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NNTP-Posting-Date: Sat, 19 Oct 2024 19:39:55 +0000
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Newsgroups: sci.math
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 <ve683o$6c2o$1@solani.org> <09d9f0df-b1bb-42a7-af9b-890bfbcfc581@att.net>
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 <eb21591a-a60a-4baf-bdbd-afef2a69c230@att.net> <vejte9$e3ds$1@solani.org>
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 19 Oct 2024 12:40:09 -0700
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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 ⦃j∈𝕆:j<k⦄ of priors is finite.
>>
>> ⦃j∈𝕆:j<k⦄∩S ⊆ ⦃j∈𝕆:j<k⦄
>> ⦃j∈𝕆:j<k⦄∩S is a finite set
>> ⦃j∈𝕆:j<k⦄∩S holds its first or is empty.
>>
>> ⎛ If Priors.in.S ⦃j∈𝕆:j<k⦄∩S is empty
>> ⎝then k is first.in.S
>>
>> ⎛ If Priors.in.S ⦃j∈𝕆:j<k⦄∩S is not empty
>> ⎜ then i is first.in.⦃j∈𝕆:j<k⦄∩S
>> ⎜
>> ⎜⎛ For i and m ∈ S, i≠m,
>> ⎜⎜ consider set {i,m} of finite ordinals
>> ⎜⎜ {i,m} holds first.in.{i,m}
>> ⎜⎜ i<m ∨ m<i
>> ⎜⎜
>> ⎜⎜ i<m
>> ⎜⎜⎛ Otherwise, m<i  and
>> ⎜⎜⎜ m ∈ ⦃j∈𝕆:j<k⦄∩S  and
>> ⎜⎝⎝ i isn't first.in.⦃j∈𝕆:j<k⦄∩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 ω
>> The least.upper.bound of doubled finites is ω
>>
>>
>
> 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
the search results these days are much reduced
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