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From: joes <noreply@example.org>
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 20:06:24 -0000 (UTC)
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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.