Terence Tao

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

Compactness and Contradiction

Compactness and Contradiction

Terence Tao

Compactness and Contradiction

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

Compactness and Contradiction

Terence Tao

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

http://dx.doi.org/10.1090/mbk/081

2010 Mathematics Subject Classification. Primary 00B15.

For additional information and updates on this book, visitwww.ams.org/bookpages/mbk-81

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See www.loc.gov/publish/cip/

ISBN: 978-0-8218-9492-7

Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

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c© 2013 Terence Tao. All rights reserved.Printed in the United States of America.

©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13

To Garth Gaudry, who set me on the road;

To my family, for their constant support;

And to the readers of my blog, for their feedback and contributions.

Contents

Preface xi

A remark on notation xi

Acknowledgments xii

Chapter 1. Logic and foundations 1

§1.1. Material implication 1

§1.2. Errors in mathematical proofs 2

§1.3. Mathematical strength 4

§1.4. Stable implications 6

§1.5. Notational conventions 8

§1.6. Abstraction 9

§1.7. Circular arguments 11

§1.8. The classical number systems 12

§1.9. Round numbers 15

§1.10. The “no-self-defeating object” argument, revisited 16

§1.11. The “no-self-defeating object” argument, and the vaguenessparadox 28

§1.12. A computational perspective on set theory 35

Chapter 2. Group theory 51

§2.1. Torsors 51

§2.2. Active and passive transformations 54

§2.3. Cayley graphs and the geometry of groups 56

§2.4. Group extensions 62

vii

viii Contents

§2.5. A proof of Gromov’s theorem 69

Chapter 3. Analysis 79

§3.1. Orders of magnitude, and tropical geometry 79

§3.2. Descriptive set theory vs. Lebesgue set theory 81

§3.3. Complex analysis vs. real analysis 82

§3.4. Sharp inequalities 85

§3.5. Implied constants and asymptotic notation 87

§3.6. Brownian snowflakes 88

§3.7. The Euler-Maclaurin formula, Bernoulli numbers, the zetafunction, and real-variable analytic continuation 88

§3.8. Finitary consequences of the invariant subspace problem 104

§3.9. The Guth-Katz result on the Erdos distance problem 110

§3.10. The Bourgain-Guth method for proving restriction theorems 123

Chapter 4. Non-Standard analysis 133

§4.1. Real numbers, non-standard real numbers, and finite precisionarithmetic 133

§4.2. Non-Standard analysis as algebraic analysis 136

§4.3. Compactness and contradiction: the correspondence principlein ergodic theory 137

§4.4. Non-Standard analysis as a completion of standard analysis 150

§4.5. Concentration compactness via non-standard analysis 168

Chapter 5. Partial differential equations 181

§5.1. Quasilinear well-posedness 181

§5.2. A type diagram for function spaces 189

§5.3. Amplitude-frequency dynamics for semilinear dispersiveequations 194

§5.4. The Euler-Arnold equation 203

Chapter 6. Miscellaneous 217

§6.1. Multiplicity of perspective 217

§6.2. Memorisation vs. derivation 220

§6.3. Coordinates 222

§6.4. Spatial scales 227

§6.5. Averaging 228

§6.6. What colour is the sun? 231

Contents ix

§6.7. Zeno’s paradoxes and induction 232

§6.8. Jevons’ paradox 233

§6.9. Bayesian probability 236

§6.10. Best, worst, and average-case analysis 242

§6.11. Duality 244

§6.12. Open and closed conditions 246

Bibliography 249

Index 255

Preface

In February of 2007, I converted my “What’s new” web page of researchupdates into a blog at terrytao.wordpress.com. This blog has since grownand evolved to cover a wide variety of mathematical topics, ranging from myown research updates, to lectures and guest posts by other mathematicians,to open problems, to class lecture notes, to expository articles at both basicand advanced levels. In 2010, I also started writing shorter mathematicalarticles on my Google Buzz feed at

profiles.google.com/114134834346472219368/buzz .

This book collects some selected articles from both my blog and my Buzzfeed from 2010, continuing a series of previous books [Ta2008], [Ta2009],[Ta2009b], [Ta2010], [Ta2010b], [Ta2011], [Ta2011b], [Ta2011c] basedon the blog.

The articles here are only loosely connected to each other, although manyof them share common themes (such as the titular use of compactness andcontradiction to connect finitary and infinitary mathematics to each other).I have grouped them loosely by the general area of mathematics they pertainto, although the dividing lines between these areas is somewhat blurry, andsome articles arguably span more than one category. Each chapter is roughlyorganised in increasing order of length and complexity (in particular, the firsthalf of each chapter is mostly devoted to the shorter articles from my Buzzfeed, with the second half comprising the longer articles from my blog).

A remark on notation

For reasons of space, we will not be able to define every single mathematicalterm that we use in this book. If a term is italicised for reasons other than

xi

xii Preface

emphasis or for definition, then it denotes a standard mathematical object,result, or concept, which can be easily looked up in any number of references.(In the blog version of the book, many of these terms were linked to theirWikipedia pages, or other on-line reference pages.)

I will, however, mention a few notational conventions that I will usethroughout. The cardinality of a finite set E will be denoted |E|. We willuse the asymptotic notation X = O(Y ), X � Y , or Y � X to denote theestimate |X| ≤ CY for some absolute constant C > 0. In some cases we willneed this constant C to depend on a parameter (e.g., d), in which case weshall indicate this dependence by subscripts, e.g., X = Od(Y ) or X �d Y .We also sometimes use X ∼ Y as a synonym for X � Y � X.

In many situations there will be a large parameter n that goes off toinfinity. When that occurs, we also use the notation on→∞(X) or simplyo(X) to denote any quantity bounded in magnitude by c(n)X, where c(n)is a function depending only on n that goes to zero as n goes to infinity. Ifwe need c(n) to depend on another parameter, e.g., d, we indicate this byfurther subscripts, e.g., on→∞;d(X).

Asymptotic notation is discussed further in Section 3.5.

We will occasionally use the averaging notation

Ex∈Xf(x) :=1

|X|∑x∈X

f(x)

to denote the average value of a function f : X → C on a non-empty finiteset X.

If E is a subset of a domain X, we use 1E : X → R to denote theindicator function of X, thus 1E(x) equals 1 when x ∈ E and 0 otherwise.

Acknowledgments

I am greatly indebted to many readers of my blog and buzz feed, includingRex Cheung, Dan Christensen, David Corfield, Quinn Culver, Tim Gow-ers, Greg Graviton, Zaher Hani, Bryan Jacobs, Bo Jacoby, Sune KristianJakobsen, Allen Knutson, Ulrich Kohlenbach, Diego Maldona, Mark Meckes,David Milovich, Timothy Nguyen, Michael Nielsen, Matthew Petersen, An-thony Quas, Pedro Lauridsen Ribeiro, Jason Rute, Americo Tavares, WillieWong, Qiaochu Yuan, Pavel Zorin, and several anonymous commenters, forcorrections and other comments, which can be viewed online at

terrytao.wordpress.com

The author is supported by a grant from the MacArthur Foundation, byNSF grant DMS-0649473, and by the NSF Waterman award.

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Index

a priori estimate, 185active transformation, 54Archimedean principle, 133Arzela-Ascoli diagonalisation trick, 139asymptotic notation, xii

Balog-Szemeredi-Gowers lemma, 230barrier, 107Bayes’ formula, 236Bayesian probability, 236Bernoulli numbers, 96Bolzano-Weierstrass theorem, 160Burgers’ equation, 185busy beaver function, 27

Cantor’s theorem, 21, 32Cartan-Killing form, 211Cayley graph, 57cell decomposition, 121characteristic subgroup, 67Christoffel symbols, 205cogeodesic flow, 205continuity method, 232coordinate system, 222

decomposition into varieties, 163difference equation, 186differentiating the equation, 185direct product, 66Duhamel’s formula, 182

elemengary convergence, 154energy, 197equipartition of energy, 203

Erdos distance problem, 110Euclid’s theorem, 19Euler equations of incompressible fluids,

213Euler-Arnold equation, 209Euler-Maclaurin formula, 98explicit formula, 102extension problem, 124

Faulhaber formula, 90finitely generated group, 57friendship paradox, 229Furstenberg correspondence principle,

137Furstenberg recurrence theorem, 143,

164

G-space, 51Godel incompleteness theorem, 25Godel sentence, 24Godel’s universe, 34Grandi’s series, 91Gromov’s theorem, 69, 140growth function, 105

harmonic function, 70Heine-Borel theorem, 161hereditary property, 68homogeneous space, 51

impredicativity of truth, 24indicator function, xiiinteresting number paradox, 31invariant subspace problem, 104

255

256 Index

Jordan’s theorem, 76

Klein geometry, 112Kleiner’s theorem, 70

lamplighter group, 52length contraction, 226Loeb measure, 165

mean ergodic theorem, 149metabelian group, 66metacyclic group, 66modus ponens, 239Morawetz inequality, 201

nilpotent group, 67non-standard universe, 158nonlinear wave equation, 194Notation, xinull hypothesis, 238

omnipotence paradox, 28oracle, 39overspill principle, 162

passive transformation, 54phase polynomial, 148Picard iteration, 181Poincare inequality, 75polycyclic group, 66polynomial ham sandwich theorem, 120problem of induction, 238product rule, 220profile decomposition, 174, 177

quasilinear equation, 183Quining trick, 24quotient rule, 220

regulus, 114restriction problem, 124

semi-direct product, 66semilinear equation, 182sequential Banach-Alaoglu theorem, 169Simpson’s paradox, 230smoothed sums, 91solvable group, 67sorites paradox, 30split exact sequence, 62standard part, 172stationary process, 142supersolvable group, 66Szemeredi regularity lemma, 166

Szemeredi’s theorem, 142, 164

Tarski’s undefinability theorem, 24torsor, 51tragedy of the commons, 235transfer principle, 159transport equation, 183trapezoidal rule, 94tropical algebra, 80Turing’s halting theorem, 26

ultrapower, 158underspill principle, 162uniquely transitive, 51universal set, 22

van der Waerden theorem, 163virtual properties, 68vorticity, 214vorticity equation, 214

wave equation, 194wave packet, 190word metric, 57

Zorn’s lemma, 34

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