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Re: Some results about unit fractions

Newsgroups sci.logic
Date 2023-04-18 04:58 -0700
References (11 earlier) <87sfd0u0vc.fsf@bsb.me.uk> <31df6707-2b15-4f26-bc8d-c099dd5f319fn@googlegroups.com> <87sfczsef4.fsf@bsb.me.uk> <6f0eeea3-00f9-484e-8ac4-beb5490b342cn@googlegroups.com> <877cuaot6r.fsf@bsb.me.uk>
Message-ID <548ae661-a85e-4cfb-9842-79b02cbb485en@googlegroups.com> (permalink)
Subject Re: Some results about unit fractions
From WM <askasker48@gmail.com>

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Ben Bacarisse schrieb am Dienstag, 18. April 2023 um 03:49:53 UTC+2:
> WM <askas...@gmail.com> writes: 

> > You said: Speculation about something in (0,1] for which NUF(x) =/= ℵo 
> > can be ruled out

> You know that there are no unit fractions smaller than all x in (0,1] 
> and I know that there are no unit fractions smaller than all x in (0,1]. 
> Everyone knows that there are no unit fractions smaller than all x in 
> (0,1].

Then your above statement is wrong. "Something" in (0,1] can be every x > 0.

> > But if you now want to withdraw this statement, then NUF(x) is not 
> > constant over the whole interval.
> Specifically, NUF(x) = ℵ₀ for every x in (0,1] (the set of reals >0 and 
> <=1). This is trivial to prove. For any real x in (0,1]

Any real x = eps that you can prove something for has many unit fractions in (0, eps). But every real x that is existing has not many unit fractions in (0, x). That is the difference between visible and dark.

> >> So 
> >> start over and try again to do this "simple" thing: characterise 
> >> everything in (0,1] and prove ∃x in (0,1] with NUF(e) =/= ℵ₀. 
> > 
> > Impossible,
> So you are withdrawing the claim that it's simple to do what I asked?

Why should I? I use dark numbers, not only your rough epsilontics. For every eps, there are infinitely many smaller distances 1/(n(n+1)) between unit fractions. But if you believe it or not, these distances are real numbers although you cannot define every of them. This is a mathematical proof:

Your claim is NUF(x) is infinite for all x in (0, 1] which you can define. If however this were true for all distances 1/(n(n+1)) then you'd get this contradiction: If there are many unit fractions at the beginning of the interval, then there is also at least one unit fraction 1/n among them. Then by
∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0
the next one, 1/(n+1) sits inside the interval, not at the beginning. But this is true for each of these infinitely many unit fractions. Therefore only one can sit at the beginning before every x. Your claim is true for all eps that you can define, not for all distances between unit fractions.

Regards, WM

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Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-12 07:06 -0700
  Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-12 07:42 -0700
    Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-12 07:49 -0700
      Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-12 08:26 -0700
      Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-12 08:27 -0700
      Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-12 23:55 -0700
        Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-13 09:44 -0700
  Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-12 16:11 +0100
    Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-12 09:37 -0700
      Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-12 09:56 -0700
      Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-12 09:58 -0700
        Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-12 14:22 -0700
      Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-12 20:43 +0100
        Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-12 14:20 -0700
          Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-13 01:44 +0100
            Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-13 09:44 -0700
              Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-13 10:47 -0700
              Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-14 00:00 +0100
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-13 18:04 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-14 21:27 +0100
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-14 19:25 -0700
                Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-15 00:12 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-15 11:19 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-16 01:29 +0100
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-15 18:50 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-16 05:41 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-16 22:30 +0100
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-17 12:02 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-18 02:50 +0100
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-18 05:05 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-21 03:39 +0100
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-04-21 03:41 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-21 13:20 +0100
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-21 06:42 -0700
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-04-21 08:34 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-21 12:52 -0700
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-04-21 13:08 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-22 06:45 -0700
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-04-22 15:30 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-23 06:31 -0700
                Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-04-22 06:48 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-22 11:56 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-22 13:29 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-22 16:36 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-23 06:30 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-23 15:23 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-23 13:43 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-24 00:35 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-24 07:35 -0700
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-04-24 07:53 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-24 12:59 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-24 12:15 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-24 17:00 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-25 11:44 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-22 15:03 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-22 15:06 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-22 15:07 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-23 06:57 -0700
                Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-04-23 14:45 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-24 07:25 -0700
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-04-24 07:57 -0700
                Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-04-24 09:37 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-24 11:56 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-24 15:51 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-25 11:26 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-25 17:16 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-26 04:47 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-26 04:50 -0700
                Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-04-24 13:44 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-25 10:51 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-21 17:12 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-22 06:49 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-22 15:24 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-23 06:22 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-16 06:02 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-16 07:28 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-16 16:50 +0100
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-16 09:15 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-16 17:26 +0100
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-14 07:38 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-15 01:17 +0100
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-14 19:30 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-15 11:31 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-16 01:28 +0100
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-16 05:29 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-16 22:30 +0100
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-17 12:23 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-17 13:17 -0700
                Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-04-18 02:49 +0100
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-18 04:58 -0700
                Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-18 05:06 -0700
                Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-18 05:07 -0700
                Re: Some results about unit fractions Mostowski Collapse <bursejan@gmail.com> - 2023-04-18 06:54 -0700
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-04-19 05:17 -0700
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-04-18 04:32 -0700
                Re: Some results about POLLUTING RETARDED CUNTS Julio Di Egidio <julio@diegidio.name> - 2023-04-18 04:48 -0700
        Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-12 14:24 -0700
          Re: Some results about unit fractions Python <python@invalid.org> - 2023-04-12 23:44 +0200
            Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-13 10:01 -0700
  Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-12 08:20 -0700
  Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-12 08:30 -0700
    Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-12 09:39 -0700
      Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-12 09:57 -0700
  Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-12 15:55 -0400
    Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-12 14:17 -0700
      Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-12 18:32 -0400
        Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-13 09:48 -0700
          Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-13 18:50 -0400
            Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-14 07:15 -0700
              Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-14 07:27 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-14 07:39 -0700
                Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-14 08:36 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-14 09:38 -0700
                Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-14 09:45 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-14 09:56 -0700
                Re: Some results about unit fractions Julio Di Egidio <julio@diegidio.name> - 2023-04-14 10:08 -0700

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