Re: More complex numbers than reals?

Le 14/07/2024 à 03:30, Ben Bacarisse a écrit :

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

Le 13/07/2024 à 02:12, Ben Bacarisse a écrit :

WM <wolfgang.mueckenheim@tha.de> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen"

>
and "Kleine Geschichte der Mathematik"

 Optional, I hope.
 

at Hochschule Augsburg.)

>
Meanwhile Technische Hochschule Augsburg.

 A sound name change that reflects the technical college's focus.
 

Le 11/07/2024 à 02:46, Ben Bacarisse a écrit :

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

>

{a, b, c} vs { 3, 4, 5 }
>
Both have the same number of elements,

That will fall down for infinite sets unless, by decree, you state that
your meaning of "more" makes all infinite sets have the same number of
elements.

>
There are some rules for comparing sets which are not subset and superset,
namely symmetry:

Still nothing about defining set membership, equality and difference in
WMaths though.

>
Are my rules appearing too reasonable for a believer in equinumerosity of
prime numbers and algebraic numbers?

 You can define equinumerosity any way you like.

And I can prove that Cantor's way leads astray.

 But you can't claim the
"surprising" result of WMaths that E in P and P \ {E} = P

That refers to potential infinity and dark elements. Visible elements form only a potentially infinite collection.

 Presumably that's why you teach history courses now -- you can avoid
having to write down even the most basic definitions of WMaths sets.

At the end of the course I talk about the present state of the art. For instance this argument which is in general understood without ado:
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... ... XXXX...
XOOO... OOOO... XOOO... XOOO... ... XXXX...
XOOO... XOOO... OOOO... OOOO... ... XXXX...
XOOO... XOOO... XOOO... OOOO... ... XXXX...
...     ... ... ... ...
  M(0)  M(2)     M(3) M(4) M(∞)
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?
Regards, WM