Do you trust Cantor's set/infinity theory?

https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download     
Theorem 7: For a set S of infinite symbols (with finite symbol length), if the
         symbols in S can be sorted and have a 'smallest' element symbol, then S
         can correspond 1-1 to the elements of the natural number set.
         Example1: There is a 1-1 correspondence between integers and
             natural numbers, because integer elements can be arranged as 0, 1,
             -1, 2, -2, ... Similar to the above example, two-dimensional
             numbers (p, q), fractions (p/q), etc. can be sorted by p*∞+q (the
             smallest is 0).
         Example2: Suppose S contains O(2^N) elements (or any order), then as
             long as the elements of S are sortable (e.g., can be represented by
             an array), they can form a 1-1 correspondence with the set of
             natural numbers.
....
  Note: Real numbers can also be defined as (very intuitive and simple):
        EN::= Same as the natural numbers defined by Peano Axioms, but the
              numbers (elements) can be infinitely long.
        ℝ::= {x| x=p/q or -p/q, q≠0, p,q∈EN }
-------------