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| From | Jim Burns <james.g.burns@att.net> |
|---|---|
| Newsgroups | sci.logic |
| Subject | Re: Some results about unit fractions |
| Date | 2023-04-26 17:36 -0400 |
| Organization | A noiseless patient Spider |
| Message-ID | <f40e0248-9ea1-396c-1a75-e5f976632d59@att.net> (permalink) |
| References | (13 earlier) <46855194-6b79-42a7-853c-327f5125e468n@googlegroups.com> <d2d13be9-3f8f-de2e-df9e-7601cfa165fa@att.net> <a4a69ca6-9d4c-4962-b5d9-7debcba88cedn@googlegroups.com> <096f452e-90aa-defd-391b-d3c71d8caace@att.net> <d7baa947-d6af-4be9-b6d1-b4c89dfd0127n@googlegroups.com> |
On 4/26/2023 7:50 AM, WM wrote:
>> On 4/25/2023 2:26 PM, WM wrote:
>>> Therefore
>>> NUF(x) grows in single steps of units.
>
>> No.
>
> Then you dismiss
> ∀n ∈ ℕ:
> 1/n - 1/(n+1) = 1/(n(n+1)) > 0.
> I don't.
∀n ∈ ℕ:
1/n - 1/(n+1) = 1/(n(n+1)) > 0.
is fine.
However,
that doesn't have the consequences
to which you feel entitled.
n ∈ ℕ⁺ ⟺ 1/n ∈ UF
ℕ⁺ is 1×1 1-ended.
A second end not-exists.
∀n ∈ ℕ⁺: n+1 ∈ ℕ⁺
UF has the same order as ℕ⁺
but reversed.
UF is 1×1 1-ended.
Each x ∈ (0,1) splits UF into
⟨ ... 1/⌈1/x⌉ ⟩ ⟨ 1/⌈1/x⌉⁻⁻ ... 1/1 ⟩
Each ⟨ ... 1/⌈1/x⌉ ⟩ is 1×1 1-ended.
ℕ⁺ and
any end segment of ℕ⁺
UF and
any fore segment of UF
can be matched.
|⟨ ... 1/⌈1/x⌉ ⟩| = |UF| = |N| = ℵ₀
NUF(x) = ℵ₀
>> Where NUF(x) = ℵ₀
>
> it must first have acquired
> ℵ₀ unit fractions.
For each natural n
for each split from 0 to n
a step across that split exists
from some last-before i
to some first-after j = i+1
However,
for ℵ₀
that's not true.
For the split {n < ℵ₀} {ℵ₀}
a last-before not-exists.
I don't say "dark but existing"
I say "not-existing".
A last-before ℵ₀ not-exists
because
each n < ℵ₀ is not last before ℵ₀
because
if
n < ℵ₀
then
for each split from 0 to n
a step across that split exists
from some last-before i
to some first-after j = i+1
and
for each split from 0 to n+1
a step across that split exists
from some last-before i
to some first-after j = i+1
and
n < n+1 < ℵ₀
and
n is not last before ℵ₀
For the split {n < ℵ₀} {ℵ₀}
a last-before not-exists.
ℵ₀ is not
a reallyreallyreallyreallyreallyreally large
natural number.
>> Where NUF(x) = ℵ₀
>
> it must first have acquired
> ℵ₀ unit fractions.
Using your terms, as best I can tell,
NUF(x) does not _acquire_ ℵ₀
x ≤ 0 ⟹ NUF(x) = 0
x > 0 ⟹ NUF(x) = ℵ₀
> It must have gone over ℵ₀ distances
> 1/(n(n+1)).
Each distance in (0,x] is above average.
ℵ₀ is not
a reallyreallyreallyreallyreallyreally large
natural number.
Informally,
each distance has finitely-many larger
distances to its right and
infinitely-many smaller distances
to its left,
because they're 1×1 1-ended.
For each distance, _on average_
they're to the left.
Each distance in (0,x] is above average.
More formally,
|
| Assume otherwise.
| Assume aₓ > 0 is the average distance
| between nearest unit fractions in (0,x]
|
| For mₓ := ⌈1/aₓ¹ᐟ²⌉
| 1/mₓ - 1/(mₓ+1) < aₓ
|
| ∀n > mₓ:
| 1/n - 1/(n+1) < aₓ
| and
| some distances are not above aₓ
|
| However,
| For
| nₓ := ⌈1/x⌉
| kₓ := ⌈1/(nₓaₓ)⌉+1
| in ⟨ 1/kₓ ... 1/nₓ ⟩
| the average distance is
| (1/nₓ - 1/kₓ)/(kₓ - nₓ) =
| 1/(nₓkₓ)
| and
| 1/(nₓkₓ) < aₓ
| and
| ∀i ∀j: j ≥ kₓ > i ⟹
| 1/j - 1/(j+1) < 1/i - 1/(i+1)
| and
| the average of _all_ unit fractions
| in (0,1] can't be larger than
| the average of all unit fractions
| in ⟨ 1/kₓ ... 1/nₓ ⟩
| which can't be as large as aₓ > 0
| Contradiction.
Therefore,
each distance in (0,x] is above average.
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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 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 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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