Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: Moebius Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Tue, 14 Jan 2025 09:02:47 +0100 Organization: A noiseless patient Spider Lines: 90 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> Reply-To: invalid@example.invalid MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Tue, 14 Jan 2025 09:02:48 +0100 (CET) Injection-Info: dont-email.me; posting-host="637db88d7b8ef9efaea5539726d204cd"; logging-data="2456064"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18XawHo+g9Zv4A6nnZqOG35" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:zqnGFz7IIXycvNya80hcjkYx5xM= In-Reply-To: Content-Language: de-DE Am 13.01.2025 um 21:55 schrieb Chris M. Thomasson: > On 1/13/2025 9:17 AM, WM wrote: >> doubling of all natural numbers creates numbers larger than ω. Idiotic nonsense. > 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... Indeed. (At least in the context of classical mathematics/set theory.) Using symols: An e IN: n*2 e IN . (*) Now, by definition (of ω): An e IN: n < ω . Hence (with (*)): An e IN: n*2 < ω . That's why WM's claim "doubling of all natural numbers [results in] numbers larger than ω" is idiotic nonsense. For this -as you can see- we do not even have to consider "numbers larger than ω". A simple diagram might be helpful (except for WM): 1 < 1*2 < 3 < 2*2 < 5 < 3*2 < 7 < ... < ω . >>> ω is first infiniteᵒʳᵈ. >>> No infiniteᵒʳᵈ is before ω >>> No finiteᵒʳᵈ is after ω Indeed! >>> distances in ⦅0,ω⦆ multiplied by 2 remain regular distances in ⦅0,ω⦆, not.in ⟦ω,2ω⦆ I guess, this should read: "not.in ⟦ω,ω2⦆". >> Doubling of all n deletes the odd numbers Mückenheim, Du redest wieder mal saudummen Scheißdreck daher. Nichts wird "deleted", Du Depp. Die Menge G = {n*2 e IN : n e IN} ist einfach eine (echte) Teilmenge der natürlichen Zahlen, die (per definitionem) genau die sog. /geraden (natürlichen) Zahlen/ enthält. >> but cannot change the number of numbers, Das wiederum ist richtig. Es gilt nämlich (wegen IN ~ G): |IN| = |G| bzw. card(IN) = card(G) . >> therefore creates even numbers. Nö, die werden nicht "created", Du Depp, sondern lediglich mit Hilfe des sog. Aussonderungsaxioms aus der Menge der natürlichen Zahlen, IN, "ausgesondert": D. h. die waren vorher schon da und in IN enthalten (Hint: G c IN.) Das "Aussonderungsaxiom" sichert die Existenz der Menge G = {n*2 e IN : n e IN}. >> They do not fit below ω Doch das tun sie, weil sie als natürliche Zahlen natürlich (sic!) kleiner als ω sind. Hint: 1 < 1*2 < 3 < 2*2 < 5 < 3*2 < 7 < ... < ω . >> Remember: [no ordinal number] fits between them and ω. In der Tat: ~Eo e ORD: An e IN: n*2 < o < ω . .. .. ..