Re: Simple enough for every reader?

On 26.05.2025 12:44, Mikko wrote:

On 2025-05-26 10:17:27 +0000, WM said:
 

On 26.05.2025 02:52, Ben Bacarisse wrote:

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

>

With pleasure:
For every n ∈ ℕ that can be defined, i.e., ∀n ∈ ℕ_def:

>
I can't comment on an argument that is based on a set you have not
defined.

>
Can you understand my proof by induction?
The resulting set is ℕ_def. (According to set theory however it is not a set but a potentially infinity collection.)

 A proof by inductin does not define. It proves.

Here it proves that the natural numbers accessible by induction are not all natural numbers.

 For example: if we assume that
  0 ∈ ℕ_def
and that
  ∀n (n ∈ ℕ_def → (n + 1) ∈ ℕ_def)
then we can apply induction and prove that
  ℕ ⊆ ℕ_def .

Then you get a contradiction because by induction the natural numbers accessible by induction are not all natural numbers.
{1} has infinitely many (ℵo) successors.
If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, 3, ..., n, n+1} has infinitely many (ℵo) successors. For every n that can be defined.
Do you accept this proof?
Regards, WM