How much choice is needed for the transfer principle?

Question

To construct the hyperreals via ultrapower the Boolean prime ideal theorem apparently suffices. However, to prove the transfer principle for the extension $\mathbb{R}\subset{}^\ast\mathbb{R}$ apparently a stronger version of choice is needed. Does anybody have the details?

Answer 1

The transfer principle, if I understand your question correctly, is the localized version of Łoś theorem for ultraproducts.

How much choice is needed in general for Łoś theorem? It was shown that BPI+Łoś imply choice over $\sf ZF$. But in models where there are no free ultrafilters, Łoś theorem is trivially true.

If you localize it to the case of $\Bbb R$ with the language of ordered rings, and only consider ultrapowers by $\Bbb N$ as an index set, then the answer is probably going to be something like "It is sufficient that Łoś theorem holds for ultrapowers of $\Bbb R$ by free ultrafilters - which are assumed to exist - over $\Bbb N$".

It's a disappointing answer, I agree.

If you want something sufficient and not optimal, then $|\Bbb R|=\aleph_1$ would certainly suffice.