Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 11/19/2024 6:01 AM, WM wrote:

On 18.11.2024 20:22, Jim Burns wrote:

An infinite set
can match some proper supersets without growing  and
can match some proper subsets without shrinking.
 Sets which can't aren't infinite.

The set of even numbers is
a proper subset of the set of integers,
AND
the set of even numbers can match
the set of integers without either set changing.

>
That implies that
our well-known intervals

Sets with different intervals are different.
Our sets do not change.

our well-known intervals can
cover the real line or
reduce the average covering to 1/1000000000.

Sets of our well.known.intervals
can match some proper supersets without growing  and
can match some proper subsets without shrinking.
So they're infinite.
Thank you for clearing that up.
So, it's time to move on, right?

But every finite translation of
any finite subset of intervals maintains
the relative covering 1/5.

Each finite subset is finite.
Some subsets of infinite sets aren't finite.

If the infinite set has
the relative covering 1 (or more or less),
then you claim that
the sequence 1/5, 1/5, 1/5, ... has
limit 1 (or more or less).

Relative covering isn't measure.
Each of those interval.unions has measure +∞
You haven't defined 'relative covering'.
Giving examples isn't a definition.
You (WM) don't know what you (WM) mean.
However,
let's keep going.
I claim that there are functions f:ℝ→ℝ
such that
⟨ f(⅟1) f(⅟2) f(⅟3) ... ⟩  =
⟨ ⅟5    ⅟5    ⅟5    ... ⟩
and  f(0)  =  1
lim.⟨ f(⅟1) f(⅟2) f(⅟3) ... ⟩  ≠
f(lim.⟨ ⅟1 ⅟2 ⅟3 ... ⟩)
f is discontinuous at 0
Yes, I claim there are functions like f

So you deny analysis or / and geometry.

I deny what you think analysis and geometry are.
I accept infinite sets
and discontinuous functions
and similar triangles in proportion.
What is it you (WM) accuse infinite sets of,
other than not being finite?
Note:
An infinite set
can match some proper supersets without growing  and
can match some proper subsets without shrinking.
Sets which can't aren't infinite.