Re: Simple enough for every reader?

On 27.05.2025 01:57, Ben Bacarisse wrote:

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

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?

 Not without knowing what the set N_def is, since the argument starts
"For all n in N_def".

It starts: For every n ∈ ℕ that can be defined.
Then it is proved that not every n ∈ ℕ can be defined.

The resulting set is ℕ_def. (According to set theory however it is not a
set but a potentially infinity collection.)

 So you are not asking me to verify a proof at all but rather to accept a
definition?

I am asking you to understand a proof by induction.

One that starts from claims about the thing being defined?

Not claims, but a proof: It starts: For every n ∈ ℕ that can be defined. And it proves that not every n ∈ ℕ can be defined.
You can also start: For every n ∈ ℕ. The you find a contradiction.

Your textbook defies N

>
It defines ℕ_def.

 It claims to define N.

Since this is the set used in applied mathematics.

It's very poor form to tell students you are
defining N when you are not.

It is the set defined by Peano and many others.

 In another reply (please don't split threads -- you may have time to
discuss this stuff endlessly but I don't) you say:
 

Your textbook defies N (incorrectly)

>
My textbook defines the classical natural numbers, ℕ, meanwhile more
precisely called ℕ_def, correctly.

 So when you write N and N_def you are referring to the same thing?

The book was written 10 to 20 years ago. At that time I did not bother about Cantor's ℕ and did not mention it.
But that is peanuts. Can you understand that this proof leads to a contradiction?
For every n ∈ ℕ:
{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.
Regards, WM