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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