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

Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:

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.

That makes all of them >=omega.

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}.

Try it with infinite sets.

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.

Infinitely so!

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.

Except to you. For finite sets you can just use cardinality.

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.

Yes, every natural number has a FIS. "Undefined numbers" aren't naturals.

Tend to yes, but not reaching it.

I thought you just said you had a degree in maths.

No, I asked him for the title.

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.

There is no real number other than 0.

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" are based upon potential infinity.

Yes, nobody refers to "actual infinity".

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.)

What we refer to as infinite isn't variable.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.