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

On 11/13/2024 11:31 AM, WM wrote:

On 13.11.2024 10:08, Jim Burns wrote:

On 11/12/2024 4:38 PM, WM wrote:

On 12.11.2024 20:01, Jim Burns wrote:

On 11/12/2024 11:43 AM, WM wrote:

But the rationals are more in the sense that
they include all naturals and 1/2.

>
These intervals
{[i/j-⅒,i/j+⅒]: i/j∈ℕ⁺/ℕ⁺}
cover all fractions ℕ⁺/ℕ⁺

>
But these are more intervals.

>
Are there more, though?
Or are there fewer?
i/j ↦ (i+j-1)(i+j-1)+2⋅i
>
⟨ 1/1 1/2 2/1 1/3 2/2 3/1 1/4 2/3 ... ⟩
↦
⟨ 2   4   6   8   10  12  14  16 ... ⟩

>
or
⟨ 1/1 1/2 2/1 1/3 2/2 3/1 1/4 2/3 ... ⟩
↦
⟨ 2   3   5   7   11  13  17  19 ... ⟩
>
or
⟨ 1/1        1/2       2/1  ... ⟩
↦
⟨ 10^10   10^10^10  10^10^10^10  ... ⟩

Yes,
or those, too.
_Without giving infinity much thought_
they each make it appear that
  ⟨ 1/1 1/2 2/1 1/3 2/2 3/1 1/4 2/3 ... ⟩
is strictly smaller than
  ⟨ 1   2   3   4   5   6   7   8   ... ⟩
My modest proposal is that
we stop declaring conclusions about infinity
without giving infinity much thought.

Or do infinite sets have different rules
than finite sets do?

>
If infinite sets obey the rules sketched above,

... _and are finite_ ...

then set theorists must discard geometry
because
 by shifting intervals
 the relative covering 1/5 of ℝ+ becomes oo*ℝ,
and analysis
because
 the constant sequence 1/5, 1/5, 1/5, ...
 has limit oo,
and logic
because of
 Bob.

----

 by shifting intervals
 the relative covering 1/5 of ℝ+ becomes oo*ℝ,

By definition,
the value of a measure is an extended real≥0
An extended real≥0 is either
Archimedean == having a countable.to bound,  or
non.Archimedean == not.having a countable.to bound.
The extended reals≥0 have only
the standard reals≥0, which are Archimedean, and
a single non.Archimedean point≥0  +∞
Neither the measure of the union of unshifted intervals
nor the measure of the union of shifted intervals
are Archimedean == neither has a countable.to bound.
Both the measure of the interval.union before shifting
and the measure of the interval.union after shifting
are the single non.Archimedean value  +∞
No,
the measure doesn't _become_ +∞
It has the same value +∞ before and after shifting.
----

 the constant sequence 1/5, 1/5, 1/5, ...
 has limit oo,

If
f(x) = y is continuous at xₗᵢₘ
then
the limit yₗᵢₘ of the value equals
  the value f(xₗᵢₘ) of the limit
⎛ x₁ x₂ x₃ ... → xₗᵢₘ
⎜ f(xₙ) = yₙ
⎜ y₁ y₂ y₃ ... → yₗᵢₘ
⎝ ⇒  f(xₗᵢₘ) = yₗᵢₘ
If
any function which jumps
(which crosses a line without intersecting it)
cannot be continuous everywhere,
then
there are uncountably.many points,
more points than names.for.points.
If
any function which jumps
cannot be continuous everywhere,
but
there aren't uncountably.many points,
then
it's impossible for what's described to exist.
----

 Bob.

KING BOB!
https://www.youtube.com/watch?v=TjAg-8qqR3g
If,
  in a set A which
  can match one of its proper subsets B,
  A ⊃≠ B  ∧  |A| = |B|
  (B can overwrite A one.for.one),
and,
  before overwriting, Bob is in A\B
then
  after overwriting, Bob isn't in overwritten.A = B