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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: Wed, 26 Mar 2025 20:36:40 +0100
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On 26.03.2025 00:39, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:

>> substance is in every non-empty set.
> 
> Seems doubtful.  What you seem to be saying is that every set has a
> superset, you embed the set in that superset, then a portion of the
> superset is the original set.  That portion, a number between 0 and 1,
> then becomes the "substance".

No, that is not what I meant. Substance is simply the elements of the 
set. The amount of substance is the number of elements. This number 
exists also for actually infinity sets but cannot be expressed by 
natural numbers.

We only know that ∀k,n ∈ ℕ_def: |ℕ|/k > n.
> 
> You're saying that the "substance" isn't a property of a set as such,
> it's a property of a relationship between a superset and a subset.

The relative amount of substance can be determined. The set {1, 2, 3} 
has more substance than the set {father, mother}.
> 
> For example, to get the "substance" of N with respect to Q, you could embed
> it in the superset Q: You'd get something like: {0, 1, 1/2, 2, 1/3, 3,
> 1/4, 2/3, 3/2, 4, 1/5, 5, ....}.  Then this "substance" would come out as
> zero.

Nearly. It is smaller than any definable fraction.
> 
> So, to come back to my original example, the "substance" of {0, 4, 8, 12,
> 16, ...} wrt N is 1/4.

Yes.

> The substance of {1, 3, 5, 7, 9, ...} wrt {0,
> 1/2, 1, 3/2, 2, 5/2, 3, ....} is also 1/4.

Yes.

> Their "subtances" are thus
> the same.

Yes. Their amounts of substance, to be precise.

> I haven't come across this notion of "substance"/"Realität" before, and
> it doesn't feel like solid maths.  It all feels as though you are making
> it up as you go along.

Reality is Cantor's expression, Substance is Fritsche's (better) 
expression. For all finite sets, it is solid maths. Limits are 
well-known from analysis.
> 
>>> Countably infinite sets all have the same cardinality.
> 
>> That proves that cardinality is rather uninteresting.
> 
> On the contrary, it is fascinating.

If you consider it with cool blood, then you will recognize that all 
pairs of a bijection with ℕ are defined within a finite initial segment 
[0, n]. That is true for every n. But the infinity lies in the 
successors which are undefined.

>> Tend to yes, but not reaching it.
> 
> I thought you just said you had a degree in maths.  But you don't seem to
> understand the process of limits (a bit like John Gabriel didn't when he
> was still around).

0/oo = 0. 1/oo is smaller than every definable fraction.
>> Every theorem in analysis. This has not much changed since Cantor and
>> Hilbert.
> 
> Theroems in analysis require the infinite yes.  They don't require the
> confusing notion of "potentially infinite".

They have been created using only this notion. And also Cantor's 
"bijections" bare based upon potential infinity.

> In my undergraduate studies,
> the term "potentially infinite" wasn't used a single time.  The first
> time I came across it was in this newsgroup just a few years ago.

The Bourbakis have tried to exorcize the potential infinite from 
mathematics. Your teachers have been taught by them or their pupils.

> What everybody else refers to as infinte, you seem to want to call
> "potentially infinite".

The potential infinite is a variable finite. Cantor's actual infinity is 
not variable but fixed. (Therefore Hilbert's hotel is potential infinity.)

Regards, WM
>