Terence Tao
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| 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, visit www.ams.org/bookpages/mbk-81 Library of Congress Cataloging-in-Publication Data Library of Congress Cataloging-in-Publication Data has been applied for by the AMS. See www.loc.gov/publish/cip/ ISBN: 978-0-8218-9492-7 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to reprint-permission@ams.org. 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 guidelines established 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 vagueness paradox 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 zeta function, and real-variable analytic continuation 88 §3.8. Finitary consequences of the invariant subspace problem 104 §3.9. The Guth-Katz result on the Erd˝os 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 precision arithmetic 133
§4.2. Non-Standard analysis as algebraic analysis 136
§4.3. Compactness and contradiction: the correspondence principle in 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 dispersive equations 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 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 |E|. We will use the asymptotic notation X = O(Y ), X Y , or Y X to denote the estimate |X| ≤ 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 = O d (Y ) or X d Y .
We also sometimes use X ∼ Y as a synonym for X Y X.
infinity. When that occurs, we also use the notation o n →∞ (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., o n →∞;d (X). Asymptotic notation is discussed further in Section 3.5. We will occasionally use the averaging notation E x ∈X f (x) :=
1 |X|
x ∈X 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 1 E : X
→ R to denote the indicator function of X, thus 1 E (x) equals 1 when x ∈ E and 0 otherwise. Acknowledgments I am greatly indebted to many readers of my blog and buzz feed, including Rex Cheung, Dan Christensen, David Corfield, Quinn Culver, Tim Gow- ers, Greg Graviton, Zaher Hani, Bryan Jacobs, Bo Jacoby, Sune Kristian Jakobsen, Allen Knutson, Ulrich Kohlenbach, Diego Maldona, Mark Meckes, David Milovich, Timothy Nguyen, Michael Nielsen, Matthew Petersen, An- thony Quas, Pedro Lauridsen Ribeiro, Jason Rute, Am´ erico Tavares, Willie Wong, Qiaochu Yuan, Pavel Zorin, and several 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. Bibliography [Ar1966] V. Arnold, Sur la g´ eom´ etrie diff´ erentielle des groupes de Lie de dimension in- finie et ses applications ` a l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 1966 fasc. 1, 319–361. [ArKh1998] V. Arnold, B. Khesin, Topological methods in hydrodynamics. (English sum- mary) Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998. [AuTa2010] T. Austin, T. Tao, On the testability and repair of hereditary hypergraph properties, Random Structures and Algorithms 36 (2010), 373–463. [AvGeTo2010] J. Avigad, P. Gerhardy, H. Towsner, Local stability of ergodic averages, Trans. Amer. Math. Soc. 362 (2010), no. 1, 261–288. [BeCaTa2006] J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261–302. [BeHoMcPa2000] V. Bergelson, B. Host, R. McCutcheon, F. Parreau, Aspects of unifor- mity in recurrence. Colloq. Math. 84/85 (2000), part 2, 549–576. [BeTaZi2010] V. Bergelson, T. Tao, T. Ziegler, An inverse theorem for the uniformity seminorms associated with the action of F p , Geom. Funct. Anal. 19 (2010), no. 6, 1539–1596. [Bo1991] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. [Bo1995] J. Bourgain, Estimates for cone multipliers, Geometric aspects of functional analysis (Israel, 1992–1994), 41–60, Oper. Theory Adv. Appl., 77, Birkh¨ auser, Basel, 1995. [BoGu2010] J. Bourgain, L. Guth, Bounds on oscillatory integral operators based on mul- tilinear estimates, Geom. Funct. Anal., 21 (2011), no. 6, 1239–1295. [BoSm1975] J. Bona, R. Smith, The initial-value problem for the Korteweg-de Vries equa- tion, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601. [CeMaPeRa2003] H. Cendra, J. Marsden, S. Pekarsky, T. Ratiu, Variational principles for Lie-Poisson and Hamilton-Poincar´ e equations, Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday. Mosc. Math. J. 3 (2003), no. 3, 833-867, 1197–1198. 249
250 Bibliography [ChSzTr1992] F. Chung, E. Szemer´ edi, W. T. Trotter, The number of different distances determined by a set of points in the Euclidean plane, Discrete Comput. Geom. 7 (1992), no. 1, 1–11. [CoMi1997] T. Colding, W. Minicozzi, II. Harmonic functions on manifolds, Ann. Math. 146 (1997), 725–747. [Co1982] A. C´ ordoba, Geometric Fourier analysis, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vii, 215–226. [Co1985] A. C´ ordoba, Restriction lemmas, spherical summation, maximal functions, square functions and all that, Recent Progress in Fourier Analysis (El Escorial, 1983), 57–64, North-Holland Math. Stud., 111, North-Holland, Amsterdam, 1985, [De1912] M. Dehn, Transformation der Kurven auf zweiseitigen Fl¨ achen, Math. Ann. 72 (1912), no. 3, 413–421. [Dv2009] Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), no. 4, 1093–1097. [EbMa1970] D. Ebin, J. Marsden, Groups of diffeomorphisms and the motion of an in- compressible fluid, Ann. of Math. 92 (1970) 102–163. [ElSz2007] G. Elek, B. Szegedy, Limits of Hypergraphs, Removal and Regularity Lemmas. A Non-standard Approach, arXiv:0705.2179 [ElSh2010] G. Elekes, M. Sharir, Incidences in three dimensions and distinct distances in the plane, Computational geometry (SCG’10), 413–422, ACM, New York, 2010. [Er1946] P. Erd˝ os, On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–250. [FrVi1990] S. Friedlander, M. Vishik, Lax pair formulation for the Euler equation, Phys. Lett. A 148 (1990), no. 6-7, 313–319. [Ge1936] G. Gentzen, Die Widerspruchfreiheit der reinen Zahlentheorie, Mathematische Annalen 112 (1936), 493–565. [Go2008] W. T. Gowers, How can one equivalent statement be stronger than another?, gowers.wordpress.com/2008/12/28 [Go2010] W. T. Gowers, Decompositions, approximate structure, transference, and the Hahn-Banach theorem, Bull. Lond. Math. Soc. 42 (2010), no. 4, 573–606. [Gr2005] B. Green, Roth’s theorem in the primes, Ann. of Math. (2) 161 (2005), no. 3, 1609–1636. [GrTa2008] B. Green, T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167 (2008), 481–547. [Gr1981] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes ´ Etudes Sci. Publ. Math. No. 53 (1981), 53–73. [Gu2010] L. Guth, The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya con- jecture, Acta Math. 205 (2010), no. 2, 263–286. [GuKa2010] L. Guth, N. Katz, Algebraic methods in discrete analogs of the Kakeya prob- lem, Adv. Math. 225 (2010), no. 5, 2828–2839. [GuKa2010b] L. Guth, N.H. Katz, On the Erd˝ os distinct distance problem in the plane, arXiv:1011.4105v1 [math.CO] [HeJeKoSt2009] S. Herrmann, A. Jensen, M. Joswig, B. Sturmfels, How to draw tropical planes, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Bjorner, Research Paper 6, 26 pp. Bibliography 251
[KaTa2004] N. Katz, G. Tardos, A new entropy inequality for the Erdos distance problem, Towards a theory of geometric graphs, 119–126, Contemp. Math., 342, Amer. Math. Soc., Providence, RI, 2004. [KiVa2008] R. Killip, M. Visan, Nonlinear Schr¨ odinger Equations at Critical Regularity, www.math.ucla.edu/ ∼visan/ClayLectureNotes.pdf [Kl2010] B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth, J. Amer. Math. Soc. 23 (2010), no. 3, 815–829. [KoSc1997] N. Korevaar, R. Schoen, Global existence theorems for harmonic maps to non- locally compact spaces, Comm. Anal. Geom. 5 (1997), no. 2, 333–387. [LaPi2011] M. Larsen, R. Pink, Finite Subgroups of Algebraic Groups, J. Amer. Math. Soc. 24 (2011), no. 4, 1105–1158. [Li2001] Y. Li, A Lax pair for the two dimensional Euler equation, J. Math. Phys. 42 (2001), no. 8, 3552–3553. [Lo1955] J. Los, Quelques remarques, th´ eor` emes et probl` emes sur les classes d´ efinissables d’alg` ebres, in: Mathematical Interpretation of Formal Systems, North-Holland, Am- sterdam, 1955, 98–113. [MaRa1999] J. Marsden, T. Ratiu, Introduction to mechanics and symmetry, A basic exposition of classical mechanical systems. Second edition. Texts in Applied Mathe- matics, 17. Springer-Verlag, New York, 1999. [Mi1968] J. Milnor, Growth of finitely generated solvable groups, J. Diff. Geom. 2 (1968), 447–449.
[Mo1995] N. Mok, Harmonic forms with values in locally compact Hilbert bundles, in Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay 1993), Special Issue, 1995, pp. 433–454. [Na1999] K. Nakanishi, Scattering theory for the nonlinear Klein-Gordon equation with Sobolev critical power, Internat. Math. Res. Notices 1999, no. 1, 31–60. [ReTrTuVa2008] O. Reingold, L. Trevisan, M. Tulsiani, S. Vadhan, New Proofs of the Green-Tao-Ziegler Dense Model Theorem: An Exposition, preprint. arXiv:0806.0381 [Ro1953] K.F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 245-252. [Ro2009] T. Roy, Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation, Anal. PDE 2 (2009), no. 3, 261–280. [RuSz1978] I. Ruzsa, E. Szemer´ edi, Triple systems with no six points carrying three tri- angles, Colloq. Math. Soc. J. Bolyai 18 (1978), 939–945. [Sa1915] G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2, 5th edition Hodges, Figgis and Co. Ltd. (1915). [ScTi2002] I. Schindler, K. Tintarev, An abstract version of the concentration compactness principle, Rev. Mat. Complut. 15 (2002), no. 2, 417–436. [ShTa2010] Y. Shalom, T. Tao, A finitary version of Gromov’s polynomial growth theorem, Geom. Funct. Anal. 20 (2010), no. 6, 1502–1547. [ShSt1998] J. Shatah, M. Struwe, Geometric wave equations. Courant Lecture Notes in Mathematics, 2. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. [Se1954] A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2), 60 (1954), 365–374. [Sh2011] H.-W. Shih, Some results on scattering for log-subcritical and log-supercritical nonlinear wave equation, to appear, Analysis & PDE. 252 Bibliography [St1979] E. M. Stein, Some problems in harmonic analysis. Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, pp. 3–20, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979. [Sz1975] E. Szemer´ edi, On sets of integers containing no k elements in arithmetic pro- gression, Acta Arith. 27 (1975), 299–345. [Sz1978] E. Szemer´ edi, Regular partitions of graphs, in “Problem´ es Combinatoires et Th´ eorie des Graphes, Proc. Colloque Inter. CNRS,” (Bermond, Fournier, Las Vergnas, Sotteau, eds.), CNRS Paris, 1978, 399–401. [SzTr1873] E. Szemer´ edi, W. T. Trotter Jr., Extremal problems in discrete geometry, Com- binatorica 3 (1983), 381–392. [Ta1951] A. Tarski, A decision method for elementary algebra and geometry, University of California Press, Berkeley and Los Angeles, Calif., 1951. [Ta1999] T. Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), no. 2, 363–375. [Ta2003] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–1384. [Ta2003b] T. Tao, Recent progress on the Restriction conjecture, arXiv:math.CA/0311181 [Ta2004] T. Tao, A remark on Goldston-Yıldırım correlation estimates, unpublished. www.math.ucla.edu/ ∼tao/preprints/Expository/gy-corr.dvi [Ta2006] T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, Dyn. Partial Differ. Equ. 3 (2006), no. 2, 93–110. [Ta2006b] T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS re- gional conference series in mathematics, 2006. [Ta2006c] T. Tao, Analysis Vols. I, II, Hindustan Book Agency, 2006. [Ta2007] T. Tao, Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data, J. Hyperbolic Differ. Equ. 4 (2007), no. 2, 259–265. [Ta2007b] T. Tao, A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma, J. d.Analyse Mathematique 103 (2007), 1–45. [Ta2008] T. Tao, Structure and Randomness: pages from year one of a mathematical blog, American Mathematical Society, Providence RI, 2008. [Ta2008b] T. Tao, Norm convergence of multiple ergodic averages for commuting trans- formations, Ergodic Theory and Dynamical Systems 28 (2008), 657–688. [Ta2009] T. Tao, Poincar´ e’s Legacies: pages from year two of a mathematical blog, Vol. I, American Mathematical Society, Providence RI, 2009. [Ta2009b] T. Tao, Poincar´ e’s Legacies: pages from year two of a mathematical blog, Vol. II, American Mathematical Society, Providence RI, 2009. [Ta2010] T. Tao, An epsilon of room, Vol. I, American Mathematical Society, Providence RI, 2010. [Ta2010b] T. Tao, An epsilon of room, Vol. II, American Mathematical Society, Provi- dence RI, 2010. [Ta2011] T. Tao, An introduction to measure theory, American Mathematical Society, Providence RI, 2011. [Ta2011b] T. Tao, Higher order Fourier analysis, American Mathematical Society, Provi- dence RI, 2011.
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[Ta2011c] T. Tao, Topics in random matrix theory, American Mathematical Society, Prov- idence RI, 2011. [TaVaVe1998] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), no. 4, 967–1000. [TaVu2006] T. Tao, V. Vu, Additive combinatorics. Cambridge Studies in Advanced Mathematics, 105. Cambridge University Press, Cambridge, 2006. [TaZi2010] T. Tao, T. Ziegler, The inverse conjecture for the Gowers norm over finite fields via the correspondence principle, Anal. PDE 3 (2010), no. 1, 1–20. [TaZi2011] T. Tao, T. Ziegler, The inverse conjecture for the Gowers norm over finite fields in low characteristic, Ann. Comb. 16 (2012), no. 1, 121–188. [Th1994] W. Thurston, On proof and progress in mathematics, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 161–177. [Ti1972] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. [To1975] P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. [Tz2006] N. Tzvetkov, Ill-posedness issues for nonlinear dispersive equations, Lectures on nonlinear dispersive equations, 63–103, GAKUTO Internat. Ser. Math. Sci. Appl., 27, Gakkotosho, Tokyo, 2006. [Va1959] P. Varnavides, On certain sets of positive density, J. London Math. Soc. 39 (1959), 358–360. [Wo1968] J. Wolf, Growth of finitely generated solvable groups and curvature of Riemann- ian manifolds, J. Diff. Geom. 2 (1968), 421–446. [Wo1995] T. Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–674. [Wo2001] T. Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. (2) 153 (2001), no. 3, 661–698. Index a priori estimate, 185 active transformation, 54 Archimedean principle, 133 Arzel´ a-Ascoli diagonalisation trick, 139 asymptotic notation, xii Balog-Szemer´ edi-Gowers lemma, 230 barrier, 107 Bayes’ formula, 236 Bayesian probability, 236 Bernoulli numbers, 96 Bolzano-Weierstrass theorem, 160 Burgers’ equation, 185 busy beaver function, 27 Cantor’s theorem, 21, 32 Cartan-Killing form, 211 Cayley graph, 57 cell decomposition, 121 characteristic subgroup, 67 Christoffel symbols, 205 cogeodesic flow, 205 continuity method, 232 coordinate system, 222 decomposition into varieties, 163 difference equation, 186 differentiating the equation, 185 direct product, 66 Duhamel’s formula, 182 elemengary convergence, 154 energy, 197 equipartition of energy, 203 Erd˝
os distance problem, 110 Euclid’s theorem, 19 Euler equations of incompressible fluids, 213
Euler-Arnold equation, 209 Euler-Maclaurin formula, 98 explicit formula, 102 extension problem, 124 Faulhaber formula, 90 finitely generated group, 57 friendship paradox, 229 Furstenberg correspondence principle, 137 Furstenberg recurrence theorem, 143, 164 G-space, 51 G¨ odel incompleteness theorem, 25 G¨ odel sentence, 24 G¨ odel’s universe, 34 Grandi’s series, 91 Gromov’s theorem, 69, 140 growth function, 105 harmonic function, 70 Heine-Borel theorem, 161 hereditary property, 68 homogeneous space, 51 impredicativity of truth, 24 indicator function, xii interesting number paradox, 31 invariant subspace problem, 104 255
256 Index
Jordan’s theorem, 76 Klein geometry, 112 Kleiner’s theorem, 70 lamplighter group, 52 length contraction, 226 Loeb measure, 165 mean ergodic theorem, 149 metabelian group, 66 metacyclic group, 66 modus ponens, 239 Morawetz inequality, 201 nilpotent group, 67 non-standard universe, 158 nonlinear wave equation, 194 Notation, xi null hypothesis, 238 omnipotence paradox, 28 oracle, 39 overspill principle, 162 passive transformation, 54 phase polynomial, 148 Picard iteration, 181 Poincar´ e inequality, 75 polycyclic group, 66 polynomial ham sandwich theorem, 120 problem of induction, 238 product rule, 220 profile decomposition, 174, 177 quasilinear equation, 183 Quining trick, 24 quotient rule, 220 regulus, 114 restriction problem, 124 semi-direct product, 66 semilinear equation, 182 sequential Banach-Alaoglu theorem, 169 Simpson’s paradox, 230 smoothed sums, 91 solvable group, 67 sorites paradox, 30 split exact sequence, 62 standard part, 172 stationary process, 142 supersolvable group, 66 Szemer´
edi regularity lemma, 166 Szemer´
edi’s theorem, 142, 164 Tarski’s undefinability theorem, 24 torsor, 51 tragedy of the commons, 235 transfer principle, 159 transport equation, 183 trapezoidal rule, 94 tropical algebra, 80 Turing’s halting theorem, 26 ultrapower, 158 underspill principle, 162 uniquely transitive, 51 universal set, 22 van der Waerden theorem, 163 virtual properties, 68 vorticity, 214 vorticity equation, 214 wave equation, 194 wave packet, 190 word metric, 57 Zorn’s lemma, 34
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