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

On 11/21/2024 11:24 AM, WM wrote:

On 21.11.2024 16:39, Jim Burns wrote:

On 11/21/2024 5:21 AM, WM wrote:

That means the function describing this,
1/10, 1/10, 1/10, ...
has limit 1/10.
That is the quotient of
the infinity of black intervals and
the infinity of all intervals.

>
The Paradox of the Discontinuous Function
(not a paradox):

>
It is a paradox
that only 1/10 of the real line is covered
for every finite interval (0, n]
but all is covered completely in the limit.
By what is it covered,
after all n have been proved unable?

⎛ n ↦ i/j ↦ n
⎜
⎜ (i+j) := ⌈(2⋅n+¼)¹ᐟ²+½⌉
⎜ i := n-((i+j)-1)⋅((i+j)-2)/2
⎜ j := (i+j)-i
⎜
⎝ (i+j-1)⋅(i+j-2)/2+i = n

lim.⟨ rc(1), rc(2), rc(3), ... ⟩  ≠
rc( lim.⟨ 1, 2, 3, ... ⟩ )
>
You (WM) do not "believe in"
proper.superset.matching sets
discontinuous functions

>
There is no reason to believe in magic.

⎛ Arthur C Clarke's Third Law
⎜
⎜ Any sufficiently advanced technology
⎝ is indistinguishable from magic.
What is sufficiently advanced for you (WM)?
Arithmetic.

But if you do, then
all Cantor-bijections can fail as well
"in the infinite".
Then mathematics is insufficient
to determine limits.

I am not enough of a scholar to know
that this is true of _all_ mathematics, but
I know that much knowledge of infinity,
including what I'm most familiar with,
is grounded in the _finite_
Here, I DON'T refer to finite numbers, etc.
I refer to finite sequences of CLAIMS,
each of which is true.or.not.first.false.
Because the sequence of CLAIMS is finite,
it is well.ordered.in.both.directions.
Because its subset of false CLAIMS
cannot hold a first false claim,
then, by well.order, that subset is empty.
In a sequence with no false claim,
each claim is true.
Even if a claim refers to
an indefinite one of infinitely many,
that claim is true for
that indefinite one of infinitely.many.
NOT because magically
we can check infinitely.many numbers
Because we can see FINITELY.many claims
and, for some sequences,
SEE that they are each true.or.not.false.