"This definition of the real numbers has a bigge..."

This definition of the real numbers has a bigger problem with it than just circular logic — it also runs into the 0.9999… = 1 paradox. The sets $~$\mathbb{N} \setminus \{1, 2, 3, 4, 5\}$~$ and the set $~${5}$~$ both encode the number $~$1/8$~$.

Normally the real numbers are defined using either Dedekind cuts or Cauchy sequences of rational numbers. Could we please use one of those definitions instead, as they're the standard ones used by most mathematicians?