Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Mon, 17 Mar 2025 14:38:15 +0100
Organization: A noiseless patient Spider
Lines: 63
Message-ID: <vr98k8$7mbl$1@dont-email.me>
References: <vqrbtd$1chb7$2@solani.org> <vqv62q$18mn$2@news.muc.de>
 <vr169k$18k4i$1@dont-email.me> <vr1bav$p45$1@news.muc.de>
 <vr1e8i$1er2v$1@dont-email.me> <vr1hig$5qt$1@news.muc.de>
 <vr29g3$23fi7$3@dont-email.me> <vr2d3k$jli$1@news.muc.de>
 <vr3fbu$1gbs1$3@solani.org> <vr3pvd$20r1$1@news.muc.de>
 <vr4cgl$3qbcs$3@dont-email.me> <vr6fgl$1uok$1@news.muc.de>
 <vr6tit$21dt9$1@dont-email.me> <vr71ea$qjf$1@news.muc.de>
 <vr774e$2a6rj$2@dont-email.me> <vr7b30$qjf$2@news.muc.de>
 <vr7jql$2jj8r$4@dont-email.me> <vr92l8$1pc1$1@news.muc.de>
 <vr986a$5ce0$2@dont-email.me>
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 14:38:16 +0100 (CET)
Injection-Info: dont-email.me; posting-host="1f236aeef04da6bd4e320424da101892";
	logging-data="252277"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1+83CAPoYDjHBNeFSXZD//t"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:pOe+tCeDEKLBWJCThjGGOa1piyo=
Content-Language: de-DE
In-Reply-To: <vr986a$5ce0$2@dont-email.me>
Bytes: 3640

Am 17.03.2025 um 14:30 schrieb Moebius:
> Am 17.03.2025 um 12:56 schrieb Alan Mackenzie:
>> WM <wolfgang.mueckenheim@tha.de> 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?!

It seems to me that N_def might be one of those sets E with the 
"suprising" (WM) property

                  E \ {e} = E     for an (some) e in E.

"The more things change, the more they stay the same"?

> .
> .
> .