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Failed to connect to MySQL: (1203) User howardkn already has more than 'max_user_connections' active connectionsPath: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Mon, 13 Jan 2025 12:55:19 -0800 Organization: A noiseless patient Spider Lines: 41 Message-ID: References: <66868399-5c4b-4816-9a0c-369aaa824553@att.net> <417ff6da-86ee-4b3a-b07a-9c6a8eb31368@att.net> <07258ab9-eee1-4aae-902a-ba39247d5942@att.net> <1ebbc233d6bab7878b69cae3eda48c7bbfd07f88@i2pn2.org> <4c89380adaad983f24d5d6a75842aaabbd1adced@i2pn2.org> <494bfd3b-3c70-4d8d-9c70-ce917c15fc22@att.net> <72142d82-0d71-460a-a1be-cadadf78c048@att.net> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Mon, 13 Jan 2025 21:55:24 +0100 (CET) Injection-Info: dont-email.me; posting-host="7cf9ff7908854d51543addd9ba101a96"; logging-data="2139465"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/cpsTzUxkGbVBoWnCN7Y4hwozyPK7OzRw=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:G/Tc18yRg+1+/VdZPLqiLp6l9Fs= In-Reply-To: Content-Language: en-US Bytes: 3212 On 1/13/2025 9:17 AM, WM wrote: > On 13.01.2025 17:33, Jim Burns wrote: >> On 1/12/2025 2:39 PM, WM wrote: >>> On 12.01.2025 20:33, Jim Burns wrote: >>>> On 1/12/2025 10:54 AM, WM wrote: >> >>>>> No, it depends on completeness. >>>> >>>> It is completely true >>>> that each natural number is a natural number   and >>>> that only natural numbers are natural numbers. >>> >>> and that nothing fits between them and ω. >> >> Yes. > > Therefore doubling of all natural numbers creates numbers larger than ω. No. double all of the natural numbers: { 1 * 2, 2 * 2, 3 * 2, 4 * 2, ... } = { 2, 4, 6, 8, ... } All of those results are natural numbers. They were already there... >> >> ω is first infiniteᵒʳᵈ. >> No infiniteᵒʳᵈ is before ω >> No finiteᵒʳᵈ is after ω > > Right. >> Regular distances in ⦅0,ω⦆ multiplied by 2 >> remain regular distances in ⦅0,ω⦆, not.in ⟦ω,2ω⦆ > > Doubling of all n deletes the odd numbers but cannot change the number > of numbers, therefore creates even numbers. They do not fit below ω. > Remember: nothing fits between them and ω. > > Regards, WM >