Re: Simple enough for every reader?

On 31.05.2025 02:02, 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 30.05.2025 03:08, Ben Bacarisse wrote:

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

>

I thought it might be something cumbersome and vague like that.  I can't
even tell if this is a inductive collection,

>
It is obvious and clear. Do you know a case where a natural number can be
in it and cannot be in it? No. You can only curse. It is the same as
Peano's set. If you can't understand blame it on yourself.

 Can you prove it is an inductive set/collection?

See my book. The set is defined by induction. If n is in it, then also n+1 is in it. Pascal and Fermat used it without axioms as well as Cantor: "daß die Reihe
             1, i2, i3, ..., i, ...
nur eine Permutation der Reihe
             1, 2, 3, ..., , ...
ist. Dies beweisen wir durch vollständige Induktion,"
[Cantor, collected works, p. 305]
The axiom has only been adapted because induction holds.

 

so I must decline any
request to review a proof by induction based on it.

>
Of course. There is no counter argument. So you must decline.

 No, I decline because I don't know if it is an inductive set.  Do you?

Every mathematician knows that the definable natural numbers are an inductive set.

 (I note you deleted the cumbersome and vague definition.

It has been given to be understood. Now you have or have not understood. If not, the further presence would not help, I assume.

If it really
were obvious and clear, I would have left it in to show the world how
wrong I was to call it cumbersome and vague.)

I can give you a simpler and shorter definition: Every n that can be expressed by digits is definable.

I see you've cut the incorrect definition and the claim that the axioms
directly say that 1 is in N because, presumably, you now see that they
don't.

As I said that requires an intelligent reader recognizing that without ℕ
obeying the axioms too the paragraph would be nonsense.

 That's funny!  Yes, an intelligent reader will see you've written a junk
definition

You are a dishonest liar. But that is not relevant.

All natural numbers of Cantor's set ℕ can be manipulated collectively, for
instance subtracted: ℕ \ {1, 2, 3, ...} = { }. Here all have
disappeared.

 What definition of N do you want your intelligent readers to assume?

ℕ is Cantor's infinite set. Otherwise I could not use ℵo in my proof.
ℕ \ {1, 2, 3, ...} = { }
For the set ℕ_def defined in my book we have
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
Since all numbers can be reduced to the empty set by subtracting them collectively,
ℕ \ {1, 2, 3, ...} = { }
they could also be reduced to the empty set by subtracting them individually - if this was possible. But then the well-order would force the existence of a last one. Contradiction.
Regards, WM