Special Model Axiom in Nonstandard Set Theory

We demonstrate that the special model axiom SMA of Ross admits a natural formalization in Kawai's nonstandard set theory KST but is independent of KST. As an application of our methods to classical model theory, we present a short proof of the consistency (with ZFC) of the existence of a k⁺ like k-saturated model of PA for a given cardinal k.