Walks on ordinals and their characteristics

Progress in Mathematics Volume 263

Series Editors H.Bass J. Oesterlé A. Weinstein

Stevo Todorcevic

Walks on Ordinals and Their Characteristics

Birkhäuser Basel · Boston · Berlin

Stevo Todorcevic Université Paris VII – C.N.R.S. UMR 7056 2, Place Jussieu – Case 7012 75251 Paris Cedex 05 France e-mail: [email protected]

Department of Mathematics University of Toronto Toronto M5S 2E4 Canada e-mail: [email protected]

and Mathematical Institute, SANU Kneza Mihaila 35 11000 Belgrad Serbia e-mail: [email protected]

2000 Mathematics Subject Classification 03E10, 03E75, 05D10, 06A07, 46B03, 54D65, 54A25 Library of Congress Control Number: 2007933914 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-8528-6 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2007 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-8528-6

e-ISBN 978-3-7643-8529-3

987654321

www.birkhauser.ch

Contents 1 Introduction 1.1 Walks and the metric theory of ordinals 1.2 Summary of results . . . . . . . . . . . . 1.3 Prerequisites and notation . . . . . . . . 1.4 Acknowledgements . . . . . . . . . . . .

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2 Walks on Countable Ordinals 2.1 Walks on countable ordinals and their basic characteristics . 2.2 The coherence of maximal weights . . . . . . . . . . . . . . 2.3 Oscillations of traces . . . . . . . . . . . . . . . . . . . . . . 2.4 The number of steps and the last step functions . . . . . . .

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3 Metric Theory of Countable Ordinals 3.1 Triangle inequalities . . . . . . . . . . . . . . . 3.2 Constructing a Souslin tree using ρ . . . . . . . 3.3 A Hausdorff gap from ρ . . . . . . . . . . . . . 3.4 A general theory of subadditive functions on ω1 3.5 Conditional weakly null sequences based on subadditive functions . . . . . . . . . . . . . . . 4 Coherent Mappings and Trees 4.1 Coherent mappings . . . . . . . . . . . . . . 4.2 Lipschitz property of coherent trees . . . . . 4.3 The global structure of the class of coherent 4.4 Lexicographically ordered coherent trees . . 4.5 Stationary C-lines . . . . . . . . . . . . . . 5 The 5.1 5.2 5.3

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Square-bracket Operation on Countable Ordinals The upper trace and the square-bracket operation . . . . . . . . . 133 Projecting the square-bracket operation . . . . . . . . . . . . . . . 139 Some geometrical applications of the square-bracket operation . . . . . . . . . . . . . . . . . . . . . . . . 144

vi

Contents

5.4 5.5

A square-bracket operation from a special Aronszajn tree . . . . . 152 A square-bracket operation from the complete binary tree . . . . . 157

6 General Walks and Their Characteristics 6.1 The full code and its application in characterizing Mahlo cardinals . . . . . . . . . . . . . . . . . . . . 161 6.2 The weight function and its local versions . . . . . . . . . . . . . . 174 6.3 Unboundedness of the number of steps . . . . . . . . . . . . . . . . 178 7 Square Sequences 7.1 Square sequences and their full lower traces . . . . . . . . 7.2 Square sequences and local versions of ρ . . . . . . . . . . 7.3 Special square sequence and the corresponding function ρ 7.4 The function ρ on successors of regular cardinals . . . . . 7.5 Forcing constructions based on ρ . . . . . . . . . . . . . . 7.6 The function ρ on successors of singular cardinals . . . . . 8 The 8.1 8.2 8.3

Oscillation Mapping and the Square-bracket Operation The oscillation mapping . . . . . . . . . . . . . . . . . The trace filter and the square-bracket operation . . . Projections of the square-bracket operation on accessible cardinals . . . . . . . . . . . . . . . . . . 8.4 Two more variations on the square-bracket operation .

9 Unbounded Functions 9.1 Partial square-sequences . . . . . . . . . . . . . . 9.2 Unbounded subadditive functions . . . . . . . . . 9.3 Chang’s conjecture and Θ2 . . . . . . . . . . . . 9.4 Higher dimensions and the continuum hypothesis 10 Higher Dimensions 10.1 Stepping-up to higher dimensions . . . 10.2 Chang’s conjecture as a 3-dimensional Ramsey-theoretic statement . . . . . . 10.3 Three-dimensional oscillation mapping 10.4 Two-cardinal walks . . . . . . . . . . .

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187 195 202 205 213 220

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271 273 277 283

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Chapter 1

Introduction 1.1 Walks and the metric theory of ordinals This book is devoted to a particular recursive method of constructing mathematical structures that live on a given ordinal θ, using a single transformation ξ
→ Cξ which assigns to every ordinal ξ < θ a set Cξ of smaller ordinals that is closed and unbounded in the set of ordinals < ξ. The transfinite sequence Cξ (ξ < θ) which we call a ‘C-sequence’ and on which we base our recursive constructions may have a number of ‘coherence properties’ and we shall give a detailed study of them and the way they influence these constructions. Here, ‘coherence’ usually means that the Cξ ’s are chosen in some canonical way, beyond the already mentioned and natural requirement that Cξ is closed and unbounded in ξ for all ξ. For example, choosing a canonical ‘fundamental sequence’ of sets Cξ ⊆ ξ for ξ < ε0 , relying on the specific properties of the Cantor normal form for ordinals below the first ordinal satisfying the equation x = ω x , is a basis for a number of important results in proof theory. In set theory, one is interested in longer sequences as well and usually has a different perspective in applications, so one is naturally led to use some other tools besides the Cantor normal form. It turns out that the sets Cξ can not only be used as ‘ladders’ for climbing up in recursive constructions but also as tools for ‘walking’ from an ordinal β to a smaller one α, β = β0 > β1 > · · · > βn−1 > βn = α, where the ‘step’ βi → βi+1 is defined by letting βi+1 be the minimal point of Cβi that is bigger than or equal to α. This notion of a ‘walk’ and the corresponding ‘characteristics’ and ‘distance functions’ constitute the main body of study in this book. We show that the resulting ‘metric theory of ordinals’ is a theory of considerable intrinsic interest which provides not only a unified approach to a

2

Chapter 1. Introduction

number of classical problems in set theory but is also easily applicable to other areas of mathematics. For example, highly applicable characteristics of the walk are defined on the basis of the corresponding ‘traces’. The most natural trace of the walk is its ‘upper trace’ defined simply to be the set Tr(α, β) = {β0 > β1 > · · · > βn−1 > βn } of places visited along the way, which is of course most naturally enumerated in decreasing order. Another important trace of the walk is its ‘lower trace,’ the set Λ(α, β) = {λ0 ≤ λ1 ≤ · · · ≤ λn−2 ≤ λn−1 },
i where λi = max( j=0 Cβj ∩ α) for i < n. The traces are usually used in defining various binary operations on ordinals < θ, the most prominent of which is the ‘square-bracket operation’ that gives us a way to transfer the quantifier ‘for every unbounded set’ to the quantifier ‘for every closed and unbounded set’. It is perhaps not surprising that this reduction of quantifiers has proven to be quite useful in constructions of mathematical structures on θ where one needs to have some grip on substructures of cardinality θ. From the metric theory of ordinals based on analysis of walks, one also learns that the triangle inequality of an ultrametric (α, γ) ≤ max{ (α, β), (β, γ)} has three versions, depending on the natural ordering between the ordinals α, β and γ. The three versions of the inequality are in fact of a quite different character and occur in quite different places and constructions in set theory. For example, the most frequent occurrence is the case α < β < γ, when the triangle inequality becomes something that one can call ‘transitivity’ of . Considerably more subtle is the case α < γ < β of this inequality1 . It is this case of the inequality that captures most of the coherence properties found in this article. It is also an inequality that has proven to be quite useful in applications. A large portion of the book is organized as a discussion of four basic characteristics of the walk ρ, ρ0 , ρ1 , ρ2 and ρ3 . The reader may choose to follow the analysis of any of these functions in various contexts. The characteristic ρ0 (α, β) codes the entire walk β = β0 > β1 > · · · > βn−1 > βn = α by simply listing the positions of βi+1 in the set Cβi for i < n. While this looks simple-minded, the resulting mapping ρ0 is a rather remarkable object. For example, in the realm of the space ω1 of countable ordinals, it gives us a canonical example of a special Aronszajn tree of increasing sequences of rationals which has the additional remarkable property that, when ordered lexicographically, its cartesian square can be covered by countably many chains. In other words, the single characteristic ρ0 of walks on countable ordinals gives two critical structures, one in the class of 1 It

appears that the third case β < α < γ of this inequality is rarely a reasonable assumption to be made in this context.

1.1. Walks and the metric theory of ordinals

3

so-called Lipschitz trees and the other in the class of linear orderings. For higher cardinals θ, analysis of ρ0 leads us to some interesting finitary characterizations of hyper inaccessible cardinals. This is given in some detail in Chapter 6 of this book. The characteristic ρ1 (α, β) loses a considerable amount of information about the walk as it records only the maximal order type among the sets {Cβ0 ∩ α, Cβ1 ∩ α, . . . , Cβn−1 ∩ α}. Nevertheless it gives us the first example of what we call a ‘coherent mapping’. The class of coherent mappings and trees in the case θ = ω1 exhibits an unexpected structure that we study in great detail in Chapter 4 of the book. The fine structure in the class of ‘coherent trees’ is based on the metric notion of a ‘Lipschitz mapping’ between trees. The profusion of such mappings between coherent trees eventually leads us to the so-called ‘Lipschitz Map Conjecture’ that has proven crucial for the final resolution of the basis problem for uncountable linear orderings and that is presented in the same chapter. For higher cardinals θ the characteristic ρ1 and its local versions offer a rich source of so-called ‘unbounded functions’ that have some applications. The characteristic ρ2 (α, β) simply counts the number of steps of the walk from β to α. While this also looks rather simple minded, the remarkable properties of the corresponding function ρ2 become especially apparent on higher cardinals θ. Important properties of this characteristic are its coherence and its unboundedness. The coherence property of ρ2 requires the corresponding C-sequence Cξ (ξ < θ) to be ‘coherent’ in the sense that Cα = Cβ ∩ α whenever α is a limit point of Cβ . On the other hand, the unboundedness of ρ2 translates into a requirement that the corresponding C-sequence Cξ (ξ < θ) be ‘nontrivial’2 , a condition that eventually leads us to a simple and natural characterization of weakly compact cardinals that we choose to reproduce in some detail in Chapter 6. Finally, the characteristic ρ3 (α, β) attaches one of the digits 0 or 1 to the walk according to the behavior of the last step βn−1 → βn = α. The full analysis of this characteristic is currently available only in the reals of the space ω1 , where ρ3 becomes a rather canonical example of a sequence-coherent mapping with values in {0, 1} and with properties reminiscent of those appearing in the well-known notion of a Hausdorff gap in the quotient algebra P(ω)/fin (another critical object that shows up in many problems about this quotient structure). The true ‘metric theory of ordinals’ comes only with development of the characteristic ρ(α, β) of the walk that takes advantage of the so-called ‘full lower trace’ of the walk. The depth of this characteristic is apparent even in the space ω1 of countable ordinals, but its full power comes at higher cardinals θ and especially at θ that are successors of singular cardinals. The full analysis of the characteristic ρ requires Cξ (ξ < θ) to be a so-called ‘square sequence’ or in other words requires 2 We

say that Cξ (ξ < θ) is nontrivial if there is no closed and unbounded set C ⊆ θ such that. for all limit points α of C, there is β ≥ α such that C ∩ α ⊆ Cβ .

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Chapter 1. Introduction

the most widely known coherence condition on this sequence, which says that if α is a limit point of Cβ , then Cα = Cβ ∩α. It is not surprising that this characteristic has the largest number of applications, many of which are reproduced in this book. We have already mentioned that its development in the case θ = ω1 was the initial impulse for development of the so-called metric theory of countable ordinals that has already a rich spectrum of applications. At higher cardinals θ the characteristic ρ can be used in facilitating set-theoretic forcing constructions of rather special objects and we shall reproduce some of these constructions in Chapter 7 of this book. While, as said above, the full development of ρ requires Cξ (ξ < θ) to be a ‘square sequence’, the function ρ itself holds considerable information about the notion of ‘square sequences.’ For example in Chapter 7, we use ρ to turn an arbitrary square sequence Cξ (ξ < θ) into a non-special one by expressing the usual order relation among ordinals < θ as an increasing union of tree-orderings that come from square sequences on θ themselves. This again leads us to some applications that we choose to reproduce in detail at the end of Chapter 7. We have already mentioned that one of the important outcomes of our study of walks on ordinals is the ‘square-bracket operation’, a transformation which to every pair α < β of ordinals < θ assigns an ordinal [αβ] belonging to the upper trace Tr(α, β) of the walk from β to α. We have also mentioned that the choice of [αβ] has to be rather careful in order to reduce an arbitrary unbounded subset A of θ to the corresponding set of values {[αβ] : α, β ∈ A and α < β} that contains a closed and unbounded subset of θ relative to some fixed stationary set Γ ⊆ θ, which the C-sequence Cξ (ξ < θ) avoids3. We present several variations on the way [αβ] is chosen, each of which works best in some particular context. The common feature of these definitions of [αβ] is that they are all based on the oscillation mapping osc : P(θ)2 → Card defined by osc(x, y) = |x \ (sup(x ∩ y) + 1)/ ∼ |, where ∼ is the equivalence relation on x \ (sup(x ∩ y) + 1) defined by letting α ∼ β if and only if the closed interval determined by the ordinals α and β contains no point from y. In other words, osc(x, y) is simply the number of convex pieces that the set x \ (sup(x ∩ y) + 1) is split into by the set y. The original theory of the oscillation mapping osc has been developed in the realm of partial functions from θ into θ. In other words, there is a well-developed theory of the oscillation mapping osc(s, t) = |{ξ : s(ξ) ≤ t(ξ) but s(ξ + ) > t(ξ + )}|, 4 3 C (ξ ξ 4 Here,

< θ) avoids Γ if Cα ∩ Γ = ∅ for all limit ordinals α < θ. ξ + is the immediate successor of ξ in the common domain of s and t.

1.1. Walks and the metric theory of ordinals

5

(see, for example, [111]), but the general theory works equally well and it will be in part reproduced here in Chapters 8 and 9. The common feature of all results of such a theory is identification of the notion of ‘unbounded’, in either of the two contexts, in such a way that the typical oscillation result would say that the set of values osc(x, y) the oscillation mapping takes when x and y run inside two ‘unbounded’ sets is in some sense rich. In our context of defining the squarebracket operation, the sets x and y in osc(x, y) are members of our C-sequence Cξ (ξ < θ) on which we base the notion of walk, and the notion of ‘unbounded’ becomes the familiar notion of nontriviality of Cξ (ξ < θ). This makes the squarebracket operation [αβ] well defined in a wide variety of contexts and therefore quite applicable. Judging from the applications found so far, it appears that in order to make a particular variation of the oscillation mapping or the square-bracket operation useful, one needs to be be able to give a quite precise estimate of its behavior, not only on unbounded subsets of θ but also on families of θ pairwise-disjoint finite subsets of θ. It is for this reason that definite results about any particular variation of osc and [αβ] presented in this book will typically be about families A of pairwise-disjoint finite subsets of θ. Given such a family A of cardinality θ, by going to a subfamily, we may assume that elements of A have some fixed finite cardinality n. So, given a in A one can view it enumerated increasingly as a(0), . . . , a(n − 1). All variations of the square-bracket operation that we present will have the property that the set of ordinals ξ < θ that can be represented as ξ = [a(0)b(0)] = [a(1)b(1)] = · · · = [a(n − 1)b(n − 1)], for some a < b in A, contains a closed and unbounded set relative to some fixed stationary set Γ ⊆ θ which the C-sequence Cα (α < θ) avoids. Many applications however require that we know the values [a(i)b(j)] when i = j < n and when a and b run through A. It turns out that modulo taking a ‘projection’ [[··]] of [··], (or in other words, modulo composing [··] with a map from θ into θ) for many of the square-bracket operations that we define in this book, the particular set {[[a(i)b(j)]] : i, j < n and i = j} of values will be independent of the choice of a = b in A. This turns out to be crucial in several applications of [··] presented in this book. Naturally, one would also like to know whether one can define a variation on [··] where we would have freedom of getting arbitrary values of the form [a(i)b(j)] independently of whether i = j or not. It turns out that this is indeed possible for some choices of θ, though the corresponding definitions are necessarily less general as they do not apply in the case θ = ω1 since otherwise one would be able to prove that the countable chain condition is not productive without appealing to additional axioms of set theory.5 In case θ = ω1 , we do have symmetric binary operations with some degree 5 Recall

that the countable chain condition is a productive property under MAω1 (see [36], 41E).

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Chapter 1. Introduction

of freedom in that direction (see Chapter 2), but the exact breaking point between this and what requires additional axioms of set theory has yet to be determined. This book will also present some higher-dimensional characteristics of the walk, though in that context the full theory is yet to be developed. For example, in Chapter 10, we consider the characteristic τ (α, β, γ) which to any given three ordinals α < β < γ < θ assigns the place where the walk from γ to α branches from the walk from γ to β. It turns out that when Cξ (ξ < θ+ ) is a square sequence, the characteristic τ can be used to ‘step-up’ objects living on θ to objects on its successor θ+ . For example, one application of this characteristic is found in the proof that Chang’s Conjecture6 is equivalent to a 3-dimensional Ramsey-theoretic statement saying that, for every coloring of [ω2 ]3 with ω1 colors, there is an uncountable set B ⊆ ω2 which misses at least one of the colors. The 3-dimensional characteristic χ(α, β, γ) that simply measures the length of the common parts of the walks γ → α and γ → β can be used for detecting when a subset Γ of θ admits a rich restriction of the 3-dimensional version of the oscillation mapping, osc : [θ+ ]3 −→ ω, defined on the basis of its 2-dimensional version as follows: osc(α, β, γ) = osc(Cβs \ α, Cγt \ α), where s = ρ0 (α, β)
χ(α, β, γ) and t = ρ0 (α, γ)
χ(α, β, γ). Here βs is the member βi of the trace of the walk β = β0 > · · · > βl = α whose code is the sequence s, i.e., ρ0 (βi , β) = s, and similarly γt is the term γj of the walk γ = γ0 > · · · > γk = α whose code is the sequence t. In Chapter 10 we show that our analysis of the 3-dimensional version of the oscillation mapping leads naturally to a squarebracket operation in that dimension, though the full analogy is yet to be completed as one still needs to determine the behavior of this operation on families of pairwise-disjoint finite subsets of θ+ (which at the moment seems elusive). The final section of Chapter 10 is concerned with generalizing the basic notion of walks to the context of sets of ordinals rather than ordinals themselves, with the goal of obtaining two-cardinal versions of the square-bracket operation. For example, we show that for every pair of infinite cardinals κ < λ with κ regular, there is a mapping c : [[λ]κ ]2 → λ such that, for every cofinal subset U of [λ]κ and every ξ ∈ λ, there exist x ⊂ y in U such that c(x, y) = ξ. Fuller analogues of the square-bracket operation are however obtained only in certain cases. For example, assuming that there is a stationary subset S of [λ]ω which is equinumerous with a locally countable7 subset of [λ]ω , one can define a square-bracket operation [··]S : [[λ]ω ]2 → [λ]ω 6 Recall

that Chang’s conjecture is the model-theoretic statement claiming that every model of a countable signature that has the form (ω2 , ω1 , β, and fβ (α) = min{1, osc∗1 (α, β)} for α < β. Consider F = {fβ : β < ω1 } as of a subspace of {0, 1}ω1 . Clearly F is not separable. That F is hereditarily Lindel¨ follows easily from Lemma 2.3.10.  Remark 2.3.12. The projection osc∗0 of the oscillation mapping osc0 appears in [109] as the historically first such map with more than four colors that takes all of its values on every symmetric square of an uncountable subset of ω1 . The variations osc1 and osc∗1 are on the other hand very recent and are due to J.T. Moore [78] who made them in order to obtain the conclusion of Corollary 2.3.11. Concerning Corollary 2.3.11 we note that the dual implication behaves quite differently since, assuming the Proper Forcing Axiom all, hereditarily separable regular spaces are hereditarily Lindel¨ of (see [111]).

2.4 The number of steps and the last step functions In this section we show that a very natural characteristic associated to the minimal walks between countable ordinals lead to functions that have coherence and nontriviality properties very much reminiscent of the Hausdorff gap phenomenon that will be a subject of our study in Section 3.1 below. Definition 2.4.1. The number of steps of the minimal walk is the two-place function ρ2 : [ω1 ]2 −→ ω defined recursively by ρ2 (α, β) = ρ2 (α, min(Cβ \ α)) + 1, with the boundary condition ρ2 (γ, γ) = 0 for all γ. This is an interesting mapping which is particularly useful on higher cardinalities and especially in situations where the more informative mappings ρ0 , ρ1 and ρ lack their usual coherence properties. Later on we shall devote a whole section to ρ2 but here we list only few of its basic properties. We start with the coherence property that this function enjoys. Lemma 2.4.2. sup{|ρ2 (ξ, α) − ρ2 (ξ, β)| : ξ < α} < ∞ for all α < β < ω1 . Proof. Suppose the conclusion of the lemma fails for some α < β < ω1 . Then for every k < ω, we can find ξk < α such that |ρ2 (ξk , α) − ρ2 (ξk , β)| > k. We may assume that the sequence of ξk ’s is strictly increasing and let δ = supk ξk . Then δ is a limit ordinal ≤ α, so the lower traces of walks from α to δ and β to δ have a common upper bound γ < δ. Then by Lemma 2.1.6, for every ordinal ξ ∈ [γ, δ), we have that ρ0 (ξ, α) = ρ0 (α, δ)
ρ0 (δ, ξ) and ρ0 (ξ, β) = ρ0 (β, δ)
ρ0 (δ, ξ).

(2.4.1)

48

Chapter 2. Walks on Countable Ordinals

It follows that for every ξ ∈ [γ, δ), |ρ2 (ξ, α) − ρ2 (ξ, β)| ≤ |ρ2 (δ, α) − ρ2 (δ, β)|,

(2.4.2)

and so in particular, ξk ∈ / [γ, δ) for all k such that k > |ρ2 (δ, α) − ρ2 (δ, β)|, a contradiction.  We mention also the following unboundedness property of this function which introduces another theme to be explored fully in later sections of this book. Lemma 2.4.3. For every uncountable family A of pairwise-disjoint finite subsets of ω1 , all of some fixed size n, and for every integer k, there exist an uncountable subfamily B of A such that for all a < b in B, we have ρ2 (a(i), b(j)) ≥ k for all i, j < n.26 Proof. The proof is by induction on k. So suppose that our given family A already satisfies that ρ2 (a(i), b(j)) ≥ k for all i, j < n and all a < b in A. We shall find uncountable B ⊆ A such that ρ2 (a(i), b(j)) ≥ k + 1 for all i, j < n and all a < b in B. To this end, for each limit ordinal δ < ω1 , we fix bδ > δ in A. Then there is ηδ < δ such that ρ0 (ξ, β) = ρ0 (β, δ)
ρ0 (δ, ξ) for all ξ ∈ [ηδ , δ) and β ∈ bδ .

(2.4.3)

Find a stationary set Γ ⊆ ω1 and γ < ω1 such that ηδ = η for all δ ∈ Γ. Choose a stationary subset Ξ of Γ such that γ < bγ < δ < bδ for all γ < δ in Ξ.

(2.4.4)

Form the family A∗ = {{γ} ∪ bγ : γ ∈ Ξ} of pairwise-disjoint sets of size n + 1. By the inductive hypothesis there is uncountable B ∗ ⊆ A∗ such that ρ2 (a(i), b(j)) ≥ k for all i, j < n + 1 and all a < b in B ∗ . Then the uncountable subfamily B = {b \ {min(b)} : b ∈ B ∗ } of A has the property that ρ2 (a(i), b(j)) ≥ k + 1 for all i, j < n and all a < b in B.  The unboundedness property of ρ2 given by Lemma 2.4.3 shows, in particular, that while the fiber maps ρ2 (·, α) (α < ω1 ) are at a finite distance from each other, there is no global map with domain ω1 that has a finite distance from all of these fibers. Corollary 2.4.4. For every g : ω1 −→ ω there is α < ω1 such that sup{|ρ2 (ξ, α) − g(ξ))| : ξ < α} = ∞. 26 Here,

a < b signifies the fact α < β whenever α ∈ a and β ∈ b, while a(i) denotes the ith element of a relative to its increasing enumeration {a(0), a(1), . . . , a(n − 1)}.

2.4. The number of steps and the last step functions

49

Proof. Otherwise we can find an integer k and an unbounded set Γ ⊆ ω1 such that sup{|ρ2 (ξ, α) − g(ξ))| : ξ < α} ≤ k for all α ∈ Γ. Find unbounded Ξ ⊆ ω1 and an integer l such that g takes the constant value l on Ξ. Forming a family of pairs by taking an element from Ξ and the other from Γ and applying Lemma 2.4.3, we can find uncountable sets Ξ0 ⊆ Ξ and Γ0 ⊆ Γ such that ρ2 (ξ, α) > k + l + 1 for all ξ ∈ Ξ0 , α ∈ Γ0 such that ξ < α.

(2.4.5)

Consider an α ∈ Γ0 that is above ξ0 = min Ξ0 . Then k ≥ |ρ2 (ξ0 , α) − g(ξ0 ))| > k + l + 1 − l = k + 1,

(2.4.6) 

a contradiction.

The main object of study in this section, however, is another natural characteristic of the minimal walk. This new mapping ρ3 not only has strong coherence properties but its full analysis will lead us naturally to some finer requirements that one can put on the given C-sequences on which we base our walks. Definition 2.4.5. The last step function is the characteristic ρ3 : [ω1 ]2 −→ 2 of the walk defined by letting ρ3 (α, β) = 1 iff ρ0 (α, β)(ρ2 (α, β) − 1) = ρ1 (α, β). In other words, we let ρ3 (α, β) = 1 just in case the last step of the walk β → α comes with the maximal weight. Lemma 2.4.6. {ξ < α : ρ3 (ξ, α) = ρ3 (ξ, β)} is finite for all α < β < ω1 . Proof. It suffices to show that for every infinite Γ ⊆ α there exists ξ ∈ Γ such that ¯ ∈ F (α, β) ρ3 (ξ, α) = ρ3 (ξ, β). Shrinking Γ we may assume that for some fixed α and all ξ ∈ Γ: α ¯ = min(F (α, β) \ ξ), (2.4.7) ρ1 (ξ, α) = ρ1 (ξ, β),

(2.4.8)

ρ1 (ξ, α) > ρ1 (¯ α, α),

(2.4.9)

ρ1 (ξ, β) > ρ1 (¯ α, β).

(2.4.10)

It follows (see 2.1.9) that for every ξ ∈ Γ: α, α)
ρ0 (ξ, α), ¯ ρ0 (ξ, α) = ρ0 (¯

(2.4.11)

ρ0 (ξ, β) = ρ0 (¯ α, β)
ρ0 (ξ, α). ¯

(2.4.12)

¯ is its maximal term iff So for any ξ ∈ Γ, ρ3 (ξ, α) = 1 iff the last term of ρ0 (ξ, α) ρ3 (ξ, β) = 1. 

50

Chapter 2. Walks on Countable Ordinals

The sequence (ρ3 )α : α −→ 2 (α < ω1 )27 is therefore coherent in the sense that (ρ3 )α =∗ (ρ3 )β
α whenever α < β. We need to show that the sequence is not trivial, i.e., that it cannot be uniformized by a single total map from ω1 into 2. In other words, we need to show that ρ3 still contains enough information about the C-sequence Cα (α < ω1 ) from which it is defined. For this it will be convenient to assume that Cα (α < ω1 ) satisfies the following natural condition: (d) If α = λ + ω for some limit ordinal λ, then Cα = {λ + n : m < n < ω} for some non-negative integer m; if α is a limit of limit ordinals and if ξ occupies the nth place in the increasing enumeration of Cα , then ξ = λ + m for some limit ordinal λ and integer m > n. Definition 2.4.7. Let Λ denote the set of all countable limit ordinals and for an integer n ∈ ω, let Λ + n = {λ + n : λ ∈ Λ}. Lemma 2.4.8. ρ3 (λ + n, β) = 1 for all but finitely many n with λ + n < β. Proof. Clearly we may assume that α = λ + ω ≤ β. Then there is n0 < ω such that for every n ≥ n0 the walk β → λ + n passes through α. By (d) we know that Cα = {λ + n : m < n < ω} for some non-negative integer m, so in any such walk β → λ + n there is only one step from α to λ + n. So choosing n > n0 , m, ρ1 (α, β) we will ensure that the last step of β → λ + n comes with the maximal weight,  i.e., ρ3 (λ + n, β) = 1. Lemma 2.4.9. For all β < ω1 , n < ω, the set {λ ∈ Λ : λ + n < β and ρ3 (λ + n, β) = 1} is finite. Proof. Given an infinite subset Γ of (Λ + n) ∩ β we need to find a λ + n ∈ Γ such that ρ3 (λ + n, β) = 0. Shrinking Γ if necessary assume that ρ1 (λ + n, β) > n + 2 for all λ + n ∈ Γ. So if ρ3 (λ + n, β) = 1 for some λ + n ∈ Γ, then the last step of β → λ + n would have to be of weight > n + 2 which is impossible by our  assumption (d) about Cα (α < ω1 ). The meaning of these properties of ρ3 is perhaps easier to comprehend if we reformulate them in a way that resembles the original formulation of the existence of Hausdorff gaps. It is not surprising that this sort of variation on the classical Hausdorff gap phenomenon has appeared first in a topological study that analyses how the space of subuniform 28 ultrafilters on ω1 is embedded into the space of all ultrafilters on ω1 . 27 Recall the way one always defines the fiber functions from a two-variable function applied to the context of ρ3 : (ρ3 )α (ξ) = ρ3 (ξ, α). 28 An ultrafilter on ω is subuniform if it is nonprincipal and if it concentrates on a countable 1 subset of ω1 . The space SU(ω1 ) of subuniform ultrafilters on ω1 is an open subspace of the ˇ Cech–Stone remainder of the discrete space on ω1 . Lemma 2.4.10 says that there is a continuous {0, 1}-valued function on SU(ω1 ) which does not extend to a continuous function defined on the ˇ whole Cech–Stone remainder (see [131] and [23]).

2.4. The number of steps and the last step functions

51

Lemma 2.4.10. Let Bα = {ξ < α : ρ3 (ξ, α) = 1} for α < ω1 . Then: 1. Bα =∗ Bβ ∩ α for α < β, 2. (Λ + n) ∩ Bβ is finite for all n < ω and β < ω1 , 3. {λ + n : n < ω} ⊆∗ Bβ whenever λ + ω ≤ β.



In particular, there is is no uncountable Γ ⊆ ω1 such that Γ ∩ β ⊆∗ Bβ for all β < ω1 , and so the tree T (ρ3 ) = {(ρ3 )β
α : α ≤ β < ω1 } contains no uncountable chains. On the other hand, the P-ideal29 I generated by Bβ (β < ω1 ) is large as it contains all intervals of the form [λ, λ + ω). The following general dichotomy about P-ideals shows that here indeed we have quite a canonical example of a P-ideal on ω1 . Definition 2.4.11. The P-ideal dichotomy. For every P-ideal I of countable subsets of some set S either: 1. there is uncountable X ⊆ S such that [X]ω ⊆ I, or 2. S can be decomposed into countably many sets orthogonal to I. Remark 2.4.12. It is known that the P-ideal dichotomy is a consequence of the Proper Forcing Axiom and moreover that it does not contradict the Continuum Hypothesis (see [122]). This is an interesting dichotomy which will be used in this article for testing various notions of coherence as we encounter them. In fact, this was the original and still most important reason for isolating this dichotomy from the rest of the consequences of PFA (see[111], Chapter 8). Let us now turn our attention to the lexicographical ordering given by ρ3 , or more precisely, the lexicographical ordering among the fibers (ρ3 )α of ρ3 induced by the global lexicographical ordering of the coherent tree T (ρ3 ). Definition 2.4.13. Consider the linear ordering