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:

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.

By which, you mean that translation changes intervals.
Intervals do not change.
"After" translation of [4-⅒,4+⅒] to [1/3-⅒,1/3+⅒]
[4-⅒,4+⅒] will continue being [4-⅒,4+⅒]
[1/3-⅒,1/3+⅒] will have never been [4-⅒,4+⅒]

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

The set of intervals remains constant. Absolutely.
Sets do not change.
Intervals do not change.
Mathematical objects do not change.
⎛ But what about descriptions of change?
⎝ Matheologians are famous for those.
The description of a cannon ball's arc,
the description of the beat of a pendulum,
these are what matheologians are famous for.
But these are maps, not journeys.
The maps do not change.
Matheologians make true claims about
each one of infinitely.many journeys,
and then, they augment the description with
further true.or.not.first.false claims.
Without shooting cannons or swinging pendulums,
we know that the further claims must be true.
If the matheologians are clever enough or lucky enough,
then the further true claims eliminate
all but one description, and
we have a map (which has never changed).
The description and the further claims
are finite. Our finiteness is no barrier to saying them,
although we might not be clever or lucky enough
to eliminate all but one.
The description and the further claims
are always and everywhere _about_
the same infinitely.many, not.changing.