Theories with EF-Equivalent Non-Isomorphic Models

TL;DR

This paper investigates the classification of countable first-order theories based on the existence of non-isomorphic models that are indistinguishable by Ehrenfeucht-Fraïssé games across all ordinal lengths, aiming to identify a dichotomy similar to the main gap.

Contribution

It introduces a framework for characterizing theories where non-isomorphic models are EF-equivalent at all levels, and proves the consistency of the non-structure side for certain classes of theories.

Findings

Established the consistency of the non-structure side for aleph_0-independent theories.

Proposed a dichotomy in the classification of theories based on EF-equivalence of models.

Connected the results to the broader context of the main gap in model theory.

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 lambda,M_1,M_2 such that M_1,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 we get a strong dichotomy, so in the non-structure side we have stronger, better examples, and in the structure side we have a parallel of [Sh:c,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).