Dependence and isolated extensions
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- by Vincent Guingona PDF
- Proc. Amer. Math. Soc. 139 (2011), 3349-3357 Request permission
Abstract:
In this paper, we show that if $\varphi (x; y)$ is a dependent formula, then all $\varphi$-types $p$ have an extension to a $\varphi$-isolated $\varphi$-type, $p’$. Moreover, we can choose $p’$ to be an elementary $\varphi$-extension of $p$ and $|\mathrm {dom}(p’) - \mathrm {dom}(p)| \le 2 \cdot \mathrm {ID}(\varphi )$. We show that this characterizes $\varphi$ being dependent. Finally, we give some corollaries of this theorem and draw some parallels to the stable setting.References
- Saharon Shelah, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 513226
- —, Dependent theories and the generic pair conjecture, preprint, September 2009. Communications in Contemporary Mathematics, submitted.
Additional Information
- Vincent Guingona
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 942387
- Email: vincentg@math.umd.edu
- Received by editor(s): November 6, 2009
- Received by editor(s) in revised form: August 16, 2010
- Published electronically: January 21, 2011
- Additional Notes: Special thanks to Chris Laskowski
The author was partially supported by Laskowski’s NSF grants DMS-0600217 and 0901336 - Communicated by: Julia Knight
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3349-3357
- MSC (2010): Primary 03C45
- DOI: https://doi.org/10.1090/S0002-9939-2011-10739-1
- MathSciNet review: 2811289