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NNTP-Posting-Date: Tue, 05 Nov 2024 18:28:16 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (re-Vitali-ized)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Tue, 5 Nov 2024 10:28:14 -0800
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On 11/05/2024 10:15 AM, Jim Burns wrote:
> On 11/5/2024 12:25 PM, Jim Burns wrote:
>> On 11/4/2024 12:32 PM, WM wrote:
>
>>> [...]
>>
>> ⎛ i/j ↦ kᵢⱼ = (i+j-1)(i+j-2)/2+i
>> ⎜ k ↦ iₖ+jₖ = ⌈(2⋅k+¼)¹ᐟ²+½⌉
>> ⎜  iₖ = k-(iₖ+jₖ-1)(iₖ+jₖ-2)/2
>> ⎝  jₖ = k-iₖ
>
> jₖ = (iₖ+jₖ)-iₖ
>
>> proves that
>> the rationals are countable.
>>
>>
>

Hausdorff even made for that all the
constructible is a countable union of countable.


Hausdorff was a pretty great geometer and
versed in set theory, along with Vitali he
has a lot going on with regards to "doubling
spaces" and "doubling measures", where there's
that Vitali made the first sort of example known
about "doubling measure", with splitting the
unit line segment into bits and re-composing
them length 2, then Vitali and Hausdorff also
made the geometric equi-decomposability of a ball.


Then, later, it's called Banach-Tarski for the
usual idea in measure theory that a ball can be
decomposed and recomposed equi-decomposable into
two identical copies, that it's a feature of
the measure theory and continuum mechanics actually.
Their results are ordinary-algebraic, though.

Then, it's said that von Neumann spent a lot
of examples in the equi-decomposable and the planar,
the 2D case, where Vitali wrote the 1D case and
Vitali and Hausdorff the 3D case, then I'd wonder
what sort of summary "von" Neumann, as he preferred
to be called, would make of "re-Vitali-ized"
measure theory.

There are also some modern theories about
"Rationals are HUGE" with regards to them
in various meaningful senses being much,
much larger than integers, among the integers.


Vitali and Hausdorff are considered great geometers,
and well versed in set theory. That's where
"non-measurable" in set theory comes from, because
Vitali and Hausdorff were more geometers than set theorists.