Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"]

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.