EF equivalent not isomorphic models

TL;DR

This paper constructs non-isomorphic models of size aleph_1 that are indistinguishable by Ehrenfeucht-Fraisse games of length less than omega_1, highlighting limitations of these games in model comparison.

Contribution

It introduces a method to build non-isomorphic models that cannot be distinguished by certain Ehrenfeucht-Fraisse games, advancing understanding of model equivalence.

Findings

Models of size aleph_1 can be non-isomorphic yet Ehrenfeucht-Fraisse indistinguishable for games of length less than omega_1

Demonstrates limitations of Ehrenfeucht-Fraisse games in distinguishing non-isomorphic models

Provides explicit constructions of such models

Abstract

We construct non-isomorphic models M, N, e.g. of cardinality aleph_1 such that in the Ehrenfeucht-Fraisse game of length zeta < omega_1 the isomorphism player wins