Re: Cantor Diagonal Proof --- PLO

On Sat, 2025-04-05 at 14:40 -0700, Keith Thompson wrote:

Lawrence D'Oliveiro <ldo@nz.invalid> writes:
[...]

The cardinality of the set of integers (and therefore also the set of
computer programs, and of the set of computable numbers) is conventionally
written as ℵ₀. The cardinality of the set of reals is written as ℵ₁. Both
are infinite, but ℵ₁ is supposed to be a larger infinity than ℵ₀ -- at
least, that’s what the Cantor diagonal construction is supposed to prove.
 
In this thread I am trying to point out why the proof doesn’t work. For a
start, in general, the diagonal construction never converges to an answer.

 
Which is more likely, that you've found a flaw in a proof that's
been accepted by mathematicians for over a century, or that you've
reached an incorrect conclusion?
 
There's nothing wrong with trying to find flaws in established
proofs.  It can be a great way to understand the proof more deeply.
But please consider the possibility that you're mistaken and everyone
else is right.
 
Cantor's construction proves that, given a list of all real numbers,
there is a number that is not in the list -- and therefore, by
contradiction, that no such list is possible.
 
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
 

IMO, Cantor Diagonal Proof (along with Cantor set theory) is a trick of magic for convenience.

... Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download
Theorem 5: The set of elements composed of finite discrete symbols and the set
         of elements composed of infinite discrete symbols cannot form 1-1
         correspond.
  Proof: According to the meaning of finite/infinite, during the correspondence
         process, the elements of the finite set will be exhausted (according to
         the definition), while the elements of the infinite set won't.

Theorem 6: There is no 1-1 correspondence procedure between the sets ℝ and ℕ.
  Proof: It can be proved according to Theorem 5.
.... 

The wording looks not good, but you should get the idea.
All is that simple. Just infinite and finite.