〈 Berger | Dillon 〉

How does this work exactly, since you can pick a real number to be as close to zero as you want?

Informally, you can think of this as saying "no you actually get as close as you want. The closest you can get is infinitesimally close" That is there is a "smallest" number that is nonzero in the hyperreals

Does this imply that hyperreals don't have the property that given two hyperreals a and b, I can always find a hyperreal c such that a < c < b?

If a,b,c are all infinitesimals I believe that is correct. However, through something called the transfer principle all "regular" notions of the reals gets carried over, except those having to do with sets of numbers

I mean, if it werent correct then there would exist a hyperreal number x such that 0 < x < dx.

My understanding is that the hyperreals are non-Archimedian, and so I'm trying to tread softly with the infinitesimals, but I think that's correct