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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The reality of sets, on a scale of 1 to 10 [Was: The
 non-existence of "dark numbers"]
Date: Tue, 25 Mar 2025 16:59:41 +0100
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On 24.03.2025 21:52, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:

> How is it disastrous to "lump every [infinite] countable set together"?
> Does it lead to a mathematical contradiction?  It doesn't that I'm aware
> of.

It doesn't. It is simply a property of potentially infinite initial 
segments of actually infinite set. Disastrous is that some naive minds 
are lead to believe that the actually infinite sets have "in fact" same 
substance. Assisted imbecility.
> 
>>> The cardinality of N is aleph-0.
> 
>>> What is the "reality" (in this sense) of N?
> 
>> The substance of ℕ is |ℕ|. It is larger than every finite set. The
>> substance of the set of prime numbers is far less than |ℕ| ....
> 
> By how much is its "substance" supposedly smaller?  Quantify it!

It cannot be quantified yet. That would be a rewarding subject of future 
research.
> 
>> .... but larger than every finite set. These are useful mathematical
>> findings.
> 
> Are they?  What use are they? 

Some researchers may be interested.

> What mathematical theorems do they enable
> the proof of?

Mathematical theorems can only be proved by use of potential infinity.

Regards, WM