A THEOREM ON THE ISOMORPHISM PROPERTY

An L-structure is called internally presented in a nonstandard universe if its base set and interpretation of every symbol in L are internal. A nonstandard universe is said to satisfy the kappa-isomorphism property if for any two internally presented L-structures A and B, where L has less than kappa many symbols, A is elementarily equivalent to B implies that A is isomorphic to B. In this paper we prove that the aleph1-isomorphism property is equivalent to the alpeh0-isomorphism property plus aleph1-saturation.