We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.

Mathematicians justify axioms of set theory “intrinsically”, by reference to the universe of sets of their intuition, and “extrinsically”, for example, by considerations of simplicity or usefullness for mathematical practice. Here we apply the same kind of justifications to Nonstandard Analysis and argue for acceptance of BNST+ . BNST+ has nontrivial consequences for standard set theory; for example, it implies existence of inner models with measurable cardinals. We also consider how to practice Nonstandard Analysis in BNST+, and compare it with (...) other existing nonstandard set theories. (shrink)

We study combinatorial principles related to the isomorphism property and the special model axiom in nonstandard analysis.

Currently the two popular ways to practice Robinson’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.