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| Newsgroups | sci.logic |
|---|---|
| Date | 2023-06-16 06:25 -0700 |
| References | (24 earlier) <adb974b4-f192-4a39-a9e0-a5314df6f99fn@googlegroups.com> <d3d96577-746b-4b6c-8505-c5d44734bcd6n@googlegroups.com> <fed0e46e-8421-4007-9381-9469f04d959cn@googlegroups.com> <86b2b992-195a-4350-bc93-4d83861cf59cn@googlegroups.com> <46943df9-46c2-423c-b15d-f13d98fe3bc3n@googlegroups.com> |
| Message-ID | <ea8fd240-89ea-4b5d-bbba-e9762ac8fcb8n@googlegroups.com> (permalink) |
| Subject | Re: Some results about unit fractions |
| From | WM <askasker48@gmail.com> |
Fritz Feldhase schrieb am Freitag, 16. Juni 2023 um 02:06:20 UTC+2:
> On Thursday, June 15, 2023 at 6:26:26 PM UTC+2, WM wrote:
> > Fritz Feldhase schrieb am Donnerstag, 15. Juni 2023 um 15:01:04 UTC+2:
> > > On Thursday, June 15, 2023 at 1:53:14 PM UTC+2, WM wrote:
> > > >
> > > > Can you imagine two terms of an inclusion-monotonic sequence (M_n): M(n) ⊇ M(n+1) which have no element in common?
> > > >
> > > Ja, Wenn für zwei Mengen A, B, A ⊇ B gilt, dann ist A n B = B. (*)
> > >
> > > Fordern wir also A n B = { }, so folgt B = { }. (Where A may be any set.)
> > >
> > [...] A last question: Why does this not hold for all terms of the sequence of endsegments?
>
> Because it does not hold for all terms of the sequence of endsegments.
It does.
> ______________________________
>
> An the other hand, the result from above can be generalized to any (finite) number of terms.
>
> Not a straight proof, but a reasonable explanation:
>
> Assume (for any n e IN) that we have A_1 ⊇ A_2 ⊇ ... ⊇ A_n. Then by (*) (...(((A_1 n A_2) n A_3) n A_4) ... ) n A_n = (...((A_2 n A_3) n A_4) ... ) n A_n = (...(A_3 n A_4) ... ) n A_n = ... = A_n-1 n A_n = A_n. In short: A_1 n A_2 n ... n A_n = A_n
>
> Hence if we require A_1 n A_2 n ... n A_n = { }, this implies A_n = { }. Where A_1, ..., A_n-1 may be any sets.
>
> Now this generalization does not "work" for _infinitely_ many terms A_1, A_2, A_3, ..., since there is no "last" term (A_n, in the "proof" above) in this case.
There you cannot require the empty intersection.
>
> You see, there's no last/biggest natural number, and an infinite sequene (A_n)_(n e IN) does not have a last term.
>
> ______________________________
>
> Actually, A_1 n A_2 n A_3 ... = { } does NOT imply that there is a k e IN such that A_k = { }, even though A_n ⊇ A_(n+1) may hold (for all n e IN).
>
> The sequence (E_n)_(n e IN) with E_n = {m e IN : m >= n} (for all n e IN) is a "counterexample".
>
> 1. We have E_n ⊇ E_(n+1) for all n e IN.
> 2. E_n =/= { } for all n e IN.
> 3. INTERSECTION_(n e IN) E_n = { }.
That is blatantly wrong.
The intersection loses in every step the same as the sequence of endsegments.
The intersection is the last endsegment. If there is no last endsegment, then there is no (last) intersection.
By simplest logic: If E_n =/= { } for all n e IN, then the intersection of all E_n is not empty.
But for every infinite endsegment the majority of numbers is contents and not indices.
Regards, WM
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Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-04 20:39 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-05 07:12 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-05 07:41 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 07:46 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-06 07:55 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 11:06 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-06 13:18 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 13:33 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-06 13:52 -0400
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-05 13:25 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 11:04 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-05 15:49 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-06 11:11 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-07 12:25 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-08 03:46 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-08 04:13 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-08 05:51 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-08 07:15 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-08 09:34 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-08 16:48 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-08 16:50 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-09 03:43 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-08 14:48 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-08 13:50 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-08 18:39 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-09 03:48 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-09 12:26 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-09 13:36 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-09 18:25 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-10 05:05 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-10 15:10 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-11 06:47 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-11 10:24 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-11 11:37 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-11 11:53 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-11 13:24 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-11 15:21 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-11 15:38 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-12 11:52 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-12 11:52 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-12 13:16 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-12 14:03 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-12 14:45 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-13 07:20 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-13 07:35 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-13 14:02 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-13 14:50 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-13 20:12 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-12 14:48 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-11 15:11 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-11 13:28 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-11 17:12 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-12 11:52 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-13 13:25 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-13 14:07 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-13 20:11 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-14 04:55 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-14 05:58 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-14 11:30 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-14 12:55 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 04:53 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-15 13:22 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 05:40 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-16 01:17 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:32 -0700
Re: Some results about unit fractions Python <python@invalid.org> - 2023-06-16 16:27 +0200
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-18 02:09 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-18 07:15 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-20 01:34 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-20 08:34 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-20 10:09 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-20 10:18 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-21 07:17 -0700
Re: Some results about unit fractions Python <python@invalid.org> - 2023-06-21 16:52 +0200
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-22 05:48 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-22 13:20 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-22 00:51 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-22 05:58 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-23 01:05 +0100
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-22 17:17 -0700
Re: Some results about unit fractions Ben Bacarisse <ben.usenet@bsb.me.uk> - 2023-06-23 02:26 +0100
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-23 09:01 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-23 08:53 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-23 09:52 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-15 06:01 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 09:26 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-15 09:52 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 10:59 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-15 12:40 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:07 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-16 06:22 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:41 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-16 08:54 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-16 12:37 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-17 05:54 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-17 08:46 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-18 06:57 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-17 11:56 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-18 06:59 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-18 14:40 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-19 07:01 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-19 07:34 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-20 08:30 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-20 10:30 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-19 13:55 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-20 08:36 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-20 10:24 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-20 14:03 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-21 07:22 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-21 12:26 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-22 05:54 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-22 09:38 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-23 08:44 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-23 12:37 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-23 09:53 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-23 15:39 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-24 05:43 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-16 08:53 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-15 17:06 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-16 06:18 -0700
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:53 -0700
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-17 08:10 -0700
Re: Some results about unit fractions Python <python@invalid.org> - 2023-06-17 17:17 +0200
Re: Some results about unit fractions Fritz Feldhase <franz.fritschee.ff@gmail.com> - 2023-06-17 08:30 -0700
Re: Some results about unit fractions Python <python@invalid.org> - 2023-06-17 17:33 +0200
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-16 06:25 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-14 08:03 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-14 10:58 -0700
Re: Some results about unit fractions Jim Burns <james.g.burns@att.net> - 2023-06-14 18:30 -0400
Re: Some results about unit fractions WM <askasker48@gmail.com> - 2023-06-15 04:51 -0700
Re: Some results about unit fractions Gus Gassmann <horand.gassmann@gmail.com> - 2023-06-09 14:18 -0700
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