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

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

On 08.11.2024 00:29, Jim Burns wrote:

On 11/7/2024 2:33 PM, WM wrote:

It is impossible however to cover
the real axis (even many times) by
the intervals
J(n) = [n - 1/10, n + 1/10].

>
Those are not the cleverly.re.ordered intervals.

>
They are the intervals that we start with.

Starting intervals
  {[n-⅒,n+⅒]: n∈ℕ⁺}
with midpoints
  { 1, 2, 3, 4, 5, ... }
and cleverly.shifted intervals
  {[iₙ/jₙ-⅒,iₙ/jₙ+⅒]: n∈ℕ⁺}
with midpoints
  { 1/1, 1/2, 2/1, 1/3, 2/2, ... }
⎛ n ↦ iₙ/jₙ
⎜  iₙ+jₙ = ⌈(2⋅n+¼)¹ᐟ²+½⌉
⎜  iₙ = n-((iₙ+jₙ)-1)((iₙ+jₙ)-2)/2
⎜  jₙ = (iₙ+jₙ)-iₙ
⎝  (iₙ+jₙ-1)(iₙ+jₙ-2)/2+iₙ = n

No boundaries are involved because
every interval of length 1/5 contains
infinitely many rationals and
therefore is essentially covered by
infinitely many intervals of length 1/5
- if Cantor is right.

>
I haven't claimed anything at all about
your all.1/5.length intervals.

>
Then consider the two only alternatives:
Either by reordering
(one after the other or simultaneously)
the measure of these intervals

https://en.wikipedia.org/wiki/Measure_(mathematics)
⎛ Let X be a set and Σ a σ-algebra over X.
⎜ A set function μ from Σ to
⎜  the extended real number line [!]
⎜ is called a measure
⎜ if the following conditions hold:
⎜ • Non-negativity: For all E ∈ Σ, μ(E) ≥ 0.
⎜ • μ(∅) = 0.
⎜ • Countable additivity (or σ-additivity):
⎜   For all countable collections {Eₖ}ₖ⃛₌₁ of
⎜   pairwise disjoint sets in Σ,
⎝   μ(⋃ₖ⃛₌₁Eₖ) = ∑ₖ⃛₌₁μ(Eₖ).
https://en.wikipedia.org/wiki/Extended_real_number_line
⎛ In mathematics, the extended real number system
⎜ is obtained from the real number system ℝ
⎜ by adding two[!] elements denoted +∞ and −∞
⎜ that are respectively greater and lower than
⎝ every real number.
The value of a measure is 0 or +∞ or
a point (in ℝ⁺) between fore and hind of
a split (of ℚ⁺) of all ratios of
numbers (in ℕ⁺) the countable.to from.1
(AKA finite).

the measure of these intervals can grow
from 1/10 of the real axis

≠ 0
≠ a point between a split of
  all ratios of the countable.to from.1
= +∞

to infinitely many times the real axis,

≠ 0
≠ a point between a split of
  all ratios of the countable.to from.1
= +∞

or not.

The clever re.ordering does not
increase the measure.
It is +∞ before and +∞ after.
+∞ has properties different from
  anything in (0,+∞)
In particular,
∀x ∈ (0,+∞): ∃x′ ∈ (0,x): 5⋅x′ = x
However,
+∞ ∉ (0,+∞)
¬∃x′ ∈ (0,+∞): 5⋅x′ = +∞

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"

But there is also an analytical proof:
Every reordering of
any finite set of intervals
does not increase their measure.
The limit of a constant sequence is
this constant however.

You are assuming that the measure of
the all.⅕.intervals at their starting positions
is some value outside the extended reals.
Otherwise,
for midpoints in {...,-2,-1,0,1,2,...}
the measure is in [0,+∞]
and not.in [0,+∞)
and thus equals +∞

This geometrical consequence of Cantor's theory
has, to my knowledge, never been discussed.
By the way I got the idea after a posting of yours:
Each of {...,-3,-2,-1,0,1,2,3,...} is
the midpoint of an interval.