A Note on Axioms for Infinite-generic Structures

In this note we give some natural examples of theories T in countable logics L such that the class GT of infinite-generic structures for T is not axiomatizable by a single sentence of LW1> w. The point of the examples is to provide a contrast to a certain phenomenon recently observed in model-theory, concerning the assignment to L-theories T of classes GT of L-structures. In many interesting cases it has been verified that GT is axiomatizable by a sentence of L(ou m, and indeed by a sentence of some countable infinitary fragment constructive from T. Two typical cases occur when Gr is the class of T-existentially closed structures [9] or the class of finitegeneric structures for T [6]. Other examples, and remarks on this absoluteness phenomenon, are to be found in [1]. The results of this note show in particular that GT does not enjoy this absoluteness property. We are grateful to Paul Bacsich who prevailed upon us to write up our results.