Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Stability theory, permutations of indiscernibles, and embedded finite models
HTML articles powered by AMS MathViewer

by John Baldwin and Michael Benedikt
Trans. Amer. Math. Soc. 352 (2000), 4937-4969
DOI: https://doi.org/10.1090/S0002-9947-00-02672-6
Published electronically: July 21, 2000

Abstract:

We show that the expressive power of first-order logic over finite models embedded in a model $M$ is determined by stability-theoretic properties of $M$. In particular, we show that if $M$ is stable, then every class of finite structures that can be defined by embedding the structures in $M$, can be defined in pure first-order logic. We also show that if $M$ does not have the independence property, then any class of finite structures that can be defined by embedding the structures in $M$, can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let $I$ be a set of indiscernibles in a model $M$ and suppose $(M,I)$ is elementarily equivalent to $(M_1,I_1)$ where $M_1$ is $|I_1|^+$-saturated. If $M$ is stable and $(M,I)$ is saturated, then every permutation of $I$ extends to an automorphism of $M$ and the theory of $(M,I)$ is stable. Let $I$ be a sequence of $<$-indiscernibles in a model $M$, which does not have the independence property, and suppose $(M,I)$ is elementarily equivalent to $(M_1,I_1)$ where $(I_1,<)$ is a complete dense linear order and $M_1$ is $|I_1|^+$-saturated. Then $(M,I)$-types over $I$ are order-definable and if $(M,I)$ is $\aleph _1$-saturated, every order preserving permutation of $I$ can be extended to a back-and-forth system.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C45, 68P15, 03C40
  • Retrieve articles in all journals with MSC (2000): 03C45, 68P15, 03C40
Bibliographic Information
  • John Baldwin
  • Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607
  • Email: jbaldwin@math.uic.edu
  • Michael Benedikt
  • Affiliation: Bell Laboratories, 1000 E. Warrenville Rd., Naperville, Illinois 60566
  • Email: benedikt@research.bell-labs.com
  • Received by editor(s): March 17, 1998
  • Published electronically: July 21, 2000
  • Additional Notes: The first author was partially supported by DMS-9803496
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 4937-4969
  • MSC (2000): Primary 03C45; Secondary 68P15, 03C40
  • DOI: https://doi.org/10.1090/S0002-9947-00-02672-6
  • MathSciNet review: 1776884