Skip to main content

Some Themes Around First Order Theories Without the Independence Property

  • 1580 Accesses

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 2111)

Abstract

The aim of these notes (as well as the course of lectures they are based on) is to describe some current work around theories with NIP (not the independence property). This is a broad class of first order theories, including natural examples such as algebraically closed fields, differentially closed fields (both of which are stable) as well as real closed fields, p-adically closed fields and algebraically closed valued fields (which are unstable).This is really a paper on “pure” model theory, but I will comment here and there on applications and connections.

Keywords

  • Morley Sequence
  • Home Sort
  • Uniform Definability
  • Shelah
  • Regular Probability Measure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. N. Alon, D.J. Kleitman, Piercing convex sets and the Hadwiger-Debrunner (p, q)- problem. Adv. Math. 96, 103–112 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. A. Chernikov, I. Kaplan, Forking and dividing in NTP 2 theories. J. Symb. Log. 77, 1–20 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. A. Chernikov, P. Simon, Externally definable sets and dependent pairs. Isr. J. Math. 194, 409–425 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. A. Chernikov, P. Simon, Externally definable sets and dependent pairs II. Trans. AMS (to appear)

    Google Scholar 

  5. A. Chernikov, I. Kaplan, S. Shelah, On nonforking spectra (preprint, 2012)

    Google Scholar 

  6. E. Hrushovski, F. Loeser, Non-archimedean Tame Topology, and Stably Dominated Types. Princeton Monograph Series (to appear)

    Google Scholar 

  7. E. Hrushovski, Y. Peterzil, A. Pillay, Groups, measures and the NIP. J. AMS 21, 563–596 (2008).

    Google Scholar 

  8. E. Hrushovski, A. Pillay, On NIP and invariant measures. J. Eur. Math. Soc. 13, 1005–1061 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. E. Hrushovski, A. Pillay, P. Simon, On generically stable and smooth measures in NIP theories. Trans. AMS 365, 2341–2366 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. H.K. Keisler, Measures and forking. Ann. Pure Appl. Log. 45, 119–169 (1987)

    CrossRef  MathSciNet  Google Scholar 

  11. J. Matousek, Bounded VC-dimension implies a fractional Helly theorem. Discrete Comput. Geom. 31, 251–255 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. A. Pillay, Geometric Stability Theory (Oxford University Press, Oxford, 1996)

    MATH  Google Scholar 

  13. A. Pillay, Externally definable sets and a theorem of Shelah, in Felgner Festchrift (College Publications, London, 2007)

    Google Scholar 

  14. A. Pillay, On weight and measure in NIP theories. Notre Dame J. Formal Log. (to appear)

    Google Scholar 

  15. S. Shelah, Dependent first order theories, continued. Isr. J. Math. 173, 1–60 (2009)

    CrossRef  MATH  Google Scholar 

  16. S. Shelah, Dependent dreams and counting types (preprint 2012)

    Google Scholar 

  17. S. Shelah, Strongly dependent theories. Isr. J. Math. (to appear)

    Google Scholar 

  18. V. N. Vapnik, A.Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities. Theory Prob. Appl. 16, 264–280 (1971)

    CrossRef  MATH  Google Scholar 

Download references

Acknowledgements

Supported by EPSRC grant EP/I002294/1.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anand Pillay .

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Pillay, A. (2014). Some Themes Around First Order Theories Without the Independence Property. In: Model Theory in Algebra, Analysis and Arithmetic. Lecture Notes in Mathematics(), vol 2111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54936-6_2

Download citation