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Re: Some results about unit fractions

From Jim Burns <james.g.burns@att.net>
Newsgroups sci.logic
Subject Re: Some results about unit fractions
Date 2023-05-01 15:05 -0400
Organization A noiseless patient Spider
Message-ID <3caf0814-d41b-87bf-b7e5-70e623b59552@att.net> (permalink)
References (13 earlier) <337b5c2b-3464-4db6-a590-09cc97f9e839n@googlegroups.com> <2cd267cf-31b0-35bc-c7fb-4f06318e875d@att.net> <e2d27509-ae70-4248-92d9-8cd28884e316n@googlegroups.com> <5cc671b4-a3a7-72c0-3e79-b321367b8ec6@att.net> <6064be44-bdf0-424e-a7b0-cca9760ac04en@googlegroups.com>

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On 4/30/2023 8:51 AM, WM wrote:
> Jim Burns schrieb am Sonntag,
> 30. April 2023 um 05:42:03 UTC+2:
>> On 4/29/2023 6:30 AM, WM wrote:

>>> That is geometry.
>>
>> Consider the line segment [0,1]
>> from pt 0 to pt 1
>
> No, I am not willing to follow [geometry].

I agree, you aren't willing
to follow geometry.

> The sum of the internal intervals
> defined by
> ∀n ∈ ℕ:
> 1/n - 1/(n+1) = 1/(n(n+1)) > 0
> is needed to accumulate
> as many unit fractions
> as intervals are summed.
> This is more than nothing
> for more than zero unit fractions.

⟨ ... ⅟(n+2) ⟩,  ⟨ ⅟(n+1) ⅟n ... ⅟1 ⟩
⟨ ... ⅟(n+2) ⅟(n+1) ⟩,  ⟨ ⅟n ... ⅟1 ⟩
1×1 1-ended, 1×1 2-ended
1×1 1-ended, 1×1 2-ended

|⟨ ... ⅟(n+2) ⟩|,  |⟨ ⅟(n+1) ⅟n ... ⅟1 ⟩|
|⟨ ... ⅟(n+2) ⅟(n+1) ⟩|,  |⟨ ⅟n ... ⅟1 ⟩|
ℵ₀,  n+1
ℵ₀,  n

> Therefore
> ∀x ∈ (0, 1]: NUF(x) = ℵ
> is blatantly wrong.

| Assume you're correct.
| Assume ∃x ∈ (0,1]: NUF(x) < ℵ₀
|
| For totally-ordered set S ≠ ∅
| define
| |S| < ℵ₀  :⇔
| S is 1×1 2-ended
|
| NUF(x)  :=  |(0,x]∩⅟ℕ⁺|
|
| ⅟ℕ⁺ is the set of unit fractions.
|
| Assume (0,x]∩⅟ℕ⁺  =
| ⟨ ⅟ℵ₀⁻⁻ ... ⅟⌈1/x⌉ ⟩
| is 1×1 2-ended.
|
| ⅟ℵ₀⁻⁻  ∈  (0,x]∩⅟ℕ⁺
| ℵ₀⁻⁻  ∈  ℕ⁺
|
| ⟨ 1 ... ℵ₀⁻⁻ ⟩ exists
| such that,
| for any split between 1 and ℵ₀⁻⁻
| ⟨ 1 ... i ⟩,  ⟨ i⁺⁺ ... ℵ₀⁻⁻ ⟩
| some i is last-before
| and i⁺⁺ is first-after.
|
| Therefore,
| ⟨ 1 ... ℵ₀⁻⁻ ℵ₀ ⟩ exists
| such that,
| for any split between 1 and ℵ₀⁻⁻
| ⟨ 1 ... i ⟩,  ⟨ i⁺⁺ ... ℵ₀⁻⁻ ℵ₀ ⟩
| some i is last-before
| and i⁺⁺ is first-after,
| and,
| for the split ⟨ 1 ... ℵ₀⁻⁻ ⟩, ⟨ ℵ₀ ⟩
| ℵ₀⁻⁻ is last-before
| and ℵ₀ is first-after.
|
| ℵ₀  ∈  ℕ⁺
|
| (0,x]∩⅟ℕ⁺   ⊇
| ⟨ ⅟ℵ₀ ⅟ℵ₀⁻⁻ ... ⅟⌈1/x⌉ ⟩
|
| ⟨ ⅟ℵ₀⁻⁻ ... ⅟⌈1/x⌉ ⟩  ⊉
| ⟨ ⅟ℵ₀ ⅟ℵ₀⁻⁻ ... ⅟⌈1/x⌉ ⟩
|
| (0,x]∩⅟ℕ⁺  ≠  ⟨ ⅟ℵ₀⁻⁻ ... ⅟⌈1/x⌉ ⟩
| Contradiction.

Therefore,
¬∃x ∈ (0,1]: NUF(x) < ℵ₀

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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 WM <askasker48@gmail.com> - 2023-05-06 13:22 -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 WM <askasker48@gmail.com> - 2023-05-06 13:18 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-06 23:41 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-07 07:17 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-07 23:50 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-08 06:06 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-08 10:04 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-08 14:59 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-08 13:59 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-08 21:02 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-09 05:45 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-09 11:46 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-09 13:53 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-10 08:47 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-10 06:07 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-10 13:58 -0400
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-08 10:10 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-08 10:13 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-08 10:19 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-08 15:32 -0400
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-08 13:13 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-08 14:02 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-08 13:57 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-08 13:54 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-08 14:24 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-09 05:49 -0700
                Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-05-09 12:41 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-09 14:03 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-09 17:09 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-10 06:05 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-10 09:19 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-10 13:05 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-10 13:21 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-11 04:37 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-11 09:51 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-12 05:31 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-11 14:00 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-12 05:31 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-12 15:37 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-13 06:22 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-13 14:30 -0400
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-13 11:41 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-13 23:24 -0400
                Re: Some results about unit fractions JVR <jrennenkampff@googlemail.com> - 2023-05-14 02:20 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-14 14:04 -0700
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-14 14:44 -0700
                Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-05-13 15:44 -0700
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-14 14:09 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-13 19:41 -0400
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-13 23:41 -0400
                Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-05-14 14:11 -0700
                Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-05-14 18:17 -0400
                Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-05-14 16:16 -0700
                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

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