Re: Simple enough for every reader?

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

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.

"i.e. ∀n ∈ ℕ_def:".

Then it is proved that not every n ∈ ℕ can be defined.

The "proof" starts with an undefined collection.  Pretending the magic
words define the collection is just a red herring.

We both know that you can't define N_def so you need to find some way of
waffling about it that starts by assuming it is known.

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.

So it's /not/ a definition.  OK.  A proof by induction can't be over an
unknown set.

...

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.

It sounds as if you are saying that it (your book) defines N_def, and
that it (the set defined in your textbook) is the set defined by Peano
and many others.  That would make N_def and N the same.  Really?

No, I think what you mean is that your book defines N_def even though it
misleadingly calls it N, the name of the set defined by Peano and many
others.  Is that what you mean?  That would make sense because your
definition is certainly /not/ a definition of N as Peano would
understand the term.  That's because it is /wrong/, as has been pointed
out more than once, by more than one poster.

I see you cut the request to prove that 1 is in N (or it is N_def?)
using your junk "definition".  Of course you cut it.  You can't do it!

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.

Can you even prove that 1 is in N using your definition?  Why does this
matter?  Because if you can't even prove that 1 is in N, everything that
follows is vacuously true.

If you can prove that at least 1 is in N (as you define it) you then
need to prove the base case of your induction.  How you prove that {1}
"has ℵo" successors.  I'd like to see the base case proved.  Induction
does not work by assertion alone!

--
Ben.