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

Am 27.03.2025 um 12:18 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

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.

Not at all! Cardinality concerns only definable elements.

 

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.

It is undefined *as a finite number*. But it is defined by
|N|/(k-1) > |N|/k > |N|/(k+1).

 

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 finite sets cardinality is a meaningful notion expressing the number of elements. But I dislike it because it is often extended to cover infinite sets where it does not describe the number of elements.

 

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.

Ideas surpassing your knowledge by far.

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.

No, to be imprecise. Their numbers of elements differ but their cardinality is the same.

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.

It is correct and shows that cardinality is nonsense.

 

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.

Study surreal and hyperreal numbers which appear even in modern mathematics.

 

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.

Look, you don't know much. Not even surreal and hyperreal numbers. Why should I take your word on other topics as fact?>

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".

Then its number of guests and of rooms could not change.
Regards, WM