Internal approach to external sets and universes

Studia Logica 56 (3):293 - 322 (1996)
In this article we show how the universe of HST, Hrbaek set theory (a nonstandard set theory of external type, which includes, in particular, the ZFC Replacement and Separation schemata for all formulas in the language containing the membership and standardness predicates, and Saturation for standard size families of internal sets, but does not include the Power set axiom) admits a system of subuniverses which keep the Replacement, model Power set and Choice (in fact all of ZFC, with the exception of the Regularity axiom, which indeed is replaced by the Regularity over the internal subuniverse), 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.
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DOI 10.1007/BF00372770
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Frederik S. Herzberg (2008). A Definable Nonstandard Enlargement. Mathematical Logic Quarterly 54 (2):167-175.

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