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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Mon, 17 Mar 2025 14:30:49 +0100
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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?!

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