  In this paper we provide a (negative) solution to a problem posed by
Stanis{\l}aw 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 $\ge$ 2. We show that, for any finite extension $\alpha$ of U in the
expanded language that is conservative over U, there is a conservative
extension $\beta$ of U in the expanded language, such that $\alpha\vdash\beta$
and $\beta\nvdash\alpha$. 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 $\vdash$ 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.
