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 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.

You seem to mean the cardinality of the set.

This number exists also for actually infinite sets but cannot be
expressed by natural numbers.

We only know that ∀k,n ∈ ℕ_def: |ℕ|/k > n.

|N|/k is undefined.

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}.

You mean it has a larger cardinality.

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.

Crank talk.  You don't understand limits, as I've already said.  Have you
really got a degree in mathematics?  It seems unlikely.

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.

Or their cardinality, to be even more 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.

That's pure baloney.  Every element of a bijection is an ordered pair of
an element of set 1 and an element of set 2.  Each element of these sets
occurs in exactly one ordered pair.  There is no need to obfuscate this
definition with considerations of finite initial segments or infinity or
what have you.

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.

More crank talk.  Ordinary arithmetic is not defined on infinity.  And
"smaller than every definable fraction" is zero.

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.

But there is no theorem requiring "potentially infinite" for its proof that
isn't equally valid using the simpler notion of "infinite".  "Potentially
infinite" is a needless complication, if it's even well defined.

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.

"Potentially infinite" doesn't belong in mathematics.  It's not of any
use, and causes only obfuscation and confusion, not illumination.

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.)

Hilbert's hotel is infinite, not "variably finite".

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).