Relationship between nonstandard analysis and nonstandard models of arithmetic

Question

So I've been reading through some introductory texts on nonstandard analysis, basically through the ultrafilter construction of hyperreals and the transfer principle. It seems that much of the time, the notation is directly copied over from the standards in nonstandard models of arithmetic (a la Skolem), with $\omega$ representing an infinite nonstandard value, etc.

This got me thinking: is it possible to reverse this process? ie. take tuples from some appropriate nonstandard model of arithmetic, then apply an analogue to dedekind cuts in order to construct $^*\mathbb{R}$?

This SEEMS like the obvious approach to take, which makes me suspect that either we won't end up with a field, or it's difficult to figure out which nonstandard model of arithmetic is needed, or the resulting set will be somehow "smaller" than the hyperreals and not sufficient to prove the transfer principle.

Anyone know the answer or some good resources for clearing up why this wouldn't work?

Answer 1

This is not going to work because the hyperreal field ${}^\ast\mathbb R$ is not Dedekind-complete. Also, since the nonstandard models of arithmetic a la Skolem are constructed in ZF, a canonical procedure (not relying on any version of Choice) such as taking the Dedekind completion is not going to work, since it is known that some version of AC is required to specify an ${}^\ast \mathbb R$ with the required properties.

Answer 2

In my understanding of the topic, the hard problem is determining what the internal sets should be.

The better framework for the problem is to take as a starting point a nonstandard model of set theory (whether ZFC or something more like ETCS), rather than merely a nonstandard model of arithmetic.

Since it's a nonstandard model of set theory, you can do all of the usual things internally, such as construct its set of real numbers via (internal) Dedekind cuts of its set of rational numbers.

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My impression is that there cannot be a good way to start from a model of arithmetic and generate a universe of sets with the property that the given model of arithmetic corresponds to its set of natural numbers. But I'm nowhere near an expert on that sort of question.