Quote:
Originally Posted by Happy5214
Note the difference. The axioms for the rational numbers (an ordered field) are the same as the real numbers (essentially the only Dedekindcomplete ordered field) minus the least upper bound property/Dedekind completeness, so if your book has axioms for the real numbers, you can derive the axioms for the rational numbers from those.

This isn't quite correct. The axioms of the reals minus the least upper bound property gives you an ordered field, but it is not sufficient to characterize the rationals. Specifically, it need not be Archimedean, which the rationals are. E.g. see the surreal numbers, for which these axioms apply, but which are certainly not isomorphic to the rationals.