Tbilisi Mathematical Journal

Theories with Ehrenfeucht-Fraïssé equivalent non-isomorphic models

Saharon Shelah

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Our “long term and large scale” aim is to characterize the first order theories $T$ (at least the countable ones) such that for every ordinal $\alpha$ there are $\lambda$, $M_1$, $M_2$ such that $M_1$ and $M_2$ are non-isomorphic models of $T$ of cardinality $\lambda$ which are EF$^+_{\alpha,\lambda}$-equivalent. We expect that as in the main gap [11, XII], we get a strong dichotomy, i.e., on the non-structure side we have stronger, better examples, and on the structure side we have an analogue of [11, XIII]. We presently prove the consistency of the non-structure side for $T$ which is $\aleph_0$-independent (= not strongly dependent), even for PC$(T_1,T)$.


The author would like to thank the Israel Science Foundation for partial support of this research (Grant Number 242/03), Alice Leonhardt for the beautiful typing and the three anonymous referees for many helpful remarks. Particular thanks are due to one referee who pointed out that an earlier version of this paper did not contain anything new beyond [2]; Definition 2.5 now fixes this problem. This paper is publication number 897 of the list of the author's publications.

Article information

Tbilisi Math. J., Volume 1 (2008), 133-164.

Received: 16 March 2008
Revised: 15 August 2008
Accepted: 25 October 2008
First available in Project Euclid: 12 June 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C55: Set-theoretic model theory
Secondary: 03C68: Other classical first-order model theory 03E40: Other aspects of forcing and Boolean-valued models

Ehrenfeucht-Fraïssé games Isomorphism Model Theory Classification Theory


Shelah, Saharon. Theories with Ehrenfeucht-Fraïssé equivalent non-isomorphic models. Tbilisi Math. J. 1 (2008), 133--164. doi:10.32513/tbilisi/1528768827. https://projecteuclid.org/euclid.tbilisi/1528768827

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