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NNTP-Posting-Date: Sat, 19 Oct 2024 20:43:00 +0000
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Newsgroups: sci.math
References: <vb4rde$22fb4$2@solani.org> <ve6329$19d5$1@news.muc.de>
 <ve64kl$2m0nm$4@dont-email.me> <ve66f3$19d5$2@news.muc.de>
 <ve683o$6c2o$1@solani.org> <09d9f0df-b1bb-42a7-af9b-890bfbcfc581@att.net>
 <b0fa9a1c-8375-4523-a15e-65789688660e@tha.de>
 <3f63bc22-83b2-4d56-9837-849551170c77@att.net>
 <50ac7044-f8c1-47d9-947f-9fa6044e1848@tha.de>
 <68b8be64-7fe8-47e7-a991-7adf14713af5@att.net> <vejmkm$e069$1@solani.org>
 <eb21591a-a60a-4baf-bdbd-afef2a69c230@att.net> <vejte9$e3ds$1@solani.org>
 <53460f91-4542-4a92-bc4b-833c2ad61e52@att.net> <ventec$255vi$2@dont-email.me>
 <venunr$2533b$4@dont-email.me> <29ce40e9-f18a-44d4-84d9-23e587cf9dea@att.net>
 <veor6u$2asus$1@dont-email.me> <2b6f9104-a927-49ee-9cf0-6ee3f82edc23@att.net>
 <verkkk$2r6kk$1@dont-email.me> <22f95ff7-c361-4d8a-943c-1df76abb98cc@att.net>
 <vevpsl$3pi3s$2@dont-email.me> <ed1862ff-3679-4175-bb25-c317be9713b2@att.net>
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 <7sacnUBFsJKWlon6nZ2dnZfqn_WdnZ2d@giganews.com>
From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 19 Oct 2024 13:43:27 -0700
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On 10/19/2024 12:40 PM, Ross Finlayson wrote:
> On 10/19/2024 11:04 AM, Ross Finlayson wrote:
>> On 10/19/2024 09:04 AM, Jim Burns wrote:
>>> On 10/19/2024 4:16 AM, WM wrote:
>>>> On 18.10.2024 00:34, Jim Burns wrote:
>>>>> On 10/1v7/2024 2:22 PM, WM wrote:
>>>>>> On 17.10.2024 00:39, Jim Burns wrote:
>>>
>>>>>>> The only set of natural numbers with no first
>>>>>>> is the empty set..
>>>>>>
>>>>>> No, the set of dark numbers is
>>>>>> another set without smallest element.
>>>>>
>>>>> A nonempty set without a first element
>>>>> is not a set of only finite ordinals.
>>>>
>>>> The set of dark numbers contains
>>>> only natural numbers.
>>>
>>> There is a general rule not open to further discussion:
>>> Things which aren't natural numbers
>>> shouldn't be called natural numbers.
>>>
>>>> What you call a "set of finite ordinals" is
>>>> not a set
>>>> but a potentially infinite collection.
>>>
>>> There is a general rule not open to further discussion:
>>> Finite sets aren't potentially infinite collections.
>>>
>>> ----
>>> Consider nonempty S of only finite ordinals:
>>> only ordinals with only finitely.many priors.
>>>
>>> k ∈ S is a finite ordinal
>>> Its set ⦃jβˆˆπ•†:j<k⦄ of priors is finite.
>>>
>>> ⦃jβˆˆπ•†:j<kβ¦„βˆ©S βŠ† ⦃jβˆˆπ•†:j<k⦄
>>> ⦃jβˆˆπ•†:j<kβ¦„βˆ©S is a finite set
>>> ⦃jβˆˆπ•†:j<kβ¦„βˆ©S holds its first or is empty.
>>>
>>> βŽ› If Priors.in.S ⦃jβˆˆπ•†:j<kβ¦„βˆ©S is empty
>>> ⎝then k is first.in.S
>>>
>>> βŽ› If Priors.in.S ⦃jβˆˆπ•†:j<kβ¦„βˆ©S is not empty
>>> ⎜ then i is first.in.⦃jβˆˆπ•†:j<kβ¦„βˆ©S
>>> ⎜
>>> βŽœβŽ› For i and m ∈ S, iβ‰ m,
>>> ⎜⎜ consider set {i,m} of finite ordinals
>>> ⎜⎜ {i,m} holds first.in.{i,m}
>>> ⎜⎜ i<m ∨ m<i
>>> ⎜⎜
>>> ⎜⎜ i<m
>>> βŽœβŽœβŽ› Otherwise, m<i  and
>>> ⎜⎜⎜ m ∈ ⦃jβˆˆπ•†:j<kβ¦„βˆ©S  and
>>> ⎜⎝⎝ i isn't first.in.⦃jβˆˆπ•†:j<kβ¦„βˆ©S
>>> ⎜
>>> ⎜ for i and m ∈ S, i≀m
>>> ⎝ i is first.in.S
>>>
>>> Nonempty S of only finite ordinals
>>> holds first.in.S
>>>
>>>>>> No, the set of dark numbers is
>>>>>> another set without smallest element.
>>>>>
>>>>> A nonempty set without a first element
>>>>> is not a set of only finite ordinals.
>>>>
>>>> The set of dark numbers contains
>>>> only natural numbers.
>>>
>>> If dark numbers 𝔻 doesn't hold first.in.𝔻
>>> then
>>> either 𝔻 is empty
>>> or 𝔻 isn't only finite ordinals.
>>>
>>>> Proof:
>>>> If you double all your finite ordinals
>>>> you obtain only finite ordinals again,
>>>
>>> Yes.
>>>
>>>> although the covered interval is
>>>> twice as large as the original interval
>>>> covered by "all" your finite ordinals.
>>>
>>> No.
>>> The least.upper.bound of finites is Ο‰
>>> The least.upper.bound of doubled finites is Ο‰
>>>
>>>
>>
>> The washing of dishes is one of those things
>> where the basic idea is, when it's deemed
>> necessary to wash a dish, and for some it's
>> right away and that's a good way of doing things,
>> that the idea is that once it's put away,
>> then you don't go hauling it out and washing it
>> again just for fun.
>>
>> What I'm saying is that WM never introduces
>> anything new so there's no reason to reply,
>> because, the readership here is already having
>> the benefit of any needful knowledge about it
>> otherwise.
>>
>>
>> Then though besides where it's like neither of
>> "countable cardinality" nor "asymptotic density"
>> need attack nor defense, each being a thing,
>> then the only amusement is that AP is an abstract
>> thinker with a langauge like Leonardo in the mirror
>> though it's broken, so a generous reading has to
>> be particularly generous and even a contrived sort
>> of way - then that what possible meaning the
>> infinite numbers or "the high side" of the integers,
>> may have, they're not "dark numbers" they're infinite
>> numbers, then there are simple theories where it's
>> so that "half the naturals are infinitely-grand each"
>> or "one of the naturals is infinitely-grand" or
>> "none of the naturals are infinitely-grand" then
>> usual Archimedean aspect, and usual enough non-Archimedean.
>>
>>
>> I have a job washing dishes one summer when what it
>> is: is that when one turns 16, then they could get a job,
>> and it was expected, because it was, so anyways I washed
>> dishes for a couple months, and got pretty good at it,
>> I'm a pro. Then I got some computer work, yet, that's
>> because most anybody should know how to do usual menial
>> things with acceptable quality like manual/manuel labor.
>>
>> There was this one song in the '80's called "On the Dark Side",
>> it got very heavy radio rotation for sure, one-hit wonder
>> of a sort.
>>
>>
>>
>>
>>
>
>
> The only Google hit for "non-Archimedean integer" is this
> Bottazi et alia about Robinsin's useless hyper-reals, ...,
> mostly seeming to shill "Easwaran and Towsner, ET", ...
>
>
> "Earlier in their text, ET do admit an uncountable number
> system for reasons of elegance, so as to be able to defend
> the use of R as the basic number system. But their insistence
> on trimming the language to countable size does not deliver
> the desired disqualification of non-Archimedean systems,
> since such countable systems can be constructed that admit
> no automorphisms, i.e., are rigid ...."
>
> So, the "non-standard countable" is a usual thing.
>
> The only search hit was a cross-mention of Skolem
> about non-Archimedean integer, yet apparently
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