Path: ...!news.mixmin.net!proxad.net!feeder1-2.proxad.net!usenet-fr.net!news.gegeweb.eu!gegeweb.org!pasdenom.info!from-devjntp Message-ID: <6WIT-GYNvuMQ6ADdNvBdVKBkQ1c@jntp> JNTP-Route: news2.nemoweb.net JNTP-DataType: Article Subject: Re: Replacement of Cardinality References: <6b837540-3d9a-4b8e-9a70-88d52e81a1a4@att.net> <9822f5da-d61e-44ba-9d70-2850da971b42@att.net> <4f606ef2-ef6c-487b-b959-d109e374929f@att.net> Newsgroups: sci.logic,sci.math JNTP-HashClient: uO04DEw7HcEfbv2KOFgnD472h10 JNTP-ThreadID: KFm3f7lT2HjaTSiMfnv5xqZoSBw JNTP-Uri: http://news2.nemoweb.net/?DataID=6WIT-GYNvuMQ6ADdNvBdVKBkQ1c@jntp User-Agent: Nemo/0.999a JNTP-OriginServer: news2.nemoweb.net Date: Sun, 04 Aug 24 18:13:22 +0000 Organization: Nemoweb JNTP-Browser: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/127.0.0.0 Safari/537.36 Injection-Info: news2.nemoweb.net; posting-host="82b75c1d0a83e677ff646b52485f72f8b23749df"; logging-data="2024-08-04T18:13:22Z/8975610"; posting-account="217@news2.nemoweb.net"; mail-complaints-to="julien.arlandis@gmail.com" JNTP-ProtocolVersion: 0.21.1 JNTP-Server: PhpNemoServer/0.94.5 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit X-JNTP-JsonNewsGateway: 0.96 From: WM Bytes: 3862 Lines: 77 Le 04/08/2024 à 18:39, Jim Burns a écrit : > On 8/4/2024 11:29 AM, WM wrote: >> Le 03/08/2024 à 21:54, Jim Burns a écrit : >>> On 8/3/2024 10:23 AM, WM wrote: > >>>> I recognized lately that you use >>>> the wrong definition of NUF. >>>> >>>> Here is the correct definition: >>>> There exist NUF(x) unit fractions u, such that >>>> for all y >= x: u < y. >> >>> Here is an equivalent definition: >>> There exist NUF(x) unit fractions u, such that >>> u < x >>> >>>> Note that the order is ∃ u ∀ y. >>> >>> The order is ∀x ∃u ∀y When all x are involved, the universal quantifier is usually not written. > > The order of the claim which you (WM) address > in an attempt to "prove" dark numbers is > ∀ᴿx > 0: > ∃U ⊆ ⅟ℕ ∧ |U| = ℵ₀: > ∀ᴿy ≥ x: > y >ᵉᵃᶜʰ U > > That claim and the following claim are > either both true or both false. > ∀ᴿx > 0: > ∃U ⊆ ⅟ℕ ∧ |U| = ℵ₀: > x >ᵉᵃᶜʰ U More of interest are these two claims which are not both true or both false: For every x there is u < x. There is u < x for every x. The latter is close to my function: There are NUF(x) u < x. > Your recently corrected definition of NUF is > NUF(x) = > |{u ∈ ⅟ℕ: ∀ᴿy ≥ x: y > u}| > > That definition is equivalent to > NUF(x) = > |{u ∈ ⅟ℕ: x > u}| > > Note that, > for x > 0, {u ∈ ⅟ℕ: x > u} > is maximummed and down.stepped and non.max.up.stepped. > For x > 0: |{u ∈ ⅟ℕ: x > u}| = ℵ₀ > > The claim you (WM) use > ∃U ⊆ ⅟ℕ ∧ |U| = ℵ₀: > ∀ᴿx > 0: > ∀ᴿy ≥ x: > y >ᵉᵃᶜʰ U > > is an unreliable quantifier shift from > the claim we make What claim you make is not of interest to me. I express that no u can be smaller than all x but that some u can be smaller than many x. You express that for all x, there is a smaller u. Both are very different. > Only you (WM) think that ∃u ∀x>0: u < x > follows from ∀x>0 ∃u: u < x, Not at all! Please spare these insults! Your claim concerns only definable x. For ℵo*2^ℵo undefinable points x it is wrong. My claim concerns all x. Regards, WM