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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Do you trust Cantor's set/infinity theory?
Date: Mon, 14 Apr 2025 23:39:05 +0800
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https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/d=
ownload     =20
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, t=
hen 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*=E2=
=88=9E+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 represent=
ed 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::=3D Same as the natural numbers defined by Peano Axioms, but th=
e
              numbers (elements) can be infinitely long.
        =E2=84=9D::=3D {x| x=3Dp/q or -p/q, q=E2=89=A00, p,q=E2=88=88EN }
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