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

WM <wolfgang.mueckenheim@tha.de> wrote:

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}.

So by "substance" you mean cardinality?

For many sets the relative substance cannot be determined. But this
drawback is less disastrous than to lump every countable set together.

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.

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!

.... but larger than every finite set. These are useful mathematical
findings.

Are they?  What use are they?  What mathematical theorems do they enable
the proof of?

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).