This article establishes the existence of a definable , countably saturated nonstandard enlargement of the superstructure over the reals. This nonstandard universe is obtained as the union of an inductive chain of bounded ultrapowers . The underlying ultrafilter is the one constructed by Kanovei and Shelah [10].

In this article ‡ we show how the universe of HST, Hrbaček set theory admits a system of subuniverses which keep the Replacement, model Power set and Choice, and also keep as much of Saturation as it is necessary. This gives sufficient tools to develop the most complicated topics in nonstandard analysis, such as Loeb measures.

In this article we show how the universe of BST, bounded set theory can be enlarged by definable subclasses of sets so that Separation and Replacement are true in the enlargement for all formulas, including those in which the standardness predicate may occur. Thus BST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop topics in nonstandard analysis inaccessible in the framework of a purely internal approach, such as Loeb measures.