Path: 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: Sun, 5 Jan 2025 14:18:32 -0800 Organization: A noiseless patient Spider Lines: 54 Message-ID: References: <8e95dfce-05e7-4d31-b8f0-43bede36dc9b@att.net> <53d93728-3442-4198-be92-5c9abe8a0a72@att.net> <9c18a839-9ab4-4778-84f2-481c77444254@att.net> <8ef20494f573dc131234363177017bf9d6b647ee@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sun, 05 Jan 2025 23:18:33 +0100 (CET) Injection-Info: dont-email.me; posting-host="0a9b3b8b2325ec9f2ae161f5c632ce0f"; logging-data="1298991"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19rMThWOo9xxVbXZ2sdGT762XIlNSqqNT4=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:xaEn2XK3MaV9OyCtnDwlQbBJ8bU= In-Reply-To: Content-Language: en-US On 1/5/2025 9:52 AM, WM wrote: > On 05.01.2025 18:35, Alan Mackenzie wrote: >> WM wrote: > >> Maybe Cantor did say this, as a pioneer early on in the development of >> set theory.  Things have developed since then, and we see that there is >> nothing to be gained by construing infinite sets as "fixed quantities"; > > It is a precondition of set theory. > >> there is no mathematical proof where such a concept makes the slightest >> difference. > > Every proof in set theory is based on the invariability of sets. >> >>> But all finite initial segments of natural numbers FISONs {1, 2, 3, >>> ..., n} cover less than 1 % of ℕ. >> >> That is a thoroughly unmathematical statement.  To talk about 1% of an >> infinite set is meaningless. > > It is not meaningless but an abbreviation for the fact that > multiplication by 100 does not reach or surpass ℕ. > >> Finally, it is >> wrong, absurdly wrong.  The union of all FISONs _is_ N. > > That is a a dogma of matheologioy disproved by the fact that every union > of FISONs which stay below a certain threshold stays below that > threshold. Every FISON stays below 1 % of ℕ. > >> No, not a mathematical proof. > > You mathematical "proofs" contradict the simple theorem stated above. > Therefore they are invalid. >>> {1, 2, 3, ..., 100n} is less than ℕ. That means the set of FISONs will >>> never cover ℕ, nor will its union reach the invariable quantity. >> >> No, it doesn't mean that at all.  The set of FISONs does indeed "cover" >> N, in the sense that their union is equal to N.  A proof of this is >> trivial, well within the understanding of a school student studying >> maths. > > My theorem is better understood by school students Oh my! > not yet stultified by > set theory. > > Regards, WM > >