Path: ...!Xl.tags.giganews.com!local-3.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail
NNTP-Posting-Date: Tue, 05 Nov 2024 21:52:02 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (re-Vitali-ized)
Newsgroups: sci.math
References: <vg7cp8$9jka$1@dont-email.me>
 <0e67005f-120e-4b3b-a4d2-ec4bbc1c5662@att.net> <vgab11$st52$3@dont-email.me>
 <ecffc7c0-05a2-42df-bf4c-8ae3c2f809d6@att.net> <vgb0ep$11df5$4@dont-email.me>
 <35794ceb-825a-45df-a55b-0a879cfe80ae@att.net>
 <772154a8-da85-49bc-a401-e217778e76b9@att.net>
 <B4adnZ06Ye9d_rf6nZ2dnZfqn_udnZ2d@giganews.com>
From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Tue, 5 Nov 2024 13:51:46 -0800
User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101
 Thunderbird/38.6.0
MIME-Version: 1.0
In-Reply-To: <B4adnZ06Ye9d_rf6nZ2dnZfqn_udnZ2d@giganews.com>
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Message-ID: <o4adnVXcobceDrf6nZ2dnZfqnPg1yJ2d@giganews.com>
Lines: 67
X-Usenet-Provider: http://www.giganews.com
X-Trace: sv3-yoi71ULxzmv9JKsTROhqjTYHSbJ/MZQ6cZ+d99wY8W94Ej8A3YslS4+Oe328OOytvepycoUdi8rtflV!3ZUXvvbPUmiGQFINg12umiFVZZypTtoIPsxhtYeLq8oe7BW44Hm0g4zMUWp7OQ1tVEOm1phV62Ft
X-Complaints-To: abuse@giganews.com
X-DMCA-Notifications: http://www.giganews.com/info/dmca.html
X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers
X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly
X-Postfilter: 1.3.40
Bytes: 3646

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.