On Ehrenfeucht-Fraïssé equivalence of linear orderings

Abstract

AbstractC. Karp has shown that if α is an ordinal with ωα = α and A is a linear ordering with a smallest element, then α and α ⊗ A are equivalent in L∞ω up to quantifer rank α. This result can be expressed in terms of Ehrenfeucht-Fraïssé games where player ∀ has to make additional moves by choosing elements of a descending sequence in α. Our aim in this paper is to prove a similar result for Ehrenfeucht-Fraïssé games of length ω1. One implication of such a result will be that a certain infinite quantifier language cannot say that a linear ordering has no descending ω1-sequences (when the alphabet contains only one binary relation symbol). Connected work is done by Hyttinen and Oikkonen in [H] and [O].