Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T21:12:38.244Z Has data issue: false hasContentIssue false
  • English
  • Français

Adding linear orders

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah  and
Pierre Simon
Saharon Shelah
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University Of Jerusalem, Jerusalem, 91904, Israel, E-mail: [email protected], URL: http://shelah.logic.at
Pierre Simon
Affiliation:
Département De Mathématiques et Applications 45, Rue d'Ulm, 75005 Paris, France, E-mail: [email protected], URL: http://www.normalesup.org/˜simon/
Get access Rights & Permissions [Opens in a new window]

Abstract

We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets pseudofinite arithmetic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 717 - 725
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Baldwin, John and Benedikt, Michael, Stability theory, permutations of indiscernibles, and embedded finite models, Transactions of the American Mathematical Society, vol. 352 (2000), no. 11. pp. 4937–4969.CrossRefGoogle Scholar
[2] Chernikov, Artem and Simon, Pierre, Externally definable sets and dependent pairs, submitted. Modnet preprint 265.Google Scholar
[3] Graham, R., Rothschild, B., and Spencer, J.H., Ramsey theory, 2nd ed., New York: John Wiley and Sons. 1990.Google Scholar
[4] Hoeffding, Wassily, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association, vol. 58 (1963), no. 301, pp. 13–30.CrossRefGoogle Scholar
[5] Schmerl, James H., Decidability and N 0-categoricity of theories of partially ordered sets, this Journal, vol. 45 (1980), no. 3, pp. 585–611.Google Scholar
[6] Tsuboi, Akito. Random amalgamation of simple theories, Mathematical Logic Quarterly, vol. 47 (2001), no. 1, pp. 45–50.3.0.CO;2-A>CrossRefGoogle Scholar