Re: Simple enough for every reader?

On 05.06.2025 23:51, Ben Bacarisse wrote:

WM <wolfgang.mueckenheim@tha.de> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte
des Unendlichen" and "Kleine Geschichte der Mathematik" at Technische
Hochschule Augsburg.)
 

On 04.06.2025 02:35, Ben Bacarisse wrote:

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

On 02.06.2025 03:56, Ben Bacarisse wrote:

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

>

Not all natural numbers of Cantor's set can be individually defined:

Not an answer.  Is b not injective?  Is b not surjektiv?

>
It is for the set of definable numbers, it is not for the dark
numbers.

 No, the topic is your exam papers and the nonsense

You cannot disprove any of my results.

that a student must
accept to get full marks.

Have you something substantial to say about dark numbers? Then do it. Note that my book Evidence of dark numbers has been published by Elsevier SSRN in four different eJournals without my application (I have only submitted it once).
SSRN Discrete Mathematics & Combinatorics eJournal 3,1 (20 Jan 2025)
SSRN Logic eJournal 2,13 (25 Nov 2024)
SSRN Mathematics Educator: Courses, Cases & Teaching eJournal 2,17 (26 Nov 2024)
SSRN Numerical Analysis eJournal 3,11 (24 Mar 2025)

b is both injective and surjective.

No.
(...(((ℕ \ {1}) \ {2}) \ {3}) ...) = { }
is wrong for all singletons of definable numbers. It is impossible however to replace them with the singletons of *all* natural numbers in order to make the result true. This is the proof that not all natural numbers are definable.

Your students need to say thing like "Das Cantorsche Diagonalargument
ist falsch" to get full marks.

>
So it is. The reason is what you refuse to answer:

(...(((ℕ \ {1}) \ {2}) \ {3}) ...) = { }
is wrong for all singletons of definable numbers.

Not all natural numbers of Cantor's set can be individually defined:
All natural numbers can be thought as defining the diagonal but not
individually. The well-order would force the existence of a last
one. Contradiction.
>
Therefore most indices of the diagonal elements are undefined, dark.

You cannot contradict even one of many proofs.

Not to your satisfaction, no.

Not at all. Then you would try it.

>
But I have shown my students how it goes.

 I feel for any student who knows how mathematics works.  With luck they
know how German exams work as well and will just write stuff they know
to be wrong so they get the marks.

You "know" what is wrong without being able to disprove it. That is not the way to pass an exam.
Regards, WM