Path: ...!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail 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:

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<1o2dnS1j2ssgaY76nZ2dnZfqnPqdnZ2d@giganews.com> <7sacnUBFsJKWlon6nZ2dnZfqn_WdnZ2d@giganews.com> From: Ross Finlayson Date: Sat, 19 Oct 2024 13:43:27 -0700 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: <7sacnUBFsJKWlon6nZ2dnZfqn_WdnZ2d@giganews.com> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 247 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-rOYeahnmEZW0LIRgrMpNYVuY1tVICv5vyjnH6dog4gUPkAiHdTSXN82RvKBEj5Dj3mJQVogfehMVran!Zl776T1aYUhG0Oh8nEhQ3nqL0Tn3+nxEm5VXY6t6pLGbarDGyzuRSnRLRNU0UquA+/NiEyEyAvJa X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 10663 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 ⦃j∈𝕆:j>> >>> ⦃j∈𝕆:j>> ⦃j∈𝕆:j>> ⦃j∈𝕆:j>> >>> ⎛ If Priors.in.S ⦃j∈𝕆:j>> ⎝then k is first.in.S >>> >>> ⎛ If Priors.in.S ⦃j∈𝕆:j>> ⎜ then i is first.in.⦃j∈𝕆:j>> ⎜ >>> ⎜⎛ For i and m ∈ S, i≠m, >>> ⎜⎜ consider set {i,m} of finite ordinals >>> ⎜⎜ {i,m} holds first.in.{i,m} >>> ⎜⎜ i>> ⎜⎜ >>> ⎜⎜ i>> ⎜⎜⎛ Otherwise, m>> ⎜⎜⎜ m ∈ ⦃j∈𝕆:j>> ⎜⎝⎝ i isn't first.in.⦃j∈𝕆:j>> ⎜ >>> ⎜ 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 ========== REMAINDER OF ARTICLE TRUNCATED ==========