Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: WM Newsgroups: sci.logic Subject: Re: Simple enough for every reader? Date: Sat, 31 May 2025 16:04:37 +0200 Organization: A noiseless patient Spider Lines: 85 Message-ID: <101f29k$14h5f$3@dont-email.me> References: <100a8ah$ekoh$1@dont-email.me> <878qmt1qz6.fsf@bsb.me.uk> <100fu5r$1oqf5$1@dont-email.me> <87plg4yujh.fsf@bsb.me.uk> <100ho1d$272si$1@dont-email.me> <87ecwizrrj.fsf@bsb.me.uk> <100kbsj$2q30f$1@dont-email.me> <874ixbxy26.fsf@bsb.me.uk> <100s897$lkp7$1@dont-email.me> <87r00dv5s4.fsf@bsb.me.uk> <100ukdf$19g96$1@dont-email.me> <87ldqkura6.fsf@bsb.me.uk> <1011f3m$1uskr$1@dont-email.me> <871psbudqe.fsf@bsb.me.uk> <1014ade$2jasc$3@dont-email.me> <87jz61txrm.fsf@bsb.me.uk> <1017beq$39rdc$2@dont-email.me> <874ix4tg8m.fsf@bsb.me.uk> <1019qbq$3sv8u$1@dont-email.me> <877c1ysy5f.fsf@bsb.me.uk> <101ceht$gl20$1@dont-email.me> <871ps5sl4m.fsf@bsb.me.uk> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sat, 31 May 2025 16:04:36 +0200 (CEST) Injection-Info: dont-email.me; posting-host="1c760c873f92e67a7cfa5f1bcf4a03bf"; logging-data="1197231"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/CiZK1xLd4YSdpBTmvfTn+8D6ysU479ZM=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:5ikPFY/Wr1BaI3CmkDTBCC1T87M= Content-Language: en-US In-Reply-To: <871ps5sl4m.fsf@bsb.me.uk> On 31.05.2025 02:02, Ben Bacarisse wrote: > WM writes: > (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte > des Unendlichen" and "Kleine Geschichte der Mathematik" at Technische > Hochschule Augsburg.) > >> On 30.05.2025 03:08, Ben Bacarisse wrote: >>> WM writes: >> >>> I thought it might be something cumbersome and vague like that. I can't >>> even tell if this is a inductive collection, >> >> It is obvious and clear. Do you know a case where a natural number can be >> in it and cannot be in it? No. You can only curse. It is the same as >> Peano's set. If you can't understand blame it on yourself. > > Can you prove it is an inductive set/collection? See my book. The set is defined by induction. If n is in it, then also n+1 is in it. Pascal and Fermat used it without axioms as well as Cantor: "daß die Reihe 1, i2, i3, ..., i, ... nur eine Permutation der Reihe 1, 2, 3, ..., , ... ist. Dies beweisen wir durch vollständige Induktion," [Cantor, collected works, p. 305] The axiom has only been adapted because induction holds. > >>> so I must decline any >>> request to review a proof by induction based on it. >> >> Of course. There is no counter argument. So you must decline. > > No, I decline because I don't know if it is an inductive set. Do you? Every mathematician knows that the definable natural numbers are an inductive set. > > (I note you deleted the cumbersome and vague definition. It has been given to be understood. Now you have or have not understood. If not, the further presence would not help, I assume. > If it really > were obvious and clear, I would have left it in to show the world how > wrong I was to call it cumbersome and vague.) I can give you a simpler and shorter definition: Every n that can be expressed by digits is definable. > I see you've cut the incorrect definition and the claim that the axioms > directly say that 1 is in N because, presumably, you now see that they > don't. >> As I said that requires an intelligent reader recognizing that without ℕ >> obeying the axioms too the paragraph would be nonsense. > > That's funny! Yes, an intelligent reader will see you've written a junk > definition You are a dishonest liar. But that is not relevant. >> All natural numbers of Cantor's set ℕ can be manipulated collectively, for >> instance subtracted: ℕ \ {1, 2, 3, ...} = { }. Here all have >> disappeared. > > What definition of N do you want your intelligent readers to assume? ℕ is Cantor's infinite set. Otherwise I could not use ℵo in my proof. ℕ \ {1, 2, 3, ...} = { } For the set ℕ_def defined in my book we have ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo Since all numbers can be reduced to the empty set by subtracting them collectively, ℕ \ {1, 2, 3, ...} = { } they could also be reduced to the empty set by subtracting them individually - if this was possible. But then the well-order would force the existence of a last one. Contradiction. Regards, WM