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

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.
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.
⋃{[⅟(k+√2),1-⅟(k+√2)]: k∈ℕ} = (0,1)
Recall that
0 = glb.{⅟(k+√2): k∈ℕ}
0 ∉ {⅟(k+√2): k∈ℕ}
1 = lub.{1-⅟(k+√2): k∈ℕ}
1 ∉ {1-⅟(k+√2): k∈ℕ}

Something of your theory is inconsistent.

>
Your intuition is disturbed by
an almost.all boundary.

>
No. Your boundary is nonsense.

"My" boundary is a definition.
It states what we take the term "boundary" to mean.
The open sets involved exist or not.exist,
independently of whether we use or not.use
  the term "boundary".
The term "boundary" helps clarify
what I admit is a confusing situation.
_Of course_ you (WM) object to clarity.
Clarity is your nemesis.
https://en.wikipedia.org/wiki/Boundary_(topology)
⎛ It is the set of points p ∈ X such that
⎜ every neighborhood of p contains
⎜ at least one point of S and
⎝ at least one point not of S :