Deutsch English Français Italiano |
<vmo4s0$22am$1@dont-email.me> View for Bookmarking (what is this?) Look up another Usenet article |
Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: WM <wolfgang.mueckenheim@tha.de> Newsgroups: sci.math Subject: Re: The set of necessary FISONs Date: Tue, 21 Jan 2025 13:44:50 +0100 Organization: A noiseless patient Spider Lines: 19 Message-ID: <vmo4s0$22am$1@dont-email.me> References: <vmo1bs$1rnl$1@dont-email.me> <3a603a4009f4bdb24c23fc0851757c687e136bc8@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Tue, 21 Jan 2025 13:44:49 +0100 (CET) Injection-Info: dont-email.me; posting-host="19b83162cdb171373f09e1ba2e31b5fc"; logging-data="67926"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+ZuxirDMXv1yKiOEjhCAmoFzZxL1LwEJQ=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:WE/+4a5YhQJbGm5oJPCGZS2L2UY= In-Reply-To: <3a603a4009f4bdb24c23fc0851757c687e136bc8@i2pn2.org> Content-Language: en-US Bytes: 1927 On 21.01.2025 13:17, Richard Damon wrote: > On 1/21/25 6:45 AM, WM wrote: >> All finite initial segments of natural numbers, FISONs F(n) = {1, 2, >> 3, ..., n} as well as their union are less than the set ℕ of natural >> numbers. >> >> Proof: Assume UF(n) = ℕ. The small FISONs are not necessary. What is >> the first necessary FISON? There is none! All can be dropped. But >> according to Cantor's Theorem B, every non-empty set of different >> numbers of the first and the second number class has a smallest >> number, a minimum. This proves that the set of indices n of necessary >> F(n), by not having a first element, is empty. > Which is a proof of ANY, not ALL together, It is a proof of not any. The proof that not all together are necessary is this: U{F(1), F(2), F(3), ...} = U{F(2), F(3), F(4), ...}. Regards, WM