Path: ...!weretis.net!feeder9.news.weretis.net!news.quux.org!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: WM Newsgroups: sci.math Subject: Re: The set of necessary FISONs Date: Wed, 22 Jan 2025 11:34:09 +0100 Organization: A noiseless patient Spider Lines: 43 Message-ID: References: <3a603a4009f4bdb24c23fc0851757c687e136bc8@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 22 Jan 2025 11:34:09 +0100 (CET) Injection-Info: dont-email.me; posting-host="d8975555c6cdd0ee1f87064ff6013368"; logging-data="1018731"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/KAnajONq0mbofVjhoBOI1AmeFvX2+L2w=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:OyxGV+Npxfql4A1FW3iv1NZpco0= Content-Language: en-US In-Reply-To: Bytes: 2999 On 22.01.2025 00:41, Richard Damon wrote: > On 1/21/25 7:44 AM, WM wrote: >> 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), ...}. > which doesn't prove your claim about the Natural Numbers. It proves what I said: not all are required. > But this doesn't say that the infinite doesn't exist, and that we can't > make the Natural Numbers from a union of an infinite set of FISONs. 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 FISONs, by not having a first element, is empty. > > And, because FISONs are finite, no less than an infinite number of them > should be expected to be needed. Infinitely many fail like infinitely many traiangles would fail. > > This doesn't mean we need ALL of them, just an infinite number of them. Contradicted by Cantor's theorem. Regards, WM