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?

## Comments

Michael Cohen

Sorry about my inactivity on Arbital, and thanks for going ahead and fixing it!