Re: Incompleteness of Cantor's enumeration of the rational numbers

On 11/11/2024 3:33 PM, WM wrote:

On 11.11.2024 19:23, Jim Burns wrote:

On 11/11/2024 3:41 AM, WM wrote:

But it will never complete
an infinite set of claims.

>
We do not need an infinite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 completed.
We do not want an infinite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 completed.

>
But you claim that
_all_ fractions are in bijection with
all natural numbers,
don't you?

>
Yes, I claim that.

>
For that claim you need an infinite set of claims.

For that claim, I need
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀, in which
each 𝗰𝗹𝗮𝗶𝗺 is true.or.not.first.false,
and in which that is one of the 𝗰𝗹𝗮𝗶𝗺𝘀.

This is one 𝗰𝗹𝗮𝗶𝗺:
⎛ All fractions are in bijection with
⎝ all natural numbers.

 It is wrong.

It is one claim.
A conceivable argument that
  claims.like.that are infinite and
  therefore are not expressible finitely
fetches up hard against the bare fact of
it having been expressed, here, a few lines up.
I won't re.present here
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀, in which
each 𝗰𝗹𝗮𝗶𝗺 is true.or.not.first.false,
and in which that is one of the 𝗰𝗹𝗮𝗶𝗺𝘀.
But I could.
Suppose I did.
Would you (WM) accept that 𝗰𝗹𝗮𝗶𝗺?

My intervals I(n) = [n - 1/10, n + 1/10]
must be translated to all the midpoints
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5,
2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
if you want to contradict my claim.

>
Your 𝗰𝗹𝗮𝗶𝗺𝘀 start with "Sets change".

>
No, I claim that intervals can be translated.

Which means what?
Apparently, to you, it means that
  sets change.
However,
our sets do not change.

(The set of intervals remains constant
in size and multitude.)

Our sets do not change.

For every finite subset this is possible.

Our sets do not change.