Expressiveness and definability in circumscription

we investigate expressiveness and definability issues with respect to minimal models, particularly in the scope of circumscription. first, we give a proof of the failure of the l？wenheim-skolem theorem for circumscription. then we show that, if the class of p; z-minimal models of a first-order sentence is δ-elementary, then it is elementary. that is, whenever the circumscription of a first-order sentence is equivalent to a first-order theory, then it is equivalent to a finitely axiomatizable one. this means that classes of models of circumscribed theories are either elementary or not δ-elementary. finally, using the previous result, we prove that, whenever a relation pi is defined in the class of p; z-minimal models of a first-order sentence φ and whenever such class of p; z-minimal models is δ-elementary, then there is an explicit definition ψ for pi such that the class of p; z-minimal models of φ is the class of models of φ ∧ ψ. in order words, the circumscription of p in φ with z varied can be replaced by φ plus this explicit definition ψ for pi.