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Compactness and contradiction Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: [email protected] 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 x1.1. Material implication 1 x1.2. Errors in mathematical proofs 2 x1.3. Mathematical strength 4 x1.4. Stable implications 6 x1.5. Notational conventions 8 x1.6. Abstraction 9 x1.7. Circular arguments 11 x1.8. The classical number systems 12 x1.9. Round numbers 15 x1.10. The \no self-defeating object" argument, revisited 16 x1.11. The \no self-defeating object" argument, and the vagueness paradox 28 x1.12. A computational perspective on set theory 35 Chapter 2. Group theory 51 x2.1. Torsors 51 x2.2. Active and passive transformations 54 x2.3. Cayley graphs and the geometry of groups 56 x2.4. Group extensions 62 vii viii Contents x2.5. A proof of Gromov's theorem 69 Chapter 3. Analysis 79 x3.1. Orders of magnitude, and tropical geometry 79 x3.2. Descriptive set theory vs. Lebesgue set theory 81 x3.3. Complex analysis vs. real analysis 82 x3.4. Sharp inequalities 85 x3.5. Implied constants and asymptotic notation 87 x3.6. Brownian snowflakes 88 x3.7. The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation 88 x3.8. Finitary consequences of the invariant subspace problem 104 x3.9. The Guth-Katz result on the Erd}osdistance problem 110 x3.10. The Bourgain-Guth method for proving restriction theorems 123 Chapter 4. Nonstandard analysis 133 x4.1. Real numbers, nonstandard real numbers, and finite precision arithmetic 133 x4.2. Nonstandard analysis as algebraic analysis 135 x4.3. Compactness and contradiction: the correspondence principle in ergodic theory 137 x4.4. Nonstandard analysis as a completion of standard analysis 150 x4.5. Concentration compactness via nonstandard analysis 167 Chapter 5. Partial differential equations 181 x5.1. Quasilinear well-posedness 181 x5.2. A type diagram for function spaces 189 x5.3. Amplitude-frequency dynamics for semilinear dispersive equations 194 x5.4. The Euler-Arnold equation 203 Chapter 6. Miscellaneous 217 x6.1. Multiplicity of perspective 218 x6.2. Memorisation vs. derivation 220 x6.3. Coordinates 223 x6.4. Spatial scales 227 x6.5. Averaging 229 x6.6. What colour is the sun? 231 Contents ix x6.7. Zeno's paradoxes and induction 233 x6.8. Jevons' paradox 234 x6.9. Bayesian probability 237 x6.10. Best, worst, and average-case analysis 242 x6.11. Duality 245 x6.12. Open and closed conditions 247 Bibliography 249 Index 255 Preface In February of 2007, I converted my \What's new" web page of research updates into a blog at terrytao.wordpress.com. This blog has since grown and evolved to cover a wide variety of mathematical topics, ranging from my own research updates, to lectures and guest posts by other mathematicians, to open problems, to class lecture notes, to expository articles at both basic and advanced levels. In 2010, I also started writing shorter mathematical articles on my Google Buzz feed at profiles.google.com/114134834346472219368/buzz . This book collects some selected articles from both my blog and my Buzz feed from 2010, continuing a series of previous books [Ta2008], [Ta2009], [Ta2009b], [Ta2010], [Ta2010b], [Ta2011], [Ta2011b], [Ta2011c] based on the blog. The articles here are only loosely connected to each other, although many of them share common themes (such as the titular use of compactness and contradiction to connect finitary and infinitary mathematics to each other). I have grouped them loosely by the general area of mathematics they pertain to, although the dividing lines between these areas is somewhat blurry, and some articles arguably span more than one category. Each chapter is roughly organised in increasing order of length and complexity (in particular, the first half of each chapter is mostly devoted to the shorter articles from my Buzz feed, 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 mathematical term 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 their Wikipedia pages, or other on-line reference pages.) I will however mention a few notational conventions that I will use throughout. The cardinality of a finite set E will be denoted jEj. We will use the asymptotic notation X = O(Y ), X Y , or Y X to denote the estimate jXj ≤ CY for some absolute constant C > 0. In some cases we will need this constant C to depend on a parameter (e.g. d), in which case we shall 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 to infinity. When that occurs, we also use the notation on!1(X) or simply o(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. If we need c(n) to depend on another parameter, e.g. d, we indicate this by further subscripts, e.g. on!1;d(X). Asymptotic notation is discussed further in Section 3.5. 1 P We will occasionally use the averaging notation Ex2X f(x) := jXj x2X f(x) to denote the average value of a function f : X ! C on a non-empty finite set X. If E is a subset of a domain X, we use 1E : X ! R to denote the indicator function of X, thus 1E(x) equals 1 when x 2 E and 0 otherwise. Acknowledgments I am greatly indebted to many readers of my blog and buzz feed, includ- ing Dan Christensen, David Corfield, Quinn Culver, Tim Gowers, Greg Graviton, Zaher Hani, Bryan Jacobs, Bo Jacoby, Sune Kristian Jakobsen, Allen Knutson, Ulrich Kohlenbach, Mark Meckes, David Milovich, Timothy Nguyen, Michael Nielsen, Anthony Quas, Pedro Lauridsen Ribeiro, Jason Rute, Am´ericoTavares, Willie Wong, Qiaochu Yuan, Pavel Zorin, and sev- eral anonymous commenters, for corrections and other comments, which can be viewed online at terrytao.wordpress.com The author is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award. Chapter 1 Logic and foundations 1.1. Material implication The material implication \If A, then B" (or \A implies B") can be thought of as the assertion \B is at least as true as A" (or equivalently, \A is at most as true as B"). This perspective sheds light on several facts about the material implication: (1) A falsehood implies anything (the principle of explosion). Indeed, any statement B is at least as true as a falsehood. By the same token, if the hypothesis of an implication fails, this reveals nothing about the conclusion. (2) Anything implies a truth. In particular, if the conclusion of an implication is true, this reveals nothing about the hypothesis. (3) Proofs by contradiction. If A is at most as true as a falsehood, then it is false. (4) Taking contrapositives. If B is at least as true as A, then A is at least as false as B. (5) \If and only if" is the same as logical equivalence.\A if and only if B" means that A and B are equally true. (6) Disjunction elimination. Given \If A, then C" and \If B, then C", we can deduce \If (A or B), then C", since if C is at least as true as A, and at least as true as B, then it is at least as true as either A or B. (7) The principle of mathematical induction. If P (0) is true, and each P (n + 1) is at least as true as P (n), then all of the P (n) are true. (Note, though, that if one is only 99% certain of each implication 1 2 1. Logic and foundations \P (n) implies P (n + 1)", then the chain of deductions can break down fairly quickly. It is thus dangerous to apply mathematical in- duction outside of rigorous mathematical settings. See also Section 6.9 for further discussion.) (8) Material implication is not causal. The material implication \Is A, then B" is a statement purely about the truth values of A and B, and can hold even if there is no causal link between A and B. (e.g. \If 1 + 1 = 2, then Fermat's last theorem is true.".) 1.2. Errors in mathematical proofs Formally, a mathematical proof consists of a sequence of mathematical state- ments and deductions (e.g. \If A, then B"), strung together in a logical fashion to create a conclusion. A simple example of this is a linear chain of deductions, such as A =) B =) C =) D =) E, to create the con- clusion A =) E. In practice, though, proofs tend to be more complicated than a linear chain, often acquiring a tree-like structure (or more generally, the structure of a directed acyclic graph), due to the need to branch into cases, or to reuse a hypothesis multiple times.
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