Path: ...!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 non-existence of "dark numbers" Date: Thu, 13 Mar 2025 11:35:30 +0100 Organization: A noiseless patient Spider Lines: 82 Message-ID: References: MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 13 Mar 2025 11:35:31 +0100 (CET) Injection-Info: dont-email.me; posting-host="4c743e27c1816064b655fe5558154685"; logging-data="3353956"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/4ZUT9W3JfOukAiWq3ijFxQwwYw/6ubM0=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:TPzRKkD9/RZREdSUDX6H1Enajjg= Content-Language: en-US In-Reply-To: Bytes: 4271 On 12.03.2025 22:31, Alan Mackenzie wrote: > WM wrote: >> On 12.03.2025 18:42, Alan Mackenzie wrote: >>> WM wrote: > >>>> If the numbers are definable. > >>> Meaningless. Or are you admitting that your "dark numbers" aren't >>> natural numbers after all? > >> They > > They? > >>>> Learn what potential infinity is. > >>> I know what it is. It's an outmoded notion of infinity, popular in the >>> 1880s, but which is entirely unneeded in modern mathematics. > >> That makes "modern mathematics" worthless. > > What do you know about modern mathematics? I know that it is self-contradictory because it cannot distinguish potential and actual infinity. When |ℕ| \ |{1, 2, 3, ..., n}| = ℵo, then |ℕ| \ |{1, 2, 3, ..., n+1}| = ℵo. This holds for all elements of the inductive set, i.e., all FISONs F(n) or numbers n which have more successors than predecessors. Only those contribute to the inductive set! Modern mathematics must claim that contrary to the definition ℵo vanishes to 0 because ℕ \ {1, 2, 3, ...} = { }. That is blatantly wrong and shows that modern mathematicians believe in miracles. Matheology. You may recall me challenging > others in another recent thread to cite some mathematical result where > the notion of potential/actual infinity made a difference. There came no > coherent reply (just one from Ross Finlayson I couldn't make head nor > tail of). Potential infinity isn't helpful and isn't needed anymore. >>>>> 3. The least element of the set of dark numbers, by its very >>>>> definition, has been "named", "addressed", "defined", and >>>>> "instantiated". It is named but has no FISON. That is the crucial condition. > >>> So you counter my proof by silently snipping elements 4, 5 and 6 of it? >>> That's not a nice thing to do. > >> They were based on the mistaken 3 and therefore useless. > > You didn't point out any mistake in 3. I doubt you can. I told you that potential infinity has no last element, therefore there is no first dark number. > >>>> Try to remove all numbers individually from the harmonic series such >>>> that none remains. If you can't, find the first one which resists. > >>> Why should I want to do that? > >> In order to experience that dark numbers exist and can't be manipulated. > > Dark numbers don't exist, as Jim and I have proven. When |ℕ| \ |{1, 2, 3, ..., n}| = ℵo, then |ℕ| \ |{1, 2, 3, ..., n+1}| = ℵo. How do the ℵo dark numbers get visible? >> Induction cannot cover all natural numbers but only less than remain >> uncovered. > > The second part of that sentence is gibberish. Nobody has been talking > about "uncovering" numbers, whatever that might mean. Induction > encompasses all natural numbers. Anything it doesn't cover is not a > natural number, by definition. Every defined number leaves ℵo undefined numbers. Try to find a counterexample. Fail. Regards, WM