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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: Mon, 24 Mar 2025 20:17:45 +0100
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On 23.03.2025 22:28, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:
>> On 22.03.2025 12:31, joes wrote:
> 
>>> for infinite sets "reality" and cardinality don't have to correspond.
> 
>> So it is! Substance and cardinality have nothing in common because sets
>> of very different substance have same cardinality.
> 
> What is this "reality" or "substance" of which you speak?

Substance is by far the better word. It denotes the number of elements. 
The set {1, 2, 3} has more substance than the set {7, 14}. For many sets 
the relative substance cannot be determined. But this drawback is less 
disastrous than to lump every countable set together.
> 
> 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| but larger 
than every finite set. These are useful mathematical findings.

Regards, WM