The Isomorphism Property in Nonstandard Analysis and Its Use in the Theory of Banach Spaces

The basic setting of nonstandard analysis consists of a set-theoretical structure 4' together with a map * from 4' into another structure *X' of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make *X' into an enlargement of 4' [13]. The structures 4' and *X' may be type-hierarchies as in [11] and [13] or they may be cumulative structures with co levels as in [14]. The assumption that *4' is an enlargement of 4' has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that *4' has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11]. This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number K, *4' satisfies the K-isomorphism property (as an enlargement of 4) if the following condition holds: For each first order language, L with fewer than K nonlogical symbols, if 21 and Z are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to *4' and .X), then 21 and 23 are isomorphic. These properties bring the language and tools of model theory even more firmly into nonstandard analysis than before. In particular, they provide a vehicle for applying such results as the downward Lowenheim-Skolem theorem [16] in topology and analysis. In ?1 it is shown that for each 4' and K there exists an enlargement *4' of 4' which has the K-isomorphism property and that any such enlargement of 4' is necessarily K-saturated. The rest of ?1 is devoted to exploring the effects of the various isomorphism properties on the structure of internal sets and functions in *X.' For example, it is shown that if *4' has the g0-isomorphism property and if A and B are infinite, internal sets in *4', then there are bijectionsf and g of A onto B with the following properties: (1) Cis an internal subset of A if and only if {fx: x E C} is an internal subset of B; (2) C is an internal subset of A if and only if {gx: x E C} or its complement is a *-finite subset of B. In particular, such bijections exist when A is the *-finite set {1, 2,** , co} for some c E *N N and B is *N. (Note that a *-finite set can be infinite.) In ?2 the isomorphism properties are applied in the theory of Banach spaces. Some of the results given here concern the structure of nonstandard hulls of