Re: Incompleteness of Cantor's enumeration of the rational numbers (doubling-spaces)

On 11/12/2024 05:02 PM, Chris M. Thomasson wrote:

On 11/12/2024 3:13 PM, Ross Finlayson wrote:

On 11/12/2024 01:36 PM, Jim Burns wrote:

On 11/12/2024 12:40 PM, Ross Finlayson wrote:

On 11/11/2024 12:59 PM, Ross Finlayson wrote:

On 11/11/2024 12:09 PM, Jim Burns wrote:

On 11/11/2024 2:04 PM, Ross Finlayson wrote:

On 11/11/2024 11:00 AM, Ross Finlayson wrote:

On 11/11/2024 10:38 AM, Jim Burns wrote:

>

Our sets do not change.
Everybody who believes that
  intervals could grow in length or number
is deeply mistaken about
  what our whole project is.

>
How about Banach-Tarski equi-decomposability?

>
The parts do not change.

>

any manner of partitioning said ball or its decomposition,
would result in whatever re-composition,
a volume, the same.

>

So, do you reject the existence of these?

>
No.
>
What I mean by "The parts do not change" might be
too.obvious for you to think useful.to.state.
Keep in mind with whom I am primarily in discussion.
I am of the strong opinion that
"too obvious" is not possible, here.
>
Finitely.many pieces of the ball.before are
  associated.by.rigid.rotations.and.translations to
finitely.many pieces of two same.volumed balls.after.
>
They are associated pieces.
They are not the same pieces.
>
Galileo found it paradoxical that
each natural number can be associated with
its square, which is also a natural number.
But 137 is associated with 137²
137 isn't 137²
>
I don't mean anything more than that.
I hope you agree.
>

Mathematics doesn't, ....

>
Mathematics thinks 137 ≠ 137²
>
>

>
1 = 1^2
0 = 0^2

[...]
>
Don't forget the i... ;^)
>
sqrt(-1) = i
i^2 = -1
>
?

Nah, then the quotients according to the
definition of division don't have unique quotients.
That is to say, complex numbers are given a definition
of division, that though is just an opinion, because
there are at least two ways it could be.
The above there instead are about _arithmetic's_
unique cases.
Now, of course 0^0 is a special case and said to
be either 0 or 1 depending on whether it's a
removable discontinuity or not the trigonometric,
that in powers and roots a removable discontinuity,
usually that either are replace-able.
How about instead making do with less and making
the x = y = x in positive real numbers stand in
as the original dimension, then framing all the
functions usually living in the positive quadrant
fit in an octant, and that all the line's crossing
that make for the "roots of zero" and more cases
where there are answers to "divide by zero".
I don't use complex numbers any day,
while 0 meters/second is always infinity seconds/meter.
Though, if I wanted to implement SIMD then
get into the FPU and make it up into complex
or hypercomplex to model reflections and rotations
and line up the Markov and Wiener series and use
that to make Kohonen and Hopfield on one of these
new dirt-cheap machines, yeah I could see why
having it hard-coded on the FPU would make
it dirt-cheap.
Though, I can always just hire a guy to flip a coin.
Which one arrives at the red dot?