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

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

On 12.11.2024 16:58, Jim Burns wrote:

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.

>
No, the intervals remain constant
in size and multitude.

Intervals which are constant _only_
in size and multitude
are not constant absolutely.

(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.

>
Therefore
the intervals covering all naturals
cannot cover more.

These intervals
{[n-⅒,n+⅒]: n∈ℕ⁺}
cover all naturals ℕ⁺  and
do not cover all fractions ℕ⁺/ℕ⁺

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⁺
These intervals
{[n-⅒,n+⅒]: n∈ℕ⁺}
and these intervals
{[i/j-⅒,i/j+⅒]: i/j∈ℕ⁺/ℕ⁺}
are different intervals.
Our intervals do not change.
Our sets do not change.
Our mathematical objects do not change.

By your argument
Cantor has been falsified.

_What you understand_ as Cantor
has been falsified.
I'd say "misunderstand", but
I do not find it attractive to
dress up as Jacques Derrida and
deconstruct Georg Cantor.
Let someone else do it, if they choose.
That thing you think Cantor says,
whether or not Cantor says it,
is false.
Our sets and other things of their ilk
  do not change.
Our finite sequences of claims which
  are each true.or.not.first.false
are each true.