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

On 10.11.2024 18:49, Jim Burns wrote:

On 11/10/2024 4:35 AM, WM wrote:

The set
   {3,4,5}
does not _change_ to the set
   {6,7,8}
because
our sets do not change.

But points or intervals in geometry can be translated on the real axis.

Our sets do not change.

But points or intervals in geometry can be translated on the real axis.

 

>
In the first case, with the not.changing sets,
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 which
  has only true.or.not.first.false 𝗰𝗹𝗮𝗶𝗺𝘀
has only true 𝗰𝗹𝗮𝗶𝗺𝘀.

>
But it

 "It" refers to who or what?

To that finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲.

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?

It will forever remain in the status nascendi.
Therefore
irrelevant for actual or completed infinity.

 A finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀, each of which
is true.or.not.first.false,
will forever remain
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀, each of which
is true.or.not.first.false.

Therefore such a sequence does not entitle you to claim infinite mappings.

 

Infinite sets can correspond to
other infinite sets which,
without much thought about infinity,
would seem to be a different "size".

>
But they cannot become such sets.

 Our sets do not change.

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.

 

But they cannot be completely transformed

< into each other.
 Our sets do not change.

But intervals can be shifted.

Consider again the two sets of midpoints
⟨ 1, 2, 3, 4, 5, ... ⟩ and
⟨ 1/1, 1/2, 2/1, 1/3, 2/2, ... ⟩
 They both _are_
And their points correspond
by i/j ↦ n = (i+j-1)(i+j-2)/2+i

The first few terms do correspond or can be made c orresponding. That can be proven by translating the due intervals. But the full claim is nonsense because it is impossible to satisfy.
Regards, WM