It is a well-known fact that the real numbers are the only complete totally ordered field. So, if we perform the Dedekind cut construction on the hyperreals, then the result must either be:
- The real numbers, or
- Something that isn't a complete totally ordered field.
I tried doing this construction to see what goes wrong, and I couldn't seem to prove that additive inverses exist (without using the Archimedean principle, which of course doesn't hold for the hyperreals). But perhaps I'm just not clever enough. So, my question is: is this the axiom that fails, or is it another one, or do we get a complete ordered field, ie. the real numbers?