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NNTP-Posting-Date: Wed, 20 Nov 2024 03:59:02 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-standard)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Tue, 19 Nov 2024 19:59:05 -0800
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On 11/19/2024 07:45 PM, Ross Finlayson wrote:
> On 11/19/2024 02:38 PM, FromTheRafters wrote:
>> Jim Burns was thinking very hard :
>>> On 11/19/2024 4:38 PM, Ross Finlayson wrote:
>>>> On 11/19/2024 11:56 AM, Jim Burns wrote:
>>>>> On 11/19/2024 12:52 PM, Ross Finlayson wrote:
>>>
>>>>>> The "bait-and-switch" and "back-slide"
>>>>>> don't go well together.
>>>>>>
>>>>>> In either order, ....
>>>>>
>>>>> ⎛ Necessary and sufficient conditions for finiteness
>>>>> ⎜
>>>>> ⎜ 3. (Paul Stäckel)
>>>>> ⎜ S can be given a total ordering which is
>>>>> ⎜ well-ordered both forwards and backwards.
>>>>> ⎜ That is, every non-empty subset of S has both
>>>>> ⎝ a least and a greatest element in the subset.
>>>>>
>>>>> https://en.wikipedia.org/wiki/Finite_set
>>>>
>>>> Yeah we looked at that before also,
>>>> and I wrote another, different, definition of finite.
>>>
>>> Thank you for admitting that.
>>>
>>> However, you (RF) might NOT be bait.and.switch.ing
>>> if the definitions are equivalent.
>>>
>>> What was the definition you wrote before?
>>> I didn't see it in the rest of your post.
>>
>> Didn't he use not.ultimately.untrue instead of not.first.false? Is that
>> just an inversion? It seems to imply an ultimate or last instead of a
>> first or least. Just like WM when he inverted the naturals to unit
>> fractions.
>
> Bourbaki: is a French panel of "algebraic geometers".
>
> Now, you should know that algebra and geometry are
> two _different_ theories, besides for what all and
> where they may agree, in part, in parts.
>
> So, some for example Lefschetz make for being much
> more algebraic GEOMETERs, than, ALGEBRAIC, geometers.
>
> One of the concepts out of Bourbaki is, "strictly",
> with regards to their vacillations about "positive",
> whether "positive always means non-zero", "strictly".
>
>
> No, not like "WM". We're algebraic GEOMETERS.
>
> Then, besides models of potential, practical, effective,
> and/or actual infinity, with regards what is "finite"
> and what is "infinite", is here for what would be
> matters of "the infinite limit", that results finites.
> (Finite quantities.)
>
> Otherwise it's gratifying you might recall that
> remark in passing, because, it's a thing.
>
>
> In "Replacement of Cardinality (infinite middle)", 8/19 2024, this was:
>
>>
>
> I mean it's a great definition that finite has that
> there exists a normal ordering that's a well-ordering
> and that all the orderings of the set are well-orderings.
>
> That's a great definition of finite and now it stands
> for itself in enduring mathematical definition in defense.
>
> Why is it you think that Stackel's definition of finite
> and "not Dedekind's definition of countably infinite"
> don't agree?
>
> The entire idea here that there's a particular _regularity_
> due dispersion and modularity only courtesy division down
> from a fixed-point, that "Peano's axioms" don't give integers,
> they only give increments, i.e. not necessarily constant increments,
> that there's more than one _regularity_, REQUIRED, is another
> little fact of mathematics missing from your neat little hedgerow.
>
>
>


This went on for some time, ....



 >>>> Why is it you think that Stackel's definition of finite
 >>>> and "not Dedekind's definition of countably infinite"
 >>>> don't agree?
 >>
 >> I don't think they disagree, normally.
 >>
 >> Note: If you mean Dedekind's definition of infinite,
 >> it isn't limited to countably.infinite.
 >>
 >>>> The entire idea here that there's a particular _regularity_
 >>>> due dispersion and modularity only courtesy division down
 >>>> from a fixed-point, that "Peano's axioms" don't give integers,
 >>>> they only give increments, i.e. not necessarily constant increments,
 >>>> that there's more than one _regularity_, REQUIRED, is another
 >>>> little fact of mathematics missing from your neat little hedgerow.
 >>>
 >>> ..., REQUIRED, ....
 >>
 >> Things missing from my neat little hedgerow are
 >> missing because I intend for them to be missing.
 >> My neat little hedgerow has no weeds.
 >> It has not had and will not have weeds.
 >> And weeds would not be an improvement.
 >>
 >> My neat little hedgerow is well.ordered;
 >> each non.empty subset holds a minimum.
 >>
 >> In my neat little hedgerow,
 >> each Little Bunny Foo Foo has a successor,
 >> scooping up the field mice and bopping them on the head,
 >> and is a successor, except the first, named 0.
 >>
 >> Successors are non.0 non.doppelgänger non.final.
 >>
 >> You are welcome to talk about something else, Ross.
 >> Note, though, that,
 >> if you are talking about something else,
 >> then you are talking about something else.
 >> Non.triangles are not counter.examples to triangles.
 >> Non.Bunny.Foo.Foos are not counter.examples to Bunny.Foo.Foos.
 >>
 >> Have a nice day.
 >>
 >>
 >
 > The other day I read or leafed through and enjoyed
 > this pretty good little book called "Us & Them: The
 > Science of Identity", by a D. Berreby. Now, I don't
 > necessarily adhere to any same opinions, yet it's
 > rather didactic and establishes a sort of discourse
 > about what is so and considered so and what's not
 > and considered not.
 >
 > Then, the idea that that sort of reflexivity is or isn't
 > symmetrical, about the usual notions of conservation
 > and symmetry in this sort of world, is explored as
 > for matters of Berreby's opinion and lens about
 > how science that isn't physics or "mathematical",
 > i.e. that it's "non-logical", at all, isn't science.
 >
 > So, for nominalist fictionalists of the formalist
 > sort, while there may be strong mathematical
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