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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (re-Vitali-ized)
Date: Tue, 5 Nov 2024 14:50:35 -0800
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On 11/5/2024 1:51 PM, Ross Finlayson wrote:
> On 11/05/2024 10:28 AM, Ross Finlayson wrote:
>> 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.
>>
>>
>>
>>
> 
> Of course "ye olde Pythagoreans" had all rational.
> 

A fun animation about the Pythagoreans:

https://youtu.be/4m_ROtUQ5Vk

;^)