Constructing nonstandard hulls and Loeb measures in internal set theories

TL;DR

This paper demonstrates that internal set theories can effectively construct nonstandard hulls and Loeb measures, challenging claims that they cannot handle external set constructions.

Contribution

It shows that internal frameworks can successfully develop nonstandard hulls and Loeb measures, bridging a gap in nonstandard analysis methods.

Findings

Internal frameworks can construct nonstandard hulls.

Internal frameworks can develop Loeb measures.

Ultrapowers are isomorphic to subuniverses in internal theories.

Abstract

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.