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NNTP-Posting-Date: Thu, 05 Dec 2024 02:52:06 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Wed, 4 Dec 2024 18:52:04 -0800
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On 12/04/2024 04:42 PM, Chris M. Thomasson wrote:
> On 12/4/2024 4:00 PM, Ross Finlayson wrote:
>> On 12/04/2024 03:05 PM, FromTheRafters wrote:
>>> Chris M. Thomasson used his keyboard to write :
>>>> On 12/4/2024 10:22 AM, FromTheRafters wrote:
>>>>> Ross Finlayson laid this down on his screen :
>>>>>> On 12/04/2024 02:33 AM, FromTheRafters wrote:
>>>>>>> WM formulated the question :
>>>>>>>> On 03.12.2024 21:34, Jim Burns wrote:
>>>>>>>>> On 12/3/2024 8:02 AM, WM wrote:
>>>>>>>>
>>>>>>>>>> E(1)∩E(2)∩...∩E(n) = E(n).
>>>>>>>>>> Sequences which are identical in every term
>>>>>>>>>> have identical limits.
>>>>>>>>>
>>>>>>>>> An empty intersection does not require
>>>>>>>>>   an empty end.segment.
>>>>>>>>
>>>>>>>> A set of non-empty endsegments has a non-empty intersection. The
>>>>>>>> reason is inclusion-monotony.
>>>>>>>
>>>>>>> Conclusion not supported by facts.
>>>>>>
>>>>>> Is it "pair-wise" inclusion, or "super-task" inclusion?
>>>>>>
>>>>>> Which inclusion is of this conclusion?
>>>>>>
>>>>>> They differ, ....
>>>>>
>>>>> I like to look at it as {0,1,2,...} has a larger 'scope' of natural
>>>>> numbers than {1,2,3,...} while retaining the same set size.
>>>>
>>>> { 1 - 1, 2 - 1, 3 - 1, ... } = { 0, 1, 2, ... }
>>>>
>>>> { 0 + 1, 1 + 1, 2 + 1, ... } = { 1, 2, 3, ... }
>>>>
>>>> A direct mapping between them?
>>>
>>> Yes, which more than just hints at a bijection. A bijection doesn't care
>>> about the symbols, only some idea of 'same size' or 'just as many'. An
>>> intersection requires knowing what symbols are in each set in order to
>>> 'find' matches. His infinite intersection of all endsegment sets is
>>> doomed to failure in the first iteration.
>>
>> The 'isomorphism" is a very generous term, usually,
>> indicating mutual structure.
>
> For some reason this is making me think of a conformal map...
>
>>
>> You know, the direct sum of infinitely many copies
>> of the naturals is defined one way while inductively
>> it's the other way, because it would otherwise see arrival
>> at this sort of "doom" you mention.
>>
>> Maybe instead you should figure it out that, for example,
>> in function theory there are non-Cartesian functions,
>> courtesy the domains of course, while there are a many
>> and a wide variety of topologies, with regards to what's
>> often relevant "continuous", topologies.
>>
>> The "function theory" and "topology" over time have seen
>> the most flexibility in, "definition", say. The,
>> probability theory, probably has the most lit-rature
>> on "non-standard probability", and for example they
>> say things like "erm let's not talk about functions
>> and just says distributions instead like Dirac delta
>> in case it would make our colleagues up the hall
>> stew at the luncheon".
>>
>>
>

In my podcasts I spent a lot of time talking
about "function theory" and "topology" as after
geometry, number theory, arithmetic, algebra,
function theory and topology, and operator calculus,
that being about it or much of it, what it is.

Of course, many things I maintain here,
I also mention there.

What you got there seems more about a "convolutional setting",
which intends to entail a usually left-right symmetry, with
regards to a usual idea of variables in x drawn left-right.

It reminds me when applying transformation matrices, and,
multiplying to result a true 90 degree rotation, which
is like pi/2, when the formulas into the transformation
matrix would reflect floating point values just... the
bit... off... that instead for rational cases are put
rational values or otherwise to result what results
alignment with the grid, say, the lattice.