Of course, this makes all of this rather abstract. It looks to me like for almost any two hyperreals (e.g. a, b as above), the answer to “which of them is larger?” is “It depends on the ultrafilter. Also, I can not tell you if a set is part of any specific ultrafilter. But fear not, for any given ultrafilter, the hyperreals are well-ordered.”
Basically for any usable theorem, one would have to prove that the result is independent of the actual ultrafilter used, which means that numbers such as a and b will probably not feature in them a lot.
I can not fault my analysis 1 professor for opting to stick to the reals (abstract as they are already are) instead.
Thanks, this is helpful to point out.
Of course, this makes all of this rather abstract. It looks to me like for almost any two hyperreals (e.g. a, b as above), the answer to “which of them is larger?” is “It depends on the ultrafilter. Also, I can not tell you if a set is part of any specific ultrafilter. But fear not, for any given ultrafilter, the hyperreals are well-ordered.”
Basically for any usable theorem, one would have to prove that the result is independent of the actual ultrafilter used, which means that numbers such as a and b will probably not feature in them a lot.
I can not fault my analysis 1 professor for opting to stick to the reals (abstract as they are already are) instead.