The special model axiom in nonstandard analysis

Abstract

κ-saturation [SL] is probably the single most useful property of nonstandard models of analysis. For some applications, however, stronger saturation hypotheses seem necessary. Henson formulated his elegant κ-isomorphism property (see [H1], and §2 below) to address this need. This property, though well-suited to certain situations (notably those arising in Banach space theory), is often difficult to apply in practice (see [SL, §7.7]).In this paper I describe an alternative to κ-isomorphism which is much easier to use; in particular, a proof which assumes that the nonstandard model is fully saturated can usually be converted directly to one using this special model axiom.Precise definitions of both these axioms appear in §2. In §3 I prove some simple properties of the special model axiom, one of which is that it is at least as strong as κ-isomorphism. In §§4, 5, and 6 the axiom is used to construct a few examples, many of which are pathological, or at the very least counterintuitive. (These examples are given primarily to illustrate use of the axiom; the only one of independent interest is Theorem 5.5.) In §7 some alternative axioms and open problems are discussed.Many of the results in this paper grew out of discussion and correspondence with C. Ward Henson, to whom I am consequently most grateful.