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

On 25.03.2025 19:27, Alan Mackenzie wrote:

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

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.

 According to one of your other posts today, this "substance" is a
property only of subsets of N.

They supply the simplest explanation. But substance is in every non-empty set.

Countably infinite sets all have the same cardinality.

That proves that cardinality is rather uninteresting.

 

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.

 It can indeed by quantified.  The assymptotic distribution of prime
numbers is known: the probability of a number near n being prime is
1/log(n).  So the proportion of numbers in {1, ..., n} which are prime
will tend to zero as n tends to infinity.

Tend to yes, but not reaching it.

 

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

 

Are they?  What use are they?

 

Some researchers may be interested.

 Maybe.  On the other hand, maybe not.
 

What mathematical theorems do they enable the proof of?

 

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

 That's a very bold statement.  Many theorems can be proven without regard
to the infinite.

Of course I meant theorems using the infinite.

 Many others do in fact use the infinite.
 But theorems which require the concept of "potentially infinite", over
and above plain infinite, for their proof?  I've asked you before for an
example, and you've yet to come up with one.

Every theorem in analysis. This has not much changed since Cantor and Hilbert.
Regards, WM