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| From | wm <wolfgang.mueckenheim@tha.de> |
|---|---|
| Newsgroups | sci.math |
| Subject | Cardinalities of sets |
| Date | 2026-02-22 18:55 +0100 |
| Organization | tha |
| Message-ID | <10nffup$kq1d$1@solani.org> (permalink) |
| References | (18 earlier) <10n93er$6196$5@dont-email.me> <9bGdnZo4hL-aDQX0nZ2dnZfqnPqdnZ2d@giganews.com> <10na9a8$kc8a$1@dont-email.me> <10nabhd$k7kl$1@dont-email.me> <10nci9d$7c6h$2@gwaiyur.mb-net.net> |
Am 21.02.2026 um 16:16 schrieb Thomas 'PointedEars' Lahn:
> [X-Post & F'up2 sci.math]
>
> Jeroen Belleman wrote:
>> On 2/20/26 19:31, Bill Sloman wrote:
>>> I know enough to know that the infinite number of integers is a smaller
>>> number than the infinite number of rational numbers,
Of course. There are reals which are not integers but every integer is a
real.
>
> "One of the great challenges in this world is knowing enough about
> a subject to think you're right, but not enough about the subject
> to know you're wrong."
That's the case with all conbvinced set theorists. They claim actual
infinity (or deny to think about that at all) but don't know that their
bijections concern only potential infinity. In every case they claim
that their sets are complete.
>
> --Neil deGrasse Tyson, astrophysicist and science communicator
> (in his MasterClass promotion video:
> <https://www.youtube.com/watch?v=io6QdGcoWMU>)
>
> (SCNR)
>
>>> but I don't get excited about it.
>>
>> I don't think that is correct. Both the sets of natural and rational
>> numbers are aleph-0 in size,
>
> More precisely, their _cardinality_ is ℵ₀ (strictly: _alef_-0).
Even more precisely, ℵ₀ is a useless notion.
>
>> because it's possible to create a one-to-one mapping of every rational
>> number to every integer.
That is impossible.
Proof: According to Cantor all positive fractions
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
...
can be indexed by the Cantor function k = (m + n - 1)(m + n - 2)/2 + m
which attaches the index k to the fraction m/n in Cantor's sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2,
5/1, 1/6, 2/5, 3/4, ... .
Its terms can be represented by matrices. When we attach all indeXes k =
1, 2, 3, ..., for clarity represented by X, to the integer fractions m/1
and indicate missing indexes by hOles O, then we get the matrix M(0) as
starting position:
XOOO... XXOO... XXOO... XXXO...
XOOO... OOOO... XOOO... XOOO...
XOOO... XOOO... OOOO... OOOO...
XOOO... XOOO... XOOO... OOOO...
... ... ... ...
M(0) M(2) M(3) M(4) ...
M(1) is the same as M(0) because index 1 remains at 1/1. In M(2) index 2
from 2/1 has been attached to 1/2. In M(3) index 3 from 3/1 has been
attached to 2/1. In M(4) index 4 from 4/1 has been attached to 1/3.
Successively all fractions of the sequence get indexed. In the limit,
denoted by M(∞), we see no fraction without index remaining. Note that
the only difference to Cantor's enumeration is that Cantor does not
render account for the source of the indices.
Every X, representing the index k, when taken from its present fraction
m/n, is replaced by the O taken from the fraction to be indexed by this
k. Its last carrier m/n will be indexed later by another index.
Important is that, when continuing, no O can leave the matrix as long as
any index X blocks the only possible drain, i.e., the first column. And
if leaving, where should it settle?
As long as indexes are in the drain, no O has left. The presence of all
O indicates that almost all fractions are not indexed. And after all
indexes have been issued and the drain has become free, no indexes are
available which could index the remaining matrix elements, yet covered by O.
It should go without saying that by rearranging the X of M(0) never a
complete covering can be realized. Lossless transpositions cannot suffer
losses. The limit matrix M(∞) only shows what should have happened when
all fractions were indexed. Logic proves that this cannot have happened
by exchanges. The only explanation for finally seeing M(∞) is that there
are invisible matrix positions, existing already at the start. Obviously
by exchanging O and X no O can leave the matrix, but the O can disappear
by moving without end, from visible to invisible positions.
The number of not indexed fractions remains |ℕ|*(|ℕ|-1) for all
definable terms of the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3,
3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... .
Hence |ℕ|*(|ℕ|-1) fractions cannot be indexed by definable indices.
Regards, WM
Back to sci.math | Previous | Next — Previous in thread | Next in thread | Find similar
Cardinalities of sets (was: energy and mass) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-21 16:16 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-21 08:11 -0800
Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-22 19:04 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-22 20:32 +0000
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-22 22:41 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-22 22:08 +0000
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-22 21:07 -0800
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-23 13:32 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-23 13:45 +0000
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-23 16:51 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-23 17:16 +0000
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-23 18:27 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-23 09:38 -0800
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-23 21:10 +0000
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-24 19:33 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-25 12:52 +0000
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-25 21:28 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-26 08:59 -0800
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-24 18:40 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-21 08:14 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-21 08:30 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-21 08:51 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-19 20:59 -0700
Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-22 18:55 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-22 22:52 +0100
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-23 13:24 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-23 23:41 +0100
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-24 18:46 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-25 18:10 +0100
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-25 21:32 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-26 00:36 +0100
Re: Cardinalities of sets "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-02-26 01:10 -0800
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-26 16:36 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-26 22:46 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-26 18:08 -0800
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-27 16:52 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-28 00:39 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-27 20:03 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-27 20:12 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-01 08:12 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-04 10:15 -0800
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-28 19:29 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-03-17 15:57 +0100
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-03-17 18:11 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-17 14:54 -0700
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-03-22 01:19 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-21 18:24 -0700
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-03-22 15:42 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-22 15:10 -0700
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-23 09:32 -0700
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-03-23 23:24 +0100
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-03-24 20:44 +0100
Re: Cardinalities of sets Maximilian Takashita <atiia@kiimia.jp> - 2026-03-23 20:18 +0000
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