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.

(We had a great long thread over on sci.logic
about Banach-Tarski and Vitali-Hausdorff, there's
quite a bit about the historical and technical arrival,
including references and links to Hausdorff's original.

Vitali's doubling-space reflects on "Zeno's graduation
course", where Zeno also has a doubling-space or
doubling-measure argument, since about 2300 years ago.

These are considered part of "mathematics", if
your project is wider than "bumbing- or dumbing-down W.M.".)

My project is potentially (dare I say it?) much wider
than explanations that sets do not change.

But what do I accomplish by talking about more,
if I say one thing, and a different thing is heard?

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On 11.11.2024 21:09, 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.

Neither do my intervals [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒].

Regards, WM

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On 11/11/2024 3:40 PM, WM wrote:

On 11.11.2024 21:09, 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.

Neither do my intervals [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒].

When I first read that,
I thought you meant [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒]
Later,
I thought you meant [4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

I feel that there's a good chance that
you mean one or the other.

Would you (WM) mind saying which?

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On 12.11.2024 05:32, Jim Burns wrote:

On 11/11/2024 3:40 PM, WM wrote:

On 11.11.2024 21:09, Jim Burns wrote:

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

How about Banach-Tarski equi-decomposability?

The parts do not change.

Neither do my intervals [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒].

When I first read that,
I thought you meant [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒]
Later,
I thought you meant [4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

Both intervals are one and the same, only shifted a bit. Or is it by
accident that you used n = 4 to cover q = 1/3?
1/1, 1/2, 2/1, 1/3, ...

Regards, WM

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On 11/12/2024 9:10 AM, WM wrote:

On 12.11.2024 05:32, Jim Burns wrote:

On 11/11/2024 3:40 PM, WM wrote:

On 11.11.2024 21:09, Jim Burns wrote:

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

How about Banach-Tarski equi-decomposability?

The parts do not change.

Neither do my intervals [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒].

When I first read that,
I thought you meant [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒]
Later,
I thought you meant [4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

Both intervals are one and the same,
only shifted a bit.

No.
[4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

⎛ Assume otherwise.
⎜ 4 ∈ [4-⅒,4+⅒]
⎜ Translateᵂᴹ.
⎜ 4 ∉ [4-⅒,4+⅒] = [1/3-⅒,1/3+⅒]
⎜
⎜ ’Twas brillig, and the slithy toves
⎜ Did gyre and gimble in the wabe:
⎜ All mimsy were the borogoves,
⎝ And the mome raths outgrabe.

The borogove.claims are not.first.false.
It doesn't matter where your interval is,
one of those prior claims is false.

But
we can't accept mimsy borogroves and slithy toves.
They're nonsense, of course.
We must surrender our telescope into infinity:
finite sequences of claims which are
only true.or.not.first.false claims.

One can't have both
transmogrifying intervals and
finite sequences giving knowledge of infinity.

Or is it by accident that
you used n = 4 to cover q = 1/3?
1/1, 1/2, 2/1, 1/3, ...

No accident.
You got my point.

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On 12.11.2024 17:47, Jim Burns wrote:

On 11/12/2024 9:10 AM, WM wrote:

[4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

|[4-⅒,4+⅒]| = |[1/3-⅒,1/3+⅒]| and only that is important for my argument.

Or is it by accident that
you used n = 4 to cover q = 1/3?
1/1, 1/2, 2/1, 1/3, ...

No accident.
You got my point.

Then you will get my point, hopefully.

Regards, WM

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On 11/12/2024 1:06 PM, WM wrote:

On 12.11.2024 17:47, Jim Burns wrote:

[4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

|[4-⅒,4+⅒]| = |[1/3-⅒,1/3+⅒]|
and only that is important for my argument.

Yes,
μ[4-⅒,4+⅒] = μ[1/3-⅒,1/3+⅒]

⎛ Also, |[0,1]| = |[0,2]|
⎝ so I think you mean 'measure', not 'cardinality'.

The value of a measure is defined to be
in the extended reals≥0.
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ } ∈ [0,+∞]

The extended reals≥0 are
the Archimedean reals≥0 [0,+∞)
plus one non.Archimedean point +∞
[0,+∞] = [0,+∞)∪{+∞}

The value of μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }
is not any Archimedean point, that is,
there is no finite m ∈ ℕ⁺ such that
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ } ≤ m

The value of μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }
is not Archimedean, so must be +∞
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ } = +∞

Also,
μ⋃{ [i/j-⅒,i/j+⅒]:i/j∈ℕ⁺/ℕ⁺ } = +∞

Or is it by accident that
you used n = 4 to cover q = 1/3?
1/1, 1/2, 2/1, 1/3, ...

No accident.
You got my point.

Then you will get my point, hopefully.

Your point is that
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }
isn't in the extended reals.

I get it.
Getting it and
agreeing with your changes to our definitions,
changes which turn the discussion into gibberish,
are different.

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On 12.11.2024 21:03, Jim Burns wrote:

On 11/12/2024 1:06 PM, WM wrote:

On 12.11.2024 17:47, Jim Burns wrote:

[4-⅒,4+⅒] ≠ [1/3-⅒,1/3+⅒]

|[4-⅒,4+⅒]| = |[1/3-⅒,1/3+⅒]|
and only that is important for my argument.

Yes,
μ[4-⅒,4+⅒] = μ[1/3-⅒,1/3+⅒]

⎛ Also, |[0,1]| = |[0,2]|
⎝ so I think you mean 'measure', not 'cardinality'.

Yes.

Your point is that
μ⋃{ [n-⅒,n+⅒]:n∈ℕ⁺ }
isn't in the extended reals.

I get it.

Here is the final solution of the problem:

1) If Cantor was right, then the initial intervals would be sufficient
to cover all rationals in the order 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4,
2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1,
... (these rationals are then centres of intervals).

2) If all rationals could be covered, then this could be accomplished in arbitrary order, not only along Cantors sequence. Because only the
measure is important, not the identity of an interval.

3) Then we could first cover all naturals and then all halves and then
all quarters and so on. But we know that already after covering all
naturals no further intervals are available.

Regards, WM

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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

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Am Tue, 12 Nov 2024 22:38:51 +0100 schrieb WM:

On 12.11.2024 21:03, Jim Burns wrote:

On 11/12/2024 1:06 PM, WM wrote:

On 12.11.2024 17:47, Jim Burns wrote:

3) Then we could first cover all naturals and then all halves and then
all quarters and so on. But we know that already after covering all
naturals no further intervals are available.

Well, because this order has the type
omega + omega + … omega = omega*omega = omega^2.
This amounts to saying that the naturals are a subset of the rationals.
It goes back to the lines (or columns) of your tired matrix.
That is not a bijection between N and Q. That doesn’t prove there is none.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.

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Am 13.11.2024 um 03:37 schrieb Chris M. Thomasson:

[...] It's fun that all real numbers are complex numbers but not
all complex numbers are real numbers...

Actually, this is a quite fascinating state of affairs. (In math the
there's a whole (and rather important) "branch" which relies on complex numbers: "function theory" or "complex analysis".*)

Many technical "approaches" (say in electrical engineering or physics)
DEPEND on complex numbers.

In fact, there's something quite remarkable in connection with quantum
theory: https://www.nature.com/articles/s41586-021-04160-4

_____________________________________________________________________

*) "Complex analysis, traditionally known as the theory of functions of
a complex variable, is the branch of mathematical analysis that
investigates functions of complex numbers. It is helpful in many
branches of mathematics, including algebraic geometry, number theory,
analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum
mechanics, and twistor theory. By extension, use of complex analysis
also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering." (Wikipedia)

--- SoupGate-Win32 v1.05
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Am 13.11.2024 um 03:37 schrieb Chris M. Thomasson:

[...] It's fun that all real numbers are complex numbers but not
all complex numbers are real numbers...

Actually, this is a quite fascinating state of affairs. (In math there's
a whole (and rather important) "branch" which relies on complex numbers: "function theory" or "complex analysis".*)

Many technical "approaches" (say in electrical engineering or physics)
DEPEND on complex numbers.

In fact, there's something quite remarkable in connection with quantum
theory: https://www.nature.com/articles/s41586-021-04160-4

_____________________________________________________________________

*) "Complex analysis, traditionally known as the theory of functions of
a complex variable, is the branch of mathematical analysis that
investigates functions of complex numbers. It is helpful in many
branches of mathematics, including algebraic geometry, number theory,
analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum
mechanics, and twistor theory. By extension, use of complex analysis
also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering." (Wikipedia)

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On 12.11.2024 23:03, joes wrote:

Am Tue, 12 Nov 2024 22:38:51 +0100 schrieb WM:

On 12.11.2024 21:03, Jim Burns wrote:

On 11/12/2024 1:06 PM, WM wrote:

On 12.11.2024 17:47, Jim Burns wrote:

3) Then we could first cover all naturals and then all halves and then
all quarters and so on. But we know that already after covering all
naturals no further intervals are available.

Well, because this order has the type
omega + omega + … omega = omega*omega = omega^2.
This amounts to saying that the naturals are a subset of the rationals.
It goes back to the lines (or columns) of your tired matrix.
That is not a bijection between N and Q. That doesn’t prove there is none.

The geometry of covering intervals does not depend on Cantor's trick.
The impossibility to cover the real line by intervals
I(n) = [n - 1/10, n + 1/10]
proves that there is no bijection possible.

By the way, when in the matrix
XOOO...
XOOO...
XOOO...
XOOO...
...
the X are accumulated around the upper left corner, we recognize the
same: By re-ordering the X there is no covering of the matrix by X possible.

Believers in matheology must doubt geometry.

Regards, WM

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