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Re: Some results about unit fractions

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From	Jim Burns <james.g.burns@att.net>
Newsgroups	sci.logic
Subject	Re: Some results about unit fractions
Date	Mon, 24 Apr 2023 15:51:29 -0400
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On 4/24/2023 2:56 PM, WM wrote:
> Gus Gassmann schrieb am Montag,
> 24. April 2023 um 18:37:34 UTC+2:
>> On Monday, 24 April 2023
>> at 11:26:00 UTC-3, WM wrote:

>>> One of them is next to zero.
>>
>> Nope.
>> To every 1/n
>> there is another 1/(n+1) closer to,
>> but not equal to zero, and so on.
>
> NUF(x) is zero at x = 0 and
> ℵ for every eps that can be chosen.

NUF(x) is zero at x = 0 and
ℵ₀ for every eps > 0 which is
the between-point of a split of ℚ

We describe a point x
| x > 0 is a between-point of a slit of ℚ

We augment the description with
visibly-not-first-possibly-false claims.

We know that the augmenting claims are
as true as the augmented claims, about
whatever they're about.
Choosing is unnecessary for this knowledge.

> But it is impossible that NUF grows
> from 0 other than in units because
> all unit fraction sit at different places:
> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0.

The minimum number of unit fractions in
any neighborhood of 1/6.02214076×10²³
is 1
NUF() cannot grow by more than 1 near
1/6.02214076×10²³

Also, since
NUF(1/6.02214076×10²³) = ℵ₀ = ℵ₀+1
it cannot grow by 1, either.

The minimum number of unit fractions in
any neighborhood of zero
is ℵ₀
NUF() cannot grow by more than ℵ₀ near
zero.

----
x > 0  ⟹  NUF(x) = ℵ₀


2x > 1/n  ⟹  x/2 > 1/(4n)

NUF(2x) > 0  ⟹  NUF(x/2) > 0

0 < β ≤ᵋ {x: NUF(x) > 0}  ⟹
β/2 ∈∉ {x: NUF(x) > 0}  ⟹
0 ≥ β

0 ≥ β  ⟹
x > 0 ≥ β  ⟹
NUF(x) > 0  ⟹
x > 1/n  ⟹
⟨ ... 1/n ⟩ ⊆ (0,x]  ⟹
|⟨ ... 1/n ⟩| = NUF(x) = ℵ₀

x > 0  ⟹  NUF(x) = ℵ₀

> But it is impossible that NUF grows
> from 0 other than in units

> Do you deny this?

NUF() near x
can grow by up to the minimum number
of unit fractions near x.

Near 1/6.02214076×10²³
that minimum-present/maximum-growth
is 1

Near zero
that minimum-present/maximum-growth
is ℵ₀

Since ℵ₀+1 = ℵ₀
NUF is constant to right  of zero.

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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 Jim Burns <james.g.burns@att.net> - 2023-04-26 17:36 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-27 06:31 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-27 08:11 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-27 12:21 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-27 14:14 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-27 12:36 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-27 17:59 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-28 04:53 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-28 15:11 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-29 03:30 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-29 17:25 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-30 05:46 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-04-29 23:41 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-30 05:51 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-01 15:05 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-02 05:14 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-01 15:06 -0400
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-01 12:19 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-02 05:08 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-02 05:06 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-02 12:22 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-02 09:49 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-02 14:49 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-03 03:20 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-03 13:38 -0400
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-03 19:43 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-04 05:16 -0700
                Re: Some results about unit fractions Jeff Barnett <jbb@notatt.com> - 2023-05-04 09:43 -0600
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-05 05:01 -0700
                Re: Some results about unit fractions Jeff Barnett <jbb@notatt.com> - 2023-05-06 00:15 -0600
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-06 07:21 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-04 15:24 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-05 05:01 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-05 13:01 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-06 07:22 -0700
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-05-06 07:56 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-06 13:02 -0400
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-27 17:03 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-28 04:53 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-04-28 07:22 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-04-28 08:39 -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

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