Ehrenfeucht–Fraïssé games in finite set theory

Abstract

In this article, with Ehrenfeucht–Fraïssé games we prove that Δ 1 ≠ Δ 0 on BFR , which implies Δ ≠ Δ 0 on BFR , and thus solve an open problem raised by Albert Atserias in his dissertation ( Δ 0 , Δ 1 , Δ are fragments of first order logic and BFR is a class of finite sets which in essence is equivalent to a class of finite pure arithmetic structures with built-in BIT predicate).