Re: Cantor Diagonal Proof

On 04/04/2025 20:13, Lawrence D'Oliveiro wrote:

On Fri, 4 Apr 2025 11:20:28 +0300, Mikko wrote:
 

On 2025-04-04 08:03:44 +0000, Lawrence D'Oliveiro said:
>

On Fri, 4 Apr 2025 11:00:36 +0300, Mikko wrote:
>

<snip>

>
Can you prove that it computes an incomputable number?

>
It’s trying to come up with a number that cannot fit into a set with
cardinality ℵ₀. The cardinality of the computable numbers is the same
as that of the integers, which is ℵ₀.

>
It is also the cardinality of rationals but not of reals. Cantor's proof
is that for each list of reals one can construct a real that is not in
the list.

 That proof doesn’t quite work, because at any point in the construction,
the number can be shown to match some later item in the list.

Then show it. I've already explained why you're mistaken, but don't let that stop you.

The mismatch
with all items in the list is only demonstrated when the proof completes.
But the proof never completes. Therefore there is no mismatch. QED.

The proof was completed by 1891, when it was published. To claim that it "never completes" is like claiming that Achilles never catches the tortoise.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within