Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. A Guide to NIP Theories
  • Cited by 121
Publisher:
Cambridge University Press
Online publication date:
July 2015
Print publication year:
2015
Online ISBN:
9781107415133
DOI:
https://doi.org/10.1017/CBO9781107415133
Subjects:
Mathematics, Logic, Categories and Sets, Programming Languages and Applied Logic
Series:
Lecture Notes in Logic (44)
Information

Book description

The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to be written on NIP theories, is an introduction to the subject that will appeal to anyone interested in model theory: graduate students and researchers in the field, as well as those in nearby areas such as combinatorics and algebraic geometry. Without dwelling on any one particular topic, it covers all of the basic notions and gives the reader the tools needed to pursue research in this area. An effort has been made in each chapter to give a concise and elegant path to the main results and to stress the most useful ideas. Particular emphasis is put on honest definitions, handling of indiscernible sequences and measures. The relevant material from other fields of mathematics is made accessible to the logician.

Reviews

'This book presents NIP theories as a rich and coherent subject, showing a field with a considerable degree of development, particularly taking into account how recent most of the results are. Also, the author made a clear effort in presenting the most elegant proofs he could find, making this a very valuable book and accessible for any reader who understands model theory …'

Alf Onshuus Source: Mathematical Reviews

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
REFERENCES
[1] HANS, ADLER, >Kim's Lemma in NTP2theories, handwritten notes available on author's webpage: www.logic.univie.ac.at/˜adler/.
[2] HANS, ADLER, Strong theories, burden, and weight, draft, available on the author's webpage.
[3] HANS, ADLER, Theories controlled by formulas of Vapnik-Chervonenkis codimension 1, draft, available on the author's webpage.
[4] HANS, ADLER, An introduction to theories without the independence property, Archive of Mathematical Logic, (2008), to appear.
[5] NOGA, ALON, On the density of sets of vectors, Discrete Mathematics, vol. 46 (1983), no. 2, pp. 199–202.
[6] NOGA, ALON, and DANIEL J., KLEITMAN, Piercing convex sets and the Hadwiger-Debrunner (p, q) problem, Advances in Mathematics, vol. 96 (1992), pp. 103–112.
[7] NOGA, ALON and JOEL H., SPENCER, The Probabilistic Method, Wiley-Interscience series in discrete mathematics and optimization, Wiley, 2000.
[8] MATTHIAS, ASCHENBRENNER, ALF, DOLICH, DEIRDRE, HASKELL, DUGALD, MACPHERSON, and Sergei, Starchenko, Vapnik-Chervonenkis density in some theories without the independence property, I, to appear in the Transactions of the American Mathematical Society, 2013.
[9] MATTHIAS, ASCHENBRENNER, Vapnik-Chervonenkis density in some theories without the independence property, II, Notre Dame Journal of Formal Logic, vol. 54 (2013), no. 3–4, pp. 311–363.
[10] JAMES, AX and SIMON, KOCHEN, Diophantine problems over local fields II. a complete set of axioms for p-adic number theory, American Journal of Mathematics, vol. 87 (1965), no. 3, pp. 631–648.
[11] YERZHAN, BAISALOV and BRUNO, POIZAT, Paires de structures o-minimales, The Journal of Symbolic Logic, vol. 63 (1998), pp. 570–578.
[12] JOHN T., BALDWIN, Fundamentals of Stability Theory, Perspectives in Mathematical Logic, Springer-Verlag, 1988.
[13] JOHN T., BALDWIN and MICHAEL, BENEDIKT, Stability theory, permutations of indiscernibles, and embedded finite models, Transactions of the American Mathematical Society, vol. 352 (2000), no. 11, pp. 4937–4969.
[14] JOHN T., BALDWIN and JAN, SAXL, Logical stability in group theory, Journal of the Australian Mathematical Society (Series A), vol. 21 (1976), no. 3, pp. 267–276.
[15] LUC BÉLAIR, Types dans les corps valués munis d'applications coefficients, Illinois Journal of Mathematics, vol. 43 (1999), no. 2, pp. 410–425.
[16] OLEG, BELEGRADEK, YA'ACOV, PETERZIL, and FRANK, WAGNER, Quasio- minimal structures, The Journal of Symbolic Logic, vol. 65 (2000), no. 3, pp. 1115–1132.
[17] ITAï, BEN YAACOV, Continuous and random Vapnik-Chervonenkis classes, Israel Journal of Mathematics, vol. 173 (2009), pp. 309–333.
[18] ITAï|BEN YAACOV and Artem, Chernikov, An independence theorem for NTP2 theories, to appear in the Journal of Symbolic Logic.
[19] ALEXANDER, BERENSTEIN, ALFRED, DOLICH, and ALF, ONSHUUS, The independence property in generalized dense pairs of structures, The Journal of Symbolic Logic, vol. 76 (2011), no. 2, pp. 391–404.
[20] Gareth, Boxall, NIP for some pair-like theories, Archive for Mathematical Logic, vol. 50 (2011), pp. 353–359.
[21] ENRIQUE, CASANOVAS, Simple Theories and Hyperimaginaries, Lecture Notes in Logic, Cambridge University Press, 2011.
[22] ZOÉ, CHATZIDAKIS, Théorie des modèles des corps valuÉs, http://www.logique.jussieu.fr/˜zoe/.
[23] GREGORY L., CHERLIN, Model-Theoretic Algebra: Selected Topics, Lecture Notes in Mathematics, vol. 52, Springer-Verlag, Berlin, 1976.
[24] ARTEM, CHERNIKOV, Theories without the tree property of the second kind, Annals of Pure and Applied Logic, vol. 165 (2014), pp. 695–723.
[25] ARTEM, CHERNIKOV and ITAY, KAPLAN, Forking and dividing in NTP2 theories, The Journal of Symbolic Logic, vol. 77 (2012), no. 1, pp. 1–20.
[26] ARTEM, CHERNIKOV, ANAND, PILLAY, and PIERRE, SIMON, External definability and groups in NIP theories, to appear in the Journal of the London Mathematical Society.
[27] ARTEM, CHERNIKOV and PIERRE, SIMON, Externally definable sets and dependent pairs II, to appear in the Transactions of the American Mathematical Society.
[28] ARTEM, CHERNIKOV, Externally definable sets and dependent pairs, Israel Journal of Mathematics, vol. 194 (2013), no. 1, pp. 409–425.
[29] RAF, CLUCKERS and IMMANUEL, HALUPCZOK, Quantifier elimination in ordered abeliangroups, Confluentes Mathematici, vol. 3 (2011), pp. 587–615.
[30] ANNALISA, CONVERSANO and ANAND, PILLAY, Connected components of definable groups and o-minimality I, Advances in Mathematics, vol. 231 (2012), pp. 605–623.
[31] SARAH, COTTER and SERGEI, STARCHENKO, Forking in VC-minimal theories, The Journal of Symbolic Logic, vol. 77 (2012), no. 4, pp. 1257–1271.
[32] FRANC¸OISE, DELON, Types sur C((X)), Study Group on Stable Theories, Second year: 1978/79, Secrétariat Math., 1981.
[33] JAN, DENEF and LOU VAN DEN DRIES, p-adic and real subanalytic sets, The Annals of Mathematics, vol. 128 (1988), no. 1, pp. 79–138.
[34] MAX, DICKMANN, Elimination of quantifiers for ordered valuation rings, The Journal of Symbolic Logic, vol. 52 (1987), no. 1, pp. 116–128.
[35] ALFRED, DOLICH, DAVID, LIPPEL, and JOHN, GOODRICK, Dp-minimal theories: basic facts and examples, Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 267–288.
[36] CLIFTON, EALY, KRZYSZTOF, KRUPIŃNSKI, and ANAND, PILLAY, Superrosy dependent groups having finitely satisfiable generics, Annals of Pure and Applied Logic, vol. 151 (2008), pp. 1–21.
[37] CLIFTON, EALY and ALF, ONSHUUS, Characterizing rosy theories, The Journal of Symbolic Logic, vol. 72 (2007), no. 3, pp. 919–940.
[38] A. J., ENGLER and A., PRESTEL, Valued Fields, Springer Monographs in Mathematics, Springer, 2006.
[39] JOSEPH, FLENNER and VINCENT, GUINGONA, Convexly orderable groups and valued fields, to appear in the Journal of Symbolic Logic.
[40] SALLY, FLOYD and MANFRED, WARMUTH, Sample compression, learnability, and the Vapnik-Chervonenkis dimension, Machine Learning, 1995, pp. 269–304.
[41] PETER, FRANKL, On the trace of finite sets, Journal of Combinatorial Theory, Series A, vol. 34 (1983), no. 1, pp. 41–45.
[42] JAKUB, GISMATULLIN, Model theoretic connected components of groups, Israel Journal of Mathematics, vol. 184 (2011), pp. 251–274.
[43] JAKUB, GISMATULLIN and KRZYSZTOF, KRUPIŃNSKI, On model-theoretic connected components in some group extensions, preprint.
[44] JAKUB, GISMATULLIN, DAVIDE, PENAZZI, and ANAND, PILLAY, Some model theory of SL(2,R), preprint.
[45] JOHN, GOODRICK, A monotonicity theorem for dp-minimal densely ordered groups, The Journal of Symbolic Logic, vol. 75 (2010), no. 1, pp. 221–238.
[46] VINCENT, GUINGONA, Dependence and isolated extensions, Proceedings of the American Mathematical Society, vol. 139 (2011), pp. 3349–3357.
[47] VINCENT, GUINGONA, On uniform definability of types over finite sets, The Journal of Symbolic Logic, vol. 77 (2012), no. 2, pp. 499–514.
[48] VINCENT, GUINGONA and M.C., LASKOWSKI, On VC-minimal theories and variants, Archive for Mathematical Logic, vol. 52 (2013), no. 7, pp. 743–758.
[49] AYHAN, GÜNAYDIN and PHILIPP, HIERONYMI, Dependent pairs, The Journal of Symbolic Logic, vol. 76 (2011), no. 2, pp. 377–390.
[50] YURI, GUREVICH and PETER H., SCHMITT, The theory of ordered abelian groups does not have the independence property, Transactions of the American Mathematical Society, vol. 284 (1984), no. 1, pp. 171–182.
[51] DEIRDRE, HASKELL, EHUD, HRUSHOVSKI, and DUGALD, MACPHERSON, Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries, Journal für die reine und angewandte Mathematik, vol. 597 (2006), pp. 175–236.
[52] DEIRDRE, HASKELL, Stable Domination and Independence in Algebraically Closed Valued Fields, Lecture Notes in Logic, vol. 30, Association for Symbolic Logic, Chicago, IL, 2008.
[53] DEIRDRE, HASKELL and DUGALD, MACPHERSON, Cell decompositions of C-minimal structures, Annals of Pure and Applied Logic, vol. 66 (1994), no. 2, pp. 113–162.
[54] ASSAF, HASSON and ALF, ONSHUUS, Stable types in rosy theories, The Journal of Symbolic Logic, vol. 75 (2010), no. 4, pp. 1211–1230.
[55] ASSAF, HASSON, Unstable structures definable in o-minimal theories, Selecta Mathematica, vol. 16 (2010), no. 1, pp. 121–143.
[56] WILFRID, HODGES, A Shorter Model Theory, Cambridge University Press, 1997.
[57] WILFRID, HODGES, A. H., LACHLAN, and SAHARON, SHELAH, Possible orderings of an indiscernible sequence, Bulletin of the London Mathematical Society, vol. 9 (1977), no. 2, pp. 212–215.
[58] EHUD, HRUSHOVSKI and DAVID, KAZHDAN, Integration in valued fields, Algebraic Geometry and Number Theory (Victor Ginzburg, editor), Progress in Mathematics, vol. 253, Birkhäauser Boston, 2006, pp. 261–405.
[59] EHUD, HRUSHOVSKI and FRANCOIS, LOESER, Non-archimedean tame topology and stably dominated types, preprint.
[60] EHUD, HRUSHOVSKI, YA'ACOV, PETERZIL, and ANAND, PILLAY, Groups, measures, and the NIP, Journal of the American Mathematical Society, vol. 21(2008), no. 2, pp. 563–596.
[61] EHUD, HRUSHOVSKI and ANAND, PILLAY, On NIP and invariant measures, Journal of the European Mathematical Society, vol. 13 (2011), pp. 1005–1061.
[62] EHUD, HRUSHOVSKI, Anand, Pillay, and Pierre, Simon, A note on generically stable measures and fsg groups, Notre Dame Journal of Formal Logic, vol. 53 (2012), no. 4, pp. 599–605.
[63] EHUD, HRUSHOVSKI, Generically stable and smooth measures in NIP theories, Transactions of the American Mathematical Society, vol. 365 (2013), pp. 2341–2366.
[64] HUNTER R., JOHNSON and MICHEAL C., LASKOWSKI, Compression schemes, stable de. nable families, and o- minimal structures, Discrete & Computational Geometry, vol. 43 (2010), no. 4, pp. 914–926.
[65] ITAY, KAPLAN, ALF, ONSHUUS, and ALEX, USVYATSOV, Additivity of the dp-rank, Transactions of the American Mathematical Society, vol. 365 (2013), pp. 5783–5804.
[66] ITAY, KAPLAN, THOMAS, SCANLON, and FRANK O., WAGNER, Artin-Schreier extensions in NIP and simple fields, Israel Journal of Mathematics, vol. 185 (2011), no. 1, pp. 141–153.
[67] ITAY, KAPLAN and SAHARON, SHELAH, A dependent theory with few indiscernibles, to appear in the Israel Journal of Mathematics.
[68] ITAY, KAPLAN, Examples in dependent theories, to appear in the Journal of Symbolic Logic.
[69] ITAY, KAPLAN, Chain conditions in dependent groups, Annals of Pure and Applied Logic, vol. 164 (2013), no. 12, pp. 1322–1337.
[70] ITAY, KAPLAN and ALEX, USVYATSOV, Strict independence in dependent theories, to appear in the Journal of Mathematical Logic.
[71] H. JEROME, KEISLER, Measures and forking, Annals of Pure and Applied Logic, vol. 34 (1987), no. 2, pp. 119–169.
[72] KRZYSZTOF, KRUPIŃnski, Fields interpretable in rosy theories, Israel Journal of Mathematics, vol. 175 (2010), pp. 421–444.
[73] KRZYSZTOF, KRUPIŃnski, Fields interpretable in superrosy groups with NIP (the nonsolvable case), The Journal of Symbolic Logic, vol. 75 (2010), pp. 372–386.
[74] MICHAEL, C. LASKOWSKI, Vapnik-Chervonenkis classes of definable sets, Journal of the London Mathematical Society, vol. 45 (1992), pp. 377–384.
[75] ROI, LIVNI and PIERRE, SIMON, Honest compressions and their application to compression schemes, Proceedings of the 26th Conference on Learning Theory (COLT), 2013.
[76] NEWELSKI, LUDOMIR, Topological dynamics of definable group actions, The Journal of Symbolic Logic, vol. 74 (2009), pp. 50–72.
[77] ANGUS, MACINTYRE, On definable subsets of p-adic fields, The Journal of Symbolic Logic, vol. 41 (1976), no. 3, pp. 605–610.
[78] DUGALD, MACPHERSON, DAVID, MARKER, and CHARLES, STEINHORN, Weakly o-minimal structures and real closed fields, Transactions of the American Mathematical Society, vol. 352 (2000), no. 12, pp. 5435–5483.
[79] DUGALD, MACPHERSON and CHARLES, STEINHORN, On variants of ominimality, Annals of Pure and Applied Logic, vol. 79 (1996), no. 2, pp. 165–209.
[80] DUGALD, MACPHERSON and KATRIN, TENT, Pseudofinite groups with NIP theory and de.nability in finite simple groups, Groups and Model Theory, Contemp. Math. Series, vol. 576, AMS, 2012.
[81] DAVID, MARKER, Model Theory: An Introduction, Springer, 2002.
[82] JIRĺ, MATOUSEK, Bounded VC-dimension implies a fractional Helly theorem, Discrete & Computational Geometry, vol. 31 (2004), pp. 251–255.
[83] WILLIAM, MITCHELL, Aronszajn trees and the independence of the transfer property, Annals of Mathematical Logic, vol. 5 (1972), pp. 21–46.
[84] LUDOMIR, NEWELSKI, Model theoretic aspects of the Ellis semigroup, Israel Journal of Mathematics, vol. 190 (2012), no. 1, pp. 477–507.
[85] ALF, ONSHUUS, Properties and consequences of thorn-independence, The Journal of Symbolic Logic, vol. 71 (2006), no. 1, pp. 1–21.
[86] ALF, ONSHUUS and ALEXANDER, USVYATSOV, On dp-minimality, strong dependence and weight, The Journal of Symbolic Logic, vol. 76 (2011), no. 3, pp. 737–758.
[87] JOHAN, PAS, Uniform p-adic cell decomposition and local zeta functions, Journal für die reine und angewandte Mathematik, vol. 399 (1989), pp. 137–172.
[88] YA'ACOV, PETERZIL, Pillay's conjecture and its solution— a survey, Logic Colloquium 2007, Cambridge University Press, 2010.
[89] ANAND, PILLAY, Geometric Stability Theory, Oxford Logic Guides, Clarendon Press, 1996.
[90] ANAND, PILLAY, Topological dynamics and definable groups, The Journal of Symbolic Logic, vol. 78 (2013), no. 2, pp. 657–666.
[91] ANAND, PILLAY and CHARLES, STEINHORN, Definable sets in ordered structures I, Transactions of the American Mathematical Society, vol. 295 (1986), no. 2, pp. 565–592.
[92] ANAND, PILLAY and PREDRAG, TANOVIĆC, Generic stability, regularity, and quasi-minimality, preprint.
[93] BRUNO, POIZAT, Théories|instables, The Journal of Symbolic Logic, vol. 46 (1981), no. 3, pp. 513–522.
[94] BRUNO, POIZAT, Post-Scriptum á “Théories Instables”, The Journal of Symbolic Logic, vol. 48 (1983), no. 1, pp. 60–62.
[95] BRUNO, POIZAT, Cours de théorie des modèles, Nur al-Mantiq wal-Mari'fah, 1985, (English translation: A Course in Model Theory, Springer.).
[96] BRUNO, POIZAT, Groupes stables, Nur al-Mantiq wal-Ma'rifah, 1987.
[97] ABRAHAM, ROBINSON, Complete Theories, North-Holland, Amsterdam, 1956.
[98] N., SAUER, On the density of families of sets, Journal of Combinatorial Theory, Series A, vol. 13 (1972), no. 1, pp. 145–147.
[99] P. H., SCHMITT, Model Theory of Ordered Abelian Groups, Universitäat Heidelberg, 1982, Habilitationsschrift.
[100] ALEXANDER, SCHRIJVER, Theory of Linear and Integer Programming, Wiley, New York, 1986.
[101] SAHARON, SHELAH, Definable groups for dependent and 2-dependent theories, preprint.
[102] SAHARON, SHELAH, Strongly dependent theories, preprint.
[103] SAHARON, SHELAH, Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory, Annals of Mathematical Logic, vol. 3 (1971), no. 3, pp. 271–362.
[104] SAHARON, SHELAH, A combinatorial problem; stability and order for models and theories in in finitary languages, Pacific Journal of Mathematics, vol. 41 (1972), pp. 247–261.
[105] SAHARON, SHELAH, Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), no. 3, pp. 177–203.
[106] SAHARON, SHELAH, Classification Theory and the Number of Nonisomorphic Models, second ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990.
[107] SAHARON, SHELAH, Classification theory for elementary classes with the dependence property—a modest beginning, Scientiae Mathematicae Japonicae, vol. 59 (2004), no. 2, pp. 265–316, special issue on set theory and algebraic model theory.
[108] SAHARON, SHELAH, Minimal bounded index subgroup for dependent theories, Proceedings of the American Mathematical Society, vol. 136 (2008), no. 3, pp. 1087–1091.
[109] SAHARON, SHELAH, Dependent first order theories, continued, Israel Journal of Mathematics, vol. 173 (2009), no. 1, pp. 1–60.
[110] SAHARON, SHELAH, A dependent dream and recounting types, (2010), preprint, 950 in Shelah archive.
[111] SAHARON, SHELAH, Dependent theories and the generic pair conjecture, Communications in Contemporary Mathematics, vol. submitted (2010), math.LO/0702292.
[112] PIERRE, SIMON, On dp-minimal ordered structures, The Journal of Symbolic Logic, vol. 76 (2011), no. 4, pp. 448–460.
[113] PIERRE, SIMON, Finding generically stable measures, The Journal of Symbolic Logic, vol. 77 (2012), no. 1, pp. 263–278.
[114] PIERRE, SIMON, Distal and non-distal theories, Annals of Pure and Applied Logic, vol. 164 (2013), no. 3, pp. 294–318.
[115] KATRIN, TENT and MARTIN, ZIEGLER, A Course in Model Theory, Lecture Notes in Logic, Cambridge University Press, 2012.
[116] ALEXANDER, USVYATSOV, On generically stable types in dependent theories, The Journal of Symbolic Logic, vol. 74 (2009), no. 1, pp. 216–250.
[117] LOU VAN DEN, DRIES, Remarks on Tarski's problem concerning (R, +, *, exp), Logic Colloquium '82 (G. Lolli, G. Longo, and A. Marcja, editors), Studies in Logic and the Foundations of Mathematics, vol. 112, Elsevier, 1984, pp. 97–121.
[118] LOU VAN DEN, DRIES, A generalization of the Tarski-Seideberg theorem, and some nonde finability results, Bulletin of the American Mathematical Society, vol. 15 (1986), pp. 189–193.
[119] LOU VAN DEN, DRIES, Tame Topology and O-minimal Structures, London Mathematical Society Lecture Note Series, Cambridge University Press, 1998.
[120] VLADIMIR, N. VAPNIK and ALEXEY, YA. CHERVONENKIS, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and its Applications, vol. 16 (1971), no. 2, pp. 264–280.
[121] FRANK, O. WAGNER, Stable Groups, London Mathematical Society Lecture Note Series, Cambridge University Press, 1997.
[122] FRANK, O. WAGNER, Simple Theories, Mathematics and its Applications, vol. 503, Kluwer Academic Publishers, Dordrecht, 2000.
[123] ALEX, WILKIE, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, Journal of the American Mathematical Society, vol. 9 (1996), pp. 1051–1094.
[124] ALEX, WILKIE, O-minimal structures, Séminaire Bourbaki (2007/08), exp. n° 985, AstÉrisque, vol. 326 (2009), pp. 131–142.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.