On a question of Krajewski's

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In this paper we provide a (negative) solution to a problem posed by Stanisław Krajewski. Consider a recursively enumerable theory U and a finite expansion of the signature of U that contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extension α of U in the expanded language that is conservative over U, there is a conservative extension β of U in the expanded language, such that αβ and βα. The result is preserved when we consider either extensions or model-conservative extensions of U in stead of conservative extensions. Moreover, the result is preserved when we replace as ordering on the finitely axiomatized extensions in the expanded language by a special kind of interpretability, to wit interpretability that identically translates the symbols of the U-language. We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U.