Walks on Ordinals and Their Characteristics

327 Pages·2007·2.295 MB·English

by  Stevo Todorcevic (auth.)

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. Department of Mathematics
UMR 7056 University of Toronto
2, Place Jussieu – Case 7012 Toronto M5S 2E4
75251 Paris Cedex 05 Canada
France e-mail: [email protected]
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 Biblio-
thek 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
9 8 7 6 5 4 3 2 1 www.birkhauser.ch
Contents
1 Introduction
1.1 Walks and the metric theory of ordinals . . . . . . . . . . . . . . . 1
1.2 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Prerequisites and notation . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Walks on Countable Ordinals
2.1 Walks on countable ordinals and their basic characteristics. . . . . 19
2.2 The coherence of maximal weights . . . . . . . . . . . . . . . . . . 29
2.3 Oscillations of traces . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 The number of steps and the last step functions . . . . . . . . . . . 47
3 Metric Theory of Countable Ordinals
3.1 Triangle inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Constructing a Souslin tree using ρ . . . . . . . . . . . . . . . . . . 58
3.3 A Hausdorff gap from ρ . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 A general theory of subadditive functions on ω . . . . . . . . . . 66
1
3.5 Conditional weakly null sequences based on
subadditive functions . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Coherent Mappings and Trees
4.1 Coherent mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Lipschitz property of coherent trees . . . . . . . . . . . . . . . . . . 95
4.3 The global structure of the class of coherent trees . . . . . . . . . . 108
4.4 Lexicographically ordered coherent trees . . . . . . . . . . . . . . . 124
4.5 Stationary C-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5 The Square-bracket Operation on Countable Ordinals
5.1 The upper trace and the square-bracketoperation . . . . . . . . . 133
5.2 Projecting the square-bracketoperation . . . . . . . . . . . . . . . 139
5.3 Some geometrical applications of the
square-bracketoperation . . . . . . . . . . . . . . . . . . . . . . . . 144
vi Contents
5.4 A square-bracketoperation from a special Aronszajn tree . . . . . 152
5.5 A square-bracketoperation 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 . . . . . . . . . . . . . 187
7.2 Square sequences and local versions of ρ . . . . . . . . . . . . . . . 195
7.3 Special square sequence and the corresponding function ρ . . . . . 202
7.4 The function ρ on successors of regular cardinals . . . . . . . . . . 205
7.5 Forcing constructions based on ρ . . . . . . . . . . . . . . . . . . . 213
7.6 The function ρ on successors of singular cardinals . . . . . . . . . . 220
8 The Oscillation Mapping and the Square-bracket Operation
8.1 The oscillation mapping . . . . . . . . . . . . . . . . . . . . . . . . 233
8.2 The trace filter and the square-bracketoperation . . . . . . . . . . 243
8.3 Projections of the square-bracketoperation
on accessible cardinals . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.4 Two more variations on the square-bracketoperation . . . . . . . . 257
9 Unbounded Functions
9.1 Partial square-sequences . . . . . . . . . . . . . . . . . . . . . . . . 271
9.2 Unbounded subadditive functions . . . . . . . . . . . . . . . . . . . 273
9.3 Chang’s conjecture and Θ . . . . . . . . . . . . . . . . . . . . . . 277
2
9.4 Higher dimensions and the continuum hypothesis . . . . . . . . . . 283
10 Higher Dimensions
10.1 Stepping-up to higher dimensions . . . . . . . . . . . . . . . . . . . 289
10.2 Chang’s conjecture as a 3-dimensional
Ramsey-theoretic statement . . . . . . . . . . . . . . . . . . . . . . 294
10.3 Three-dimensional oscillation mapping . . . . . . . . . . . . . . . . 298
10.4 Two-cardinalwalks . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Chapter 1
Introduction
1.1 Walks and the metric theory of ordinals
Thisbookis devotedtoaparticularrecursivemethodofconstructingmathemati-
calstructuresthat live ona givenordinalθ, using a single transformationξ (cid:1)→C
ξ
whichassignsto everyordinalξ <θ asetC ofsmallerordinalsthatisclosedand
ξ
unbounded in the set of ordinals <ξ. The transfinite sequence
C (ξ <θ)
ξ
whichwecalla‘C-sequence’andonwhichwebaseourrecursiveconstructionsmay
haveanumberof‘coherenceproperties’andweshallgiveadetailedstudyofthem
and the way they influence these constructions. Here, ‘coherence’ usually means
thattheC ’sarechoseninsomecanonicalway,beyondthealreadymentionedand
ξ
natural requirement that C is closed and unbounded in ξ for all ξ. For example,
ξ
choosing a canonical ‘fundamental sequence’ of sets C ⊆ ξ for ξ < ε , relying
ξ 0
on the specific properties of the Cantor normal form for ordinals below the first
ordinal satisfying the equation x = ωx, 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
C cannotonlybeusedas‘ladders’forclimbingupinrecursiveconstructionsbut
ξ
also as tools for ‘walking’ from an ordinal β to a smaller one α,
β =β0 >β1 >···>βn−1 >βn =α,
where the ‘step’ β →β is defined by letting β be the minimal point of C
i i+1 i+1 βi
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},
(cid:1)
where λ =max( i C ∩α) for i< n. The traces are usually used in defining
i j=0 βj
various binary operations on ordinals < θ, the most prominent of which is the
‘square-bracketoperation’that gives us a wayto transfer the quantifier ‘for every
unboundedset’tothequantifier‘foreveryclosedandunboundedset’.Itisperhaps
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.
Fromthemetrictheoryofordinalsbasedonanalysisofwalks,onealsolearns
that the triangle inequality of an ultrametric
(cid:9)(α,γ)≤max{(cid:9)(α,β),(cid:9)(β,γ)}
has three versions, depending on the natural ordering between the ordinals α, β
andγ.Thethreeversionsoftheinequalityareinfactofaquitedifferentcharacter
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
becomessomethingthatonecancall‘transitivity’of(cid:9).Considerablymoresubtleis
thecaseα<γ <β ofthisinequality1.Itisthiscaseoftheinequalitythatcaptures
mostof the coherence properties found in this article.It is also aninequality that
has proven to be quite useful in applications.
A large portion of the book is organized as a discussion of four basic char-
acteristics of the walk ρ,ρ ,ρ ,ρ and ρ . The reader may choose to follow the
0 1 2 3
analysis of any of these functions in various contexts. The characteristic ρ (α,β)
0
codes the entire walk β = β0 > β1 > ··· > βn−1 > βn = α by simply listing the
positions of β in the set C for i < n. While this looks simple-minded, the
i+1 βi
resulting mapping ρ is a rather remarkable object. For example, in the realm of
0
the space ω of countable ordinals, it gives us a canonical example of a special
1
Aronszajn tree of increasing sequences of rationals which has the additional re-
markable property that, when ordered lexicographically, its cartesian square can
be covered by countably many chains. In other words, the single characteristic
ρ of walks on countable ordinals gives two critical structures, one in the class of
0
1Itappearsthatthethirdcaseβ<α<γ ofthisinequalityisrarelyareasonableassumptionto
bemadeinthiscontext.
1.1. Walksand themetric theory of ordinals 3
so-called Lipschitz trees and the other in the class of linear orderings. For higher
cardinals θ, analysis of ρ leads us to some interesting finitary characterizations
0
of hyper inaccessible cardinals. This is given in some detail in Chapter 6 of this
book.
The characteristicρ (α,β)losesaconsiderableamountofinformationabout
1
the walk as it records only the maximal order type among the sets
{C ∩α,C ∩α,...,C ∩α}.
β0 β1 βn−1
Neverthelessitgivesusthefirstexampleofwhatwecalla‘coherentmapping’.The
class of coherent mappings and trees in the case θ = ω exhibits an unexpected
1
structurethatwestudyingreatdetailinChapter4ofthebook.Thefinestructure
intheclassof‘coherenttrees’isbasedonthemetricnotionofa‘Lipschitzmapping’
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 finalresolutionof the basis problemfor uncountable linearorderingsandthat
is presented in the same chapter. For higher cardinals θ the characteristic ρ and
1
its local versions offer a rich source of so-called ‘unbounded functions’ that have
some applications.
The characteristic ρ (α,β) simply counts the number of steps of the walk
2
from β to α. While this also looks rather simple minded, the remarkable proper-
ties of the corresponding function ρ become especially apparent on higher car-
2
dinals θ. Important properties of this characteristic are its coherence and its un-
boundedness.Thecoherencepropertyofρ requiresthecorrespondingC-sequence
2
C (ξ < θ) to be ‘coherent’ in the sense that C = C ∩α whenever α is a limit
ξ α β
pointofC .Ontheotherhand,theunboundednessofρ translatesintoarequire-
β 2
ment that the corresponding C-sequence C (ξ < θ) be ‘nontrivial’2, a condition
ξ
that eventually leads us to a simple and natural characterization of weakly com-
pact cardinals that we choose to reproduce in some detail in Chapter 6.
Finally,thecharacteristicρ (α,β)attachesoneofthedigits0or1tothewalk
3
according to the behavior of the last step βn−1 → βn = α. The full analysis of
thischaracteristicis currentlyavailableonlyinthe realsofthe spaceω ,where ρ
1 3
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
notionofaHausdorffgapinthequotientalgebraP(ω)/fin(anothercriticalobject
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’ofthewalk.Thedepthofthischaracteristicisapparenteveninthespaceω
1
ofcountableordinals,butitsfullpowercomesathighercardinalsθ andespecially
atθ thataresuccessorsofsingularcardinals.Thefullanalysisofthecharacteristic
ρrequiresC (ξ <θ)tobeaso-called‘squaresequence’orinotherwordsrequires
ξ
2Wesaythat Cξ (ξ<θ)isnontrivial ifthereisnoclosedandunbounded set C ⊆θ suchthat.
foralllimitpointsαofC,thereisβ≥αsuchthatC∩α⊆Cβ.