Path: csiph.com!news.mixmin.net!gandalf.srv.welterde.de!eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Jim Burns Newsgroups: sci.logic Subject: Re: Some results about unit fractions Date: Sat, 17 Jun 2023 11:56:14 -0400 Organization: A noiseless patient Spider Lines: 130 Message-ID: <7332a9ff-5d3f-bbe8-99f1-a7763c8a4cee@att.net> References: <5ecf2c26-d125-4ba6-ac58-55acd906e111n@googlegroups.com> <19ad04ee-38d5-47c6-b9d8-1a453c4f6b0fn@googlegroups.com> <86b2b992-195a-4350-bc93-4d83861cf59cn@googlegroups.com> <312b0287-f7aa-422c-bf4d-2671310a286dn@googlegroups.com> <5014348b-dc24-4c62-bfc1-c7a251e46779n@googlegroups.com> <7c5b1af0-08cd-4289-be3f-0bc57e199e30n@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="bd4e8ad075c3a5b8a658fa0e094e74f3"; logging-data="1365278"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18SxVdPiZFqPKVpJmuY7LB4pQjjME02yuI=" User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:102.0) Gecko/20100101 Thunderbird/102.12.0 Cancel-Lock: sha1:axuW4wS1Am1ZRAaZqJil0ffoqZo= Content-Language: en-US In-Reply-To: <7c5b1af0-08cd-4289-be3f-0bc57e199e30n@googlegroups.com> Xref: csiph.com sci.logic:254497 On 6/17/2023 8:54 AM, WM wrote: > Jim Burns schrieb am Freitag, > 16. Juni 2023 um 18:37:25 UTC+2: >> On 6/16/2023 9:41 AM, WM wrote: >>> Fritz Feldhase schrieb am Freitag, >>> 16. Juni 2023 um 15:22:02 UTC+2: >>>> Die Menge der/aller Endsegmente ist >>>> unendlich und jedes Endsegment ist selbst >>>> eine unendliche Menge. > >>> That is a self-contradicting statement. >> >> Those are true statements > > Every infinite endsegment has > an infinite set in common with > every predecessor and > every infinite successor. No. ⟨finite⟩₀ ⁺⁺ing 1×1 2-ended from 0 Define ℕ := ⋃⟨finite⟩₀ For each j ∈ ℕ := ⋃⟨finite⟩₀ ⟨finite⟩₀ exists such that j ∈ ⟨finite⟩₀ ⟨semifinite⟩ⱼ₊₊ ⁺⁺ing 1×1 1-ended from j⁺⁺ ¬(⟨semifinite⟩ⱼ₊₊ = ∅) ℕ = ⟨semi-finite⟩₀ ...because each j ∈ ℕ is non-final, because each ⟨finite⟩₀ is non-final. For each j ∈ ℕ := ⋃⟨finite⟩₀ a split ⟨finite⟩₀∥⟨semifinite⟩ⱼ₊₊ exists such that j ∈ ⟨finite⟩₀ ¬(j ∈ ⟨semifinite⟩ⱼ₊₊) ¬(⟨semifinite⟩ⱼ₊₊ = ∅) ⟨semifinite⟩ⱼ₊₊ ⊇ {common} ¬(j ∈ {common}) For each j ∈ ℕ := ⋃⟨finite⟩₀ ¬(j ∈ {common}) {common} = ∅ The set in common with every predecessor and every infinite successor has no elements and is not infinite. > The set of indices ends at > the minimum common infinite set. The minimum common infinite set is not infinite and not-exists. > An ending set is not infinite. A description can exist of something which not-exists. An ending set not-exists. >>> As long as the first set is finite, >>> 1, 2, 3, ..., n | n, n+1, n+2, ..., >>> the second set is infinite and >>> larger than the first. > >>> The infinity of the endsegments prevents >>> the infinity of their cardinal number. >> >> No. > > Yes. No. >> Each j ∈ ⟨0...⟩ splits ℕ into >> ⟨0...j⟩∥⟨j⁺⁺...⟩ >> and > each ⟨0...j⟩ is non-final. Also, each ⟨j⁺⁺…⟩ is non-final. ⟨⟨1…⟩...⟩ is semifinite, not finite. > finite. There is no exception. > ∀j ∈ ⟨0...j⟩: |ℕ \ {1, 2, 3, ..., j}| = ℵo Better: ∀j ∈ ⟨0...j′⟩: {1,2,3,...,j} = ⟨finite⟩₁ ℕ⁺ := ⋃⟨finite⟩₁ ℕ⁺ = ⟨semifinite⟩₁ ℕ⁺\{1,2,3,...,j} = ⟨semifinite⟩ⱼ₊₊ ¬(j ∈ ℕ⁺\{1,2,3,...,j}) ¬(j ∈ {common}) {common} = ∅ >> A step from finite to infinite >> not-exists, >> because >> anything which is a step from finite >> is not infinite. > > That shows that > all stepable sets are finite. Each ⁺⁺ing 1×1 2-ended from 0 is finite. The union of all (finite) ⁺⁺ing 1×1 2-endeds from 0 is ⁺⁺ing 1×1 1-ended from 0 and is not finite ...because each ⟨finite⟩₀ is non-final.