Path: csiph.com!weretis.net!feeder8.news.weretis.net!eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: Jim Burns <james.g.burns@att.net>
Newsgroups: sci.logic
Subject: Re: Some results about unit fractions
Date: Fri, 23 Jun 2023 12:37:08 -0400
Organization: A noiseless patient Spider
Lines: 74
Message-ID: <935029c9-dd60-d452-e73b-c9bda9a95516@att.net>
References: <5ecf2c26-d125-4ba6-ac58-55acd906e111n@googlegroups.com> <c14f72f7-21bd-4e1e-83c2-47ccff361d54n@googlegroups.com> <a4b94e13-7d86-4e25-a158-cfeaa15ec4aen@googlegroups.com> <f8313e7d-292b-48d5-bb1c-ad6212c42e9dn@googlegroups.com> <a968c1f2-41f5-0e5b-7f37-c3f6179391f2@att.net> <7c5b1af0-08cd-4289-be3f-0bc57e199e30n@googlegroups.com> <7332a9ff-5d3f-bbe8-99f1-a7763c8a4cee@att.net> <4b598837-4840-4932-9944-0b16d264c43bn@googlegroups.com> <a7ffa792-df05-f96a-b9b0-05620f0226f5@att.net> <b8a4f329-873f-48df-be83-9e433f52757cn@googlegroups.com> <4ec24863-35b9-fc54-7e96-23f2677e1c2c@att.net> <4ddad321-cb5c-467f-a59e-f4d624eb4a61n@googlegroups.com> <2fb4c5a1-3d9d-9b02-f6f8-8e800f2f8ea3@att.net> <d41fe914-4fa1-448d-ac93-05e6c6293362n@googlegroups.com> <8363ca78-8496-8d6d-765e-50d4854bfd4c@att.net> <6115c3b7-20f0-49f2-a676-03adb4b8eb5cn@googlegroups.com> <456c8e66-730b-ce76-fe7a-1927dbf6fa85@att.net> <173f0674-3927-4d71-8392-db499056a751n@googlegroups.com>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Info: dont-email.me; posting-host="9aadc5e021debf909fdb31f05433aefb"; logging-data="3973129"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1+qJ6nIDhCFKQarcxGi61/MC5x/63inF0E="
User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:102.0) Gecko/20100101 Thunderbird/102.12.0
Cancel-Lock: sha1:VHUd1rxjpqLOYWQHtfmnnlOH4hA=
In-Reply-To: <173f0674-3927-4d71-8392-db499056a751n@googlegroups.com>
Content-Language: en-US
Xref: csiph.com sci.logic:254878

On 6/23/2023 11:44 AM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 22. Juni 2023 um 15:38:24 UTC+2:
>> On 6/22/2023 8:54 AM, WM wrote:

>>>> E(1) is a counter-example to your claim.
>>>> | Every infinite endsegment has
>>>> | an infinite set in common with
>>>> | every predecessor and
>>>> | every infinite successor.
>>>
>>> No.
>>
>> An element in common with each successor
>> not-exists.
> 
> An element in common with
> each *infinite* successor exists.

No.

ℕ := ⋃⟨ ⟨0…k⟩ ⟩

Each ⟨0…k⟩ is ⁺⁺ing 1×1 2-ended from 0

For each j ∈ ⋃⟨⟨0…k⟩⟩
⟨0…j⟩ exists, is ⁺⁺ing 1×1 2-ended
⟨0…j,j⁺⁺⟩ exists, is ⁺⁺ing 1×1 2-ended
j⁺⁺ ∈ ⋃⟨⟨0…k⟩⟩

For each j ∈ ⋃⟨⟨0…k⟩⟩
j is non-final

⋃⟨⟨0…k⟩⟩ is ⁺⁺ing 1×1 1-ended

For each j ∈ ⋃⟨⟨0…k⟩⟩
⋃⟨⟨0…k⟩⟩ =
⟨0…j⟩∥⟨j⁺⁺…⟩ =
⟨2-ended⟩∥⟨1-ended⟩
⟨j⁺⁺…⟩ ⊇ {common}
j ∉ ⟨j⁺⁺…⟩
j ∉ {common}

¬∃j ∈ {common}

> Successors contain only elements of E(1).

Each element of E(1) is non-final.
Each element of each successor of E(1)
is non-final.

>>>> An _infinite_ set common to
>>>> each (infinite) end segment
>>>> not-exists.
>>>
>>> Where should infinite endsegments have
>>> gained numbers not inherited from E(1)?
>>
>> That does not have the consequences
>> to which you feel entitled.
>
> It has the consequence:
> An element in common with
> each *infinite* successor exists.

No.
j ∈ ⟨0…j⟩∥⟨j⁺⁺…⟩ ⊇ {common}

> Successors contain only elements of E(1).

Each element in each successor
is non-final.