Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: Moebius Newsgroups: sci.math Subject: Re: The non-existence of "dark numbers" Date: Mon, 17 Mar 2025 14:30:49 +0100 Organization: A noiseless patient Spider Lines: 50 Message-ID: References: Reply-To: invalid@example.invalid MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Mon, 17 Mar 2025 14:30:50 +0100 (CET) Injection-Info: dont-email.me; posting-host="1f236aeef04da6bd4e320424da101892"; logging-data="176576"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+5YhNIlihGGBmErfFeLpMY" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:Q7uYgLG8meXphSvBeyD0biQssJc= Content-Language: de-DE In-Reply-To: Am 17.03.2025 um 12:56 schrieb Alan Mackenzie: > WM wrote: >> On 16.03.2025 21:08, Alan Mackenzie wrote: >>> N is defined as the smallest inductive set. Indeed! Or rather, can be defined as such. :-P >> But that definition is impossible to satisfy. Sets are fixed, inductive >> "sets" are variable collections. Huh?! > Wrong. An inductive set exists by the axiom of infinity. Yeah, I already told this fucking asshole full of shit that fact in dsm. "Das Unendlichkeitsaxiom besagt, dass es eine induktive Menge gibt." ["The axiom of infinity states that there is an inductive set."] Source: https://de.wikipedia.org/wiki/Induktive_Menge > But just how do you think inductive sets vary? Do they vary by the day > of the week, the phases of the moon, or what? Can you give two > "variations" of an inductive set, and specify an element which is in one > of these variations, but not the other? Good question! Hint@Mückenheim: There are many different inductive sets which we usually consider as "fixed" (i.e. not varying). for example, IR, Q, Z, IN, etc. Hint@Mackenzie: WM calls IN in his ~textbook~ "für die ersten Semester" a "potentially infinite" set (whatever that may mean). Consequently he states an "axiom system for IN" which does not even allow to derive that 0 e IN (and for all n e IN: n+1 e IN). Hmmm... He may now call this IN (mentioned in his book) "IN_def", who knows? > [...] your N_def, as you have "defined" it, is satisfied by any proper subset of N. Indeed! > Or in a different interpretation, N_def = N, since An e N, N\{n} [as well as N\{1, ..., n} --moebius] is non-empty. See?! .. .. ..