Earlier quoted context omitted.
What axiom do you have to add, are you talking about the axiom of infinity, defining the set of natural numbers? Because you do need that, but without that wouldn't even be able to define uncountability. After that, you can define the reals using Dedekind cuts or Cauchy sequences, which was known at the time (I believe Cantor actually worked on the Cauchy sequence construction). But there's an even simpler uncountabl…
To construct R you need to repeat Dedekind cut uncountable number of times, i.e. you need to iterate over all real numbers, which would make them countable.
1. A is not the empty set
2. A is not the set of all rational numbers
3. A is closed downwards, meaning if x is in A and y 4. A has no greatest element, meaning for all x in A, there is a y in A such that y > x.
And each Dedekind cut is in one-to-one correspondence with a real number. You can define the usual arithmetic operations on them, show that every rational number has a corresponding Dedekind cut (for any rational q, we have {x in Q | y But as you can see, you don't need any sort of repetition to do the construction, it's just a set of sets of rational numbers satisfying a few simple properties. For more information, check out https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...