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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: Tue, 25 Mar 2025 21:23:32 +0100
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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