Re: The non-existence of "dark numbers"

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"?

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