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

On 11/4/2024 12:32 PM, WM wrote:

On 04.11.2024 15:37, Jim Burns wrote:

On 11/4/2024 6:26 AM, WM wrote:

On 03.11.2024 23:18, Jim Burns wrote:

On 11/3/2024 3:38 AM, WM wrote:

Further there are never
two irrational numbers
without a rational number between them.

>

(Even the existence of neighbouring intervals
is problematic.)

>
There aren't any neighboring intervals.
Any two intervals have intervals between them.

>
That is wrong in geometry.
The measure outside of the intervals is infinite.
Hence there exists at least one point outside.
This point has two nearest intervals

>
This point,
which is on the boundary of two intervals,
is not two irrational points.

>
You are wrong.
The intervals together cover a length of less than 3.
The whole length is infinite.
Therefore there is plenty of space for
a point not in contact with any interval.

⎛ Assuming the covering intervals are translated
⎜ to where they are end.to.end.to.end,
⎜ there is plenty of space for
⎝ not.in.contact exterior points.
I mean 'exterior' in the topological sense.
For a point x in the boundary ∂A of set A
each open set Oₓ which holds x
holds points in A and points not.in A
An interior point of A is
is in A and not.in ∂A
An exterior point of A is
not.in A and not.in ∂A
⎛ Assuming end.to.end.to.end intervals,
⎜ there are exterior points
⎝ a distance 10¹⁰⁰⁰⁰⁰ from any interval.
However,
the intervals aren't end.to.end.to.end.
Their midpoints are
the differences of ratios of countable.to,
and not any other points.
Each of {...,-3,-2,-1,0,1,2,3,...} is
the midpoint of an interval.
There can't be any exterior point
a distance 1 from any interval.
There can't be any exterior point
a distance ⅟2 from any interval.
Nor ⅟3. Nor ⅟4. Nor any positive distance.
An exterior point which is not
a positive distance from any interval
is not an exterior point.
Therefore,
in what is _almost_ your conclusion,
there are no exterior points.
Instead, there are boundary points.
  For each x not.in the intervals,
  each open set Oₓ which holds x
  holds points in the intervals and
  points not.in the intervals.
x is a boundary point.
All of the line except at most 2³ᐟ²⋅ε is
boundary of the intervals.
This is a figure.ground inversion of
how we are used to thinking about boundaries,
the expanse of a square's interior with
the boundary _line_ around it, ...
⎛ In the limit, there is no interior, too.
⎜ And no exterior.
⎝ It's only boundary.
However, ℚ is not a square, nor is it
close to anything else we could call a figure.
⎛ This is a nice example of the difference between
⎝ what is intuitive and what is true.

Further there are never
two irrational numbers
without an interval between them.

 Not in reality. But in the used model.

What you're saying is:
⎛ I (WM) am not.talking about
⎝ what you all are talking about.
Which, in itself, is fine.
Billions of other people are not.talking about
what we all are talking about.
However, those people talking about
pop stars, or cosmology, or the rainy season
aren't imagining that they're talking about
what we all are talking about.
Which is what you are imagining, apparently.

The rationals are dense

Yes.
Each multi.point interval [x,x′] holds
rationals.

but the intervals are not.

No.
Each multi.point interval [x,x′] holds
ε.cover intervals.

This proves that
the rationals are not countable.

⎛ i/j ↦ kᵢⱼ = (i+j-1)(i+j-2)/2+i
⎜ k ↦ iₖ+jₖ = ⌈(2⋅k+¼)¹ᐟ²+½⌉
⎜  iₖ = k-(iₖ+jₖ-1)(iₖ+jₖ-2)/2
⎝  jₖ = k-iₖ
proves that
the rationals are countable.