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

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.

 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.

"My" boundary is a definition.

But it is irrelevant here like the offside rule in soccer.

The term "boundary" helps clarify
what I admit is a confusing situation.

No,it is not confusing at all. For every interval we can decide whether a poin is inside or outside.

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 :

But there are no points outside of intervals because, if Cantor has enumerated all rationals, then all rationals are caught and irrationals cannot be outside. Therefore a boundary is excluded. Points p can only exist insideof intervals.
Further: For every point x and for every interval we can find x - e where e is the endpoint of an interval. No boundary is useful, in particular because the infinite positive real axis except 3 unit intervals cannot be covered by the blurred boundary.
Regards, WM