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 {isomorphism property if for any two internally presented L{structures A and B, where L has less than many symbols, A is elementarily equivalent to B implies that A is isomorphic to B. In this paper we prove that the @1{isomorphism property is equivalent to the @0{isomorphism property plus @1{saturation. Throughout this paper we use L for some rst{order language and A,B for models (or structures) of some L with base sets A, B respectively. By a standard universe we mean the superstructure V!(N) = S n2! Vn with the \2" relation, where V0 = N , a set of all positive integers as urelements, and Vn+1 = Vn S P(Vn). By a nonstandard universe we mean the image of V under Mostowski collapse, where V is an elementary extension of the standard universe truncated at 2{rank !. The author refers to [CK] for details of model theory and nonstandard universes. In this paper we use V (together with the \2" relation) for a nonstandard universe and N for the nonstandard version of N in V . A set A is called internal in V if A is an element of V . An L{structure A is called internally presented in V if both base set A and the interpretation under A of every symbol in L are internal in V . A nonstandard universe V is said to satisfy the {isomorphism property (IP for short) for an in nite cardinal if the following is true in V , For any two internally presented L{structures A and B with jLj < , if A B, then A = B. A direct corollary of the de nition is that IP =) IP if > . IP was given in [H1] by C. Ward Henson. It is a very strong property for a nonstandard universe. It implies {saturation. In Henson's several papers (cf.[H1], [H2], [H3], etc.) he applied the isomorphism property to nonstandard functional analysis and used it to get isometries between nonstandard hulls of Banach spaces.