabberwocky)

Set Theory, besides having its origins^{1} in Cantor’s investigations into the convergence of trigonometric series, has been closely linked with attempts to formulate a viable Foundation of Mathematics. At the turn of the century Zermelo, with the purpose of clearly exhibiting the assumptions used in his proof of the Well-Ordering Theorem, provided the basis for what has now become the standard axiomatisation of Set Theory, namely ZF. From the beginning, the question of the consistency of this (or any) system of axioms loomed large. Hilbert [Hil00] had included the consistency of Arithmetic in his famous list of open problems at the international congress of mathematicians in Paris in 1900. The rest is history, one is tempted to say: Gödel, in his 1931 paper [Göd31], showed that Hilbert’s goal, i. e., proving the consistency of ZF by purely finitary means, could not be achieved. The consistency of any system strong enough to code Peano arithmetic cannot be proved even by means of the system itself, let alone by finitary reasoning, as demanded by Hilbert. In fact, for any such system, there will be undecidable sentences. The prime example is, of course, the Continuum Hypothesis, which dates back to Cantor’s original investigations of the universe of sets. Gödel [Göd38] gave one half of the independence proof, proving the consistency of ZF+ CH from Con(ZF), and Cohen [Coh63], [Coh64] showed that the negation of CH is equally relatively consistent.

1For a detailed survey of the early history of Set Theory and its ramifications — which we have not attempted here — cf. [Kan96].

These two proofs already contain the two main methods for arriving at relative consistency results: Gödel constructed an inner model, i. e., a class M ⊆ V which satisfies all the axioms of ZF, together with CH. His model, L, can be seen as the prototype of the whole family of core models that were to be developed later on. Cohen, on the other hand, by his method of forcing, constructed an extension of the universe V[G], in which the ZF-axioms hold, as well as the negation of CH.

All consistency proofs must be relative, i. e., they have to assume the consistency of some set of axioms at the outset, possibly ZF, possibly ZF+Γ, where Γ is some (set of) sentence(s). Then they can construct a new model in which the statement in question, say CH or Φ, holds. One class of extensions of ZF, that of large cardinal axioms, has proved to be particularly fruitful for these investigations.^{2} They are perhaps best characterized by positing the existence of some ordinal with special properties or that of some elementary map from the universe of sets to some structure M. What makes them attractive is the fact that they form a nearly linear scale against which one can gauge the consistency strength of various other, say, combinatorial statements. Thus, taking ZFC as the base system, if Λ_{1} and Λ_{2} are large cardinal axioms and Φ is some combinatorial statement, then Λ_{1} is an upper bound to the consistency strength of Φ if one can show that Con(ZFC+Λ_{1}) implies Con(ZFC+Φ). Analogously, Λ_{2} will be a lower bound to the consistency strength of Φ if one can construct, starting from a model of ZFC+Φ, a model of ZFC+Λ_{2}. As exemplified by Gödel’s and Cohen’s results on CH (although their proofs did not require any axioms beyond ZF), upper bounds are usually constructed by the method of forcing, where the forcing exploits the properties of the large cardinals, while the lower bounds make use of inner models.

2[Kan94] gives a comprehensive introduction to large cardinals including numerous historical remarks.

Core models, a special type of inner models, where invented by Jensen and Dodd [DJ81]. They go back to Jensen’s investigation of Gödel’s model L of constructible sets. Gödel’s fundamental idea was to construct the model layer by layer, adding sets which are definable from parameters previously constructed at each stage. Jensen [Jen72] took a closer look at this layering or definability and thus arrived at more precise statements about the (fine) structure of the resulting model. Using this finestructure, he succeeded together with Dodd in constructing K^{DJ}, the first core model. It lies somewhere between L and L[U], the canonical inner model for a measurable cardinal.

The building blocks of this model are initial segments of the final model called mice. They play a central rôle in all of core model theory. Subsequently, extensions of this concept of mouse led to new core models encompassing ever larger large cardinal hypotheses.^{3} Some common properties are their construction from the bottom up along the ordinals, their rigidity (the existence of an elementary embedding from K to K being the least large cardinal axiom inconsistent with K) and some sort of covering lemma, which asserts that — assuming the absence of the corresponding large cardinal in V — the model K is close to V. Mitchell [Mit84] described the core model for sequences of measures, Jensen [Jen8x] among others that for measures of order zero. Schindler [Sch96], building on work of Jensen and Koepke [Koe89], constructed a core model up to a strong cardinal. Transcending the boundary of linear iterations using iteration trees, Steel [MS94], [Ste96] extended the theory so as to accomodate even larger hypotheses, currently somewhere in the region of limits of Woodin cardinals.

A common approach to consistency strength analysis using core models is the following. Suppose we have some combinatorial statement Φ for which we want to give a lower bound of its consistency strength, say Λ, where Λ is some large cardinal axiom. Assume that Λ does not hold in V. Then a covering lemma will hold for a suitable core model K. One then tries to show that the close connection between the core model K and the surrounding universe V contradicts Φ, in the sense that Φ disrupts the “constructible” nature of V imposed by K.

3For an introduction and an approximate definition of this class of models cf. [Mit94].

In Chapter 2 we review the theory of the so-called short core models. This theory has been developed in detail in [Koe83], and [Koe88] contains an introduction to the main concepts and basic results, which we summarise without proof. Short core models allow inner models for sequences of measurable cardinals, provided that the length of the sequence is less than the least measurable. This assumption greatly simplifies the theory of iterations, as the number of measures in a mouse does not change (increase, that is) when one takes an ultrapower.

Chapter 3 takes another look at Chang’s Conjecture, CC, and shows that it is equivalent to an apparently stronger version, which we call Chang’s Conjecture at club-many points, CC^{club}. Silver [Sil71] had shown that CC can be forced to hold in a generic extension of the universe, using a variant of the Levy collapse, now known as the Silver collapse. Later, Donder and Levinski [DL89] and Baumgartner [Bau91] showed that the same result can be achieved using the original Levy collapse. We modify this approach to show that in the same generic extension also CC^{club} holds.

In Chapter 4 a combinatorial principle closely related to Chang’s Conjecture is considered, the so-called Transversal Hypothesis, TH. Usually, the interest lies in the negation of this hypothesis, as this is a consequence of CC. The exact consistency strength of ¬ TH at ω_{1} has been determined by Donder and Levinski [DL89] in terms of certain game principles. However, at higher cardinals the situation is less clear. In fact, there is a large gap between the lower and upper bounds for the consistency of CC at higher cardinals. So far, a huge cardinal (cf. [For82]) has been needed to force, say, ⟨ℵ_{4}, ℵ_{3}⟩ ⟨ℵ_{3}, ℵ_{2}⟩, whereas Schindler [Sch96] gets a strong cardinal as a lower bound for the consistency strength of this statement (together with 2^{ℵ_{1}} = ℵ_{2}). We in turn consider a variant of TH, TH^{stat}, concentrating on a stationary set, and give a lower bound for the consistency strength of ¬ TH^{stat} at cardinals beyond ℵ_{1}, using the short core models of Chapter 2. Just as ¬ TH is implied by CC, ¬ TH^{stat} is a consequence of CC^{club}, considered in Chapter 3. That chapter thus serves to make the consistency (relative to some large cardinal, of course) of TH^{stat} plausible.

Chapter 5 lays the ground for another consistency strength result. To get higher lower bounds than just 0^{long} (the “sharp” for short core models), we need to consider higher core models. For our purposes, core models up to a strong cardinal will do nicely. The finestructure for these models has been developed in [Koe89], and the actual core model construction is carried through in [Sch96]. As in Chapter 2, we present a brief summary of the main concepts and results.

In Chapter 6 we calculate an estimate for the length of certain short iterations of mice. This bound will play a crucial rôle in the following chapter. We are only able to give this bound under a more restrictive assumption than just the absence of an inner model for a strong cardinal, which limits the possible application in the subsequent consistency strength analysis. However, even though our assumption is perhaps more restrictive than ultimately necessary, some bound strictly below a strong cardinal is necessary, as a short “counterexample” at the end of the chapter shows.

Finally, in Chapter 7 we compute a new lower bound for the consistency strength of the existence of irregular ultrafilters. Regular ultrafilters were first considered in Model Theory. Keisler showed that they yield ultrapowers of maximal cardinality. However, not all ultrafilters need to be regular. A measure on a measurable cardinal, for example, will always be fully irregular. More interesting, though, is the question whether irregular ultrafilters can exist on “small” or successor cardinals, ℵ_{1} or ℵ_{2}, say. Prikry [Pri70] showed that in L, all ultrafilters on ω_{1} (in fact, on any successor cardinal) are regular. Ketonen [Ket76] then showed that if there is, on some regular cardinal, a weakly normal ultrafilter which is fully irregular, then 0^{#} exists. This result was later improved by Jensen [DJK81], who proved that the existence of an inner model for a measurable cardinal can be deduced. Kanamori [Kan76] showed that if the ultrafilter lives on a successor cardinal, then the weak normality requirement can be dropped. Donder [Don88] investigated the matter at singular cardinals: if there is a uniform, non-regular ultrafilter on κ and κ is singular, then there is an inner model for a measurable cardinal. Also, if κ is regular and (κ^{+} )^{K^{DJ}} = κ^{+} , then the same conclusion holds. Thus in the Dodd-Jensen core model K^{DJ}, all uniform ultrafilters are regular.

Not too much is known about the upper bound for the consistency strength of (fully) irregular ultrafilters over small cardinals. Magidor [Mag79] used a forcing starting from a huge cardinal to get a model in which ω_{3} carries an ultrafilter that is not (ω_{1}, ω_{3})-regular. Also, he constructed a model in which ω_{2} carries an ultrafilter that is not (ω, ω_{2})-regular. This not quite full irregularity, though. Laver [Lav82] showed that in a model constructed by Woodin, ω_{1} carries an irregular ultrafilter. The hypothesis used for this construction was “ZF+ AD_{}+Θ regular”^{4}. Finally, Foreman, Magidor, and Shelah [FMS88] got an upper bound from large cardinals only: Given an infinite cardinal μ and κ > μ huge, it is consistent that μ^{+} carries a fully irregular ultrafilter.^{5}

We are able to improve the lower bound of the Jensen result [DJK81] to “There exists an inner model with o^{M}(μ) ω”. Were it not for the restriction of the estimate of the length of short iterations from Chapter 6, our proof would be able to cope with any hypothesis up to a strong cardinal. It is for this reason that we have written up the proof in such a way that it would go through for any such large hypothesis without change, provided this estimate were improved. It is in this vein that we have also included a presentation of the so-called Gitik Game, which permits the reconstruction of certain extenders used in “long” iterations (cf. [Git93]). Given our assumptions it would not have been truly necessary. However, we consider it an interesting method in its own right, apart from its value in pinpointing the exact location of the limitation of our proof. One should also note that our current assumptions would permit further simplifications to the proof, e. g., there is obviously no need to consider overlapped cardinals at all.

A word on notation: our notation adheres to the set theoretic standard, such as that employed in [Kan94]. Thus H_{μ} denotes the collection of the sets of hereditary cardinality less than μ, where the hereditary cardinality of a set is the cardinality of its transitive closure. The cofinality of an ordinal α is denoted by cf(α), the ordertype of a set x by otp(x), its cardinality by card(x), or, in some cases, by \overline{\overline{x}}. The least upper bound of a set of ordinals is lub(x). On denotes the class of all ordinals, Lim that of all limit ordinals, Card that of all cardinals, Reg that of all regular cardinals.

4Here, Θ = sup{ξ | ∃f : ^{ω} ω → ξ onto}. The theory “AD_{}+Θ regular” is the strongest determinacy type hypothesis known, much stronger than AD_{}alone (whose consistency strength is within the limits of current inner model theory). Woodin has shown that the axiom I1 is an upper bound for the strength of “AD_{}+Θ regular”, where I1 is the statement that there is an elementary embedding j : V_{δ+1} → V_{δ+1} .

5Huberich [Hub94] extended this result to show that a measurable cardinal alone is sufficient to force the existence of a fully irregular ultrafilter on a non-measurable weakly compact cardinal. This puts a bound on the possibility of irregular ultrafilters on limit cardinals having high consistency strengths.

A set x ⊆ κ is called γ-club if it is unbounded in κ and closed under limits of cofinality at least γ. The set of all subsets of κ of size λ is denoted by [κ]^{λ}, that of all subsets of κ of size less than λ by [κ]^{<λ} , accordingly.

If f is a function (a fact denoted by fun(f)), then dom(f) stands for its domain and rge(f) its range. f′′x = {f(y) | y ∈ x} is the image of x under f and f x = {⟨y, f(y)⟩ | y ∈ x} is the restriction of f to x. If π is an embedding from M to N, then crit(π) denotes the critical point of π, that is the least ordinal moved by π.

U ⊆ P (κ) is an ultrafilter if i) it is closed under finite intersections and supersets, and if ii) it contains κ, and if iii) for any x ⊆ κ it contains either x or κ \ x.

U is γ-complete, for some γ κ, if U contains the intersection of any β-many elements of U, for any β < γ.

8 CHAPTER 1. INTRODUCTION

Short Core Models

Alice thought this must be the right way of speaking to a mouse: she had never done such a thing before, but she remembered having seen in her brother’s Latin Grammar, “A mouse – of a mouse – to a mouse – a mouse – O mouse!”

(Lewis Carroll, Alice’s Adventures in Wonderland, Chapter II)

The theory of short core models has been developed in [Koe83]. The paper [Koe88] contains a concise introduction to the main concepts and basic results, of which we cite a number without proof for future reference. The numbering from [Koe88] is given in parentheses.

J_{α}[A] denotes the α-th level of the relativized Jensen-hierarchy, and <_{J_{α} [A]} its canonical well-ordering. A class D is simple iff every element x of D is of the form ⟨κ, a⟩, where κ is an ordinal and a ⊆ κ, and ⟨κ, a⟩∈D implies ⟨κ, κ⟩∈D. For simple D define dom(D) := {κ | ⟨κ, κ⟩ ∈ D}, D(κ) := {a | ⟨κ, a⟩ ∈ D}, and D X := {⟨κ, a⟩ | κ∈X}.

U is a measure on κ iff U is a non-principal, κ-complete, normal ultrafilter on κ. U is a sequence of measures iff U is simple and U(κ) is a measure on κ for every κ∈dom(U).

2.1 Definition (2.1) Let D be a simple predicate. M = J_{α}[U, D] is a premouse over D iff U is simple, supdom(D) < mindom(U), and M U is a sequence of measures. Then meas(M) := dom(U) ∩ M is the set of measurables in M, and lp(M) := H_{κ}, where κ = minmeas(M), is the low part of M (setting lp(M) := M if meas(M) = ∅).

We omit the detailed definitions of ultrapowers, iterations, and iterability, as well as the relevant criteria for iterability ([Koe88, Definitions and Lemmas 2.2 – 2.8]).

2.2 Definition (2.9) Let M = J_{α}[U, D] be a premouse over D. M is called short if i) either D = ∅ and otpmeas(M ∩ γ) < minmeas(M) for all γ < ωα or ii) D ≠ ∅ and otpmeas(M) mindom(D).

A D-premouse is a short premouse over D, and a D-mouse is an iterable short premouse over D.

The main advantage of dealing with short premice is that the number of measures present in a premouse does not change when one takes an ultrapower: meas(Ult(M, U)) = π′′(meas(M)).

2.3 Definition (2.13) “0^{long} exists” means that there is an iterable premouse over ∅ which is not short. ¬0^{long} is taken to abbreviate the statement “0^{long} does not exist”.

2.4 Definition (3.1) Let D be simple such that D = ∅ or otpdom(D) mindom(D). Define the class K[D] as

K[D] := ⋃ {lp(M) | M is a D-mouse}.

For α∈On set K_{α}[D] := H_{α}^{K[D]}.

2.5 Theorem (3.2) K[D] is a transitive inner model of ZFC. D̄ := D ∩ K[D]∈K[D] and K[D] V = K[D̄]. If α > supdom(D) is an uncountable cardinal in K[D], then K_{α}[D] V = K[D̄].

2.6 Lemma (3.3) Let Q be a transitive model of a sufficiently large finite part of ZFC+ V = K[D̄], where D̄ = D ∩ Q ∈ Q. Assume dom(D̄) = dom(D) and that D̄-mice are absolute for Q. Then Q ⊆ K_{α}[D], where α = On∩Q ∞.

2.7 Theorem (3.4) For x, y ∈ K[D] set x _{D} y iff x _{M} y for every D-mouse M such that x, y ∈lp(M). Then _{D} is a well-ordering of K[D].

2.8 Definition (3.6) A model K[U] is called a short core model if K[U] U is a sequence of measures. A set U such that K[U] is a short core model is called strong.

2.9 Definition (3.8) Let U, U′ be strong. Set U _{e} U′ if U = U′ η for some η ∈On, i. e., if U′is an end-extension of U. Set U <_{e} U′if U _{e} U′and U ≠ U′. U is maximal if U is strong and there is no strong U′ end-extending U.

2.10 Theorem (3.9) Assume ¬0^{long}. Let U be strong, and γ ∈On. Then

i) U γ is strong.

ii) K[U γ] ⊆ K[U], indeed, K[U γ] = (K[Ū γ])^{K[U]}, where Ū =

U ∩ K[U]. iii) P (γ) ∩ K[U γ] = P (γ) ∩ K[U].

2.11 Theorem (3.11) Assume ¬0^{long}. Let U be strong. Then there is a maximal U′ _{e} U.

2.12 Lemma (3.12) Assume ¬0^{long}. Let U be strong. Then for every regular cardinal η, sup(dom(U) ∩ η) < η.

2.13 Theorem (3.14) Assume ¬0^{long}. Let K[U], K[U′] be core models with dom(U) = dom(U′). Then |K[U]| = |K[U′]| and U ∩K[U] = U′∩K[U′].

2.14 Definition (3.15) Let U_{can} be the unique maximal strong sequence satisfying

i) U_{can} ⊆ K[U_{can}], ii) if κ ∈ dom(U_{can}), then κ is the minimal ordinal ζ such that there is

some U′>_{e} U_{can} κ with ζ = mindom(U′\ (U_{can} κ). U_{can} is called the canonical sequence and K[U_{can}] is the canonical core model.

2.15 Theorem (3.18) Assume ¬0^{long}. Let K[U] be a core model and let j : K[U] → K[U] be elementary with critical point κ > supdom(U). Assume δ is a regular cardinal greater than κ which is a limit cardinal in K[U]. Then there exists a strong U′ >_{e} U with τ := mindom(U′\ U) satisfying τ κ and τ = δ, if δ = ω_{1}, and τ < δ, if δ ω_{2}.

2.16 Theorem (3.20) Assume ¬0^{long}. Let τ be an ordinal such that supdom(U_{can} (τ + 1)) < τ. Then

i) If τ ω_{2} is a limit ordinal and cf(τ) < card(τ), then τ is singular in K[U_{can}].

ii) If τ is a singular cardinal in V, then τ is singular in K[U_{can}] and τ^{+} = τ^{+K[U_{can}]}.

2.17 Theorem (3.24) Let D be a simple predicate with otpdom(D) mindom(D) or D = ∅. Let Q be a transitive model of a sufficiently large finite part of ZFC+ V = K[D̄], where D̄ := D ∩ Q ∈ Q. Let ω_{1} ⊆ Q and dom(D) = dom(D̄). Then

i) Let M be a D-mouse, meas(M) ≠ ∅, and let κ = minmeas(M) be singular in Q. Then lp(M) ⊆ Q.

ii) Let λ ⊆ Q be a cardinal greater than supdom(D) and assume the following condition is satisfied: If C ⊆ λ is closed unbounded in λ, then there exists a κ ∈ C which is singular in Q. Then K_{λ}[D] ⊆ Q.

2.18 Theorem (3.25) Assume ¬0^{long}. Let K[U] be a core model. Let λ be a cardinal greater than γ := supdom(U). Assume π : K_{λ}[U] → W is elementary, W is transitive, and π has critical point α, greater than γ. Then there is an elementary embedding π̄ : K[U] → K[U] with critical point α.

Chang’s Conjecture at club-many Points

Hotspur Will this content you, Kate? Lady Percy It must of force. [Exeunt]

(William Shakespeare, King Henry IV, Part I, Act 2, Scene 3)

3.1 Definition Chang’s Conjecture, CC, is the statement that any structure 𝔄 = ⟨A, P, …⟩ of countable type such that card(A) = ω_{2} and card(P) = ω_{1} has an elementary substructure 𝔅 ≺ 𝔄, 𝔅 = ⟨B, R, …⟩, such that card(B) = ω_{1} and card(R) = ω. This is also written as ⟨ω_{2}, ω_{1}⟩ ⟨ω_{1}, ω⟩. For regular cardinals κ, λ, μ, ν, ⟨κ, λ⟩ ⟨μ, ν⟩ denotes the obvious generalisation. Instead of requiring card(R) = ν one can also consider demanding card(R) < ν, denoted by ⟨κ, λ⟩ ⟨μ, <ν⟩.

14 CHAPTER 3. CHANG’S CONJECTURE AT CLUB-MANY POINTS

Silver [Sil71] showed that the existence of an ω_{1}-Erdős cardinal (cf. Definition 3.7) implies that CC holds in some forcing extension of the universe. He used a modification of the Levy collapse as partial order for his forcing, which is now aptly known as the Silver collapse. Later, Donder and Levinski [DL89] and Baumgartner [Bau91] showed that, in fact, the Levy collapse will also do. On the other hand, Donder [DJK81] established that if CC holds, then ω_{2} is ω_{1}-Erdős in an inner model, so that this is really the exact consistency strength of CC. We now want to consider what seems at first sight to be a strengthening of CC, requiring that there exist a closed unbounded set C such that the ordertype of the predicate R of 𝔅 can assume any value from this set. It will turn out that this is actually equivalent to the original CC.

Note first that by relabeling, if necessary, one can assume that A = ω_{2} and P = ω_{1} (or respectively κ and λ).

3.2 Definition CC^{club} is the statement that for any structure 𝔄 = ⟨A, P, …⟩ of countable type such that A = ω_{2} and P = ω_{1}, there exists a closed unbounded set C ⊆ ω_{1} such that for all α ∈ C there exists an elementary substructure 𝔅^{α} ≺ 𝔄, 𝔅^{α} = ⟨B^{α}, R^{α}, …⟩, such that card(B^{α}) = ω_{1} and R^{α} = α. We denote this by ⟨ω_{2}, ω_{1}⟩ _{club} ⟨ω_{1}, <ω_{1}⟩.

For higher cardinals, one has to be slightly more careful, as the example in [Sch96, Lemma 6.2] shows that not all cofinalities for α are always possible. Thus, assuming ν = ϱ^{+} , let ⟨κ, λ⟩ _{club} ⟨μ, <ν⟩ denote the obvious generalization of the above statement, except that we now require C to be only ϱ-club in ν, and hence w.l.o.g. concentrating on α with cf(α) = ϱ.

Note that if π_{α} denotes the inverse of the transitive collapse of 𝔅^{α} to ℭ^{α}, then α will be the critical point of π_{α}, and π_{α}(α) = λ.

It is known that CC has a combinatorial equivalent in terms of a partition property.

3.3 Definition Let κ → [μ]_{λ,ν}^{<ω} denote the statement that for any function f : [κ]^{<ω} → λ there exists some H ⊆ κ, card(H) = μ, such that card(f′′[H]^{<ω} ) = ν.

Let κ → [μ]_{λ,<ν}^{<ω} denote the statement that for any function f : [κ]^{<ω} → λ there exists some H ⊆ κ, card(H) = μ, such that card(f′′[H]^{<ω} ) < ν.

3.4 Lemma Let κ λ, κ μ ν > ω. Then

⟨κ, λ⟩ ⟨μ, <ν⟩  iff  κ → [μ]_{λ,<ν}^{<ω} .

Proof Cf. [Kan94, Theorem 8.1], or the proof of Lemma 3.6 below.

Note that ⟨ω_{2}, ω_{1}⟩ ⟨ω_{1}, ω⟩ is equivalent to ⟨ω_{2}, ω_{1}⟩ ⟨ω_{1}, < ω_{1}⟩, and thus to ω_{2} → [ω_{1}]_{ω_{1},<ω_{1}}^{<ω} , as an infinite predicate can surely not be reduced to a finite one in a substructure.

It is thus natural to seek a combinatorial equivalent for CC^{club}:

3.5 Definition Let κ →_{club} [μ]_{λ,<ν}^{<ω} denote the statement that for any function f : [κ]^{<ω} → λ which is onto there exists a ϱ-club set C ⊆ ν, where ν = ϱ^{+} , such that for any α ∈ C there is a set H^{α} ⊆ κ, card(H^{α}) = μ, such that f′′[H^{α}]^{<ω} = α.

3.6 Lemma Let κ λ, κ μ ν = ϱ^{+} > ω. Then

⟨κ, λ⟩ _{club} ⟨μ, <ν⟩  iff  κ →_{club} [μ]_{λ,<ν}^{<ω} .

Proof Assume ⟨κ, λ⟩ _{club} ⟨μ, < ν⟩. Let f : [κ]^{<ω} → λ be onto. Let 𝔄 := ⟨κ, λ,∈ , (f [κ]^{n})_{n∈ω} ⟩. Let C ⊆ ν be a ϱ-club set guaranteed by the assumption. Choose α ∈ C, and let 𝔅^{α} = ⟨B^{α}, R^{α},∈ , …⟩ be as guaranteed by the assumption. 𝔅^{α} is an elementary substructure of 𝔄, and f′′[κ]^{<ω} = λ, so that f′′[B^{α}]^{<ω} = λ ∩ B^{α} = R^{α} = α, and card(B^{α}) = μ.

On the other hand, assume that κ →_{club} [μ]_{λ,<ν}^{<ω} . Let 𝔄 = ⟨κ, λ, …⟩. Let {h_{n} | n ∈ ω} be a complete set of Skolem functions for 𝔄 such that h_{n} is k(n)-ary, k(n) n. Define a function f : [κ]^{<ω} → λ by setting

f(ξ_{1}, …, ξ_{n}) := \{ \begin{matrix}h_{n}(ξ_{1}, …, ξ_{k(n)}) & if this is less than λ, \\ 0 & else. \end{matrix}

Note that f′′[κ]^{<ω} = λ, as e. g. h_{∃v_{1}(v_{1}=v_{0})} (ξ) = ξ, for ξ < λ. Thus we can apply the assumption to get some ϱ-club set C ⊆ ν. Let α ∈ C and let H^{α} ⊆ κ, card(H^{α}) = μ, f′′[H^{α}]^{<ω} = α. Let B^{α} := ⋃ _{n∈ω} h_{n}′′[H^{α}]^{k(n)}, so that card(B^{α}) = μ. Let 𝔅^{α} := ⟨B^{α}, λ ∩ B^{α}, …⟩. Then 𝔅^{α} ≺ 𝔄. Also, λ ∩ B^{α} ⊆ f′′[H^{α}]^{<ω} = α. In fact, the converse is true, too: f′′[H^{α}]^{<ω} is a subset of λ by the definition of f, and a subset of B^{α}, by the definition of B^{α}. Thus R^{α} = α.

3.7 Definition β → (α)_{δ}^{<ω} denotes the statement that for any f : [β]^{<ω} → δ there exsits an H ⊆ β with card(H) = α such that H is homogeneous for f, i. e., card(f′′[H]^{n}) 1 for every n ∈ ω. For α > ω, any λ that satisfies λ → (α)_{2}^{<ω} is called α-Erdős, and the least such λ is denoted by κ(α).

3.8 Theorem Suppose κ = κ(ω_{1}), the least ω_{1}-Erdős cardinal. Then

Col(ω_{1},κ) κ = ω̇_{2} ∧ CC^{club} .

Proof The proof is based on that of that of [Kan00, Theorem 33.10]. Let κ = κ(ω_{1}) and let P := Col(ω_{1}, κ) be the Levy collapse, i. e.,

P := {p | fun(p) ∧ dom(p) ⊆ κ × ω_{1} ∧ card(p) < ω_{1} ∧ ∀⟨α, ξ⟩∈dom(p) (α > 0 → p(α, ξ) < α)}.

Then P is ω_{1}-closed, and as κ is inaccessible, P has the κ-chain condition (κ-c.c.). P collapses κ to ω_{2}, preserving ω_{1} and cardinals greater than κ (cf. [Kan94, §10]). Assume that p_{0}∈P is a forcing condition such that

p_{0} ḟ : [κ]^{<ω} → ω_{1} onto.

Let G be a P -generic filter over V. We now seek a closed unbounded set C ⊆ ω_{1} such that, in V[G],

∀α∈C ∃Y^{α} ⊆ κ (card(Y^{α}) = ω_{1} ∧ ḟ^{G} ′′[Y^{α}]^{<ω} = α).

We will actually find such a set C independently of G. Note that if C ∈V is club in V, then C is club in V[G], too.

Since κ is the least ω_{1}-Erdős cardinal, for each γ < κ there exists a function f_{γ} : [γ]^{<ω} → 2 witnessing γ ↛ (ω_{1})_{2}^{<ω} . For n ∈ ω, let g_{n} : [κ]^{n+1} → 2 be defined by

g_{n}(α_{1}, …, α_{n}, γ) := f_{γ}(α_{1}, …, α_{n}).

Let W : V_{κ} −→^{∼} κ be a bijection (recall that κ is inaccessible), such that for all δ < κ such that δ is inaccessible, too, W δ : V_{δ} −→^{∼} δ is also bijective. Finally, code ḟinto a predicate R ⊆ P × [κ]^{<ω} × ω_{1} by setting

⟨p, x, β⟩∈R ↔ p p_{0} ∧ p ḟ(x̌) = β.

Let 𝔄 := ⟨V_{κ},∈, ω_{1}, {ω_{1}}, P, _{P}, {p_{0}}, W, R, (g_{n})_{n∈ω} ⟩, and let ⟨h_{n} | n∈ω⟩ be a complete set of definable (using W) Skolem functions for 𝔄. Assume w.l.o.g. that h_{n} is k(n)-ary with k(n) n. Let ⟨φ_{ξ} | ξ < ω_{1}⟩ be an enumeration of the formulae of L_{𝔄} ∪ {c_{β} | β < ω_{1}}. Define a function g : [κ]^{<ω} → ω_{1} by setting

g(ξ_{1}, …, ξ_{n}) := \{ \begin{matrix}h_{n}(ξ_{1}, …, ξ_{k(n)}) \\ if m = 2n + 1 and h_{n}(ξ_{1}, …, ξ_{k(n)}) < ω_{1}, \\ 0 & if m = 2n + 1 and h_{n}(ξ_{1}, …, ξ_{k(n)}) ω_{1}, \\ 0 & if m = 2n and ⟨ξ_{1}, …, ξ_{n}⟩ and ⟨ξ_{n+1} , …, ξ_{2n}⟩ \\ realize the same type over ⟨𝔄, (c_{β})_{β<ω_{1}} ⟩, \\ ξ & if m = 2n and φ_{ξ} is a witness to the fact that \\ ⟨ξ_{1}, …, ξ_{n}⟩ and ⟨ξ_{n+1} , …, ξ_{2n}⟩ do not \\ realize the same type over ⟨𝔄, (c_{β})_{β<ω_{1}} ⟩. \end{matrix}

As κ is ω_{1}-Erdős, let E ⊆ κ, card(E) = ω_{1}, be homogeneous for g. Let E = {η_{ζ} | ζ < ω_{1}} be the increasing enumeration and assume w.l.o.g. that η_{ω} is minimal. Let

H := H_{0} := {h_{n}(ζ_{1}, …, ζ_{k(n)}) | n∈ω ∧ ζ_{1} < … < ζ_{k(n)} ∈E}
H_{α} := {h_{n}(ζ_{1}, …, ζ_{k(n)}) | n∈ω ∧ ζ_{1} < … < ζ_{k(n)} ∈E ∪ α},

and let ℌ, ℌ_{α} be the corresponding substructures of respectively 𝔄 and ⟨𝔄, (c_{β})_{β<α} ⟩. To enhance legibility, for any α ω_{1}, let 𝔄_{α} denote ⟨𝔄, (c_{β})_{β<α} ⟩.

Claim 1 For any α∈ω_{1}, E is a set of indiscernibles for ℌ_{α}.

Proof Let α ∈ ω_{1}, and ⟨ξ_{1}, …, ξ_{n}⟩ and ⟨ζ_{1}, …, ζ_{n}⟩ from [E]^{<ω} . Choose another n-tuple ⟨ϑ_{1}, …, ϑ_{n}⟩ from [E]^{n}, such that ϑ_{1} > max(ξ_{n}, ζ_{n}). E is homogeneous for g, so

g(ξ_{1}, …, ξ_{n}, ϑ_{1}, …, ϑ_{n}) = g(ζ_{1}, …, ζ_{n}, ϑ_{1}, …, ϑ_{n}) = ξ,

for some ξ ∈ ω_{1}. If ξ = 0, then ⟨ξ_{1}, …, ξ_{n}⟩ and ⟨ζ_{1}, …, ζ_{n}⟩ realize the same type over 𝔄_{ω_{1}} and so, a fortiori, over 𝔄_{α}, so that

ℌ_{α} φ(ξ_{1}, …, ξ_{n}) ↔ ℌ_{α} φ(ζ_{1}, …, ζ_{n}).

If, on the other hand, ξ > 0, then find another n-tuple ⟨ι_{1}, …, ι_{n}⟩ from [E]^{n}, such that ι_{1} > ϑ_{n}. Then

g(ξ_{1}, …, ξ_{n}, ι_{1}, …, ι_{n}) = g(ϑ_{1}, …, ϑ_{n}, ι_{1}, …, ι_{n}) = ξ,

too. But this gives the following absurdity:

𝔄_{ω_{1}} φ_{ξ}(ξ_{1}, …, ξ_{n}) ↔

¬𝔄_{ω_{1}} φ_{ξ}(ϑ_{1}, …, ϑ_{n}) ↔ 𝔄_{ω_{1}} φ_{ξ}(ι_{1}, …, ι_{n})

↔ ¬𝔄_{ω_{1}} φ_{ξ}(ξ_{1}, …, ξ_{n}).

Note that any η ∈ E must be at least ω_{1}, as else 𝔄_{η+1} c_{η} = η, which no other element ζ ∈E can possibly satisfy. (Claim 1)

Claim 2 For any α∈ω_{1}, H_{α} ∩ ω_{1} is countable.

Proof Let α ∈ ω_{1}, and β_{1}, …, β_{l} ∈ α, ξ_{1}, …, ξ_{n} ∈ E. Let h be one of the Skolem functions and assume that h(β_{1}, …, β_{l}, ξ_{1}, …, ξ_{n}) < ω_{1}. If l = 0, then by the homogeneity of E for g, we get

h(ξ_{1}, …, ξ_{n}) = h(ζ_{1}, …, ζ_{n})

for any n-tuple ⟨ζ_{1}, …, ζ_{n}⟩ from E. Thus for l = 0, only countably many different values are possible. If l > 0, then let γ := h(β_{1}, …, β_{l}, ξ_{1}, …, ξ_{n}) < ω_{1}. Assume w.l.o.g. that α > γ. If ⟨ζ_{1}, …, ζ_{n}⟩ is another n-tuple from E, then by Claim 1 (and as the Skolem functions are definable) we may conclude that

𝔄 h(β_{1}, …, β_{l}, ξ_{1}, …, ξ_{n}) = γ
↔ 𝔄_{α} h(c_{β_{1}} , …, c_{β_{l}} , ξ_{1}, …, ξ_{n}) = c_{γ}
↔ 𝔄_{α} h(c_{β_{1}} , …, c_{β_{l}} , ζ_{1}, …, ζ_{n}) = c_{γ}
↔ 𝔄 h(β_{1}, …, β_{l}, ζ_{1}, …, ζ_{n}) = γ.

Now [α]^{<ω} is countable and there are only countably many Skolem functions, so that, indeed, H_{α} ∩ ω_{1} is countable. (Claim 2)

Claim 3 The set C := {α ∈ ω_{1} | H_{α} ∩ ω_{1} = α} is closed and unbounded in ω_{1}.

Proof To see that C is closed, let ⟨α_{i} | i < γ⟩ be a sequence from C, γ < ω_{1}. Let α := ⋃ _{i<γ} α_{i}. Let ζ ∈ H_{α} ∩ ω_{1}. Then there are some Skolem function h, some β_{1}, …, β_{l}∈α, and some ζ_{1}, …, ζ_{n} ∈E such that

ζ = h(β_{1}, …, β_{l}, ζ_{1}, …, ζ_{n}).

Obviously, there exists some i_{0} < γ such that β_{1}, …, β_{l}∈α_{i_{0}} , whence

ζ ∈H_{α_{i_{0}}} ∩ ω_{1} = α_{i_{0}} ⊆ α.

Thus H_{α} ∩ ω_{1} ⊆ α, and the converse is trivially true, so that α∈C.

To see that C is unbounded in ω_{1}, let α_{0} < ω_{1} be arbitrary. Define inductively α_{i+1} := sup(H_{α_{i}} ∩ω_{1}) < ω_{1} for i < ω and set α := ⋃ _{i<ω} α_{i}. Again α ⊆ H_{α} ∩ω_{1} is trivially true. Let ζ ∈ H_{α} ∩ ω_{1}. Then there are some Skolem function h, some β_{1}, …, β_{l}∈α, and some ζ_{1}, …, ζ_{n} ∈E such that

ζ = h(β_{1}, …, β_{l}, ζ_{1}, …, ζ_{n}).

Again, there exists some i_{0} < ω such that β_{1}, …, β_{l}∈α_{i_{0}} , so that now

ζ ∈H_{α_{i_{0}}} ∩ ω_{1} ⊆ α_{i_{0}+1} ⊆ α.

(Claim 3)

Claim 4 For any α∈ω_{1} the set E is α-remarkable, i. e., for any (l + m + n)- ary Skolem function h, any β_{1} < … < β_{l} < α, and any ζ_{1} < … < ζ_{m+n} < ω_{1},

if h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{m+n}} )∈η_{ζ_{m+1}} , then for all ζ_{m} < δ_{1} < … < δ_{n} < ω_{1} h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{m+n}} ) = h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{m}}, η_{δ_{1}}, …, η_{δ_{n}}).

Proof Assume that for some α ∈ ω_{1}, some Skolem function h, some β_{1} < … < β_{l} < α, and some ζ_{1} < … < ζ_{m+n} < ω_{1},

h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{m+n}} )∈η_{ζ_{m+1}} ,

but

h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{m+n}} ) ≠ h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{m}}, η_{δ_{1}}, …, η_{δ_{n}})

for some (and hence by indiscernibility for all) n-tuple δ_{1}, …, δ_{n} from ω_{1} such that δ_{1} > ζ_{m}. Now partition E into successive bits s_{γ} of length n, with the exception of s_{0}, having length m. I. e., s_{0} = {η_{1}, …, η_{m}}, s_{1} = {η_{m+1} , …, η_{m+n} }, and s_{ω} = {η_{ω}, …, η_{ω+n−1} }. By indiscernibility, for γ < δ < ω_{1},

h(β⃗, s_{0}, s_{γ}) ≠ h(β⃗, s_{0}, s_{δ}).

But surely h(β⃗, s_{0}, s_{γ}) > h(β⃗, s_{0}, s_{δ}) is impossible, as this would give an infinite descending chain of ordinals. Thus γ < δ must imply h(β⃗, s_{0}, s_{γ}) < h(β⃗, s_{0}, s_{δ}). Note that by assumption, and indiscernibility, we have

h(β⃗, s_{0}, s_{ω})∈η_{ω}.

But {h(β⃗, s_{0}, s_{γ}) | γ < ω_{1}} is a set of indiscernibles for 𝔄 of size ω_{1}, and hence homogeneous for g, contradicting the minimality of η_{ω}. (Claim 4)

Claim 5 For any α ∈ ω_{1}, Condition II of the theory of 0^{#} holds for E ∪ α, i. e., for any (l + n)-ary Skolem function h, any β_{1} < … < β_{l} < α, and any ζ_{1} < … < ζ_{n+1} < ω_{1},

if h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{n}})∈On then h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{n}})∈η_{ζ_{n+1}} .

Proof If the claim is false, then using indiscernibility one sees that

E′:= E \ η_{ζ_{n+1}} ⊆ γ := h(β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{n}}) + 1.

E′ has cardinality ω_{1}, is a subset of γ, and the functions g_{n} ensure that it is homogeneous for f_{γ}: let m∈ω, ι∈{0, 1}, and ⟨η_{ϑ_{1}} , …, η_{ϑ_{m}} ⟩, ⟨η_{κ_{1}} , …, η_{κ_{m}} ⟩∈ [E′]^{m}. Then

f_{γ}(η_{ϑ_{1}} , …, η_{ϑ_{m}} ) = ι
↔ 𝔄 g_{n}(η_{ϑ_{1}} , …, η_{ϑ_{m}} , γ) = ι
↔ 𝔄 φ(ι, β_{1}, …, β_{l}, η_{ζ_{1}}, …, η_{ζ_{n}}, η_{ϑ_{1}} , …, η_{ϑ_{m}} )
↔ 𝔄_{α} φ(ι, c_{β_{1}} , …, c_{β_{l}} , η_{ζ_{1}}, …, η_{ζ_{n}}, η_{ϑ_{1}} , …, η_{ϑ_{m}} )
↔ 𝔄_{α} φ(ι, c_{β_{1}} , …, c_{β_{l}} , η_{ζ_{1}}, …, η_{ζ_{n}}, η_{κ_{1}} , …, η_{κ_{m}} )
↔ f_{γ}(η_{κ_{1}} , …, η_{κ_{m}} ) = ι.

But this contradicts the choice of f_{γ} as a witness to γ ↛ (ω_{1})_{2}^{<ω} . (Claim 5)

Claim 6 If η ∈E, then η is inaccessible.

Proof By indiscernibility, it suffices to show that η_{0} is inaccessible. So let j be an order-preserving injection from E to E with j(η_{0}) > η_{0}. This map easily extends to an elementary embedding j̃ : ℌ → ℌ. By remarkability, we must have crit(j̃) = η_{0}: Let γ = h(η_{ζ_{1}}, …, η_{ζ_{n}}) < η_{0}. Remarkability implies that we can substitute for η_{ζ_{1}}, …, η_{ζ_{n}} any other n-tuple of indiscernibles (we have α = m = 0), so in particular

γ = h(η_{ζ_{1}}, …, η_{ζ_{n}}) = h(j(η_{ζ_{1}}), …, j(η_{ζ_{n}})) = j̃(γ).

From this, we can conclude that η_{0} is totally indescribable, whence inaccessible: Assume that for some formula φ

ℌ (R ⊆ V_{η_{0}} ∧ ⟨V_{η_{0}},∈, R⟩ φ).

Then

ℌ ∃α < j̃(η_{0}) (R ⊆ V_{α} ∧ ⟨V_{α},∈, R⟩ φ),

and since j̃ is an elementary embedding from ℌ to ℌ,

ℌ ∃α < η_{0} (R ⊆ V_{α} ∧ ⟨V_{α},∈, R⟩ φ),

so η_{0} is totally indescribable (and thus inaccessible) in ℌ. Since ℌ ≺ 𝔄, η_{0} is really inaccessible. (Claim 6)

Consider now the partial order P. Let P_{γ} := Col(ω_{1}, γ). Then P ≃ P_{γ} × Col(ω_{1}, κ \ γ), by [Kan94, Lemma 10.17 b)].

Claim 7 If A is a maximal antichain in P_{γ}, then it is a maximal antichain in P.

Proof Let A be a maximal antichain in P_{γ}. Let p ∈ P \ A. Then p γ ∈ P_{γ}, so that there is some q ∈ A such that p γ and q are compatible. But surely p (κ\γ) and q are compatible, too, so that in the end p and q are compatible. Thus A is maximal in P, too. (Claim 7)

Let S := {x∈[E]^{<ω_{1}} \{0} | x has no last element}, and define a partial order on S by setting

x <_{i} y ↔ ∃γ < ∪y (x = y ∩ γ),

i. e., x <_{i} y iff x is a proper initial segment of y.

From now on, let α∈C. For x∈S, let N_{x}^{α} be the domain of the Skolem hull of x ∪ α in 𝔄. Then N_{x}^{α} is a countable model of ZFC.

For p ∈ P, say that p is P -generic over N_{x}^{α} iff {q ∈ P ∩ N_{x}^{α} | q p} is P -generic over N_{x}^{α}, and similarly define P_{γ}-generic for γ ∈N_{x}^{α} ∩ κ.

Claim 8 Let x, y ∈ S, x <_{i} y, γ := min(y \ x). Assume that p is P -generic over N_{x}^{α}. Then p is P_{γ}-generic over N_{y}^{α}.

Proof Let A∈N_{y}^{α} be a maximal antichain in P_{γ}. Then

A = h(β_{1}, …, β_{n}, η_{ζ_{1}}, …, η_{ζ_{m}}, η_{ζ_{m+1}} , …, η_{ζ_{m+n}} ),

for some β_{1}, …, β_{n} ∈α, η_{ζ_{1}}, …, η_{ζ_{m}} ∈x, and η_{ζ_{m+1}} , …, η_{ζ_{m+n}} ∈y\x. By Claim 6, γ is inaccessible, so P_{γ} has the γ-c.c. Thus card(A) < γ, and hence A∈V_{γ}. Recall that W V_{γ} : V_{γ} −→^{∼} γ is bijective. Applying α-remarkability, one can find η_{δ_{1}}, …, η_{δ_{n}} ∈ x (!) such that A = h(β_{1}, …, β_{n}, η_{ζ_{1}}, …, η_{ζ_{m}}, η_{δ_{1}}, …, η_{δ_{n}}). This shows that A ∈ N_{x}^{α}. Since A is a maximal antichain in P_{γ}, it is also a maximal antichain in P, by Claim 7. Since p was assumed to be P -generic over N_{x}^{α}, there must exist some q ∈ P ∩ N_{x}^{α} such that q p and q ∈ A. But N_{x}^{α} ⊆ N_{y}^{α}, so that {q ∈P_{γ} ∩ N_{y}^{α} | q p} ∩ A ≠ ∅. Thus p is P_{γ}-generic over N_{y}^{α}. (Claim 8)

Claim 9 Suppose x∈S, and p is P -generic over N_{x}^{α}. Then there exists some y ∈S, y >_{i} x, and some p′ p such that p′ is P -generic over N_{y}^{α}.

Proof Let τ := sup{α < sup(E) | ∃ξ (⟨α, ξ⟩∈dom(p))}. Then τ < sup(E), as p is countable. Choose some y ∈ S, y >_{i} x, such that y \ x ⊆ E \ (τ + 1). Set γ := min(y \ x). By Claim 8,

G_{0} := {q ∈P_{γ} ∩ N_{y}^{α} | q p}

is P_{γ}-generic over N_{y}^{α}. Apply the product analysis ([Kan94, Lemma 10.17 b)]) to P ≃ P_{γ} × Col(ω_{1}, κ \ γ) to find some G_{1} such that G_{0} × G_{1} is P - generic over N_{y}^{α}. (Recall that N_{y}^{α} is countable, so that generics exist.) Note that On∩N_{y}^{α} ⊆ sup(E), by Claim 5, so that if r ∈ G_{1}, then dom(r) ⊆ ( sup(E) \ (τ + 1)) × ω_{1}. Thus if r ∈ G_{1}, then p and r are compatible, since dom(p) ⊆ ((τ + 1) ∪ (κ \ sup(E))) × ω_{1}. Finally, set p′ := p ∪ ∪G_{1}. Then p′∈ P, as G_{1} is countable and P is ω_{1}-closed, and p′ is P -generic over N_{y}^{α}.

(Claim 9)

We will now proceed to show that whenever G is P -generic over V such that p_{0}∈G, then there exists some set Y^{α} ∈V[G] of size ω_{1} such that ḟ^{G} ′′[Y^{α}]^{<ω} = α.

Define recursively in V[G] a sequence ⟨y_{ξ} | ξ < ω_{1}⟩ of elements of S, such that ξ < ζ implies y_{ξ} <_{i} y_{ζ}, and a sequence ⟨q_{ξ} | ξ < ω_{1}⟩ of elements of G, such that q_{ξ} p_{0} and such that q_{ξ} is P -generic over N_{y_{ξ}}^{α}.

Set y_{0} := {η_{n} | n∈ω}, the first ω-many elements of E. Note that p_{0}∈N_{y}^{α}, as it was included as a constant in 𝔄. Now find some q_{0}∈G, q_{0} p_{0}, such that q_{0} is P -generic over N_{y_{0}}^{α}. To this end, let ⟨D_{n} | n ∈ ω⟩ be an enumeration of the dense subsets of P from N_{y_{0}}^{α}. Note that if N_{y_{0}}^{α} D dense in P, then by elementarity, 𝔄 D dense in P, too. So inductively find r_{0} p_{0} and r_{n} r_{n−1} such that r_{n} ∈ G ∩ D_{n}. As P is ω_{1}-closed, there exists q_{0}∈ G such that ∀n∈ω(q_{0} r_{n}). Then q_{0} is as desired.

At successor steps, assume y_{ξ} and q_{ξ} to be given. Then the set {q q_{ξ} | q p_{0} ∧ ∃y (y >_{i} y_{ξ} ∧ q is P -generic over N_{y}^{α})} is dense in P below q_{ξ}: For let r q_{ξ}. Then r is P -generic over N_{y_{ξ}}^{α}, too, and by Claim 9 there exists some y_{r} >_{i} y_{ξ} and some q_{r} r such that q_{r} is P -generic over N_{y_{r}}^{α}. But now, since G is P -generic and q_{ξ} ∈ G, there must exist some q_{ξ+1} q_{ξ}, q_{ξ+1} ∈ G ∩ {q q_{ξ} | q p_{0} ∧ ∃y (y >_{i} y_{ξ} ∧ q is P -generic over N_{y}^{α})}. Let y_{ξ+1} be an appropriate witness.

At limit stages δ < ω_{1}, set y_{δ} := ⋃ _{ξ<δ} y_{ξ} and, using the ω_{1}-closure of P and the genericity of G, find some q_{δ} ∈ G such that for all ξ < δ, q_{δ} q_{ξ}. Then obviously q_{δ} is P -generic over N_{y_{δ}}^{α}.

Finally, set Y^{α} := ⋃ _{ξ<ω_{1}} N_{y_{ξ}}^{α} ∩ κ. All N_{y}^{α} are countable, so card(Y^{α}) ω_{1}, and as y_{ξ} <_{i} y_{ξ+1} , in fact, one has card(Y^{α}) = ω_{1}. Now

ḟ^{G} ′′[Y^{α}]^{<ω} ⊆ ω_{1} ∩ ⋃_{ξ<ω_{1}} N_{y_{ξ}}^{α} ⊆ ω_{1} ∩ H^{α} = α,

for if ⟨γ_{1}, …, γ_{n}⟩∈[Y^{α}]^{n}, then ⟨γ_{1}, …, γ_{n}⟩∈ ⋃ _{ξ<δ} N_{y_{ξ}}^{α} = N_{y_{δ}}^{α}, for some δ < ω_{1}, so that ḟ^{G} (γ_{1}, …, γ_{n}) ∈ N_{y_{δ}}^{α}, too. But the converse is true, too. Let β < α.

Then β ∈N_{y_{0}}^{α}, and D_{β} := {q p_{0} | ∃x∈[κ]^{<ω} (q ḟ(x̌) = β)} is an element of N_{y_{0}}^{α} and dense in P below p_{0} (as p_{0} ḟonto). Now q_{0} is P -generic over N_{y_{0}}^{α}, and q_{0}∈G. So there is indeed some x∈[κ]^{<ω} ∩N_{y_{0}}^{α} such that ḟ^{G} (x̌^{G}) = β. But then x∈[Y^{α}]^{<ω} , as N_{y_{0}}^{α} ∩ κ ⊆ Y^{α}. Thus β ∈ḟ^{G} ′′[Y^{α}]^{<ω} .

Thus ḟ^{G} ′′[Y^{α}]^{<ω} = α, as desired.

We have shown that given any p_{0} such that p_{0} ḟ : [κ]^{<ω} → ω_{1} onto, there exists some closed unbounded set C ⊆ ω_{1} such that for any α∈C and for any P -generic filter G with p_{0} ∈ G, there is a set Y^{α} ⊆ κ, card(Y^{α}) = ω_{1}, such that V[G] ḟ^{G} ′′[Y^{α}]^{<ω} = α. That is, _{Col(ω_{1},κ)} κ = ω̇_{2} ∧ CC^{club}.

Note that the proof will only work for CC^{club} and not for higher transfer properties, say ⟨κ^{+} , κ⟩ _{club} ⟨κ, < ν⟩, where κ = ν^{+} > ω_{1}: The countability of the N_{x}^{α} is essential to get the required generic objects G in the proof of Claim 9. This is no coincidence. For example, [Sch96, Theorem 6.2] shows that one needs at least a strong cardinal to get a model for ⟨ℵ_{4}, ℵ_{3}⟩ ⟨ℵ_{3}, ℵ_{2}⟩ and 2^{ℵ_{1}} = ℵ_{2}. So far, any forcing to get this uses, in fact, a huge cardinal [For82].

The Transversal Hypothesis

“Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.”

(Lewis Carroll, Through the Looking-Glass, Chapter IV)

We will now consider a combinatorial principle closely related to Chang’s Conjecture.

4.1 Definition Let κ be a regular cardinal. The Transversal Hypothesis, TH(κ), is the statement: There exists a sequence of functions ⟨g_{ξ} | ξ < κ^{+} ⟩ such that for all ξ, g_{ξ} is regressive, and that for all ξ < ζ, {α < κ | g_{ξ}(α) = g_{ζ}(α)} has cardinality less than κ. TH^{*}(κ) is the same statement, with “has cardinality less than κ” replaced by “is not stationary”. In this case, the set {α < κ | g_{ξ}(α) ≠ g_{ζ}(α)} contains a club set.

Assume κ = ϱ^{+} . Let TH^{stat}(κ) be the statement: There exists a set S ⊆ κ, a sequence of functions ⟨g_{ξ} | ξ < κ^{+} ⟩ and sequences of sets ⟨D_{ξ} | ξ < κ⟩ and ⟨D_{ξζ} | ξ < ζ < κ⟩ such that

i) S is stationary and S ⊆ CF_{ϱ} := {α < κ | cf(α) = ϱ}.

ii) ∀ξ < κ^{+} (D_{ξ} is club in κ, D_{ξ} ∩ S ⊆ dom(g_{ξ}), and g_{ξ} is regressive).

iii) ∀ξ < ζ < κ^{+} (D_{ξζ} is club in κ and
D_{ξζ} ∩ S ⊆ {α∈D_{ξ} ∩ D_{ζ} ∩ S | g_{ξ}(α) ≠ g_{ζ}(α)}).

Obviously, TH(κ) → TH^{*}(κ) → TH^{stat}(κ). Jensen observed (cf. [Don84]) that TH^{*}(κ) → TH(κ). It suffices that the g_{ξ} only be defined on club sets themselves. TH^{stat}(κ) intends to capture the essence of TH^{*}(κ) when restricted to a fixed stationary set S. Shelah [She86] has shown that without the requirement that S ⊆ CF_{ϱ}, TH^{stat}(ℵ_{n}) (for 2 n ω) is a theorem of ZFC. Usually, the interest lies in the negations of these hypotheses, as, e. g., Chang’s Conjecture at κ, with κ = ϱ^{+} , implies ¬ TH(κ). Correspondingly, ¬ TH^{stat}(κ) is a consequence of ⟨κ^{+} , κ⟩ _{club} ⟨κ, <κ⟩, cf. Lemma 4.2 below. The previous chapter, in showing the consistency of ⟨ω_{2}, ω_{1}⟩ _{club} ⟨ω_{1}, < ω_{1}⟩ from the existence of an ω_{1}-Erdős cardinal, hence serves to make the consistency of ¬ TH^{stat}(κ) plausible for κ > ω_{1} (although surely larger cardinals are needed than just Erdős-cardinals). In Theorem 4.3, we give a lower bound for the consistency strength of this hypothesis.

4.2 Lemma Let κ = ϱ^{+} be a cardinal, and assume that ⟨κ^{+} , κ⟩ _{club} ⟨κ, <κ⟩. Then ¬ TH^{stat}(κ) holds.

Proof Assume ⟨κ^{+} , κ⟩ _{club} ⟨κ, < κ⟩. Assume S, ⟨g_{ξ} | ξ < κ^{+} ⟩, ⟨D_{ξ} | ξ < κ⟩

and ⟨D_{ξζ} | ξ < ζ < κ⟩ satisfied the conditions of TH^{stat}(κ). Code the sequences by some predicates G ⊆ κ^{+} × κ × κ, and D^{1}, D^{2} ⊆ κ^{+} × κ^{+} × κ.

Set 𝔄 := ⟨κ^{+} , κ, G, D^{1}, D^{2}⟩. Now apply ⟨κ^{+} , κ⟩ _{club} ⟨κ, < κ⟩ to find a set

C ⊆ κ such that C is ϱ-club, and such that for all γ ∈ C there exists some 𝔅^{γ} ≺ 𝔄, card(B^{γ}) = κ, B^{γ} ∩ κ = γ. As S is stationary and S ⊆ CF_{ϱ}, there exists a γ ∈C ∩ S.

We claim that for any ξ < ζ, both from B ∩ κ^{+} , γ is an element of D_{ξζ}: D_{ξζ} is club in κ, thus as 𝔅^{γ} ≺ 𝔄 and B^{γ} ∩ κ = γ, it follows that D_{ξζ} ∩ γ is club in γ. But then sup(D_{ξζ} ∩ γ) = γ ∈D_{ξζ}, as D_{ξζ} is closed.

Now if γ is both in D_{ξζ} and in S, then g_{ξ}(γ) ≠ g_{ζ}(γ) (provided of course that ξ, ζ are both from B ∩ κ^{+} ). Thus x := {g_{ξ}(γ) | ξ ∈B} has cardinality κ. But all the g_{ξ} are regressive, so that x ⊆ γ < κ, contradiction! Hence TH^{stat}(κ) must fail.

4.3 Theorem Let κ be a successor cardinal, κ = ϱ^{+} ω_{2}, 2^{κ} = κ^{+} . Assume that ¬ TH^{stat}(κ). Then 0^{long} exists.

Proof The proof of this theorem adapts techniques from [Koe88] and [DK83], especially the proof of Theorem B there, as well as those of Ketonen’s proof that the existence of an irregular ultrafilter entails the existence of 0^{#} (cf. also Chapter 7).

Let κ = ϱ^{+} ω_{2}, and assume ¬ TH^{stat}(κ). We have to show that 0^{long} exists. So assume to the contrary that 0^{long} does not exist and work for a contradiction.

Let U_{can} be the canonical, maximal strong sequence of filters from Definition 2.14. Let U := U_{can} κ^{+} , ν := supdom(U). Then by Lemma 2.12, ν < κ^{+} .

For τ ∈(κ, κ^{+} ) let f_{τ} be a surjection from κ onto τ, and let F := {⟨ξ, ζ, τ⟩ | f_{τ}(ξ) = ζ}. Let F τ := F ∩ (κ × τ × τ).

For τ ∈ (κ, κ^{+} ) such that cf(τ) = κ, let g_{τ} : κ → τ monotone cofinally. If cf(τ) < κ, let g_{τ} : κ → τ be arbitrary. Let G := {⟨ξ, ζ, τ⟩ | g_{τ}(ξ) = ζ}. Let G τ := G ∩ (κ × τ × τ).

Let h : κ^{+} → H_{κ^{+}} be a bijection. Let H = ⟨H_{κ^{+}} ,∈, h, U, F, G⟩, W := K_{κ^{+}} [U]. For τ ∈(κ, κ^{+} ), let H̃_{τ} := H (h′′τ), H_{τ} := h′′τ = |H̃_{τ} |, W_{τ} := H_{τ} ∩ W. Let

E := {τ ∈(κ, κ^{+} ) | H̃_{τ} ≺ H ∧ On∩H_{τ} = τ ∧ H_{τ} transitive ∧ H̃_{τ} W_{τ} = K[U]}

Then E is a club subset of κ^{+} .

For τ ∈E let

K_{τ}[U]:= W_{τ}
Q_{τ} := ⟨K_{τ}[U],∈, U τ, F τ, f_{τ}, G τ⟩
Q̃_{α}^{τ} := the smallest Q ≺ Q_{τ} s. t. α ⊆ Q
C_{τ} := {α < κ | Q̃_{α}^{τ} ∩ κ = α}

If x is an element of K_{τ}[U], then {α ∈ C_{τ} | x ∈ Q̃_{α}^{τ}} is a final segment of C_{τ}. Assume w.l.o.g. that min(C_{τ}) > ϱ, where ϱ^{+} = κ. Then for α ∈ C_{τ}, card(α)^{+} = κ. Also, by Lemma 2.12, γ := supdom(U κ) < κ, so that we may also assume w.l.o.g. that min(C_{τ}) > γ. It is straightforward to show that C_{τ} is club in κ.

Let Q_{α}^{τ} := ⟨K_{α}^{τ},∈, U_{α}^{τ}, F_{α}^{τ}, (f_{τ})_{α}^{τ}, G_{α}^{τ}⟩ be the transitive model isomorphic to Q̃_{α}^{τ} and let π̃_{α}^{τ} : Q_{α}^{τ} → Q̃_{α}^{τ} be the inverse of the collapsing isomorphism. Let K_{α}^{τ} := (K[U_{α}^{τ}])^{Q_{α}^{τ}} , τ_{α} := On∩K_{α}^{τ}, and π_{α}^{τ} := π̃_{α}^{τ} K_{α}^{τ}. Then π_{α}^{τ} : K_{α}^{τ} → K_{τ}[U] is an elementary embedding. We refer to this setup as a Ketonen diagramme, cf. Figure 4.1.

[figure: −; K; −; κ^{+}; −; σ; −; τ; −; κ; −; α; −; K_{α}^{τ}, K_{α}^{σ}; κ; −; σ_{α}; −; τ_{α}; −; α; π_{α}^{τ}; π_{α}^{σ}]

Figure 4.1: The basic structure of the Ketonen diagramme.

Let τ, σ ∈ E, τ < σ, τ ∈ rge(π_{α}^{σ}), and α ∈ C_{σ}. Then α ∈ C_{τ} and Q_{α}^{τ} = (π̃_{α}^{σ})^{−1} (Q_{τ}) ∈ Q_{α}^{σ}, π̃_{α}^{τ} = π̃_{α}^{σ} Q_{α}^{τ}, and hence K_{α}^{τ} = (π_{α}^{τ})^{−1} (K_{τ}[U]) ∈ K_{α}^{σ}, π_{α}^{τ} = π_{α}^{σ} K_{α}^{τ}, and K_{α}^{τ} ≺ K_{α}^{σ}.

The following lemma, adapted from [DJK81, Lemma 2.3], will play a key rôle in the proof:

4.4 Lemma Let S ⊆ CF_{ϱ} be stationary, f ∈ ∏ _{α∈S} card(α)^{+} . Let τ ∈ E. Then there is a σ ∈E, σ > τ, such that

{α∈S ∩ C_{σ} | f(α) < σ_{α}} is stationary.

Proof Assume the statement to be false. Then for all σ ∈ E, σ > τ, there exists a set D′_{σ} ⊆ κ such that D′_{σ} is club in κ and such that D′_{σ} ∩ {α ∈ S ∩ C_{σ} | f(α) < σ_{α}} = ∅. Set D_{σ} := D′_{σ} ∩ C_{σ}. Then D_{σ} is also club in κ, and for all α ∈ D_{σ} ∩ S, f(α) σ_{α}. For each α ∈ S, let h^{α} be an injection from f(α) + 1 into card(α). For σ ∈ E, σ > τ, define a function g_{σ} with dom(g_{σ}) = D_{σ} ∩ S by setting g_{σ}(α) := h^{α}(σ_{α}). Surely, g_{σ} is regressive. For σ < σ′ set D_{σσ′} := D_{σ} ∩ D_{σ′} ∩ C_{σσ′}. Then if α ∈ D_{σσ′} ∩ S, σ_{α} ≠ σ′_{α} and hence g_{σ}(α) ≠ g_{σ′}(α), i. e., ⟨g_{σ} | σ ∈ E \ (τ + 1)⟩ is a TH^{stat}(κ)-sequence, contradiction. (Lemma 4.4)

We now start to distinguish several cases. One of two things will happen: Either, as in Case 1, we can define, from one of the embeddings π_{α}^{τ}, an ultrafilter on some α which allows us to get an embedding from K[U] to K[U], contradicting the rigidity of K[U]. Or we can, as in Case 2.1, show that some ultrafilter has been omitted in the definition of the canonical sequence, equally a contradiction.

Case 1 dom(U) ∩ [κ, κ^{+} ) = ∅.

In this case, there is no ultrafilter from the canonical sequence in the interval from κ to κ^{+} . By Lemma 2.12 we now know that ν = supdom(U) = supdom(U κ) < κ. By restricting to the right sets of τ’s and α’s, we can mimick the original Ketonen proof, as the K_{α}^{τ} condensate more or less to initial segments of K[U].

For the rest of this case, assume that for all τ ∈ E, min(C_{τ}) > ν. Also, as d := dom(U) ⊆ ν < κ < κ^{+} , d ∈ K[U], we have that d ∈ Q̃_{α}^{τ} for a final segment of C_{τ}, so again assume w.l.o.g. that this holds for all α∈C_{τ}. Then, as ν < α, we have that dom(U_{α}^{τ}) = dom(U). Similarly argue that U ∈Q̃_{α}^{τ}, so U_{α}^{τ} = (π_{α}^{τ})^{−1} (U), and if σ ∈E, τ < σ, and τ ∈rge(π_{α}^{σ}), then U_{α}^{τ} = U_{α}^{σ}.

The following assertion (⋆)_{κ} is a variation of the property (∗)_{λ} from the proof of [DK83, Theorem B].

(⋆)_{κ} Let τ ∈ E, S ⊆ C_{τ} ∩ CF_{ϱ}, S stationary, and let ⟨M_{α} | α ∈ S⟩ be a sequence of short, iterable U_{α}^{τ}-premice, card(M_{α}) < card(α)^{+} = κ.

Then there is σ ∈E, σ > τ, and S′ stationary, S′⊆ S ∩ C_{σ}, such that
∀α∈S′ (τ ∈rge(π_{α}^{σ}) ∧ lp(M_{α}) ⊆ K_{α}^{σ}).

Claim 1 (⋆)_{κ} holds.

Proof By Theorem 2.16 we have that for limit ordinals η ∈ (max(ν, κ), κ^{+} ) with cf(η) < card(η)

K[U_{can}] “η is singular”.

Thus

K_{κ^{+}} [U] “η is singular”.

By our choice of G we then get that

∀η ∈(max(ν, κ), κ^{+} ) ⟨K_{κ^{+}} [U], F, G⟩ “g_{η} does not map κ monotone cofinally into η ⟹ η is singular”.

Let τ, S, ⟨M_{α} | α ∈ S⟩ be given, and let μ_{α} := minmeas(M_{α}). By iterating M_{α} with its least measure if necessary, we can assume w.l.o.g. that μ_{α} > α and thus μ_{α} > ν, while still keeping μ_{α} < κ. Then the ω_{1}-st iterate μ_{α}^{(ω_{1})} of μ_{α} is also less than κ = card(α)^{+} . Thus by Lemma 4.4 there is a σ ∈ E, σ > τ, and S′ stationary, S′ ⊆ S ∩ C_{σ}, such that for α ∈ S′ we have that τ ∈ rge(π_{α}^{σ}) and μ_{α}^{(ω_{1})} < σ_{α}. Now set μ_{α}^{∗} := μ_{α}^{(ω)} or := μ_{α}^{(ω_{1})} and M_{α}^{∗} := M_{α}^{(ω)} or := M_{α}^{(ω_{1})}, ensuring that cf(μ_{α}^{∗}) ≠ cf(α). Then, setting η := π_{α}^{σ}(μ_{α}^{∗}),

⟨K_{α}^{σ}, F_{α}^{σ}, G_{α}^{σ}⟩ “(g_{α}^{σ})_{μ_{α}^{∗}} does not map α monotone cofinally into μ_{α}^{∗}”,

so

⟨K_{κ^{+}} [U], F, G⟩

“g_{η} does not map κ monotone cofinally into η”.

As η ∈(max(ν, κ), κ^{+} ) , it follows that

⟨K_{κ^{+}} [U], F, G⟩

“η is singular”,

and so

⟨K_{α}^{σ}, F_{α}^{σ}, G_{α}^{σ}⟩ “μ_{α}^{∗} is singular”.

As U_{α}^{σ} = U_{α}^{τ}, Theorem 2.17 now implies that lp(M_{α}) ⊆ K_{α}^{σ}. (Claim 1)

Now pick some τ ∈E.

Claim 2 The set S_{1} := {α∈C_{τ} ∩ CF_{ϱ} | U_{α}^{τ} is strong} is stationary.

Proof Assume not. Then there is some closed unbounded set D such that D ∩ S_{1} = ∅, and for any α ∈ D ∩ C_{τ} ∩ CF_{ϱ}, K[U_{α}^{τ}] “U_{α}^{τ} is not a sequence of measures”. This is witnessed, for some ζ ∈ dom(U_{α}^{τ}), by some x ∈ P (ζ)∩ K[U_{α}^{τ}], or x ∈^{ζ}ζ ∩ K[U_{α}^{τ}], or x ∈^{ξ}P (ζ)∩ K[U_{α}^{τ}], for some ξ < ζ. x_{α} := x is either a set not measured by U_{α}^{τ}(ζ), or a non-constant regressive function, or a sequence of sets of the filter whose intersection is not in the filter. In any case, x ∈ K_{ζ^{+K[U^{τ}_{α} ]}} [U_{α}^{τ}], and ζ^{+K[U_{α}^{τ}]} κ. Thus, for α ∈ D ∩ C_{τ} ∩ CF_{ϱ}, we can find a short iterable U_{α}^{τ}-premouse M_{α}, card(M_{α}) < κ, such that x_{α} ∈lp(M_{α}). (⋆)_{κ} implies that there is a σ ∈E, σ > τ, and S′stationary, S′⊆ D∩C_{τ}∩CF_{ϱ}, such that for all α ∈ S′, τ ∈ rge(π_{α}^{σ}) and x_{α} ∈ lp(M_{α}) ⊆ K_{α}^{σ}. As U_{α}^{σ} = U_{α}^{τ}, K_{α}^{σ} “ U_{α}^{σ} is not a sequence of measures“. But this is a contradiction.

(Claim 2)

For α ∈ S_{1}, U_{α}^{τ} is strong and dom(U_{α}^{τ}) = dom(U), so that Theorem 2.13 implies that U_{α}^{τ} = U. This is what we meant when we said that the K_{α}^{τ} condensate more or less to initial segments of K[U].

Claim 3 κ is inaccessible in K[U].

Proof Assume to the contrary that κ = λ^{+K[U]}. We can assume that min(C_{τ}) > λ. Thus for each α ∈ S_{1}, K[U] “α is not a cardinal”. Pick for each such α a short iterable U-premouse M_{α}, card(M_{α}) < κ, such that there is a surjection h_{α} : λ → α, h_{α} ∈lp(M_{α}). By Lemma 4.4 there is a σ ∈E, σ > τ, and S stationary, S′⊆ S_{1}∩C_{σ}, such that for all α∈S′, τ ∈rge(π_{α}^{σ}) and lp(M_{α}) ⊆ K_{α}^{σ}. But then K_{α}^{σ} “α is not a cardinal”, clearly a contradiction, as π_{α}^{σ}(α) = κ. (Claim 3)

Claim 4 The set S_{2} := {α∈S_{1} | P (α) ∩ K[U]∈K_{α}^{τ}} is stationary.

Proof By the last claim, P (α) ∩ K[U] ∈ K_{κ}[U] for α < κ. Pick a sequence ⟨M_{α} | α ∈ S_{1}⟩ of short iterable U-premice such that P (α) ∩ K[U] ∈ lp(M_{α}). Then, again, there is a σ ∈ E, σ > τ, and S′ stationary, S′ ⊆ S_{1} ∩ C_{σ}, such that for α ∈ S′, τ ∈ rge(π_{α}^{σ}) and P (α) ∩ K[U] ∈ K_{α}^{σ}. As K_{α}^{σ} ⊆ K[U], we get that P (α) ∩ K[U] = P (α)^{K_{α}^{σ}} . Since K_{α}^{τ} ≺ K_{α}^{σ}, K_{α}^{τ} “P (α) exists”, and so P (α)^{K_{α}^{τ}} = P (α)^{K_{α}^{σ}} = P (α) ∩ K[U]∈K_{α}^{τ}. S_{2} is obviously a superset of S′and hence is also stationary. (Claim 4)

For α∈S define K[U]-ultrafilters V_{α}^{τ} by

x∈V_{α}^{τ} :↔ α∈π_{α}^{τ}(x).

Claim 5 There is an α∈S_{2} such that Ult(K_{κ}[U], V_{α}^{τ}) is well-founded.

Proof Assume not. Then for each α ∈ S_{2} there are functions f_{0}^{α}, f_{1}^{α}, … ∈ K_{κ}[U] such that for all i ∈ ω, {ξ < α | f_{i+1}^{α} (ξ) ∈ f_{i}^{α}(ξ)} ∈ V_{α}^{τ}. Pick short iterable U-premice M_{α}, card(M_{α}) < κ, such that f_{i}^{α} ∈ lp(M_{α}). Find σ ∈ E, σ > τ, and S′ stationary, S′ ⊆ S_{2} ∩ C_{σ}, such that for all α ∈ S′, τ ∈ rge(π_{α}^{σ}) and lp(M_{α}) ⊆ K_{α}^{σ}. Then

α ∈π_{α}^{τ} ({ξ < α | f_{i+1}^{α} (ξ)∈f_{i}^{α}(ξ)})
= π_{α}^{σ} ({ξ < α | f_{i+1}^{α} (ξ)∈f_{i}^{α}(ξ)})
= {ξ < κ | π_{α}^{σ}(f_{i+1}^{α} )(ξ)∈π_{α}^{σ}(f_{i}^{α})(ξ)}.

This gives an infinite descending sequence of ordinals,

…∈π_{α}^{σ}(f_{i+1}^{α} )(α)∈π_{α}^{σ}(f_{i}^{α})(α)∈…∈π_{α}^{σ}(f_{0}^{α})(α).

(Claim 5)

Let α ∈ S_{2} be such that Ult(K_{κ}[U], V_{α}^{τ}) is well-founded, and let π denote the canonical embedding of this ultrapower. We thus have κ ω_{1}, κ > supdom(U), and π : K_{κ}[U] → W, crit(π) = α > ν. Theorem 2.18 implies that there is an elementary embedding π̃ : K[U] → K[U] with the same critical point. As κ is inaccessible in K[U] and κ ω_{2}, Theorem 2.15 implies that there is U′ >_{e} U with η := mindom(U′\ U) satisfying η α > ν and η < κ. But this contradicts the definition of U = U_{can} κ. (Case 1)

Case 2 dom(U) ∩ [κ, κ^{+} ) ≠ ∅.

Again let ν := supdom(U).

Case 2.1 κ^{+} > ν^{+K[U]}.

We have U ∈ K_{κ^{+}} [U]. W.l.o.g. assume that for all τ ∈ E, U ∈ K_{τ}[U]. Furthermore, assume that for all α ∈ C_{τ}, U ∈ Q̃_{α}^{τ}. Then U_{α}^{τ} = (π_{α}^{τ})^{−1} (U), and if τ ∈rge(π_{α}^{σ}), then U_{α}^{σ} = U_{α}^{τ}. Pick some τ ∈E.

Claim 6 (⋆)_{κ} holds.

Proof Assume w.l.o.g. that for α ∈ C_{τ}, ν ∈ Q̃_{α}^{τ}, and let ν_{α}^{τ} := (π_{α}^{τ})^{−1} (ν). Note that ν κ. Then as before we have

∀η ∈ [ν + 1, κ^{+} ) ⟨K_{κ^{+}} [U], F, G⟩ “g_{η} does not map κ monotone cofinally into η ⟹ η is singular”.

Given the sequence ⟨M_{α} | α ∈ S⟩, proceed as before. Iterate M_{α} initially with its least measure to ensure not only μ_{α} > α but also μ_{α} > ν_{α}^{τ}. Then η := π_{α}^{σ}(μ_{α}^{∗}) will be larger than π_{α}^{σ}(ν_{α}^{τ}) = ν, so we can continue as in Case 1.

(Claim 6)

Claim 7 The set S_{1} := {α∈C_{τ} ∩ CF_{ϱ} | U_{α}^{τ} is strong} is stationary.

Proof As before. (Claim 7)

Now pick an α from S_{1}. U_{α}^{τ} is strong, so U_{α}^{τ} α is strong, too. Furthermore, dom(U_{α}^{τ} α) = dom(U α) = dom(U κ), as we assured α > γ = supdom(U κ) at the outset. Thus by Theorem 2.13, U_{α}^{τ} α = U κ. Hence U_{α}^{τ} >_{e} U κ. Now mindom(U_{α}^{τ}\U κ) = mindom(U_{α}^{τ}\U_{α}^{τ} α) < κ, as dom(U)∩[κ, κ^{+} ) ≠ ∅. This contradicts the definition of U = U_{can} κ^{+} (cf. Figure 4.2). (Case 2.1)

Case 2.2 κ^{+} = ν^{+K[U]}.

Case 2.2.1 ν ∉ dom(U).

This is similar to Case 2.1. We have d := dom(U) ⊆ ν, so d ∈ K_{κ^{+}} [U], hence w.l.o.g. d∈Q̃_{α}^{τ} for all relevant τ and α. Set d_{α}^{τ} := (π_{α}^{τ})^{−1} (d), so d_{α}^{τ} = dom(U_{α}^{τ}), and if τ ∈ rge(π_{α}^{σ}), then d_{α}^{σ} = d_{α}^{τ}. But then U_{α}^{σ} = U_{α}^{τ}, too. For if ξ ∈ d_{α}^{τ}, then ζ := π_{α}^{τ}(ξ) = π_{α}^{σ}(ξ) ∈ d, and U_{α}^{τ}(ξ) = (π_{α}^{τ})^{−1} (U(ζ)) = (π_{α}^{σ})^{−1} (U(ζ)) = U_{α}^{σ}(ξ). Note that U(ζ)∈Q̃_{α}^{τ} as it is definable from ζ and U.

The rest of the proof is as Case 2.1. Note that we need neither the functions g_{η} nor Theorem 2.16 to show that (⋆)_{κ} holds, as K_{κ^{+}} [U] “η is singular” for any η ∈(ν, κ^{+} ), anyway. (Case 2.2)

Case 2.2.2 ν ∈dom(U).

In this case, there is an ultrafilter “at ν”. To proceed, we will just cut this ultrafilter away from the sequence and try to continue as before: If there is

[figure: −; K; −; κ^{+}; −; τ; −; ν; −; U_{β}; −; κ; −; α; −; γ; −; K_{α}^{τ}; −; τ_{α}; −; α; −; (U_{α}^{τ})_{β_{α}}; −; γ; =; π_{α}^{τ}]

Figure 4.2: The condensated substructure K_{α}^{τ} provides an ultrafilter (U_{α}^{τ})_{β_{α}} that should have been included in the canonical sequence, as indicated by the dotted circle.

some ultrafilter left, at some ζ ∈ [κ, ν), we copy Case 2.1, which will happen in Case 2.2.2.1. Otherwise, in Case 2.2.2.2, there is no ultrafilter with critical point between κ and ν, which makes it possible to apply the method of Case 1.

Case 2.2.2.1 ∃ζ ∈dom(U) ∩ [κ, ν).

Set U′:= U ν. Note that as in Case 2.2.1 K_{κ^{+}} [U] “η is singular” for any η ∈(ν, κ^{+} ). By Theorem 2.10, P (ν)∩K[U] = P (ν)∩K[U ν], so K_{κ^{+}} [U ν] “η is singular”, too. Now consider the “′-version” of the Ketonen-diagramme, i. e., define Q′_{τ}, E′, Q̃′_{α}^{τ}, U′_{α}^{τ}, etc., starting out from K[U′] instead of K[U]. Then show that the corresponding property (⋆′)_{κ} holds, noting that by the above remark K[U′] “η is singular”. Continue as in Case 2.1 to see that U′_{α}^{τ} is strong for sufficiently many α and then derive a contradiction as before.

(Case 2.2.2.1)

Case 2.2.2.2 dom(U) ∩ [κ, ν] = {ν}.

The proof of this case is a combination of the proofs of Case 2.2.2.1 and Case 1. Again, build the “ ′-version” of the Ketonen-diagramme and show that (⋆′)_{κ} holds. Prove that U′_{α}^{τ} is strong for sufficiently many α. We cannot, as in Case 2.2.2.1, derive a contradiction from this just yet, as there is no ultrafilter on the U′_{α}^{τ}-sequence above α. But we can show, returning to Case 1, that if U′_{α}^{τ} is strong and α > ν′ := supdom(U′) (ν′ < κ by our assumption), then U′_{α}^{τ} = U′, as dom(U′_{α}^{τ}) = dom(U′). Thus we can continue as in Case 1 to prove that κ is inaccessible in K[U′]. The definition of the K[U′]-ultrafilters V_{α}^{τ} also goes through. Note that supdom(U′) < κ. Thus as before we get an elementary embedding π : K_{κ}[U′] → W with crit(π) = α > ν′. This leads to the same contradiction as in Case 1. (Case 2.2.2.2)

This completes the proof of the theorem.

¬ TH implies the weak Chang’s Conjecture (cf. [DK83]). One could try to extend the above proof to start from the assumption of wCC(κ) instead of ¬ TH^{stat}(κ). However, this fails for the same reason as the attempt to start from just ¬ TH instead of ¬ TH^{stat}: the proof requires a repeated application of the assumption (or rather of Lemma 4.4). But ¬ TH needs as an “input” at least a club set, yielding only a stationary set as “output”; wCC, in fact, gives even just one single α. In contrast, although ¬ TH^{stat} also just yields a stationary set, that is also enough as “input” for a second application (or, indeed, a finite number of them). In Chapter 7, we will show how infinite applications can yield even stronger results, albeit starting from a stronger premise, the existence of an irregular ultrafilter.

36 CHAPTER 4. THE TRANSVERSAL HYPOTHESIS

Core Models up to a Strong Cardinal

“What do you keep that mouse for?” I said. “You should bury it, or throw it into the lake.” “Why, it’s to measure with!” cried Bruno.

(Lewis Carroll, Bruno’s Revenge)

To cope with inner models for larger large cardinals, in our case anything up to a strong cardinal, we need to extend our concept of mouse. Every year seems to bring a new definition of this fundamental concept of core model theory, encompassing ever larger hypotheses. For our purposes, the finestructure developed in [Koe89] will suffice completely. [Sch96] goes on to show how one can build a decent core model from these mice. As in Chapter 2, we now proceed to review some basic definitions and theorems. The original numbering from [Sch96] is given in parentheses. Any terms not defined here can be found there or in [Koe89].

5.1 Definition A predicate E is natural iff

i) ∀z ∈ E ∃ν ∈ Lim ∃κ < ν ∃a ∈ [ν]^{<ω} ∃x ∈ P ([κ]^{card(a)}) (z =

⟨ν, κ, a, x⟩).
ii) ∀⟨ν, κ, a, x⟩, ⟨ν′, κ′, a′, x′⟩∈E (ν = ν′ → κ = κ′).

κ(ν) = κ^{E}(ν) is the unique κ such that, for some a and x, ⟨ν, κ, a, x⟩ is in E. For ν ∈ On set E_{ν} := {⟨a, x⟩ | ⟨ν, κ(ν), a, x⟩ ∈ E}, and for a ∈ [ν]^{<ω} , set E_{ν,a} := {x | ⟨a, x⟩∈E_{ν}}. Let dom(E) := {ν ∈On | E_{ν} ≠ ∅}, and for X ⊆ On let E X := {⟨ν, κ, a, x⟩ | ν ∈X}.

5.2 Definition Let E be a natural predicate and M := ⟨J_{α}[E], F ⟩, for some α ∈ On (or α = On), be an acceptable model. For ν < α, let E_{ν}^{M} := E_{ν} and set E_{α}^{M} := F. Then M is called an extender structure iff

i) ∀ν ∈dom(E) (E_{ν} ⊆ J_{ν}[E] is a 0-folded extender at κ(ν), ν on J_{ν}[E]), ii) E is coherent, i. e., for any ν ∈dom(E), if N := Ult(J_{ν}[E], E_{ν}), then

E^{N} ν = E ν and E_{ν}^{N} = ∅, iii) ∀ν ∈ dom(E) ∀ξ ∈ [κ(ν)^{+ J_{ν}[E]}, ν] (ξ ∈ dom(E) and E_{ξ} is an extender

at κ(ν), ξ on J_{ξ}[E]), iv) E is non-overlapping, i. e., for any ν ∈dom(E) and any λ less than

κ(ν), sup{ξ < κ(ν) | κ(ξ) < λ} < κ(ν).

5.3 Definition Let α ∈ On and let M = ⟨J_{α}[E], E_{α}⟩ be an extender structure. Let κ∈Card^{M}. Then define

i) O^{M}(κ) to be the smallest limit ordinal greater than or equal to κ^{+M} which is a strict upper bound of {ξ α | E_{ξ} ≠ ∅}, ii) o^{M}(κ) := (O^{M}(κ) − κ^{+M} )/ω, the Mitchell order of κ in M.

M is called topless if E_{α} = ∅. For β α set M|β := ⟨J_{β}[E], E_{β}⟩ and M ↓ β := ⟨J_{β}[E], ∅⟩. These are both extenderstructures, M ↓ β being the topless version of M|β.

Let N be another extenderstructure. Then M is an initial segment of N and N an end-extension of M, denoted by M ⊆ N, iff M = N|(On∩M) or M = N ↓ (On∩M). If M ⊆ N and M ≠ N, then M is a proper initial segment of N.

5.4 Definition (1.2) An extender structure M = ⟨J_{α}[E], E_{α}⟩ is called weakly amenable iff E_{α} ≠ ∅ and if for all sequences ⟨x_{ξ} | ξ < κ(α)⟩ from J_{α}[E] and for any a∈[α]^{<ω} the set {ξ < κ(α) | x_{ξ}∈E_{α,a} } is in J_{α}[E].

5.5 Lemma (1.2) Assume M = ⟨J_{α}[E], E_{α}⟩ is an extender structure such that N := Ult(M, E_{α}) is transitive. Then

M is weakly amenable ↔ P (κ(α)) ∩ N = P (κ(α)) ∩ M.

Premice and (coarse and fine) ultrapowers are defined in a straightforward way. An iteration is given by the sequence of indices ⟨ν_{i} | i < ϑ⟩, indicating which extender to use to compute the next ultrapower, the sequence of cutback-points ⟨α_{i} | i < ϑ⟩, and the sequence of indicators ⟨r_{i} | i < ϑ⟩, determining whether coarse or fine ultrapowers should be taken. An iteration is non-degenerate if only finitely many α_{i} are less than On∩M^{i}. A premouse M is iterable above τ if any iteration with indices at least τ can be freely continued. M is a mouse if it is iterable above 0. An iteration is simple if no cutbacks occur, i. e., if for all i less than ϑ, α_{i} is equal to On∩M^{i}.

Mice are finely coiterable. If M^{∗} and N^{∗} denote the coiterates of respectively M and N and if M^{∗} ⊆ N^{∗}, then we say M ^{∗} N. Also, the iteration of M to M^{∗} is simple in this case.

If M is a mouse above τ, then there exists a unique premouse M_{0} such that M_{0} is τ-sound and M is a simple fine iterate of M_{0} above max{τ, ϱ_{M}^{ω}}. This is the τ-core of M, denoted by core_{τ}(M). The core of M is its 0-core.

5.6 Definition (1.14) Let M = J_{α}[E] be an extender structure. Then let κ^{M} be defined as follows:

i) If there exists some ξ α such that E_{ξ} is an extender at κ, ξ on M and for all ζ ∈ [ξ, α), E_{ζ} is an extender at κ, ζ on M, then set κ^{M} := κ,

ii) otherwise, let κ^{M} := sup(Card^{M}). Thus, if E_{α} ≠ ∅, then κ^{M} = crit(E_{α}).

5.7 Definition (1.15) Let M = ⟨J_{α}[E], E_{α}⟩ be a premouse such that κ^{M} < α. A collapsing mouse for M is a premouse N = ⟨J_{β}[E′], E′_{β}⟩ such that

i) M ⊆ N,

ii) sq(N) is a κ^{M}-sound mouse above κ^{M}, iii) P (κ^{M} ) ∩ N ⊆ M, but iv) ϱ_{sq(N)}^{ω} κ^{M}.

In this case, sq(N) is called a squashed collapsing mouse for M (for details on squashed and stretched mice, see [Sch96]). N is thus a “minimal” endextension of M over which (κ^{M})^{+M} is collapsed onto κ^{M}. M is presolid if κ^{M} < α and there is no collapsing mouse for M.

5.8 Lemma (1.16) Collapsing mice are unique: let M be a premouse such that N and Q are collapsing mice for M. Then N = Q.

5.9 Corollary (1.17) Let M and N be coarse mice such that for some κ < On∩M ∩ N, M ↓ κ = N ↓ κ, and E_{κ}^{M} = E_{κ}^{N} = ∅. Let ν := min{κ^{+M} , κ^{+N} }. Then M ↓ν = N ↓ν.

5.10 Definition (1.19) Let M = ⟨J_{α}[E], E_{α}⟩ be a premouse, β α. M is called neat beyond β provided

i) ∀γ < α (M|γ is a mouse),

ii) ∀γ ∈[β, α] (E_{γ}^{M} ≠ ∅ → E_{γ}^{M} is countably complete).

M is neat if it is neat beyond 0.

Any neat premouse is a mouse [Sch96, Lemma 1.20].

5.11 Definition (1.21) Let M = ⟨J_{α}[E], E_{α}⟩ be a premouse, β α. An iteration I of M is called beyond β if all indices ν used in the iteration are greater than or equal to β. M is called (coarsely, finely) iterable beyond β if any non-degenerate (coarse, fine) iteration of M beyond β can be freely continued beyond β.

M is called prestrong provided the following holds: Any premouse N endextending M which is finely iterable beyond α + ω (or α, if E_{α}^{M} = ∅) is, in fact, a mouse.

M is called strong if M is prestrong and for any mouse N end-extending M, M is an initial segment of core(N).

M is called solid if M is presolid and prestrong.

Hence every prestrong premouse is a mouse, and every strong mouse is sound. Moreover, neat mice are prestrong [Sch96, Lemma 1.22].

5.12 Definition (1.26) L^{strong} denotes the statement that there exists an inner model with a strong cardinal. ¬ L^{strong} denotes the negation of this statement.

5.13 Definition (1.27) A J-model M = ⟨J_{α}[E], U⟩ is called a pistol-pre-mouse (p-premouse) if i) M̄ := J_{α}[E] is a topless premouse such that κ := κM̄ is the largest

cardinal in M̄. ii) There is some λ < κ such that OM̄(λ) κ (in which case one says

that κ is overlapped^{1}. iii) M “U is a non-trivial κ-complete normal ultrafilter on κ”.

M is called neat if M̄ is neat above λ and U is countably complete.

5.14 Lemma (1.27) If there is a neat p-premouse, then L^{strong} holds.

Next, we present a result on upward extensions of embeddings. The construction itself is defined in detail in [Sch96, Chapter 4], which also contains the proofs of the following statements. Starting from a premouse N = ⟨J_{β}[A], F ⟩ and some η ∈ Card^{N} ∪{β}, set M := N ↓η. Assume that π : M →_{Σ_{0}} Q is a cofinal map, Q some topless premouse. Then one can construct an ultrapower of N using π, yielding some map π̃ : N → R, such that if R is transitive, then R is a premouse, π̃ ⊇ π, and Q = R↓η̃, where η̃ = On∩Q = supπ′′η. π̃ is then called the fine upward extension of π with respect to N, N is called M-based, and R is denoted by Ult^{∗}(N, π).

5.15 Definition (4.6) Let ϑ > ω be regular and let π : H̄ →_{Σ_{ω}} H_{ϑ}, where H̄ is transitive. Let Q ∈ rge(π) be a topless mouse and set M := π^{−1} (Q). Then π M : M →_{Σ_{ω}} Q and for any α γ := On∩M, π (M ↓ α) : M ↓ α →_{Σ_{0}} Q (supπ′′α) cofinally.

Let α γ, M′ := M ↓ α. π is called 1-lousy at α with respect to M if κ := κ^{M′} is the largest cardinal in M′ and there is a M′-based mouse N above κ such that Ult^{∗}(N, π M′) is not a mouse above π(κ) = κ^{Q↓(supπ′′α)} .

π is called 2-lousy at α with respect to M if there is a M′-based coarse mouse (or mouse above α) N such that, if π̃ : N → Ult^{∗}(N, π M′) = R is the upward extension of π M′ : M′ → Q π′′α, then R is not a coarse mouse (or mouse above π̃(α)). π is called 3-lousy at α with respect to M if there exists a stretched M′- based Σ_{0}-mouse N such that Ult(N, π M′) is a stretched premouse but not a stretched Σ_{0}-mouse.

1In keeping with [Sch96], although it is a somewhat unfortunate piece of terminology, for any premouse M, we will call any ξ such that there exists λ < ξ with OM̄ (λ) ξ overlapped.

This is not the same as saying that the extender-sequence E^{M} is overlapping: this would only be the case were ξ measurable itself.

Finally, π is called perfect with respect to M if it is neither 1-, 2-, nor 3-lousy at α with respect to M for any α γ.

5.16 Lemma (4.7) Let ϑ be a regular uncountable cardinal, and let π : H̄ → H_{ϑ} be perfect with respect to M, where M ∈ H̄, H̄ is transitive, and Q := π(M) is a mouse such that κ^{M} is the largest cardinal of M. Suppose that π M : M → Q cofinally. Then

i) if Q is presolid, then so is M, and ii) if Q is solid, then so is M.

5.17 Lemma (4.8) Let ϑ be a regular uncountable cardinal, π : H̄ →_{Σ_{ω}} H_{ϑ} where H̄ is transitive and ^{ω} H̄ ⊆ H̄. Let Q∈rge(π) be a topless mouse. Then π is perfect with respect to M := π^{−1} (Q).

Finally, we conclude our summary of strong core model basics by some remarks on a covering lemma.

5.18 Definition (2.8) A weasel is an extender structure W = J[E] such that for every α ∈ On, W |α is a mouse. W is called universal if the fine coiteration of W with any coiterable premouse terminates. W is called weakly universal if the fine coiteration of W with any mouse terminates.

5.19 Definition (5.1) Let M be a mouse or a weasel. Let α < On∩M. M is called weakly full above α if for all β ∈ [α, On∩M), if N ⊇ M ↓ β is a collapsing mouse for M ↓β then N ⊆ M. M is weakly full if it is weakly full above 0.

Weakly universal weasels are weakly full, as are solid mice [Sch96, Lemma 5.8]. Also, strong mice are weakly full [Sch96, Lemma 5.9].

5.20 Definition (5.2) Let M = ⟨J_{ξ}[Ē], ∅⟩ be a strong topless mouse, and let Γ be a non-empty set or class of uncountable regular cardinals. A weasel W = L[E] is called Γ-full over M if i) L[E] end-extends M, and W is weakly full above ξ. ii) For any α ξ, if E_{α} ≠ ∅ is an extender at κ, α on W, then κ > ξ,

E_{α} is countably complete, and cf(κ)∈Γ. iii) For any α > ξ, if ξ < κ := κ^{W↓α} < α, W ↓(κ^{+W↓α} ) is presolid, cf(κ)∈

Γ, and F is a countably complete extender at κ, α on W ↓ α such that Ult(W ↓ α, F)|α = W ↓ α and ⟨W ↓ α, F ⟩ is weakly amenable, then F = E_{α}.

W is called full over M if it is Γ-full over M when Γ is the class of all uncountable regular cardinals. W is called Γ-full (or full) if it is Γ-full (or full) over ∅.

5.21 Lemma (5.3) For any strong topless mouse M = ⟨J_{ξ}[Ē], ∅⟩ and any non-empty set or class Γ of uncountable regular cardinals there is a unique Γ-full weasel over M, denoted by W^{Γ}(M).

K^{c}, the countably complete core model, is the unique Γ-full weasel over ∅, where Γ is the class of all uncountable regular cardinals.

5.22 Lemma (5.10) Assume ¬ L^{strong}. Let κ ∈ Card^{K^{c}} be such that κ is not overlapped in K^{c}. Let ν := O^{K^{c}}(κ). Let x ⊆ ν be a set such that card(x)^{ℵ_{0}} < card(κ). Then there is some y ∈ K^{c} such that x ⊆ y and card^{K^{c}}(y) κ.

44 CHAPTER 5. CORE MODELS UP TO A STRONG CARDINAL

Short Iterations

“Mine is a long and a sad tale!” said the Mouse, turning to Alice, and sighing. “It is a long tail, certainly,” said Alice, looking down with wonder at the Mouse’s tail; “but why do you call it sad?”

(Lewis Carroll, Alice’s Adventures in Wonderland, Chapter III)

In this chapter, we show how under considerably more restrictive assumptions than just the absence of an inner model for a strong cardinal one can give an upper bound for the length of certain short iterations. This will be done by considering an in a sense complete iteration into which any short iteration may be embedded.

Let us introduce the following definition:

6.1 Definition Let M̄ be a mouse, and let I be an iteration of M̄, of length ϑ, with indices ⟨ν_{i} | i < ϑ⟩ and iteration maps ⟨π_{ij} | i j ϑ⟩. I is short if no extender has been used in I for ω_{1} + 1 many times in a row, i. e., if for no j < ϑ with cf(j) = ω_{1} the set

P (j, ν_{j}) := {i < j | π_{ij} (ν_{i}) = ν_{j}}

is unbounded in j.

Given a mouse M̄, we want to find an upper bound for the ordinal height of short iterates of M̄.

6.2 Lemma Let M̄ be a sound, topless mouse, card(M̄) < κ, κ some regular cardinal greater than ω_{1}. Assume that there is no inner model for o^{M}(μ) ω. Then there is an ordinal s < κ, denoted by s^{(ω_{1})}(M̄), which is an upper bound for the ordinal height of short iterates of M̄:

s^{(ω_{1})}(M̄) sup{(On∩M) + 1 | M is a short, simple, fine iterate of M̄ }.

Proof We take s^{(ω_{1})} to be the ordinal height of one specially defined iterate of M̄. This iterate, M^{ϑ^{(ω_{1})}} , will be defined recursively. The lemma will be proved if we can show that any arbitrary short, simple, fine iterate M^{∗} of M̄ can be embedded into M^{ϑ^{(ω_{1})}} . Let us note that the iteration yielding M^{ϑ^{(ω_{1})}} will, in fact, not be short itself.

For a mouse M, let

c(M) := {μ∈On∩M | μ is the critical point of some extender E_{ν}^{M} },
d(M, μ) := {ν ∈On∩M | E_{ν}^{M} is at μ, ν }.

Note that we assumed M̄ to be topless, so that these definitions really capture all possible extenders of M̄.

Set M^{0} := M̄. Let Λ = ω_{1}^{ω} (using ordinal exponentiation). Define bookkeeping functions m^{0} and m̃^{0} : c(M^{0}) → Λ + 1, setting m^{0}(ξ) = m̃^{0}(ξ) = 0 for all ξ.

Let j be a limit ordinal and assume ⟨M^{i} | i < j⟩ had been defined, with iteration maps ⟨π_{ii′} | i i′ < j⟩. Then let M^{j} be the direct limit of this system, with corresponding maps ⟨π_{ij} | i < j⟩. Define the auxiliary bookkeeping function m̃^{j} as follows

m̃^{j} (ξ) := sup{m^{i}(ξ̄_{i}) | i < j ∧ π_{ij} (ξ̄_{i}) = ξ}

By induction, if all m^{i} take values less than Λ + 1, then so does m̃^{j}. Note that if ξ ∈c(M^{j}) and ξ = π_{ij} (ξ̄_{i}), then ξ̄_{i}∈c(M^{i}) and hence m^{i}(ξ̄_{i}) is defined.

If j is a successor ordinal i + 1, then set

m̃^{j} (ξ) := \{ \begin{matrix}m^{i}(ξ̄_{i}) & if ξ = π_{ij} (ξ̄_{i}) \\ 0 & if ξ ∉ rge(π_{ij}). \end{matrix}

Again, note that if ξ ∈c(M^{j}) ∩ rge(π_{ij}), then m^{i}(ξ̄_{i}) is defined.

To construct M^{j+1} , for arbitrary j, let first be

μ_{j} := the least μ∈c(M^{j}) such that m̃^{j} (μ) < Λ and ∀i < j (ν_{i} < μ).

We are thus looking for the first critical point which has not been used up Λ-many times yet, and which would allow us to continue the iteration in a normal way.

Case 0 If no such μ exists, then set ϑ^{(ω_{1})} := j and terminate the construction.

Case 1 cf(j) ≠ ω_{1} (including the case that j is a successor ordinal).

Let ν_{j} := mind(M^{j}, μ_{j}), and set M^{j+1} := Ult(M^{j}, E_{ν_{j}}^{M^{j}} ), π_{j,j+1} := the canonical embedding, and for i < j let π_{i,j+1} := π_{j,j+1} ∘ π_{ij}. Set m^{j} := m̃^{j}. I. e., we take the least extender associated with this critical point and take the ultrapower with it. The bookkeeping function remains unchanged.

Case 2 cf(j) = ω_{1}.

Let ν_{j} be the least ν ∈d(M^{j}, μ_{j}) such that the set

P (j, ν) := {i < j | π_{ij} (ν_{i}) = ν}

is bounded in j.

Case 2.1 If such ν_{j} exists, again set M^{j+1} := Ult(M^{j}, E_{ν_{j}}^{M^{j}} ), π_{j,j+1} := the canonical embedding, π_{i,j+1} accordingly, and m^{j} := m̃^{j}. Thus instead of just taking the least extender with the chosen critical point, we insist that it had also only been used boundedly often before j.

Case 2.2 If no such ν exists, i. e., if for all ν ∈d(M^{j}, μ_{j}), P (j, ν) is unbounded in j, then set ν_{j} := mind(M^{j}, μ_{j}), and let M^{j+1} := Ult(M^{j}, E_{ν_{j}}^{M^{j}} ). In this case, we have exhausted all extenders with critical point μ_{j}, i. e., used them all cofinally up to j, and now start all over again. Informally, we will call this a block of the iteration. To keep track of how often we do this, we now increase the bookkeeping function at μ_{j}, setting

m^{j}(ξ) := \{ \begin{matrix}m̃^{j} (ξ) + 1 & if ξ = μ_{j} \\ m̃^{j} (ξ) & else. \end{matrix}

This completes the definition of the iteration.

Claim 1 Let j < ϑ^{(ω_{1})}, and assume m̃^{j} (μ) < Λ. Then

∀μ′ μ (m̃^{j} (μ′) < Λ),

i. e., not only is μ_{j} the least μ such that m̃^{j} (μ) < Λ, it is also the least upper bound (in c(M^{j})) of those μ that have m̃^{j} (μ) = Λ (as well as satisfying the normality requirement ∀i < j (ν_{i} < μ)). Since m^{j}(ξ) m̃^{j} (ξ) + 1 for any ξ, this claim is also true if one replaces m̃^{j} by m^{j}.

Proof Assume the claim were false. Let j be the least index such that this happens. Recall that μ_{j} was defined to be the least μ such that m̃^{j} (μ) < Λ. Thus there must be some μ > μ_{j}, μ∈c(M^{j}), such that m̃^{j} (μ) = Λ. As j was chosen minimally,

∀i < j (μ_{i} = minc(M^{i}) \ {μ′∈c(M^{i}) | m̃^{i}(μ′) = Λ ∧ ∀i′< i (ν_{i′} < μ′)}).

First note that j must be a limit ordinal. Assume to the contrary that j = i + 1. Then we must have μ ∈ rge(π_{ij}), as otherwise m̃^{j} (μ) = 0. So let μ = π_{ij} (μ̄). Obviously, μ̄ μ_{i}, as otherwise μ = π_{ij} (μ̄) = μ̄ < μ_{i} < μ_{j} < μ, contradiction. But then, by the minimal choice of j, m̃^{i}(μ̄) < Λ, whence m^{i}(μ̄) < Λ, as well, so that m̃^{j} (μ) = m^{i}(μ̄) < Λ. Contradiction.

So j is a limit ordinal. Find some i_{0} < j such that μ, μ_{j} ∈ rge(π_{i_{0}j}), and set μ̄_{i} := π_{ij}^{−1} (μ) and μ̃_{i} := π_{ij}^{−1} (μ_{j}), for i ∈ [i_{0}, j). As Λ > ξ_{0} := m̃^{j} (μ_{j}) = sup{m^{i}(μ̃_{i}) | i < j}, we know that

∀i∈[i_{0}, j) (m^{i}(μ̃_{i}) ξ_{0} < Λ).

Note that again we must have that for all i ∈ [i_{0}, j) (μ̄_{i} > μ̃_{i} μ_{i}): the first inequality is immediate from μ > μ_{j}, whereas the failure of the second would imply that μ_{j} = π_{ij} (μ̃_{i}) = μ̃_{i} < μ_{i} < μ_{j}, a contradiction. But then we have that

∀i∈[i_{0}, j) (m^{i}(μ̄_{i}) = m̃^{i}(μ̄_{i})),

as this holds for all μ′, unless we are in Case 2.2 of the definition of the iteration and μ′ = μ_{i} (whereas we just showed that μ̄_{i} > μ_{i}). Thus inductively ones sees that

∀i∈[i_{0}, j) (m^{i}(μ̄_{i}) = m̃^{i_{0}} (μ̄_{i_{0}} )).

As the claim is true at i_{0}, we conclude that

m^{i_{0}} (μ̄_{i_{0}} ) < Λ

since μ̄_{i_{0}} > μ̃_{i_{0}} and m^{i_{0}} (μ̃_{i_{0}} ) ξ_{0} < Λ. Thus

m̃^{j} (μ) = sup{m^{i}(μ̄_{i}) | i∈[i_{0}, j)} = m^{i_{0}} (μ̄_{i_{0}} ) < Λ,

contradicting the assumption. (Claim 1)

Claim 2 The construction terminates at some ϑ^{(ω_{1})} < κ.

Proof Assume to the contrary that ϑ^{(ω_{1})} κ. We will show first that the set

C := {j ∈κ | cf(j) = ω_{1} and P (j) := P (j, ν_{j}) is unbounded in j }

is stationary in κ. Assume to the contrary that there is some club subset D ⊆ κ such that C ∩ D = ∅. Let C_{0} := κ \ C and C_{1} := C_{0} ∩ {j ∈κ | cf(j) = ω_{1}}. Then C_{1} is stationary in κ: let D′ be a club subset of κ. We have to show C_{1} ∩ D′ ≠ ∅. Since both D and D′ are club, so is D ∩ D′. Let i be a limit point of D ∩ D′ of cofinality ω_{1}. As i ∈ D and D ⊆ C_{0}, we must have i∈C_{1}. Thus i∈C_{1} ∩ D′, and so C_{1} must, indeed, be stationary.

Let

E := {j ∈κ | j is a limit ordinal and j is closed under Γ and Γ_{3} and (i < j → On∩M^{i} < j)}, where Γ denotes the Gödel Pairing Function and Γ_{3} the derived function for triples. Then E is club in κ. Hence C_{2} := C_{1} ∩ E is stationary in κ. Define a function f : C_{2} → κ by letting

f(j) := Γ_{3}(i, ν, b),

where i is the least i such that ν_{j} ∈ rge(π_{ij}), ν = π_{ij}^{−1} (ν_{j}), and b is the least upper bound of P (j). Note that i < j as j is a limit ordinal, ν < j as On∩M^{i} < j (j ∈ E!), and b < j as j ∈ C_{1}. Thus, as j is closed under Gödel Pairing, f(j) < j, i. e., f is regressive. By Fodor’s Theorem, there is a stationary set C_{3} ⊆ C_{2}, and ī, ν̄, and b̄ auch that

∀j ∈C_{3} (f(j) = Γ_{3}(ī, ν̄,b̄)).

Now choose i < j ∈C_{3}. Then π_{īi}(ν̄) = ν_{i} and π_{ij} (ν_{i}) = π_{ij} (π_{īi}(ν̄)) = π_{īj}(ν̄) = ν_{j}. Thus i ∈ P (j), so that i < b̄ = the least upper bound of P (j). On the other hand, i > b̄, as b̄ is also the least upper bound of P (i), which must be less than i as i∈C_{0}. This is a contradiction. Hence C really is stationary in κ.

Assume w.l.o.g. that C ⊆ E. Now again define a function f : C → κ by letting

f(j) := Γ(i, ν)

where i and ν are chosen as before. Again, there will be a stationary set C_{4} ⊆ C and ī, ν̄ such that

∀j ∈C_{4} (f(j) = Γ(ī, ν̄)).

Pick a sequence ⟨j_{ξ} | ξ Λ⟩ of elements of C_{4}. Thus for all ξ Λ, P (j_{ξ}) is unbounded in j_{ξ}. Note that this can only happen if we are in Case 2.2 of the definition of the iteration. (Not only is P (j_{ξ}) = P (j_{ξ}, ν_{j_{ξ}}) unbounded in j_{ξ} but one also uses the extender E_{ν_{j_{ξ}}}^{M^{j_{ξ}}} to construct the next ultrapower.) In this case, the bookkeeping function m^{j_{ξ}} is incremented by one at μ_{j_{ξ}}. Thus inductively one sees that

∀ξ < Λ(m^{j_{ξ}} (μ_{j_{ξ}}) ξ).

and so

m^{j_{Λ}} (μ_{j_{Λ}} ) sup{m^{j_{ξ}} (μ_{j_{ξ}}) | ξ < Λ} + 1 Λ + 1

which is absurd, as no m^{j} ever takes values greater than Λ. (Note that, of course, if π_{ij} (ν_{i}) = ν_{j}, then π_{ij} (μ_{i}) = μ_{j}, too.) Thus the iteration cannot have had length ϑ^{(ω_{1})} κ. (Claim 2)

Claim 3 Let N^{∗} = N^{ϑ} be a short, simple, fine, normal iterate of N^{0} := M̄, with indices ⟨η_{j} | j < ϑ⟩, critical points ⟨λ_{j} | j < ϑ⟩, and maps ⟨π_{ij}^{N} | i j ϑ⟩. Then there exists an embedding h : N^{∗} → M^{ϑ^{(ω_{1})}} , and hence On∩N^{∗} On∩M^{ϑ^{(ω_{1})}} .

Proof We will inductively construct a sequence of maps h^{j}, j < ϑ, such that the limit of this sequence will be the desired map h. Each h^{j} will map N^{j} into some M^{α_{j}} . Set α_{0} := 0, h^{0} := id|M̄.

Basically, if we have embedded N^{j} into M^{α_{j}} and are given the task of embedding N^{j+1} = Ult(N^{j}, E_{η_{j}}^{N^{j}} ) into some M^{α_{j+1}} , first consider the image of E_{η_{j}}^{N^{j}} under h^{j}. Set η̂_{j} := h^{j}(η_{j}) and λ̂_{j} := h^{j}(λ_{j}), where λ_{j} = crit(E_{η_{j}}^{N^{j}} ). By elementarity of h^{j}, EM^{α_{j}} ; _{η_{j}} will be an extender at λ̂_{j} , η̂_{j}. Now ν_{α_{j}} will not necessarily be equal to η̂_{j}. But possibly there will be a (least) β_{j} α_{j} such that π_{α_{j} β_{j}}^{M} (η̂_{j}) = ν_{β_{j}} . In this case, let g^{j} := π_{α_{j} β_{j}}^{M} ∘ h^{j}. We can then use [Koe89, Theorem 14.2] to get the desired map h^{j+1} : N^{j+1} → M^{α_{j+1}} , where α_{j+1} := β_{j} + 1, completing the diagramme in Figure 6.1.

\xymatrix{M^{α_{j}} \ar[r]_{π_{α_{j} β_{j}}^{M}} & M^{β_{j}} \ar[r]_{E_{ν_{β_{j}}}^{M^{β_{j}}}} & M^{β_{j} +1} \\ N^{j} \ar[u]_{h^{j}} \ar[ur]_{g^{j}} \ar[rr]_{E_{η_{j}}^{N^{j}}}_{π_{j,j+1}^{N}} &  & N^{j+1} \ar@{-->}[u]_{h^{j+1}}}

Figure 6.1: The map h^{j+1} is the canonical completion of the diagramme.

At limits, the obvious completion is used. The problem thus reduces to ensuring that at every stage j < ϑ, there exists some β_{j} α_{j} such that π_{α_{j} β_{j}}^{M} (η̂_{j}) = ν_{β_{j}} , so that we can construct h^{j+1} . Furthermore, we have to do some bookkeeping. Inductively, we prove the following statement

\begin{matrix}(1) & ∀λ λ_{j} ( ñ^{j} (λ) = m̃^{α_{j}} (h^{j}(λ))) \\ (2) & ∀λ λ_{j} (n^{j}(λ) = m^{α_{j}} (h^{j}(λ)) = m^{β_{j}} (g^{j}(λ))), \end{matrix} (∗)

where the functions ñ^{j} and n^{j} will be defined similarly to m̃^{j} and m^{j}.

Case 1 j is a successor ordinal, j = i + 1.

Set

ñ^{j} (ξ) := \{ \begin{matrix}n^{i}(ξ̄_{i}) & if ξ = π_{ij}^{N}(ξ̄_{i}) \\ 0 & if ξ ∉ rge(π_{ij}^{N}). \end{matrix}

We claim that μ_{α_{j}} λ̂_{j} . We first show that the normality requirement is fulfilled, i. e., that

∀α < α_{j} (ν_{α} < λ̂_{j} ).

It is of course sufficient to show ν_{β_{i}} < λ̂_{j} , as α_{j} = β_{i} + 1 and I^{M} is normal. Assume to the contrary that λ̂_{j} ν_{β_{i}}. By construction of M^{β_{i}+1} we have that o^{M^{β_{i} +1}} (μ_{β_{i}}) = ν_{β_{i}}. Since we are in a situation where extender sequences do not overlap, we conclude that then λ̂_{j} μ_{β_{i}}, as by elementarity of h^{j}, λ̂_{j} is measurable in M^{β_{i}+1} . Using the fact that g^{i} and h^{j} agree up to η_{i} we now reach the following contradiction:

λ̂_{j} μ_{β_{i}} = g^{i}(λ_{i}) = h^{j}(λ_{i}) < h^{j}(λ_{j}) = λ̂_{j} .

By Claim 1, the bookkeeping function m^{β_{i}} takes values less than Λ for arguments greater than or equal to μ_{β_{i}} = π_{α_{i}β_{i}}^{M} (λ̂_{i}). But at successor steps (in the iteration of M) the bookkeeping function does not increase, so that

∀ξ μ_{β_{i}} (m^{β_{i}+1} (ξ) < Λ)

This is true for those ξ ∈ rge(π_{β_{i},β_{i+1}}^{M} ), ξ π_{β_{i},β_{i+1}}^{M} (μ_{β_{i}}), as they keep the mvalues of their pre-images, as well as for those ξ ∉ rge(π_{β_{i},β_{i+1}}^{M} ), as they take m-value 0 by definition. (We are not interested in ξ less than μ_{β_{i}}. In fact, we are not even interested in ξ less than ν_{β_{i}}, as we have λ̂_{j} > ν_{β_{i}}. The m-values for these ξ might indeed be Λ. In fact, they are bound to be Λ, as otherwise μ_{β_{i}} would not have been chosen correctly!)

Now, since λ̂_{j} > μ_{β_{i}}, we conclude that

m^{α_{j}} (λ̂_{j} ) < Λ.

We know inductively that α_{j} = α_{i+1} = β_{i} + 1, i. e., α_{j} is itself a successor ordinal. Hence P^{M}(α_{j}, η̂_{j}) is bounded in j (by i, trivially). Thus ν_{α_{j}} η̂_{j}. This implies that the extender EM^{α_{j}} ; _{η_{j}} will be used later on in the iteration, i. e., there does exist some least β_{j} α_{j} such that π_{α_{j} β_{j}}^{M} (η̂_{j}) = ν_{β_{j}} .

Thus we can now set α_{j+1} := β_{j} + 1 and define h^{j+1} : N^{j+1} → M^{α_{j+1}} as described above. Set n^{j} := ñ^{j}

Case 2 j is a limit ordinal.

Note that cf(α_{j}) = cf(j). Set

ñ^{j} (ξ) := sup{n^{i}(ξ̄_{i}) | i < j ∧ π_{ij}^{N}(ξ̄_{i}) = ξ}

Case 2.1 μ_{α_{j}} λ̂_{j} .

We must have m^{α_{j}} (λ̂_{j} ) < Λ, by Claim 1.

Case 2.1.1 cf(j) ≠ ω_{1}.

As cf(α_{j}) ≠ ω_{1}, there will be a least β_{j} α_{j} such that π_{α_{j} β_{j}}^{M} (η̂_{j}) = ν_{β_{j}} .

As in Case 1, set α_{j+1} := β_{j} +1 and define h^{j+1} : N^{j+1} → M^{α_{j+1}} as described above. Note that we must have m^{α_{j}} = m̃^{α_{j}} , as this is true for all α except those of cofinality ω_{1}, in the special Case 2.2 of the definition of the iteration of M. Set n^{j} := ñ^{j}.

Case 2.1.2 cf(j) = ω_{1}.

Case 2.1.2.1 ν_{α_{j}} > η̂_{j}.

In this case, we first go to M^{α_{j} +1} . Note that there, P^{M}(α_{j}+1, π_{α_{j} ,α_{j} +1}^{M} (η̂_{j})) will be bounded in α_{j}+1 (by α_{j}, trivially). Also, μ_{α_{j} +1} π_{α_{j} ,α_{j} +1}^{M} (μ_{α_{j}} ), as this last term satisfies both conditions for the definition of μ_{α_{j} +1} : m̃^{α_{j} +1} (π_{α_{j} ,α_{j} +1}^{M} (μ_{α_{j}} )) = m^{α_{j}} (μ_{α_{j}} ) by definition, and this last term must be less than Λ, by the choice of μ_{α_{j}} . But then there will be some least β_{j} α_{j} +1 such that ν_{β_{j}} = π_{α_{j} ,β_{j}}^{M} (η̂_{j}) and we are done. Set n^{j} := ñ^{j}.

Case 2.1.2.2 ν_{α_{j}} η̂_{j} and P^{M}(α_{j}, ν_{α_{j}} ) is bounded in α_{j}.

In this case the existence of a least suitable β_{j} is immediate. Set n^{j} := ñ^{j}.

Case 2.1.2.3 ν_{α_{j}} η̂_{j} and P^{M}(α_{j}, ν_{α_{j}} ) is unbounded in α_{j}.

In this case we have reached, on the M-side of the iteration, a point where the critical point μ_{α_{j}} has been exhausted, i. e., we are in the situation of Case 2.2 of the definition of the iteration of M. For in no other case will an extender E_{ν}^{M^{α}} be used although P^{M}(α, ν) is unbounded in α and cf(α) = ω_{1}. Note that this implies that for all ν ∈d(M^{α_{j}} , μ_{α_{j}} ) we must have that P^{M}(α_{j}, ν) is unbounded in α_{j}, and that ν_{α_{j}} = mind(M^{α_{j}} , μ_{α_{j}} ).

However, there will none the less exist some least β_{j} α_{j} such that ν_{β_{j}} = π_{α_{j} ,β_{j}}^{M} (η̂_{j}), so that the construction of h^{j+1} can proceed as before. The only difference is that we have to adjust the bookkeeping function, provided that h^{j}(λ_{j}) = μ_{α_{j}} . In this case, set

n^{j}(ξ) := \{ \begin{matrix}ñ^{j} (ξ) + 1 & if ξ = λ_{j} \\ ñ^{j} (ξ) & else, \end{matrix}

Note that m^{α_{j}} (μ_{α_{j}} ) increases by one at this point, too. Otherwise (i. e., if h^{j}(λ_{j}) > μ_{α_{j}} ) let n^{j} = ñ^{j}. The following subclaim will be used later on.

Subclaim If h^{j}(λ_{j}) = μ_{α_{j}} , then the set Q^{N}(j) := {i < j | π_{ij}^{N}(λ_{i}) = λ_{j}} is unbounded in j.

This says that we can conclude that one and the same critical point must have been used cofinally up to j on the N-side. Unfortunately, we cannot tell which of the extenders (with this critical point) was responsible, i. e., we cannot show that P^{N}(j, η_{j}) is unbounded in j, as well. However, by later on taking sufficiently closed limits, we will be able to remedy the situation.

Proof j is a limit ordinal, so there exists some i_{0} < j such that λ_{j} ∈rge(π_{i_{0}j}^{N}). For i ∈ [i_{0}, j), let λ̄_{i} := (π_{ij}^{N})^{−1} (λ_{j}). Note that then λ̄_{i} λ_{i}, as otherwise λ_{j} = π_{ij}^{N}(λ̄_{i}) = λ̄_{i} < λ_{i}, contradicting the normality of I^{N}. As a consequence, we can conclude that π_{ij}^{N}(λ_{i}) λ_{j}. Also, set μ̄_{α} := (π_{αα_{j}}^{M} )^{−1} (h^{j}(λ_{j})), for α α_{i_{0}} , so that μ̄_{α_{i}} = h^{i}(λ̄_{i}).

Now assume for a contradiction that Q^{N}(j) were bounded in j. Then there exists some i_{1} (w.l.o.g. i_{1} i_{0}) such that for all i∈[i_{1}, j) (π_{ij}^{N}(λ_{i}) < λ_{j}), i. e., λ_{i} < λ̄_{i}. Subsequently,

∀i∈[i_{1}, j) (h^{i}(λ_{i}) < μ̄_{α_{i}} ).

Since P^{M}(α_{j}, h^{j}(η_{j})) is unbounded in α_{j}, it follows that also Q^{M}(α_{j}, h^{j}(λ_{j})) := {α < α_{j} | π_{αα_{j}}^{M} (μ_{α}) = h^{j}(λ_{j})} is unbounded in α_{j}. (If some extender is used cofinally often, then, a fortiori, its critical point is used cofinally often.) Recall that α_{i} β_{i} and μ_{β_{i}} = π_{α_{i}β_{i}}^{M} (h^{i}(λ_{i})). Thus

μ_{β_{i}} < π_{α_{i}β_{i}}^{M} (h^{i}(λ̄_{i})) = μ̄_{β_{i}} .

Now for α ∈ Q^{M}(α_{j}, h^{j}(λ_{j})), μ_{α} = μ̄_{α} . Thus it cannot hold that for all sufficiently large i, α_{i} = β_{i}. For then we would have that for all sufficiently large α, there exists some i such that α = α_{i} = β_{i}, and thus μ_{α} = μ_{β_{i}} < μ̄_{β_{i}} . This would imply α ∉ Q^{M}(α_{j}, h^{j}(η_{j})), so that this set would be bounded in α_{j}, contradiction. This is just a complicated way of saying that if on the M-side the critical point corresponding to λ_{j} is used cofinally often below α_{j}, and if the upwards maps h^{i} from N^{i} always require the use of critical points corresponding to λ_{i} < λ̄_{i} = (π_{ij}^{N})^{−1} (λ_{j}), then sometimes we have to skip from α_{i} to β_{i} to make room for possible uses of the higher critical point (cf. Figure 6.2).

\xymatrix{M^{α_{i}} \ar[r] & ⋯ \ar[r] & M^{α} \ar[r]_{crit =μ̄_{α}} & M^{α+1} \ar[r] & … \ar[r] & M^{β_{i}} \ar[r]_{crit =μ_{β_{i}}} & M^{β_{i} +1} = M^{α_{i+1}} \\ N^{i} \ar[u]_{h^{i}} \ar[urrrrr]_{g^{i}} \ar[rrrrrr]_{crit =λ_{i} <λ̄_{i}} &  &  &  &  &  & N^{i+1} \ar[u]_{h^{i+1}}}

Figure 6.2: α_{i} cannot always be equal to β_{i} if the pre-image of λ_{j} is not used cofinally below j.

But this will yield a contradiction. Pick some i ∈ [i_{1}, j) such that α_{i} < β_{i}, and there is α∈[α_{i}, β_{i}) ∩ Q^{M}(α_{j}, h^{j}(λ_{j})), i. e., μ_{α} = μ̄_{α} . Then

π_{α_{i}α}^{M} (h^{i}(λ_{i})) < π_{α_{i}α}^{M} (h^{i}(λ̄_{i}))
= π_{α_{i}α}^{M} (μ̄_{α_{i}} )
= μ̄_{α} = μ_{α} = crit(π_{α,α+1}^{M} ) < crit(π_{α+1,β_{i}}^{M} )

so that

π_{α_{i}β_{i}}^{M} (h^{i}(λ_{i})) = π_{α+1,β_{i}}^{M} ∘ π_{α,α+1}^{M} ∘ π_{α_{i}α}^{M} (h^{i}(λ_{i}))
= π_{α+1,β_{i}}^{M} (π_{α_{i}α}^{M} (h^{i}(λ_{i})))
= π_{α_{i}α}^{M} (h^{i}(λ_{i}))
< μ_{α},

while at the same time

π_{α_{i}β_{i}}^{M} (h^{i}(λ_{i})) = μ_{β_{i}} > μ_{α},

by the normality of I^{M}. Contradiction. (Subclaim)

Case 2.2 μ_{α_{j}} > λ̂_{j} or α_{j} = ϑ^{(ω_{1})}.

In this case, no suitable β_{j} exists. Let the construction break down. (Construction of h^{j})

We now turn to the proof of (∗). Properly, this proof should be part of the same induction, but to avoid confusion, we separate the two. Say (∗) had been shown to hold for all i < j.

Case 1 j = i + 1 a successor ordinal.

Equation (1) “at j” follows from (2) “at i”: Let λ λ_{j}. If λ ∈ rge(π_{ij}^{N}), say λ = π_{ij}^{N}(λ̄), then

h^{j}(λ) = h^{j}(π_{ij}^{N}(λ̄)) = π_{β_{i},β_{i}+1}^{M} (g^{i}(λ̄)).

Thus m̃^{α_{j}} (h^{j}(λ)) = m^{β_{i}} (g^{i}(λ̄)) = n^{i}(λ̄) = ñ^{j} (λ), applying the definition of m̃^{α_{j}} , (∗) (2) at i, and finally the definition of ñ^{j}. If λ∉rge(π_{ij}^{N}), then ñ^{j} (λ) = 0, by definition. It suffices to show that h^{j}(λ)∉rge(π_{β_{i},β_{i}+1}^{M} ) as well, as then m̃^{α_{j}} (h^{j}(λ)) = 0, again by definition. So assume to the contrary that h^{j}(λ) = π_{β_{i},β_{i}+1}^{M} (ξ̄). Now λ = π_{ij}^{N}(f)(a) for some a∈[μ_{i}]^{<ω} , f : [μ_{i}]\overline{\overline{a}} → N^{i}, and by the construction of h^{j} (cf. [Koe89, Theorem 14.2]), h^{j}(λ) = π_{β_{i},β_{i}+1}^{M} (g^{i}(f))(g^{i}(a)). Now

N^{i} f is not constant modulo E_{a},

where E := E_{η_{i}}^{N^{i}}, whence

M^{β_{i}} g^{i}(f) is not constant modulo Ē_{a},

where Ē := E_{ν_{β_{i}}}^{M^{β_{i}}} . But then h^{j}(λ) = π_{β_{i},β_{i}+1}^{M} (g^{i}(f))(g^{i}(a)) is not an element of rge(π_{β_{i},β_{i}+1}^{M} ) either. So m̃^{α_{j}} (h^{j}(λ)) = 0 = ñ^{j} (λ).

This proves (∗) (1) “at j”. As for (2), note that α_{j} = β_{i} + 1 is a successor ordinal, whence m^{α_{j}} = m̃^{α_{j}} , and that we set n^{j} = ñ^{j}. So it remains to show that m^{β_{j}} (g^{j}(λ)) = m^{α_{j}} (h^{j}(λ)) for all λ λ_{j}. But if this were not true, then we would have, somewhere between α_{j} and β_{j}, exhausted a critical point greater than or equal to h^{j}(λ). But this is absurd, since then we would have used E_{h^{j} (λ)}^{M^{α_{j}}} not only once but in fact many times before β_{j}, contradicting the choice of β_{j} as the first index where this extender is used.

Case 2 j is a limit ordinal.

As in Case 1, (∗) (1) follows from (2) at earlier stages: Let λ λ_{j}. Then λ = π_{ij}^{N}(λ̄), for some i < j and some λ̄. Thus h^{j}(λ) = π_{α_{i}α_{j}}^{M} (h^{i}(λ̄)) = π_{β_{i}α_{j}}^{M} (g^{i}(λ̄)). Note that α_{j} = sup{α_{i} | i < j} = sup{β_{i} | i < j}, so that using (2) at i, one concludes that

m̃^{α_{j}} (h^{j}(λ)) = sup{m^{α}(ξ̄) | α < α_{j} ∧ π_{αα_{j}}^{M} (ξ̄) = h^{j}(λ)}
= sup{m^{β_{i}} (ξ̄) | i < j ∧ π_{β_{i}α_{j}}^{M} (ξ̄) = h^{j}(λ)}
= sup{m^{β_{i}} (ξ̄) | i < j ∧ ξ̄ = g^{i}(λ̄) ∧ π_{ij}^{N}(λ̄) = λ}
= sup{n^{i}(λ̄) | i < j ∧ π_{ij}^{N}(λ̄) = λ}
= ñ^{j} (λ)

To see that (2) holds, note that n^{j}(λ) = m^{α_{j}} (h^{j}(λ)) for all λ λ_{j}. For λ > λ_{j} this follows from n^{j}(λ) = ñ^{j} (λ) = m̃^{α_{j}} (h^{j}(λ)) = m^{α_{j}} (h^{j}(λ)), noting that then h^{j}(λ) > μ_{α_{j}} , too. It is also true for λ_{j} itself, as n^{j}(λ_{j}) = ñ^{j} (λ_{j}), unless we are in Case 2.1.2.3 and h^{j}(λ_{j}) = μ_{α_{j}} . But in this case n^{j}(λ_{j}) = ñ^{j} (λ_{j}) + 1 = m̃^{α_{j}} (μ_{α_{j}} ) + 1 = m^{α_{j}} (μ_{α_{j}} ), so (2) holds anyway.

It remains to show that m^{β_{j}} (g^{j}(λ)) = m^{α_{j}} (h^{j}(λ)). But β_{j} was chosen to be the least β such that ν_{β} = π_{α_{j} ,β}^{M} (h^{j}(η_{j})), i. e., the first place in the iteration of M where the extender EM^{α_{j}} ; _{η_{j}} is used. This will happen long before the next point at which this critical point will be exhausted and hence m would be increased.

Note that (1) remains true when the construction of h^{j} breaks down in Case 2.2. (Proof of (∗))

Subclaim The construction goes through for all j ϑ.

Proof Assume the construction breaks down at some j, i. e., h^{j} is defined, but h^{j+1} cannot be constructed. (Recall that if j is a limit ordinal and all h^{i} are constructed, it is a rather trivial matter to find h^{j}.) This can only happen in Case 2.2, whence j must be a limit ordinal. So either μ_{α_{j}} > λ̂_{j} or α_{j} = ϑ^{(ω_{1})}.

Note first that λ̂_{j} satisfies the normality requirement of the definition of μ_{α_{j}} : For assume that there existed some α < α_{j} such that ν_{α} λ̂_{j} . As j is a limit ordinal and α_{j} = sup{α_{i} | i < j}, there must be some i < j such that α_{i} > α and thus μ_{α_{i}} > ν_{α} (by the normality of I^{M}). Pick i large enough such that λ_{j} ∈rge(π_{ij}^{N}), say λ_{j} = π_{ij}^{N}(λ̄). By normality of I^{N}, λ̄ λ_{i}. But then

λ̂_{j} = h^{j}(π_{ij}^{N}(λ̄)) = π_{α_{i},α_{j}}^{M} (h^{i}(λ̄))
π_{α_{i},α_{j}}^{M} (h^{i}(λ_{i}))
π_{α_{i},α_{j}}^{M} (μ_{α_{i}} )
> π_{α_{i},α_{j}}^{M} (ν_{α}) = ν_{α}
λ̂_{j} ,

a contradiction! Thus the only reason why μ_{α_{j}} might be larger than λ̂_{j} or not defined at all is that m̃^{α_{j}} (λ̂_{j} ) = Λ.

By (∗) we know that then ñ^{j} (λ_{j}) = Λ, too. This will lead to a contradiction. We will find a sequence ⟨i_{ξ} | ξ < Λ⟩ of indices where n^{i} was increased at λ_{j} (or rather its pre-image). At each of these points, one extender with critical point (corresponding to) λ_{j} must have been used cofinally before. By sufficiently thinning out the sequence, we will show that at i_{ω_{1}^{n}} (where n = o^{N^{j}} (λ_{j})), all available extenders must have been used up cofinally, leaving none to continue the iteration in a short way.

Fix some notation: for i < j such that λ_{j} ∈ rge(π_{ij}^{N}), let λ̄_{i} := (π_{ij}^{N})^{−1} (λ_{j}). For ξ < Λ, let i_{ξ+1} be the least i < j, i > i_{ξ}, such that n^{i}(λ̄_{i}) = ñ^{i}(λ̄_{i}) + 1, and for limit ordinals ζ < Λ, let i_{ζ} := sup{i_{ξ} | ξ < ζ}. Such i must exists, as Λ = ñ^{j} (λ_{j}) = sup{n^{i}(λ̄_{i}) | i < j}. Note that then λ_{i_{ξ+1}} = λ̄_{i_{ξ+1}} and that μ_{α_{i_{ξ+1}}} = h^{i_{ξ+1}} (λ_{i_{ξ+1}} ). By the subclaim of Case 2.1.2.3 of the construction of h^{j}, we know that

Q^{N}(i_{ξ+1} ) = {i′< i_{ξ+1} | π_{i′i_{ξ+1}}^{N} (λ_{i′}) = λ_{i_{ξ+1}} } is unbounded in i_{ξ+1} .

But it is also true that at the limit points i_{ζ} of the sequence the same critical point is used, i. e., λ_{i_{ζ}} = λ̄_{i_{ζ}} : We cannot have λ_{i_{ζ}} > λ̄_{i_{ζ}} , for then one would get the absurdity of

λ_{j} = π_{i_{ζ}j}^{N} (λ̄_{i_{ζ}} ) = λ̄_{i_{ζ}} < λ_{i_{ζ}} < λ_{j},

where the last inequality stems from the normality of the iteration I^{N}. On the other hand, the normality also implies λ_{i_{ζ}} λ̄_{i_{ζ}} , for this critical point has been used cofinally up to i_{ζ}: Surely λ_{i_{ζ}} > λ_{i} for any i < i_{ζ}. Now assume λ_{i_{ζ}} < λ̄_{i_{ζ}} . Then, for some i < i_{ζ}, λ_{i_{ζ}} = π_{ii_{ζ}}^{N} (λ̃_{i}). Since i_{ζ} = sup{i_{ξ} | ξ < ζ}, assume w.l.o.g. that i = i_{ξ+1} for some such ξ. But then, keeping in mind that π_{ii_{ζ}}^{N} (λ̄_{i}) = λ̄_{i_{ζ}} and λ̄_{i} = λ_{i}, we must have λ̃_{i} λ_{i}, as otherwise λ_{i_{ζ}} = π_{ii_{ζ}}^{N} (λ̃_{i}) = λ̃_{i} < λ_{i}, contradicting normality, and also λ̃_{i} λ_{i}, as otherwise λ_{i_{ζ}} = π_{ii_{ζ}}^{N} (λ̃_{i}) > π_{ii_{ζ}}^{N} (λ_{i}) = λ̄_{i_{ζ}} , which we excluded earlier on.

Thus we know that at every point of the sequence ⟨i_{ξ} | ξ < Λ⟩, λ_{i_{ξ}} = λ̄_{i_{ξ}} , i. e., we use the same critical point at all these stages. Also, this same critical point has been used cofinally leading up to each i_{ξ}:

Q^{N}(i_{ξ}) is unbounded in i_{ξ}

for any ξ < Λ (and not only the successor ordinals).

We will now show that, setting Ω := ω_{1}^{n}, every possible index k < n := o^{N^{j}} (λ_{j}) will be used cofinally leading up to i_{Ω} . But then we will arive at a contradiction: at stage i_{Ω} , we must again use some extender with the right critical point. But no matter which index is used, the iteration cannot have been short any more, contradicting the assumption at the outset that I^{N}, indeed, was short.

To this end we will define sets and sequences of sets as well as (finite) ordinals and sequences of finite ordinals. The idea is that at any stage of the iteration from a prescribed set we can tell which of the (finitely many) extenders has been used. We consider Ω as a limit of ordinals of cofinality ω_{1} such that each of the ordinals is again a limit ordinal of cofinality ω_{1} such that etc., n-levels deep. In a first step, we choose one extender (i. e., an index k^{1} < n) which will be used cofinally up to i_{Ω} at the limit points of the first level. Then, for each of these points leading up to i_{Ω} on the first level, we look at the limit points of the second level. Again, one extender must have been used cofinally, depending only on the point of the first level we are leading up to. By thinning out the first level now, we can assure that all these “secondlevel” extenders are, in fact, the same. Thus, after the second step of our construction, we will have shown that already two different extenders must have been used cofinally up to i_{Ω} , provided of course that the iteration I^{N} is short. Repeating this n-times shows that, in fact, all n-many extenders with critical point λ_{i_{Ω}} must have been used cofinally up to i_{Ω} , leaving none to continue I^{N} in a short way.

For the first step, in which A_{1}^{1} and k^{1} are defined, note that

ω_{1}^{n} = lim_{ζ_{1}→ω_{1}} ω_{1}^{n−1} (ζ_{1} + 1),

and since the sequence ⟨i_{ξ} | ξ < Λ⟩ is a normal sequence, also

i_{ω_{1}^{n}} = lim_{ζ_{1}→ω_{1}} i_{ω_{1}^{n−1} (ζ_{1}+1)} .

To facilitate notation, let e_{ξ} := η_{i_{ξ}}, the index of the extender used at stage i_{ξ} of the iteration I^{N}. Now there are only n possible values of e_{ξ}, so there must exist some k^{1} < n such that the extender with index k^{1} has been used stationarily often, i. e., the set

A_{1}^{1} := {ζ_{1} < ω_{1} | e_{ω_{1}^{n−1} (ζ_{1}+1)} = k^{1}}

is stationary in ω_{1}. Note that then

P^{N}(i_{Ω} , k^{1}) = {i < i_{Ω} | π_{ii_{Ω}}^{N} (η_{i}) = k^{1}}

is unbounded in i_{Ω} , as it contains {i_{ω_{1}^{n−1} (ζ_{1}+1)} | ζ_{1}∈A_{1}^{1}}.

For the second step, first fix some ζ_{1} ∈ A_{1}^{1}. Now consider ω_{1}^{n−1} ζ_{1} + ω_{1}^{n−1} = lim_{ζ_{2}→ω_{1}} (ω_{1}^{n−1} ζ_{1} + ω_{1}^{n−2} (ζ_{2} + 1)). Thus,

i_{ω_{1}^{n−1} (ζ_{1}+1)} = lim_{ζ_{2}→ω_{1}} i_{ω_{1}^{n−1} ζ_{1}+ω_{1}^{n−2} (ζ_{2}+1)} .

As before, there must now exist some k_{1}^{2}(ζ_{1}) such that the set

A_{2}^{2}(ζ_{1}) := {ζ_{2} < ω_{1} | e_{ω_{1}^{n−1} ζ_{1}+ω_{1}^{n−2} (ζ_{2}+1)} = k_{1}^{2}(ζ_{1})}

is stationary in ω_{1}. Again, one concludes that

P^{N}(i_{ω_{1}^{n−1} (ζ_{1}+1)} , k^{2}(ζ_{1})) is unbounded in i_{ω_{1}^{n−1} (ζ_{1}+1)} .

The iteration I^{N} was assumed to be short, so that P^{N}(i_{ω_{1}^{n−1} (ζ_{1}+1)} , e_{ω_{1}^{n−1} (ζ_{1}+1)} ) must be bounded in i_{ω_{1}^{n−1} (ζ_{1}+1)} . And since ζ_{1} was chosen from A_{1}^{1}, one has e_{ω_{1}^{n−1} (ζ_{1}+1)} = k^{1}. Thus it follows that k_{1}^{2}(ζ_{1}) ≠ k^{1}.

Next, notice that there must be some k^{2} such that the set

A_{2}^{1} := {ζ_{1}∈A_{1}^{1} | k_{1}^{2}(ζ_{1}) = k^{2}}

is stationary in ω_{1}. Obviously, k^{2} ≠ k^{1}. Furthermore,

P^{N}(i_{Ω} , k^{2}) is unbounded in i_{Ω} ,

as this set contains {i_{ω_{1}^{n−1} ζ_{1}+ω_{1}^{n−2} (ζ_{2}+1)} | ζ_{1}∈A_{1}^{1} ∧ ζ_{2}∈A_{1}^{2}(ζ_{1})} and this latter set is unbounded in i_{Ω} .

Before writing down the induction step in full generality, we present the case n = 3 in some detail, believing that this will much better serve the purpose of shedding some light onto this rather unwieldy profusion of indices.

Let q^{1}(ζ_{1}) := ω_{1}^{n−1} ζ_{1} and q^{2}(ζ_{1}, ζ_{2}) := ω_{1}^{n−1} ζ_{1} + ω_{1}^{n−2} ζ_{2}. Fix some ζ_{1}∈A_{2}^{1}, and some ζ_{2}∈A_{2}^{2}(ζ_{1}). Then the second step ensured that

e_{q^{1}(ζ_{1}+1)} = k^{1}  and  e_{q^{2}(ζ_{1},ζ_{2}+1)} = k_{1}^{2}(ζ_{1}) = k^{2}.

Set q^{3}(ζ_{1}, ζ_{2}, ζ_{3}) := ω_{1}^{n−1} ζ_{1} + ω_{1}^{n−2} ζ_{2} + ω_{1}^{n−3} ζ_{3} and note that

q^{2}(ζ_{1}, ζ_{2} + 1) = lim_{ζ_{3}→ω_{1}} q^{3}(ζ_{1}, ζ_{2}, ζ_{3} + 1),

so that, since ⟨i_{ξ} | ξ < Λ⟩ is a normal sequence, also

i_{q^{2}(ζ_{1},ζ_{2}+1)} = lim_{ζ_{3}→ω_{1}} i_{q^{3}(ζ_{1},ζ_{2},ζ_{3}+1)} .

Thus there must be some k_{2}^{3}(ζ_{1}, ζ_{2}) < n such that the set

A_{3}^{3}(ζ_{1}, ζ_{2}) := {ζ_{3} < ω_{1} | e_{q^{3}(ζ_{1},ζ_{2},ζ_{3}+1)} = k_{2}^{3}(ζ_{1}, ζ_{2})}

is stationary in ω_{1}. Thus, as before,

P^{N}(i_{q^{2}(ζ_{1},ζ_{2}+1)} , k_{2}^{3}(ζ_{1}, ζ_{2})) is unbounded in i_{q^{2}(ζ_{1},ζ_{2}+1)} ,

which implies, together with the shortness of I^{N}, that

k_{2}^{3}(ζ_{1}, ζ_{2}) ≠ k_{1}^{2}(ζ_{1}) = k^{2}.

Now let ζ_{2} range over elements of A_{2}^{2}(ζ_{1}). Surely there must exist some k_{1}^{3}(ζ_{1}) such that

A_{3}^{2}(ζ_{1}) := {ζ_{2}∈A_{2}^{2}(ζ_{1}) | k_{2}^{3}(ζ_{1}, ζ_{2}) = k_{1}^{3}(ζ_{1})}

is a stationary subset of A_{2}^{2}(ζ_{1}). Note that for a fixed ζ_{1}, the set

{i_{q^{3}(ζ_{1},ζ_{2},ζ_{3}+1)} | ζ_{2}∈A_{3}^{2}(ζ_{1}) ∧ ζ_{3}∈A_{3}^{3}(ζ_{1}, ζ_{2})}

is unbounded in i_{q^{1}(ζ_{1}+1)} , so that

P^{N}(i_{q^{1}(ζ_{1}+1)} , k_{1}^{3}(ζ_{1})) is unbounded in i_{q^{1}(ζ_{1}+1)} ,

whence by the shortness of I^{N}

k_{1}^{3}(ζ_{1}) ≠ e_{q^{1}(ζ_{1}+1)} = k^{1}.

(The last equation holds as ζ_{1} was chosen from A_{2}^{1} ⊆ A_{1}^{1}.) Finally (for the third step), we need to thin out the set A_{2}^{1} to get one value of k_{1}^{3} working for all ζ_{1}: find some k^{3} < n such that the set

A_{3}^{1} := {ζ_{1}∈A_{2}^{1} | k_{1}^{3}(ζ_{1}) = k^{3}}

\xymatrix{A_{1}^{1} \ar@{-->}[drr] & ⊇ & A_{2}^{1} \ar@{-->}[ddrr] & ⊇ & A_{3}^{1} \ar@{-->}[dddrr] & ⊇ & A_{4}^{1} & … & ⊇ & A_{n}^{1} \\  &  & A_{2}^{2}(ζ_{1}) \ar@{-->}[u] & ⊇ & A_{3}^{2}(ζ_{1}) \ar@{-->}[u] & ⊇ & A_{4}^{2}(ζ_{1}) \ar@{-->}[u] & … & ⊇ & A_{n}^{2}(ζ_{1}) \ar@{-->}[u] \\  &  &  &  & A_{3}^{3}(ζ_{1}, ζ_{2}) \ar@{-->}[u] & ⊇ & A_{4}^{3}(ζ_{1}, ζ_{2}) \ar@{-->}[u] & … & ⊇ & A_{n}^{3}(ζ_{1}, ζ_{2}) \ar@{-->}[u] \\  &  &  &  &  &  & ⋮ \ar@{-->}[u] &  &  & ⋮ \ar@{-->}[u] \\  &  &  &  &  &  &  &  &  & A_{n}^{n}(ζ_{1}, …, ζ_{n−1} ) \ar@{-->}[u]}

Figure 6.3: Arrows indicate the order in which the sets are chosen in the construction. Sets in the l-th row contain “good” values for ζ_{l}. Sets in the mth column represent combinations ζ_{1}, …, ζ_{m} which simultaneously “exhaust” the extenders with indices from {k^{1}, …, k^{m}}.

is stationary in A_{2}^{1}. Obviously, we must have k^{3}∈\{k^{1}, k^{2}}. Furthermore,

{i_{q^{3}(ζ_{1},ζ_{2},ζ_{3}+1)} | ζ_{1}∈A_{3}^{1} ∧ ζ_{2}∈A_{3}^{2}(ζ_{1}) ∧ ζ_{3}∈A_{3}^{3}(ζ_{1}, ζ_{2})}

is unbounded in i_{Ω} , and at any such stage e_{q^{3}(ζ_{1},ζ_{2},ζ_{3}+1)} = k^{3}, so that

P^{N}(i_{Ω} , k^{3}) is unbounded in i_{Ω} .

Let l n, and assume that all relevant (sequences of) sets and ordinals had been chosen (cf. Figure 6.3). Define q^{l}(ζ_{1}, …, ζ_{l}) := Σ_{m=1}^{l} ω_{1}^{n−m} ζ_{m}. Fix ζ_{1}∈A_{l−1}^{1} , ζ_{2}∈A_{l−1}^{2} (ζ_{1}), …, ζ_{l−1} ∈A_{l−1}^{l−1} (ζ_{1}, …, ζ_{l−2} ). Note that

i_{q^{l−1} (ζ_{1},...,ζ_{l−2} ,ζ_{l−1} +1)} = lim_{ζ_{l}→ω_{1}} i_{q^{l}(ζ_{1},...,ζ_{l−1} ,ζ_{l}+1)} .

Thus there is some k_{l−1}^{l} (ζ_{1}, …, ζ_{l−1} ) < n such that

A_{l}^{l}(ζ_{1}, …, ζ_{l−1} ) := {ζ_{l} < ω_{1} | e_{q^{l}(ζ_{1},...,ζ_{l−1} ,ζ_{l}+1)} = k_{l−1}^{l} (ζ_{1}, …, ζ_{l−1} )}

is stationary in ω_{1}. This also implies that

P^{N}(i_{q^{l−1} (ζ_{1},...,ζ_{l−2} ,ζ_{l−1} +1)} , k_{l−1}^{l} (ζ_{1}, …, ζ_{l−1} )) is unbounded in i_{q^{l−1} (ζ_{1},...,ζ_{l−2} ,ζ_{l−1} +1)} , whence by the shortness of I^{N} and the fact that e_{q^{l−1} (ζ_{1},...,ζ_{l−2} ,ζ_{l−1} +1)} = k^{l−1} it follows that

k_{l−1}^{l} (ζ_{1}, …, ζ_{l−1} ) ≠ k^{l−1} .

Assume that m < l, k_{m}^{l}(ζ_{1}, …, ζ_{m}) < n is unequal to k^{m}, and

A_{l}^{m+1} (ζ_{1}, …, ζ_{m}) = {ζ_{m+1} ∈A_{l−1}^{m+1} (ζ_{1}, …, ζ_{m}) | k_{m+1}^{l} (ζ_{1}, …, ζ_{m}, ζ_{m+1} ) = k_{m}^{l}(ζ_{1}, …, ζ_{m})}

(if m < l − 1, or

A_{l}^{l}(ζ_{1}, …, ζ_{l−1} ) = {ζ_{l}∈ω_{1} | e_{q^{l}(ζ_{1},...,ζ_{l−1} ,ζ_{l}+1)} = k_{l−1}^{l} (ζ_{1}, …, ζ_{l−1} )}

if m = l − 1) is a stationary subset of A_{l−1}^{m+1} (ζ_{1}, …, ζ_{m}) (if m < l − 1, or ω_{1}, if m = l − 1). Then there must be some k_{m−1}^{l} < n such that

A_{l}^{m}(ζ_{1}, …, ζ_{m−1} ) := {ζ_{m} ∈A_{l−1}^{m} (ζ_{1}, …, ζ_{m−1} ) | k_{m}^{l}(ζ_{1}, …, ζ_{m}) = k_{m−1}^{l} (ζ_{1}, …, ζ_{m−1} )}

is a stationary subset of A_{l−1}^{m} (ζ_{1}, …, ζ_{m−1} ). Again one concludes that k_{m−1}^{l} (ζ_{1}, …, ζ_{m−1} ) cannot be equal to e_{q^{m−1} (ζ_{1},...,ζ_{m−1} +1)} = k^{m−1} . Repeating this procedure l-times finally gives some k^{l} ∈ n \ {k^{m} | m < l} such that for any ζ_{1}∈A_{l}^{1}, ζ_{2}∈A_{l}^{2}(ζ_{1}), …, ζ_{l}∈A_{l}^{l}(ζ_{1}, …, ζ_{l−1} ),

e_{q^{l}(ζ_{1},...,ζ_{l−1} ,ζ_{l}+1)} = k^{l},

and consequently

P^{N}(i_{Ω} , k^{l}) is unbounded in i_{Ω} .

At the end of this procedure, one will have chosen n-many k^{l}, so that {k^{l} | l < n} = n. By the pigeon hole principle, we must have that e_{Ω} = k^{l}, for some l < n. But then, at stage i_{Ω} , the iteration I^{N} cannot have been short. Contradiction. Thus the construction of the h^{j} cannot have broken down at some j < ϑ and we can indeed embed N^{ϑ} into M^{ϑ^{(ω_{1})}} . (Subclaim)

(Claim 3)

Our original goal was to prove in Chapter 7 that the existence of an irregular ultrafilter on κ implied the existence of an inner model for a strong cardinal. Currently, the restrictions on Lemma 6.2 prevent this. This Lemma is central to the proof of Claim 8 in the proof of Theorem 7.5. The following example shows that it cannot be extended straightforwardly to accomodate inner models for, say, strong cardinals.

Let M be a mouse of size ϱ, where κ = ϱ^{+} , with a measurable cardinal μ with o^{M}(μ) = ϱ. Then for any ζ < κ there are iterations of M of length ζ in which any extender is used at most once (not even < ω_{1}-many times). Just consider a well-ordering r of ϱ of ordertype ζ (assuming w.l.o.g. that ζ ϱ), and use the (image of the) r(ξ)-th extender on μ to construct the ξ-th step of the iteration. Hence the bound of the ordinal height of short iterates of M cannot be less than κ.

66 CHAPTER 6. SHORT ITERATIONS

Irregular Ultrafilters

If there wasn’t anything to find out, it would be dull. Even trying to find out and not finding out is just as interesting as trying to find out and finding out; and I don’t know but more so.

(Mark Twain, Eve’s Diary)

Recall some basic ultrafilter definitions.

7.1 Definition Let V be an ultrafilter on some cardinal κ. Then V is called uniform iff every element of V has cardinality κ. V is called normal iff it is closed under diagonal intersections iff every regressive function (mod V) is constant (mod V). V is called weakly normal iff every regressive function (mod V) is bounded below κ (mod V).

From now on, V will always denote an ultrafilter on some cardinal κ.

7.2 Definition An ultrafilter V is regular iff there is a sequence ⟨a_{ξ} | ξ < κ⟩ of elements of V such that the intersection of any ω-many of them is empty. Such a sequence is then called regularity sequence.

More generally, V is (γ, κ)-regular (for some γ < κ) iff the intersection of any γ-many of them is empty. V is (γ, τ)-regular if the regularity sequence has length τ instead of κ. (This last concept will not be considered here.)

Finally, for the purpose of this thesis, call an ultrafilter V weakly (γ, κ)- regular iff the intersection of any γ-many elements of the sequence ⟨a_{ξ} | ξ < κ⟩ is bounded in κ.

So regularity is a (strong) form of incompleteness. Trivially, if V is (γ, κ)- regular, then it is (γ′, κ)-regular for any γ′ > γ. Recall that an ultrafilter V is γ-complete iff the intersection of any β-many sets from V, β < γ, is again an element of V.

Note the following (easy) equivalence, illustrated in Figure 7.1: V is (γ, κ)- regular iff there exists a (γ, κ)-covering sequence for V, i. e., a family ⟨x_{η} | η < κ⟩ of subsets of κ, each of cardinality less than γ, such that for all ξ < κ, {η < κ | ξ ∈ x_{η}} ∈ V.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; a_{ξ}∈V; −; ξ; x_{η} = {ξ < κ | η ∈a_{ξ}}; +; η; κ; κ]

Figure 7.1: The two versions of (γ, κ)-regularity are just two ways of looking at the same diagramme. “Horizontal” sets a_{ξ} are elements from the regularity sequence, i. e., elements of V, whereas “vertical” sets are elements from the covering sequence, i. e., of cardinality less than γ.

We are interested in the connection between weak regularity and regularity.

7.3 Lemma Let ϱ ω_{1} be a cardinal such that ϱ^{ℵ_{0}} = ϱ. Let κ = ϱ^{+} . Assume V is a uniform ultrafilter on κ which is weakly (ω, κ)-regular. Then V is (ω_{1}, κ)-regular.

Proof Let V be a uniform ultrafilter on κ > ω_{1} which is weakly (ω, κ)- regular, and let ⟨a_{ξ} | ξ < κ⟩ witness this fact. We have to construct an (ω_{1}, κ)-regularity sequence. This is done by induction on ξ < κ. The idea is to successively cut off the a_{ζ} so that they have empty intersection with any ω-many a_{ξ} preceeding them in the sequence. The assumption that ϱ^{ℵ_{0}} = ϱ thus seems to be essential to the proof.

For ξ < ω, set b_{ξ} := a_{ξ}∈ V. Now assume ⟨b_{ξ} | ξ < ζ⟩ had been constructed, for some ζ < κ. For any sequence x = ⟨ξ_{i} | i∈ω⟩ of ordinals less than ζ, we know that ⋂ _{i∈ω} a_{ξ_{i}} is bounded in κ, since the sequence ⟨a_{ξ} | ξ < κ⟩ witnesses weak (ω, κ)-regularity. So let

α_{x} := sup ⋂_{i∈ω} a_{ξ_{i}} < κ

and let

α_{ζ} := sup{α_{x} + 1 | x∈^{ω}ζ}.

Since, by assumption, card(ζ) ϱ and ϱ^{ℵ_{0}} = ϱ < κ it follows that α_{ζ} < κ. Set b_{ζ} := a_{ζ} \ α_{ζ}. b_{ζ}∈V, as V is uniform.

It remains to show that ⟨b_{ξ} | ξ < κ⟩ is an (ω_{1}, κ)-regularity sequence. So let x = ⟨ξ_{i} | i ∈ ω_{1}⟩ be a sequence of ordinals less than κ. By construction, b_{ξ_{i}} ⊆ a_{ξ_{i}}, and thus, considering the intersection of the first ω-many of these sets,

⋂_{i∈ω} b_{ξ_{i}} ⊆ ⋂_{i∈ω} a_{ξ_{i}} ⊆ α_{⟨ξ_{i} | i∈ω⟩} =: ᾱ.

But ⟨ξ_{i} | i∈ω⟩∈^{ω}ξ_{ω}, so by the definition of α_{ξ_{ω}}, ᾱ < α_{ξ_{ω}} and so

( ⋂_{i∈ω} b_{ξ_{i}}) ∩ b_{ξ_{ω}} ⊆ ᾱ ∩ (κ \ α_{ξ_{ω}}) = ∅.

A fortiori, ⋂ _{i∈ω_{1}} b_{ξ_{i}} = ∅, and ⟨b_{ξ} | ξ < κ⟩ is an (ω_{1}, κ)-regularity sequence. In fact, the proof shows that already the intersection of (ω + 1)-many sets from the sequence is empty.

7.4 Definition An ultrafilter V is fully irregular iff it is not (γ, κ)-regular for any γ < κ.

7.5 Theorem Let κ = ϱ^{+} be successor cardinal, κ ω_{3}. Assume 2^{ϱ} = ϱ^{+} and ϱ^{ℵ_{0}} = ϱ. Let V be a fully irregular ultrafilter on κ. Then there is an inner model N satisfying o^{N}(μ) = ω for some μ.

Proof Assume for a contradiction that there is no such inner model. As we have mentioned before, this proof is written up so as to go through for the assumption that there was no inner model for a strong cardinal. It is only Lemma 6.2, which in turn is used in the proof of Claim 8 below, that forces us to make the much stronger assumption that there is no inner model satisfying o^{N}(μ) = ω. A proof based completely on this latter assumption would, of course, permit numerous simplifications, e. g., in the discussion of the various subcases, some of which are outright impossible under the stronger assumption.

Note that by [BK74, Corollary 2.2], cited as [Ket76, Theorem 1.11], we can conclude that 2^{κ} = κ^{+} .

By a theorem of Kanamori and Ketonen ([Kan76], [Ket76]), we may assume that V is weakly normal. By [Kan76, Theorem 2.3], we may also assume that {α∈κ | cf(α) ω_{2}}∈V. Thus, if x ⊆ κ is ω_{2}-club, then x∈V.

Let h : κ^{+} → H_{κ^{+}} be a bijection. For τ ∈(κ, κ^{+} ) let f_{τ} be a surjection from κ onto τ, and let

F := {⟨ξ, ζ, τ⟩ | f_{τ}(ξ) = ζ}.

Write F τ for F ∩ (κ × τ × τ).

Let H = ⟨H_{κ^{+}} ,∈, h, F ⟩, W := K^{c} ↓ κ^{+} . For τ ∈ (κ, κ^{+} ), let H̃_{τ} := H (h′′τ), H_{τ} := h′′τ = |H̃_{τ} |, W_{τ} := H_{τ} ∩ W. Let

I := {τ ∈(κ, κ^{+} ) | cf(τ) = ω_{2} ∧ H̃_{τ} ≺ H ∧ On∩H_{τ} = τ ∧ H_{κ} ⊆ H_{τ} ∧ H_{τ} transitive ∧ ^{ω_{1}} H_{τ} ⊆ H_{τ} ∧ W_{τ} = W ↓τ}

7.6 Lemma I is unbounded in κ.

Proof The usual construction of building a tower of substructures of height ω_{2}, where each successive structure is the closure of the previous one under all relevant operations, yields the result. Note that card(H_{κ}) = 2^{ϱ} = κ, so that the initial step requiring H_{κ} ⊆ H is feasible.

For each τ ∈ I, let ⟨η_{ξ}^{τ} | ξ < ω_{2}⟩ be a cofinal sequence in τ. Let η̄_{ξ}^{τ} := f_{τ}^{−1} (h^{−1} (η_{ξ}^{τ})) < κ. Since κ is regular, κ > ω_{2}, we will have η̄^{τ} := sup{η̄_{ξ}^{τ} | ξ < ω_{2}} < κ.

For τ ∈ I let H_{τ} := ⟨H_{τ},∈, h τ, F τ, f_{τ}⟩. For α ∈ κ let H̃_{α}^{τ} := H_{τ} (h ∘ f_{τ}′′α) and H̃_{α}^{τ} := |H̃_{α}^{τ} | = h ∘ f_{τ}′′α. Let

C_{τ} := {α∈(ϱ, κ) | H̃_{α}^{τ} ≺ H_{τ} ∧ κ ∩ H̃_{α}^{τ} = α ∧ ^{ω_{1}} H̃_{α}^{τ} ⊆ H̃_{α}^{τ} ∧ α > η̄^{τ}}

and for α∈C_{τ}

π̃_{α}^{τ} : H_{α}^{τ} −→^{∼} H̃_{α}^{τ} , H_{α}^{τ} transitive
H_{α}^{τ} := |H_{α}^{τ}|,  τ_{α} := On∩H_{α}^{τ}
K_{α}^{τ} := (K^{c})^{H_{α}^{τ}} ,  π_{α}^{τ} := π̃_{α}^{τ} K_{α}^{τ}

Since α = crit(π_{α}^{τ}) > η̄^{τ}, we have f_{τ}′′η̄^{τ} ⊆ H̃_{α}^{τ}, so On∩H̃_{α}^{τ} is cofinal in τ. This implies π_{α}^{τ} : K_{α}^{τ} →_{e} W_{τ} cofinally.

7.7 Lemma i) For τ ∈I, C_{τ} is ω_{2}-club, hence C_{τ} ∈V.

ii) ^{ω_{1}} H_{α}^{τ} ⊆ H_{α}^{τ}.

iii) π_{α}^{τ} : K_{α}^{τ} →_{e} W_{τ} ≺ W iv) crit(π_{α}^{τ}) = α.

v) K_{α}^{τ}↓α = K↓α.

Proof This is straightforward.

7.8 Lemma Let σ, τ ∈I, τ ∈rge(π_{α}^{σ}), α∈C_{τ} ∩ C_{σ}. Then

H_{α}^{τ} ≺ H_{α}^{σ}  π̃_{α}^{τ} = π̃_{α}^{σ} H_{α}^{τ}
K_{α}^{τ} ≺ K_{α}^{σ}  π_{α}^{τ} = π_{α}^{σ} K_{α}^{τ}

and, in fact, H_{α}^{τ} = (π̃_{α}^{σ})^{−1} (H_{τ})∈H_{α}^{σ} and K_{α}^{τ} = (π_{α}^{σ})^{−1} (W_{τ}).

Proof Note that H_{τ} is definable from τ in H_{σ}, since H_{τ} = h′′τ = (h H_{σ})′′τ etc. Thus H_{τ} ∈ H_{σ}. But then H̃_{α}^{σ} ∩ H_{τ} ≺ H_{σ}. Since every H ≺ H_{σ} is uniquely determined by |H| ∩ κ via h, we get that H̃_{α}^{τ} = H̃_{α}^{σ} ∩ H_{τ}. But H̃_{α}^{σ} is an end-extension of H̃_{α}^{σ} ∩ H_{τ}, by which the rest follows.

The following lemma [DJK81, Lemma 2.3] (cf. also Lemma 4.4) will frequently be used.

7.9 Lemma Let C ∈V and f ∈ ∏ _{α∈C} α^{+} . Then there is a σ ∈I such that

{α ∈ C ∩ C_{σ} | f(α) < σ_{α}}∈V.

If {α ∈ κ | φ(α)} ∈ V, say that φ(α) holds for V-almost all α, or also ∀^{V}α (φ(α)).

Claim 1 ∀τ ∈I ∀^{V}α∈C_{τ} (K_{α}^{τ} is not weakly full).

Proof This proof is modelled after the proof of a similar claim in [Sch96, Lemma 5.6]. Pick some τ ∈I.

Case 1 κ not overlapped in W.

In this case, we have that ∀μ < κ (O^{W}(μ) < κ) and thus E_{κ}^{W} = ∅.

Case 1.1 ∀^{V}α∈C_{τ} (E_{α}^{W} = ∅).

Pick such an α. By elementarity of π_{α}^{τ} we have E_{α}^{K_{α}^{τ}} = ∅, and also E^{W} α = E^{K_{α}^{τ}} α. It follows from 5.9 that W ↓ min{α^{+W} , α^{+K_{α}^{τ}} } = K_{α}^{τ} ↓ min{α^{+W} , α^{+K_{α}^{τ}} }. We claim that α^{+K_{α}^{τ}} α^{+W} . Assume not. Then α^{+K_{α}^{τ}} > α^{+W} , and an initial segment of K_{α}^{τ}↓α^{+K_{α}^{τ}} is a collapsing mouse for W ↓α^{+W} . (Note that κ^{W↓α^{+W}} = α.) Since W is weakly full, this initial segment would be a subset of W, clearly a contradiction.

If α^{+K_{α}^{τ}} < α^{+W} , we can similarly deduce that K_{α}^{τ} is not weakly full: an initial segment of W ↓ α^{+W} is a collapsing mouse for K_{α}^{τ} ↓ α^{+K_{α}^{τ}} which cannot be contained in K_{α}^{τ}.

So assume that α^{+K_{α}^{τ}} = α^{+W} . This will lead to a contradiction. We can conclude P (α) ∩ W = P (α) ∩ K_{α}^{τ}. As κ is not overlapped in W, ν := O^{W}(α) < κ = π_{α}^{τ}(α). Thus we can define an extender F at α, ν on W: For a∈[ν]^{<ω} , x∈P ([α]\overline{\overline{a}}) ∩ W set

x∈F_{a} ↔ a∈π_{α}^{τ}(x).

Since H_{α}^{τ} is ω-complete and P (α) ∩ W = P (α) ∩ K_{α}^{τ}, the usual argument shows that F is countably complete: Let ⟨x_{n} | n ∈ ω⟩ be a sequence such that for n ∈ ω, x_{n} ∈ F_{a_{n}}, where a_{n} ∈ [ν]^{<ω} . We need to show that there is a function h : ⋃ _{n∈ω} a_{n} → α such that for all n ∈ ω, h′′a_{n} ∈ x_{n}. Let ξ = otp(⋃ _{n∈ω} a_{n}) < ω_{1}, and let f : ξ → ⋃ a_{n} be the monotone enumeration. Since P (α) ∩ W ⊆ K_{α}^{τ}, we have x_{n} ∈K_{α}^{τ}, too. And since H̃_{α}^{τ} is closed under ω-sequences, it follows that ⟨⟨x_{n}, f^{−1} ′′a_{n}⟩ | n ∈ ω⟩ ∈ H̃_{α}^{τ}. Notice that the function f is an element of H_{τ}: ξ < ω_{1} < κ and also sup(⋃ _{n∈ω} a_{n}) < κ, since ν < κ, a_{n} ⊆ ν, and cf(κ) = κ > ω. So f ∈ H_{κ} ⊆ H_{τ}. Also, ν < κ = π_{α}^{τ}(α), and a_{n} ∈π_{α}^{τ}(x_{n}) by the definition of F, so

H_{τ} ∃g (g : ξ → π_{α}^{τ}(α) ∧ g is orderpreserving ∧ ∀n∈ω (g′′(f^{−1} ′′a_{n})∈π_{α}^{τ}(x_{n})) ),

since f is a possible candidate. By elementarity, one concludes that

H̃_{α}^{τ} ∃g (g : ξ → α ∧ g is orderpreserving ∧ ∀n∈ω (g′′(f^{−1} ′′a_{n})∈x_{n}) ).

Pick any such g and set h := g∘f^{−1} . Then h : ⋃ _{n∈ω} a_{n} → α is orderpreserving and for n∈ω, h′′a_{n} ∈x_{n}.

To reach a contradiction, it now suffices to show that if j : W ↓ν →_{F} Ult(W ↓ ν, F) =: W̃, then W̃|ν = W ↓ν and ⟨W ↓ν, F ⟩ is weakly amenable. For then, by the definition of W, one concludes that E_{ν}^{W} = F, contradicting the fact that ν = O^{W}(α). For what follows, refer to Figure 7.2. Since W ↓ α^{+W} = K_{α}^{τ} ↓ α^{+K_{α}^{τ}} , we have W̃ ↓ j(α) = Ult(K_{α}^{τ} ↓ α^{+K_{α}^{τ}} , F) ↓ j(α). F was derived from π_{α}^{τ}, so there is a canonical map k : Ult(K_{α}^{τ} ↓ α^{+K_{α}^{τ}} , F) →_{Σ_{1}} W ↓ κ^{+W} such that k ∘ j (K_{α}^{τ} ↓ α^{+K_{α}^{τ}} ) = π_{α}^{τ} (K_{α}^{τ} ↓ α^{+K_{α}^{τ}} ) and crit(k) ν. But this now immediately implies that W̃|ν = W |ν = ⟨W ↓ ν, ∅⟩ and also that ⟨W ↓ ν, F ⟩ is weakly amenable, as desired. (Note that ν α^{+W} , so that P (α) ∩ W = P (α) ∩ W |ν = P (α) ∩ W̃|ν.)

Case 1.2 ∀^{V}α∈C_{τ} (E_{α}^{W} ≠ ∅).

For such α, E_{α}^{W} is an extender at some μ_{α}, α on W ↓α.

Case 1.2.1 ∀^{V}α∈C_{τ} (α < μ_{α}^{+W} ).

Notice that then α ∉ Card^{W} and κ^{K_{α}^{τ}↓α} = μ_{α}. Take γ_{α} minimal such that ωϱ_{W ∥γ_{α}}^{ω} μ_{α}, but P (μ_{α}) ∩ W ∥γ_{α} ⊆ W ↓ α = K_{α}^{τ} ↓ α. Then W ∥γ_{α} is a collapsing mouse for K_{α}^{τ} ↓α (κ^{K_{α}^{τ}↓α} = μ_{α} < α!). Notice γ_{α} < μ_{α}^{+W} κ = α^{+} . We claim that for V-almost all α, W ∥γ_{α} ⊈ K_{α}^{τ}. To see this, use Lemma 7.9 to find σ ∈ I such that τ ∈ rge(π_{α}^{σ}) and γ_{α} < σ_{α} for V-almost all α. Then

[figure: W̃ = Ult(W ↓ν, F); W; K_{α}^{τ}; Ult(K_{α}^{τ} ↓α^{+K_{α}^{τ}} , F); j(α); −; τ; −; ν = O^{W}(α); −; κ^{+}; −; α; −; α^{+W}; −; κ; τ_{α}; −; α; −; α^{+K_{α}^{τ}}; j(α); π_{α}^{τ}; k; Σ_{1}; j; F; F]

Figure 7.2: F is an extender at α, ν , so that j(α) ν and thus W̃|ν = W ↓ν.

we would have K_{α}^{σ} α ∉ Card, a contradiction. Thus in this case, too, we can conclude that K_{α}^{τ} is not weakly full (for V-almost all α). Note that we need to go up to σ as the assumption W ∥γ_{α} ⊆ K_{α}^{τ} does not already imply α∉Card^{K_{α}^{τ}} .

Case 1.2.2 ∀^{V}α∈C_{τ} (α μ_{α}^{+W} ).

Then O^{W}(μ_{α}) > α. Note that π_{α}^{τ}(μ_{α}) = μ_{α}. If μ_{α}^{+K_{α}^{τ}} α, then

μ_{α}^{+W} = π_{α}^{τ}(μ_{α})^{+K_{α}^{τ}} = π_{α}^{τ}(μ_{α}^{+K_{α}^{τ}} ) π_{α}^{τ}(α) = κ,

implying the obvious contradiction α κ. Thus we must have μ_{α}^{+K_{α}^{τ}} < α. Since K_{α}^{τ} ↓ α = W ↓ α, it follows that O^{K_{α}^{τ}} (μ_{α}) α. This in turn implies O^{W}(μ_{α}) π_{α}^{τ}(O^{K_{α}^{τ}} (μ_{α})) κ, contradicting the assumption of Case 1 that κ is not overlapped in W.

Case 2 κ overlapped in W.

Say, O^{W}(μ) κ, for some μ < κ.

Case 2.1 ∀^{V}α∈C_{τ} (P (α) ∩ W ⊆ K_{α}^{τ}).

Pick some such α and define U := {x ∈ P (α) ∩ K_{α}^{τ} | α ∈ π_{α}^{τ}(x)}. Let N := ⟨K_{α}^{τ}↓α^{+K_{α}^{τ}} , U⟩. Then

N “U is a non-trivial, α-complete ultrafilter on α”.

U is countably complete: Let ⟨x_{n} | n ∈ ω⟩ be a sequence of elements of U. As H_{α}^{τ} is ω-closed, ⟨x_{n} | n∈ω⟩∈H_{α}^{τ}. By the definition of U one has that

∀n∈ω (α∈π_{α}^{τ}(x_{n})),

whence

H_{κ^{+}} ∃β ∀n∈ω (β ∈π_{α}^{τ}(x_{n})),

and by elementarity of π_{α}^{τ}

H_{α}^{τ} ∃β ∀n∈ω (β ∈x_{n}),

so that ⋂ _{n∈ω} x_{n} ≠ ∅. Furthermore, N is amenable: Let x ∈ K_{α}^{τ} ↓ α^{+K_{α}^{τ}} be arbitrary. Pick some function f ∈K_{α}^{τ} mapping α onto x ∩ P (α) ∩ K_{α}^{τ}. Then

x ∩ U = {y ∈P (α) ∩ K_{α}^{τ} | y ∈x ∧ ∃ξ < α (α∈π_{α}^{τ}(f)(ξ)) }.

Since x ∩ P (α) ∩ K_{α}^{τ} ∈ W, x ∩ U ∈ W. Thus x ∩ U ∈ P (α) ∩ W ⊆ K_{α}^{τ}, and by acceptability of K_{α}^{τ}, x ∩ U ∈ K_{α}^{τ} ↓ α^{+K_{α}^{τ}} . Thus N is a neat p-premouse and L^{strong} exists by 5.14. This is a contradiction.

Case 2.2 ∀^{V}α∈C_{τ} (P (α) ∩ W ⊈ K_{α}^{τ}).

Case 2.2.1 μ^{+W} = κ.

In this case we get the required collapsing mice for K_{α}^{τ} fairly soon: By elementarity of π_{α}^{τ}, μ^{+K_{α}^{τ}} = α < κ = μ^{+W} . So let γ_{α} be minimal such that ωϱ_{W ∥γ_{α}}^{ω} μ, γ_{α} α. Note that κ^{K_{α}^{τ}↓α} = μ, and K_{α}^{τ} ↓ α = W ↓ α. By the minimality of γ_{α}, we have P (μ) ∩ W ∥γ_{α} ⊆ K_{α}^{τ} ↓α. Thus W ∥γ_{α} is a collapsing mouse for K_{α}^{τ} ↓ α. If W ∥γ_{α} were contained in K_{α}^{τ} then we would get a contradiction as in Case 1.2.1. Thus K_{α}^{τ} is not weakly full for V-almost all α.

Case 2.2.2 μ^{+W} < κ.

W.l.o.g. α > μ^{+W} , and so μ^{+K_{α}^{τ}} = π_{α}^{τ}(μ^{+K_{α}^{τ}} ) = μ^{+W} , P (μ) ∩W = P (μ) ∩K_{α}^{τ}, and O^{K_{α}^{τ}} (μ) α.

Case 2.2.2.1 O^{W}(μ) < κ^{+} .

We can assume w.l.o.g. that for all τ ∈I, O^{W}(μ) < τ. Thus ∀^{V}α∈C_{τ} (O^{K_{α}^{τ}} (μ) < τ_{α} ). Pick τ and α, and let ν_{α} := O^{K_{α}^{τ}} (μ) < τ_{α} < κ O^{W}(μ). Then by the Condensation Lemma [Koe89, Theorem 22.3] we have W ↓ν_{α} = K_{α}^{τ}|ν_{α}.

As in Case 1.1, we aim to derive a contradiction from defining an extender F at μ, η := O^{W}(μ) from (more or less) π_{α}^{τ}. However, π_{α}^{τ} has critical point α and π_{α}^{τ}(α) is equal to κ, which in turn is less than or equal to η, so the direct approach from Case 1.1 will fail. We will thus first take an ultrapower of W with the extender at μ, ν_{α} = O^{K_{α}^{τ}} (μ), with embedding i, and then lift π_{α}^{τ} to this ultrapower, giving some map j. The composition k of i and j will then suffice to define an appropriate extender F (see Figure 7.3).

[figure: W^{∗∗}; W^{∗} = Ult(W, E_{ν}^{W}); W; K_{α}^{τ}; ν^{+W^{∗}}; −; κ^{+}; −; ν = O^{K_{α}^{τ}} (μ); −; μ; −; μ^{+W}; −; α; −; κ; −; η = O^{W}(μ); −; τ; τ_{α}; −; μ^{+K_{α}^{τ}}; −; α; −; ν = O^{K_{α}^{τ}} (μ); π_{α}^{τ}; E_{ν}^{W}; E_{ν}^{W}; i; j = lifting of π_{α}^{τ}; k = j ∘ i; M = Ult(W ↓ν, E_{ν}^{W})]

Figure 7.3: Let i : W → W^{∗} = Ult(W, E_{O^{K^{τ}_{α}} (μ)}^{W}). Lift π_{α}^{τ} to W^{∗}, giving j, and let k := j ∘ i. Then define an extender F at μ, η from k and go for a contradiction.

Let M_{α} := Ult(W ↓ ν_{α}, E_{ν_{α}}^{W}). Then M_{α}|ν_{α} = W ↓ ν_{α} = K_{α}^{τ}|ν_{α}, and E_{ν_{α}}^{M_{α}} = E_{ν_{α}}^{K_{α}^{τ}} = ∅. Let η_{α} := min{ν_{α}^{+K_{α}^{τ}} , ν_{α}^{+M_{α}} }. Then by Corollary 5.9

M_{α} ↓η_{α} = K_{α}^{τ}↓η_{α}.

If ν_{α}^{+M_{α}} > ν_{α}^{+K_{α}^{τ}} , then an initial segment of M_{α} is a collapsing mouse omitted in K_{α}^{τ}, and K_{α}^{τ} is not weakly full, as claimed.

So assume ν_{α}^{+M_{α}} ν_{α}^{+K_{α}^{τ}} . Then, since M_{α} ↓ν_{α}^{+M_{α}} = K_{α}^{τ}↓ν_{α}^{+M_{α}} , P (ν_{α})∩ M_{α} ⊆ K_{α}^{τ}. This will lead to a contradiction. For the sake of legibility, denote ν_{α} by ν, M_{α} by M etc. Let i : W →_{E_{ν}^{W}} W^{∗}. Then W^{∗} ↓ ν = M ↓ ν and E_{ν}^{W^{∗}} = E_{ν}^{M} = ∅. In fact, the two ultrapowers agree up to ν^{+M} = ν^{+W^{∗}} , since ν is the index of the extender used, and P (μ) ∩ W ↓ν = P (μ) ∩ W.

Since ν^{+M} ν^{+K_{α}^{τ}} , we can thus construct the upward extension of π_{α}^{τ} (K_{α}^{τ} ↓ (ν^{+M} )) to W^{∗}, say j : W^{∗} → W^{∗∗}. Notice that ν^{+M} ∈ Card^{W^{∗}}. Let k := j ∘ i denote the composition, k : W → W^{∗∗}. Let η := O^{W}(μ). Then k(μ) = j(i(μ)) j(ν) = j(O^{K_{α}^{τ}} (μ)) = π_{α}^{τ}(O^{K_{α}^{τ}} (μ)) = O^{W}(μ) = η. So we can derive an extender F at μ, η on W by setting, for a∈[η]^{<ω} and x∈P ([μ]\overline{\overline{a}}),

x∈F_{a} ↔ a∈k(x).

Noting that P (μ) ∩ W = P (μ) ∩ K_{α}^{τ}, one sees that the same argument as in Case 1.1 shows that F must be countably complete. In fact, ⟨W ↓η, F ⟩ is weakly amenable, and if h : W ↓η →_{F} W̃ denotes the ultrapower map then W̃|η = W ↓η (see Figure 7.4).

[figure: W^{∗∗}; W̃ = Ult(W ↓η, F); W; K_{α}^{τ}; Ult(W ↓μ^{+W} , F); k(μ)^{+W^{∗∗}}; h(μ); −; κ^{+}; −; η = O^{W}(μ); −; μ^{+W}; −; α; −; κ; −; τ; τ_{α}; −; α; −; ν^{+K_{α}^{τ}}; h(μ); π_{α}^{τ}; k; h; F; F; Σ_{1}; g]

Figure 7.4: W̃|η = W ↓η.

We have W̃ ↓ h(μ) = Ult(W ↓ μ^{+W} , F) ↓ h(μ). Since F was derived from k, there is the canonical map g : Ult(W ↓μ^{+W} , F) →_{Σ_{1}} W^{∗∗}↓k(μ)^{+W^{∗∗}} such that g ∘ h (W ↓μ^{+W} ) = k and crit(g) η. Thus W̃↓η = W^{∗∗} ↓η. Furthermore, O^{W^{∗∗}}(μ) = j(O^{W^{∗}}(μ)) = j(ν) = j(O^{K_{α}^{τ}} (μ)) = π_{α}^{τ}(O^{K_{α}^{τ}} (μ)) = O^{W}(μ) = η, so E_{W} ; η = ∅. Also, j was the upward extension of π_{α}^{τ} (K_{α}^{τ} ↓ ν^{+K_{α}^{τ}} ), so W^{∗∗} is an extension of W ↓sup(π_{α}^{τ}′′ν^{+K_{α}^{τ}} ). Since sup(π_{α}^{τ}′′ν^{+K_{α}^{τ}} ) > π_{α}^{τ}(ν) = O^{W}(μ) = η, we get W̃|η = W^{∗∗}|η = W |η. Since η κ > μ^{+W} , we conclude that P (μ) ∩ W ⊆ W ↓η = W̃↓η, so ⟨W ↓η, F ⟩ is indeed weakly amenable.

But taking all this together, one concludes that by the definition of W one should have E_{η}^{W} = F, whereas, by the choice of η = O^{W}(μ), E_{η}^{W} = ∅ — contradiction.

Case 2.2.2.2 O^{W}(μ) = κ^{+} .

In this case, for all τ ∈ I, ∀^{V}α ∈ C_{τ} (O^{K_{α}^{τ}} (μ) = τ_{α} ). By the Condensation Lemma [Koe89, Theorem 22.3] we have K_{α}^{τ} = W ↓ τ_{α}. First we show κ to be inaccessible in W. Assume to the contrary that κ = λ^{+W} and pick for each α ∈ C_{τ} a surjection h_{α} : λ → α, h_{α} ∈ W. Then h_{α} ∈ W ↓ γ_{α} for some γ_{α} < κ = α^{+} . Now pick σ ∈ I such that, for V-almost all α, τ ∈ rge(π_{α}^{σ}) and γ_{α} < σ_{α}. Note that, since also ∀^{V}α ∈ C_{σ} (O^{K_{α}^{σ}} (μ) = σ_{α} ), K_{α}^{σ} = W ↓ σ_{α}, again by the Condensation Lemma. Then for such α we will have K_{α}^{σ} “α is not a cardinal”, obviously a contradiction.

Hence κ is inaccessible in W. Thus for each α < κ there is γ_{α} < κ = α^{+} such that P (α) ∩ W ∈ W ↓ γ_{α}. Using Lemma 7.9, find σ ∈ I such that, for V-almost all α, τ ∈ rge(π_{α}^{σ}) and σ_{α} > γ_{α}. Since we have full condensation it follows that P (α) ∩ W ∈K_{α}^{σ}. By elementarity, P (α) ∩ W ∈K_{α}^{τ}, too. But the assumption of Case 2.2 is that P (α) ∩ W ⊈ K_{α}^{τ}, contradiction.

This proves the claim. (Claim 1)

Assume w.l.o.g. that Claim 1 holds for all τ ∈I and all α∈C_{τ}. Note that since H_{α}^{τ} is closed under ω-sequences K_{α}^{τ} is neat and thus prestrong. If κ^{K_{α}^{τ}} < τ_{α}, then K_{α}^{τ} must have a collapsing mouse: otherwise it would be presolid, and since a presolid, prestrong mouse is solid, and since solid mice are weakly full, it would also be weakly full. Thus either κ^{K_{α}^{τ}} = τ_{α} or K_{α}^{τ} has a collapsing mouse.

Case A First, let us analyse the case that there is some τ ∈I such that for V-almost all α we have κ^{K_{α}^{τ}} < τ_{α} and hence K_{α}^{τ} has a collapsing mouse. By elementarity, we must have κ^{W} < κ^{+} and so, in fact, this is true for all σ. Let N_{α}^{τ} denote the collapsing mouse for K_{α}^{τ}.

Denote κ^{K_{α}^{τ}} by κ_{α}^{τ}. Then if α ∈ C_{τσ}, it follows that κ_{α}^{τ} = κ_{α}^{σ} by elementarity.

Claim 2 ∀τ ∈I (κ_{α}^{τ} is the largest cardinal in K_{α}^{τ}).

Proof We first show that by increasing τ, one can always increase the power set of κ_{α}^{τ}:

∀τ ∈I ∃σ ∈I ({α∈C_{τσ} | P (κ_{α}^{τ}) ∩ K_{α}^{τ} P (κ_{α}^{σ}) ∩ K_{α}^{σ}}∈V).

Assume to the contrary that τ is such that

∀σ ∈I, σ > τ ({α∈C_{τσ} | P (κ_{α}^{τ}) ∩ K_{α}^{τ} = P (κ_{α}^{σ}) ∩ K_{α}^{σ}}∈V).

But then for any σ > τ in I, if N_{α}^{σ} is a collapsing mouse for K_{α}^{σ}, it is also a collapsing mouse for K_{α}^{τ}. Since, by Lemma 5.8, collapsing mice are unique it follows that N_{α}^{σ} = N_{α}^{τ}. Pick σ ∈ I such that, for V-almost all α, σ_{α} > On∩N_{α}^{τ}. Notice that now On∩N_{α}^{σ} On∩K_{α}^{σ} = σ_{α} > On∩N_{α}^{τ}, contradicting N_{α}^{σ} = N_{α}^{τ}.

Assume that κ_{α}^{τ} is not the largest cardinal in K_{α}^{τ}. Pick some σ as above. Then κ_{α}^{τ} ^{+K_{α}^{τ}} exists, and by elementarity so does κ_{α}^{τ} ^{+K_{α}^{σ}} . But since P (κ_{α}^{τ}) ∩ K_{α}^{τ} P (κ_{α}^{τ}) ∩ K_{α}^{σ}, κ_{α}^{τ} ^{+K_{α}^{τ}} < κ_{α}^{τ} ^{+K_{α}^{σ}} , contradicting K_{α}^{τ} ≺ K_{α}^{σ}. (Claim 2)

Claim 3 κ^{+} is a successor cardinal in W.

Proof This is immediate by the elementarity of π_{α}^{τ}. (Claim 3)

So we have κ^{+} = λ^{+W} . For α ∈ C_{τ}, let λ_{α}^{τ} = π_{α}^{τ} ^{−1} (λ). Then λ_{α}^{τ} = κ_{α}^{τ} is the largest cardinal in K_{α}^{τ}.

Claim 4 There is a set C ∈ V and a sequence ⟨M_{α} | α ∈ C⟩ of mice of size less than κ such that for all τ ∈ I, and for V-almost all α, if M_{α} and K_{α}^{τ} coiterate to respectively M_{α}^{∗} and K_{α}^{τ} ^{∗}, then K_{α}^{τ} ^{∗} ⊆ M_{α}^{∗} and the coiteration is simple on the K-side.

The point of this claim is that the sequence ⟨M_{α} | α ∈ C⟩ is independent of τ.

Proof To prove the claim, we distinguish several different cases.

Case 1 κ is not overlapped, i. e., ∀μ < κ (O^{W}(μ) < κ).

Case 1.1 κ is a successor in W, say κ = ϑ^{+W} .

Then we must have ō := sup{O^{W}(μ) | μ < κ} < κ: Assume contrariwise that ō = κ. Then the measurables μ have to be cofinal in κ, since any measurable μ is not overlapped, i. e., μ > O^{W}(μ′) for μ′< μ. But we assumed κ to be a successor cardinal in W — contradiction.

Let C := (max{ō, ϑ}, κ) ∈ V. Then for any α∈C there is a minimal γ_{α} such that, setting M_{α} := W ↓γ_{α}, we have

M_{α} ∃f_{α} (f_{α} : ϑ → α onto).

Pick some arbitrary τ ∈ I, and let α ∈ C_{τ} ∩ C. Then E_{κ}^{W} = ∅ and thus E_{α}^{K_{α}^{τ}} = ∅, too. Coiterate M_{α} and K_{α}^{τ} to get respectively M_{α}^{∗} and K_{α}^{τ} ^{∗}. The iteration is obviously beyond α on both sides. Since α is a cardinal in K_{α}^{τ} and E_{α}^{K_{α}^{τ}} = ∅, the coiteration is, in fact, above α on the K-side. Let F := E_{ν}^{M_{α}} be the first extender used on the M-side, let μ = crit(F). If μ were less than ϑ, then we would have ν α > ϑ μ^{+W} , so F would be a full extender on W and ν < O^{W}(μ) ō < α, contradicting the fact that the iteration is beyond α. Thus μ ϑ, i. e., the iteration is above ϑ.

Assume that the coiteration is not simple on the K-side. Then it must be simple on the M-side and M_{α}^{∗} ⊆ K_{α}^{τ} ^{∗}. Assume that K_{α}^{τ} ^{∗} ⊈ M_{α}^{∗}. Then M_{α}^{∗}∈K_{α}^{τ} ^{∗}, and again, the coiteration must be simple on the M-side. In both cases, one can conclude that

P (ϑ) ∩ M_{α} = P (ϑ) ∩ M_{α}^{∗}
⊆ P (ϑ) ∩ K_{α}^{τ} ^{∗} ⊆ P (ϑ) ∩ K_{α}^{τ}.

But this implies K_{α}^{τ} α ∉ Card, a contradiction. Thus the coiteration must be simple on the K-side and K_{α}^{τ} ^{∗} ⊆ M_{α}^{∗}, as claimed.

Case 1.2 κ is a limit cardinal in W.

In this case set C := Card^{W} ∩κ∈V. For α∈C, there exists γ_{α} < κ such that P (α) ∩ W ∈ W ↓γ_{α} =: M_{α}.

Let τ ∈ I and pick α ∈ C_{τ} ∩ C. Again we must have E_{κ}^{W} = ∅, and thus E_{α}^{K_{α}^{τ}} = ∅. As before, the coiteration is thus above (and not only beyond) α on the K-side.

In fact, the coiteration will also be above α on the M-side for V-almost all α. Otherwise, some E_{ξ}^{M_{α}} with critical point less than α and index ξ α will be used. By coherency, and since α∈Card^{W}, E_{α}^{M_{α}} = E_{α}^{W} is also an extender, with the same critical point μ_{α}, say. Then the function mapping α to μ_{α} is regressive and monotone increasing on a set in V, hence constant on a set in V. But then one concludes that for some μ < κ, O^{W}(μ) κ, contradicting the assumption that κ is not overlapped.

So assume w.l.o.g. that the coiteration is above α for all α ∈ C_{τ} ∩ C, and pick any such α. Let K_{α}^{τ} ^{∗} and M_{α}^{∗} be the coiterates of respectively K_{α}^{τ} and M_{α}. As in the first case, if either the coiteration is not simple on the K-side or if K_{α}^{τ} ^{∗} ⊈ M_{α}^{∗}, then the coiteration must be simple on the M-side and M_{α}^{∗} ⊆ K_{α}^{τ} ^{∗}. Then

P (α) ∩ W = P (α) ∩ M_{α} = P (α) ∩ M_{α}^{∗}
⊆ P (α) ∩ K_{α}^{τ} ^{∗} ⊆ P (α) ∩ K_{α}^{τ}.

Note that, since κ is not overlapped in W, O^{W}(α) < κ = π_{α}^{τ}(α). Thus one can define an extender at α, O^{W}(α) on W as in Case 1.1 of Claim 1, and reach exactly the same contradiction as there.

Hence, for V-almost all α, the coiteration is simple on the K-side and K_{α}^{τ} ^{∗} ⊆ M_{α}^{∗}.

Case 2 κ is overlapped in W

Then there is some μ < κ such that O^{W}(μ) κ. If κ were a limit cardinal in W then, since κ is also regular (in V and hence in W), we would get W ↓κ ZFC and thus get a set model for a strong cardinal.

So κ is a successor in W, say κ = ϑ^{+W} , as in Case 1.1. We can assume that for all τ ∈I and all α∈C_{τ}, α > ϑ.

Case 2.1 μ = ϑ.

Let C := (ϑ, κ) ∈ V. As in Case 1.1, for any α∈C there is a minimal γ_{α} such that, setting M_{α} := W ↓γ_{α}, we have

M_{α} ∃f_{α} (f_{α} : ϑ → α onto).

Let τ ∈ I, α ∈ C_{τ} ∩ C, and coiterate K_{α}^{τ} and M_{α} to get K_{α}^{τ} ^{∗} and M_{α}^{∗}. Then the coiteration is beyond α and thus above ϑ on both sides. Now conclude, literally as in Case 1.1, that the coiteration must be simple on the K-side and K_{α}^{τ} ^{∗} ⊆ M_{α}^{∗}, as claimed.

Case 2.2 μ < ϑ.

In this case we must have O^{W}(μ) < κ^{+W} . Otherwise, pick the γ_{α} as in Case 2.1. If O^{W}(μ) κ^{+W} , then, for any τ ∈ I and any α ∈ C_{τ}, α > ϑ, one has O^{K_{α}^{τ}} (μ) = τ_{α}, and by the Condensation Lemma K_{α}^{τ} = W ↓ τ_{α}. If τ is now chosen so that for V-almost all α, τ_{α} γ_{α}, one gets the contradiction that there is, in K_{α}^{τ}, a function f_{α} mapping ϑ onto α.

Let τ̄ := min(I), C := Cτ̄. For α ∈ C, let ν_{α} := O^{Kτ̄ ; α} (μ), and set M_{α} := Ult(W |ν_{α}, E_{ν_{α}}^{W}).

Let τ ∈I, and let α∈C_{τ̄τ} ∩ C. Then, since O^{Kτ̄ ; α} (μ) = ν_{α} < τ̄_{α} and Kτ̄ ; α ≺ K_{α}^{τ}, O^{K_{α}^{τ}} (μ) = ν_{α}, and by the Condensation Lemma K_{α}^{τ}|ν_{α} = W ↓ν_{α}, so that also M_{α}|ν_{α} = K_{α}^{τ}|ν_{α}.

The assumption that P (ν_{α}) ∩ M_{α} ⊆ P (ν_{α}) ∩ K_{α}^{τ} leads to a contradiction, literally as in the second part of Case 2.2.2.1 of the proof of Claim 1. All of the relevant assumptions of that case hold here, too.

So we can conclude that P (ν_{α}) ∩ M_{α} ⊈ P (ν_{α}) ∩ K_{α}^{τ}. Coiterate K_{α}^{τ} and M_{α}, yielding respectively M_{α}^{∗} and K_{α}^{τ} ^{∗}. Notice that the coiteration is above ν_{α} on both sides: The coiteration is above μ and beyond ν_{α}, and ν_{α} = O^{M_{α}} (μ) = O^{K_{α}^{τ}} (μ) — we are assuming that there are no overlapping extender sequences, so ν_{α} cannot be a critical point on either side.

Assume that the K-side of the coiteration is non-simple or that K_{α}^{τ} ^{∗} ⊈ M_{α}^{∗}. Then the M-side must be simple and M_{α}^{∗} ⊆ K_{α}^{τ} ^{∗}. But then

P (ν_{α}) ∩ M_{α} = P (ν_{α}) ∩ M_{α}^{∗}
⊆ P (ν_{α}) ∩ K_{α}^{τ} ^{∗} ⊆ P (ν_{α}) ∩ K_{α}^{τ},

which we concluded not to hold. So if P (ν_{α}) ∩ M_{α} ⊈ P (ν_{α}) ∩ K_{α}^{τ}, then the coiteration must be simple on the K-side and K_{α}^{τ} ^{∗} ⊆ M_{α}^{∗}, as desired.

(Claim 4)

Take the set C ∈ V and the sequence ⟨M_{α} | α ∈ C⟩ guaranteed by Claim 4. For τ ∈ I, assume w.l.o.g. that the conclusion of that claim holds for all α ∈ C_{τ}. Denote the coiteration of K_{α}^{τ} and M_{α} by respectively I_{α}^{τ} and J_{α}^{τ}, and let β_{α}^{τ} be the length of this coiteration and let i_{α}^{τ} ^{(ξ,ζ)}, j_{α}^{τ} ^{(ξ,ζ)} denote the iteration maps, for ξ ζ β_{α}^{τ}. Let μ_{α}^{τ} ^{(ξ)} be the critical point at stage ξ < β_{α}^{τ} (cf. Figure 7.5). Recall that λ_{α}^{τ} = π_{α}^{τ} ^{−1} (λ) is the largest cardinal of K_{α}^{τ}, and κ = λ^{+W} .

[figure: −; −; λ_{α}^{τ}; −; α; I_{α}^{τ}; −; K_{α}^{τ}; −; −; _{K_{α}^{τ}_{(ξ)}}; …; −; −; _{K_{α}^{τ}_{(ζ)}}; …; ^{i_{α}^{τ} ^{(ξ,ζ)}}; −; −; _{K_{α}^{τ}_{(β)}}; …; −; −; α; J_{α}^{τ}; −; M_{α}; −; −; M_{α}^{(ξ)}; …; −; −; M_{α}^{(ζ)}; …; ^{j_{α}^{τ} ^{(ξ,ζ)}}; −; ^{μ_{α}^{τ} ^{(ξ)}}; −; −; M_{α}^{(β)}; −; …; =]

Figure 7.5: The coiteration of M_{α} and K_{α}^{τ}, as guaranteed by Claim 4. I_{α}^{τ} is a simple iteration, no truncations are performed.

Claim 5 There is a τ ∈ I such that for V-almost all α, there is ξ < β_{α}^{τ} such that μ_{α}^{τ} ^{(ξ)} i_{α}^{τ} ^{(0,ξ)}(λ_{α}^{τ}).

Proof First, pick some τ ∈I and let α∈C_{τ}. To enhance legibility, we drop the indices τ and α occasionally. E. g., let I, J be the coiteration of K_{α}^{τ} and M_{α}, and let μ^{(ξ)} be the critical point at stage ξ, where ξ < β = β_{α}^{τ}, the length of the coiteration. Since K_{α}^{τ} ^{(β)} ⊆ M_{α}^{(β)}, it follows that E^{M_{α}^{(β)}} δ_{α} = E^{K_{α}^{τ} ^{(β)}} , where δ_{α} := On∩K_{α}^{τ} ^{(β)}. Let λ_{α}^{τ} ^{(β)} := i^{(0,β)}(λ_{α}^{τ}). Then λ_{α}^{τ} ^{(β)} is the largest cardinal in K_{α}^{τ} ^{(β)}.

We can assume that for ξ < β, μ^{(ξ)} < λ_{α}^{τ} ^{(ξ)}. As in all coiterations, β < max(cardK_{α}^{τ}, cardM_{α})^{+} = κ. Thus there exists σ ∈I such that for V-almost all α∈C_{τσ}, σ_{α} > On∩M_{α}^{(β)} (cf. Figure 7.6).

For such α, consider next the coiteration of K_{α}^{σ} and M_{α}, denoted by I′, J′ etc. Then I is an initial segment of I′. This is proved by a straightforward induction on ξ β = length of the iteration I, using the fact that the coiterations are simple on K-side in both cases, so that no truncations have to be taken into account there. For the successor step, use [Koe89, Lemma 14.2], which applies here as K_{α}^{τ} ≺ K_{α}^{σ}.

Then On∩K_{α}^{σ} ^{(β)} σ_{α} > On∩M_{α}^{(β)}, so β′> β. But actually nothing happens on the K-side: Assume for a contradiction that some extender F is used to build an ultrapower of K_{α}^{σ} ^{(β)}. This extender would have to have index at least δ_{α} = On∩K_{α}^{τ} ^{(β)}. As λ_{α}^{τ} ^{(β)} is the largest cardinal in K_{α}^{τ} ^{(β)} and in K_{α}^{σ} ^{(β)} and no truncations are allowed on the K-side, the critical point of F would have to be less than λ_{α}^{τ} ^{(β)}. So by coherency K_{α}^{τ} ^{(β)} has extenders with this critical point all the way up to δ_{α}. Then by elementarity, K_{α}^{τ} has extenders with some critical point μ < λ_{α}^{τ}, all the way up to τ_{α}. Thus by the definition of κ_{α}^{τ} = κ^{K_{α}^{τ}} , one gets κ_{α}^{τ} = μ < λ_{α}^{τ}, contradicting Claim 2.

Effectively, the coiteration on the K-side halts, and all that happens is that M_{α}^{(β)} is iterated till we have agreement. The index ν^{(β)} of the extender used in the next step has to be at least δ_{α}, of course.

If the coiteration of K_{α}^{σ} and M_{α} terminates after the (β + 1)-th step, find ϱ ∈ I to make sure that the coiteration of K_{α}^{ϱ} and M_{α} goes at least one step further, by the same argument that gave σ in the first place. Assume w.l.o.g. that already the coiteration of K_{α}^{σ} and M_{α} takes at least (β +2)-many steps. But again, nothing happens on the K-side after the β-th step, and so i^{(0,β+1)} (λ_{α}^{σ}) = i^{(0,β)}(λ_{α}^{τ}). Fine coiterations are normal, so μ^{(β+1)} > ν^{(β)} δ_{α} > i^{(0,β)}(λ_{α}^{τ}) = i^{(0,β+1)} (λ_{α}^{σ}), so σ and β + 1 are as desired. (Claim 5)

[figure: −; K_{α}^{σ}; −; σ_{α}; −; λ_{α}^{τ}; −; τ_{α}; −; −; _{K_{α}^{σ} _{(β)}}; −; δ_{α}; −; λ_{α}^{τ} ^{(β)} = λ_{α}^{σ} ^{(β)}; −; −; M_{α}^{(β)} ∩ On; −; _{K_{α}^{σ} _{(β+1)}}; −; −; ^{λ_{α}^{σ} ^{(β+1)}}; −; _{K_{α}^{σ} _{(β′)}}; −; ^{i_{α}^{τ} ^{(0,β)}}; ^{i_{α}^{σ} ^{(0,β)}}; I_{α}^{τ} = I_{α}^{σ} β; =; =; −; M_{α}; −; −; M_{α}^{(β)}; −; δ_{α}; −; ^{λ_{α}^{τ} ^{(β)}}; −; μ^{(β)}; −; −; ν^{(β)}; −; M_{α}^{(β+1)}; −; −; μ^{(β+1)}; −; j_{α}^{τ} ^{(β,β+1)} (μ^{(β)}); −; ^{λ_{α}^{σ} ^{(β+1)}}; −; M_{α}^{(β′)}; −; ^{K_{α}^{σ} ^{(β′)}}; −; −; K_{α}^{τ}; −; ^{j_{α}^{σ} ^{(0,β)}}; J_{α}^{τ} = J_{α}^{σ} β; ^{j_{α}^{σ} ^{(β,β+1)}}; ^{i_{α}^{τ} ^{(0,β)}}]

Figure 7.6: By choosing an appropriate σ one can ensure that at some point of the coiteration of K_{α}^{σ} and M_{α} the critical point μ^{(ξ)} is greater than the largest cardinal λ_{α}^{σ} ^{(ξ)} of K_{α}^{σ} ^{(ξ)}. It happens here at ξ = β + 1 (see the dotted circles), where β is the length of the coiteration of K_{α}^{τ} and M_{α}, and where σ was chosen such that σ_{α} is greater than M_{α}^{(β)} ∩ On. β′ denotes the length of the coiteration of K_{α}^{σ} and M_{α}.

This now quickly leads to a contradiction: μ^{(ξ)} is a cardinal in M_{α}^{(β)}. By choosing a suitable σ, one can arrange that On∩K_{α}^{σ} ^{(ξ)} > μ^{(ξ)}, and hence K_{α}^{σ} ^{(ξ)} (and so K_{α}^{σ} ^{(β)}) thinks μ^{(ξ)} has cardinality λ_{α}^{σ} ^{(ξ)} < μ^{(ξ)}. This contradicts K_{α}^{σ} ^{(β)} ⊆ M_{α}^{(β)} (for V-almost all α).

This concludes the discussion of Case A. (Case A)

Case B Recall that we had proved in Claim 1 that w.l.o.g. for all τ ∈I and all α ∈ C_{τ}, K_{α}^{τ} is not weakly full. We now want to lead the assumption that κ^{K_{α}^{τ}} = τ_{α} to a contradiction. As before, by elementarity, if this is true for one τ ∈ I and V-almost all α ∈ C_{τ}, then it is true for all σ ∈ I (and V-almost all α∈C_{σ}). As before, K_{α}^{τ} is neat and thus prestrong. As K_{α}^{τ} is not weakly full, it cannot be strong, since strong mice are weakly full by [Sch96, Lemma 5.9]. Thus there exists a mouse M_{α}^{τ} end-extending K_{α}^{τ} such that K_{α}^{τ} ⊈ core(M_{α}^{τ}). We aim to show that, given some α, many of these core mice for varying τ are identical. This allows us to conclude that the iteration from this core to some suitable M_{α}^{τ} must be long (in the sense of Definition 6.1). This in turn is necessary for the Gitik Game to go through, which will be be final step towards the desired contradiction.

We have to proceed with some care in the choice of M_{α}^{τ}.

Claim 6 For each τ ∈I and each α∈C_{τ}, there is a mouse M_{α}^{τ} end-extending K_{α}^{τ} such that K_{α}^{τ} ⊈ M̄_{α}^{τ} := core(M_{α}^{τ}), P(ϱ_{M_{α}^{τ}}^{ω} ) ∩ M_{α}^{τ} ⊆ K_{α}^{τ} and such that for no intermediate stage M′ in the iteration from M̄_{α}^{τ} to M_{α}^{τ} does K_{α}^{τ} ⊆ M′. Let ϱ_{α}^{τ} denote ϱ_{M_{α}^{τ}}^{ω}.

Proof Let M be any mouse end-extending K_{α}^{τ} such that K_{α}^{τ} ⊈ core(M) =: M̄. Such M exists as K_{α}^{τ} is not weakly full. We want to arrange

P (ϱ_{M}^{ω}) ∩ M ⊆ K_{α}^{τ}.

Note that ϱ_{M}^{ω} < τ_{α}. Otherwise, since the iteration from M̄ to M is simple and above ϱ_{M}^{ω}, we would have K_{α}^{τ} ⊆ M ↓τ_{α} = M̄ ↓τ_{α} ⊆ M̄, a contradiction. Let

η := max{ξ On∩M | P (ϱ_{M}^{ω}) ∩ M ↓ξ ⊆ K_{α}^{τ}}.

Let N := M ↓η. Then surely N is a mouse and K_{α}^{τ} ⊆ N. Note that ϱ_{N}^{ω} ϱ_{M}^{ω} by the choice of η. Thus P(ϱ_{N}^{ω}) ∩ N ⊆ K_{α}^{τ}.

If K_{α}^{τ} ⊈ core(N), then we have a good candidate for M_{α}^{τ}: Let M_{α}^{τ} be the first stage of the iteration from core(N) to N such that K_{α}^{τ} ⊆ M_{α}^{τ}. i. e., if N^{(0)} = core(N), N^{(ϑ)} = N ⊇ K_{α}^{τ}, then let ξ ϑ be minimal such that N^{(ξ)} ⊇ K_{α}^{τ} and set M_{α}^{τ} := N^{(ξ)}. Since the iteration is simple and above ϱ_{N}^{ω}, one has ϱ_{M_{α}^{τ}}^{ω} = ϱ_{N}^{ω} and P(ϱ_{M_{α}^{τ}}^{ω} ) ∩M_{α}^{τ} = P (ϱ_{N}^{ω})∩N ⊆ K_{α}^{τ}, and K_{α}^{τ} ⊈ core(M_{α}^{τ}) =: M̄_{α}^{τ}.

Otherwise, K_{α}^{τ} ⊆ core(N). Note that this implies N ≠ M and so η < On∩M. We first claim that ϱ_{N}^{ω} = ϱ_{M}^{ω}: If ϱ_{N}^{ω} < ϱ_{M}^{ω}, then pick some a∈P (ϱ_{N}^{ω})∩Σ^{∗}(N)\ N. As η < On∩M, we conclude

a∈P (ϱ_{N}^{ω}) ∩ M = P (ϱ_{N}^{ω}) ∩ M̄
⊆ M̄ ↓(ϱ_{N}^{ω})^{+M̄} ⊆ M̄ ↓ϱ_{M}^{ω} = M ↓ϱ_{M}^{ω} = K_{α}^{τ}↓ϱ_{M}^{ω} ⊆ N,

a contradiction. (This uses the fact that (ϱ_{N}^{ω})^{+M̄} ϱω ; M̄ = ϱ_{M}^{ω} and that the iteration from M̄ to M is simple and above ϱ_{M}^{ω}.)

Next we claim that η = On∩M, which immediately gives a contradiction. Otherwise, i. e., if η < On∩M, as before pick some a ∈ P (ϱ) ∩ Σ^{∗}(N) \ N, where ϱ denotes ϱ_{N}^{ω} = ϱ_{M}^{ω}. As in the previous paragraph, a∈M̄. Thus either a∈M̄ ↓ϱ^{+M̄} or ϱ is the largest cardinal of M̄. In this latter case, since M is a simple iterate of M̄ above ϱ, we must have M̄ = M, contradicting K_{α}^{τ} ⊆ M and K_{α}^{τ} ⊈ M̄. But the former case also leads to a contradiction: Again as M is a simple iterate of M̄ above ϱ, M and M̄ agree up to ϱ^{+M} = ϱ^{+M̄} . Thus a ∈ M ↓ ϱ^{+M} . Now if ϱ^{+M} τ_{α}, then a ∈ K_{α}^{τ} ⊆ N, contradiction. But if ϱ^{+M} > τ_{α}, then K_{α}^{τ} ⊆ M ↓ϱ^{+M} = M̄ ↓ϱ^{+M} , also a contradiction. (Claim 6)

Claim 7 Let τ < σ both be elements of I and let α∈C_{τσ}. Then either

M̄_{α}^{σ} ^{∗} M̄_{α}^{τ} or ϱ_{α}^{σ} < ϱ_{α}^{τ},

where ^{∗} denotes the canonical pre-well-ordering of mice.

Proof Assume that ϱ_{α}^{σ} ϱ_{α}^{τ}. We first claim that the coiteration of M̄_{α}^{τ} and M̄_{α}^{σ} is above ϱ := min{ϱ_{α}^{σ}, ϱ_{α}^{τ}} = ϱ_{α}^{τ}. Otherwise, since coiterations are normal, there is a first disagreement between M̄_{α}^{τ} and M̄_{α}^{σ} at some μ, ξ, where μ < ϱ.

Say EM̄_{α}^{τ} ; ξ is at μ, ξ. Since I_{M̄_{α}^{τ} ,M_{α}^{τ}} is above ϱ_{α}^{τ} = ϱ, it follows that E_{ξ}^{M_{α}^{τ}} = EM̄_{α}^{τ} ; ξ. If ξ < τ_{α}, then

EM̄_{α}^{τ} ; ξ = E_{ξ}^{K_{α}^{τ}} = E_{ξ}^{K_{α}^{σ}} = E_{ξ}^{M_{α}^{σ}} = EM̄_{α}^{σ} ; ξ ,

where the last equality holds as I_{M̄_{α}^{σ} ,M_{α}^{σ}} is above ϱ_{α}^{σ} ϱ. But this is a contradiction. So ξ τ_{α}. But then EM̄_{α}^{τ} τ_{α} = E^{M_{α}^{τ}} τ_{α} = E^{K_{α}^{τ}} , contradicting the choice of M_{α}^{τ}.

Thus EM̄_{α}^{τ} ; ξ = ∅, and EM̄_{α}^{σ} ; ξ is at μ, ξ. As before, EM̄_{α}^{σ} ; ξ = E_{ξ}^{M_{α}^{σ}} , EM̄_{α}^{τ} ; ξ = E_{ξ}^{M_{α}^{τ}} and ξ

must be greater than τ_{α}. By the argument of the previous paragraph, ξ must be less than σ_{α}. Thus E_{ξ}^{K_{α}^{σ}} is at μ, ξ, and hence by coherence E_{τ_{α}}^{K_{α}^{σ}} is at μ, τ_{α}, and E^{K_{α}^{τ}} τ_{α} = E^{K_{α}^{σ}} τ_{α} (recall that K_{α}^{τ} does not have a largest cardinal, so μ^{+K_{α}^{τ}} = μ^{+K_{α}^{σ}} exists and is less than τ_{α}). But this implies κ_{α}^{τ} = κ^{K_{α}^{τ}} = μ < τ_{α}, whereas the general assumption of Case B is that κ_{α}^{τ} = τ_{α}.

Thus the coiteration of M̄_{α}^{τ} and M̄_{α}^{σ} is above ϱ_{α}^{τ} ϱ_{α}^{σ}. Let M̃_{α}^{τ} and M̃_{α}^{σ} be the comparable coiterates and assume M̃_{α}^{τ} ∈M̃_{α}^{σ}. Then

P (ϱ_{α}^{τ}) ∩ Σ^{∗}(M_{α}^{τ}) = P (ϱ_{α}^{τ}) ∩ Σ^{∗}(M̄_{α}^{τ})  as I_{M_{α}^{τ} ,M̄_{α}^{τ}} is simple above ϱ_{α}^{τ}
= P (ϱ_{α}^{τ}) ∩ Σ^{∗}(M̃_{α}^{τ})  as I_{M̄_{α}^{τ} ,_{M_{α}^{τ}}} is simple above ϱ_{α}^{τ}
⊆ P (ϱ_{α}^{τ}) ∩ M̃_{α}^{σ}  as M̃_{α}^{τ} ∈M̃_{α}^{σ}
⊆ P (ϱ_{α}^{τ}) ∩ M̄_{α}^{σ}  as I_{M̄_{α}^{σ} ,_{M_{α}^{σ}}} is above ϱ_{α}^{σ} ϱ_{α}^{τ}
= P (ϱ_{α}^{τ}) ∩ M_{α}^{σ}  as I_{M̄_{α}^{σ} ,M_{α}^{σ}} is simple above ϱ_{α}^{σ}
= P (ϱ_{α}^{τ}) ∩ K_{α}^{σ}  as P (ϱ_{α}^{σ}) ∩ M_{α}^{σ} ⊆ K_{α}^{σ}
= P (ϱ_{α}^{τ}) ∩ K_{α}^{τ}
⊆ M_{α}^{τ},

a contradiction! The last equation holds since K_{α}^{τ} has no largest cardinal: Let ξ < τ_{α}. Then ξ^{+K_{α}^{τ}} exists and is less than τ_{α}, so P (ξ)∩ K_{α}^{τ} ∈K_{α}^{τ} and as K_{α}^{τ} ≺ K_{α}^{σ}, P (ξ)∩ K_{α}^{τ} = P (ξ)∩ K_{α}^{σ}.

Thus the assumption M̃_{α}^{τ} ∈ M̃_{α}^{σ} was wrong and M̃_{α}^{σ} ⊆ M̃_{α}^{τ} after all, i. e., M̄_{α}^{σ} ^{∗} M̄_{α}^{τ}. (Claim 7)

We are now in a position to prove that all the prerequisites of the Gitik Game are satisfied by some K_{α}^{σ}.

Claim 8 There exists a σ ∈ I and an α ∈ C_{σ} (in fact, unboundedly many such α) such that the iteration I_{M̄_{α}^{σ} ,M_{α}^{σ}} from M̄_{α}^{σ} to M_{α}^{σ} is not short (in the sense of Definition 6.1).

Proof We will inductively choose a κ-long sequence of elements of I, corresponding sets in V and functions from ^{κ}κ.

Let τ^{0} be arbitrary, and set A^{0} := C_{τ^{0}} ∈V, f^{0}∈^{κ}κ, where

f^{0}(α) = \{ \begin{matrix}max\{s^{(ω_{1})}(M̄_{α}^{τ^{0}}), τ_{α}^{0}\} + 1 & if α∈A^{0}, \\ 0 & else. \end{matrix}

By Lemma 6.2, f^{0} really takes values less than κ.

Assume ⟨τ^{i} | i < ζ⟩ had been chosen. Define h^{ζ}∈^{κ}κ by setting

h^{ζ}(α) = sup{f^{i}(α) + 1 | i < α ∩ ζ}.

Then since inductively all the f^{i} take values less than κ, so does h^{ζ}. Using Lemma 7.9 find τ^{ζ} ∈ I such that τ^{ζ} > sup{τ^{i} | i < ζ} and such that for V-almost all α τ_{α}^{ζ} > h^{ζ}(α). Set A^{ζ} := {α∈C_{τ^{ζ}} | τ^{ζ} > h^{ζ}(α)}∈V and let

f^{ζ}(α) = \{ \begin{matrix}max\{s^{(ω_{1})}(M̄_{α}^{τ^{ζ}}), τ_{α}^{ζ}\} + 1 & if α∈A^{ζ}, \\ 0 & else. \end{matrix}

Since V is not (ϱ, κ)-regular and ϱ ω_{1}, it is not weakly (ω, κ)-regular by Lemma 7.3. Thus ⟨A^{i} | i < κ⟩ is not a weak regularity sequence, and so there are indices ⟨i_{j} | j ∈ω⟩ such that ⋂ _{j∈ω} A^{i_{j}} is unbounded in κ.

Recall that C_{τσ} is a final segment of C_{σ}. So let β_{τσ} be (least) such that

∀α > β_{τσ} (α∈C_{τ} ∩ C_{σ} → α∈C_{τσ}).

Let, for j < k < ω, β_{jk} := β_{τ^{i_{j}} τ^{i_{k}}} and set

ᾱ := sup{i_{j} | j < ω} ∪ sup{β_{jk} | j < k < ω} < κ.

Now pick some α∈ ( ⋂ _{j∈ω} A^{i_{j}} ) \ ᾱ.

Let j < k < ω. Then α > i_{k}, so h^{i_{j}} (α) > f^{i_{k}} (α) > max{s^{(ω_{1})}(M̄_{α}^{τ^{i_{k}}} ), τ_{α}^{i_{k}} }, as also α∈A^{i_{k}}. Thus

τ_{α}^{i_{j}} > h^{i_{j}} (α) > τ_{α}^{i_{k}} ∪ s^{(ω_{1})}(M̄_{α}^{τ^{i_{k}}} ).

and so by Claim 7

∀j < k < ω ( \underbrace{M̄_{α}^{τ^{i_{j}}} = M̄_{α}^{τ^{i_{k}}}}_{(A)} or \underbrace{M̄_{α}^{τ^{i_{j}}} >^{∗} M̄_{α}^{τ^{i_{k}}}}_{(B)} or \underbrace{ϱ_{α}^{τ^{i_{j}}} > ϱ_{α}^{τ^{i_{k}}}}_{(C)}).

By Ramsey’s Theorem, there must be a homogeneous set J ⊆ ω of size ω such that one of (A), (B), (C) holds for all j < k ∈ J. Obviously, if J were homogeneous of “type (C)”, then we would get an infinite descending sequence of ordinals. Similarly, a homogeneous set of “type (B)” would give an infinite descending sequence of mice, also a contradiction. Thus there must be ω-many indices τ^{i} that share the same core mouse M̄. In fact, we are only interested in two of them. I. e., we have τ, σ ∈I such that

On∩M_{α}^{σ} σ_{α} > h^{σ}(α) > max{s^{(ω_{1})}(M̄_{α}^{τ}), τ_{α}}.

and M̄_{α}^{σ} = M̄_{α}^{τ}. But then the iteration I_{M̄_{α}^{σ} ,M_{α}^{σ}} from M̄_{α}^{σ} to M_{α}^{σ} cannot be short. (Claim 8)

We are now ready to play the Gitik Game. Our presentation follows closely that of [Sch96, Chapter 6].

Pick σ as in Claim 8 and some α such that the iteration I_{M̄,M} from M̄ := M̄_{α}^{σ} to M := M_{α}^{σ} is not short. Let I_{M̄,M} have length ϑ, iteration maps ⟨π̃_{ij} | i j ϑ⟩, and indices ⟨ν̃_{i} | i < ϑ⟩. As the iteration is not short, there is a sequence ⟨i_{ξ} | ξ ω_{1}⟩ of ordinals less than ϑ such that

∀ξ ζ ω_{1} ( π̃_{i_{ξ}i_{ζ}} (ν̃_{i_{ξ}}) = ν̃_{i_{ζ}} ).

Let us introduce some further notational simplifications. We will subsequently only be interested in stages of the iteration determined by the sequence ⟨i_{ξ} | ξ ω_{1}⟩, thus set, for ξ ζ ω_{1},

N^{ξ} := M^{i_{ξ}}  ν_{ξ} := ν̃_{i_{ξ}}
π_{ξζ} := π̃_{i_{ξ}i_{ζ}}  μ_{ξ} := crit(E_{ν_{ξ}}^{N^{ξ}})

Let F := E_{ν_{ω_{1}}}^{N^{ω_{1}}} . The point of the Gitik Game is to reconstruct this extender inside the collapsed substructure H̄ := H_{α}^{σ}. (For convenience, denote K_{α}^{σ} by K̄, too.)

Claim 9 F ∈H̄.

Before we proceed with the proof of this claim, let us show how to conclude the discussion of Case (B) and hence the proof of the theorem, given Claim Claim 10 F is countably complete.

Proof Let μ := μ_{ω_{1}} , ν := ν_{ω_{1}} . Let ⟨a_{n} | n ∈ ω⟩ be a sequence of elements from [ν]^{<ω} and let ⟨x_{n} | n ∈ ω⟩ be a sequence of sets such that for all n ∈ ω, x_{n} ∈ F_{a_{n}}. We have to show that there is an orderpreserving map t : ⋃ _{n∈ω} a_{n} → μ such that for all n∈ω, t′′a_{n} ∈x_{n}.

First, find i < ω_{1} such that there a sets y_{n} and b_{n}, n∈ω, in N^{i} such that

x_{n} = π_{iω_{1}} (y_{n}) a_{n} = π_{iω_{1}} (b_{n}).

Let t := π_{iω_{1}}^{−1} ⋃ _{n∈ω} a_{n}. Then t is an orderpreserving map from ⋃ _{n∈ω} a_{n} to ⋃ _{n∈ω} b_{n}, since π_{iω_{1}} is elementary. Also note that the iteration I_{N^{0}N^{ω_{1}}} is normal, so μ > ν^{i}. ⋃ _{n∈ω} a_{n} ⊆ ν and π_{iω_{1}} (ν_{i}) = ν, so ⋃ _{n∈ω} b_{n} ⊆ ν_{i} ⊆ μ.

Finally, x_{n} ∈ F_{a_{n}} = (E_{π_{iω_{1}} (ν_{i})}^{N^{ω_{1}}} )_{π_{iω_{1}} (b_{n})}, thus y_{n} ∈ (E_{ν_{i}}^{N^{i}} )_{b_{n}}. By the normality of I_{N^{0}N^{ω_{1}}} , b_{n} ∈π_{iω_{1}} (y_{n}), i. e., t′′a_{n} = b_{n} ∈x_{n}. (Claim 10)

Now recall that N^{ω_{1}} was just a stage in the iteration from M̄_{α}^{σ} to M_{α}^{σ}. We have

N^{ω_{1}} ↓ν_{ω_{1}} = M_{α}^{σ} ↓ν_{ω_{1}}  since I_{M̄_{α}^{σ} ,M_{α}^{σ}} is normal
= K_{α}^{σ} ↓ν_{ω_{1}} .

The second equation holds since ν_{ω_{1}} must be less than σ_{α}. Otherwise an earlier stage of the iteration I_{M̄_{α}^{σ} ,M_{α}^{σ}} would already have satisfied the conditions of Claim 6. Thus we have that

H_{α}^{σ} ⟨K_{α}^{σ} ↓ν_{ω_{1}} , F ⟩ is a premouse such that F is countably complete.

By definition, K_{α}^{σ} = (K^{c})^{H_{α}^{σ}} . Thus it follows that E_{ν_{ω_{1}}}^{K_{α}^{σ}} = F. On the other hand, F is an extender used in the iteration from M̄_{α}^{σ} to M_{α}^{σ}, an end-extension of K_{α}^{σ}, and thus E_{ν_{ω_{1}}}^{K_{α}^{σ}} = E_{ν_{ω_{1}}}^{M_{α}^{σ}} = ∅, a contradiction.

This concludes the proof of the theorem, save for Claim 9.

Proof of Claim 9

Our aim is to show that the extender F = E_{ν_{ω_{1}}}^{N^{ω_{1}}} is an element of the structure H̄ = H_{α}^{σ}. By our choice of extenders (or rather indices) we have

∀ξ ζ ω_{1} (π_{ξζ}(ν_{ξ}) = ν_{ζ})

and so it will be enough to reconstruct in H̄ a sufficiently good approximation to the direct limit system ⟨N^{ξ} | ξ ω_{1}⟩. More precisely, it will be enough to define in H̄ a thread b_{β} for each β < ν_{ω_{1}} , where the thread b_{β} is taken to represent β in the direct limit N^{ω_{1}} . Thus b_{β} will be a sequence of ordinals ⟨β_{i} | i < ω_{1}⟩, such that β_{i} < μ (since β_{i} < ν_{i} < μ_{i+1} < μ) and such that

∃j < ω_{1} ∀i∈[j, ω_{1}) (π_{iω_{1}} (β_{i}) = β).

Given such threads in H̄, one can reconstruct F as follows: Let a ∈ [ν_{ω_{1}} ]^{<ω} and x ∈ P ([μ]\overline{\overline{a}}) ∩ N^{ω_{1}} . Assume a = \{β^{1}, …, β^{n}\}. Then x is in F_{a} if a final sequence of the thread for a is in x, i. e.,

x∈F_{a} ↔ ∃j < ω_{1} ∀i∈[j, ω_{1}) ({β_{i}^{1}, …, β_{i}^{n}}∈x).

Thus if the threads b_{β} are all definable in H̄, then F is definable in H̄ and hence an element of H̄.

The threads will be defined via winning strategies for the second player in the following game, due to Gitik ([Git93]).

The Gitik Game 𝔊 is played by two players, say Alice and Bob, who alternatingly make ω-many moves. Alice is the first to play. On her n-th move Alice plays a pair ⟨B_{n}, h_{n}⟩ such that

B_{n} ⊆ ν_{ω_{1}} , card(B_{n}) ℵ_{1}, and B_{n−1} ⊆ B_{n}, (if n > 0),

and

h_{n} ∈^{μ}ν_{ω_{1}} ∩ K̄.

Note that N^{i} ↓ ν_{i} = M ↓ ν_{i} = K̄ ↓ ν_{i}, as on the one hand the iteration after index i is beyond ν_{i} and on the other hand K̄ = K_{α}^{σ} ⊆ M_{α}^{σ} = M.

Bob responds to Alice’s move by playing a sequence t^{n} = ⟨t_{i}^{n} | i < ω_{1}⟩ satisfying the following three conditions:

i) For each i < ω_{1}, t_{i}^{n} maps a submodel of N^{i} ↓ ν_{i} of size at most ℵ_{1} elementarily into N^{ω_{1}} ↓ ν_{ω_{1}} , i. e., t_{i}^{n} : N^{i} ↓ ν_{i} ↪_{Σ_{ω}} N^{ω_{1}} ↓ ν_{ω_{1}} and

card(t_{i}^{n}) ℵ_{1}.
ii) If n > 0, then for each i < ω_{1}, t_{i}^{n} extends t_{i}^{n−1} , i. e., t_{i}^{n−1} ⊆ t_{i}^{n}.

iii) For sufficiently large i, rge(t_{i}^{n}) “covers” B_{n} ∩h_{n}′′μ_{i}, i. e., ∃j < ω_{1} ∀i∈

[j, ω_{1}) (B_{n} ∩ h_{n}′′μ_{i} ⊆ rge(t_{i}^{n})).

If Bob is unable to find such a sequence, then Alice wins the game. Otherwise, i. e., if the number of moves is infinite, Bob wins. Thus Bob tries to construct, in the course of the game, an approximation to the direct limit system ⟨N^{i} | i < ω_{1}⟩, ⟨π_{ij} | i j < ω_{1}⟩. Alice can require certain points (given by B_{n} and h_{n}) to be considered in this approximation, thus making the task harder for Bob.

Note that H̄ is closed under ω_{1}-sequences and also that N^{i}↓ν_{i} = K̄ ↓ν_{i} ⊆ H̄, for any i ω_{1}. Thus any move that either Alice or Bob makes is an element of H̄: they all have cardinality at most ℵ_{1} and are subsets of H̄. Again by ω_{1}-closure, the whole run of the game is then also an element of H̄. Also note that the game 𝔊 is closed and hence determined. Thus either Alice or Bob must have a winning strategy. This last statement holds relativized to H̄, too.

Claim 11 Bob has a winning strategy τ for 𝔊 in H̄, i. e.,

H̄ ∃τ (τ is a winning strategy for Bob).

Proof Suppose not. Then Alice has a winning strategy σ for 𝔊 in H̄. Consider a run of the game where Alice plays ⟨B_{n}, h_{n}⟩ according to σ and where Bob responds with the following sequence t^{n}. For i < ω_{1}, let H_{i} be a hull of π_{iω_{1}}^{−1} ′′B_{n} in K̄ ↓ν_{i}, card(H_{i}) ℵ_{1}, constructed in V. Let t_{i}^{n} := π_{iω_{1}} H_{i}. Then by the above remark, t^{n} = ⟨t_{i}^{n} | i < ω_{1}⟩∈H̄.

But these choices are always legal moves for Bob: Condition i) is satisfied by making H_{i} ≺ K̄ ↓ ν_{i} = N^{i} ↓ ν_{i}. Since Alice always chooses sets B_{n} of size ℵ_{1}, card(H_{i}) ℵ_{1} can also always be achieved. Condition ii) is fulfilled as Alice chooses B_{n} ⊇ B_{n−1} . Finally, note that the iteration from N^{ω_{1}} to M ⊇ K̄ is beyond ν_{ω_{1}} . Thus if h_{n} ∈ ^{μ}ν_{ω_{1}} ∩ K̄, then h_{n} ∈ N^{ω_{1}} , by [Koe89, Lemma 17.2 (iii)]. Subsequently, find some j < ω_{1} such that there exists h̄ ∈ N^{j} satisfying π_{jω_{1}} (h̄) = h_{n}. Now let i ∈ [j, ω_{1}) and set h̄_{i} := π_{ji}(h̄). For any ξ ∈B_{n} ∩h_{n}′′μ_{i}, let ξ̄_{i} < μ_{i} be such that h_{n}(ξ̄_{i}) = ξ. Then ξ = h_{n}(ξ̄_{i}) = π_{iω_{1}} (h̄_{i})(ξ̄_{i}) = π_{iω_{1}} ( h̄_{i}(ξ̄_{i})) ∈ rge(π_{iω_{1}} ). Thus B_{n} ∩ h_{n}′′μ_{i} ⊆ rge(π_{iω_{1}} ). Furthermore, ξ ∈ B_{n} ∩ rge(π_{iω_{1}} ) implies that π^{−1} (ξ) ∈ H_{i} = hull of π_{iω_{1}}^{−1} ′′B_{n}, so that ξ = t_{i}^{n}(π_{iω}^{−1} (ξ)) ∈ rge(t_{i}^{n}). So for sufficiently large i, rge(t_{i}^{n}) indeed covers B_{n} ∩ h_{n}′′μ_{i}, and condition iii) is also satisfied.

Thus this run of the game 𝔊 is infinite, so that Bob wins although Alice was using her winning strategy — contradiction. (Claim 11)

Let τ ∈ H̄ be any winning strategy for Bob. Given β < ν_{ω_{1}} , we want to show that this will yield a candidate for a thread for β. Consider a run of the game 𝔊 where Bob plays according to τ and where Alice starts out by playing the singleton B_{0} := {β} together with h_{0} := c_{β}, the function with constant value β (and domain μ, of course). Then, by clause iii) for Bob’s moves, β ∈rge(t_{i}^{0}) for sufficiently large i. Thus there is j < ω_{1} and there are β_{i}, for i ∈ [j, ω_{1}), such that t_{i}^{0}(β_{i}) = β. Choose some arbitrary β_{i}, for i < j, and let τ(β) denote the sequence ⟨β_{i} | i < ω_{1}⟩. Thus τ(β) is a thread for β in the sense of Bob’s approximation t^{0} to the real direct limit system. We will show next that in a certain sense the true thread for β is minimal among these approximations.

Consider the ordering of sequences of ordinals by eventual dominance:

7.10 Definition Given sequences c = ⟨γ_{i} | i < ω_{1}⟩ and d = ⟨δ_{i} | i < ω_{1}⟩, let c d (or c ≺ d) iff for sufficiently large i, γ_{i} δ_{i} (or γ_{i} < δ_{i}), i. e., ∃j < ω_{1} ∀i∈[j, ω_{1}) (γ_{i} δ_{i}) (or γ_{i} < δ_{i}).

For β < ν_{ω_{1}} let ⟨β̌_{i} | i < ω_{1}⟩ denote the (unique maximal) thread for β in N^{ω_{1}} .

Claim 12 Let τ be some winning strategy for Bob and let β < ν_{ω_{1}} . Then τ(β) ⟨β̌_{i} | i < ω_{1}⟩.

Proof Let τ(β) = ⟨β_{i} | i < ω_{1}⟩, where τ is some winning strategy for Bob. Assume for a contradiction that τ(β) ⟨β̌_{i} | i < ω_{1}⟩, and consider the run of game 𝔊 where Alice starts out by playing B_{0} := {β} and h_{0} := c_{β}. Bob responds by playing t^{n} = ⟨t_{i}^{n} | i < ω_{1}⟩ according to τ. For n > 0, Alice chooses B_{n} and h_{n} as follows:

B_{n} := {β} ∪ ⋃_{i<ω_{1}} π_{iω_{1}} ′′( dom(t_{i}^{n−1} ) ∩ On ).

Since all t_{i}^{n−1} have cardinality at most ℵ_{1}, so does B_{n}, so that B_{n} ∈ [ν_{ω_{1}} ]^{ω_{1}} as required. By induction it also follows that B_{n} ⊇ B_{n−1} , using the fact that t_{i}^{n−1} extends t_{i}^{n−2} (if n > 1). Note that by the ω_{1}-closure of H̄ B_{n} is an element of H̄. We now wish to apply the weak covering lemma 5.22 to choose h_{n}. μ = μ_{ω_{1}} is not overlapped in K̄ = (K^{c})H̄, as it is measurable and by ¬ L^{strong} there are no overlapping extendersequences. Also, ν_{ω_{1}} = OK̄(μ), since an end-extension of K̄ is produced by iterating N^{ω_{1}} simply, beginning with E_{ν_{ω_{1}}}^{N^{ω_{1}}} . B_{n} ⊆ ν_{ω_{1}} and (card(B_{n})^{ℵ_{0}} < card(μ))H̄. Thus we can conclude that there is some h∈K̄, h : κ → ν_{ω_{1}} such that B_{n} ⊆ rge(h_{n}).

This effectively forces Bob to respect all of B_{n} with his next move, for sufficiently large i. This would not be a priori impossible, were it not for our assumption that τ(β) ⟨β̌_{i} | i < ω_{1}⟩, i. e., that the true thread for β unboundedly often lies strictly above Bob’s approximation to it. This latter fact is now used to show that Bob cannot win the game:

Suppose to the contrary that Bob does win, i. e., that the number of moves is infinite. Let n ∈ ω. Then h_{n} ∈^{μ}ν_{ω_{1}} ∩ K̄ ⊆ M. M is a simple fine iterate of N^{ω_{1}} beyond ν_{ω_{1}} , so that we can use [Koe89, Lemma 17.2 iii)] again to conclude that h_{n} ∈ N^{ω_{1}} . Thus there is some j < ω_{1} and some h̄ ∈ N^{j} such that h_{n} = π_{jω_{1}} (h̄). For any i∈[j, ω_{1}), set h̄_{i} := π_{ji}(h̄). We claim that

B_{n} ∩ rge(π_{iω_{1}} ) ⊆ h_{n}′′μ_{i}.

To see this, let ξ ∈ B_{n} ∩ rge(π_{iω_{1}} ), say ξ = π_{iω_{1}} (ξ̄). By construction B_{n} ⊆ rge(h_{n}), so that

N^{ω_{1}} ∃ξ′< μ (h_{n}(ξ′) = ξ).

By elementarity,

N^{i} ∃ξ′< μ_{i} ( h̄_{i}(ξ′) = ξ̄).

So pick some ξ′< μ_{i} such that

N^{i} h̄_{i}(ξ′) = ξ̄.

Since the iteration from N^{i} to N^{ω_{1}} is normal and above μ_{i}

N^{ω_{1}} h_{n}(ξ′) = ξ,

i. e., ξ ∈h_{n}′′μ_{i}, as claimed.

By condition iii) for Bob’s moves, there is j_{n} < ω_{1} such that

∀i∈[j_{n}, ω_{1}) (B_{n} ∩ rge(π_{iω_{1}} ) ⊆ B_{n} ∩ h_{n}′′μ_{i} ⊆ rge(t_{i}^{n})).

Since this is true for all n∈ω, let j := sup{j_{n} | n∈ω} < ω_{1}. Then

∀n∈ω ∀i∈[j, ω_{1}) (B_{n} ∩ rge(π_{iω_{1}} ) ⊆ rge(t_{i}^{n})). (1)

W.l.o.g. pick j large enough so that also

∀i∈[j, ω_{1}) (t_{i}^{0}(β_{i}) = β). (2)

This is possible by definition of ⟨β_{i} | i < ω_{1}⟩ as τ(β). (Recall that Alice started out by playing B_{0} = {β} and Bob answered according to τ.) Finally, we want to make use of our assumption that τ(β) ⟨β̌_{i} | i < ω_{1}⟩. This ensures that for some i > j β_{i} < β̌_{i}, i. e.,

∃i∈[j, ω_{1}) (π_{iω_{1}} (β_{i}) < π_{iω_{1}} (β̌_{i}) = β). (3)

We can now inductively go back and forth between N^{i} and N^{ω_{1}} and construct a descending sequence of ordinals, as shown in Figure 7.7.

Fix some i as in equation (3). Let η_{0} := π_{iω_{1}} (β_{i}) < β. Note that η_{0} ∈ B_{1} ∩ rge(π_{iω_{1}} ): By equation (2), β_{i} ∈ dom(t_{1}^{0}), so that η_{0} = π_{iω_{1}} (β_{i}) ∈ π_{iω_{1}} ′′(dom(t_{1}^{0}) ∩ On) ⊆ B_{1}. Assume that η_{n−1} had been defined and shown to be an element of B_{n} ∩ rge(π_{iω_{1}} ). Then set

η_{n} := π_{iω_{1}} ((t_{i}^{n})^{−1} (η_{n−1} )).

This is well-defined, as by equation (1) we have that B_{n} ∩rge(π_{iω_{1}} ) ⊆ rge(t_{i}^{n}), so that ζ_{n} := (t_{i}^{n})^{−1} (η_{n−1} ) is defined. Also η_{n} = π_{iω_{1}} (ζ_{n}) ∈ π_{iω_{1}} ′′(dom(t_{i}^{n}) ∩ On) ⊆ B_{n+1} , by the definition of B_{n+1} . Thus the inductive definition goes through.

We claim that ∀n ∈ ω (η_{n+1} < η_{n}). First, note that η_{0} = π_{iω_{1}} (β_{i}) < β = t_{i}^{0}(β_{i}), so that (t_{1}^{0})^{−1} (η_{0}) < β_{i}. This implies that

η_{1} = π_{iω_{1}} ((t_{i}^{0})^{−1} (η_{0})) < π_{iω_{1}} (β_{i}) = η_{0}.

[figure: i; −; N^{i}; ⋮; −; (t_{i}^{2})^{−1} (η_{1}); −; (t_{i}^{1})^{−1} (η_{0}); −; β_{i}; −; β̌_{i} = π_{iω_{1}}^{−1} (β); −; −; N^{ω_{1}}; ω_{1}; ⋮; −; η_{2} := π_{iω_{1}} ((t_{i}^{2})^{−1} (η_{1})); −; η_{1} := π_{iω_{1}} ((t_{i}^{1})^{−1} (η_{0})); −; η_{0} := π_{iω_{1}} (β_{i}); −; β = t_{i}^{0}(β_{i}); −; ν; π_{iω_{1}}; t_{i}^{0}; t_{i}^{1}; t_{i}^{2}; π_{iω_{1}}]

Figure 7.7: To construct an infinite descending sequence of ordinals, start with β in N^{ω_{1}} , take its pre-image under t_{i}^{0} to go to N^{i}, then use π_{iω_{1}} to return to N^{ω_{1}} and start over again, using t_{i}^{1} next.

Inductively, if η_{n} < η_{n−1} , n > 0, then

η_{n+1} = π_{iω_{1}} ((t_{i}^{n+1} )^{−1} (η_{n}))
< π_{iω_{1}} ((t_{i}^{n+1} )^{−1} (η_{n−1} )),  since η_{n−1} ∈rge(t_{i}^{n}) ⊆ rge(t_{i}^{n+1} ),
= π_{iω_{1}} ((t_{i}^{n})^{−1} (η_{n−1} )),  since t_{i}^{n} ⊆ t_{i}^{n+1} ,
= η_{n}  by the definition of η_{n}.

This is a contradiction. Thus Alice wins the game. But Bob was playing according to a winning strategy τ. Contradiction yet again! Thus our initial assumption that τ(β) ⟨β̌_{i} | i < ω_{1}⟩ must have been wrong, and the claim is proved. (Claim 12)

Now the argument for Claim 11 shows that, for any β < ν_{ω_{1}} , Bob has a winning strategy τ ∈ H̄ such that τ(β) is a thread for β in N^{ω_{1}} . Thus if we let

b_{β} := the -minimum of {τ(β) | τ is a winning strategy for Bob}, then b_{β} is a thread for β in N^{ω_{1}} , defined inside H̄: The set of possible threads τ(β) consists only of threads greater or equal than the canonical thread for β, by Claim 12, and it contains one which is eventually equal to this thread. But this proves Claim 9, and with it the theorem. (Claim 9)

As we already noted at the end of Chapter 6, our original goal was to prove that the existence of an irregular ultrafilter entailed the existence of an inner model for a strong cardinal. Under the present circumstances, the Gitik Game is not strictly necessary: all measures involved are ultrafilters, no recourse to extenders is needed, and ultrafilters are reconstructible in an ω-closed model after an ω-long iteration, just from their Prikry-sequence. However, we considered the Gitik Game interesting enough to merit a detailed exposition, and it also serves to locate the exact point where the proof breaks down for higher orders of measurability, namely Lemma 6.2.

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Index

0^{long}, 10 CC, 13 CC^{club}, 14 Card, 7 card(x), 7 cf(α), 7 CF_{ϱ}, 25 Col(ω_{1}, κ), 16 crit(π), 7 D(κ), 9 D X, 9 dom(D), 9 dom(E), 38 dom(f), 7 E X, 38 E_{ν}, 38 E_{ν,a}, 38 f′′x, 7 f x, 7 fun(f), 7 H_{μ}, 6 J_{α}[A], 9 K[D], 10 K^{c}, 43 L^{strong}, 40 Lim, 7 lp(M), 10 lub(x), 7 M ↓β, 38 M|β, 38 meas(M), 10 O^{M}(κ), 38 o^{M}(κ), 38 On, 7 otp(x), 7 Reg, 7 rge(f), 7 s^{(ω_{1})}(M̄), 46 TH(κ), 25 TH^{*}(κ), 25 TH^{stat}(κ), 25 U_{can}, 11 wCC(κ), 35 W^{Γ}(M), 43 ∀^{V}α, 72 β → (α)_{δ}^{<ω} , 16 ⟨κ, λ⟩ ⟨μ, ν⟩, 13

⟨κ, λ⟩ _{club} ⟨μ, < ν⟩, 14

κ → [μ]_{λ,ν}^{<ω}, 14 κ → [μ]_{λ,<ν}^{<ω} , 14

κ →_{club} [μ]_{λ,<ν}^{<ω} , 15

κ(α), 16 κ(ν), 38 [κ]^{λ}, 7 [κ]^{<λ} , 7 κ^{M}, 39 ∗, 39 D, 11 e, 11 , 94 <_{i}, 22 <_{J_{α} [A]}, 9 \overline{\overline{x}}, 7

above τ, 39 V-almost all, 72

M-based, 41 beyond β, 40

canonical core model, 11 sequence, 11

Changs’s Conjecture, 13

weak, 35 γ-club, 7 coherent, 38 collapsing mouse, 39 γ-complete, 7 Condition II, 20 core, 39 core model

canonical, 11

countably complete, 43 short, 11

covering sequence, 68

end-extension, 11, 38 α-Erdős, 16 extender structure, 38

full, 43 Γ-, 42 weakly, 42

fully irregular, 69 P -generic, 22

initial segment, 38

iterable above τ, 39

beyond β, 40

iteration, 39 non-degenerate, 39 short, 45 simple, 39

Ketonen diagramme, 28

Levy collapse, 16

lousy, 41 low part, 10

maximal, 11 measurables set of, 10

measure, 9

sequence of ∼s, 9 Mitchell order, 38 mouse, 39

collapsing, 39

D-, 10

natural, 37 neat, 40

beyond β, 40 p-premouse, 41 non-degenerate, 39 non-overlapping, 38 normal, 67

overlapped, 41

perfect, 42 premouse

D-, 10 over D, 10 p-, 40 short ∼ over D, 10

presolid, 39 prestrong, 40

regular, 67

weakly, 68 regularity sequence, 67 α-remarkable, 19

sequence canonical, 11 covering, 68 of measures, 9 regularity, 67 strong, 11

short core model, 11 iteration, 45 premouse over D, 10

simple class, 9 iteration, 39

solid, 40 strong

mouse, 40

sequence of measures, 11

topless, 38 Transversal Hypothesis, 25

ultrafilter, 7 uniform, 67 universal, 42 upward extension, 41

weakly amenable, 38 weakly full, 42 weakly normal, 67 weakly regular, 68 weakly universal, 42 weasel, 42