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Re: Cardinalities of sets

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Am 23.02.2026 um 22:10 schrieb Alan Mackenzie:
> WM <wolfgang.mueckenheim@tha.de> wrote:

>>> They also know that operators are applied, not numbers.
> 
>> For enumerating the algebraic numbers natural numbers are applied.
> 
> You just aren't familiar with mathematical usage.  There is no meaning
> of "apply" when the thing being applied is supposedly a set.

Enumerating is done applying natural numbers as indices.
> 
>>>> But kind of Freudian repression prevents most to understand that the
>>>> whole set ℕ can only be manipulated collectively.  Most numbers are
>>>> dark such that Dedekind-Cantor's way of enumerating the algebraics must
>>>> break down.
> 
>>> Mathematicians don't think in these terms.
> 
>> Dedekind and Cantor did.
> 
> No,

Yes, try to learn a bit at least. It is boring to have to repeat the 
basics again and again!

Es lassen sich alsdann die Zahlen des Inbegriffes (), d. h. sämtliche 
algebraischen reellen Zahlen folgendermaßen anordnen: man nehme als 
erste Zahl 1 die eine Zahl mit der Höhe N = 1; lasse auf sie, der Größe 
nach steigend, die (2) = 2 algebraischen reellen Zahlen der Höhe N = 2 
folgen, bezeichne sie mit 2, 3; an diese mögen sich die (3) = 4 
Zahlen mit der Höhe N = 3, ihrer Größe nach aufsteigend, anschließen;

Do you see Zahl and Zahlen again and again!

> I'm a graduate mathematician whereas you're 

a professor having written five books about mathematics, one of them 
with seven editions, one with four editions, published by De Gruyter.

>>>> Please try to learn the stuff before you play the expert.
>>>> The list is considered complete by using all natural numbers for
>>>> enumerating the entries.
> 
>>> Or, more mathematically expressed, a bijection is created between the
>>> natural numbers and the rational numbers, or whatever countable set
>>> you're thinking about.  Completeness, whatever you mean by that, doesn't
>>> come into it.

Invariability of sets is the basic of set theory.

> And as you ought to be aware, Cantor's diagonal number doesn't exist.

True.

> It is a purely hypothetical construct posited on the existence of a
> complete list of real numbers.

Enumerated by the complete set of natural numbers!

Regards, WM>

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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

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