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NNTP-Posting-Date: Thu, 05 Dec 2024 00:00:50 +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 16:00:59 -0800
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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.

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".