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From: Alan Mackenzie <acm@muc.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 18:27:15 -0000 (UTC)
Organization: muc.de e.V.
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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 awa=
re
>> of.

> It doesn't. It is simply a property of potentially infinite initial=20
> segments of actually infinite set. Disastrous is that some naive minds=20
> are lead to believe that the actually infinite sets have "in fact" same=
=20
> substance. Assisted imbecility.

According to one of your other posts today, this "substance" is a
property only of subsets of N.  Other countable sets, or even subsets of
N not being considered as such, cannot use the concept "substance".

Countably infinite sets all have the same cardinality.

>>>> The cardinality of N is aleph-0.

>>>> What is the "reality" (in this sense) of N?

>>> The substance of =E2=84=95 is |=E2=84=95|. It is larger than every fi=
nite set. The
>>> substance of the set of prime numbers is far less than |=E2=84=95| ..=
...

>> By how much is its "substance" supposedly smaller?  Quantify it!

> It cannot be quantified yet. That would be a rewarding subject of futur=
e=20
> 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.

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

>> Are they?  What use are they?=20

> 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.  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.  I don't believe there are
such theorems, though I'm willing to be persuaded otherwise.

> Regards, WM

--=20
Alan Mackenzie (Nuremberg, Germany).