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

On 11/6/2024 2:50 PM, WM wrote:

On 06.11.2024 20:08, Jim Burns wrote:

On 11/6/2024 11:24 AM, WM wrote:

On 06.11.2024 15:22, Jim Burns wrote:

On 11/6/2024 5:35 AM, WM wrote:

The intervals are closed with irrational endpoints.

>
'Exterior' seems like a good way to say
'not in contact'.

>
Every point outside is not an endpoint

>
Yes.
>

and is not in contact.

>
You don't know that.

>
From every positive point we know that
it is not 0 and
not in contact with (-oo, 0].
Same for every point not in an interval.

Is 0 "not in contact with" [-1,0) ⊆ ℝ
where
ℝ is points.between.splits of ℚ
ℚ is differences of ℚ⁺
ℚ⁺ is ratios of ℕ⁺
ℕ⁺ is countable.to from 1
?

The _union_ of
arbitrarily.many _open_ sets
is an open set.
However,
the _union_ of
these infinitely.many _closed_ intervals
  with irrational endpoints
is an open interval
  with rational endpoints.

>
We use only intervals, not limits.
A point is in an interval or it is not.

A point can be not.in the closure of each
of infinitely.many sets
and also in the closure of their union.
An ε.cover of ℚ holds infinitely.many sets.