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

On 11/9/2024 6:45 AM, WM wrote:

On 08.11.2024 19:01, Jim Burns wrote:

On 11/8/2024 5:18 AM, WM wrote:

My understanding of mathematics and geometry
is that
reordering cannot increase the measure
(only reduce it by overlapping).
This is a basic axiom which
will certainly be agreed to by
everybody not conditioned by matheology.

>
By
"everybody not conditioned by matheology"
you mean
"everybody who hasn't thought much about infinity"

>
Everybody who believes that the intervals
I(n) = [n - 1/10, n + 1/10]
could grow in length or number
to cover the whole real axis
is a fool or worse.

Our sets do not change.
The set
   {[n-⅒,n+⅒]: n∈ℕ⁺}
with the midpoints at
   ⟨ 1, 2, 3, 4, 5, ... ⟩
does not _change_ to the set
   {[iₙ/jₙ-⅒,iₙ/jₙ+⅒]: n∈ℕ⁺}
with the midpoints at
   ⟨ 1/1, 1/2, 2/1, 1/3, 2/2, ... ⟩
----
Either
all instances of a 𝗰𝗹𝗮𝗶𝗺 about a set
are _only_ true or _only_ false
or
a set changes.
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.
Even though
we are _not_ physically able to check, for each number
  in an infinite set of numbers,
  that a 𝗰𝗹𝗮𝗶𝗺 is true about it,
we _are_ physically able to check, for each 𝗰𝗹𝗮𝗶𝗺
  in a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀,
  that it is not.first.false in that 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲.
Also, we already know some 𝗰𝗹𝗮𝗶𝗺𝘀 to be true.
Some finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲𝘀 of 𝗰𝗹𝗮𝗶𝗺𝘀 are
  known to be only true.or.not.first.false 𝗰𝗹𝗮𝗶𝗺𝘀,
and thus known to be only true 𝗰𝗹𝗮𝗶𝗺𝘀.
Un.physically.checkable numbers do not
  prevent us from knowing they're true 𝗰𝗹𝗮𝗶𝗺𝘀.
In the second case, with the changing sets,
who knows?
Perhaps something else could be done,
but not that.
In any case,
what.we.do is the first case, with
its not.changing sets and
its known.about infinity.
For that reason (and more, I suspect),
our sets do not change.

Everybody who believes that
the intervals
I(n) = [n - 1/10, n + 1/10]
could grow in length or number
to cover the whole real axis
is a fool or worse.

Our sets do not change.
Infinite sets can correspond to
other infinite sets which,
without much thought about infinity,
would seem to be a different "size".
Consider geometry.
Similar triangles have
corresponding sides in the same ratio.
Consider these points, line.segments, and triangles
in the ⟨x,y⟩.plane
A = ⟨0,-1⟩
B = ⟨0,0⟩
C = ⟨x,0⟩  with 0 < x < 1
D = ⟨1,0⟩
E = ⟨1,y⟩  with points A C E collinear.
△ABC and △EDC are similar
△ABC ∼ △EDC
μA͞B = 1
μB͞C = x
μE͞D = y
μD͞C = 1-x
Similar triangles.
μA͞B/μB͞C = μE͞D/μD͞C
1/x = y/(1-x)
y = 1/x - 1
x = 1/(y+1)
To each point C = ⟨x,0⟩ in (0,1)×{0}
there corresponds
exactly one point E = ⟨1,y⟩ in {1}×(0,+∞)
and vice versa.
(0,1)×{0} is not stretched over {1}×(0,+∞)
{1}×(0,+∞) is not shrunk to (0,1)×{0}
They both _are_
And their points correspond
by line A͞C͞E through point A.
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