## Tbilisi Mathematical Journal

### Theories with Ehrenfeucht-Fraïssé equivalent non-isomorphic models

Saharon Shelah

#### Abstract

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$(T_1,T)$.

#### Note

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

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

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

https://projecteuclid.org/euclid.tbilisi/1528768827

Digital Object Identifier
doi:10.32513/tbilisi/1528768827

Mathematical Reviews number (MathSciNet)
MR2563810

Zentralblatt MATH identifier
1158.03315

#### Citation

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