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Failed to connect to MySQL: (1203) User howardkn already has more than 'max_user_connections' active connectionsPath: ...!news.mixmin.net!proxad.net!feeder1-2.proxad.net!usenet-fr.net!pasdenom.info!from-devjntp Message-ID: JNTP-Route: news2.nemoweb.net JNTP-DataType: Article Subject: Re: Replacement of Cardinality References: <980a0ec7476c9dc5823e59b2969398bd39d9b91d@i2pn2.org> <8d5b0145-b30d-44d2-b4ff-b01976f7ca66@att.net> Newsgroups: sci.math JNTP-HashClient: vFDSzDRvKtf-vf_bBxYYSzKGg3s JNTP-ThreadID: KFm3f7lT2HjaTSiMfnv5xqZoSBw JNTP-Uri: http://news2.nemoweb.net/?DataID=WtY45o_2xWeSk0s5VFpkO7OO7b0@jntp User-Agent: Nemo/0.999a JNTP-OriginServer: news2.nemoweb.net Date: Sat, 24 Aug 24 14:11:53 +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="2024-08-24T14:11:53Z/8998811"; 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: 2782 Lines: 44 Le 24/08/2024 à 00:04, Jim Burns a écrit : > On 8/22/2024 8:16 AM, WM wrote: >> Le 21/08/2024 à 20:34, Moebius a écrit : > >>> Yeah, it's like claiming: >>> "There is an end >>> (to the natural numbers), >>> because at and above omega >>> there is no natural number. >> >> Of course, but >> omega is somewhat ghostly. >> Do the natnumbers reach till omega? > > ω is an upper.bound of ℕᵈᵉᶠ. > Of all upper.bounds of ℕᵈᵉᶠ, the lowest is ω. Yes. Nevertheless almost all natural numbers are bewteen ℕᵈᵉᶠ and ω. > > Each element of ℕᵈᵉᶠ is not upper.bound of ℕᵈᵉᶠ. > No upper.bound of ℕᵈᵉᶠ is in ℕᵈᵉᶠ Correct. > >> Do the natnumbers reach till omega? > > Define > the natnumbers reach k ⇔ > (∀ᵒʳᵈj≤k:(∃ᵒʳᵈi:j=i∪{i} ⇐ j≠0) ∧ 0 > k ∈ ω ⇔ the natnumbers reach k > The natnumbers only reach elements of ℕᵈᵉᶠ. No. > > ω, an upper.bound of ℕᵈᵉᶠ, is not.in ℕᵈᵉᶠ. > The natnumbers do not reach ω ω - 1 is the greates naturak number. Easy to understand by the smallest unit fraction. Regards, WM