Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (repleteness)
Date: Wed, 18 Sep 2024 12:37:33 -0700
Organization: A noiseless patient Spider
Lines: 66
Message-ID: <vcfa5t$67r6$1@dont-email.me>
References: <vb4rde$22fb4$2@solani.org> <vbums6$8kdn$1@dont-email.me>
 <vbv9i7$bpjh$2@dont-email.me>
 <13c08e96ad635f8142b38d89863a80caf17a32a8@i2pn2.org>
 <vc1mfe$u3ec$2@dont-email.me>
 <4faa63d0ff8c163f01a38736aeb5732184218a29@i2pn2.org>
 <vc1uu8$u3ec$9@dont-email.me> <vc2gfb$130uk$1@dont-email.me>
 <mrednT-A88H9JHj7nZ2dnZfqn_ednZ2d@giganews.com>
 <y8idnRDAGsRvm3r7nZ2dnZfqnPGdnZ2d@giganews.com>
 <vc77he$29vl8$1@dont-email.me>
 <7-ycnVjnAIKXynr7nZ2dnZfqn_qdnZ2d@giganews.com>
 <vc7lrp$2dd4k$1@dont-email.me>
 <jmudndRL4M71F3r7nZ2dnZfqn_ednZ2d@giganews.com>
 <a8b5743b-1bcf-49c1-936b-c4e92af9dec4@att.net>
 <3906cb72-4bad-4a2a-97c7-4da857adc7a4@att.net>
 <Q3udnQzBXvkGcnX7nZ2dnZfqn_QAAAAA@giganews.com>
 <05bf9c39-b715-41d6-b349-87ddc67941ff@att.net>
 <jpCcnchGB4mbTHT7nZ2dnZfqnPqdnZ2d@giganews.com>
 <1b54c6c9-8b85-4c59-865a-fb601eaf4e1f@att.net>
 <NHadnZwmjuswfnT7nZ2dnZfqnPGdnZ2d@giganews.com>
 <a9b31ddd-1084-45bd-bf18-1246f3d4dd1e@att.net>
 <vJadnS6Nw_pN3Hf7nZ2dnZfqnPcAAAAA@giganews.com>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Wed, 18 Sep 2024 21:37:33 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="22866179a6b7ed6c688d4ac17c30f9bb";
	logging-data="204646"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18PMK2ZgsNz2+o1H2+n1tzGLfmuOyVlhXs="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:lfMbS9I6hPy+7t6SH90XD8R09rU=
Content-Language: en-US
In-Reply-To: <vJadnS6Nw_pN3Hf7nZ2dnZfqnPcAAAAA@giganews.com>
Bytes: 3691

On 9/17/2024 7:58 PM, Ross Finlayson wrote:
> On 09/17/2024 03:20 PM, Jim Burns wrote:
>> On 9/17/2024 4:16 PM, Ross Finlayson wrote:
>>> On 09/17/2024 01:11 PM, Jim Burns wrote:
>>>> On 9/17/2024 2:57 PM, Ross Finlayson wrote:
>>
>>>> Unlike ℕ and ℤ,  ℚ and ℝ do not 'next'.
>>
>>> Then, for initial segments or n-sets of naturals,
>>> the LUB of {f{n < m)} is "next": f(m+1).
>>
>> Yes.
>> (Presumably, you mean lub.{n:n≤m} = m. f()=? )
>>
>> You might enjoy this:
>>
>> ⎛ Define ℕ as well.ordered and nexted.
>> ⎜ well.ordered (A ⊆ ℕ holds min.A or is empty)
>> ⎝ nexted (m ∈ ℕ has m+1 m-1 next, except 0=min.ℕ)
>>
>> ⎛ In a finite order,
>> ⎝ each nonempty subset is 2.ended.
>>
>> Consider upper.bounded nonempty A ⊆ ℕ and
>> its set UB[A] ⊆ ℕ of upper.bounds
>> A ᵉᵃᶜʰ≤ᵉᵃᶜʰ UB[A]
>>
>> A is upper.bounded.
>> UB[A] ⊆ ℕ is nonempty.
>> UB[A] holds min.UB[A]
>>
>> A holds min.UB[A]
>> Otherwise,
>> (min.UB[A])-1 is a less.than.least upper.bound
>> (that is, what.it.is is gibberish)
>>
>> A holds min.UB[A] which upper.bounds A
>> min.UB[A] = max.A
>> Upper.bounded nonempty A holds max.A
>>
>> Upper.bounded nonempty A ⊆ ℕ holds min.A
>> (well.order)
>>
>> Upper.bounded nonempty A is 2.ended.
>>
>> And, similarly,
>> each (also.bounded) nonempty S ⊆ A is 2.ended.
>>
>> Upper.bounded nonempty A ⊆ ℕ is finite,
>> because
>> ℕ is well.ordered and nexted.
>>
>>
> 
> Yet, didn't you just reject, "infinite middle"?
> 
> 

Mid point:

p0 = (-1, 0)
p1 = (1, 1)
pdif = p1 - p0
pmid = p0 + pdif / 2