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 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.

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".

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.

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.

So, to come back to my original example, the "substance" of {0, 4, 8, 12,
16, ...} wrt N is 1/4.  The substance of {1, 3, 5, 7, 9, ...} wrt {0,
1/2, 1, 3/2, 2, 5/2, 3, ....} is also 1/4.  Their "subtances" are thus
the same.  Or could be made the same.  Or the notion of "substance" in
ill thought out and undefined.

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.

Countably infinite sets all have the same cardinality.

That proves that cardinality is rather uninteresting.

On the contrary, it is fascinating.

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.

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).  Those two things appear to contradict eachother.

.... 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.

If you have a degree in maths you should be able to say what you mean
clearly and concisely.  Maybe you've lost this skill over the decades
since your graduation.

 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.

Theroems in analysis require the infinite yes.  They don't require the
confusing notion of "potentially infinite".  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.

Or again thinking back to John Gabriel, he had his own non-standard
vocabulary, where he would use "number" to mean what everybody else
called rational number, and "incomensurate magnitude" to mean irrational
number.  It didn't get him anywhere.

What everybody else refers to as infinte, you seem to want to call
"potentially infinite".  That won't get you anywhere either.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).